Tilburg University

Pricing and hedging in the VIX derivative market

Kozarski, R.

Publication date:2013

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Citation for published version (APA):Kozarski, R. (2013). Pricing and hedging in the VIX derivative market. CentER, Center for Economic Research.

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Pricing and Hedging in the VIX

Derivative Market

Pricing and Hedging in the VIXDerivative Market

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan Tilburg University opgezag van de rector magnificus, prof. dr. Ph. Eijlander, in hetopenbaar te verdedigen ten overstaan van een door het college voorpromoties aangewezen commissie in de aula van de Universiteit opvrijdag 17 mei 2013 om 14.15 uur door

ROBERT KOZARSKI,

geboren op 13 september 1977 te Tomaszów Lubelski, Polen.

Promotor: prof. dr. Bertrand MelenbergCopromotor: dr. Feico C. Drost

Moim Rodzicom dedykuję(To my parents)

Preface

First of all, I would like to send my special gratitude towards my supervisors BertrandMelenberg and Feico Drost, for their patience, time, and support while working on thethesis. They helped me a lot in shaping my reasoning and following a scientific rigour inmy work, which happens to be also very helpful in a daily life. I would like to thank allthe committee members, professors: Bas Werker, Ronald Mahieu, and Joost Driessen fortheir helpful suggestions and time spent on reading the manuscript. My special thanks goto professors Hans Schumacher and Arthur van Soest, and all the people from the CentERwho made the time of being a doctoral student a nice and stimulating period of life.

Living and studying in Tilburg gave me a great opportunity to meet and share timewith my fellow Ph.D. students and folks from outside the University. In particular,Mieszko, Marta and Maciej, Sebastian, Patrycja and Krzysztof, Esen and Baris, Conny,Manon, Maria, David, Jeremie, Leon, Nathan, Renxiang, Salvatore, Tunga, and manyothers which have been around but not mentioned here.

This a good place to express my gratitude towards my colleagues from CliCIR, Uni-versity of Hertfordshire: David, John, Kirsten, Karin, Hema, Supri, Jonathan, Lewis, andSam for their kindness and support while living and working in the UK.

Last but not least, I would like to thank my parents Henryka and Jan for their un-shakeable faith in me, and my beloved wife Anna for her patience and constant supportin all of my plans and activities. I do really appreciate all of this.

Robert KozarskiMarch 2013, Hatfield (UK)

Contents

1 Introduction 1

2 Literature Review 11

3 Classical Models for the Volatility Derivative Pricing 193.1 Pricing setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Pricing Models Based on Diffusions . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Whaley model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Grünbichler-Longstaff (GL) model . . . . . . . . . . . . . . . . . . . 213.2.3 Detemple-Osakwe (DO) model . . . . . . . . . . . . . . . . . . . . . 22

3.3 Pricing Models Based on Jump-Diffusions . . . . . . . . . . . . . . . . . . 243.3.1 Merton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Mean-reverting models with jumps . . . . . . . . . . . . . . . . . . 26

3.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 P -process estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Market-implied Q-estimates . . . . . . . . . . . . . . . . . . . . . . 31

4 Data 354.1 S&P 500 volatility index VIX . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 VIX derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Market size and value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Role of the Market Information in the Empirical Performance of theDiffusion Based Volatility Derivative Pricing Models 455.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.2 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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viii

5.3.1 Parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3.2 Pricing performance . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 The Volatility Derivative Hedging Effectiveness with the Diffusion BasedModels 716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2.1 S&P 500 options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.2 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3 Hedging setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3.1 VIX options hedging . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3.2 VIX futures hedging . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.4.1 VIX option hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4.2 VIX futures hedging and VIX replication . . . . . . . . . . . . . . . 84

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Volatility Derivative Pricing Models Based on the Jump-Diffusions. Spec-ification and Pricing Performance 977.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2 Model performance assessment . . . . . . . . . . . . . . . . . . . . . . . . . 987.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3.1 Numerical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3.2 Estimates over the whole VIX sample . . . . . . . . . . . . . . . . . 1027.3.3 Estimates over the 252-day VIX subsamples . . . . . . . . . . . . . 104

7.4 Pricing performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.4.1 In-sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4.2 Out-of-sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8 Hedging Assessment of the Volatility Derivative Pricing Models Basedon Jump-Diffusions 1298.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.2 Hedging setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.2.1 Delta strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.2.2 Delta-jump strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 130

ix

8.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338.3.1 VIX option hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.3.2 VIX futures hedging . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9 Summary and Conclusions 145

A Appendix 151A.1 Bakshi-Madan pricing approach . . . . . . . . . . . . . . . . . . . . . . . . 151A.2 Characteristic function for the mean-reverting models with jumps . . . . . 152A.3 VIX calculation methodology . . . . . . . . . . . . . . . . . . . . . . . . . 153A.4 Hedge ratio derivations: VIX options . . . . . . . . . . . . . . . . . . . . . 155A.5 Hedge rates derivations: VIX futures . . . . . . . . . . . . . . . . . . . . . 156

A.5.1 Hedging with the S&P 500 options . . . . . . . . . . . . . . . . . . 156A.5.2 Hedging with the S&P 500 options and the S&P 500 asset . . . . . 157

A.6 Delta hedging ratios: jump-diffusion models . . . . . . . . . . . . . . . . . 158

x

Chapter 1

Introduction

Market volatility plays an important role in the theoretical and empirical financial anal-ysis. One of the main widely recognized problems is that volatility is clearly changingstochastically over time (Figlewski [1997]). It varies in response to changes on a macroe-conomic level (e.g., inflation, unemployment), political situation, or shifts in the companiesequity structures and their performance announcements. The measurement and modelingof this belong to one of the more crucial and intensively evolving areas of modern portfolioanalysis and derivative pricing (Andersen et al. [2010]).

After the 1987 financial crisis experiences, Brenner and Galai [1989] noticed that overthe entire history of the US stock indices the percentage volatility changes were usuallymuch greater than changes in the level of the stock indices, and an efficient tool to hedgeagainst the volatility-of-volatility is in demand among investors. They proposed for thefirst time the idea of a volatility index, which would serve as the aggregated market forwardlooking volatility measure, and the derivatives written on such an index, as a tool to hedgeagainst the volatility risk.

In September 1993, the Chicago Board of Option Exchange (CBOE) introduced thefirst volatility index (VIX), calculated from the Black-Scholes (Black and Scholes [1973])volatilities implied from the S&P 100 index option prices, following the methodologyproposed by Whaley [1993]. This methodology was changed in September 2003 and nowthe VIX relies on the volatilities implied from the S&P 500 index options, providing a muchbroader and robust measure of the market volatility (Whaley [2009]). This approach wasthen adopted as a main reference in developing the volatility indexes characterizing theother U.S. and European stock and even commodity markets.1

1In the USA, the CBOE disseminated the Nasdaq-100 Volatility Index (VXN), the DJIA VolatilityIndex (VXD), the Russell 2000 Volatility Index (RVX), and the S&P 500 3-month Volatility Index (VXV).The CBOE commodity volatility indexes are: the Crude Oil Volatility Index (OVX) based on United States

1

2

On March 26, 2004, the Chicago Futures Exchange (CFE) introduced the VIX futurescontracts, and two years later, on February 24, 2006, the CBOE extended the set ofVIX derivatives by the European-type options. The VIX futures and options were thefirst market-traded volatility derivative products.2 The market extended and the VIXderivatives are not the only traded volatility contracts these days.3 Besides the volatilityderivatives traded on the market, there are other volatility products available in over-the-counter (OTC) circulation, like variance and volatility swaps.4

The VIX derivative market size expanded rapidly since its introduction. After the firstfour months in circulation (June 2006), the VIX options were the fifth most actively tradedCBOE product. In 2007, the VIX option total trading volume increased 4.6 times over5.1 million in 2006. For the next two years, the volume increased further up to 26 and 33million contracts in 2008 and 2009, respectively. For the VIX futures, the year 2007 wasthe fourth trading year, when the total volume was 2.4 times higher than in 2006 (425,353contracts), and 11.4 times higher than in the year of inception. Over the next two years, itleveled-off at around 1 to 1.15 million traded contracts. The VIX derivatives were amongthe most actively traded contracts at the CBOE and CFE, averaging close to 260,000contracts combined per day in 2010, with much larger volumes on certain days. Thisvalue stood for almost 40% of the S&P 500 options daily average trading volume (695,000

Oil Fund options, the Gold Volatility Index (GVZ) based on the SPDR Gold Shares (GLD) options, andthe EuroCurrency Volatility Index (EVZ) based on the Currency Shares Euro Trust (FXE) options. Inaddition, on November 12, 2007, the CBOE disseminated 3-month S&P 500 Volatility Index (VXV), basedon the same calculation methodology as the 1-month implied VIX. The European stock markets deliverthe following volatility indexes: the EURO STOXX 50 Volatility Index (VSTOXX), the DAX VolatilityIndex (VDAX) on the German Deutche Börse, the CAC40 Volatility Indexes (VX1, VX6) on the FrenchMONEP, and the SMI volatility index (VSMI) index on the SIX Swiss Exchange.

2Some European stock exchanges listed volatility contracts in the first decade after the VIX inception.For example, at the beginning of 1997, the OMLX (the London-based subsidiary of the Swedish exchangeOM) launched the SEK volatility futures for the Swedish market. A year after, the Deutsche Terminbörse(DTB) launched the VOLAX futures. However, after a poor market response, these contracts werewithdrawn from trading (Daouk and Guo [2004], Carr and Lee [2009]).

3On April 2005, the CFE introduced the futures contracts on the VXD index, but delisted them onAugust 13, 2009. The RVX and VXN futures were jointly introduced on July 6, 2007. One year later, onJuly 1, 2008, the CBOE extended the VIX option set by the VIX binary options (BVZ), and the mini-VIXfutures on March 2, 2009. Besides the VIX option contracts, there are the RVX options introduced bythe CBOE on September 27, 2007, together with the VXN options launched one day later. Recently in2011, the first derivatives on the Gold Volatility Index were introduced: the futures on March 25, andoptions on April 12.

4See Demeterfi et al. [1999] for the contract specification. Broadie and Jain [2008] quote recent es-timates for daily trading volume around 30 up to 35 million USD, however, an assessment of the totaltrading volume and value of any OTC market is problematic by nature.

3

contracts) observed in 2010. All this evidence suggests that the VIX derivative marketbecame an important element of the stock market structure (Shu and Zhang [2012]). Thenegative correlation between the S&P 500 and VIX indexes is a well-documented fact(Whaley [2009]) and corresponds to a so called ‘leverage effect’ (Black [1976b]). Such aproperty makes the VIX derivatives useful in hedging against volatility risk, especially inmarket downturns. Daigler and Rossi [2006] indicate possible benefits from using a longposition in the VIX options and futures in the S&P 500 portfolio diversification. Szado[2009] shows that positions in the VIX derivatives are more efficient in alleviating thevolatility exposure, than the S&P 500 put option positions during the 2007-2008 yearcrisis. Alexander and Korovilas [2011] analyze similar strategies over a more extensiveperiod (2006–2010 years), and found that, besides clear advantages during the marketcrisis, effective use of the VIX derivatives within tranquil periods is more difficult andremains a domain of more experienced traders. Despite the VIX derivatives extending theset of possible investment and hedging strategies, in practice, their efficient applicationoften relies on mathematical models, which capture the volatility index dynamics andallow to price these contracts. In this context, development of new frameworks, as wellas a thorough empirical assessment of the available models is crucial for the volatilityinvestment decisions.

Chapter 2 provides an extensive review of the existing studies, which analyze the VIXderivative market from different aspects. The majority of these concentrate on develop-ment of new models for the VIX contracts valuation. The methodologies have developedmainly in two directions. First, the structural approach, where the VIX derivative pricingrelies on the stock market dynamics represented by the S&P 500 index. This stream is rep-resented by the models of Zhang and Zhu [2006], Lin [2007], Sepp [2008a], Lin and Chang[2009, 2010], Lu and Zhu [2010], Zhu and Lian [2011], and Lian and Zhu [2011]. The spotVIX itself is usually derived from the market index stochastic volatility dynamics (e.g.,Heston [1993] model), and such an outcome is used as the underlying for the VIX deriva-tive pricing. The alternative reduced-form approach recognizes the current VIX level asthe ‘sufficient underlying’ for its derivatives, and relies on modeling its dynamics directlywithout incorporating the S&P 500 information. This direction is represented by themodels of Whaley [1993], Grünbichler and Longstaff [1996], Detemple and Osakwe [2000],Psychoyios et al. [2009], Dupoyet et al. [2011] and Mencía and Sentana [2012]. There isno consensus in the literature which approach is the preferred one as both structural andreduced frameworks have their advantages and disadvantages.

The diversity of the available frameworks is an obvious advantage, however, the in-tensive development of pricing methodologies is not always balanced with a sufficientempirical performance assessment. Mainly due to lack of empirical evidence, the early

4

investigations of the baseline one-dimensional diffusion models of Whaley [1993], Grün-bichler and Longstaff [1996], and Detemple and Osakwe [2000], provided in Daouk andGuo [2004] and Psychoyios and Skiadopoulos [2006], were based on simulated data. Thefirst (peer-reviewed) investigation of the Whaley and Grünbichler-Longstaff models, go-ing beyond the simulated market scenarios, is provided in Wang and Daigler [2011], whocompare their performance with the Lin and Chang [2009] stochastic volatility approachand the model-free Carr and Lee [2007] methodology. They find that none of these mod-els clearly dominates the others over the whole moneyness/expiration range, and thatmodel’s simplicity seems to pay-off in the empirical applications. Despite its importantcontributions, the study of Wang and Daigler has some important drawbacks. Bearingin mind its empirical character, the sample with the daily VIX option prices, spanningthe period between January and September 2007, is too short to be representative.5 Themodel estimation is not nested within a common market information framework: 1) theparameters are calibrated to the VIX option market prices or estimated over the historicalVIX series, 2) the models rely on different underlying series (the spot VIX or its futurescontract prices). The model assumptions about the volatility risk premia structure are notclearly specified, which precludes their relative performance evaluation in a consistent way(Chan et al. [1992]). Then, the out-of-sample horizon comprises of 1-day only and it lacksthe hedging performance assessment, which leaves the picture rather incomplete. Due toall these aspects, the Wang and Daigler study cannot be considered as fully representativefor the area. However, it makes a good starting point towards building the evidence aboutempirical properties of the VIX derivative pricing models.

The need for a more complex and structured empirically orientated assessment of theexisting VIX derivative pricing models gave an impulse to conduct the research in thisdirection. In the course of the study, we concentrate exclusively on the reduced-formspecifications, which rely directly on the VIX return dynamics in the volatility derivativespricing.

The main advantage of the structural approach derives from linking the VIX derivativesegment to the overall stock market. However, excluding the empirical VIX dimension fromthe analysis and replacing it with the S&P 500 model-derived outcome, seems to be an oddsolution, especially if the available models cannot really reproduce the spot VIX values(Mencía and Sentana [2012]). Developing the VIX derivative pricing approach from suchan underlying implies another level of complexity and yields a risk of misspecification.The models are difficult to be handled numerically and the convergence of the solutions is

5In the empirical equity option studies, like Bakshi et al. [1997] or Christoffersen et al. [2011], thederivative data sample spans the period of five to nine years of daily recordings.

5

often highly sensitive to the optimizer’s initial values (Andersen and Andreasen [2000], Zhuand Lian [2011]). This aspect can seriously undermine their practical utility, especially inthe context of extensive empirical applications and often gives more credit to the simplermodels. For example, in Wang and Daigler [2011], the numerical instability of the Linand Chang [2009] model solutions lead to its failure in some cases with respect to theone-dimensional VIX diffusion specifications. The complexity drawback is also raised inDupoyet et al. [2011] with respect to the Lin [2007] VIX futures pricing model.

These problems are not commonly encountered in the reduced-form approach, wherethe available derivative pricing solutions are less complex than those emerging from thestructural approach, and often available in closed forms. This usually simplifies the nu-merical procedures involved in the parameter estimation process, which has a great valuein the empirical applications. The main critique of the reduced approach concerns the factthat it tends to ignore the S&P 500 dynamics, which remains strongly correlated with theVIX. This relation, despite being an important aspect of the derivatives market, does notpreclude from relying on the VIX as a ‘sufficient statistic’. The VIX is a well-recognizedmeasure of the market volatility (Whaley [2000]) and its calculation formula relies on theobserved S&P 500 option prices. Thus, the equity market information is not completelyignored, but provided in an aggregated form. In addition, the direct modeling of the VIXdynamics diminishes the risk of its misspecification, as might happen in the structuralapproach. All these aspects advocate the reduced-form methodology as a reliable supportfor pricing and hedging in the VIX derivative market.

In the analysis, we concentrate on the Whaley pricing framework based on a GeometricBrownian Motion (GBM) process, the Grünbichler-Longstaff (GL) and Detemple-Osakwe(DO) specifications, which rely on the Cox-Ingersoll-Ross (CIR) (Cox et al. [1985]) and logOrnstein-Uhlenbeck (logOU) (Uhlenbeck and Ornstein [1930]) mean-reverting diffusions,respectively. All these affine-type processes correspond to the instantaneous volatilityspecification used in well-established stochastic volatility derivative pricing models (seeHull and White [1987], Heston [1993], Melino and Turnbull [1990]). However, due tothe mean-reverting nature of the market volatility, the GBM dynamics is less empiricallyrelevant in modeling the volatility index dynamics when compared to the CIR or logOUdynamics (Psychoyios et al. [2003], Kaeck and Alexander [2010]). Hence, Whaley’s modelserves mainly as a reference specification for the models, which include the empiricallyrelevant mean-reverting factor. In the literature, the logOU process gets a credit in mod-eling the VIX evolution (e.g., Psychoyios et al. [2009], Mencía and Sentana [2012]), butthe CIR dynamics is used as the volatility reference in the widely applied Heston (Heston[1993]) stochastic volatility pricing framework, where the spot VIX often serves as an in-stantaneous volatility proxy (e.g., Aït-Sahalia and Kimmel [2007]). Within this context,

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comparing these two theoretically alternative dynamics in the VIX derivative pricing andhedging context seems to be a natural step towards better understanding their practicalrelevance and more efficient decision making. Then, we recognize the importance of thejump extension in the VIX modeling (Dotsis et al. [2007], Todorov and Tauchen [2011]).All the underlying frameworks are amended by the jump component, and the relevantpricing approaches are provided, following Merton [1976] for Whaley’s model and Bak-shi and Madan [2000] for the mean-reverting specifications, respectively. The assessmentdoes not cover the pricing specifications based on the one-dimensional constant elasticityvariance (CEV) dynamics, in the form considered to be a non-affine relaxation of the CIRdynamics. So far, its utility in modeling the VIX and pricing the volatility derivatives isnot fully confirmed in the literature; see the discussion provided in Chapter 2.

All six models are extensively investigated within the VIX derivative market empiricalpricing and hedging context. In particular, we concentrate on two main aspects. First, themodel robustness with respect to different market information loads used in the parameterestimation. The market content covers the VIX derivative data and their underlying VIXseries dynamics. Then, we want to quantify how the empirically relevant model extensionsinfluence the pricing and hedging performance within different market information setups.We contribute mainly by presenting extended empirical evidence and a more complexdesign to assess the model performance, which may lay the foundations for the futureassessment framework standards. In particular, we consider a sample which consists of allthe VIX derivative prices recorded since the option introduction on February 24, 2006 upto April 30, 2010. In the model estimation framework, we use extensively the informationfrom the VIX derivative market. The parameter estimates reflect both the long-term(20-year) VIX characteristics and its more local 252-day (1-year trading) features. Indetermining the model free parameters, the daily VIX option (calls, puts) and futuresprices are jointly considered. Such a strategy turned out to be more efficient than thestandard approach with only one type of derivative, which we also investigate. In addition,each contract contribution is weighted by its daily relative trading volume, which grants agreater relative weight to more intensively traded contracts. In the hedging part, we alsoutilize the information from the S&P 500 index derivative market.

The structure of the thesis is divided into nine chapters. In Chapter 2, we reviewthe literature. It covers the publications with the seminal affine diffusion based Wha-ley, Grünbichler-Longstaff, and Detemple-Osakwe frameworks, the stochastic volatilityapproaches, up to the recent attempts to model the VIX dynamics using non-affine pro-cesses. From a theoretical aspect, we pay attention to the underlying VIX dynamics typesused in the models and their parameter estimation methodologies. Since the thesis hasmainly an empirical character, we underline the market data aspects of the studies, like

7

for example, which data types and how the market information was incorporated in theparameter estimation. The VIX derivative market is still considered to be in its develop-ment phase, thus, the data time span used in the empirical evaluations determines howrepresentative they are.

The models of interest are described in Chapter 3. We start with a brief description ofthe general pricing framework, relying on the equivalent martingale measure. Then, theparticular pricing frameworks are provided: the Whaley [1993] relying on the Geomet-ric Brownian Motion assumption for the underlying VIX, the Grünbichler and Longstaff[1996] using the square-root mean-reverting process, and the Detemple and Osakwe [2000]with the log mean-reverting VIX dynamics. Later on, this framework is extended by thejump component and relevant pricing models, based on the Merton [1976] and Bakshi andMadan [2000] methodologies. In the model description, we underline the aspects of theassumed underlying dynamics and its relevance to the VIX empirical characteristics. At-tention is also paid to the assumptions about the market price of risk used in determiningthe processes under the risk neutral measure. The model estimation relies on the jointinformation for the underlying VIX series history and its futures and European optioncontract prices. In addition, we use the daily VIX derivative trading volume as a naturalweight of the contract contribution, which extends the standard ‘equal-weight’ approach(Bates [1995]).

In Chapter 4, we describe the data set used in the analysis. It consists of the underlyingS&P 500 volatility index (VIX) series, and its derivatives’ daily trade recordings. The VIXsample spans the end-of-day values between January 2, 1990 and April 30, 2010 (5,125trading days). The derivative sample covers the period between February 24, 2006, whenthe VIX option contracts were introduced to the market, and April 30, 2010 (1,052 tradingdays). The VIX series description covers its basic statistical characteristics over the wholesample, and in the period of the option trading. The VIX contract specifications areprovided, together with description of the analyzed market with respect to its value andvolume dynamics over time.

The influence of different market information on the baseline model performance isanalyzed in Chapter 5. The common practice is to use the derivative data to calibratethe parameters under the risk-neutral measure (e.g., Broadie et al. [2007]), or even awhole set of model parameters (Bakshi et al. [1997]). However, in both approaches, thederivative information choice still remains arbitrary. Usually, it stays consistent withthe priced derivative type, such as the option-implied parameters for the option contractpricing. We investigate the outcomes for all the VIX derivative types, given that the risk-neutral parameters are implied from different VIX contract combinations. Each contractcontribution is weighted by its trading volume. The assessment concerns in- and out-of-

8

sample performance. It turns out, that it matters for the pricing outcomes which derivativeinformation is used in the parameter estimation. In particular, using the option-impliedparameter values to price calls and puts does not always do a better job, when comparedto the jointly-implied (options and futures) parameter case. When we consider the modelperformances within the same information frameworks, the Detemple-Osakwe outperformsthe other specifications, and the Whaley model provides the highest errors.

The pricing outcomes from Chapter 5 are complemented by the hedging performanceassessment in Chapter 6. The applied hedging strategies comply with the approaches pro-posed in the literature. The VIX options are hedged with the VIX futures position, bothexpiring at the same date. The VIX futures are hedged with the position in the underlyingasset VIX, or the S&P 500 options strip replicating the spot VIX as a more empiricallyrelevant procedure. In all these applications, the hedged derivative trading volume servesas a natural portfolio weight in calculating the aggregated hedging outcomes. This time,the Whaley and Detemple-Osakwe models complement each other in the majority of cross-sections and outperform the Grünbichler-Longstaff specification. The Detemple-Osakweis the preferred specification for the daily rebalanced portfolios with cheap but most inten-sively traded short-term calls. However, this role is shared by the Whaley model for thedeep out-of-the-money (DOTM) group. For the longer hedging horizons, the Detemple-Osakwe usually stays best within the DOTM group, regardless of the time to maturity. Forthe VIX futures, the mean-reverting models yield smaller than the Whaley specificationhedging errors. The empirical analysis is extended with results relying on the simulateddata.

Jumps belong to one of the S&P 500 volatility index salient characteristics (Dotsiset al. [2007], Duan and Yeh [2010]). Such an extension to the underlying VIX dynamicsincluding the pricing consequences are investigated in Chapter 7. The estimation of thejump-diffusion based models and then valuation of the derivative contracts relies on theBakshi and Madan [2000] framework, see also Psychoyios et al. [2009]. The data frameworkis kept similar to that of Chapter 5, and the outcomes are put together with the diffusionbased results. The main and most important improvement is observed for the short-termoptions, as around 81% of the market total trade concentrates on these contracts. Withinthis group, the best performing jump-extended Detemple-Osakwe model improves over itsdiffusion based counterpart by almost 42%.

In the subsequent Chapter 8, we stay with the same set of models and investigate thehedging outcomes within the new jump-diffusion framework. After diffusion (volatility),there is one more risk to be hedged: jump risk. It splits into two parts, where one relatesto the expected jump size, the other to the jump timing. We assume that the moment andsize of the jump are stochastic. However, after Pan [2002], we assume that the jump size

9

risk is the only risk priced by the market, such that within the jump-diffusion framework,we hedge the call options against the diffusion (volatility) and jump size risks. The optionhedging portfolio consists of two VIX futures contracts expiring within the same and alonger horizon. The outcomes are compared with the case when the jump risk is assumedto be non-systematic, and only the diffusion risk price is left to be valued by the market.

Finally, the main thesis outcomes and conclusions are provided in Chapter 9.

All the calculations were performed within the R statistical computing environment, ver-sion 2.10.1 (R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/).

10

Chapter 2

Literature Review

The literature focusing on the VIX derivative market has been growing fast over the yearssince its inception. The mainstream concerns mainly pricing of these derivatives withindifferent underlying model assumptions. In the provided review, we concentrate only onpeer-reviewed positions published before October 2012.

The VIX derivative pricing methodology develops in two directions. First, the reduced-form approach which recognizes the current VIX level as the ‘sufficient underlying’ forits derivatives, and relies on modeling its dynamics directly without incorporating theinformation from the equity market. The alternative structural approach, where VIXderivative pricing relies on the dynamics of the stock market represented by the S&P 500index. The spot VIX itself is derived from the market index dynamics, and such outcomeis used as the VIX derivative underlying.

Among one-dimensional reduced-form diffusion based specifications, there are threemain model frameworks. The first, proposed by Whaley [1993], utilizes the Black [1976a]futures formula to value the call option on the futures contract, where the underlyingprocess is modeled by a Geometric Brownian Motion. Then, Grünbichler and Longstaff[1996] derived a pricing framework based on a Cox-Ingersoll-Ross (CIR) (Cox et al. [1985])mean-reverting square-root process for the implied volatility index. Finally, Detemple andOsakwe [2000] developed a model where the underlying follows a mean-reverting loga-rithmic Ornstein-Uhlenbeck (logOU) (Uhlenbeck and Ornstein [1930]) process, consideredto be an alternative to the CIR specification. All of these models result in closed-formpricing formulas for the VIX futures and European option contracts. The VIX dynamicsassumed in these specifications belong to well-recognized instantaneous volatility diffu-sions in stochastic volatility derivative pricing models like Heston [1993], Hull and White[1987], and Melino and Turnbull [1990]. Psychoyios et al. [2003] provide a review of thecontinuous-time volatility affine dynamics used in the stochastic volatility option pricing

11

12

literature. Due to their seminal nature regarding volatility derivative pricing methodology,the Whaley, Grünbichler-Longstaff (GL) and Detemple-Osakwe (DO) pricing frameworkswill be referred to as the baseline ones. Notice that these models were proposed longbefore the first VIX derivative contracts were introduced to the market.

The early attempts to investigate the performance of the baseline models rely, due tolack of empirical evidence, on simulated market scenarios. Daouk and Guo [2004] investi-gate the GL model error and show that it underprices the three-month volatility optionsby about 10%. The proposed extension recognizes the regime-switching extension to thediscrete-time version of GL’s framework. Psychoyios and Skiadopoulos [2006] provide thefirst extensive study on the pricing and hedging using the baseline models, and they ad-dress two questions. Firstly, the effectiveness of the volatility contracts (Whaley, GL, DO)for hedging the volatility risk, when compared to the standard option tools (the Black-Scholes (Black and Scholes [1973]) and Heston (Heston [1993]) models). Secondly, theirpricing and hedging performance when the underlying model is misspecified. They findthat the volatility option contracts are not necessarily superior to the standard options inhedging the volatility risk. In addition, the Whaley model can compete in hedging withmore empirically relevant specifications. An et al. [2007] do a very similar analysis withrespect to the over-the-counter traded exotic options. The volatility options are found tobe more effective in this context than the standard contracts. The hedging framework isextended on the volatility index vs. straddle options as hedging instruments; both provideroughly similar performance.

Wang and Daigler [2011] is the first peer-reviewed study which provides an empiricalpricing assessment and comparison of the existing VIX derivative pricing methodologies.1

In particular, it follows the diffusion based models (Whaley, GL), the stochastic volatilitymodel of Lin and Chang [2009], and the model-free methodology proposed in Carr and Lee[2007]. The analyzed VIX option sample spans the January – September, 2007 period.The VIX dynamics parameters are estimated from the underlying series over different

1Ammann and Süss [2008] provided an empirical assessment of the diffusion based specification withrespect to the VIX calls and puts, traded over the March 1, 2006 – September 28, 2007 period. How-ever, this study exists only in a working paper version and is not mentioned as a reference in the VIXderivative pricing literature. Recently, Mencía and Sentana [2012] consider the GL and DO models withproper extensions of the latter by the underlying stochastic long-term mean dynamics and/or a stochasticvolatility factor. The models are specified in a state-space form and calibrated jointly with all the VIXderivative type prices and the spot VIX values, using the Trolle and Schwartz [2009] pseudo-likelihoodbased approach. The out-of-sample horizon usually stands for a 1- or 5-day period, and the sample coversthe VIX derivative data recordings from their inceptions up to April 30, 2009. The paper is still beingupdated and its methodology might be considered in the future extensions of the diffusion based pricingmethodology.

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time horizons (5, 30, and 184 days), or calibrated with the VIX option prices, wherethe information set used in estimation process differs among the models. They find thatany of the analyzed pricing frameworks clearly outperforms the others over the wholemoneyness/expiration range. The Heston-type approach represented by the Lin-Changapproach does not outperform the diffusion based specifications out-of-sample. The mean-reverting models are superior in forecasting deep out-of-the-money option prices within1-day horizon, where Whaley’s methodology provided better results within in-the-moneysection. In addition, the results obtained for the Lin and Chang [2009] model show someinconsistency with the original study.

One of the ways where the one-dimensional framework evolves is the jump-extendeddiffusion dynamics. Within this stream, Dotsis et al. [2007] concentrate on the jumpextension to the baseline volatility index dynamics, and the empirical pricing consequencesfor the VIX futures recorded over the March 26, 2004 – March 6, 2006 period. They runan extensive empirical study over a couple of American and European volatility indexes(including VIX) and find that the jump, next to the mean-reverting component, plays asalient role in modeling the underlying VIX dynamics.2 Within this context, Psychoyioset al. [2009] build the VIX option and futures pricing framework based on the CIR andlogOU diffusions extended by a jump component, motivated by the Bakshi and Madan[2000] methodology. The study provides theoretical considerations about general pricingand hedging patterns implied by the models. However, no empirical assessment of thisextension is provided. The underlying model estimates rely on the empirical VIX dailyseries recorded between January 2, 1990 and September 13, 2005.

Alternatively, instead of directly modeling the VIX dynamics, one can rely on the S&P500 dynamics representation in the VIX derivatives pricing. Carr and Wu [2006] showthat the spot VIX value can be interpreted as the square-root of the variance swap rate,such that it can be perfectly statistically replicated through the S&P 500 vanilla callsand puts. Thus, the VIX dynamics can be implied from the S&P 500 returns by meansof a stochastic volatility approach (e.g., Heston [1993], Bakshi et al. [1997], Eraker et al.[2004]). Majority of the pricing models developed in this framework concern the VIXfutures. Zhang and Zhu [2006] provide the first attempt to price the VIX futures, wherethe S&P 500 dynamics is described by the Heston-type stochastic volatility model. Thestructural parameters are estimated over the daily VIX series from January 2, 1990 upto March 1, 2005 period, using the maximum-likelihood (ML) approach. Alternatively,the parameters are jointly calibrated with the market VIX futures prices recorded between

2Similar studies are provided in Moraux et al. [1999] for the French Market Volatility Index (VX1),or Wagner and Szimayer [2004] for the German VDAX, which concentrate on the jump extension to thediffusion based modeling. The volatility index derivative pricing aspects are not considered.

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March 1, 2004 and March 1, 2005, by minimizing the mean-square objective function. Theempirical performance of the model is analyzed over the VIX futures prices recorded onMarch 1, 2005. Zhang and Zhu [2007] derive a static no-arbitrage model for the VIX fu-tures pricing based on the variance term structure for the S&P 500 index options market.The Heston-type stochastic volatility model, with time-dependent long-term mean levelin the volatility diffusion, is calibrated to the S&P 500 option prices using the Avellanedaet al. [2000] Monte Carlo novel approach. The empirical application concerns only par-ticular contracts traded on March 5, 2005. Lin [2007] proposed the VIX futures pricingframework based on the Heston-type stochastic volatility dynamics extended with simul-taneous returns and variance jump components, and a state-dependent jump intensity.The pricing model is provided in a (quasi) closed form and relies on a Taylor expansionconvexity correction approximation. Its empirical assessment concerns pricing and hedg-ing performance carried out on daily derivative prices recorded between May 19, 2004 andMay 16, 2006. Adding jumps in the return and volatility dynamics mainly contributesin short-term futures pricing, where the latter enhances hedging effectiveness. The studyputs much effort to the model estimation issues, which is even mentioned as its primaryfocus. The model parameters are estimated with the generalized method of moments(GMM) estimation approach (Hansen [1982]) utilizing five minutes VIX returns (April2004 – 2006) to calculate integrated volatility and its close readings for the conditionalmoments. The dynamics underlying Lin’s model is considered to be the most completewithin affine stochastic volatility framework, however, a complexity of the resulting pricingformula might be perceived as a drawback in its empirical applications (Dupoyet et al.[2011]). Zhang et al. [2010] establish the relation between the VIX futures values andthe underlying VIX, and provide a framework based on the S&P 500 stochastic volatil-ity model. The long-term mean in the instantaneous variance equation is assumed to bestochastic. The parameter values are calibrated to the market VIX futures prices mini-mizing the mean-squared criterion. The empirical sample covers the period from March26, 2004 to February 13, 2009. Lu and Zhu [2010] propose another VIX futures frame-work based on the multi-factor logOU volatility dynamics and stochastic long-term mean.The principal component analysis carried out over the VIX futures price returns from theinception (March 26, 2004) up to January 30, 2009. They found that the best modelincludes three factors, where the first one covers 91% of the price term structure move-ment and captures its level shift, and the other two correspond to the slope and curvature,respectively. All factors account for 99% of the total risk. The full model improves its one-factor counterpart’s performance, in particular for the short-term contracts. The modelsare estimates with the Kalman filer and maximum likelihood methods. Zhang and Huang[2010] study the new CBOE S&P 500 three month variance futures market launched by

15

the CBOE on May 18, 2004. They establish a linear relationship between the price of thefixed time-to-maturity variance futures with the VIX2, by a square-root mean-revertingmodel for the S&P 500 index volatility. The empirical sample spans the period betweenMay 18, 2004 and August 17, 2007. Zhu and Lian [2011] continue developing the VIXfutures pricing framework based again on a Heston-type dynamics with jumps in the S&P500 returns and volatility. They show that using a convexity correction approximationto price the VIX futures as applied in Lin [2007] may lead to large mispricing. The pro-posed formula seems to alleviate these errors. The model performance improves whenthe returns dynamics is amended with the jump component. The model parameters areestimated with the Markov Chain Monte Carlo (MCMC) methodology, which utilizes theinformation from the S&P 500 and VIX index returns, and the VIX futures. The analyzedderivative sample was recorded over the period from March 26, 2004 to July 11, 2008. Thejumps in volatility did not greatly improve the futures pricing performance. Finally, Shuand Zhang [2012] do not develop any pricing framework but investigate the VIX marketinformation efficiency and causality between the VIX futures prices and the VIX itself,based on the cointegration analysis. They found that the volatility futures market is in-formation efficient, and the futures historical prices are useful in predicting the short-termcontract values, which in turn lead to the spot VIX index.

Within the VIX option pricing literature, Sepp [2008a,b] apply the stochastic mean-reverting square-root variance model with variance constant intensity jumps to describethe joint (S&P 500, VIX) evolution, and demonstrated how to price and hedge the VIXoptions and futures. The model in-sample evaluation relies on the VIX option pricesrecorded on July 25, 2007, and its (option and futures) hedging effectiveness is assessedon simulated data scenarios. The VIX futures position is covered with different futuresexpiring at different dates, which differs from the Lin [2007] strategy involving the longposition in the underlying spot VIX. The delta hedging for options relies on a long positionin the same as option or shorter-maturity futures contract. This strategy is extended byone more futures to hedge against the risk of the infrequent and large VIX jumps (delta-jump hedging), similar to Andersen and Andreasen [2000]. Lin and Chang [2009, 2010]follow the stochastic volatility diffusion model to price the VIX option contracts based onthe Lin [2007] set of the S&P 500 index dynamics: the Heston-type dynamics extendedwith simultaneous return and volatility jumps with state-dependent intensity. In the firststudy, the empirical performance assessment, carried out over the February 24, 2006 –November 30, 2007 sample of daily option recordings, revealed that the jump amendmentenhances the pricing performance over this period. The option hedging effectiveness wasnot investigated. In the next paper, the authors propose the VIX option and futurespricing framework based on the first paper dynamics. The analyzed sample was almost

16

one year longer, spanning the period between February 24, 2006 and September 30, 2008.The model assessment concerns pricing and hedging of the VIX option contracts, where inthe latter, a long position in the VIX futures is taken and the portfolio is rebalanced dailyor weekly (five trading days). The structural parameters in Sepp’s and Lin-Chang’s modelsare calibrated to the market VIX option prices, following the mean-squared criterion.

Recently, Cheng et al. [2012] carried out a thorough theoretical re-assessment on thesefour models, and pointed out some inconsistencies in the pricing formula derivations in theLin-Chang studies. The resulting formulas are neither an exact solution of their pricingequations, nor can they serve as approximations, which is confirmed by the analysis of theHeston-type models as a special case. The Sepp [2008a,b] pricing formulas were proven tobe correct.

Despite the area of volatility derivative pricing developing fast, it stays far from beingconcluded (Carr and Lee [2009]). Recent trends indicate increasing interests in the non-affine VIX dynamics formulations, like the constant elasticity variance (CEV) dynamicsintroduced in Cox [1975] and Cox and Ross [1976],

dXt = (m− κXt)dt+ σXγt dWt, (2.1)

where Xt is the spot VIX value or its logarithm, m/κ ratio represents the long-termmean of Xt, κ is the speed of mean reversion, σ is the diffusion parameter, and W P

t is astandard Wiener process. The new element, which actually nests the affine models withinthe non-affine family is the exponent γ. For example, if Xt ≡ V IXt and γ = 0.5 then(2.1) corresponds to the square-root mean-reversion CIR model, if Xt ≡ ln(V IXt) andγ = 0, then (2.1) becomes logOU dynamics. Application of the CEV-type approach iswell-documented in modeling the interest rates (Chan et al. [1992]), however, its utility indriving the market volatility is still under discussion.

Within the equity option pricing context, Christoffersen et al. [2011] provide an exten-sive empirical study comparing different types of the stochastic volatility models, includingthe CEV-type dynamics. The non-affine specifications were superior over the square-rootbased models, however, the discrete-time transition probabilities or characteristic func-tions are not available in closed forms, which makes these models very difficult to behandled numerically. Figlewski [1997] (pp. 21 in his monograph) highlights that using theCEV approach in modeling the return volatility dynamics does not necessarily improvethe volatility forecasts, as the diffusion power will change over time yielding in realitydifferent dynamics, and makes derivative valuation substantially harder.

Indeed, so far the applications of the CEV-type dynamics concentrate mainly on mod-eling the stochastic dynamics of the instantaneous volatility and the VIX itself, ratherthan using it as the underlying dynamics for the derivative pricing. Jones [2003] investi-

17

gate the Heston-type dynamics with CEV-extended instantaneous variance to model theS&P 100 index returns. The ‘old’ VIX is used as the volatility proxy. The model is es-timated with MCMC methodology using the daily stock index returns recorded over theJanuary 2, 1986 to June 2, 2000 period. The obtained CEV parameter γ estimates overthe 1986 –, and 1988 – 2000 year subsamples are 1.33 and 1.17, respectively. Bakshi et al.[2006] similar as in Jones [2003] process the S&P 100 index series using different dynamicsnested within the Heston model, with the old VIX as a market volatility measure. Follow-ing the maximum likelihood approach the CEV estimates were found to be between 1.2and 1.5. Aït-Sahalia and Kimmel [2007] point out some inconsistencies in modeling thenew VIX with (2.1) type process. In the developed maximum likelihood framework, theyrely on the Heston type stochastic volatility approach to model the S&P 500 series. Thesquare-root power in the instantaneous volatility equation is relaxed to any γ ≥ 0 and thespot VIX is used as a volatility proxy. Concurrently, they consider the same dynamicsfor the volatility index in a one-dimensional version. The estimation carried out on theJanuary 2, 1990 – September 30, 2003 (S&P 500, VIX) daily recordings, provided the γestimates on the 0.65 and 0.94 levels, respectively. Such discrepancy was considered asan evidence of some degree of misspecification of the CEV model. Kaeck and Alexander[2010] develop the VIX dynamics based on a one-dimensional CEV diffusion (with jumps),and extended to a stochastic volatility-of-volatility approach. However, the γ parameteris constrained to the values which correspond to the affine specifications (γ = {0.5, 1}),together with Xt corresponding to the spot or log VIX level, respectively. Duan and Yeh[2010] investigate the Heston-type S&P 500 stochastic volatility approach with the CEVtype relaxation to model the instantaneous volatility dynamics augmented with the pricejumps. The study develops a closed form relation between the S&P 500 latent stochasticvolatility structure and the spot VIX information, under the risk-neutral measure. Themodel estimation methodology relies on the maximum likelihood approach and the S&P500 and VIX daily series recorded between January 2, 1990 – August 31, 2007, and theirfive-seven years long subperiods. The CEV parameter γ estimates shed some light on theCIR and logOU relevance in modeling the S&P stochastic variance, and the VIX itself.The acquired range of the estimates stayed in a 0.89 – 1.03 interval, indicating that thepopular square-root dynamics is strongly at odds with the data and gives more credit tothe logarithmic specification. Notice that these outcomes contradict with the Aït-Sahaliaand Kimmel [2007] outcomes, where the sample was four years shorter (1990 – 2003) anddid not consider jump extension.

The first attempt to apply this methodology in pricing the VIX contracts is provided inDupoyet et al. [2011]. They follow one-dimensional mean-reverting square-root (CIR) andCEV based approach, both with and without jump extensions, to deliver the VIX futures

18

pricing framework. The no-jump CEV based model outperforms the other specificationsin all VIX futures expiration groups. The model was estimated within the GMM method-ology, which utilized the VIX empirical sample recorded between March 2002 – September2006; the VIX futures prices come from the April 2004 – September 2006 period. Duringthis time, the S&P 500 volatility index is considered to be in its tranquil state (Kaeck andAlexander [2010]), and jumps take the form of spikes rather than large changes. In thissituation, the diffusion element is more prone to explain the VIX volatility rather thanthe jump component. This may also explain why the jump-extended CIR-based modelprovides worse pricing performance than its no-jump counterpart. The CEV parameterestimate was between 1.59 (CEV with jump) and 1.61 (CEV), which is much beyond thetypical outcomes in the already reviewed studies.3

3The γ ≈ 1 estimate over the January 1990 – August 2007 VIX daily series in Duan and Yeh [2010]suggests that the γ ≈ 1.6 outcome in Dupoyet et al. [2011] does not fully reflect the VIX long-termtendency. In addition, Chacko and Viceira [2003] underline that any estimates of the CEV diffusion pa-rameter, which goes substantially much beyond 1, should be considered as a sign of the model specificationrejection rather than get an economic interpretation.

Chapter 3

Classical Models for the VolatilityDerivative Pricing

3.1 Pricing setup

Following Bakshi et al. [1997], we make three basic assumptions within the derivativepricing model: the specification of the underlying process, the interest rate process, andthe market price of risk. In the basic case, the level of the underlying VIX at time t, Vt,is modeled by a one-dimensional, Markov type diffusion process

dVt = a(Vt, t, θP )dt+ b(Vt, t, θP )dW Pt , (3.1)

where a and b are known real-valued, model specific drift and diffusion functions, de-pending on a finite dimensional parameter θP , and where a standard Wiener process W P

t

captures the uncertainty in the market under the physical measure P . It is assumed thatthere is a market for instantaneous borrowing and lending at a risk-free rate r. The moneymarket account Bt follows

dBt = rBtdt. (3.2)

Assuming absence of arbitrage opportunities, the current price of derivatives is equal tothe conditional expected value under the measure Q of the payoff function at expirationdiscounted with respect to the market risk-free interest rate. We consider all availablederivatives on the underlying VIX in the CBOE market: European call and put optionsand futures contracts. In particular, the price of the VIX futures contract expiring withinτ (years) is expressed as

F (Vt, τ) = EQt (Vt+τ ). (3.3)

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20

Analogously, the price of a European VIX call option expiring within the same periodwith strike price K is derived from

C(Vt, K, τ) = EQt (Vt+τ −K)+. (3.4)

These derivatives are available for several strike prices and times to maturity. Motivatedby Girsanov’s theorem, the transition of the physical probability measure P to the risk-neutral measure Q is determined by

dWQt = λtdt+ dW P

t , (3.5)

where the market price of risk λt = λ(Vt, t) is such that WQt is a standard Wiener process

under Q. Prices of calls and puts with the same strike level K, and the same time tomaturity τ , are related via the put-call parity

C(Vt, K, τ)− P (Vt, K, τ) = Vt − e−rτK. (3.6)

However, since the underlying VIX is non-tradable, we need an additional instrument likethe VIX futures contract to adapt this relation to the VIX derivative pricing purposes,which results in (Grünbichler and Longstaff [1996])

C(Vt, K, τ)− P (Vt, K, τ) = e−rτ (F (Vt, τ)−K). (3.7)

Within the current framework, we consider the specific models of Whaley [1993], Grün-bichler and Longstaff [1996], and Detemple and Osakwe [2000]. These models allow forexplicit, closed-form pricing formulas for European call and put options, and futures con-tracts. Then, the underlying VIX diffusion dynamics is extended by the jump components,and relevant pricing methodology is provided.

3.2 Pricing Models Based on Diffusions

3.2.1 Whaley model

Underlying process

In Whaley’s (Whaley [1993]) original approach, to model the underlying log-return dy-namics of the volatility index VIX, a Geometric Brownian Motion (GBM) is used,

d ln Vt = (µ− σ2

2 )dt+ σdW Pt , (3.8)

where µ and σ are the trend and diffusion parameters, respectively. Whaley [1993] assumesthe underlying VIX to be tradable, hence, the market price of risk is completely determined

21

by the P -process parameters, and the risk-free market rate. We adapt this approach to anon-tradable VIX asset, implying the market price of risk λt as an additional free parameterand take λt = λ. Thus, under the risk-neutral measure Q,

d ln Vt = (µQ − σ2

2 )dt+ σdWQt , (3.9)

where µQ = µ−σλ. In this way, Whaley’s model becomes comparable to the other modelsin the subsequent subsections. Due to the additional market price of risk parameter λ,it does not suffer from the lack of parameters in the original model when fitting modelprices to market prices. Using GBM as a way to model the log-volatility process can alsobe found in stochastic volatility models like Hull and White [1987]. However, a GBMdynamics (3.8) may not be the most appropriate to describe the volatility: 1) the meanfluctuates around an exponential trend (E0(Vt) = V0 exp{µt}), 2) the variance increasesexponentially with time (V ar0(Vt) = V 2

0 exp{2µt}(exp{σ2t}− 1)), 3) VIX seems to reverttoward its long-term mean. Therefore, we use (3.8) mainly as a reference specification.

Derivative pricing formulas

The derivative prices are obtained as in the Black [1976a] framework keeping in mind thechanged drift term due to the non-tradability of the asset. In particular, for a futurescontract expiring at time t+ τ , this yields (Psychoyios et al. [2003])

F (Vt, τ) = Vte(µ−σλ)τ , (3.10)

and for a European call option expiring at the same time with strike K,

C(Vt, r,K, τ) = e−rτF (Vt, τ)N(d)− e−rτKN(d− σ√τ), (3.11)

where d = ln(F (Vt,τ)/K)+σ22 τ

σ√τ

, and N(.) denotes a standard normal cumulative distributionfunction (CDF). The European put option prices can be determined from the put-callparity (3.7).

3.2.2 Grünbichler-Longstaff (GL) model

Underlying process

Grünbichler and Longstaff [1996] assume that the underlying VIX follows a mean-revertingsquare-root, or a CIR type process (Cox et al. [1985])

dVt = (m− κVt)dt+ σ√VtdW

Pt , (3.12)

22

where the drift function ensures the volatility reversion towards its long-run mean m/κ,with speed determined by κ, σ is the diffusion parameter. A similar volatility specificationis used in the Heston [1993] derivative pricing model. Cox et al. [1985] show that under(3.12) and for γ = 4(κ+σλ)

σ2(1−exp{−(κ+σλ)τ}) the conditional distribution of γVt+τ given Vt is of anon-central χ2 type with ν = 4m/σ2 degrees of freedom, and non-centrality parameter ζt =γVt exp{−(κ+σλ)τ}. The process (3.12)remains strictly positive if the Feller (Feller [1951])condition 2m ≥ σ2. In the original framework, the underlying volatility is considered asa non-tradable asset, and the volatility risk assumed to be proportional, under the risk-neutral measure Q, to the current level of volatility λt = λ

√Vt. Given these assumptions,

the (3.12) process under the new measure takes the following form

dVt = (m− κQ)Vt)dt+ σ√VtdW

Qt , (3.13)

where κQ = κ + σλ. The assumption about the proportional to the underlying risk-premium is in line with the literature (e.g., Heston [1993], Dai and Singleton [2000], Pan[2002], Broadie et al. [2007]), and originates from Cox et al. [1985]. Bollen [1997] relaxedthis assumption into different parametric forms.

Derivative pricing formulas

For this framework, Grünbichler and Longstaff [1996] provide the following closed-frompricing formulas for futures and European call option contracts,

F (Vt, τ) = 1− exp{−(κ+ σλ)τ}κ+ σλ

m+ Vte−(κ+σλ)τ , (3.14)

C(Vt, r,K, τ) = e−rτ1− exp{−(κ+ σλ)τ}

κ+ σλm{D(γK; ν + 2, ζt)−D(γK; ν + 4, ζt)}

+e−rτFtD(γK; ν + 4, ζt)− e−rτKD(γK; ν, ζt). (3.15)

The European put price formula can be derived from the put-call parity (3.7).

3.2.3 Detemple-Osakwe (DO) model

Underlying process

Detemple and Osakwe [2000] use an alternative to the CIR dynamics (3.12) and modelthe log VIX returns as an Ornstein-Uhlenbeck (Uhlenbeck and Ornstein [1930]) process(logOU)

d ln Vt = (m− κ ln Vt)dt+ σdW Pt , (3.16)

23

or alternatively

dVt = (m+ 12σ

2 − κ ln Vt)Vtdt+ σVtdWPt , (3.17)

where, as in the CIR dynamics, m/κ ratio denotes the long-run volatility mean and κ

governs the speed of the mean-reversion. A similar volatility specifications are used inthe Wiggins [1987] and Melino and Turnbull [1990] stochastic volatility option pricingmodels. The log modification of the OU process, or exponential OU (Scott [1987]), keepsit positive and is featured by an easy to handle normal conditional distribution of ln Vt+τgiven ln Vt.1 The latter provides an advantage over the non-central χ2 one in the case ofCIR process, especially in numerical implementations. The discrete time version of thelogOU, the EGARCH model of Nelson [1991], turned out to be successful in capturing thevolatility clustering and the leverage effect (Psychoyios et al. [2003]). Assuming λt = λ

(constant), the process (3.16) under the measure Q takes the following form

d ln Vt = (mQ − κ ln Vt)dt+ σdWQt , (3.18)

where mQ = m − σλ. Originally, Detemple and Osakwe [2000] made the assumptionthat the market risk premium is zero. However, this assumption is relaxed here and λ isconsidered as a free parameter. In the literature, there is no standard approach about thevolatility risk structure in the OU type processes. Theoretically, the market price of riskcan be assumed constant, constrained to zero, or it can be proportional to Vt (Ball andRoma [1994], Boswijk [2001]). Besides the original DO framework, the λt = 0 approachis adopted in Psychoyios and Skiadopoulos [2006]. Ball and Roma [1994] and Psychoyioset al. [2009] apply the proportional to Vt volatility risk form. Finally, Melino and Turnbull[1990], Dai and Singleton [2000], and Lu and Zhu [2010] consider the unproportional(constant) form. We follow the last framework to preserve tractability of the model.Notice that changing the measure (after (3.5)) affects the m parameter of the drift, so thethe risk premium is becomes proportional to Vt under dynamics (3.17).

Derivative pricing formulas

Detemple and Osakwe [2000] derive the following pricing solutions for futures and plainvanilla call option contracts,

F (Vt, τ) = exp{e−κτ ln(Vt) + 1− exp{−κτ}

κ(m− σλ) + α2

2 τ}, (3.19)

C(Vt, r,K, τ) = e−rτF (Vt, τ)N(d)− e−rτKN(d− α√τ), (3.20)

1Stein and Stein [1991] adopt the non log-transformed (arithmetic) version of the Ornstein-Uhlenbeckdynamics (e.g., Vasicek [1977]) to model instantaneous volatility in stochastic volatility model, whichimplies that σt may become negative (Ball and Roma [1994]).

24

where d = ln(F (Vt,τ)/K)+ 12α

2τ

α√τ

and α = σ{

1−exp{−2κτ}2κτ

}1/2. Similar as in the Whaley or GL

model, the put price formula might be derived from the put-call parity (3.7).

3.3 Pricing Models Based on Jump-Diffusions

3.3.1 Merton model

Underlying process

The Merton [1976] model is considered as the jump extended specification with respect toWhaley’s approach. The underlying dynamics for the volatility index is assumed to follow

dVt = (µ− γEP [Jt − 1])Vtdt+ σVtdWPt + (JPt − 1)VtdqPt , (3.21)

where µ and σ are is the instantaneous expected return and volatility of the asset, respec-tively, and W P

t is a Brownian motion under the physical measure P , qPt is the P -measurePoisson process with intensity γ such that P (dqt = 1) = γdt and P (dqt = 0) = 1−γdt. The(absolute) price jump size Jt is assumed to follow, under the measure P , the log-normaldistribution

log(Jt) ∼ N(µJ , σ2J). (3.22)

The process (3.21) resembles the GBM dynamics most of the time, but on average γ timesper year the Vt jumps discretely by a random amount Jt.

The market generated by jump-diffusion dynamics is incomplete (Andersen and An-dreasen [2000]). The market incompleteness implies that the derivative price cannot beuniquely determined by a non-arbitrage assumption. There are two sources of risk to beconsidered in the assumed VIX dynamics. The first concerns the risk implied by the Brow-nian motion diffusive component and is called the volatility risk. The other one, called thejump risk, comes from the possible abrupt changes in the price level, and usually concernstime of the jump and its size. In the original Merton [1976] framework, the jump (timingand/or size) risk is assumed to be non-systematic and diversifiable. This assumption isrelaxed in the current framework. After Pan [2002], the jump-size and jump-timing riskpremiums are determined by

λµJ = EQ(Jt − 1)− EP (Jt − 1)= eσ

2J/2(eµ

QJ − eµPJ ), (3.23)

λγ = γQ − γ, (3.24)

25

assuming that no premium is given to the jump size variance risk, σQJ = σJ .2 Originally,Pan [2002] assumes that the jump-timing risk premium is absorbed by the jump-size risk,and this type of premium is not considered. However, due to some numerical problemsassociated with the jump intensity parameter estimation (Honoré [1997], He et al. [2006]),it is treated here as a free parameter, and left to be determined by the market. Thevolatility (diffusive) risk is assumed to be proportional to the current volatility dynamicsµQ = µ − λσ. Given these assumptions, the underlying dynamics under the risk-neutralmeasure Q is described by

dVt = (µQ − γQkQ)Vtdt+ σVtdWQt + (JQ − 1)VtdqQt , (3.25)

where kQ = EQ[Jt − 1].

Derivative pricing formulas

The price of a European call option expiring within time τ with strike price K is given by

C(Vt, K, τ) =∑j≥0

e−γQτ (γQτ)jj! Cj(Vt, K, τ, σj), (3.26)

where Cj denotes the Black-Scholes (Black and Scholes [1973]) European call option pricingformula, given j (possible) number of jumps, which may occur since the option inceptionup to time-t (Bates [1991]),

Cj(Vt, K, τ, σj) = e−rτ [VtebjτN(dj)−KN(dj − σj√τ)], (3.27)

where bj = µQ − γQkQ + jµQJ +σ2

J/2τ

, σj =√σ2 + j

σ2J

τ, and dj = ln(Vt/K)+(bj+σ2

J/2)τσj√τ

. TheMerton pricing formula might be interpreted as a sum of individual Black-Scholes values,weighted by the probability of j jumps in the lifetime of the option. If there are no jumps(j = 0, γ = 0) the Merton pricing formula simplifies to the standard Black-Scholes one.The value of a put option is obtained by substituting the Black-Scholes put price in (3.26).

Given the formulas for the European call and put option values, the pricing formulafor futures expiring within τ periods emerges from the put-call parity (3.7) and

F (Vt, τ) = VteµQτ−γQτ(kQ+1) ∑

j≥0

(γQτ(kQ + 1))jj! , (3.28)

2He et al. [2006] and Andersen and Andreasen [2000] follow Naik and Lee [1990] where the marketrepresentative agent is assumed to maximize the power utility function, and imply the following formulasto bind the risk-neutral parameters with historical estimates, σQJ = σJ , µQJ = µJ + (λµJ − 1)(σQJ )2 ,γQ = γ exp{(λγ − 1)(µQJ + 1

2 (1− λγ)(σQJ )2)}.

26

which reduces to

F (Vt, τ) = VteµQτ . (3.29)

The occurrence of jumps does not affect the risk-neutral expectation used in the futuresprice determination (Hillard and Reis [1998]).

3.3.2 Mean-reverting models with jumps

Underlying processes

Under the objective measure P , the jump-extended baseline dynamics

dVt = (m− κVt)dt+ σ√VtdW

Pt + JPdqPt (3.30)

will be called a CIR jump-diffusion (CIR-J) and

d ln Vt = (m− κ ln Vt)dt+ σdW Pt + JPdqPt . (3.31)

a log Ornstein-Uhlenbeck jump-diffusion (logOU-J). Alternatively, after an application ofIto’s lemma, dynamics (3.31) can be written as

dVt = (m+ 12σ

2 − κ ln Vt)Vtdt+ σVtdWPt + (eJ − 1)VtdqPt . (3.32)

In all these specifications, qPt denotes a Poisson process controlling the jump occurrence,featured by a constant jump intensity γ, such that P (dqPt = 1) = γdt and P (dqPt = 0) =1 − γdt. The Brownian motion W and Poisson process q are assumed to be mutuallyindependent. Following Kou [2002] and Dotsis et al. [2007], the random jump size J isassumed to follow an exponential distribution with density

f(J) = pη+ exp{−η+J}1J≥0 + qη− exp{−η−J}1J<0, (3.33)

with p, q ≥ 0 and p + q = 1 representing the probabilities of the upward and downwardjumps, and 1/η+ and 1/η− as the means of the upward and downwards jumps, respec-tively. An asymmetric double exponential distribution is numerically more tractable (overthe normal distribution) and empirically more realistic since it allows for upward anddownward jumps. However, in our further analysis only upward jumps are considered topreserve the positivity of Vt (Duffie et al. [2000], Bakshi and Cao [2002]), such that

f(J) = η+ exp{−η+J}1J≥0. (3.34)

27

The underlying dynamics (3.30) and (3.31), transformed up to a risk-neutral measure Q,are

dVt = [m− κQVt]dt+ σ√VtdW

Qt + JQdqQt , (3.35)

d ln Vt = [mQ − κ ln Vt]dt+ σdWQt + JQdqQt , (3.36)

whereWQt and dqQt represent Wiener and Poisson processes under the risk-neutral measure,

respectively. The volatility risk premium structure in (3.35) and (3.36) remains similar asin (3.13) and (3.18), respectively. After Pan [2002] and Broadie et al. [2007] the jump-riskpremiums is determined by

λη = ηQ+ − η+. (3.37)

These references omit, or consider as a special case, the jump-timing risk (γQ = γ) sinceits role might be taken by the jump-size risk λη.

Derivative pricing formulas

The pricing framework follows the general one for affine jump-diffusions proposed in Bak-shi and Madan [2000], which relies on the characteristic function concept spanning anyderivative payoff function. For details, see Appendix A.1. Both the CIR-UJ and logOU-UJunderlying VIX dynamics are nested within those assumed in Grünbichler-Lonstaff (GL)and Detemple-Osakwe (DO) specifications. Henceforth, the new pricing frameworks, rely-ing on the mean-reverting jump extended diffusions, will be called the GL-UJ and DO-UJmodels, respectively.

Under both (3.35) and (3.36) dynamics, the price of the call option with strike K andtime to the expiration τ is given by (Appendix A.1, formula (A.2))

C(Vt, K, τ) = e−rτ [F (Vt, τ)Π1(τ)−KΠ2(τ)], (3.38)

where price of the futures contract expiring at t+ τ , F (Vt, τ) and probability Πj, j = 1, 2are determined by:

• for GL-UJ,

F (Vt, τ) =[m+ γ/ηQ+

κQ(1− e−κQτ ) + e−κ

QτVt

], (3.39)

Πj = 12 + 1

π

∫ ∞0

Re[

exp(−iφK)× fj(Vt, τ, φ)iφ

]dφ, (3.40)

28

• for DO-UJ,

F (Vt, τ) = exp

4mQ(1− e−κτ ) + (1− e−2κτ )σ2

4κ + γ

κln(ηQ+ − e−κτ

ηQ+ − 1

)+

+ e−κτ ln Vt

, (3.41)

Πj = 12 + 1

π

∫ ∞0

Re[

exp(−iφ lnK)× fj(Vt, τ, φ)iφ

]dφ. (3.42)

The time-scaled characteristic functions fj(.) for the assumed CIR-UJ and logOU-UJdynamics are provided in the Appendix A.12 and A.13, respectively. The European putoption price formula emerges from the put-call parity relation (3.7), given the call andfutures pricing formulas.

3.4 Estimation

The parameters can be directly calibrated to the derivative data, by minimizing the dif-ferences between the model and observed market prices. Under the stochastic volatilityfundamental dynamics, this methodology is applied in Bates [1996, 2000], Bakshi et al.[1997] for the equity options, and Lin and Chang [2009, 2010] for the VIX option prices.Wang and Daigler [2011] utilize it to price the VIX option contracts under a diffusion anda stochastic volatility model. Zhang and Zhu [2006] and Lin [2007] apply it for the VIXfutures. This methodology brings a couple of important drawbacks. First of all, thereis no consensus about the form of the objective function to be minimized, such that itdoes not favor one contract over the others. Next, the objective function might be farfrom being convex with many local extrema, which makes the numerical results unstableand dependent on the initial parameter values choice (e.g., Mikhailov and Nogel [2003]).3

Then, last but not least, as the number of parameters to be calibrated increases, the opti-mization problem becomes usually more complex and the global optimum is more difficultto be reached. This may impair the jump-diffusion models performance with respect totheir diffusion based counterparts. We use this approach as a reference in the estimation

3To overcome this problem, various regulating and correction methods were proposed. Cont andTankov [2004] add to the squared-error function a relative entropy factor, which usefulness is checked byMatsuda [2005]. Mikhailov and Nogel [2003] regulate the function by the Euclidean difference betweenthe initial and implied parameter values. He et al. [2006] propose a solution based on the local volatilityconcept.

29

of the diffusion based models, where the number of parameters to be calibrated is small,but do not apply it for the jump-diffusion based models.

There are methodologies, which tend to combine the return time series and derivativedata in the estimation procedure. Chernov and Ghysels [1999] underline that calibrationto the derivative cross-section allows for a good in-sample fit, but does not cover thedynamics of the underlying series. This approach relies essentially on mixing two differentstatistical techniques: time series estimation of the objective measure diffusion parameters,and the market prices of risk implied by the derivative prices. Such a two-step approachis utilized in Heston and Nandi [2000b], Christoffersen and Jacobs [2004b], and Hsiehand Ritchken [2005] for the discrete-time GARCH-based equity pricing models. In thefirst step, the parameters are estimated from the historical asset return series using themaximum likelihood (ML) methodology (Bollerslev [1986]). If the ML approach is feasible,then it becomes the preferred estimation method (Campbell et al. [1997], pp. 358).4 Asfor the affine (jump) diffusion processes we consider, the conditional density functionsare known or they can be approximated from the corresponding conditional characteristicfunction inversion (Duffie et al. [2000]), we follow the ML methodology in the first step.Then, given these estimates, the market risk premia are implied from the derivative marketprices by minimizing the mean squared relative error criterion.

In the alternative to the two-step methodology approach, the returns and derivativecross-sectional data are combined in a so called ‘single-step’ estimation procedure. Withinthis setup, Chernov and Ghysels [2000] and Pan [2002] propose the methodologies be-ing modifications of the efficient and generalized method of moments (EMM, GMM) ofHansen [1982] and Gallant and Tauchen [1996], respectively. Despite being a promisingvenue for the further explorations, we do not apply this approach in our analysis. Thismethodology was proposed explicitly for the stochastic volatility models, and even Cher-nov and Ghysels express uncertainty how well the procedure will perform for diffusions,and their pricing and hedging performance (Chernov and Ghysels [2000], pp. 411). One ofits outstanding features is that all the parameter and risk premia estimates get standarderror values. This is a strong advantage over the calibration-based methodologies, wherea lack of appropriate statistical theory precludes the estimation of standard errors (Bates

4Due to the presence of a latent volatility structure, the maximum likelihood estimation approachis not always feasible for the stochastic volatility models. Hence, some alternative approaches can beutilized, like the Generalized Method of Moments (GMM) of Hansen [1982], the Indirect-Inference (II)based Simulated and Efficient Method of Moments (SMM, EMM) of Duffie et al. [2000] and Gallant andTauchen [1996], respectively, or the Bayesian MCMC-based frameworks (e.g., Jones [2003], Eraker [2001]).These methods allow for filtering the latent spot volatility dynamics and achieving the ML efficiency level,without knowing an explicit form of the transition density function (Chernov and Ghysels [1999]).

30

[2000]). However, due to the complexity of the objective functions and problems withstability of the optimization procedures, the authors leave the precision of the resultingstandard error values as an open question for further investigations (Chernov and Ghy-sels [2000], pp. 430). In addition, Broadie et al. [2007] point out that the computationalburden of this methodology constrains how much and what type of data can be used inthe estimation process. Instead, they advocate the already described two-step approach,which is considered as a pragmatic compromise between the computational feasibility andstatistical efficiency, if the return and derivative data are planned to be combined in thederivative pricing model estimation.

3.4.1 P -process estimation

The conditional maximum likelihood (ML) estimation procedure requires knowledge of thetransitional probability density function (PDF) of the process Xt, f(Xt+τ |Xt, θ

P ), whereτ is the sampling time interval. Given a (N + 1) long sample series {Xt}T=Nτ

t=1 , the set ofthe parameters under the real-world measure θP is estimated, such that it maximizes thelog-likelihood (logLik) function

θP = argmaxθP

T−τ∑t=1

ln f(Xt+τ |Xt, θP ). (3.43)

Under suitable regularity conditions, θP is consistent, asymptotically normally distributed,and efficient achieving the Cramer-Rao lower bound for consistent estimators (Cox andHinkley [1974], pp. 288). Hence, if the ML is feasible, then it becomes the preferredmethod of estimation (Campbell et al. [1997], pp. 358).

For the dynamics underlying the Whaley (GBM), Grünbichler-Longstaff (CIR), andDetemple-Osakwe (logOU) models, the transition densities correspond to known distribu-tions: log-normal, non-central χ2, and normal, respectively (Psychoyios et al. [2003], Hurnet al. [2007], Gourieroux and Jasiak [2001] pp. 287).

For the Merton model, the transitional PDF of ln Vt+τ given Vt in (3.21) takes thefollowing form (Matsuda [2004]),

f(lnVt+τ |Vt) =∞∑j=0

exp {−γτ} (γτ)j

j! ×exp

{−[ln Vt+τ

Vt−(µ−γkP−σ2

2 )τ−jµJ]2

2(σ2τ+jσ2J

)

}√

2π(σ2τ + jσ2J)

, (3.44)

where j denotes the number of jumps and kP = EP (Jt − 1) = eµJ+ 12σ

2J − 1.

For the mean-reverting models with the (upward) jump-diffusion dynamics, CIR-UJand logOU-UJ, the density functions will be acquired from the corresponding characteristic

31

function for the affine diffusions (Duffie et al. [2000]),

ϕ(Xt, τ, φ, θP ) = exp(α(φ, τ, θP ) + β(φ, τ, θP )Xt), (3.45)

compare with (A.12) and (A.13). The conditional densities are derived from the Fourierinversion of the relevant ϕt. In particular,

f(Xt+τ |Xt, θP ) = 1

π

∫ ∞0

Re[exp{−iφXt+τ}ϕ(Xt, τ, φ, θP )]dφ, (3.46)

where Xt ≡ Vt for CIR-UJ, and Xt ≡ ln Vt for logOU-UJ. The ML via Fourier inversionprovides asymptotically efficient estimates of the unknown parameters (Aït-Sahalia [2004],Bates [2003]). The integral in (3.46) is approximated using the Legendre type of Gauss-quadrature method (Singleton [2001]).

3.4.2 Market-implied Q-estimates

The VIX derivative pricing models we consider are complete in the sense of delivering thepricing solutions to both types of the VIX derivatives: options and futures. Thus, theirinformation can be combined in one objective function used for the purpose of the modelparameter calibration. For any date t, the observed daily cross-section of the VIX call(C), put (P ), and futures (F ) contracts are used to imply the risk-free parameters θQt byminimizing the following objective function,

argminθQ1 ,...,θ

QT

1T

∑t,i

wCt,iV olt

Ci(θPt , θQt )− Ct,iCt,i

2

+∑t,j

wPt,jV olt

Pj(θPt , θQt )− Pt,jPt,j

2

+∑t,k

wFt,kV olt

Fk(θPt , θQt )− Ft,kFt,k

2, (3.47)

where θPt are the ML estimates under the objective measure (P ) and T denotes the totalnumber of trading days in the sample. The objective function applies a weighting schemeimplied from the derivative day-t trading volume (volt) information, and where

wt,. =

volt,., if the derivative (C, P , F ) information is considered in

the parameter calibration at day-t0, otherwise,

(3.48)

V olt =∑i

wCt,i +∑j

wPt,j +∑k

wFt,k. (3.49)

32

The i and j correspond to the particular combinations of call and put strike and expira-tions, respectively, observed at the particular day t; the futures k distinguishes contractswith different maturities. The daily market prices, which in the case of options are repre-sented by the last observed mid-level between the bid and ask quotes.5

The objective function (3.47) allows for selecting the particular VIX derivative infor-mation or its combinations for the purpose of the parameter calibration. The importanceof the derivative information contribution is weighted by its daily relative trading volume.Such a scheme grants naturally more importance to the intensively traded contracts anddiminishes the influence of the cases with a smaller trading volumes.6 When combiningdifferent derivative information, one needs to consider any (possible) differences in themodel derivative price exposure to the underlying changes. Figures 3.1 and 3.2 providethe option and futures price sensitivities implied by the models. The patterns and mag-nitudes of the exposures, in the diffusion and jump diffusion based pricing frameworks,remain pretty much comparable between both derivative types. Finally, the relative errorsconsidered in the objective function (3.47) diminish the influence of the derivative pricedifferences among the moneyness and/or expiration groups.

5There is a possibly for the time-synchronization bias, reflected in different underlying prices for whichthe contract values might be recorded. However, it is beyond the scope of the analysis to consider theimpact of this issue. For further reference about the non-synchronous trading impact, see Bates [1995],Fleming et al. [1995], and references therein.

6Bates [1995] provides a short discussion about different weighting schemes in the model parametercalibration. He underlines, that the one which relies on the trading volume has a chance to become astandard approach in the future.

33

Whaley

Time to expiration (days)

Cal

l (−

Put

) D

elta

0 50 100 150 200 2500.

00.

51.

01.

5

OTMATMITM

GL

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

OTMATMITM

DO

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

OTMATMITM

Whaley

Time to expiration (days)

Fut

ures

Del

ta

0 50 100 150 200 250

0.0

0.5

1.0

1.5

V=0.10V=0.20V=0.30

GL

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

V=0.10V=0.20V=0.30

DO

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

V=0.10V=0.20V=0.30

Figure 3.1: The diffusion based model derivative price Delta(s) over different maturities. Upper row: thecall(-put) price Delta, ∂C(V )

∂V , over the out-of-the-money (OTM), at-the-money (ATM), and in-the-money(ITM) moneyness groups determined by the strike K=0.20, and the underlying VIX equal to 0.10, 0.20,and 0.30, respectively. Bottom row: the futures price Delta, ∂F (V )

∂V , for the different underlying levels.The parameter estimates: for the Whaley model [µQ,σ]=[0.17, 0.94]’, for GL [m,κQ,σ]=[0.84, 3.41, 0.45]’,and for DO [mQ,κ,σ]=[-5.36, 3.50, 0.94]’. Market risk-free rate is fixed at r=5%.

Merton

Time to expiration (days)

Cal

l (−

Put

) D

elta

0 50 100 150 200 250

0.0

0.5

1.0

1.5

OTMATMITM

GL−UJ

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

OTMATMITM

DO−UJ

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

OTMATMITM

Merton

Time to expiration (days)

Fut

ures

Del

ta

0 50 100 150 200 250

0.0

0.5

1.0

1.5

V=0.10V=0.20V=0.30

GL−UJ

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

V=0.10V=0.20V=0.30

DO−UJ

Time to expiration

0 50 100 150 200 250

0.0

0.5

1.0

1.5

V=0.10V=0.20V=0.30

Figure 3.2: The jump-diffusion based model derivative price Delta(s) over different maturities. Upperrow: the call (-put) price Delta, ∂C(V )

∂V , over the out-of-the-money (OTM), at-the-money (ATM), and in-the-money (ITM) moneyness groups determined by the strike K=0.20, and the underlying VIX equal to0.10, 0.20, and 0.30, respectively. Lower row: the futures price Delta, ∂F (V )

∂V , for the different underlyinglevels. The parameter estimates: for the Merton model [µQ,σ, γQ,µQJ ,σJ ]=[0.3, 0.67, 28.43, 2.78, 0.08]’,for GL-UJ [m,κQ,σ,γ,ηQ+ ]=[1.05, 7.01, 0.36, 16.84, 14.99]’, and for DO-UJ [mQ,κ,σ,γ,ηQ+ ]=[-9.56, 5.16,0.77, 34.5, 17.09]’. Market risk-free rate is fixed at r=5%. (Compare with Figure 3.1).

34

Chapter 4

Data

4.1 S&P 500 volatility index VIX

In September 1993, the Chicago Board Option Exchange (CBOE) started quoting the firststock market volatility index (VIX). The volatility index was designed to reflect investorsentiment and the volatility expectation over the future 30 days (22 business calendardays). Hence, it is often referred as the Investor Fear Gauge (Whaley [2000]) or as theFear Index.1

The first VIX (current ticker: VXO) calculation methodology relied on the frameworkproposed in Whaley [1993]. The index spot value was determined from the Black-Scholes(Black and Scholes [1973]) volatilities implied from eight S&P 100 (ticker: SPO) at-the-money options, such that, at any given time, the VXO represents implied volatility of ahypothetical at-the-money option with exactly 30 days to the expiration (Carr and Wu[2006]). On September 22, 2003, this methodology was replaced by a new one, consideredto be more robust and representative for the market. The major changes concerned thebaseline S&P 100 index, which was replaced by the more extensive S&P 500 (ticker: SPX)index measure. The new VIX index (ticker: VIX) uses near-term and next-term out-of-themoney SPX options with at least eight days left to expiration, and then weights them toyield a constant, 30-day measure of the expected volatility of the S&P 500 Index. TheBlack-Scholes parametric approach was replaced with a new design which uses a kernel-smoothed estimator to imply the volatility from the SPX option prices. Carr and Wu[2004] show that the squared spot VIX approximates the 30-day variance swap rate, sinceit represents the conditional risk-neutral expectation of an annualized variance return overthe next 30 calendar days. Detailed description of the VIX calculation methodologies canbe found in Whaley [1993], Carr and Wu [2006], or the VIX White Paper (CBOE [2003]).

1On Wall Street, Eyes Turn to the Fear Index, New York Times (October 21, 2008).

35

36

The old (VXO) and new (VIX) end-of-day volatility index series were calculated back toJanuary 2, 1990 by the CBOE, and their daily updates are provided by the CBOE.2 Bothindexes are highly correlated (Carr and Wu [2006]). In empirical applications, the VIXserves as a market volatility measure (e.g., Fleming et al. [1995], Corrado and Miller [2005],Engle and Gallo [2006], Banerjee et al. [2007]). The exchange-traded volatility derivatives,futures, and European call and put options rely on the spot VIX value specification.

The used sample covers the period from January 2, 1990 up to April 30, 2010, com-prising of 5,125 end-of-day VIX recordings. Figure 4.4 confirms the stylized facts of theVIX as a volatility measure; periods of high (low) volatility are followed by calmer (morevulnerable) periods, both in line with mean-reversion and clustering (Poon and Granger[2003]). Note the immediate, sharp upturns in the volatility index during several marketcrises. The provided picture clearly indicates that the VIX is negatively correlated withthe S&P 500 index, reflecting the so called leverage effect (Black [1976b]).3 The highervolatility (VIX) periods are usually initiated by sharp upward VIX spikes (jumps). Theseare the 1990-1991 period of the Gulf-War I, the Asian Crisis in 1997, the Russian restruc-turing year after, the ‘9/11’ terrorist attack, the Gulf War II in 2003, or the sub-primemortgage financial crisis in July 2007. In these periods, the VIX return volatility showstendency to cluster. Figure 4.5 provides the end-of-day VIX series used in the parameterestimation. The period a year before and after the VIX options trade started (February24, 2006) was rather calm. Spikes occurring around June 13, 2006, reverted its 12% localmean before the end of the year. The increase in the interest rates announced by theChinese government in February 2007, brought a sharp upward jump up to the 20% level.The next period, between October 2007 and 2008, strongly undermined by the sub-primemortgage crisis turmoil, brought the VIX mean up to the 25% level and increased itsvolatility. The large upsurge after October 2008, reflected in the sub-prime crisis, with theVIX above the 50% level. The more VIX goes up, the more volatile it becomes. Then,the closer it gets to its long-run mean the less volatile it is.

Table 4.1 provides summary statistics, both for the whole sample and for the optiontrading period February 24, 2006 – April 30, 2010, where call, puts, and futures are tradedsimultaneously. The standard deviation is slightly higher during the sub-period. TheJarque-Bera statistics clearly indicate non-normality of both the VIX levels and its log-returns. Some positive skewness seems to be present in the data as well. Autocorrelation

2www.cboe.com/micro/vix/historical.aspx3Fountain et al. [2008] analyze this dependence thoroughly after removing all the autocorrelated and

heteroscedastic predictive structures. Reported Pearson’s, Kendall’s, and Spearman’s correlation measureswere -0.661, -0.476, and -0.653, respectively. The sample spanned the period January, 1990 and December2004.

37

of the log-returns decreases as time span gets longer, but remains significantly negativefor shorter ranges. The positional statistics are almost comparable in both samples forthe log-return series. Skewness and kurtosis are less pronounced than for the level itself.

VIX(%) ∆ logVIXStatistics whole option trading whole option trading

Mean 20.28 23.72 0.00 0.00SD 8.29 12.69 0.06 0.071st Qr 14.27 14.46 -0.03 -0.04Median 18.76 21.17 0.00 -0.013rd Qr 23.90 26.31 0.03 0.03Min 9.31 9.89 -0.30 -0.30Max 80.86 80.86 0.50 0.50Skewness 2.06 1.70 0.65 0.70Kurtosis 7.41 3.11 4.22 4.42Jarque-Bera 15,373.44 936.07 4,163.63 950.46Autocorr1(L-B) - - -0.09 (42.27) -0.14 (19.36)Autocorr5(L-B) - - -0.01 (90.50) -0.02 (35.34)Autocorr20(L-B) - - 0.00 (150.67) -0.03 (63.05)N 5,125 1,052

Table 4.1: Summary statistics of daily VIX levels and returns in two sample periods: the ‘whole’ VIXlisting period January 2, 1990 – April 30, 2010, and the VIX ‘option trading’ period February 24, 2006 –April 30, 2010. The Ljung-Box (L-B) statistics up to the indicated autocorrelation lag (1, 5, and 20) areprovided in brackets, and N denotes the series length (trading days).

4.2 VIX derivatives

The VIX futures and options are the only market traded volatility index derivatives. Thefutures’ historical data are provided by the CBOE.4 The option data came from the MarketData Express.5 The VIX futures (ticker: VX) are the first listed derivative contracts onthe CBOE volatility index VIX; its trading started at March 26, 2004. Reported valuesuntil March 26, 2007, use the Increased-Value Index (ticker: VBI) as underlying (beingthe spot VIX × 10). After this date the CFE abolished the associated VBI in favor of theunderlying VIX and publishes the corresponding rescaled futures prices. The Exchangemay list for trading up to nine near-term serial months and five months on the Februaryquarterly cycle for the VIX futures contract. The futures are settled on the Wednesday,which is 30 days prior to the third Friday of the subsequent calendar month. The VIXcall and put option trading started on February 24, 2006. Options on the VIX will havecontract months of two near-term contract months plus one additional month on the

4www.cboe.com/micro/vix/vixfuturesprices.aspx5www.marketdataexpress.com

38

February quarterly cycle. Strike price intervals can be 1 or 0.5 point. The options arecash settled European-style exercise, expiring on the Wednesday that is 30 days priorto the third Friday of the month subsequent to expiration of the applicable VIX futurescontract. Therefore, the last day of options and futures trading is usually a Tuesday andsettlement is determined by Wednesday’s open. The derivative sample spans over theperiod from February 24, 2006 up to April 30, 2010, comprising of 1,052 trading days.To diminish an influence of the microstructure noise the bid-ask quotes are averaged toobtain the option prices. Options with ask price more than zero and not expiring withinat most 14 days were considered. In addition, due to possible rounding errors, cases witha mid-quote value below 0.3 ($30) were excluded (George and Longstaff [1993]). Finally,after Lin and Chang [2008], the remaining VIX call and put option cases were selectedaccording to the arbitrage criteria,

C(Vt, K, τ) ∈ (e−rτ (F (Vt, τ)−K)+, e−rτF (Vt, τ)), (4.1)P (Vt, K, τ) ∈ (e−rτ (K − F (Vt, τ))+, e−rτK), (4.2)

for the VIX calls and puts, respectively. Following the selection criteria, out of the originaloption sample (224,633 recordings) only 85,440 were trade recordings (55,341 calls and30,099 puts). Among the traded contracts, 73,599 expired within at least 14 days (48,269calls, 25,330 puts). The price and arbitrage criteria excluded a further 31,838 observations,yielding the final sample of 57,244 VIX option trade recordings: 37,306 calls, and 19,938puts, comprising of 1,960 call and 1,864 put types featured by a unique combination ofthe strike and expiration date.

The expiration and arbitrage criteria also apply to the futures recordings. Similarly,as for options, recordings expiring within less than 14 trading days were excluded. Anon-arbitrage criterion was determined by the put-call parity (3.7), where, after Wangand Daigler [2011], the lower(upper) bound is determined using a bid(ask) of the VIXcall and ask(bid) of the put. The futures arbitrage criterion only applies when the optioncounterparts exist, which fulfil the inclusion criteria (4.1) and (4.2). Cases which donot match and/or are outside the arbitrage bounds were excluded from further analysis.Since the sample period starts on the VIX option introduction date, the futures tradedbefore February 24, 2006, are not considered. The primary VIX futures sample (8,895recordings) reduced, after applying the expiration and volume filters, down to 7,093 traderecords comprising of 56 contract types with unique expiration dates.

Table 4.2 provides summary statistics about the VIX derivatives data. On a dailybasis, the average number of the futures contract types traded is lower than for calls orputs. However, the average value of the futures contract ($24.32× 1, 000) is much highercompared to options (e.g., for calls $3.34 × 100). Calls are more actively traded than

39

puts, and also the number of traded call contract types is usually higher than the ones forputs. Concerning the time to expiration, available futures have longer time to expirationcompared to options, with the average of 100 days, which is almost 50% longer than theoptions.

The option and futures contracts are grouped with respect to the time to expiration(τ) criterion. Contracts expiring within less than 60 days are called the short-term ones.If maturity is within more than 60 and less than 120 days, then these contracts are themid-term ones. Finally, expiration within more than 120 days is considered as a long-termone. In addition, the options are divided into several categories with respect to either mon-eyness or time to expiration (τ). The VIX call(put) moneyness is defined as time-t +(-)log(strike/spot VIX) ratio. If at a given t, the option moneyness is ≥ 0.15, (0.15, 0.034],(0.034,−0.034], (−0.034,−0.15], or < −0.15, then such contracts are called the deep-out-of-money (DOTM), out-of-money (OTM), at-the-money (ATM), in-the-money (ITM), ordeep-in-the-money (DITM), respectively.6 Table 4.3 provides basic summary statisticscharacterizing the VIX derivative sample with respect to the moneyness/expiration divi-sion.

4.3 Market size and value

The VIX derivative market shows expansion since its origin. Its size can be measured intotal volume and value, as well as the number of all contracts traded over certain periodof time. The market value is understood as a sum of the particular contracts’ tradingvolumes×their market prices.7 In 2007, the second VIX option trading year, around 23.5million VIX option contracts changed hands, over 4.6 times more than in the first year2006. In 2008, the volume increased slightly up to 26 million contracts. In 2009, thenumber stopped at 33 million. The great majority of the VIX options trade concernedcall contracts, comprising of 17.7 in 2007 and almost 21 million in 2009. Up to the end ofApril 2010, almost 20 million VIX option contracts changed hands, where 12.6 belongedto calls. For the VIX futures, the year 2005 was the second trading year and the totalvolume was 1.5 times higher than the 88.7 thousand in 2004. In 2007, the volume reachedthe first time 1 million and stayed around this level over the next two years.

The total VIX option market value was around 713 million USD (615 for calls) in the6The existing literature about the VIX option pricing does not provide one standard way of money-

ness/expiration categorization. We use the same moneyness definition as in Wang and Daigler [2011], butmodify the moneyness and expiration bounds, such that the average volume and number of recordingswithin the groups become more evenly allocated.

7For the VIX options, the price is represented by the end of the day mid bid-ask spread.

40

Volume Type Strike Moneyness Maturity Price B-A spread

Call options (N=37,306)Min 1 1 10 -2.09 14 0.3 0.05Mean 1,190 36 29.7 0.13 69 3.34 0.311st Qr 13 26 19 -0.06 35 0.98 0.15Median 71 35 26 0.15 58 2.1 0.253rd Qr 505 46 35 0.36 93 4.15 0.4Max 180,035 99 130 1.28 314 56.5 56.8Put options (N=19,938)Min 1 1 10 -1.59 14 0.3 0.05Mean 945 19 28.4 -0.01 59 4.86 0.371st Qr 10 12 20 -0.19 30 1.03 0.15Median 46 19 25 -0.02 47 2.52 0.253rd Qr 391 25 32.5 0.16 77 5.75 0.4Max 105,783 71 130 1.39 309 77.75 114.2Futures (N=7,093)Min 1 2 - - 14 11.18 -Mean 457 7 - - 100 24.32 -1st Qr 27 6 - - 50 16.37 -Median 116 7 - - 90 23.69 -3rd Qr 421 8 - - 134 28.05 -Max 18,575 10 - - 521 66.4 -

Table 4.2: Statistics for selected daily VIX derivative records, which changed the owner. The secondcolumn corresponds to the number of contract types: options with the unique strike and expiration date,futures expiring on the same date. The moneyness is defined as log(strike/spot VIX) ratio. The fifthcolumn provides time to maturity (days) statistics. The option price being the end-of-day mid bid-ask(B-A) spread (×100); futures price (×1, 000). The sample spans February 24, 2006 – April 30, 2010. Ndenotes the total number of contract records in sample.

inception year and rose to 3.3 billion a year after (2.7 for calls). Over the next two years,it stayed at the 5.2 million USD level. In 2010, the market value over the first four monthsreached 50% of the 2009 level. The VIX futures market value was at the level of 5.4 billionUSD in 2006. This number was almost four and six times higher in 2007 and 2008-2009,respectively. Up to April 2010, the total market value was already on the 2007 year level(19.9 billion USD).

Figure 4.2 reports the market daily volumes and values with respect to different VIXderivatives, over the analyzed period. Figure 4.3 reports the daily number of traded optionswith unique strike-expiration date combination. For the futures, unique expiration datesapply. The call options were usually most intensively traded contracts. The averagedaily trading volume for calls was 83,548 contracts, then 39,019 and 45,059 for puts andfutures, respectively. In June 2007, the average daily volume of the VIX options was95,283 contracts, making the VIX the second most actively traded index and the fifthmost actively traded product on the CBOE.8 The expansion of the VIX derivative market

8The CBOE, CBSX, & CFE press release, July 11, 2007.

41

was partly reflected in an increasing number of traded contract types.

Figure 4.1: The daily trading volume (upper row) and value (bottom row) of the individual VIX deriva-tives contracts recorded over the February 24, 2006 – April 30, 2010 period. The value of the contractcorresponds to its price × daily trading volume, multiplied by the contract size: 100 (options) and 1,000(futures).

42

0 200 400 600 800 1000

010

020

030

040

050

060

0

Calls

Trading days

Trad

ing

volu

me

(1,0

00 c

ontr

acts

)

0 200 400 600 800 1000

010

020

030

040

050

060

0

Puts

Trading days

0 200 400 600 800 1000

010

020

030

040

050

060

0

Futures

Trading days

0 200 400 600 800 1000

020

040

060

080

0

Calls

Trading days

Mar

ket v

alue

(m

ln U

SD

)

0 200 400 600 800 1000

020

040

060

080

0Puts

Trading days

0 200 400 600 800 1000

020

040

060

080

0

Futures

Trading days

Figure 4.2: The daily size and value of the VIX derivative market over the February 24, 2006 – April 30,2010 period. Upper row: daily total trading volumes of the VIX derivatives. Lower row: daily marketvalues (in $106) calculated as the trading volume × contract price. The option price is represented by theend of the day mid bid-ask spread.

Calls

Trading days

Num

ber

of c

ontr

acts

0 200 400 600 800 1000

020

4060

8010

0

Puts

Trading days

Num

ber

of c

ontr

acts

0 200 400 600 800 1000

020

4060

8010

0

Futures

Trading days

Num

ber

of c

ontr

acts

0 200 400 600 800 1000

24

68

10

Figure 4.3: The daily numbers of traded VIX derivative contract types, recorded over the period betweenFebruary 24, 2006 and April 30, 2010. The options with the unique strike and expiration date combination,and the futures with the unique expiration date.

43

1990 1995 2000 2005 2010

Trading days

400

600

800

1000

1200

1400

1600

020

4060

8010

0

SPXVIX(%)

Gulf−War I

1992 President election

Asian Currency Crisis

Russian Default

9/11

Gulf War II

Subprime default rate increases

Subprime crisis

VIX futures

VIX options

1990 1995 2000 2005 2010

−15

−10

−5

05

1015

Trading days

VIX returns

Figure 4.4: Daily closing levels of the SPX and VIX and the VIX returns over the period January, 21990 – April, 30 2010. The vertical lines indicate the introduction dates of the VIX futures and options,respectively.

1020

3040

5060

7080

Trading days

2005 2006 2007 2008 2009 2010

VIX(%)

1st 252−day series

February 24, 2006

June 13, 2006 FTSE/Xinhua25 falls 10%

Economy deteriorates subprime default rates increase

Subprime crisis

Inauguration Day

Congress passes stimulus package

Large U.S. Bank fall rumors

Figure 4.5: The historical evolution of the daily closing VIX index levels, over the February 24, 1990 –April 30, 2010 period. The dotted line indicates the first 252-day subsample used in the estimation.

44

Calls

Puts

Futuresτ

Moneyness:

DOTM

OTM

ATM

ITM

DIT

MDOTM

OTM

ATM

ITM

DIT

M–

<60

mean

price(10

2USD

)0.96

2.132.93

3.215.97

1.241.21

1.411.99

3.57247.24

median

price(10

2USD

)0.83

1.982.63

3.084.75

0.700.85

0.981.73

3.08242.00

totalvalue(10

6USD

)2,197.84

1,584.00814.74

762.011,312.21

387.97437.60

404.05827.70

717.9360,423.01

%value

26.7619.29

9.929.28

15.9811.43

12.9011.91

24.3921.16

77.03mean

volume

2,4622,599

1,7461,174

6261,191

1,9731,969

1,776485

1,095median

volume

394559

270131

35200

355324

13620

571totalvolum

e(10

3)22,784

7,4232,779

2,3712,199

3,1223,604

2,8714,153

2,0132,444

%volum

e51.32

16.726.26

5.344.95

16.5619.12

15.2322.03

10.6875.32

n9,256

2,8561,592

2,0203,515

2,6221,827

1,4582,339

4,1522,232

[60−

120)mean

price(10

2USD

)1.51

3.333.72

3.705.29

1.491.69

1.992.05

3.82226.86

median

price(10

2USD

)1.28

2.903.60

3.504.55

0.951.23

1.301.50

3.05228.90

totalvalue(10

6USD

)548.89

276.97112.37

121.09200.58

120.5872.12

55.53117.82

158.0814,466.61

%value

6.683.37

1.371.47

2.443.55

2.131.64

3.474.66

18.44mean

volume

516431

273247

167387

443436

658298

248median

volume

3734

2321

1421

2724

2210

109totalvolum

e(10

3)3,642

831302

327379

810426

279574

414638

%volum

e8.20

1.870.68

0.740.85

4.302.26

1.483.05

2.2019.65

n7,058

1,9271,108

1,3232,262

2,093963

641873

1,3872,571

≥120

mean

price(10

2USD

)1.37

3.043.25

3.894.46

1.141.20

1.361.94

2.96218.14

median

price(10

2USD

)1.20

2.882.80

4.253.80

1.130.68

0.751.90

2.03224.50

totalvalue(10

6USD

)124.46

55.9521.71

48.5630.93

6.2030.92

8.8525.33

22.483,555.75

%value

1.520.68

0.260.59

0.380.18

0.910.26

0.750.66

4.53mean

volume

399277

183306

104132

1,106375

498151

71median

volume

2520

1820

1212

1215

1210

23totalvolum

e(10

3)910

18467

12569

54257

65131

76163

%volum

e2.05

0.410.15

0.280.16

0.291.36

0.340.69

0.405.02

n2,284

664365

408668

411232

173262

5052,290

Table4.3:

The

mean

price(×100)

andtrading

volumestatistics

ofthe

VIX

optionand

futurescontracts.

The

optionsare

groupedwithin

themoneyness

andtim

eto

expirationτ(trading

days)cross-section.

The

moneyness

attim

etfor

calls(puts)is

definedas

+(-)

log(strike/spotVIX

)ratio.

Then,the

call(put)option

iscalled

thedeep-out-of-the-m

oney(D

OTM),out-of-m

oney(O

TM),at-the-m

oney(AT

M),in-the-m

oney(IT

M),

anddeep-in-the-m

oney(D

ITM),

ifits

moneyness

attim

et,

definedas

+(-)

log(strike/spotVIX

)ratio,

is≥

0.15,(0.15,0.035],

(0.035,−

0.035],

(−0.035

,−0.15],and

for<−

0.15,respectively.

The

maturity

divisionconcerns

threecategories:

theshort-

(τ<

60),mid-

([60-120)),andlong-

(≥120

days)term

contracts.

Chapter 5

Role of the Market Information inthe Empirical Performance of theDiffusion Based Volatility DerivativePricing Models

5.1 Introduction

The VIX call and/or futures market studies concentrate mainly on the contract pricingimprovements acquired from a better underlying model of the VIX dynamics (see Chap-ter 2). However, little attention is paid to the influence of the choice of different marketinformation used in the estimation procedure on the models’ empirical performance. Asin the case of the standard index options, all parameters can be implied directly from theobserved derivative data by minimizing a mean-squared criterion. Alternatively, the avail-able derivative data can be combined with historical asset return series when estimatingthe parameters. See Chapter 3, section 3.4 for references and discussion. In all the empir-ical studies available so far, the choice of derivative type used in the parameter calibrationprocedures remains usually consistent with the type of the priced derivative. For example,the option pricing relies on parameter values calibrated to call and/or put values. How-ever, there is no clear reasoning provided, whether this is the best way to follow in- and/orout-of-sample. Similar questions apply to the asset returns. Is it better for pricing to uselong return series and rely on the general features of the asset dynamics (Campbell et al.[1997], pp. 364), or to estimate parameters on the shorter backward return samples, takinginto account the local characteristics (e.g., Heston and Nandi [2000b])? Existing literaturedoes not provide clear answers, leaving these issues as a subject to further explorations.

45

46

Similar questions are also relevant to the VIX derivative pricing models, since thesestudies follow the methodology used in the standard derivative analysis. They concernthe ability to discriminate and/or improve the performance of the existing specificationsby using different market information. In this Chapter, we try to answer these questionswith respect to the existing VIX derivative pricing models. In particular, we investigatethe in-sample fit and out-of-sample pricing performance of three basic models: the oneof Whaley, Grünbichler-Longstaff (GL), and the Detemple-Osakwe (DO) model. Besidethe closed-form pricing formulas, all three models have known return transition densities,which is an advantage in the estimation numerical procedures. More flexibility is allowedby extending the set of the parameters left to be determined by the market to see whetherthe pricing results improve. In the parameter estimation procedure, we follow the approachwhich combines both the asset returns and all traded VIX derivatives data. Hence, theinference is not limited to one type of market information only (e.g., futures or call andput options), but takes advantage of a more complete view. Each contract contribution isweighted by its daily relative trading volume.

Combining the information from the derivative market might also be relevant for in-vestment decisions. The models covering a wider market spectrum are more representative,than those reflecting only one of its segments. Their parameters, which aggregate differ-ent derivative information, can be used as the market composite measures and reduce itscomplexity down to standardized quantitative outcomes.

The Chapter consists of three main sections. The first one provides the descriptionof the parameter estimation/calibration methodology, and the models assessment criteria.The next section concentrates on the description and discussion over the empirical in- andout-of-sample pricing outcomes. The last section sums up and concludes the main findingsof the Chapter.

47

5.2 Setup

5.2.1 Parameter Estimation

The model parameters are estimated following the two step framework described in section3.4. In the first step, we use the (conditional) ML procedure to estimate the P -parametervector θP on the basis of the VIX return series. There are two sample selection schemesused in the estimation to verify the robustness of our methodology. First, all availableVIX series are used, such that the θP estimates reflect the VIX features aggregated overthe whole VIX history (January 2, 1990 – April 30, 2010). Henceforth, these are called the‘whole sample’ estimates and they are kept constant over the evaluation period, implyingthat the process properties do not change over time. However, such an assumption mightbe considered too restrictive and more flexibility in the parameter dynamics could resultin a better pricing performance. Hence, as an alternative approach, we relax the constantestimates setup such that the estimation VIX sample changes every day over the past 252-day series, starting on February 24, 2006.1 Such a ‘rolling window’ framework provides1,052 daily P -estimates over the VIX derivative sample (February 24, 2006 – April 30,2010). These estimates reflect the local characteristics of the VIX dynamics.

In the second step, given the P -process estimates at specific t, the Q-parameters arere-calibrated at a daily basis using formula (3.47), with respect to the following VIXderivative information schemes:

1) jointly: calls + puts + futures,

2) derivative subsets: calls + puts, calls + futures,

3) particular derivatives: calls, puts, futures.

Each of these frameworks implies different set of the contract weights, which are specifiedby (3.48) and (3.49).

For the diffusion models, only the drift-based market price of risk parameter λ has tobe calibrated. However, precise estimation of drift parameters is known to be notoriouslydifficult for diffusions (Figlewski [1997], pp. 22, Campbell et al. [1997], pp. 364–365).

1The ML estimation over one year long backward VIX series is used in Zhang and Zhu [2006] withrespect to the VIX futures pricing. Figlewski [1997] suggests for the SPX option studies using 6–12 monthshistorical return series if forecasting up to three months the market volatility. A similar approach is usedin Heston and Nandi [2000a], Christoffersen and Jacobs [2004b], Hsieh and Ritchken [2005] with half ofthis length. Their models are estimated over the six-months underlying index intervals and the estimatesare kept fixed for the subsequent six-month period. Later on, this restriction is relaxed and the modelparameters are updated every week.

48

Therefore, the set of market-implied parameters is alternatively extended with some ofthe drift parameters for the mean-reverting diffusion based models. The P -estimate forthe diffusion parameter σ is kept, as it is expected to be quite accurate on the basis ofdaily return data. The extension of the Q-parameters set concerns only the mean-revertingmodels, in particular, m for GL (3.13) and κ for DO (3.18). Within the GL model (3.14) –(3.15), the λ parameter adjusts to a given κ value, by the κ+σλ term. A similar reasoningapplies to the DO specification (3.19) – (3.20), where κ is left to be market-implied dueto the m − σλ term. Within the Whaley model (3.10) – (3.11), λ adjusts to a given µ

value by the µ− σλ term, so it is left as the only parameter to be implied by the market.In the end, we consider two possible sets for the free Q-parameters. In the first case, θQ

includes only one parameter, the market price of risk λ, which determines the Q-measure.Then, this set is extended by the drift parameters for the mean-reverting models.

5.2.2 Performance Evaluation

To validate the models, we investigate the discrepancies between the model and marketderivative prices. After Christoffersen and Jacobs [2004a], we use a similar to (3.47) themean square relative error (MSRE) criterion to evaluate the empirical model performancewithin the time horizon of H trading days,

MSRE(H) = 1T

∑t,i

wCt,iV olt

Ci(Vt, θPt−H , θQt−H)− Ct,iCt,i

2

+∑t,j

wPt,jV olt

Pj(Vt, θPt−H , θQt−H)− Pt,jPt,j

2

+∑t,k

wFt,kV olt

Fk(Vt, θPt−H , θQt−H)− Ft,kFt,k

2 , (5.1)

where:

- T : the number of trading days included in the assessment,

- wt,. = volt,. : the daily trading volume of the particular VIX derivative listed on dayt,

- V olt = ∑iw

Ct,i +∑

j wPt,j +∑

k wFt,k.

Due to differences in the contract trading volume magnitudes (Figure 4.1), and differencesbetween futures and option contract volumes (1,000 vs. 100) (see Chapter 4 for details),the wt, are multiplied by 10 to correct the futures contribution with respect to the options

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(Figure 5.1 first row). The weights of the particular contracts (Figure 5.1 second row)correspond to their daily trading volumes (volt,.) relative to the total ones (V olt) over allthe VIX derivatives recorded on day t and considered in the assessment. These weightsdo not change between the derivative information frameworks used in the Q-parametercalibration.

The more rich parametric model structure implies the better in-sample (H=0) fit, butdoes not necessarily imply a successful out-of-sample performance, due to the possibility ofoverfitting (Bakshi et al. [1997]). Therefore, we evaluate the models in the out-of-sampleframework for H=1, 5, or 20 trading days, which correspond to the short-, mid-, andlong-term forecast horizons, respectively. The underlying value at t, Vt, is assumed to beknown. In the analysis, the joint MSRE values, which aggregate the magnitude of therelative mispricing of all the VIX derivatives considered, are decomposed into the call,put, and futures contributions.

Figure 5.1: The individual VIX derivative contract daily trading volumes (first row), and their relativeversions used as the weights in the model assessment (second row), both recorded over the period betweenFebruary 24, 2006 and April 30, 2010. Due to the differences in the unit contract size (futures: 1,000,options: 100), the futures volumes get multiplied by 10.

5.3 Empirical results

First, we present and briefly discuss the model parameter estimation outcomes. Then, weconcentrate on the performance of the option pricing models from the in- and out-of-sample

50

point of view. The section concludes with an economic assessment of the magnitude ofthe model pricing errors.

5.3.1 Parameter estimates

θP over the whole VIX sample

The upper panel in Table 5.1 provides annualized estimation results for the whole sampleperiod. All the estimates are significant at the 5% significance level. The speed of themean reversion (κ) estimates for the GL and DO models are comparable and stay around4, which is consistent with results provided by, e.g., Harvey and Whaley [1992]. In thecase of GL, the mean-reversion parameter estimates imply that on average the deviationfrom the long-run mean is eliminated by 50% within 37 days (Chacko and Viceira [2003]).The long run means (m/κ) in both models are around 0.19, and correspond to the meanvalues of the VIX levels (see Table 4.1).

θP over the 252-day VIX subsamples

The lower panel in Table 5.1 provides the averaged results over the 252-day rolling windowseries (Figure 5.2). The ‘rolling window’ drift estimates for all models reflect local featuresof the underlying series and are higher compared to the whole sample counterparts. Hestonand Nandi [2000b] consider the estimates as stable when standard deviation up to the mean(100%×SD/mean) ratio is around 18%. For the drift estimates the ratio differs from 46%(GL) up to 91% (Whaley). The ratio for the κ estimates is lower and stays around 60%.The diffusion parameters are more precisely estimated with the ratio about 15% for theWhaley and DO models, and 32% for the GL specification. A lack of stability, especiallyin the case of P -process estimates, may harm the forecasting performance.

Derivative-implied estimation

Out of the individual daily estimation results, we find that the relevant m and κ estimates,implied from the VIX derivative data, are sometimes negative or on the zero-boundary.Indeed, the shape of the objective function around the boundary region implies that thederivative based optimization algorithm may imply negative or close to zero parametervalues. In such cases, these parameters are optimized with imposing a lower bound of 10−3.It usually happens for non-homogeneous derivative samples consisting of, for example, veryshort and long term options. However, such cases have a rather marginal character andfeature the market in its early stage.

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Model Whaley GL DOParameter µ σ λ m κ σ λ m κ σ λ

whole sampleP 0.451 0.936 - 0.842 4.140 0.448 - -5.802 3.496 0.940 -SE P 0.208 0.009 - 0.123 0.644 0.004 - 1.004 0.591 0.009 -logLik 14,489 14,632 14,540Mean Q - - 0.365 - - - -1.687 - - - -0.862SD Q - - 0.776 - - - 3.208 - - - 1.390whole sample (GL, DO extended θQ)P - - - - 4.140 0.448 - -5.802 - 0.940 -Mean Q - - - 0.824 - - -0.863 - 3.937 - -0.370SD Q - - - 0.555 - - 7.427 - 2.108 - 3.589rolling windowMean P 0.752 1.089 - 3.014 16.073 0.555 - -25.384 14.279 1.104 -SD P 0.681 0.159 - 1.384 9.609 0.175 - 18.997 9.034 0.158 -Mean SE 1.090 0.049 - 1.118 5.415 0.025 - 9.182 5.268 0.051 -Mean logLik 655 783 810Mean Q - - 1.085 - - - -4.906 - - - -1.962SD Q - - 1.290 - - - 7.062 - - - 2.800rolling window (GL, DO extended θQ)Mean P - - - - 16.073 0.555 - -25.384 - 1.104 -Mean Q - - - 2.084 - - -23.360 - 7.092 - -15.850SD Q - - - 1.893 - - 30.920 - 4.290 - 20.944

Table 5.1: Basic statistics for the maximum likelihood (ML) estimates and the VIX derivative-implied(call, put, and futures) parameter values for the Whaley, Grünbichler-Longstaff (GL) and Detemple-Osakwe (DO) model dynamics. The upper panel corresponds to estimation results over the ‘whole sample’of the VIX returns spanning the period between January 2, 1990 and April 30, 2010 (5,125 trading days).The ML parameter estimates (P ) are followed by their standard errors (SE P ), and the value of log-likelihood function for these estimates. The last two rows in the ‘whole sample’ panel refer to the meanvalues of the parameters implied jointly from call, put, and futures contracts (Mean Q), followed by thestandard deviation of these estimates (SD Q). The lower panel refers to the daily estimation results overthe 252-day (overlapping) ‘rolling windows’ of the VIX returns. The sample starts on February 24, 2006and ends up on April 30, 2010. The first row presents the averages of these estimates for each model(Mean P ). They are followed by their standard deviation (SD P ), mean standard errors (Mean SE), andmean log-likelihood function (Mean logLik) values. The last two rows in the panel refer to a mean valueof the parameters implied jointly from call, put, and futures contracts, and standard deviation of theseestimates.

5.3.2 Pricing performance

The model assessment relies on the mean squared relative errors (MSRE) and mean relativeerrors (MRE), which originate from the objective function (3.47), used in the parametercalibration. These criteria rely on the squared or signed difference between the model andmarket derivative price, relative to the market value of the contract and weighted by itsdaily relative trading volume. Such a construction implies that the resulting outcomesare scalars, and serve only as relative measures of the model performance. The MSRE

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and MRE outcomes from different specifications can be compared, given that they werecalculated based on the same derivative set. The MSRE remains the primary criterion asit measures the model fit precision. The MRE is the secondary one and complements themean squared outcomes, informing about the average mispricing direction. The better themodel performance, the smaller the MSRE and the MRE closer to zero.

In-sample

Tables 5.2 and 5.3 are complementary and present results for the in-sample model fits. Thefirst one corresponds to the case when λ is the only free parameter left to be determinedby the market. The latter one concerns the framework when the free parameter set isextended by some of the P -process parameters for the GL and DO models. The left-hand side results correspond to the P -parameters estimated over the whole VIX historicalsample (January 2, 1990 – April 30, 2010). The right-hand side presents the results forthe parameters reflecting the local VIX characteristics over the 252-day series. Withineach framework, we distinguish the cases when, given the P -process estimates, the freeparameters are implied from different combinations of the VIX derivative data.

The in-sample model performance differs with respect to the information frameworksused in the parameters estimation. Within the baseline setup where λ remains the onlymarket implied parameter, the best in-sample fit in terms of the aggregated MSRE, isobtained when the daily calibration is carried out to the joint VIX derivative or theoption-based information, regardless of the underlying series used in the P -estimation.In terms of the partial MSREs, the particular VIX derivative information is not relevantfor fitting the other derivatives prices either, except the case when the parameters arecalibrated to the particular derivative itself. This implies that the choice of the derivativedata in parameter calibration seems to matter and should not be arbitrary. Notice, forexample, that following a common practice (e.g., Lin and Chang [2009], Wang and Daigler[2011]), and relying the VIX call fit on λ calibrated exclusively to the option (call and put)data, we would not outperform the outcomes relying on the call and futures information.In this case, the DO model improves by around 21% (MSRE=0.039 vs. 0.031), for thewhole VIX sample P -estimates. Similar situation is observer for the ‘rolling window’ basedestimates, where DO improves by 32% (0.07 vs. 0.103). Even more striking is the differenceobserved for puts when the model fit relies on call, or call and futures information. Theircontribution to the aggregated GL and DO MSRE values within these frameworks standsfor more than 50%, and beyond 95% for the Whaley framework. This outcome is partlydetermined by a lower (than calls and futures) put trading volume and relative weights(Figure 5.1). The call performance is worse when the λs are calibrated to the put prices.

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The influence of the VIX dynamics information differs among the particular specifica-tions. The most apparent differences are observed for the Whaley model, which fits theVIX derivatives better if the P -estimates are based on the whole VIX sample, rather thanits local dynamics. For example, given the λ calibrated to the option data, the aggregatedMSRE for this model grows from 0.388 to 1.141. For the mean-reverting specifications,the joint MSREs are usually lower for the ‘whole sample’ estimates, if only λ is jointlyimplied from all the VIX derivatives or from the option data. If the θQ set gets extended,both GL and DO fit better for the ‘rolling window’ based P -estimates in the joint andoption-implied cases. Between these two models, DO works usually better than GL interms of the aggregated and partial MSREs in almost all considered frameworks. Thebest in-sample aggregated fit (MSRE=0.036) is provided by the DO model for the jointlyand option implied extended θQ sets and the ‘rolling window’ based P -estimates.

An inclusion of the empirically relevant mean-reverting factor improves the in-samplefit over the GBM based performance, regardless of the information load considered. Thismay suggest that the GBM dynamics in the volatility modeling is irrelevant (e.g., Dotsiset al. [2007]). However, one may argue that a better fit might be partly a consequence ofhaving a larger number of structural parameters. To diminish the impact of this connectionthe relevancy of the parametric structure is checked in the out-of-sample framework.

Out-of-sample

First of all, we analyze the pricing consequences in the case when the market price of riskλ is the only parameter left to be determined by the market. Tables 5.4 and 5.5 provideresults when the P -estimation is done over the ‘whole sample’ and ‘rolling window’ sets,respectively. In terms of the aggregated MSRE for the short-term horizon, the betterpricing performance is acquired if the P -parameter estimation is carried out over thewhole VIX historical sample. Within this framework, the best short- and mid-term horizonperformance is provided for the Q-parameters implied from all the VIX derivatives andoption contracts. In the long term, the call and futures information is the preferred one,given the aggregated MSRE criterion. With respect to the particular derivatives, the VIXcalls are more precisely forecasted, given that λ is calibrated to the daily call and futuresprices jointly. This pattern, known from the in-sample performance, applies also in theout-of-sample and confirms that the choice of the option (call and put) prices to imply themarket price of the risk (e.g., Lin and Chang [2009], Wang and Daigler [2011]) may notpay-off in the short-term forecasts. For the VIX puts and futures, the best strategy is torely exclusively on these contract prices, when calibrating the λs. The same rules applyfor the mid- and long-term horizons, except the aggregated MSRE in the latter period,

54

where the call and futures based information works better than the ‘jointly implied’ one.In the all considered data frameworks and forecast horizons, the mean-reverting DO

model provides the best pricing performance followed by the GL andWhaley specifications.For example, within the 1-day horizon and given the Q-measure determined jointly from allthe VIX derivatives, the DO implied MSRE equals 0.091, which is 32.5% and 86.5% betterthan GL’s and Whaley’s performance, respectively. In the other sections, the DO over GLrate of improvement stays usually between 20 and 35%. The GL model works better onlyfor puts over the ‘rolling window’ based P -estimation framework, given that theQ-measureis determined over the call and futures or exclusively over the call contracts. The betterlog OU performance over the CIR based dynamics in pricing the VIX derivatives stays inline with Mencía and Sentana [2012] findings, but in contrast to Wang and Daigler [2011],who find these models doing almost equally well in the case of VIX call option pricing.

When comparing these results with the ‘one-dimensional’ counterparts, the extensionof the GL and DO free parameter sets leads to pricing improvements in the ‘jointly’ andoption implied Q-parameter section, for the short- and mid-term horizons. For the otherderivative information schemes, the reduced θQ framework still remains the preferred one.Tables 5.6 and 5.7 provide the relevant results. Regardless of the forecast horizon consid-ered, the best pricing performance in terms of the aggregated MSRE is provided when theQ-parameters are calibrated to the joint derivative information, slightly improving thosebased on the option data. This pattern holds for each of the P -estimation frameworks,where the one based on the whole VIX sample dynamics provides usually the more preciseoutcomes. Within this framework, the best performing DO model improves its ‘one-dimensional’ Q counterpart in the 1- and 5-day horizon by 13% (from MSRE= 0.091) and8% (from MSRE=0.114), respectively. Between the two mean-reverting specifications, theDO model outperforms again the GL for the P -estimates based on the whole and rollingVIX samples. An exception from this rule is observed in the case of put-implied infor-mation, where the GL approach proves usually to be more accurate for the call options,given the ‘whole sample’ θP estimates. An interesting fact concerns the call performancefor the mid- and long-term forecasts, where the joint VIX derivative information basedresults are superior to the ones based on the Q-parameters implied exclusively from thesederivatives. A similar rule applies to puts within the long-term out-of-sample horizon.

Economic assessment

The MSRE outcomes can be also assessed from an economic point of view. Heston andNandi [2000b], while pricing the S&P 500 option contracts, consider $1 as an economically

55

significant root mean squared error (RMSE) value,√√√√ 1N

N∑i=1

(Di −Di)2 ≥ 1, (5.2)

where D and D denote the model and market derivative price, respectively, and N is thenumber of contracts in the sample. The baseline $1 value is used with respect to thereported average S&P 500 option price D ≈ $13, which, given (5.2), implies that√√√√ 1

N

N∑i=1

(Di −Di)2

D2≥ 1

13(≈ 0.077). (5.3)

If both sides of (5.3) are squared, the above economic threshold for the root relativesquared error can be further converted into the mean square relative error (MSRE) cutoff,

1N

N∑i=1

(Di −Di)2

D2≥( 1

13

)2(≈ 0.006). (5.4)

A relative character of the error metrics implies that the derived threshold values can beconsidered as a reference for any derivative type assessment, regardless of its price range.However, since the denominators in (5.3) and (5.4) left-hand side formulas correspond tothe average contract price, instead of its particular value, both derived cutoffs should beconsidered as approximations of economically significant relative error values.

The outcome in (5.4) is particularly important for us, as the model pricing assessmentrelies primarily on the MSRE type criterion. With respect to the VIX derivative in-samplefit provided in Tables 5.2 and 5.3, only the VIX futures performance fulfils the economicsignificance criterion (MSRE < 0.006) in majority of the sections. The exceptions concernthe Whaley model performance within the put-implied parameter sections. This outcomeis mainly retained in the out-of-sample framework.

5.4 Summary

We have examined an empirical performance of the alternative VIX derivative pricingmodels, proposed by Whaley [1993], Grünbichler and Longstaff [1996] (GL), and Detempleand Osakwe [2000] (DO), which rely on three well-established volatility dynamics. Ratherthan developing new models, we have concentrated on the existing ones and investigatedhow different market data used for the parameter estimation may influence their in- andout-of-sample pricing performance. The VIX returns and all available daily VIX derivativedata were used in the estimation. This approach extends and contributes to the recent

56

literature methodologies, where usually one type of derivative is used to calibrate themodel free parameters, sometimes together with the VIX returns. The robustness of theresults is checked using different methodologies and data combinations.

First of all, it matters which information load is used in the empirical applications.The patterns we found indicate on differences in the pricing outcomes when differentderivative information is used to imply the model parameters. In particular, against acommon practice, the Q-measure determined exclusively from the VIX option (call andput) prices does not guarantee the best in- and/or out-of-sample pricing outcomes for thesecontracts, when compared to the results based on the call and futures implied information.It revealed also a potential irrelevance of the information given from one type of derivatives,when applied in pricing these or the other contracts.

The mean-reverting component turns out to be an important factor in discriminatingamong the pricing models. The Whaley approach, which relies on the GBM volatilityspecification, performs far below the mean-reverting GL or DO specifications. Betweenthese two alternative models, the latter one usually outperforms the former in- and out-of-sample. This outcome contradicts studies where these two models are used alternatively.Further improvements are possible if the P -parameters are estimated over the VIX rollingwindow series, which reflect the short-term characteristics of the underlying. However,the observed improvements have a local character and do not hold for the Whaley model,which works relatively better for the P -estimates reflecting the long-term VIX properties.Besides the jointly implied information, the put-implied estimates seem to be less relevantfor pricing compared to the one implied from the call or futures contracts. This resultis supported by a relatively lower contribution of the VIX puts to the market’s volumeand/or value.

The obtained results show that the VIX derivative pricing is not concluded yet, evenwith respect to the baseline models. The choice of the market information used in themodel parameter estimation matters, and should not be arbitrary, as in the majority ofempirical studies. The preferred framework should cover different types of the derivativedata used to calibrate the model free parameters and possibly long-term underlying seriesfor the physical dynamics assessment.

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Figure 5.4: The model estimates of θQ = λ implied jointly from the VIX call, put, and futures prices,given the remaining θP ML estimates over the whole VIX sample (January 1990 – April 30, 2010) andthe 252-day backed VIX series, starting on February 24, 2006, respectively.

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Figure 5.6: The model estimates of θQ = λ implied jointly from the VIX call and put prices, given theremaining θP ML estimates over the whole VIX sample (January 1990 – April 30, 2010) and the 252-daybacked VIX series, starting on February 24, 2006, respectively.

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060

080

0

λ

Trading days

Figure 5.13: The GL and DO model estimates of θQ (extended) implied from the VIX put option prices.

63

0 200 400 600 800 1000

−1

01

23

4

Trading days

Wha

ley

Whole sample

0 200 400 600 800 1000

−6

−4

−2

02

4

Trading days

GL

0 200 400 600 800 1000

−4

−3

−2

−1

01

2

Trading days

DO

0 200 400 600 800 1000

−2

02

4

Trading days

Wha

ley

Rolling window

0 200 400 600 800 1000

−20

−15

−10

−5

05

Trading days

GL

0 200 400 600 800 1000

−8

−6

−4

−2

02

Trading days

DO

Figure 5.14: The model estimates of θQ = λ implied from the VIX futures contract prices.

0 200 400 600 800 1000

050

100

150

200

m

Trading days

GL

Whole sample

0 200 400 600 800 1000

050

010

0015

00

λ

Trading days

0 200 400 600 800 1000

050

100

150

200

250

m

Trading days

Rolling window

0 200 400 600 800 1000

050

010

0015

00

λ

Trading days

0 200 400 600 800 1000

010

020

030

040

050

0

κ

Trading days

DO

Whole sample

0 200 400 600 800 1000

020

040

060

080

0

λ

Trading days

0 200 400 600 800 1000

010

020

030

040

050

060

0

κ

Trading days

Rolling window

0 200 400 600 800 1000

020

040

060

080

0

λ

Trading days

Figure 5.15: The GL and DO model estimates of θQ (extended) implied from the VIX futures contractprices.

64

MSRE θP over the whole VIX series θP over the 252-day VIX seriesModel aggregated call put futures aggregated call put futures

θQ = [λ]: jointly impliedWhaley 0.387 0.142 0.240 0.004 1.140 0.387 0.748 0.005GL 0.101 0.069 0.030 0.002 0.196 0.116 0.077 0.003DO 0.064 0.039 0.024 0.001 0.178 0.104 0.071 0.003

option (call and put) impliedWhaley 0.388 0.143 0.239 0.006 1.141 0.387 0.748 0.007GL 0.102 0.069 0.030 0.003 0.197 0.115 0.077 0.004DO 0.064 0.039 0.024 0.002 0.178 0.103 0.071 0.004

call and futures impliedWhaley 0.817 0.036 0.776 0.005 2.966 0.028 2.925 0.013GL 0.118 0.060 0.056 0.002 0.301 0.082 0.216 0.004DO 0.083 0.031 0.050 0.002 0.304 0.070 0.231 0.003

call impliedWhaley 0.852 0.035 0.810 0.006 3.128 0.027 3.085 0.016GL 0.120 0.060 0.057 0.003 0.305 0.081 0.219 0.005DO 0.084 0.031 0.051 0.002 0.311 0.069 0.237 0.004

put impliedWhaley 21.471 21.132 0.014 0.324 70.952 67.641 0.017 3.295GL 0.334 0.317 0.011 0.007 1.103 1.084 0.011 0.008DO 0.201 0.190 0.008 0.004 0.772 0.758 0.009 0.006

futures impliedWhaley 0.603 0.257 0.344 0.001 1.593 0.746 0.846 0.001GL 0.151 0.099 0.052 0.000 0.284 0.158 0.126 0.000DO 0.103 0.059 0.044 0.000 0.247 0.148 0.099 0.000

Table 5.2: The in-sample mean squared relative pricing errors (MSRE) with respect to different marketinformation used in the parameter estimation. Results under the ‘whole sample’ correspond to the casewhen the P -parameters are estimated over the whole end-of-day VIX series recorded between January 2,1990 and April 30, 2010. For those under the ‘rolling window’ the P -parameters are estimated every dayover the 252-day backed end-of-day VIX series, starting from February 24, 2006 up to April 30, 2010. Theset of the free parameters (θQ) consists of the market price of the volatility risk λ, which is calibratedto different sets of the VIX derivatives prices recorded at day-t, by minimizing the daily trading volumeweighted mean square relative criterion. For example, the ‘jointly implied’ panel represents the case whenall the day-t VIX derivative (call, put, futures) prices are used in the λ calibration, and the ‘call implied’one when only the call information is considered. Within each market information setup, the ‘aggregated’MSREs are decomposed into the call, put, and futures contracts contribution.

65

MSRE θP over the whole VIX series θP over the 252-day VIX seriesModel aggregated call put futures aggregated call put futures

θQ (extended): jointly impliedWhaley 0.387 0.142 0.240 0.004 1.140 0.387 0.748 0.005GL 0.075 0.060 0.014 0.002 0.054 0.040 0.013 0.001DO 0.040 0.030 0.009 0.001 0.036 0.026 0.010 0.001

option (call and put) impliedWhaley 0.388 0.143 0.239 0.006 1.141 0.387 0.748 0.007GL 0.076 0.059 0.014 0.003 0.054 0.040 0.013 0.002DO 0.040 0.030 0.009 0.001 0.036 0.026 0.010 0.001

call and futures impliedWhaley 0.817 0.036 0.776 0.005 2.966 0.028 2.925 0.013GL 0.324 0.047 0.274 0.002 0.904 0.022 0.880 0.002DO 0.196 0.023 0.171 0.001 0.361 0.015 0.345 0.001

call impliedWhaley 0.852 0.035 0.810 0.006 3.128 0.027 3.085 0.016GL 0.542 0.046 0.488 0.007 2.083 0.020 2.055 0.007DO 0.292 0.023 0.266 0.003 0.637 0.014 0.619 0.004

put impliedWhaley 21.471 21.132 0.014 0.324 70.952 67.641 0.017 3.295GL 0.404 0.395 0.003 0.005 2.222 2.200 0.005 0.018DO 0.452 0.444 0.003 0.005 2.641 2.622 0.005 0.015

futures impliedWhaley 0.603 0.257 0.344 0.001 1.593 0.746 0.846 0.001GL 0.299 0.211 0.088 0.000 0.527 0.226 0.300 0.000DO 0.232 0.164 0.068 0.000 0.325 0.177 0.147 0.000

Table 5.3: (Cont. of Table 5.2) The set of free parameters θQ for the mean-reverting models is extendedby the underlying diffusion drift parameters: m for GL and κ for DO model.

66

MS

RE

Out-of-sam

ple:1day

Out-of-sam

ple:5days

Out-of-sam

ple:20

daysModel

aggregatedcall

putfutures

aggregatedcall

putfutures

aggregatedcall

putfutures

θP

overthe

whole

VIX

series;θQ

=λ:

jointlyim

pliedW

haley0.672

0.3040.363

0.0060.836

0.4040.424

0.0081.525

0.9520.562

0.011GL

0.1350.092

0.0400.002

0.1540.106

0.0450.003

0.2020.144

0.0540.004

DO

0.0910.057

0.0330.002

0.1140.074

0.0380.002

0.1590.114

0.0420.003

option(calland

put)im

pliedW

haley0.684

0.3130.365

0.0070.849

0.4210.419

0.0091.586

1.0100.565

0.011GL

0.1370.093

0.0410.003

0.1540.107

0.0440.003

0.2060.147

0.0550.004

DO

0.0920.058

0.0330.002

0.1130.075

0.0360.002

0.1610.116

0.0420.003

callandfutures

implied

Whaley

0.8670.087

0.7730.006

0.9890.133

0.8490.008

1.3230.360

0.9540.009

GL

0.1410.084

0.0540.003

0.1560.097

0.0550.003

0.1900.131

0.0550.004

DO

0.1010.049

0.0500.002

0.1170.062

0.0520.002

0.1490.097

0.0490.003

callimplied

Whaley

0.9000.086

0.8080.007

1.0280.130

0.8900.008

1.3510.356

0.9840.010

GL

0.1450.087

0.0550.003

0.1620.100

0.0590.004

0.1950.135

0.0560.004

DO

0.1040.050

0.0520.002

0.1210.063

0.0550.002

0.1530.100

0.0500.003

putim

pliedW

haley30.665

30.3090.041

0.31522.175

21.7380.073

0.36427.247

24.9050.151

2.191GL

0.3590.328

0.0240.007

0.3970.352

0.0380.007

0.5010.377

0.1150.009

DO

0.2220.200

0.0180.004

0.2620.229

0.0290.004

0.3580.277

0.0760.006

futuresim

pliedW

haley0.673

0.3100.361

0.0030.804

0.3950.405

0.0041.363

0.8220.533

0.009GL

0.1550.101

0.0530.001

0.1650.106

0.0570.001

0.1970.126

0.0690.002

DO

0.1070.063

0.0440.000

0.1210.071

0.0500.001

0.1540.089

0.0630.002

Table5.4:

The

out-of-sample

mean

squaredrelative

pricingerrors

(MSR

E)for

theP-param

etersare

estimated

overthe

whole

end-of-dayVIX

priceseries,recorded

between

January2,1990

andApril30,2010.

The

setof

parameters

leftto

bedeterm

inedby

themarket

(θQ)

consistsof

themarket

priceofrisk

λ.Given

theP-dynam

icsestim

ates,theλparam

etersat

day-tare

calibratedto

differentVIX

derivativeprices

backedH

days,byminim

izingthe

dailytrading

volume-w

eightedmean

squarerelative

criterion.The

market

information

schemes

usedin

theλcalibration

aresim

ilaras

inTable

5.2.

67

MS

RE

Out-of-s

ample:

1da

yOut-of-s

ample:

5da

ysOut-of-s

ample:

20da

ysMod

elag

gregated

call

put

futures

aggregated

call

put

futures

aggregated

call

put

futures

θP

over

the25

2-da

yVIX

series;θQ

=λ:jointlyim

plied

Wha

ley

1.724

0.64

81.070

0.007

1.958

0.754

1.194

0.009

2.881

1.359

1.511

0.012

GL

0.26

60.16

40.099

0.004

0.314

0.208

0.102

0.004

0.45

20.328

0.119

0.006

DO

0.24

60.14

90.093

0.003

0.294

0.186

0.105

0.004

0.433

0.273

0.154

0.005

option

(calla

ndpu

t)im

plied

Wha

ley

1.735

0.65

31.074

0.007

1.967

0.764

1.193

0.010

2.948

1.418

1.519

0.012

GL

0.26

90.16

60.099

0.004

0.317

0.210

0.102

0.004

0.45

80.333

0.119

0.006

DO

0.24

90.15

10.094

0.004

0.298

0.189

0.105

0.004

0.438

0.278

0.154

0.006

callan

dfuturesim

plied

Wha

ley

2.982

0.07

12.897

0.014

3.209

0.112

3.082

0.015

3.686

0.307

3.363

0.017

GL

0.33

70.12

00.212

0.004

0.367

0.158

0.205

0.004

0.42

80.235

0.187

0.006

DO

0.33

90.10

50.230

0.004

0.385

0.138

0.243

0.004

0.469

0.199

0.264

0.006

callim

plied

Wha

ley

3.164

0.09

23.056

0.016

3.369

0.116

3.236

0.017

3.836

0.311

3.507

0.018

GL

0.34

50.12

40.216

0.005

0.376

0.162

0.209

0.005

0.43

60.241

0.189

0.006

DO

0.35

00.10

90.236

0.005

0.400

0.143

0.251

0.005

0.480

0.204

0.269

0.006

putim

plied

Wha

ley

85.380

82.864

0.042

2.474

90.383

89.227

0.068

1.088

74.529

73.196

0.117

1.217

GL

1.13

21.10

10.023

0.008

1.314

1.274

0.031

0.009

2.03

21.902

0.117

0.012

DO

0.79

60.77

20.018

0.006

0.937

0.903

0.027

0.007

1.387

1.239

0.138

0.010

futuresim

plied

Wha

ley

1.678

0.80

60.870

0.003

1.856

0.907

0.944

0.004

2.599

1.408

1.183

0.009

GL

0.28

90.16

20.127

0.001

0.312

0.181

0.130

0.001

0.40

40.233

0.168

0.003

DO

0.25

30.15

20.100

0.001

0.281

0.170

0.110

0.001

0.400

0.215

0.181

0.003

Table5.5:

(Con

t.of

Table5.4)

The

P-param

etersareestim

ated

daily

over

theba

cked

252-da

yVIX

end-of-day

serie

s,startin

gon

Februa

ry24,

2006

upto

April30,2

010.

68

MS

RE

Out-of-sam

ple:1day

Out-of-sam

ple:5days

Out-of-sam

ple:20

daysModel

aggregatedcall

putfutures

aggregatedcall

putfutures

aggregatedcall

putfutures

θP

overthe

whole

VIX

series;θQ

(extended):jointly

implied

Whaley

0.6720.304

0.3630.006

0.8360.404

0.4240.008

1.5250.952

0.5620.011

GL

0.1260.089

0.0350.002

0.1480.100

0.0450.002

0.2220.129

0.0900.003

DO

0.0790.053

0.0250.001

0.1040.069

0.0330.002

0.2000.107

0.0900.003

option(calland

put)im

pliedW

haley0.684

0.3130.365

0.0070.849

0.4210.419

0.0091.586

1.0100.565

0.011GL

0.1350.090

0.0410.003

0.1530.102

0.0480.003

0.2280.131

0.0940.004

DO

0.0800.053

0.0250.002

0.1070.070

0.0340.002

0.2070.113

0.0910.003

callandfutures

implied

Whaley

0.8670.087

0.7730.006

0.9890.133

0.8490.008

1.3230.360

0.9540.009

GL

0.3420.078

0.2600.003

0.3840.101

0.2790.004

0.5080.147

0.3560.005

DO

0.2210.048

0.1700.002

0.3070.072

0.2320.003

0.4400.106

0.3300.004

callimplied

Whaley

0.9000.086

0.8080.007

1.0280.130

0.8900.008

1.3510.356

0.9840.010

GL

0.5260.079

0.4390.008

0.5470.106

0.4330.009

0.7420.171

0.5610.009

DO

0.3320.050

0.2780.004

0.5320.077

0.4500.005

0.5800.116

0.4580.006

putim

pliedW

haley30.665

30.3090.041

0.31522.175

21.7380.073

0.36427.247

24.9050.151

2.191GL

0.4410.421

0.0130.007

0.6200.589

0.0220.009

0.5290.388

0.1320.009

DO

0.4870.470

0.0110.006

0.9200.892

0.0200.008

0.6210.496

0.1170.008

futuresim

pliedW

haley0.673

0.3100.361

0.0030.804

0.3950.405

0.0041.363

0.8220.533

0.009GL

0.3020.213

0.0880.001

0.3310.226

0.1040.001

0.4200.268

0.1490.004

DO

0.2390.165

0.0740.001

0.2730.175

0.0970.001

0.3320.194

0.1350.003

Table5.6:

The

out-of-sample

mean

squaredrelative

pricingerrors

(MSR

E)for

theP-param

etersare

estimated

overthe

whole

end-of-dayVIX

priceseries,recorded

between

January2,1990

andApril30,2010.

The

setoffree

parameters

θQfor

themean-reverting

models

isextended

bythe

underlyingdiffusion

driftparam

eters:m

forGLand

κfor

DO

model.

The

market

information

schemes

usedin

theλcalibration

aresim

ilaras

inTable

5.2.

69

MS

RE

Out-of-s

ample:

1da

yOut-of-s

ample:

5da

ysOut-of-s

ample:

20da

ysMod

elag

gregated

call

put

futures

aggregated

call

put

futures

aggregated

call

put

futures

θP

over

the25

2-da

yVIX

series;θQ

(extended):jointlyim

plied

Wha

ley

1.724

0.64

81.070

0.007

1.958

0.754

1.194

0.009

2.881

1.359

1.511

0.012

GL

0.10

60.07

00.034

0.001

0.145

0.100

0.043

0.002

0.26

90.180

0.086

0.003

DO

0.07

60.04

90.026

0.001

0.106

0.071

0.034

0.002

0.230

0.148

0.079

0.003

option

(calla

ndpu

t)im

plied

Wha

ley

1.735

0.65

31.074

0.007

1.967

0.764

1.193

0.010

2.948

1.418

1.519

0.012

GL

0.10

70.07

10.034

0.002

0.145

0.101

0.042

0.002

0.27

30.184

0.085

0.003

DO

0.07

70.05

00.026

0.001

0.107

0.072

0.034

0.002

0.234

0.152

0.079

0.003

callan

dfuturesim

plied

Wha

ley

2.982

0.07

12.897

0.014

3.209

0.112

3.082

0.015

3.686

0.307

3.363

0.017

GL

0.91

50.04

40.868

0.003

1.073

0.066

1.003

0.004

1.32

70.109

1.213

0.005

DO

0.35

90.03

40.323

0.002

0.437

0.055

0.379

0.003

0.573

0.104

0.465

0.004

callim

plied

Wha

ley

3.164

0.09

23.056

0.016

3.369

0.116

3.236

0.017

3.836

0.311

3.507

0.018

GL

2.18

80.04

42.137

0.008

2.556

0.066

2.482

0.009

2.79

10.114

2.667

0.010

DO

0.63

00.03

60.590

0.004

0.786

0.058

0.723

0.006

0.958

0.108

0.843

0.006

putim

plied

Wha

ley

85.380

82.864

0.042

2.474

90.383

89.227

0.068

1.088

74.529

73.196

0.117

1.217

GL

2.03

72.00

20.017

0.018

4.440

4.390

0.029

0.021

2.59

42.465

0.110

0.019

DO

1.86

91.83

40.015

0.020

6.068

6.021

0.027

0.020

2.248

2.119

0.112

0.017

futuresim

plied

Wha

ley

1.678

0.80

60.870

0.003

1.856

0.907

0.944

0.004

2.599

1.408

1.183

0.009

GL

0.52

40.22

30.301

0.001

0.664

0.242

0.421

0.001

0.85

60.284

0.568

0.004

DO

0.33

60.18

10.154

0.001

0.397

0.198

0.198

0.001

0.485

0.233

0.249

0.003

Table5.7:

(Con

t.of

Table5.6)

The

P-param

etersareestim

ated

daily

over

theba

cked

252-da

yVIX

end-of-day

serie

s,startin

gon

Februa

ry24,

2006

upto

April30,2

010.

70

Chapter 6

The Volatility Derivative HedgingEffectiveness with the DiffusionBased Models

6.1 Introduction

The VIX derivatives gain a growing popularity among the investors, which is reflectedin increasing trading volume and the market value (see section 4.3). They became animportant part of the modern investment portfolio (Shu and Zhang [2012]).1 A directconsequence of such trends is an increasing demand for new pricing models, as well asa more thorough empirical evaluation of the existing specifications with respect to theirstatic (pricing) and dynamic (hedging) performance. Bakshi et al. [1997] and Nandi [1998]underline that both approaches are necessary for complete model evaluation. The pricingassessment, carried out already in Chapter 5, shows how the model derivative prices matchtheir market counterparts, while the hedging errors measure model’s ability to capture thechanges in the derivative and underlying security prices. In this chapter, we concentrateon the hedging performance, following the strategies proposed in the literature. Lin [2007]hedges the VIX futures with the VIX index position, assuming that the volatility index

1The fact that the volatility derivatives depend directly on the market volatility measure, makes them abetter candidate for hedging the exposure to changing volatility. It is considered that volatility derivativesmake markets more complete since they expand the choice of investment possibilities, and allow directhedging of volatility risk (greek: Vega), without affecting the delta exposure to the underlying assetprice risk (Szado [2009]). Strategies with the portfolio of standard futures and option contracts (Carrand Madan [1998]), or with a position in the at-the-money straddle (Brenner and Galai [1989]) are notappealing from a practical point of view due to the possible high cost burden of portfolio monitoring andadjustment (Brenner et al. [2006]).

71

72

in a tradable asset. Lin and Chang [2010] hedge the VIX options position with the VIXfutures, where both contracts expire at the same date. Sepp [2008a] considers the Lin-Chang idea and extends the option hedging portfolio with another VIX futures contract,which has a longer time to maturity. This approach is supposed to hedge, next to thediffusive volatility exposure, against the risk of sudden changes (jumps) in the underlyingasset.

Evidence from the previous standard option studies (e.g., Bakshi et al. [1997], Dumaset al. [1998], Andersen and Andreasen [2000]) does not necessarily imply that increasingmodel’s complexity pays off with a better pricing or hedging performance. Hence, thescope of this Chapter does not go beyond the baseline Whaley, Grünbichler-Longstaff(GL), and Detemple-Osakwe (DO) VIX derivative pricing frameworks. Instead, we pro-vide an extended assessment of these models’ hedging performance with respect to differ-ent hedging strategies and market information used in the model parameter estimation.The analysis concerns both the VIX call option and futures contracts. In the literature,there is no other study, which provides a hedging assessment of these models for the VIXderivatives. Psychoyios and Skiadopoulos [2006] use these specifications in a hedging ap-plication, however, the scope concerns a possible superiority of the volatility derivativesover the standard options in hedging the market volatility risk, and is based on simulateddata scenarios.

The structure of the Chapter consists of four sections. The empirical and simulateddata used in the analysis are described in the first section. The theoretical hedging setup,with respect to the VIX call and futures, is provided in the second section. The third oneconcentrates on the empirical hedging outcomes concerning both derivative types. Thelast fourth section concludes the main findings.

6.2 Data

The current analysis concentrates on the hedging performance of the VIX call and futurescontracts described in Chapter 4. The empirical data set is extended by the S&P 500index options used in one of the VIX futures hedging frameworks. For both derivatives,the empirical part of the study is complemented with outcomes based on the derivativedata simulated from the models.

6.2.1 S&P 500 options

The spot S&P 500 (ticker: SPX) index values and its European call and put options areused in the analysis to replicate the spot VIX value. These contracts are traded on the

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CBOE and their daily recordings are obtained from the Market Data Express. The samplecovers the period from February 24, 2006 to April 30, 2010, which complies with the VIXoption and futures data. The CBOE’s White Paper 2003 (CBOE [2003]) specifies whichSPX option recordings are used in the spot VIX calculation. These options must havepositive bid quotations and expire within not less than nine days. Following these criteria,out of the original sample 6.7% (80,724) of the daily recordings were excluded, comprisingof 551,946 and 569,960 of the call and put cases, respectively, in the final SPX optionsample.

6.2.2 Simulated data

The analyzed sample is extended with its simulated counterpart, implied by the model-specific closed-form solutions. The simulated data concern the option and futures prices,generated for the same contract characteristics as in the real sample. At the portfoliosetup t, the simulated price of the VIX call option (C) with strike K and the futures (F )contract, both maturing at t+ τ , are denoted by

Cmodelt ≡ Cmodel(Vt, K, τ, θt), (6.1)

Fmodelt ≡ Fmodel(Vt, τ, θt), (6.2)

and at the portfolio rebalacing point (after d time periods)

Cmodelt+d ≡ Cmodel(Vt+d, K, τ − d, θt), (6.3)

Fmodelt+d ≡ Fmodel(Vt+d, τ − d, θt), (6.4)

where Vt and Vt+k correspond to the spot VIX and its d-horizon model forecast, respec-tively. To forecast the underlying value over d periods, the Euler scheme over a daily griddt=1/252 was used, with the underlying starting value and the θ parameter estimatesbacked from the portfolio inception date t.

The simulation based outcomes are complementary to those acquired from the em-pirical data. We allow for the derivative data to be generated directly from the pricingformulas, assuming that the model underlying dynamics is the correct one. The markettendencies in the derivative price sensitivities to the underlying asset are expected to bedifferent to those derived from the models, which might be considered as a main reasonfor the model hedging failures. These differences can vary among the derivative money-ness and/or expiration groups. Thus, observing the tendencies and the differences in thehedging error magnitudes between these two frameworks will help to understand betterthe model performance and reasons for its failures.

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6.3 Hedging setup

In the applied delta hedging setup, at the setup time t, the portfolio consists of a shortposition in the hedged derivative D and a long position in bt units of some other asset R,which is used as a hedging instrument. The portfolio at its inception time t has the value

πt = −Dmarkett + at + btR

markett , (6.5)

where at denotes the residual cash invested in an instantaneously maturing risk-free bond.If the portfolio is self-financing, then for a continuous delta-neutral hedge the followingequations are expected to hold for all t,

πt = 0, (6.6)∂πt∂Rt

= −∂Dmodelt

∂Rt

+ bt = 0. (6.7)

The hedge parameters are implied by the particular pricing model specification,

at = Dmarkett − btRmarket

t , (6.8)

bt = ∂Dmodelt

∂Rt

. (6.9)

In practice, transaction costs preclude a continuous-time portfolio adjustments. To derivea hedging effectiveness measure, we assume that the portfolio rebalancing takes place atintervals of length ∆t. Then, the portfolio at the rebalancing point t+ ∆t yields

πt+∆t = −Dmarkett+∆t + at(1 + r∆t) + btR

markett+∆t , (6.10)

where r is a constant risk-free market interest rate. Formula (6.10) expresses the hedgingerror, resulting from using a particular hedge instrument and model specification withina discrete time setup. At each time t+m∆t, m=1, 2, . . . , M= τ

∆t , the errors are recordedand, at the same time, the portfolio is again reconstructed, until the hedged contract Dexpires, providing a set of the hedging error values.

The hedging performance assessment is based on the mean absolute and dollar (signed)hedging error criteria defined as

MAHE(∆t) = 1T

∑t,m

∑j

wj,t+m∆t

V olj,t+m∆t|πj,t+m∆t|, (6.11)

MDHE(∆t) = 1T

∑t,m

∑j

wj,t+m∆t

V olj,t+m∆tπj,t+m∆t, (6.12)

respectively, where:

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- T : the total number of trading days included in the assessment,

- j : the hedged derivative number, which on the setup day t, falls in the particularmoneyness and/or expiration cross-section,

- wj,t+m∆t = min(volj,t, volj,t+m∆t), where volt is the hedged derivative trading volumeobserved at t,

- V olj,t+m∆t = ∑j wj,t+m∆t.

The MAHE measures the magnitude of the hedging errors provided by strategy based onthe particular model and is considered as the primary criterion. The MDHE outcomes in-form about the direction of the hedging error, however, due to a possibility of settling-offbetween the positive and negative outcomes, this criterion is considered as a comple-mentary one to the MAHE outcomes. The particular errors at the portfolio rebalancingpoints are additionally weighted by the hedged contract trading volume, chosen to be theminimum between the one observed at the portfolio setup and its termination. Such aweighting scheme contributes twofold. First, it allows for discriminating among the port-folios, which are more representative to the market. The weight is given to the cases, whenboth (setup and termination) volumes are high, over the less informative ones when thetrading volume difference is large (Bremer and Hiraki [1999], Dupont [2001], Donaldsonand Kamstra [2004]). Second, as documented by Glosten and Harris [1988] and Georgeand Longstaff [1993], it diminishes the influence of the cases when, due to large bid-askspreads, the mid-quote option prices provide rather imprecise price evaluation (see Table6.1).

For the simulated data (section 6.2.2), the mean absolute and dollar errors are averagedover all S scenarios,

simMAHE(∆t) = 1S

S∑s=1

MAHEs, (6.13)

simMDHE(∆t) = 1S

S∑s=1

MDHEs. (6.14)

The simulations are carried out for the chosen VIX option and futures hedging method-ologies, which are described in following subsections. The number of simulations S needsto be large enough to provide stability of the outcomes around the central tendency.

The proposed theoretical setup is considered to be a standard in all the hedging strate-gies considered in the study. The methodologies for the particular VIX derivative typesmay differ with respect to the choice of the hedging instruments and the model used to

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determine the portfolio weights. However, all of them follow the same rule about the port-folio structure control: the ratios get rebalanced over equal time periods, until the hedgedderivative or contracts in the hedging portfolio expires. Such a strategy can be classifiedas a type of the adaptive or dynamic hedging approach, since the portfolio structure isadjusted accordingly to the changes in the underlying asset over time. In such a scheme,the rebalancing should he carried out on the continuous time basis, which is not practi-cally feasible, and is usually replaced with its high-frequency (e.g., several times a day)version. The adaptive approach can be replaced with the static one, where the structureof the portfolio remains constant over its life, regardless the changes in the underlyingvalue. Given such a classification and the fact that the rebalancing in our study happenson a daily basis and beyond (5 and 10 days), the applied strategy can be classified asa semi-adaptive one, as not all the changes in the underlying VIX are considered in theportfolio weight adjustments, but only those recorded on the portfolio adjusting points.2

Both adaptive and static hedging frameworks have their advantages and disadvantagesin the economic terms. The first one allows for the adjustments according to changingeconomic situation, which may pay-off in smaller losses. However, in practice, the highcosts of the portfolio adjustments may settle-off the gains. The second one does not (the-oretically) generate the portfolio management costs, but stays inflexible to the changingeconomic conditions, which might not be the best strategy in the situation when the con-tract maturity is long (e.g., one year or longer) and the market expectations are hard tobe forecasted by the model. The semi-adaptive strategy seems to be a good compromisebetween the portfolio flexibility to changes in the investment environment, and low costsof the portfolio rebalancing. So far, in the available literature, this is the only frameworkconsidered for the VIX derivative hedging.

6.3.1 VIX options hedging

The applied hedging strategy is adapted from Sepp [2008a] and Lin and Chang [2009],where the VIX call option C is hedged using a long position in the VIX futures contract,both expiring within the same period of time τ . The motivation to use this frameworkcomes from a high liquidity of the VIX futures and the same underlying VIX index, whichsimplifies the derivation of the hedge parameters. The portfolio holds at its setup time t

2The choice of the portfolio hedging period is determined by the available VIX and S&P 500 marketdata sets, which are recorded on the daily basis. As the CBOE calculates the VIX spot prices on aminute-by-minute basis, the highest frequency of portfolio rebalancing for the VIX derivatives would beone minute.

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the value

πt = −Ct + at + btFt, (6.15)

where Ct and Ft denote the VIX call option and futures market prices, respectively. Con-ditions (6.6) and (6.7) applied to (6.15) allow for determining the hedge ratios

at = Ct − btFt, (6.16)

bt = ∂Cmodel(Vt, K, τ)∂Vt

/∂Fmodel(Vt, τ)

∂Vt. (6.17)

See Appendix A.4 for the particular model formulas and derivations. The portfolio afterthe ∆t period yields the value (see (6.10))

πt+∆t = −Cmarkett+∆t + at(1 + r∆t) + btF

markett+∆t . (6.18)

Recalling (6.1) – (6.4), this framework has also its simulation based counterpart with thesetup value

πt,s = −Cmodelt + at + btF

modelt . (6.19)

Notice that for each of the market simulated scenarios s, the value of the particular callhedging portfolio at t is the same. The hedging parameters are constructed similarly asfor the empirical data, using the conditions (6.6) and (6.7). Assuming the rebalancingperiod interval ∆t, the portfolio at t+ ∆t has the value

πt+∆t,s = −Cmodelt+∆t + at(1 + r∆t) + btF

modelt+∆t . (6.20)

Similar as (6.10), the formula (6.20) expresses the hedging error, resulting from the partic-ular model based hedging strategy. For all s, at each time t+m∆t, m=1, 2, . . . , M= τ

∆t ,the errors are recorded and the portfolio is reconstructed, until the hedged call contractexpires. Each of the simulation rounds provides the corresponding mean hedging errorvalues: MAHEs and MDHEs.

6.3.2 VIX futures hedging

Sepp [2008a] proposes a framework, where the VIX futures is hedged by trading the otherVIX futures contracts with different maturities. However, such a strategy is not veryefficient since the price risk is only slightly diminished by a possibly higher liquidity of theshort-term futures contracts, but the risk source remains the same. Lin [2007] replicatesthe VIX futures using a position in the VIX spot value, which is hardly applicable inpractice, as the VIX is not directly tradable. Instead, a position in the volatility indexmight be replicated using the S&P 500 options, which is in line with the market practice:

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In particular, a commonly used hedge for VIX futures involves holding theportfolio of SPX options that will be used to calculate the settlement value ofthe VIX futures contract on settlement date.

Federal Register, Vol. 73, No. 54/Wednesday, March 19, 2008/Notices

Both approaches are considered in the current study, where the one with the spot VIXposition serves as the reference strategy to the one with a strip of the VIX replicatingS&P 500 options. Selection of the relevant options goes in line with CBOE’s VIX calcula-tion methodology (see Appendix A.3). Further on, the replicating portfolio is optionallyextended with a position in the S&P 500 index itself.

VIX asset position

The VIX futures contract with time to expiration τ is hedged with the spot VIX index Vtposition. The value of such a portfolio at its setup time t is

πt = −Ft + at + btVt. (6.21)

The conditions (6.6) and (6.7) applied to (6.21) yield the hedge ratios

at = Ft − btVt, (6.22)

bt = ∂Fmodel(Vt, τF )∂Vt

. (6.23)

For the particular model formulas and derivations, see Appendix A.4. The portfolio valueat the rebalancing point (see (6.10)) becomes

πt+∆t = −Ft+∆t + at(1 + r∆t) + btVt+∆t. (6.24)

This framework is complemented with the simulation based outcomes. The (6.21) coun-terpart portfolio has the inception value

πt,s = −Fmodelt + at + btVt, (6.25)

where s corresponds to the particular simulated market scenario, and Vt is the empiricalvolatility index value. Similar as for the call options, within each of the simulation rounds,the πt,s for the particular futures contract is the same. The hedging parameters are con-structed similarly as for the empirical data using the conditions (6.6) and (6.7). Assumingthe rebalancing interval ∆t, the portfolio at time t+ ∆t yields the value

πt+∆t,s = −Fmodelt+∆t + at(1 + r∆t) + btVt+∆t, (6.26)

where Vt+∆t is the VIX forecast value implied by the model. Again, similar as for the calloptions, the corresponding MAHEs and MDHEs values are calculated from the resultinghedging errors recorded at each of time points t + m∆t, m=1, 2, . . . , M= τ

∆t , within thesimulated market scenario.

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S&P 500 options position

The VIX futures contract with time to expiration τ is hedged with a strip of the S&P500 (call and put) options, selected to replicate the spot value of the VIX at t. At theinception, the hedging portfolio has the value

πt = −Ft + at +∑i

bt,iOSt,i, (6.27)

where OSt,. denotes the spot mid-quote price of the index option within the strip. The

conditions (6.6) and (6.7) applied to (6.27) imply the following portfolio weights:

at = Ft −∑i

bt,iOSt,i, (6.28)

bt,i = ∂Fmodel(V (OSt ), τ)

∂V (OSt ) × ∂V (OS

t )∂OS

t,i

, (6.29)

where V (.) is the function used to calculate the spot VIX value from the strip of the SPXoption prices, see Appendix A.18 and A.19. The particular hedge ratio derivations areprovided in Appendix A.5.1. The portfolio value at the rebalancing point (see (6.10))becomes

πt+∆t = −Ft+∆t + at(1 + r∆t) +∑i

bt,iOSt+∆t,i. (6.30)

S&P 500 options with spot index position

The spot VIX index calculation should rely exclusively on the OTM S&P 500 call and putoption prices. However, the way the algorithm is constructed (see Appendix A.3) allowsfor one ITM call to be included. The adjustment term in the VIX formula (A.14) convertsthe contribution of such a call into the relevant put outcome, using the following put-callparity relation

Ct(Kmint , τ)− Pt(Kmin

t , τ) = e−rτ (Ft −Kt), (6.31)

where Kmint corresponds to the strike price for which the difference between the S&P 500

call and put prices in the day-t strip is minimal. In the original CBOE’s methodology, therelation (6.31) is used to identify Ft, given the relevant derivative and strike prices. Here,we modify this approach and consider the S&P 500 forward value to be determined from

Ft = e−rτSt, (6.32)

where St denotes the spot S&P 500 index value. The resulting Ft value is used to identifythe strike priceKt in the day-t option strip (such thatKt ≤ e−rτSt), and the corresponding

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ITM call contract. Its conversion to the relevant put is done by the following put-callrelation, which results from the (6.32) substitution to (6.31),

Ct(Kt, τ)− Pt(Kt, τ) = St − e−rτKt. (6.33)

Such a relation is called the put-call parity for the tradable asset (Grünbichler andLongstaff [1996], Mencía and Sentana [2012]).3 The presented modification allows forhedging the VIX futures expiring at t+ τ , with a position in the S&P 500 option strip andthe spot index St. Such an extended portfolio has the following value, on its inception dayt,

πt = −Ft + at +∑i

bt,iOSt,i + ctSt. (6.34)

However, as the S&P 500 itself is not tradabale, this framework is only a theoreticalextension to (6.27) strategy, which is based purely on the index options. The strip of theindex options used in (6.34) may differ from (6.27) as the threshold strike prices Kmin

t andKt, used to distinguish the OTM call and out prices, may be slightly different.Applying the criteria (6.6) and (6.7) to (6.34) yields the following hedge ratio formulas:

at = Ft −∑k

bt,iOSt,i − ctSt, (6.35)

bt,i =∂Fmodel(V (OS

t,i, St, τ), τ)∂V (OS

t,i, St, τ) × ∂V (OSt , St, τ)

∂OSt,i

, (6.36)

ct = ∂Fmodel(V (OSt , St, τ), τ)

∂V (OSt , St, τ) × ∂V (OS

t , St, τ)∂St

. (6.37)

For particular derivations, see Appendix A.5.2. The portfolio value at the rebalancingpoint (see (6.10)) becomes

πt+∆t = −Ft+∆t + at(1 + r∆t) +∑i

bt,iOSt+∆t,i + ctSt+∆t. (6.38)

6.4 Empirical results

In this section, the empirical hedging results are provided and discussed. The hedgingperformance for the VIX options and futures is compared between the models and es-timation frameworks. First, the results for the options are described when a short callposition is covered with a long position in the VIX futures contract. Then, a positionin the futures is hedged using the VIX or the VIX position replicated with the relevant

3Compare with the counterpart formulas (3.6) and (3.7) for the VIX.

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S&P 500 options, plus (optionally) the spot S&P 500 index value. The described setupsare applied to the empirical derivative data, as well as simulated directly from the modelunderlying dynamics and pricing formulas, given the empirical contract specifications. Us-ing different parameter estimation frameworks, we assess how sensitive the model hedgingperformance is with respect to different scopes of information and parameter flexibility.The mean absolute and dollar hedging errors are reported as the complementary criteriafor the model assessment.

6.4.1 VIX option hedging

The call VIX options are hedged using a long position in the VIX futures, expiring atthe same date as the option. If such a futures contract is not available at the portfolioinception, then the next longer- or shorter-term contracts is used, where the latter shouldnot expire before the portfolio termination. The cases, where the chosen futures contract isnot traded at the portfolio liquidation/rebalancing point, are not considered. The analysisconcentrates on the call options exclusively and the reported weighted mean absolutehedging errors (MAHE) are put together with their dollar (MDHE) counterparts. The VIXcall hedging outcomes provide some patterns of change or improvement, under differentmarket information schemes considered in the study. Thus, each framework output isconsidered separately to track the most important consequences for hedging performance.

Table 6.2 refers to the case, where the model structural parameters reflect the volatilityindex characteristics over the whole VIX historical sample, and the volatility risk premiumλ is left to be determined by the market. The best hedging performance in the MAHEterms is provided by the DO or Whaley models, which complement each other within theconsidered moneyness/expiration cross-sections. The DO specification works best in theDOTM group (all expirations), as well as across the short-term contracts, regardless ofthe portfolio rebalancing frequency. There are some local exceptions from this rule. Oneof them concerns the cheap but intensively traded short-term DOTM calls (Table 4.3),where the Whaley and DO models work equally well (MAHE=0.072), and outperform theGL specification (MAHE=0.083). For the most expensive short-term DITM calls, the DOimplied MAHE=0.114 vs. 0.126 for the Whaley model. Both outcomes concern the dailyrebalanced hedging portfolios. The Whaley model improves over DO for the portfolioswith the ATM-ITM calls, which mature in the mid- and long-term periods. This becomesalso the case for the short-maturing calls as the hedging horizon extends to a week. TheGL model provides the worse performance in the absolute hedging errors terms, exceptthe mid- and long-term expiring DOTM calls, where it works only slightly worse than theDO model.

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The MDHE values for all models stay mainly positive and strongly circulate aroundzero for daily rebalanced portfolios. As the hedging frequency goes down, the averagedollar errors tend to grow. The DO model provides usually the smaller MDHE values thanthe GL and Whaley specifications. This concerns particularly sections, where this modelprovides the best hedging performance in the MAHE terms: DOTM and short-expiringcalls. The second best performing Whaley model does not always follow this rule. TheMDHEs provided by this model are relatively higher than GL’s and DO’s outcomes, evenin the cross-sections where its MAHEs were the smallest. Such outcomes suggest thatthe magnitudes of the errors provided by the Whaley and DO models are comparable,however, the signs of the first are mainly positive. The GL model, which provides theworst performance in the MAHE terms, yields smaller than the Whaley model MDHEvalues. For example, the portfolios with the short-term DITM calls rebalanced over 10-day period, where GL’s MDHE=0.057 vs. 0.118 for Whaley, given much worse MAHEbased performance (0.393 vs. 0.353). Such outcomes suggest that the GL outcomes arecharacterized by the higher variation around zero, which increases the absolute hedgingerror magnitude, but settles off when the mean dollar errors are considered.

Table 6.3 complements the previous results with the outcomes for 252-day based P -estimates. The change of the scope to more local VIX characteristics does not influencethe model performances in any systematic manner, when compared to the whole VIXsample based outcomes. The best performing DO and Whaley models, given the moreflexible P -estimates, provide higher hedging errors in the absolute terms. Within the 1-day hedging period, the short-maturity DOTM call absolute errors for the Whaley andDO models increase from 0.072 to 0.080 and 0.077, respectively. Concurrently, within theshort-term DITM section, the DO outcome grows from 0.114 up to 0.122. For the Whaleymodel, the short-term ATM calls change from 0.372 up to 0.382, in the 10-day hedgingperiod. Within the new estimation framework, the GL model improves its performancewithin the portfolios with short-maturing calls. For the daily rebalanced portfolios, thechanges are rather small, for example, for the DOTM calls the MAHE, decreases from0.083 down to 0.080. As the hedging frequency decreases, the changes are more apparentwithin the short-term DITM group, for example, 0.245(0.393) down to 0.221(0.343) for the5(10)-day hedging period. After these changes, both mean reverting models outperformthe Whaley outcomes in the DOTM call and majority of the DITM sections, regardless ofthe hedging horizon. The DO model still remains the best performing model, even thoughthe differences with respect to the GL based outcomes become much smaller than for the‘whole sample’ based estimation framework.

Further assessment concerns methodology, when the set of the Q-measure parametersis extended only for the GL and DO models. Table 6.4 provides the relevant MAHE and

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MDHE values for the P -estimates over the whole VIX sample. The Whaley errors areprovided as a reference and remain the same as in Table 6.2. The observed GL and DOoutcomes are not much different from the baseline ones relying on the single Q-measureparameter set. The improvements in form of the smaller MAHE values are observed almostin all cross-sections, but their magnitudes are not that large and usually stay within 0.001– 0.005 bands. For example, in the cheap but most intensively traded DOTM call section(1-day), the best performing DO model improves from 0.072 (Table 6.2) down to 0.070. Inthe same expiration DITM group, the DO model improves from 0.114 to 0.113, which isalso reflected in slightly lower MDHE values in both groups. The DO model remains thepreferred one in both the DOTM and short-maturing call contracts. Despite the observedimprovements in the mean-reverting models due to a more flexible Q-measure, the Whaleyspecification provides still better performance for the mid- and long-term expiring ATM-ITM calls.

Finally, the hedging performance was checked when the GL and DO P -estimates re-flect the VIX characteristics over the 252-day series, and the Q-measure is determinedover the extended parameter set. Table 6.5 provides the relevant MAHE and MDHE out-comes. The P -estimates relaxation works in favor of the DO model, which provides thebest short-term DOTM call MAHE outcome (0.067), over all the frameworks consideredso far. Within the same section, the GL model improves over the Table 6.4 outcome0.081 down to 0.073, such that it outperforms the relevant Whaley’s value (0.08). TheDO remains the preferred specification to hedge the call options in almost all the mon-eyness/expiration cross-sections, also in the groups where the Whaley provided the bestperformance: the mid- and long-term ATM-ITM calls. Within the most expensive short-term DITM sections, the differences between the mean-reverting models become minimal,with the preference for the DO specification.

Each of the provided Tables reports also the results based on the simulated data. Wecan compare the patterns provided by the empirical data with those based on the modelassumptions. Within all the estimation frameworks, the MAHE dynamics differs as thetime to maturity increases. For the 1-day hedging horizon, the values tend to followthe U-shaped pattern: the high values for the short-maturing calls are followed by thedrop in the mid-term ones, and increase in the long-maturing sections. These outcomescontradict the tendencies in the model-simulated results, where the MAHE values over allmoneyness groups tend to decrease towards zero, as the time to maturity increases. Forthe empirical results, this tendency gets more visible as the hedging frequency decreases.This evidence shows that the models do not mimic the market price mechanisms for thelong-term maturing call and futures contracts. Figure 3.1 provides an example of themodel call price sensitivity to the passage of time for three different moneyness groups.

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The mean-reverting models imply, given the same rebalancing frequency, that the setup vs.termination contract value difference will be higher for the portfolios with the short-termVIX derivatives, when compared to the ones with the long-term VIX contracts, regardlessof the moneyness group. This difference will become zero for the long-term contractportfolios, as these models ‘push’ the VIX to its long-term mean if it deteriorates, over thetime course. However, the long-term market option prices are more sensitive to the timeto maturity changes, than the mean-reverting model prices. This pattern is not observedfor the Whaley model. Here, the sensitivity does not change much, except the OTM andITM option groups where τ < 60. Its long-term tendency seems to stabilize around theATM level. Recall that the Whaley underlying dynamics assumes the VIX fluctuatingrandomly around an exponential trend over the time course, so the sensitivity will notdrop down to zero with longer time to maturity. Thus, the observed hedging errors arerelatively higher than those implied by the mean-reverting specifications. The marketdownward tendency for this model seems to contradict with the model (almost constant)non-zero sensitivity over the time course. Notice that for the longer portfolio rebalancingperiods, the Whaley short- and mid-term option errors over the empirical data happen tobecome lower than the ones for the simulated underlying series. The reason for this mightbe a smaller difference between the market derivative price observed at the portfolio setupand its termination, compared to the counterpart differences for the model prices over thesimulated underlying data.4 High sensitivity of the Whaley model call prices and higher(than those observed in the market) expected differences between the VIX values at theportfolio setup and its termination yield larger differences in the model simulated pricescalculated at these time points.

6.4.2 VIX futures hedging and VIX replication

Similar as for the VIX (call) options, the mean hedging error outcomes for the futures aregrouped with respect to different market information used in the model estimation. Tables6.6 and 6.7 correspond to the cases, when the risk-free parameter set θQ consists of themarket risk premium λ and the P -process estimation relies on the whole VIX sample and252-day series, respectively. A similar P -estimation setup applies to Tables 6.8 and 6.9,with the extended θQ framework. The Table panels denoted A:D correspond to differenthedging frameworks. The first division concerns the source of data used in the analysis:simulated or empirical. The panel A corresponds to the simulated market data, while theoutcomes provided in the following panels B:D are based on the empirical data.

4The portfolio error (6.10) given r∆t(Dmarkett −btRmarket

t ) ≈ 0 yields (Dmarkett −Dmarket

t+∆t )−bt(Rmarkett −

Rmarkett+∆t ). The same formula applies to the model prices over the simulated data.

85

The simulation based results were averaged over 500 different market scenarios, gener-ated according to the particular model assumptions concerning the derivative prices andspot underlying values, thus, they cannot be directly compared. These outcomes might beconsidered as a reference for the results based on empirical data, with respect to the modelfutures price sensitivity as the underlying and time to the contract maturity change. TheGBM based Whaley and the mean-reverting GL and DO model properties are different inthis aspect. Figure 3.1 shows that for the first model, the futures price Delta sensitivityremains roughly the same for different underlying levels, with a slightly upward tendencyas the time to maturity gets longer. For the mean-reverting models, the Delta sensitivitygoes down towards zero with the longer time to expiration, regardless of the contract ma-turity group. These model prices become more ‘insensitive’ to the underlying changes inthe longer maturities, where Whaley’s price sensitivity tends to grow. All these tendenciesare reflected in the provided MAHE results. The magnitude of Whaley’s outcomes doesnot differ much along the maturities, with a slight tendency to grow for the longer ones.The GL and DO outcomes clearly diminish with longer time to expiration.

Roughly similar patterns are observed within the framework based on the empiricaldata, where the futures contract is hedged with the spot VIX position (panel B). Thedifferences in the absolute error magnitudes confirm that the mean-reversion propertyremains a more relevant factor for the futures hedging than for the call options. The bestperforming DO model improves over the GBM based framework (MAHE=0.747) by 35%,for the daily rebalanced portfolios and the whole VIX sample based P -estimates (Table6.6). Such a hedging strategy, due to non-tradability of the VIX index, is not applicablein practice, but it might be considered as a reference for the outcomes relying on theS&P 500 index options replicating the volatility index position (Appendix A.3). Figure6.1 provides results of such exercise, given the end-of-day quotations of the VIX and theSPX index options. The replicated volatility index outcomes seem to comply with theirempirical counterparts.5 The hedging performance was checked within this frameworkand the results reported (panel C). In the case of the mean-reverting models, the observedpatterns along the changing time to maturity are in line with the simulated data: the longertime left to expire, the smaller absolute hedging error. These models clearly dominate the

5There two reasons for possible deviations from the VIX empirical values. First, we do not knowthe exact times of the end-of-day VIX reporting and trading times of the SPX options in the portfolio.Hence, the prices used in the replication exercise may not exactly correspond to those used by the CBOEto calculate minute-to-minute VIX quotes. Second, the CBOE does not specify precisely which interestrate series should be used in the VIX formula. The one used in our replication is fixed at 5% level. Bothissues might be considered as a possible source of the deviations from the empirical VIX values and thestudy limiting factors.

86

Whaley specification performance, regardless the maturity group. This becomes especiallyapparent within the 10-day hedging period. The smallest errors in majority of the sectionsand among all the market information frameworks are observed when λ remains the onlyparameter to be implied from the derivative prices, and the P -process estimates reflectthe local VIX characteristics (Table 6.7). For the most intensively traded short-termfutures (Table 4.3), the DO works best for all revision horizons, providing MAHE=0.479for the daily rebalanced portfolios. Such an outcome is only 15% worse compared to thebaseline VIX hedging. It keeps this position for the mid- and long-term contracts too, andprovides the outcomes being close to the VIX-based ones. As the hedging horizon getslonger, the DO and GL models complement each other, and the differences with the VIX-based counterparts become less negligible, e.g., the MAHE provided by the DO within the5-day hedging period is 34% worse from its baseline MAHE=0.942. The differences arealso reflected in higher dollar errors, which were strongly circulating around zero for the1-day section.

A further extension of the methodology has only a theoretical character and concernsthe inclusion of the S&P 500 index asset in the hedging portfolio (panel D). This exten-sion is possible by using the adjustment term (see Appendix A.14). This term includes theforward index level, which can be implied from the SPX options by the put-call parity prop-erty or replaced by the index spot value. Such a modification provides a small improvementover the pure S&P 500 option based strategy. For example, for the short-term contracts,within the 5-day portfolio revision horizon, the DO model provides MAHE=1.225 overthe MAHE=1.260 for the counterpart result in the panel C. The DO and GL remaincomplementary to each other, especially, in the longer portfolio rebalancing periods.

6.5 Summary

This Chapter provides an empirical assessment of the delta hedging effectiveness of threediffusion based volatility derivative pricing models, with respect to VIX call and futurescontracts. The hedging strategies applied in the analysis are in line with methodologiesproposed in the literature. The models are considered within frameworks, which utilizedifferent scopes of the information included in the VIX dynamics, as well as the VIX andS&P 500 derivatives. The influence of less informative portfolios on the overall modelassessment measures is diminished by using trading volume of the hedged derivative as aweighting factor. The empirical outcomes are complemented with the results based on aMonte Carlo experiment.

The VIX call options are hedged by taking the VIX futures position, where both con-

87

tracts expire (possibly) at the same date. The most intensively traded short-maturity calloptions are usually most precisely hedged by the Detemple-Osakwe (DO) mean-revertingmodel. An exception within this maturity group concerns the deep-out-of-the-money(DOTM) section, where the Whaley and DO models work almost equally well, outper-forming the Grünbichler-Lonstaff (GL) mean-reverting pricing framework. This rule holdsalmost over all considered market information setups. In the other cross-sections, if theportfolios are rebalanced less frequently, the Whaley and DO complement each other, witha stronger preference to the latter model for the deep-out- and in-the-money contracts.In addition, a strong position for the Whaley model in the option hedging performance,compared to its rather poor pricing outcomes (see Chapter 5), confirms that both pricingand hedging evaluation is required for a complete model assessment. It supports the Psy-choyios and Skiadopoulos [2006] view that the simple models are not necessarily alwaysthe worse ones.

The VIX futures hedging framework relies on the spot VIX position replication, using astrip of the S&P 500 options, selected according to the CBOE VIX calculation methodol-ogy. A position in the VIX asset is pretty well replicated by these options, even though theavailable contract strips are limited to those in the sample. In the case of the VIX futures,the mean-reversion turned out to be an important factor for better hedging performance,regardless the P -estimation framework. The best results are acquired if the local VIXdynamics is considered and the price of the risk is left to be determined by the market.Between the GL and DO models, the latter works better for the daily rebalanced portfo-lios. However, as the rebalancing happens less often, both models become complementaryto each other.

88

Trading days

0 200 400 600 800 1000

1030

5070

AVIX replicatedVIX empirical

Trading days

0 200 400 600 800 1000

1030

5070

BVIX replicatedVIX empirical

Trading days

%

0 200 400 600 800 1000

−15

−5

05

1015 Relative difference

Trading days

%

0 200 400 600 800 1000−

15−

50

510

15 Relative difference

Figure 6.1: Upper row: the daily VIX closing level replication with the S&P 500 option portfolio (A)and S&P 500 option portfolio and the index position (B) on the 30-day basis. Bottom row: the relativedifferences ( V IXrep

t −V IXempt

V IXempt

× 100%).

Mness τ E(spread) = α0 + α1volC E(spread) = α0 + α1volF

α1 SE p-value α1 SE p-value< 60 -3.710E-06 3.96E-07 2.20E-16 *** -2.04E-05 4.01E-06 3.86E-07 ***

DOTM [60,120) -5.090E-06 9.06E-07 2.07E-08 *** -5.20E-06 2.73E-06 5.67E-02 .≥ 120 -4.600E-06 9.07E-07 4.17E-07 *** -5.50E-05 2.00E-05 5.92E-03 **< 60 -3.630E-06 8.65E-07 2.90E-05 *** -1.74E-05 3.29E-06 1.44E-07 ***

OTM [60,120) -9.650E-06 2.31E-06 3.09E-05 *** -1.57E-05 6.06E-06 9.67E-03 **≥ 120 -8.510E-06 1.97E-06 1.76E-05 *** -4.09E-05 3.47E-05 2.38E-01< 60 -5.060E-06 1.19E-06 2.47E-05 *** -1.76E-05 4.57E-06 1.23E-04 ***

ATM [60,120) -1.300E-05 4.60E-06 4.93E-03 ** -2.47E-05 9.79E-06 1.18E-02 *≥ 120 -3.840E-06 3.21E-06 2.33E-01 -1.32E-05 3.68E-05 7.20E-01< 60 -5.470E-06 1.50E-06 2.66E-04 *** -2.05E-05 6.42E-06 1.44E-03 **

ITM [60,120) -7.350E-06 3.70E-06 4.71E-02 * -2.94E-05 6.85E-06 1.89E-05 ***≥ 120 -9.360E-06 2.60E-06 3.31E-04 *** -3.69E-05 4.66E-05 4.29E-01< 60 -6.210E-06 2.27E-06 6.43E-03 ** -1.62E-06 8.65E-06 8.52E-01

DITM [60,120) -3.460E-05 7.10E-06 1.17E-06 *** -3.83E-05 1.51E-05 1.14E-02 *≥ 120 -6.340E-05 1.30E-05 1.15E-06 *** -1.72E-05 9.95E-05 8.63E-01

Table 6.1: The linear regression results, when the call option bid-ask spread (spread) depends on thetrading volume from the hedged VIX option (volC) or the VIX futures (volF ) (hedging instrument).The trading volumes are chosen to be minimum between the one observed at the portfolio setup andits liquidation. The standard errors estimates (SE) include the heteroskedasticity correction (Andrews[1991]). For the moneyness notion, see Table 4.3; τ denotes time left to the contract expiration (days).The symbols ***, **, and * indicate significance of the t-statistics at 1, 5, and 10%; 1E-01≡ 10−1.

89

1-da

y(∆t)

revision

5-da

yrevision

10-day

revision

market

mod

elmarket

mod

elmarket

mod

elτ<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0Mness

Mod

elM

AH

EWha

ley

0.072

0.078

0.081

0.036

0.025

0.012

0.161

0.145

0.132

0.179

0.123

0.067

0.256

0.217

0.176

0.359

0.249

0.126

DOTM

GL

0.083

0.067

0.069

0.016

0.004

0.001

0.174

0.125

0.114

0.080

0.024

0.008

0.261

0.184

0.148

0.155

0.050

0.015

DO

0.072

0.062

0.065

0.019

0.004

0.001

0.157

0.116

0.109

0.094

0.020

0.007

0.247

0.176

0.139

0.179

0.044

0.014

Wha

ley

0.094

0.080

0.079

0.050

0.025

0.009

0.202

0.146

0.123

0.234

0.121

0.048

0.327

0.226

0.153

0.444

0.249

0.095

OTM

GL

0.106

0.085

0.083

0.025

0.005

0.002

0.226

0.164

0.125

0.118

0.027

0.009

0.359

0.249

0.163

0.216

0.057

0.018

DO

0.093

0.079

0.079

0.028

0.004

0.002

0.206

0.147

0.118

0.126

0.022

0.008

0.329

0.222

0.152

0.232

0.050

0.017

Wha

ley

0.103

0.087

0.098

0.052

0.020

0.008

0.219

0.147

0.118

0.239

0.106

0.045

0.372

0.226

0.154

0.455

0.216

0.077

ATM

GL

0.114

0.096

0.107

0.025

0.004

0.002

0.247

0.167

0.138

0.118

0.024

0.010

0.426

0.255

0.173

0.210

0.049

0.018

DO

0.098

0.091

0.101

0.028

0.004

0.002

0.215

0.154

0.127

0.123

0.020

0.009

0.384

0.233

0.162

0.228

0.046

0.017

Wha

ley

0.111

0.093

0.098

0.050

0.017

0.007

0.231

0.150

0.134

0.226

0.089

0.034

0.372

0.234

0.181

0.436

0.191

0.073

ITM

GL

0.118

0.100

0.106

0.023

0.004

0.002

0.274

0.174

0.16

00.103

0.021

0.008

0.445

0.283

0.190

0.188

0.044

0.017

DO

0.103

0.096

0.101

0.025

0.003

0.001

0.228

0.163

0.152

0.110

0.018

0.007

0.384

0.252

0.184

0.202

0.040

0.016

Wha

ley

0.126

0.123

0.121

0.034

0.014

0.007

0.225

0.189

0.167

0.161

0.073

0.037

0.353

0.301

0.216

0.320

0.155

0.069

DIT

MGL

0.124

0.137

0.124

0.013

0.004

0.002

0.245

0.216

0.173

0.064

0.020

0.011

0.393

0.372

0.198

0.122

0.040

0.021

DO

0.114

0.127

0.123

0.015

0.003

0.002

0.208

0.191

0.168

0.075

0.018

0.010

0.349

0.316

0.195

0.137

0.037

0.020

MD

HE

Wha

ley

0.005

0.001

0.009

-0.001

-0.002

-0.001

0.027

0.011

0.020

-0.003

-0.008

-0.010

0.045

0.006

0.027

-0.012

-0.018

-0.016

DOTM

GL

0.006

0.002

0.008

-0.001

-0.002

-0.001

0.02

80.012

0.013

-0.004

-0.007

-0.007

0.047

0.016

0.020

-0.004

-0.012

-0.014

DO

0.004

0.002

0.008

-0.002

-0.002

-0.001

0.021

0.011

0.010

-0.006

-0.008

-0.007

0.032

0.009

0.017

-0.013

-0.016

-0.014

Wha

ley

0.011

0.005

-0.001

-0.002

-0.003

-0.001

0.047

0.021

0.025

-0.008

-0.012

-0.009

0.082

0.030

0.042

-0.020

-0.027

-0.017

OTM

GL

0.008

0.002

0.000

-0.002

-0.003

-0.002

0.031

0.019

0.025

-0.010

-0.013

-0.009

0.045

0.016

0.040

-0.016

-0.023

-0.017

DO

0.006

0.005

0.000

-0.004

-0.003

-0.002

0.030

0.019

0.021

-0.012

-0.013

-0.008

0.044

0.017

0.034

-0.025

-0.027

-0.017

Wha

ley

0.013

0.003

0.009

-0.003

-0.003

-0.001

0.068

0.028

0.004

-0.010

-0.012

-0.010

0.104

0.030

0.028

-0.025

-0.025

-0.012

ATM

GL

0.012

0.000

0.009

-0.003

-0.003

-0.002

0.051

0.022

0.007

-0.015

-0.014

-0.010

0.039

0.007

0.016

-0.019

-0.024

-0.018

DO

0.008

0.001

0.008

-0.004

-0.003

-0.002

0.047

0.023

0.012

-0.017

-0.014

-0.009

0.043

0.004

0.020

-0.036

-0.028

-0.017

Wha

ley

0.014

0.004

0.003

-0.005

-0.003

-0.001

0.068

0.023

0.011

-0.014

-0.008

-0.007

0.117

0.024

0.030

-0.036

-0.025

-0.015

ITM

GL

0.012

0.003

0.004

-0.004

-0.003

-0.002

0.033

0.020

0.023

-0.016

-0.013

-0.008

0.029

0.007

0.031

-0.026

-0.026

-0.016

DO

0.008

0.003

0.005

-0.005

-0.003

-0.001

0.043

0.021

0.020

-0.020

-0.013

-0.007

0.049

0.005

0.033

-0.038

-0.027

-0.016

Wha

ley

0.014

-0.008

0.003

-0.006

-0.003

-0.001

0.070

-0.001

-0.018

-0.016

-0.012

-0.010

0.118

-0.006

-0.005

-0.046

-0.024

-0.015

DIT

MGL

0.014

-0.008

0.005

-0.005

-0.003

-0.002

0.043

-0.018

-0.018

-0.020

-0.015

-0.011

0.057

-0.070

-0.019

-0.041

-0.031

-0.021

DO

0.010

-0.009

0.004

-0.005

-0.003

-0.002

0.051

-0.009

-0.017

-0.026

-0.015

-0.010

0.052

-0.039

-0.014

-0.046

-0.030

-0.020

Table6.2:

Meanab

solute/d

ollarhedg

ingerrors

(MAHE/M

DHE)of

theVIX

callop

tion

absolute

hedg

ingerrors,w

eigh

tedby

theminim

alop

tion

trad

ingvolumebe

tweenthe

oneob

served

atthepo

rtfolio

setupan

ditsliq

uida

tion

.The

‘market’

resultsconcernthecaseswhenthehedg

ingis

done

ontheem

piricald

aily

VIX

callop

tion

andfutures

prices

(February24

,2006–April30,2

010),w

iththepo

rtfolio

weigh

tsim

pliedfrom

theW

haley(W

),Grünb

ichler-Lon

gstaff(G

L),a

ndDetem

ple-Osakw

e(D

O)mod

els.

The

‘mod

el’r

esults

rely

onthemod

el-sim

ulated

derivative

prices.The

derivative

priceat

thepo

rtfolio

setupis

determ

ined

from

thegivenVIX

andthecontract

characteristics.

The

term

inationVIX

valueisim

pliedby

themod

eldy

namics.

The

‘mod

el’results

areaveraged

over

theS=500marketscenarios.

The

mod

elpa

rametersareestimated

over

thewho

leVIX

samplespan

ning

over

thepe

riod

from

Janu

ary2,

1990

throug

hApril30,2

010.

The

volatilityrisk

priceis

theon

lypa

rameter

left

tobe

determ

ined

bythe

market.

90

1-day(∆t)

revision5-day

revision10-day

revisionmarket

model

market

model

market

model

τ<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120Mness

Model

MA

HE

Whaley

0.0800.088

0.0830.048

0.0290.014

0.1890.166

0.1340.236

0.1460.070

0.2940.250

0.1780.465

0.2970.140

DOTM

GL

0.0800.067

0.0700.011

0.0020.001

0.1810.137

0.1290.053

0.0100.005

0.2800.207

0.1650.097

0.0210.009

DO

0.0770.066

0.0690.013

0.0020.001

0.1810.132

0.1260.063

0.0110.005

0.2790.201

0.1640.120

0.0240.009

Whaley

0.0970.084

0.0780.061

0.0290.011

0.2130.152

0.1210.289

0.1410.051

0.3420.238

0.1520.550

0.2970.107

OTM

GL

0.1040.087

0.0840.017

0.0030.002

0.2320.165

0.1240.077

0.0160.008

0.3730.250

0.1640.137

0.0340.016

DO

0.1000.083

0.0810.019

0.0030.002

0.2270.155

0.1170.087

0.0170.008

0.3680.235

0.1570.160

0.0370.015

Whaley

0.1060.091

0.0970.063

0.0250.009

0.2320.155

0.1210.296

0.1250.049

0.3820.238

0.1540.567

0.2640.090

ATM

GL

0.1060.100

0.1060.019

0.0030.002

0.2370.176

0.1450.087

0.0170.009

0.4260.262

0.1820.152

0.0360.016

DO

0.1020.095

0.1030.021

0.0030.002

0.2320.164

0.1380.097

0.0180.008

0.4150.246

0.1710.179

0.0380.016

Whaley

0.1200.098

0.0960.062

0.0210.008

0.2460.156

0.1340.289

0.1100.036

0.3890.243

0.1850.557

0.2360.083

ITM

GL

0.1110.105

0.1110.018

0.0030.002

0.2470.179

0.1680.083

0.0170.008

0.4110.270

0.2090.146

0.0360.017

DO

0.1080.101

0.1060.020

0.0030.002

0.2380.164

0.1590.090

0.0180.007

0.4000.252

0.1990.171

0.0380.016

Whaley

0.1390.126

0.1220.046

0.0180.008

0.2600.206

0.1750.225

0.0960.042

0.4020.311

0.2310.433

0.2030.085

DIT

MGL

0.1240.130

0.1290.014

0.0040.002

0.2210.189

0.1860.070

0.0190.011

0.3430.290

0.1910.124

0.0380.022

DO

0.1220.128

0.1280.015

0.0040.002

0.2150.180

0.1800.076

0.0200.011

0.3420.272

0.1870.142

0.0420.021

MD

HE

Whaley

0.0050.002

0.009-0.001

-0.002-0.002

0.0290.013

0.018-0.001

-0.008-0.005

0.0500.008

0.024-0.012

-0.018-0.017

DOTM

GL

0.0010.001

0.005-0.001

-0.001-0.001

0.0100.008

0.003-0.008

-0.007-0.005

0.0080.003

0.013-0.015

-0.013-0.009

DO

0.0010.000

0.005-0.001

-0.001-0.001

0.0130.008

0.001-0.008

-0.007-0.005

0.0150.001

0.007-0.017

-0.013-0.009

Whaley

0.0120.005

-0.002-0.003

-0.003-0.002

0.0510.022

0.021-0.006

-0.008-0.005

0.0940.034

0.032-0.026

-0.026-0.015

OTM

GL

0.0010.006

-0.002-0.003

-0.003-0.002

0.0180.020

0.031-0.016

-0.013-0.008

0.0180.015

0.038-0.031

-0.027-0.016

DO

0.0010.005

-0.001-0.003

-0.003-0.002

0.0180.018

0.025-0.014

-0.013-0.008

0.0250.017

0.030-0.030

-0.027-0.015

Whaley

0.0140.005

0.009-0.004

-0.003-0.002

0.0740.030

-0.001-0.010

-0.009-0.007

0.1240.040

0.025-0.034

-0.030-0.015

ATM

GL

0.0020.002

0.004-0.004

-0.003-0.002

0.0310.022

0.017-0.020

-0.014-0.009

0.0100.005

0.035-0.036

-0.028-0.016

DO

0.0020.002

0.004-0.004

-0.003-0.002

0.0300.023

0.020-0.018

-0.014-0.008

0.0200.014

0.032-0.037

-0.028-0.016

Whaley

0.0160.004

0.004-0.004

-0.003-0.001

0.0790.025

0.006-0.013

-0.008-0.003

0.1420.035

0.028-0.042

-0.027-0.012

ITM

GL

0.0060.004

0.004-0.005

-0.003-0.002

0.0370.022

0.047-0.024

-0.014-0.008

0.0360.013

0.058-0.041

-0.029-0.017

DO

0.0060.004

0.004-0.004

-0.003-0.002

0.0350.020

0.043-0.021

-0.014-0.007

0.0440.020

0.057-0.045

-0.028-0.016

Whaley

0.016-0.009

0.003-0.005

-0.003-0.001

0.0750.006

-0.019-0.016

-0.010-0.007

0.1290.017

0.000-0.047

-0.030-0.016

DIT

MGL

0.009-0.005

0.009-0.005

-0.003-0.002

0.0530.003

0.006-0.027

-0.017-0.011

0.058-0.006

0.036-0.049

-0.031-0.022

DO

0.009-0.004

0.008-0.005

-0.003-0.002

0.0460.004

0.007-0.025

-0.016-0.011

0.0540.003

0.045-0.051

-0.032-0.021

Table6.3:

The

VIX

calloption

hedgingresults

(setupas

inTable

6.2).The

P-process

parameters

areestim

atedover

the252-day

VIX

seriesbacked

forthe

firsttim

eon

February24,2006.

The

volatilityrisk

priceis

theonly

parameter

leftto

bedeterm

inedby

themarket.

91

1-da

y(∆t)

revision

5-da

yrevision

10-day

revision

market

mod

elmarket

mod

elmarket

mod

elτ<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0<60

[60,120)

≥12

0Mness

Mod

elM

AH

EWha

ley

0.072

0.078

0.081

0.036

0.025

0.012

0.161

0.145

0.132

0.179

0.123

0.067

0.256

0.217

0.176

0.359

0.249

0.126

DOTM

GL

0.081

0.064

0.067

0.016

0.006

0.002

0.169

0.122

0.113

0.084

0.034

0.012

0.255

0.180

0.147

0.166

0.069

0.022

DO

0.070

0.061

0.065

0.020

0.005

0.002

0.155

0.115

0.109

0.094

0.025

0.009

0.244

0.173

0.142

0.185

0.056

0.018

Wha

ley

0.094

0.080

0.079

0.050

0.025

0.009

0.202

0.146

0.123

0.234

0.121

0.048

0.327

0.226

0.153

0.444

0.249

0.095

OTM

GL

0.102

0.081

0.082

0.027

0.008

0.002

0.217

0.156

0.125

0.128

0.042

0.012

0.347

0.235

0.162

0.237

0.085

0.026

DO

0.091

0.079

0.078

0.029

0.006

0.002

0.204

0.146

0.119

0.129

0.030

0.010

0.334

0.223

0.153

0.243

0.064

0.020

Wha

ley

0.103

0.087

0.098

0.052

0.020

0.008

0.219

0.147

0.118

0.239

0.106

0.045

0.372

0.226

0.154

0.455

0.216

0.077

ATM

GL

0.105

0.094

0.107

0.028

0.007

0.002

0.231

0.154

0.132

0.133

0.037

0.013

0.409

0.240

0.170

0.243

0.077

0.021

DO

0.094

0.090

0.102

0.029

0.005

0.002

0.210

0.153

0.123

0.129

0.027

0.011

0.387

0.230

0.162

0.248

0.061

0.018

Wha

ley

0.111

0.093

0.098

0.050

0.017

0.007

0.231

0.150

0.134

0.226

0.089

0.034

0.372

0.234

0.181

0.436

0.191

0.073

ITM

GL

0.112

0.097

0.104

0.026

0.006

0.002

0.258

0.165

0.15

20.120

0.034

0.009

0.428

0.262

0.186

0.226

0.073

0.023

DO

0.099

0.094

0.101

0.026

0.005

0.002

0.221

0.156

0.147

0.118

0.026

0.008

0.387

0.246

0.182

0.227

0.058

0.019

Wha

ley

0.126

0.123

0.121

0.034

0.014

0.007

0.225

0.189

0.167

0.161

0.073

0.037

0.353

0.301

0.216

0.320

0.155

0.069

DIT

MGL

0.125

0.129

0.123

0.016

0.006

0.003

0.254

0.201

0.166

0.083

0.031

0.013

0.413

0.343

0.193

0.161

0.064

0.026

DO

0.113

0.121

0.120

0.018

0.005

0.002

0.199

0.177

0.160

0.087

0.026

0.012

0.342

0.297

0.193

0.169

0.057

0.023

MD

HE

Wha

ley

0.005

0.001

0.009

-0.001

-0.002

-0.001

0.027

0.011

0.020

-0.003

-0.008

-0.010

0.045

0.006

0.027

-0.012

-0.018

-0.016

DOTM

GL

0.005

0.002

0.008

-0.001

-0.002

-0.001

0.02

50.009

0.010

-0.007

-0.008

-0.007

0.043

0.010

0.017

-0.014

-0.014

-0.013

DO

0.003

0.002

0.007

-0.002

-0.002

-0.001

0.023

0.014

0.009

-0.007

-0.008

-0.007

0.038

0.016

0.019

-0.014

-0.016

-0.013

Wha

ley

0.011

0.005

-0.001

-0.002

-0.003

-0.001

0.047

0.021

0.025

-0.008

-0.012

-0.009

0.082

0.030

0.042

-0.020

-0.027

-0.017

OTM

GL

0.007

0.005

0.001

-0.003

-0.003

-0.002

0.027

0.017

0.028

-0.015

-0.015

-0.009

0.039

0.016

0.039

-0.025

-0.026

-0.018

DO

0.005

0.005

0.001

-0.003

-0.003

-0.002

0.027

0.020

0.023

-0.013

-0.013

-0.008

0.042

0.020

0.034

-0.027

-0.026

-0.017

Wha

ley

0.013

0.003

0.009

-0.003

-0.003

-0.001

0.068

0.028

0.004

-0.010

-0.012

-0.010

0.104

0.030

0.028

-0.025

-0.025

-0.012

ATM

GL

0.010

0.001

0.008

-0.004

-0.003

-0.002

0.042

0.018

0.008

-0.022

-0.014

-0.010

0.030

0.001

0.017

-0.029

-0.025

-0.017

DO

0.007

0.001

0.007

-0.003

-0.003

-0.002

0.038

0.020

0.008

-0.018

-0.014

-0.009

0.037

0.007

0.016

-0.037

-0.029

-0.016

Wha

ley

0.014

0.004

0.003

-0.005

-0.003

-0.001

0.068

0.023

0.011

-0.014

-0.008

-0.007

0.117

0.024

0.030

-0.036

-0.025

-0.015

ITM

GL

0.011

0.003

0.002

-0.005

-0.003

-0.002

0.022

0.013

0.020

-0.023

-0.013

-0.007

0.016

-0.002

0.033

-0.035

-0.027

-0.017

DO

0.006

0.002

0.002

-0.005

-0.003

-0.001

0.030

0.011

0.015

-0.021

-0.013

-0.008

0.036

0.001

0.029

-0.040

-0.029

-0.016

Wha

ley

0.014

-0.008

0.003

-0.006

-0.003

-0.001

0.070

-0.001

-0.018

-0.016

-0.012

-0.010

0.118

-0.006

-0.005

-0.046

-0.024

-0.015

DIT

MGL

0.011

-0.007

0.008

-0.005

-0.003

-0.002

0.020

-0.025

-0.007

-0.026

-0.016

-0.010

0.022

-0.074

0.000

-0.048

-0.029

-0.019

DO

0.008

-0.008

0.005

-0.005

-0.003

-0.002

0.037

-0.019

-0.012

-0.024

-0.015

-0.010

0.039

-0.052

-0.005

-0.048

-0.030

-0.020

Table6.4:

The

VIX

callop

tion

hedg

ingresults(setup

asin

Table6.2):The

P-process

parametersareestimated

over

thewho

leVIX

samplespan

ning

thepe

riod

from

Janu

ary2,

1990

upto

April30

,201

0.The

setof

thepa

rametersleft

tobe

determ

ined

bythemarketis

extend

edin

thecase

ofGL(+m)an

dDO(+κ)mod

el.

92

1-day(∆t)

revision5-day

revision10-day

revisionmarket

model

market

model

market

model

τ<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120<60

[60,120)≥

120Mness

Model

MA

HE

Whaley

0.0800.088

0.0830.048

0.0290.014

0.1890.166

0.1340.236

0.1460.070

0.2940.250

0.1780.465

0.2970.140

DOTM

GL

0.0730.061

0.0630.013

0.0030.002

0.1620.113

0.1080.064

0.0150.008

0.2510.171

0.1430.119

0.0310.016

DO

0.0670.059

0.0630.015

0.0030.001

0.1560.108

0.1070.069

0.0140.007

0.2450.163

0.1420.128

0.0290.015

Whaley

0.0970.084

0.0780.061

0.0290.011

0.2130.152

0.1210.289

0.1410.051

0.3420.238

0.1520.550

0.2970.107

OTM

GL

0.0930.079

0.0780.020

0.0040.002

0.2140.147

0.1140.090

0.0200.009

0.3460.227

0.1500.159

0.0410.019

DO

0.0880.075

0.0750.020

0.0030.002

0.2040.137

0.1100.091

0.0180.009

0.3360.215

0.1450.164

0.0360.017

Whaley

0.1060.091

0.0970.063

0.0250.009

0.2320.155

0.1210.296

0.1250.049

0.3820.238

0.1540.567

0.2640.090

ATM

GL

0.0960.093

0.1010.020

0.0040.002

0.2210.161

0.1230.092

0.0200.010

0.4020.237

0.1630.163

0.0400.018

DO

0.0910.088

0.0980.020

0.0030.002

0.2110.150

0.1150.091

0.0170.009

0.3880.226

0.1550.163

0.0350.017

Whaley

0.1200.098

0.0960.062

0.0210.008

0.2460.156

0.1340.289

0.1100.036

0.3890.243

0.1850.557

0.2360.083

ITM

GL

0.1020.098

0.1030.019

0.0040.002

0.2310.164

0.1480.084

0.0200.008

0.4010.258

0.1840.153

0.0410.018

DO

0.0980.094

0.0990.018

0.0030.002

0.2220.156

0.1410.082

0.0170.007

0.3880.247

0.1740.143

0.0340.016

Whaley

0.1390.126

0.1220.046

0.0180.008

0.2600.206

0.1750.225

0.0960.042

0.4020.311

0.2310.433

0.2030.085

DIT

MGL

0.1140.123

0.1190.014

0.0040.002

0.2050.181

0.1620.070

0.0210.012

0.3550.306

0.1860.129

0.0430.022

DO

0.1130.121

0.1170.013

0.0030.002

0.2020.177

0.1570.065

0.0180.010

0.3500.297

0.1870.115

0.0350.020

MD

HE

Whaley

0.0050.002

0.009-0.001

-0.002-0.002

0.0290.013

0.018-0.001

-0.008-0.005

0.0500.008

0.024-0.012

-0.018-0.017

DOTM

GL

0.0030.002

0.006-0.002

-0.001-0.001

0.0170.011

0.004-0.007

-0.007-0.006

0.0310.015

0.017-0.014

-0.014-0.011

DO

0.0040.001

0.006-0.002

-0.001-0.001

0.0220.009

0.000-0.007

-0.007-0.006

0.0420.011

0.008-0.014

-0.014-0.011

Whaley

0.0120.005

-0.002-0.003

-0.003-0.002

0.0510.022

0.021-0.006

-0.008-0.005

0.0940.034

0.032-0.026

-0.026-0.015

OTM

GL

0.0030.005

-0.001-0.003

-0.003-0.002

0.0190.017

0.022-0.015

-0.013-0.008

0.0300.017

0.033-0.028

-0.026-0.016

DO

0.0050.005

0.000-0.003

-0.003-0.002

0.0250.016

0.014-0.014

-0.013-0.008

0.0430.016

0.021-0.026

-0.024-0.015

Whaley

0.0140.005

0.009-0.004

-0.003-0.002

0.0740.030

-0.001-0.010

-0.009-0.007

0.1240.040

0.025-0.034

-0.030-0.015

ATM

GL

0.0050.001

0.007-0.004

-0.003-0.002

0.0300.017

0.006-0.019

-0.015-0.009

0.024-0.001

0.018-0.035

-0.027-0.015

DO

0.0070.001

0.006-0.003

-0.003-0.002

0.0360.017

0.005-0.017

-0.013-0.008

0.0380.003

0.010-0.031

-0.026-0.015

Whaley

0.0160.004

0.004-0.004

-0.003-0.001

0.0790.025

0.006-0.013

-0.008-0.003

0.1420.035

0.028-0.042

-0.027-0.012

ITM

GL

0.0060.001

0.002-0.005

-0.003-0.001

0.0240.007

0.018-0.021

-0.014-0.007

0.021-0.006

0.029-0.043

-0.028-0.016

DO

0.0060.002

0.001-0.004

-0.002-0.001

0.0280.007

0.010-0.022

-0.014-0.007

0.034-0.002

0.022-0.034

-0.026-0.014

Whaley

0.016-0.009

0.003-0.005

-0.003-0.001

0.0750.006

-0.019-0.016

-0.010-0.007

0.1290.017

0.000-0.047

-0.030-0.016

DIT

MGL

0.007-0.008

0.006-0.005

-0.003-0.002

0.035-0.019

-0.013-0.026

-0.016-0.011

0.027-0.050

-0.006-0.049

-0.031-0.021

DO

0.006-0.008

0.005-0.005

-0.003-0.002

0.033-0.019

-0.016-0.024

-0.015-0.010

0.024-0.045

-0.011-0.040

-0.029-0.020

Table6.5:

The

VIX

calloption

hedgingresults

(setupas

inTable

6.2):the

P-process

parameters

areestim

atedover

the252-day

VIX

seriesbacked

forthe

firsttim

eon

February24,2006.

The

setofthe

parameters

leftto

bedeterm

inedby

themarket

isextended

inthe

caseofG

L(+m)and

DO(+κ)

model.

93

1-day (∆t) revision 5-day revision 10-day revisionτ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Model MAHEA: simVX vs. simVIX

Whaley 0.059 0.058 0.061 0.295 0.293 0.303 0.589 0.587 0.582GL 0.078 0.041 0.024 0.385 0.203 0.119 0.759 0.398 0.233DO 0.079 0.038 0.018 0.387 0.186 0.091 0.759 0.361 0.178

B: VX vs. VIXWhaley 0.747 0.917 1.003 1.338 1.764 1.943 1.793 2.278 2.477GL 0.516 0.323 0.287 0.910 0.664 0.544 1.277 0.946 0.769DO 0.489 0.266 0.266 0.893 0.618 0.536 1.290 0.933 0.781

C: VX vs. SPX optionsWhaley 0.864 1.010 1.094 2.448 2.817 3.022 4.593 4.873 5.064GL 0.618 0.372 0.300 1.735 1.060 0.744 3.078 1.824 1.142DO 0.595 0.304 0.273 1.673 0.891 0.608 3.019 1.627 1.015

D: VX vs. SPX options and SPXWhaley 0.935 1.122 1.221 2.532 2.980 3.221 4.640 4.996 5.221GL 0.653 0.391 0.305 1.755 1.075 0.745 3.071 1.827 1.141DO 0.625 0.311 0.269 1.685 0.885 0.597 3.009 1.622 1.005

MDHEA: simVX vs. simVIX

Whaley 0.000 0.015 0.030 0.003 0.078 0.159 0.006 0.156 0.318GL 0.032 0.022 0.017 0.167 0.113 0.092 0.346 0.236 0.187DO 0.036 0.024 0.015 0.199 0.125 0.077 0.433 0.258 0.155

B: VX vs. VIXWhaley -0.014 -0.097 -0.133 0.048 -0.030 0.008 0.066 -0.104 -0.019GL 0.000 -0.034 -0.007 0.022 -0.103 -0.063 0.021 -0.197 -0.092DO 0.014 -0.014 0.009 0.073 -0.036 -0.009 0.106 -0.085 -0.010

C: VX vs. SPX optionsWhaley -0.080 -0.221 -0.292 -0.757 -1.006 -1.131 -0.722 -1.081 -0.829GL -0.043 -0.068 -0.031 -0.537 -0.425 -0.253 -0.679 -0.598 -0.303DO -0.039 -0.050 -0.008 -0.499 -0.319 -0.127 -0.522 -0.318 -0.097

D: VX vs. SPX options and SPXWhaley -0.074 -0.221 -0.302 -0.736 -0.991 -1.128 -0.694 -1.056 -0.801GL -0.037 -0.066 -0.031 -0.520 -0.415 -0.249 -0.661 -0.593 -0.300DO -0.032 -0.047 -0.007 -0.482 -0.311 -0.125 -0.501 -0.311 -0.093

Table 6.6: The mean absolute/dollar hedging errors (MAHE/MDHE) of the VIX futures (VX) weighted by the minimalfutures trading volume, between the one observed at the portfolio setup and its liquidation. The numbers correspond to fourdifferent hedging frameworks denoted by A:D letters. The first one denoted by ‘A’ (simVX vs. simVIX) reflects the situationwhen all the derivative data are simulated purely within the model specifications, using the same contract characteristics asin the empirical sample. Reported numbers are averaged over the mean and median results over S=500 sample replications.The framework ‘B’ (VX vs. VIX) corresponds to the portfolio, where the VIX futures contract is hedged with a position inthe VIX asset. The third framework ‘C’ considers the VIX futures position hedged with the VIX replicating SPX options.In the last setup ‘D’, the VIX futures are hedged using both a position in the SPX options and the SPX itself. The samplespans the period from February 24, 2006 through April 30, 2010. The model parameters are estimated over the whole VIXwindow spanning over the period from January 2, 1990, to April 30, 2010. The market price of the risk is the only parameterleft to be determined by the market.

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1-day (∆t) revision 5-day revision 10-day revisionτ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Model MAHEA: simVX vs. simVIX

Whaley 0.066 0.061 0.061 0.332 0.310 0.303 0.674 0.627 0.586GL 0.080 0.016 0.007 0.405 0.080 0.036 0.797 0.159 0.069DO 0.087 0.020 0.009 0.419 0.094 0.044 0.793 0.179 0.081

B: VX vs. VIXWhaley 0.732 0.861 0.924 1.309 1.671 1.803 1.759 2.158 2.301GL 0.424 0.318 0.316 0.933 0.781 0.651 1.512 1.158 0.927DO 0.416 0.308 0.313 0.942 0.769 0.645 1.537 1.161 0.920

C: VX vs. SPX optionsWhaley 0.848 0.953 1.013 2.406 2.690 2.828 4.526 4.683 4.823GL 0.481 0.331 0.318 1.260 0.833 0.658 2.039 1.242 0.937DO 0.479 0.323 0.314 1.260 0.831 0.650 2.103 1.256 0.937

D: VX vs. SPX options and SPXWhaley 0.916 1.058 1.130 2.487 2.843 3.013 4.571 4.801 4.968GL 0.473 0.324 0.315 1.218 0.820 0.654 1.990 1.233 0.935DO 0.479 0.315 0.310 1.225 0.816 0.646 2.057 1.246 0.934

MDHEA: simVX vs. simVIX

Whaley -0.015 0.003 0.020 -0.074 0.015 0.106 -0.159 0.023 0.208GL 0.034 0.004 0.004 0.158 0.023 0.023 0.281 0.051 0.049DO 0.039 0.008 0.006 0.185 0.037 0.028 0.328 0.066 0.055

B: VX vs. VIXWhaley -0.014 -0.097 -0.126 0.049 -0.031 0.009 0.067 -0.105 -0.027GL 0.001 -0.011 0.008 -0.060 -0.114 -0.035 -0.127 -0.207 -0.042DO 0.009 -0.008 0.008 -0.052 -0.108 -0.035 -0.130 -0.198 -0.041

C: VX vs. SPX optionsWhaley -0.080 -0.215 -0.276 -0.742 -0.963 -1.058 -0.687 -0.985 -0.676GL -0.011 -0.008 0.011 -0.352 -0.185 -0.055 -0.626 -0.338 -0.063DO -0.022 -0.015 0.007 -0.407 -0.201 -0.064 -0.680 -0.338 -0.066

D: VX vs. SPX options and SPXWhaley -0.074 -0.216 -0.287 -0.723 -0.953 -1.062 -0.661 -0.966 -0.658GL -0.009 -0.007 0.010 -0.343 -0.180 -0.053 -0.626 -0.338 -0.064DO -0.020 -0.015 0.006 -0.401 -0.198 -0.063 -0.683 -0.338 -0.066

Table 6.7: (Cont. of Table 6.6) The P -process parameters are estimated over 252-day VIX series backed for the first timeon February 24, 2006.

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1-day (∆t) revision 5-day revision 10-day revisionτ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Model MAHEA: simVX vs. simVIX

Whaley 0.059 0.058 0.061 0.295 0.293 0.303 0.589 0.587 0.582GL 0.074 0.037 0.024 0.361 0.183 0.118 0.717 0.359 0.230DO 0.079 0.037 0.019 0.385 0.179 0.095 0.764 0.350 0.183

B: VX vs. VIXWhaley 0.747 0.917 1.003 1.338 1.764 1.943 1.793 2.278 2.477GL 0.548 0.482 0.442 0.987 0.894 0.771 1.349 1.211 1.025DO 0.528 0.376 0.332 0.948 0.731 0.600 1.335 1.059 0.864

C: VX vs. SPX optionsWhaley 0.864 1.010 1.094 2.448 2.817 3.022 4.593 4.873 5.064GL 0.645 0.519 0.459 1.793 1.359 1.100 3.240 2.345 1.710DO 0.625 0.406 0.347 1.695 1.058 0.748 3.129 1.962 1.321

D: VX vs. SPX options and SPXWhaley 0.935 1.122 1.221 2.532 2.980 3.221 4.640 4.996 5.221GL 0.680 0.549 0.485 1.812 1.388 1.124 3.228 2.343 1.713DO 0.655 0.419 0.355 1.707 1.066 0.749 3.119 1.960 1.327

MDHEA: simVX vs. simVIX

Whaley 0.000 0.015 0.030 0.003 0.078 0.159 0.006 0.156 0.318GL 0.028 0.017 0.013 0.146 0.084 0.067 0.290 0.170 0.142DO 0.032 0.017 0.010 0.169 0.085 0.054 0.378 0.184 0.122

B: VX vs. VIXWhaley -0.014 -0.097 -0.133 0.048 -0.030 0.008 0.066 -0.104 -0.019GL 0.002 -0.032 -0.003 -0.044 -0.207 -0.173 -0.113 -0.397 -0.292DO 0.016 -0.006 0.020 0.045 -0.053 -0.008 0.028 -0.153 -0.035

C: VX vs. SPX optionsWhaley -0.080 -0.221 -0.292 -0.757 -1.006 -1.131 -0.722 -1.081 -0.829GL -0.043 -0.073 -0.028 -0.580 -0.557 -0.441 -0.763 -0.862 -0.596DO -0.029 -0.034 0.011 -0.484 -0.317 -0.143 -0.508 -0.315 -0.046

D: VX vs. SPX options and SPXWhaley -0.074 -0.221 -0.302 -0.736 -0.991 -1.128 -0.694 -1.056 -0.801GL -0.036 -0.065 -0.022 -0.561 -0.537 -0.422 -0.748 -0.855 -0.590DO -0.021 -0.027 0.016 -0.462 -0.298 -0.129 -0.486 -0.302 -0.035

Table 6.8: The VIX futures hedging results (notion as in Table 6.6). The model parameters are estimated over the wholeVIX sample, which covers the period from January 2, 1990 up to April 30, 2010. The set of the parameters left to bedetermined by the market is extended in the case of GL (+m) and DO(+κ) model.

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1-day (∆t) revision 5-day revision 10-day revisionτ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Model MAHEA: simVX vs. simVIX

Whaley 0.066 0.061 0.061 0.332 0.310 0.303 0.674 0.627 0.586GL 0.075 0.023 0.012 0.394 0.111 0.057 0.796 0.214 0.108DO 0.081 0.026 0.013 0.404 0.120 0.062 0.807 0.226 0.116

B: VX vs. VIXWhaley 0.732 0.861 0.924 1.309 1.671 1.803 1.759 2.158 2.301GL 0.488 0.365 0.337 0.949 0.818 0.669 1.441 1.186 0.949DO 0.458 0.361 0.341 0.935 0.825 0.669 1.441 1.183 0.952

C: VX vs. SPX optionsWhaley 0.848 0.953 1.013 2.406 2.690 2.828 4.526 4.683 4.823GL 0.561 0.378 0.342 1.473 0.967 0.735 2.506 1.536 1.076DO 0.538 0.375 0.351 1.444 0.948 0.729 2.458 1.520 1.067

D: VX vs. SPX options and SPXWhaley 0.916 1.058 1.130 2.487 2.843 3.013 4.571 4.801 4.968GL 0.566 0.382 0.348 1.453 0.970 0.741 2.487 1.538 1.081DO 0.547 0.380 0.359 1.430 0.957 0.739 2.437 1.520 1.073

MDHEA: simVX vs. simVIX

Whaley -0.015 0.003 0.020 -0.074 0.015 0.106 -0.159 0.023 0.208GL 0.027 0.008 0.007 0.131 0.044 0.037 0.238 0.091 0.079DO 0.035 0.016 0.011 0.166 0.073 0.051 0.326 0.138 0.102

B: VX vs. VIXWhaley -0.014 -0.097 -0.126 0.049 -0.031 0.009 0.067 -0.105 -0.027GL 0.027 -0.003 0.014 0.051 -0.045 -0.001 0.067 -0.117 0.003DO 0.031 0.000 0.020 0.081 -0.039 -0.008 0.097 -0.113 -0.017

C: VX vs. SPX optionsWhaley -0.080 -0.215 -0.276 -0.742 -0.963 -1.058 -0.687 -0.985 -0.676GL 0.012 -0.008 0.013 -0.285 -0.176 -0.065 -0.301 -0.234 -0.097DO -0.008 -0.022 0.011 -0.321 -0.189 -0.079 -0.314 -0.216 -0.099

D: VX vs. SPX options and SPXWhaley -0.074 -0.216 -0.287 -0.723 -0.953 -1.062 -0.661 -0.966 -0.658GL 0.020 -0.005 0.015 -0.266 -0.164 -0.056 -0.280 -0.227 -0.092DO -0.001 -0.018 0.013 -0.307 -0.183 -0.074 -0.296 -0.212 -0.097

Table 6.9: The VIX futures hedging results (notion as in Table 6.6). The model parameters are estimated over the 252-dayVIX series, backed for the first time on February 24, 2006. The set of the parameters left to be determined by the marketis extended in the case of GL (+m) and DO(+κ) model.

Chapter 7

Volatility Derivative Pricing ModelsBased on the Jump-Diffusions.Specification and PricingPerformance

7.1 Introduction

Next to the mean-reverting property, the empirical characteristics of the VIX indicate onanother important element of its dynamics, namely, the way it reacts to the market stress(see section 4.1). Bates [2000] argues that large volatility increases observed, for example,in the market turmoil, create a doubt about its pure diffusion dynamics assumption andsuggests a jump component amendment. Dotsis et al. [2007] carried out an extensive studyfor the U.S. volatility indexes (including VIX), in which the jump component turnedout to be the second, after the mean-reversion, most important factor in the volatilitydynamics modeling. More evidence of jumps in the implied volatility indexes is provided,for example, in Moraux et al. [1999] for the French Market Volatility Index (VX1) orWagner and Szimayer [2004] for the German VDAX. The volatility derivative pricingbased on the jump-diffusions is studied theoretically in Andersen and Andreasen [2000]and He et al. [2006]. Both rely on a Merton-type approach, where the GBM dynamics isextended with the jump component (see section 3.3.1). Psychoyios et al. [2009] apply theframework of Bakshi and Madan [2000] to price the VIX derivatives, when the underlyingfollows the log Ornstein-Uhlenbeck jump-diffusion. However, no empirical performanceassessment is provided. We refer to Chapter 2 with review of the literature about thejump-diffusion based VIX modeling and its derivative pricing.

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The primary objective of this Chapter is to allocate and empirically quantify the VIXderivative pricing consequences, provided by the jump extensions to the baseline diffusionbased specifications. The outcomes are analyzed with respect to performance within thediffusion and jump-diffusion model categories. The Whaley model performance is com-pared with the Merton GBM-with-jump based approach. The relevant pricing frameworksfor the mean-reverting GL and DO specifications extended by jumps are provided. Theanalysis captures the in- and out-of-sample pricing performance assessment.

Similar as in Chapter 5, the model robustness is assessed within different market in-formation setups. The underlying VIX dynamics estimates reflect the VIX characteristicsover almost 20-year period, since January 1990. Alternatively, the estimates change ev-ery day along short, 252-day backward VIX series, and reflect more local volatility indexcharacteristics. In determining the model free parameters, all the daily VIX derivativesinformation is taken into account. In addition, each contract contribution is weighted byits daily relative trading volume, which grants a greater weight to the more representativerecords.

The Chapter consists of four sections. The first one establishes the framework for themodel pricing assessment. In the next section, we discuss the numerical issues related tothe parameter estimation and describe its empirical results. The third section is devotedto the pricing results for the in- and out-of-sample frameworks. The last section concludesthe main findings of the Chapter.

7.2 Model performance assessment

Given the forecast horizon ofH days, the models are assessed with respect to the in-sample(H=0) fit and the out-of-sample (H=1, 5, and 20) pricing performance. We use the meanrelative error (MRE: a = 1) and mean-squared relative error (MSRE: a=2) criteria,

M(S)RE(H) = 1T

∑t,i

wCt,iV olt

Ci(Vt, θPt−H , θQt−H)− Ct,iCt,i

a

+∑t,j

wPt,jV olt

Pj(Vt, θPt−H , θQt−H)− Pt,jPt,j

a

+∑t,k

wFt,kV olt

Fk(Vt, θPt−H , θQt−H)− Ft,kFt,k

a , (7.1)

where:

- T : the number of trading days included in the assessment,

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- wt,. =

volt,., if the derivative (C, P , F ) information is considered in

the model assessment on day t0, otherwise,

volt : the daily trading volume of the particular VIX derivative listed on day t,

- V olt = ∑iw

Ct,i +∑

j wPt,j +∑

k wFt,k.

Both MRE and MSRE criteria originate from the objective function (3.47), which wasused in the calibration of the Q-parameters to the market derivative prices. The MSRE isour primary criterion, which measures the model fit precision reflected in the magnitudeof the differences between the model prices and their market counterparts. The MREindicates the average direction of mispricing, which is an important aspect if the analy-sis concerns the jump extension pricing consequences, nested within the diffusion basedframework. The MRE complements the MSRE readings, and due to possible settling-offthe opposite sign errors around zero, cannot be used as a primary criterion. The spotunderlying value at t, Vt, is assumed to be known to avoid the additional bias from theunderlying forecast error. The θQ estimates are calibrated jointly to the VIX call, put, andfutures prices recorded at day t-H. Based on Chapter 5 outcomes, the ‘jointly implied’ θQ

estimates provided the most stable and competitive model performance for all the VIXderivative types, compared to the other information frameworks. Thus, such a frameworkis chosen to be considered in the current Chapter and gets our recommendation in thefuture applications (see Mencía and Sentana [2012]).

The model performance in the MRE and MSRE terms is considered for two differentderivative frameworks:

1) Aggregated performance: similar as in Chapter 5, the daily error outcomes areaggregated over all the VIX derivatives: call, put and futures in the sample, andtheir particular contributions considered,

2) Cross-sectional performance: the VIX call and futures pricing errors are consideredwithin the moneyness/expiration and expiration cross-sections, respectively.

These two frameworks will imply different weights of the contracts, as different dailyderivative sets will be considered within these setups.

7.3 Estimation

Similar as in Chapter 5, there are two sets of empirical data used in the estimation of theP -process parameters. The first set consists of the VIX daily return series, spanning the

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period from January 2, 1990 up to April 30, 2010, and comprises of 5,125 observations. Thesecond set consists of 252-day backward daily return series starting on February 24, 2006up to April 30 2010, providing 1,052 overlapping subsets. Figure 4.5 presents the wholeVIX series and an example 252-day subset. The first choice of data used in estimationaims to include a wider picture of the underlying VIX and keeps the θP estimates constantover the analyzed sample. The second set reflects local properties of the volatility indexand allows for some flexibility in the estimation results, which vary on a daily basis.

7.3.1 Numerical issues

The numerical procedures are used in the model P -process estimation, calibration of theparameters under the risk-neutral Q-measure, and approxiamtion of the conditional den-sity functions for the jump-diffusion underlying dynamics. In this section, we discussbriefly the assumptions and choices of the optimizer parameters, as well as the issuesrelated to the results convergence.

The numerical integration procedures used in the parameter estimation need well spec-ified starting values. In Ball and Torous [1985] or Das [2002], such values come from themaximum likelihood estimates of the Bernoulli mixture model.1 Another solution is touse the estimates from non-jump specifications fixing the jump parameters at a zero level(Beinert and Trautmann [1991]). The latter approach is exercised in our study. In thecase of a rolling-window estimation procedure, a successful convergence was also acquiredfor the previous day estimates used as starting values. All the numerical procedures reliedon the gradient-based Newton-Raphson algorithm, which is recommended (or applied) forthe jump-diffusion estimation (e.g., Ramezani and Zeng [1998], Moraux et al. [1999]), orthe simplex based Nelder-Mead approach.2

Another numerical issue concerns the quality of the numerical integration used toapproximate the CIR-UJ and logOU-UJ density functions from the relevant versions of(3.46). The number of nodes is chosen arbitrarily to be 100 for both models, but othervalues were also checked. It turned out that smaller numbers caused some convergenceproblems, especially in the estimation over the rolling window VIX series, as the numberof observations might have not been sufficient to fully recover the process properties.The upper limit of the integration is set to 1,000, however, as pointed out in Singleton[2001], this limit depends on the integrated function convergence and sometimes even lower

1Two-term truncation of the Poisson-weighted sum of normal distributions conditioned on the numberof jumps. The Poisson density is approximated using the Bernoulli distribution.

2Both algorithms are implemented in the R software (2.10.1), under the general-purpose optimizationprocedure optim, provided in default stats package.

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numbers provide successful results. An example of the real part of the integrated functionfor the logOU-UJ process is provided in Figure 7.1, where the convergence is obtained atφ=70.3.

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

φ

Inte

grat

ed fu

nctio

n

Figure 7.1: The real part of the logOU-UJ process integrated function used for computing conditionaldensity of the VIX, evaluated on a daily time basis (τ=1/252) at (Vt,Vt+τ )=(0.13,0.15). The set of usedparameter values corresponds to the maximum likelihood estimates over the daily VIX closing seriesspanning period from January 2, 1990 up to April 30, 2010, see Table 7.1.

In the case of the Merton model, the density function (3.44) can be easily used toconstruct the log-likelihood, truncating the number of jumps to j=10 (Ball and Torous[1985]). The problem is that for the Merton-like Bernoulli mixture of the distributions, theglobal maximum for the log-likelihood does not exist, or becomes infinite (Kiefer [1978]).Consequently, the jump intensity parameter γ may take any value without affecting thevalues of the log-likelihood function (He et al. [2006]), thus, it is left to be implied fromthe derivative prices as a free parameter for the Merton model. In addition, Dotsis et al.[2007] indicated after Honoré [1998] that the ‘strange’ estimates like negative diffusion σor jump variance parameter δ2 estimates may occur as a warning, however, we did notencountered such cases for our data.

The evaluation of the Merton call formula (3.26) requires an infinite summation. Thepoint where the summation stops should be high enough to allow for a price convergence,and to keep its numerical tractability (Naik and Lee [1990]). Figure 7.2 presents anexample of theoretical Merton call option prices within different moneyness and expirationgroups, given different upper summation limits (j). The P -process estimates are taken

3The Gauss-Legendre type of the quadrature method, used to solve the integral numerically, is im-plemented under the R software (2.10.1), statistical modeling statmod package, under the gauss.quadprocedure.

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from Table 7.1. Different limits apply for different sections. For the short-term optionsexpiring within 30 days, j=20 seems to be enough. However, when j=20 is used forcontracts with longer maturities, the price does not reach the convergence limit. Forcontracts expiring within 252 days, j=100 seems to be sufficient. The longer maturity callprices converge for j ≥ 100. However, in the analyzed sample a negligible fraction (0.17%)of calls mature within more than 252 days, with median trading volume of 21 contracts.Hence, using the upper limit of j=100 should not significantly affect the Merton modelperformance, but will decrease the computational burden.4

0 20 40 60 80 100 120 140

0.00

00.

004

0.00

80.

012

τ = 30 252

Upper summation limit

OT

M

0 20 40 60 80 100 120 140

0.00

0.02

0.04

0.06

τ = 120 252

Upper summation limit

0 20 40 60 80 100 120 1400.

000.

040.

080.

12

τ = 252 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

0.05

0.10

0.15

τ = 300 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

00.

010

0.02

00.

030

τ = 30 252

Upper summation limit

ATM

0 20 40 60 80 100 120 140

0.00

0.02

0.04

0.06

0.08

τ = 120 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

0.05

0.10

0.15

τ = 252 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

0.05

0.10

0.15

0.20

τ = 300 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

0.02

0.04

0.06

τ = 30 252

Upper summation limit

ITM

0 20 40 60 80 100 120 140

0.00

0.04

0.08

τ = 120 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

0.05

0.10

0.15

τ = 252 252

Upper summation limit

0 20 40 60 80 100 120 140

0.00

0.10

0.20

τ = 300 252

Upper summation limit

Figure 7.2: Theoretical Merton call prices for different upper summation limits. The strike prices cor-respond to 0.25(0.20)0.15, given the underlying spot value of 0.20 for the OTM(ATM)ITM options, re-spectively. The model parameter values correspond to the P -estimates over the whole VIX sample (Table7.1), τ is the time to maturity.

7.3.2 Estimates over the whole VIX sample

The ML estimates over the whole VIX historical series, together with the standard errorsretrieved from the inverse Hessian matrix (Deskalakis et al. [2009]), for diffusion and jump-diffusion processes are provided in Table 7.1. For each model, the value of the log-likelihoodfunction and diagnostic tests are also presented. Some interesting implications can bederived from these outcomes. The jump-diffusion processes model the VIX dynamics

4Naik and Lee [1990] use arbitrary level at j=10. Bates [1991] determines the unimodal upper ‘cap’,which does not overcome j=1,000.

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better, than their non-jump counterparts, which is reflected in the likelihood ratio (LR)test values. Such an outcome confirms an importance of the jump component in modelingthe VIX index (Poon and Granger [2003], Wagner and Szimayer [2004], Dotsis et al.[2007]). Out of all these processes, the best fit is provided by the logOU-UJ dynamics andthe worst by the GBM one. The better performance of logOU-UJ (3.32) over the CIR-UJ(3.30) one might be explained by proportional to the underlying jump structure, availablein the former specification. The empirically observed fact, underlined in Psychoyios et al.[2009], is now confirmed for even larger sample. The presence of jump-diffusion parametersdecreases the level of the volatility-of-volatility parameter in all considered models andspeeds up the mean-reversion, which is in line with the results provided in Dotsis et al.[2007], who compare the empirical characteristics of the other stock volatility indexes.

Some interesting implications are observed when the mean-reverting specifications areconsidered in the pure- and jump-diffusion cases. In the drift part, the long-run mean m/κestimates of the logOU and CIR are 0.19 and 0.203, respectively, and the speed of mean-reversion estimate is around 4. In the volatility part, different estimates of σ come mainlyfrom different specifications of the models’ diffusion functions. These two models withoutthe jump part could be considered as alternative to each other. Augmenting them with thejump component changes their characteristics. A higher expected jump size (1/η+) in caseof the CIR-UJ dynamics might be explained by the unproportional to the spot VIX jumpstructure. The square-root jump dynamics implies an average VIX upward change of 2.17units once every 15 days, and there is a stronger reversion (κ=7.106) towards the long-runmean. The logOU-UJ process with a proportional to the spot VIX jump structure resultsin a smaller, than for CIR-UJ, κ estimate, and smaller but more frequent 1.12 unit jumps,happening on average every seven days.5

In the case of GBM-J, the average number of jumps (≈66) per year is highest amongall the models. A closer look at the jump structure reveals that they are characterized bya smaller, compared to the mean-reverting models, average size (µJ=0.017) and a largestandard deviation (σJ=0.077) on the other side. This model does not well distinguishthe VIX jumps from its spikes, and the jump part clearly takes a role of the σ parameterin explaining the process volatility; the value of the σ estimate falls from 0.936 (GBM)down to 0.673, which is the highest decrease observed among the considered models.

5The jump size is implied by the exp{1/η+}−1 estimate in terms of the VIX(%) long-run mean around20.

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7.3.3 Estimates over the 252-day VIX subsamples

The alternative estimation scheme relies on 252-day overlapping VIX series, which reflectthe local rather than long-term VIX dynamics characteristics. These estimates for thediffusions underlying the Whaley, GL, and DO models are provided in Figure 5.2 andfollowed by their standard errors in Figure 5.3. The first section of Figure 7.4 providesthe 252-day P -estimates of GBM-J, underlying the Merton model. The parameter experi-encing the greatest changes is the expected annual jump intensity γ. In the first 100 dayssince February 24, 2006, the γ estimates are characterized by a strong upward change,accompanied by a deep downward jump of the σ estimates. This is a typical example,when the jump component takes part of the diffusion role in capturing the VIX volatility.Figure 4.5 reveals that the index was rather low over this period, until the local spikeoccurred in June 2006. The mean jump size remained mainly between 1-3%, which isconsistent with the overall VIX sample 1.7% estimate. Since February 2007, both the dif-fusive volatility and jump intensity had increased, reflecting more intense VIX spikes andvulnerability during the sub-prime turmoil. The jump parameters, present in the GBM-Jdynamics, decrease the diffusive volatility σ estimates rather than the µ appearing in thedrift. Similar to the whole VIX sample, the local scale GBM-J recognized (on average) theVIX as the process with small (≈2%) but frequent spikes. The estimates for the mean-reverting jump-diffusions CIR-UJ and logOU-UJ are presented in the following two panelsof Figure 7.4. The average jump intensity shows the main difference between these twomodels. In the first relatively calm period, with a slightly more volatile June 2006, the γestimate for the CIR-UJ is smaller than the one for the logOU-UJ (60 vs. 100). Strongermean-reversion for CIR-UJ pulled the VIX towards its mean faster than the logOU-UJprocess with the proportional jump structure. The local fast reverting impulse was bettercaptured by the square-root process in the calm period. However, the 2-year period ofrelatively low volatile VIX was followed by the more vulnerable time after October 2007,which had a greater effect on the CIR-UJ estimates than the logOU-UJ ones. The propor-tional jump structure of the latter model makes it less sensitive to the past observations,if the current VIX values go strongly upwards. This is particularly important for the longterm forecasts in periods of the economy turmoil. The more volatile periods are bettercaptured by the logOU-UJ process also in terms of the diffusive volatility.

7.4 Pricing performance

Three volatility derivative pricing models based on well-established underlying diffusiondynamics are extended by empirically relevant jump components. The analysis aims at

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Parameter Whaley GL DO Merton GL-UJ DO-UJµ 0.451 - - 0.503 - -

(0.208) (0.217)m - 0.842 -5.802 - 1.054 -10.701

(0.123) (1.004) (0.117) (0.972)κ - 4.140 3.496 - 7.106 5.157

(0.644) (0.591) (0.666) (0.672)σ 0.936 0.448 0.940 0.673 0.363 0.775

(0.009) (0.004) (0.009) (0.028) (0.005) (0.011)γ - - - 65.544 16.838 34.503

(16.972) (2.714) (6.924)µJ - - - 0.017 - -

(0.005)σJ - - - 0.077 - -

(0.007)η+ - - - - 46.180 18.335

(4.622) (2.458)logLik 14,489 14,632 14,540 14,634 14,744 14,820LR - - - 290 224 560χ2df,α - - - 7.815 5.991 5.991AIC -28,974 -29,258 -29,074 -29,258 -29,478 -29,630BIC -28,961 -29,238 -29,054 -29,225 -29,445 -29,597

Table 7.1: The underlying diffusion and jump-diffusion parameter ML estimates with standard errors inbrackets (compare with Table 5.1). The estimation has been performed on the daily VIX closing seriesrecorded over the period from January 2, 1990 up to April 30, 2010 (5,125 trading days). Hypothe-sis H0 : θPNO-JUMP = θPJUMP is tested using the Likelihood-ratio (LR) test, where the testing statistic−2[logLik(θPNO-JUMP) − logLik(θPJUMP)] follows χ2

df distribution with the number of degrees of freedomdf = 3 for the Merton model underlying dynamics, df = 2 for the GL-UJ and DO-UJ specifications. Theprovided statistic values are for the 5% significance level α. The Akaike Information Criterion (AIC)=2k − 2 logLik(θP ), the Bayes Information Criterion (BIC)=−2 logLik + k ln(n), where k is a number ofthe model structural parameters and n denotes the sample size.

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assessing the pricing consequences of this amendment. Both frameworks are nested withindifferent market information loads. The pricing results are separated into those for theP -process (θP ) estimates, reflecting the VIX dynamics features in the whole availablehistorical series, and more locally over 252-day overlapping subsamples. Given these esti-mates, the parameters under the Q-measure are implied jointly from all VIX derivatives,under different market risk premium assumptions. The pricing assessment is carried outboth in- and out-of-sample, relying primarily on the mean-squared relative errors (MSRE)and the mean-relative errors (MRE) as a complementary criterion.

7.4.1 In-sample

We consider the model in-sample fit for all the VIX derivative types under different as-sumptions about the volatility and jump risks, given that the Q-parameters are calibratedto the joint VIX derivative information. The aggregated performance is decomposed intothe particular VIX derivative contributions. Then, we considered the in-sample modeloutcomes for the call and futures contracts, within the moneyness and/or expiration cross-sections.

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Aggregated performance

The aggregated and particular VIX derivative MSRE and MRE values are reported inTable 7.2. The results in the left and right-hand side panels correspond to the θP estimatesover the VIX whole and 252-day series, respectively. For the diffusion based specifications,the θQ consist of the market price of the volatility risk only. In the jump-diffusion pricingframework, this set is extended with the volatility and/or jump risk premia.

Among the diffusion based specifications, the mean-reverting models (GL, DO) out-perform the model of Whaley based on the GBM dynamics in all frameworks. The bestin-sample data fit (MSRE=0.064) is provided by the DO model for the whole VIX samplebased P -estimates. The best fit is provided for the futures contracts, then calls and puts,where the last ones usually stand for more than 55% of the aggregated MSRE in the ‘wholesample’ P -estimates. Such a situation might be due to differences in the trading volumes,where futures are the most traded VIX derivatives, and puts the least. The poorest fit forputs is provided by the Merton model when the 252-day P -process estimates are used.

The jump-diffusion extension improves the in-sample fit over the diffusion based modelsin the MSRE terms. As in the diffusion based models, the mean-reverting jump-diffusionspecifications (GL-UJ, DO-UJ) fit better in-sample than the Merton model. Recall thatin the latter pricing framework three parameters are left to be determined by the market,which is one more compared to the competing GL- and DO-UJ models. Again, the bestin-sample fit (MSRE=0.026) is provided by the DO-UJ specification for the P -estimatedover the whole VIX historical series. This model also dominates in the more flexible252-day based estimates, but with a slightly higher error (MSRE=0.041) compared to its‘whole sample’ counterpart. The put MSRE contribution still dominates in the aggregatederrors (at least 44%). However, the Merton put pricing improved for the 252-day basedestimates (MSRE=0.034) when compared to the diffusion based counterpart (=0.748).

The models show worse in-sample performance when the P -process estimates rely onthe shorter 252-day VIX series, rather than the whole historical series (5,125 trading days).The extension of the parametric structures by the jump-related parameters improves morethe in-sample fits based on the 252-day P -estimates, rather than on the whole VIX series.For example, for the GL model MSRE drops from 0.196 down to 0.056, compared to 0.101down to 0.036 for the ‘whole sample’ counterpart.

The jump-diffusion based pricing models provide a better in-sample fit when comparedto the diffusion based specifications, regardless of the market information used to estimatethe structural parameters. However, a better in-sample fit of the jump-diffusion basedmodels over the diffusion based specifications might be due to a higher number of structuralparameters. To resolve this argument the assessment will be extended to the out-of-sample

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framework in section 7.4.2, such that any overfitting will be penalized.From an economic point of view, the futures in-sample pricing performance with the

MSRE falling in the [0.001, 0.004] and [0.001, 0.005] intervals for the ‘whole sample’ and252-day based estimates, respectively, fulfils the economic threshold criteria (see section5.3.2). The best call option fit (MSRE=0.017) provided by the DO-UJ model does notseverely corrupt the MSRE≈0.01 threshold either.

Different risk premiums outcomes

The issue about different risk premiums in models with jumps finds references in theliterature, mainly for the stochastic volatility index option pricing (e.g., Pan [2002]). Tothe author’s knowledge, there is no study which examines this issue in the case of theone-dimensional jump-diffusion based VIX derivative pricing models. Hence, some basicand preliminary analysis is carried out, setting the stage for future investigations.

For the diffusion based models, the volatility risk premium is left to be determinedby the market. Within the jump-extended framework, there are three setups considered.First, only volatility (diffusive) risk is priced, leaving the jump risk as non-systematic (e.g.,Merton [1976], Lin and Chang [2008]). Then, only the jump risk is priced by the market(e.g., Pan [2002]). Finally, both risks are priced concurrently within the model.

The last two sections in Table 7.2 provide the MSRE based assessment of the in-samplefit, assuming that the market prices different risks. Apparently, due to a higher numberof market determined free parameters, the approach with both risk premiums providesthe best fit. With respect to the aggregated MSRE results, the hypothesis stated in Pan[2002] that the jump (size) risk premium compensates the volatility risk is rejected forthe θP estimates over the whole VIX sample and for all models. For example, for theDO-UJ specification, the MSRE for the jump risk premium stands for 0.275, compared to0.056 for the volatility one. With respect to the particular derivatives’ contributions, thisoutcome varies among the models and estimation frameworks. The hypothesis about thepremium compensation is mainly confirmed in the 252-day based estimation frameworkby the mean-reverting models for the call and futures contracts. The Merton model fitsbetter the derivative prices when the volatility risk is determined by the market, with anexception for puts for the 252-day P -estimates. The puts improve greatly when both risksare priced by the market, especially for the ‘252-day’ estimates. In this framework, thebest performing DO-UJ improves from MSRE=0.056 (volatility risk premium) or 0.079(jump risk premium) down to 0.009 (both risk premiums).

Some intuition for interpreting these results can be found in the VIX dynamics patternsimplied by the models. Merton’s estimates imply that the VIX is characterized by a

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relatively high number of small size jumps (spikes). This suggests that the jump (size andtiming) risk might be recognized as a part of the diffusion risk, instead of being a separateentity. Contrary to the Merton model, the mean-reverting jump-diffusion specificationsrecognize VIX jumps as less frequent, but with relatively higher amplitudes. Such anoutcome implies that the market will grant a greater premium to the jump risk than thediffusion-determined uncertainty.

This issue deserves further analysis, using, for example, different market informationin the risk price determination, which is left for future research. In the current scope, boththe volatility (diffusive) and the jump (size) risks are assumed to be systematic and theirpremiums determined by the market.

Cross-sectional performance

Table 7.3 extends the aggregated in-sample call and futures outcomes into the money-ness/expiration cross-sections. Both error measures are considered within the P -estimatesover the whole VIX sample and over the 252-day series. The assumption is that bothdiffusion and jump risks are priced by the market, so the presented outcomes complementthe first two sections in Table 7.2.

Calls

In terms of the MSRE outcomes, the mean-reverting models (GL, DO) outperform theGBM-based Whaley model within all the cross-sections. For the cheapest but most tradedshort-term DOTM calls, the best in-sample performance is provided by the DO model,which improves the alternative GL fit (MSRE=0.246) by around 50%, for the ‘wholesample’ P -estimates. Switching to the more flexible P -estimates yields even higher errormagnitudes. The only exception concerns the mid- and long-term ATM and (D)ITMcalls, where the more flexible estimates provide a better fit for the GL and DO pricingframeworks.

The MRE analysis provides some insight into the misfit direction. The Whaley modeltends to overprice the option contracts within all the moneyness/expiration cross-sections,for both P -estimation frameworks. This is mainly due to the model drift-dependent un-derlying dynamics, and mean-reverting property of the VIX. The GL and DO models,which take into account the VIX long-term mean reversion, tend to overprice the calloptions in majority (except DOTM) of the moneyness/expiration groups, however, theerror magnitude is usually smaller than for Whaley. For the ‘252-day’ P -estimates, theoverpricing grows for the short-term calls in all the diffusion models. Due to a strongermean-reversion, the GL and DO tend to underprice the long-term contracts.

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In the jump extended framework, the DO-UJ model outperforms the other jump mod-els in the MSRE terms, in majority of the cross-sections. The GL-UJ model works slightlybetter than the DO-UJ for the long-term ATM and (D)ITM contracts, as the P -dynamicsis determined over the whole VIX sample. The DO-UJ improvement rate over the best per-forming DO in the short-term DOTM section is around 64% from MSRE=0.130. Changingto the more flexible P -estimates results in a worse in-sample fit (with some exceptions).The direction of the misfit usually stays consistent the diffusion based patterns and theerror magnitudes are smaller. The jump extension for the GBM dynamics leads to under-pricing of the short-term (D)OTM and ATM calls.

Futures

For the diffusion based models, the VIX futures MSREs clearly go up with a longer timeto expiration, caused mainly by decreasing sensitivity of these models to the underlyingvalue (Figures 3.1 and 3.2). Among the competing diffusion based specifications, the DOmodel provides usually the best performance regardless the maturity, but its performancediffers with respect to the P -estimation framework. The short-term futures prices arebetter fitted under the fixed P -estimates, where their flexible values are preferred in themid- and long-term expirations. The Whaley model is doing relatively well in the short-term, but fails for the longer maturities. The MRE outcomes reveal that the futures areusually overpriced by the mean-reverting models, where the GBM-based Whaley modeltends to underprice in the short-term run. The overpricing rate grows with the maturity.

The extension to the jump-diffusions improves the futures fit by at least 50% in allthe expiration groups. The Merton model clearly improves over Whaley for the long-term contracts. Both mean-reverting GL-UJ and DO-UJ models work almost equallywell over all expirations, with slightly better outcomes for the latter specification. TheMRE patterns observed for the diffusion based models do not change much within theextended framework. The overpricing magnitude for the mean-reverting models is smallerthan for the diffusion based models, which confirms the better pricing precision of thejump-extended models.

7.4.2 Out-of-sample

The out-of-sample analysis complements the models’ performance assessment. It coversthree different forecast horizons H: the short- (1-day), mid- (5-day), and long- (20-day)term. Within each period, the model structural parameters are backed-out H days, andthe actual value of the underlying is assumed to be given. The predictive abilities of thesix models are examined with respect to the MSRE outcomes aggregated over all VIX

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derivative types. Then, a more detailed investigation, based on the MSRE and MREmeasures over the moneyness and/or expiration cross-sections, is carried out for the VIXcall and futures contracts.

Aggregated performance

Table 7.4 reports the out-of-sample pricing outcomes for the analyzed models. Amongthe diffusion based specifications, the DO performs best, given that the P -dynamics isbased on the whole VIX sample information. It works clearly better than the GL andWhaley models, where the latter provides the worst performance. Relaxing the constantparameter assumption does little to change this picture, with an exception of the GLand DO performances becoming more comparable. In terms of the aggregated MSRE,the whole VIX sample information yields better pricing outcomes than the dynamicsbased on the local 252-day series. The analysis of the MRE magnitudes confirms theDO improvement over the competing specifications. The call options are usually the onlycontracts, which stay underpriced by the mean-reverting models.

The jump extension clearly improves the out-of-sample performance over the corre-sponding diffusions based models, given the ‘whole sample’ P -dynamics estimates. Outof the two GBM-based frameworks, the Merton outperforms the Whaley model regardlessof the derivative contribution or the forecast horizon length. In the joint MSRE terms,the DO-UJ model works better than the other specifications. The rate of improvementover the best diffusion based performance for the short- and mid-term forecasts staysaround 30-40% for the ‘whole sample’ θP estimates, and at least 50% for the more flex-ible framework. Surprisingly, for the put options in the long run, both diffusion basedmean-reverting models work better than the corresponding jump-diffusions. In the othersections within this horizon, GL outperforms the jump-extended counterpart, but DO-UJremains usually the preferred model. The MRE outcomes, within the jump-diffusion basedframework, stay mainly positive or only slightly below zero.

Summing up, for all six models, the out-of-sample outcomes based on the whole VIXsample θP estimates outperform the 252-day based ones in terms of the aggregated MSRE.The models based on the mean-reverting diffusions with jump extension clearly outper-form their diffusion counterparts in terms of the aggregated and derivative-decomposedMSRE outcomes. The DO-UJ model forecasts best, providing at least 25% of the aggre-gated MSRE improvement over the competing specifications for all forecast and estimationframeworks. All the considered models tend to overprice the VIX derivative contracts in-sample.

Since the VIX derivative prices and trading volumes differ among particular moneyness

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and maturity groups (see Table 4.3), this analysis needs to be extended into these cross-sections to complete the picture. Due to a high market value and trading volume, onlythe VIX call and futures contracts will be included in the analysis. The cross-sectionalperformance is evaluated with respect to both the MSRE and MRE metrics.

Call cross-sectional performance (θP over the whole VIX sample)

MSRE

Table 7.5 reports the relevant out-of-sample MSRE values. The mean-reverting specifica-tions outperform those based on GBM, within the diffusion and jump-diffusion frameworks.The DO model forecasts best when compared to the other diffusion based frameworks, withsome short-term ATM exceptions favoring the GL model. The jump extension providesmuch improvement to the short-term options, regardless of the moneyness group and fore-cast horizon. For the cheapest, but one of the most traded short-term DOTM options,DO-UJ provides the most accurate forecasts among all six models. Within this group,the MSRE equals 0.089 for 1-day horizon, which is around 42% improvement over thebest diffusion based DO’s performance. For the mid- and long-term forecasts, the DO-UJimprovement rate stands for 30% and drops down to 11.7%, respectively. For the mostexpensive short-term DITM calls, the GL- and DO-UJ models forecast best, with the 1-day MSRE around 0.030 compared to 0.052 for the best diffusion based DO’s result. Sincethe short-term VIX call options market contributes significantly to the whole VIX optionmarket size and value (Table 4.3), the observed pricing improvement for these contractsis an important outcome, and gives a credit to the jump-diffusion models. Such outcomesmight be explained in terms of the volatility implied from the VIX call prices. The em-pirical evidence provided in Sepp [2008a] shows that the short-term VIX option impliedvolatilities are positively skewed with high kurtosis. Assigning a higher probability to anyabrupt VIX change in the short term allows for capturing this option price characteristicsand, in consequence, improve their pricing performance.

The pricing patterns for the mid- and long-term calls roughly come along the short-term outcomes. The jump-diffusion based models improve further their diffusion basedcounterparts over all the forecast horizons. The DO-UJ specification works usually best inall the cross-sections. The improvement rate over the best diffusion outcomes remains atleast 37% for the DOTM calls. For the DITM calls, both the GL-UJ and DO-UJ modelswork close to each other, but the latter remains usually the preferred one.

The ability to handle jumps, which might occur in the long-term, works in favor formodels based on the mean-reverting jump-diffusions. After Psychoyios et al. [2009], thejump-diffusion based models’ sensitivity to the underlying changes is decreasing for the

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long-term maturing contracts. Any jumps occurring in the long-run do not have muchinfluence on the long-term option performance. Recall that the jump extension results ina stronger mean-reversion parameter estimates (Table 7.1), which may lead to a strongerunderpricing by the jump-diffusion in the long-term contracts, compared to their diffusionbased counterparts. This hypothesis is confirmed by the outcomes provided in the followingsection.

MRE

The MRE values presented in Table 7.6 complement the MSRE outcomes for the wholeVIX sample based P -estimates (Table 7.5). Within this setup, the Whaley model con-stantly overprices the option contracts, mainly due to increasing drift based expectationsabout the VIX value. In this situation, the mean-reverting property of the VIX (see Chap-ter 4) leads to an even higher Whaley’s overpricing for the mid- and long-term contracts,and keeps on growing as the forecast horizon extends. The GL and DO models deeplyunderprice the short-term DOTM calls and provide the prices above the market ones, inthe other cross-sections. The better performance in the MSRE terms over the Whaleyoutcomes corresponds with the smaller MRE values.

The jump-diffusion based mean-reverting models tend to underprice the DOTM calls,with the magnitude going deeply below zero for the long-term contracts. Such a patterncontradicts with the one already observed for their diffusion counterparts. This phe-nomenon can be explained in terms of capturing the VIX characteristics in the short-and long-term horizon. The ‘whole sample’ estimates of the γ (jump intensity) and 1/η+

(upward jump size) parameters (Table 7.1), suggest that the VIX is characterized by fre-quent small size jumps (spikes). The diffusion based models do not capture this property,hence, they tend to underprice the short-term calls. The long-term contract outcomescould be explained in terms of the process persistency, represented by the average shockhalf-life time. Chacko and Viceira [2003] define the half-life of the process as the expectedtime it needs to return half-way to its long-term mean. The CIR process κ estimatesimply that it takes 43 days for the VIX to recover half-way, where in the GL-UJ model,which corresponds to the CIR-UJ underlying dynamics, this period shrinks to 24.5 days.6

Thus, the jump-diffusions in the long run ‘expect’ the underlying to recover faster fromthe shock than the corresponding diffusion processes, and provide the long-term optionprices relatively lower than their market equivalents. These patterns are observed over

6Following Chacko and Viceira [2003] for the CIR-UJ process, the conditional mean of the processis given by the expression (3.39). If the shock magnitude is double the process long-run mean θ =2(m+ γ/η+)/κ, then after time τ = log(2)/κ the process reverts to a value of 1.5θ.

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all forecast horizons considered in the study. Between the GBM based specifications, thejump extension pays-off with more precise option price forecasts and improvements for thelong-maturing contracts. The Merton model, similar as the other jump-diffusions, tendsto underprice the short-maturing DOTM calls.

Summing up, the DO-UJ model forecasts the cheap but intensively traded short- andmid-term DOTM VIX call prices best, regardless of the forecast horizon. This role isshared with the GL-UJ model for the most expensive short maturity (D)ITM calls. Thejump-diffusion based models tend to underprice the long-term call contracts more thanthe corresponding diffusions.

Call cross-sectional performance (θP over the 252-day VIX subsamples)

MSRE

The aggregated call MSRE values in Table 7.4 imply that using the ‘252-day’ P -estimatesleads usually to a worse aggregated pricing performance, compared to the one based on thefixed P -parameter values. Table 7.7 extends the out-of-sample results into the assumedmoneyness/expiration cross-sections.

Among the diffusion based specifications, the DO model prices best over all the cross-sections and forecast periods. Switching to the P -estimates, reflecting the more local VIXcharacteristics, leads to higher MSRE values, especially within the DOTM group. Theshort maturing DOTM options go worse by more than twice (from 0.154 up to 0.466), forthe 1-day forecast. Similar rates apply to the other two horizons. A significant change isalso observed for the long-term contracts within this group, where the DO MSRE increasesmore than twice (MSRE= 0.227 up to 0.607). In the other moneyness/expiration groups,the errors grow with diverse rates. The exceptions are the mid- and long-term ATM andDITM call options for which the DO improves, given the 252-day based P -estimates. Forthe other models, changing to more flexible estimates leads, with some exceptions, to apoorer performance over the constant P -estimates based outcomes.

Within the jump-diffusion based framework, the short- and mid-term options’ perfor-mance is again dominated by the DO-UJ model. However, when compared to the resultsfor the estimates over the whole VIX sample, a more flexible approach does not improvethe performance either. For example, the MSRE for the short maturing DOTM optionsincreases more than twice over the ‘whole sample’ DO-UJ (MSRE=0.089) level, in the1-day forecast horizon. For the ‘252-day’ P -estimates, the Merton framework yields againmore precise forecasts than the Whaley model, within all the cross-sections. The diffu-sion and jump-diffusion based mean-reverting models dominate the GBM based models,especially in the longer out-of-sample periods.

116

MRE

The MRE results based on the ‘252-day’ P -estimates are reported in Table 7.8. Recallthat the apparent consequence of switch to the local VIX characteristics for the diffusionsis a higher speed of the mean reversion. For example, in the GL model, the κ estimatechanges from 4.146 up to the levels beyond 20, in majority of the sample (Figure 5.2).This means that, at a local level, the VIX often deviates from its local long-run meanand reverts half-way within around 9 days, compared to the 43 in the ‘whole sample’.Hence, for the short-term calls, the results based on the local estimates slightly overpricethe ones based on the whole VIX sample estimates. As the time to maturity extends,the model call prices are usually pushed below their market values for the mid- and long-term expiring contracts, much more than the ones based on the fixed P -estimates. Thesepatterns apply to almost all the moneyness groups. The Whaley model overprices theVIX call contracts usually more than the constant P -estimate based results, regardlessthe moneyness/expiration cross-sections.

A similar tendency is observed for the jump-extended specifications. Recall that theestimates of the GL-UJ and DO-UJ specifications are characterized by a relatively higherjump intensity γ, compared to the one based on the VIX ‘whole sample’ (Table 7.1), anda stronger than GL’s and DO’s mean-reversion. For the GL-UJ model, the κ estimatesstay beyond 30 for around the first 400 days (Figure 7.4). The higher jump probabilityfor short-maturing calls is immediately compensated by the fast reversion to its long-runmean, thus the MREs usually stay below their ‘whole sample’ counterparts. In the mid-and long-term expiration group, the decreasing sensitivity to the underlying and a strongmean-reversion lead to even deeper underpricing. For example, the short-term DOTM callMRE implied by the GL model equals to -0.209, when after the jump extension the errormagnitude goes up to 0.024. In the other moneyness groups, the short-term contracts stayusually at the low positive MRE, with a tendency to be deeply underpriced in the long-term expirations, especially by the jump extended mean-reverting models. The Mertonmodel underprices only the short-term (D)OTM contracts and stays usually below therelevant values provided by the Whaley model.

Futures

The out-of-sample VIX futures pricing assessment is carried out within, similar as for calls,the short-, mid-, and long-term maturity categories. Table 7.9 reports jointly the MSREand MRE pricing results, with respect to two different P -parameter estimation schemes.The resulting values were multiplied by 10 for the reading clarity.

117

Performance given the P -estimates over the whole VIX series

Among the diffusion based pricing frameworks, the DO model provides the best perfor-mance in the MSRE terms, within each considered maturity and forecast horizon. Thedifferences with the GL model become more apparent as the time to maturity extends.Both mean-reverting models clearly outperform the GBM-based Whaley outcomes, espe-cially for the contracts with the long-term expirations.

The jump extension improves the diffusion based performance. Both GL-UJ and DO-UJ provide roughly comparable outcomes for the short-term maturities, where the latterworks slightly better as the forecasts horizon extends. The Merton model improves overWhaley, but still performs much worse than GL and DO, especially when the forecast hori-zon is long. This outcome, also present in the call pricing results, indicates the importantrole that the mean-reversion component plays in the volatility modeling.

The analysis of the MRE outcomes implies that the observed tendency for the MSREvalues to increase as the time to expiration grows, originates from the increasing overpric-ing and is connected with the decreasing models’ sensitivity to the underlying changes. Allspecifications with the mean-reverting component overprice the VIX futures, regardless ofthe time to maturity and forecast horizon. The magnitude of the overpricing is smallerfor the jump extended models, mainly due to stronger mean-reversion tendencies. Thelong-term contracts are the most overpriced ones.

Performance given the P -estimates over the 252-day VIX subsamples

The more flexible P -estimates provide quite similar, in the MSRE terms, patterns asthe ‘whole sample’ based counterparts. The DO and DO-UJ models provide still thebest performance within their classes, where the latter outperforms all the consideredspecifications. However, these models work still better when the P -dynamics is determinedover the VIX whole historical series.

In the MRE terms, the jump-diffusion mean-reverting specifications tend to overpricethe futures contracts. The reason for such an outcome might be a relatively stronger jumpintensity for the local VIX dynamics. The GBM based specifications provide the contractprices, which usually stay below their market values. This concerns, in particular, theshort- and mid-term expiring futures contracts.

7.5 Summary

The baseline diffusion, underlying the Whaley, Grünbichler-Lonstaff (GL), and Detemple-Osakwe (DO) VIX derivative pricing models, were extended by empirically relevant jump

118

components. The primary objective of the study was to investigate the pricing conse-quences of this amendment. We can also observe how robust the models are with respectto the pricing outcomes, when different market information is used in the model parameterestimation. The analysis concentrates on all the VIX derivative types and extends on themoneyness and/or expiration cross-sections for the call and futures contracts.

There are interesting pricing implications of the jump extension, such as the improve-ment for the short-term (deep) out-of-the-money options ((D)OTM) group. This is aparticularly important outcome since it concerns the cheap but most intensively tradedoption contracts for this market. The jump element allows for covering the short-termpositive skewness and kurtosis observed in the underlying series. The best diffusion basedDO specification is outperformed by its jump-diffusion based counterpart. Within thealready mentioned short-term (D)OTM option group, the improvement is around 42% forthe 1-day forecast horizon. For the less intensively traded, but most expensive short-term(deep) in-the-money ((D)ITM) calls, the jump-diffusion based GL- and DO-UJ modelswork almost equally well, improving the diffusion based DO by 44% in the short-termforecast horizon. The diffusion and jump-diffusion pricing frameworks work better whenthe P -estimates reflect the long-term VIX properties, rather than the local ones.

To complement the whole picture, the focus of the next Chapter concerns the hedgingperformance provided by the jump-diffusion based models, which is then compared withthe relevant diffusion outcomes.

119

MS

RE

MR

EθP

over

thewho

leVIX

sample

θP

over

the252-da

yVIX

series

θP

over

who

letheVIX

sample

θP

over

the252-da

yVIX

series

Mod

elagg.

call

put

futures

agg.

call

put

futures

agg.

call

put

futures

agg.

call

put

futures

Diffusion:

volatilityrisk

priced

Wha

ley

0.38

70.14

20.240

0.004

1.140

0.387

0.748

0.005

0.223

0.087

0.143

-0.007

0.467

0.219

0.263

-0.014

GL

0.10

10.06

90.03

00.002

0.196

0.116

0.077

0.003

-0.023

-0.036

-0.001

0.014

-0.023

-0.042

0.000

0.019

DO

0.06

40.03

90.02

40.001

0.178

0.104

0.071

0.003

0.013

-0.007

0.007

0.012

-0.015

-0.033

0.001

0.017

Jump-diffu

sion

:volatilityan

djumprisk

priced

Merton

0.08

90.05

10.03

60.002

0.082

0.046

0.034

0.002

0.009

-0.021

0.034

-0.003

-0.006

-0.024

0.023

-0.005

GL-UJ

0.03

60.026

0.00

90.001

0.056

0.043

0.012

0.001

0.013

0.000

0.004

0.009

-0.023

-0.024

-0.002

0.003

DO-U

J0.02

60.01

70.00

80.001

0.041

0.030

0.009

0.001

0.015

0.003

0.005

0.007

-0.015

-0.015

-0.001

0.001

volatilityrisk

priced

Merton

0.11

20.07

10.03

80.003

0.120

0.077

0.041

0.003

0.002

-0.035

0.036

0.000

-0.009

-0.040

0.031

0.000

GL-UJ

0.11

10.080

0.02

80.002

0.178

0.118

0.057

0.004

-0.091

-0.071

-0.034

0.015

-0.122

-0.094

-0.049

0.021

DO-U

J0.05

60.03

50.01

90.001

0.153

0.094

0.056

0.003

-0.021

-0.021

-0.012

0.013

-0.078

-0.062

-0.035

0.018

jumprisk

priced

Merton

0.21

50.13

60.06

30.016

0.317

0.157

0.083

0.076

-0.055

-0.068

-0.011

0.024

-0.119

-0.128

-0.046

0.055

GL-UJ

0.21

30.051

0.16

00.002

0.207

0.083

0.121

0.002

0.104

0.016

0.079

0.009

-0.023

-0.033

0.001

0.009

DO-U

J0.27

50.05

50.21

80.001

0.137

0.057

0.079

0.002

0.147

0.039

0.104

0.004

-0.005

-0.014

0.002

0.007

Table7.2:

The

in-sam

plepe

rforman

ceof

thediffu

sionan

djump-diffu

sionba

sedVIX

deriv

ativepricingmod

elsinterm

softhe

meansqua

redrelativ

eerrors

(MSR

E)an

dmeanrelativ

eerrors

(MRE)

.The

aggregated

(‘agg.’)

errormeasuresan

dde

compo

sedinto

thecall,

putan

dfuturescontracts

contrib

utions.The

parametersun

dertheph

ysical

measureP

wereestim

ated

over

theen

d-of-day

VIX

serie

sspan

ning

theJa

nuary2,

1990

–April

30,2

010pe

riod(‘w

hole

sample’),

orre-estim

ated

daily

over

the252-da

ylong

backwardVIX

serie

sstartin

gon

Februa

ry24,2

006un

tilApril30,

2010.The

parametersun

dertheris

k-ne

utralm

easureQ

wereda

ilyre-calibratedto

thejointVIX

deriv

ativemarketinform

ation,

byminim

izing

themean-squa

redrelativ

eris

kcrite

rion.

The

contrib

utionof

each

deriv

ativecontract

isweigh

tedby

itsda

ilytrad

ingvolume,

relativ

eto

thetotal

trad

ingvolumeob

served

intheVIX

deriv

ativemarket.

120

MS

RE

MR

EθP

overthe

whole

VIX

sample

θP

overthe

252-dayVIX

seriesθP

overthe

whole

VIX

sample

θP

overthe

252-dayVIX

seriesMness

Model

τ<

60[60,120)

≥120

<60

[60,120)

≥120

<60

[60,120)

≥120

<60

[60,120)

≥120

Diffusion

models:

VIX

callsWhaley

0.2733.335

7.4811.234

8.26511.632

0.0841.300

2.0020.548

1.8722.254

DOTM

GL

0.2460.288

0.3670.392

0.5160.613

-0.2400.017

-0.043-0.236

-0.360-0.616

DO

0.1300.180

0.2100.372

0.5560.582

-0.1460.080

-0.047-0.166

-0.296-0.613

Whaley

0.1640.935

1.8230.490

1.6462.162

0.1620.723

1.0280.382

0.9051.040

OTM

GL

0.1550.323

0.3540.304

0.2650.238

0.0950.320

0.2620.189

0.050-0.215

DO

0.0790.188

0.1900.272

0.2180.214

0.1210.264

0.1700.185

0.027-0.257

Whaley

0.1250.645

1.2410.321

1.1371.410

0.1650.594

0.8380.317

0.7310.811

ATM

GL

0.1260.315

0.3410.229

0.2170.167

0.1280.320

0.2890.248

0.134-0.112

DO

0.0730.170

0.1930.194

0.1680.147

0.1520.249

0.1780.231

0.101-0.151

Whaley

0.1360.474

0.9460.317

0.8221.080

0.1620.493

0.6940.272

0.5870.658

ITM

GL

0.0920.292

0.3290.153

0.1900.157

0.1330.322

0.3060.231

0.1960.030

DO

0.0580.153

0.1840.127

0.1330.110

0.1470.234

0.1620.206

0.148-0.040

Whaley

0.0710.215

0.5760.110

0.2880.607

0.1040.306

0.4500.130

0.3140.385

DIT

MGL

0.0600.215

0.2920.082

0.1390.142

0.1320.284

0.3240.185

0.2270.172

DO

0.0380.110

0.1590.065

0.0960.090

0.1220.203

0.1670.150

0.1760.105

Diffusion

models:

VIX

futuresWhaley

0.0390.167

0.8930.056

0.2471.043

-0.2010.067

0.708-0.358

-0.3410.036

GL

0.0300.157

0.3730.079

0.1170.172

0.2960.814

1.1390.574

0.6070.556

DO

0.0200.097

0.2030.073

0.0930.132

0.2570.672

0.8670.511

0.4930.419

Jump-diff

usionmodels:

VIX

callsMerton

0.1290.995

3.4030.124

0.7592.821

-0.2160.672

1.289-0.202

0.5461.174

DOTM

GL-U

J0.075

0.1540.264

0.1400.284

0.439-0.045

-0.185-0.391

-0.021-0.431

-0.619DO-U

J0.047

0.0940.162

0.1070.203

0.373-0.036

-0.049-0.241

0.032-0.337

-0.561Merton

0.0720.390

1.0660.092

0.2620.833

-0.1070.421

0.717-0.156

0.3090.635

OTM

GL-U

J0.093

0.0870.092

0.1110.095

0.1820.189

0.132-0.040

0.165-0.138

-0.352DO-U

J0.053

0.0780.091

0.0850.078

0.1770.136

0.128-0.012

0.143-0.114

-0.341Merton

0.0490.291

0.7990.062

0.1800.673

-0.0440.352

0.609-0.104

0.2540.561

ATM

GL-U

J0.074

0.0860.083

0.0780.079

0.1200.183

0.1720.028

0.150-0.042

-0.251DO-U

J0.048

0.0730.094

0.0560.056

0.1170.142

0.1340.020

0.118-0.051

-0.254Merton

0.0400.218

0.5830.056

0.1700.525

0.0000.301

0.519-0.051

0.2160.440

ITM

GL-U

J0.046

0.0840.073

0.0470.067

0.0720.150

0.1890.056

0.1280.037

-0.129DO-U

J0.035

0.0720.090

0.0310.043

0.0710.123

0.1330.023

0.0930.015

-0.160Merton

0.0390.134

0.3670.048

0.1000.319

0.0260.205

0.337-0.007

0.1340.235

DIT

MGL-U

J0.021

0.0580.059

0.0200.058

0.0430.090

0.1670.108

0.0750.092

0.002DO-U

J0.019

0.0500.074

0.0130.031

0.0340.078

0.1010.038

0.0450.065

-0.029Jum

p-diffusion

models:

VIX

futuresMerton

0.0300.105

0.6750.030

0.0870.583

-0.1480.183

0.909-0.191

0.0580.681

GL-U

J0.012

0.0510.103

0.0180.067

0.1450.208

0.4610.562

0.1700.084

0.059DO-U

J0.010

0.0450.096

0.0120.047

0.1100.167

0.3750.439

0.1270.084

0.068

Table7.3:

The

in-sample

mean

squaredrelative

errors(M

SRE)

andmean

relativeerrors

(MRE),for

theVIX

callsand

futurescontracts.

The

optionsareconsidered

within

themoneyness(M

ness)/expirationtim

e(τ)cross-section

andthe

VIX

futuresaregrouped

with

respecttheexpiration

time.

The

particularcallor

futurescontract

errorobserved

atacertain

daytis

weighted

byits

dailytrading

volume,relative

tothe

totaltradingvolum

eofthe

VIX

callorfutures

market

att.

The

volatilityand

jumprisks

areallow

edto

bepriced

bythe

market.

121

MS

RE

MR

EθP

over

thewho

leVIX

sample

θP

over

the252-da

yVIX

series

θP

over

thewho

leVIX

sample

θP

over

the252-da

yVIX

series

Mod

elag

g.call

put

futures

agg.

call

put

futures

agg.

call

put

futures

agg.

call

put

futures

Diffusion:

out-of-sam

ple:

1da

yW

haley

0.67

20.30

40.36

30.006

1.724

0.648

1.070

0.007

0.293

0.131

0.169

-0.007

0.562

0.272

0.304

-0.014

GL

0.135

0.09

20.04

00.002

0.266

0.164

0.099

0.004

-0.018

-0.032

0.000

0.015

-0.026

-0.044

-0.002

0.020

DO

0.09

10.05

70.03

30.002

0.246

0.149

0.093

0.003

0.019

-0.003

0.009

0.013

-0.018

-0.035

-0.001

0.017

out-of-sam

ple:

5da

ysW

haley

0.83

60.40

40.42

40.008

1.958

0.754

1.194

0.009

0.315

0.145

0.176

-0.006

0.584

0.285

0.313

-0.014

GL

0.154

0.10

60.04

50.003

0.314

0.208

0.102

0.004

-0.012

-0.029

0.003

0.015

-0.016

-0.036

0.000

0.019

DO

0.11

40.07

40.03

80.002

0.294

0.186

0.105

0.004

0.026

0.003

0.011

0.012

-0.009

-0.030

0.004

0.016

out-of-sam

ple:

20da

ysW

haley

1.52

50.95

20.56

20.011

2.881

1.359

1.511

0.012

0.381

0.189

0.195

-0.003

0.662

0.333

0.339

-0.010

GL

0.202

0.14

40.05

40.004

0.452

0.328

0.119

0.006

-0.010

-0.028

0.005

0.014

0.020

-0.007

0.012

0.015

DO

0.15

90.11

40.04

20.003

0.433

0.273

0.154

0.005

0.034

0.010

0.013

0.012

0.019

-0.014

0.021

0.012

Jump-diffu

sion

:ou

t-of-sam

ple:

1da

ysMerton

0.27

00.14

90.11

60.004

0.234

0.131

0.099

0.004

0.057

-0.001

0.060

-0.002

0.034

-0.006

0.045

-0.004

GL-UJ

0.07

20.04

90.02

20.001

0.124

0.087

0.036

0.002

0.018

-0.003

0.012

0.009

-0.010

-0.018

0.006

0.002

DO-U

J0.054

0.03

40.02

00.001

0.085

0.064

0.021

0.001

0.018

-0.001

0.011

0.007

-0.009

-0.013

0.003

0.001

out-of-sam

ple:

5da

ysMerton

0.58

40.29

20.28

60.006

0.660

0.397

0.257

0.006

0.103

0.026

0.080

-0.003

0.083

0.026

0.061

-0.004

GL-UJ

0.11

10.07

60.03

40.001

0.239

0.171

0.065

0.003

0.034

0.010

0.016

0.009

0.019

0.005

0.012

0.001

DO-U

J0.076

0.04

70.02

80.001

0.138

0.102

0.035

0.002

0.027

0.006

0.015

0.006

0.005

-0.003

0.008

0.000

out-of-sam

ple:

20da

ysMerton

0.92

10.61

20.29

90.010

1.487

1.168

0.309

0.010

0.151

0.052

0.098

0.002

0.143

0.057

0.086

0.000

GL-UJ

0.24

80.15

90.08

70.003

0.536

0.349

0.181

0.006

0.074

0.037

0.028

0.008

0.084

0.052

0.032

0.000

DO-U

J0.163

0.09

20.06

90.002

0.361

0.193

0.164

0.005

0.048

0.017

0.025

0.006

0.053

0.021

0.034

-0.003

Table7.4:

The

aggregated

out-of

samplepe

rforman

ceof

thesix

mod

elsis

exam

ined

usingdiffe

rent

marketinform

ationsche

mes.The

P-process

parametersestim

ated

over

thewho

leVIX

sample(Jan

uary

2,1990

–April30,2

010)

or252-da

yoverlapp

ingserie

s(startingon

Februa

ry24,2

006).

The

Q-m

easure

parametersareim

pliedfrom

allt

rade

dVIX

deriv

ativeprices,u

nder

theassumptionthat

both

thevo

latility

(diffusive)

andjump

risks

arepriced

bythemarket.

The

MSR

Eis

calculated

over

allV

IXde

rivatives,a

ndde

compo

sedinto

particular

deriv

ativecontrib

utions.The

parameter

estim

ates

areba

cked

out1,

5,an

d20

days.The

samplepe

riodextend

sfrom

Februa

ry24,2

006up

toApril30,2

010,

which

stan

dsfor

1,052trad

ingda

ys.

122

MSRE Out-of-sample: 1 day Out-of-sample: 5 days Out-of-sample: 20 daysMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion models:Whaley 0.376 4.145 9.868 0.639 5.156 14.006 0.945 8.572 15.065

DOTM GL 0.268 0.328 0.416 0.278 0.345 0.469 0.331 0.497 0.545DO 0.154 0.197 0.227 0.181 0.233 0.255 0.266 0.391 0.303

Whaley 0.256 1.040 2.356 0.423 1.868 3.799 2.153 5.842 5.229OTM GL 0.170 0.339 0.379 0.204 0.370 0.457 0.270 0.534 0.500

DO 0.097 0.198 0.212 0.137 0.232 0.243 0.226 0.384 0.282Whaley 0.162 0.775 1.761 0.504 1.466 3.048 1.527 4.797 8.333

ATM GL 0.142 0.372 0.401 0.171 0.372 0.456 0.250 0.542 0.538DO 0.091 0.189 0.219 0.134 0.222 0.237 0.224 0.360 0.276

Whaley 0.255 0.730 1.348 0.444 1.109 2.220 2.964 12.372 4.399ITM GL 0.108 0.324 0.368 0.140 0.354 0.380 0.223 0.477 0.445

DO 0.075 0.167 0.216 0.115 0.202 0.211 0.196 0.313 0.245Whaley 0.126 0.304 0.829 0.245 0.561 1.477 0.776 2.433 2.484

DITM GL 0.078 0.237 0.323 0.104 0.265 0.338 0.179 0.362 0.371DO 0.052 0.118 0.181 0.081 0.143 0.195 0.140 0.225 0.205

Jump-diffusion models:Merton 0.281 1.526 4.784 0.858 2.073 7.185 0.652 4.592 9.111

DOTM GL-UJ 0.149 0.179 0.283 0.218 0.225 0.305 0.431 0.382 0.353DO-UJ 0.089 0.109 0.176 0.126 0.133 0.190 0.235 0.246 0.234Merton 0.177 0.505 1.413 0.716 1.048 2.649 1.174 3.939 3.897

OTM GL-UJ 0.128 0.103 0.107 0.190 0.141 0.139 0.321 0.296 0.218DO-UJ 0.073 0.086 0.101 0.107 0.108 0.117 0.191 0.218 0.168Merton 0.094 0.438 1.225 0.493 0.887 2.290 1.055 3.444 4.729

ATM GL-UJ 0.092 0.103 0.102 0.124 0.133 0.127 0.241 0.258 0.198DO-UJ 0.062 0.084 0.107 0.089 0.106 0.111 0.171 0.197 0.159Merton 0.111 0.346 0.947 0.571 0.642 1.701 2.135 8.816 2.759

ITM GL-UJ 0.058 0.090 0.085 0.088 0.118 0.108 0.168 0.234 0.190DO-UJ 0.047 0.079 0.098 0.074 0.102 0.105 0.136 0.185 0.153Merton 0.084 0.196 0.617 0.219 0.412 1.226 0.731 2.152 2.137

DITM GL-UJ 0.028 0.061 0.072 0.048 0.082 0.090 0.104 0.157 0.136DO-UJ 0.029 0.055 0.082 0.049 0.075 0.090 0.096 0.138 0.108

Table 7.5: The out-of-sample mean squared relative errors (MSRE) for the VIX call option pric-ing within the moneyness(Mness) and maturity (τ) cross-sections. The moneyness is defined as alog ratio of the strike up to the spot VIX level (ln(K/V IXt)) and the values ≥ 0.15, (0.15, 0.034],(0.034,−0.034],(−0.034,−0.15], < −0.15 correspond to the deep-out-of-money (DOTM), out-of-money(OTM), at-the-money (ATM), in-the-money (ITM), deep-in-the-money (DITM), respectively. The P -process parameter maximum likelihood (ML) estimates are acquired over the whole VIX series (January2, 1990 – April 30, 2010). The daily updated Q-measure parameters are implied jointly from all tradedVIX derivative prices observed at the particular date, assuming that both volatility (diffusive) and jumprisk are priced by the market. On each day, the model parameters are backed out by H=1, 5, and 20

days, and the particular strike and expiration call contract pricing error: wCtV olt

(C(θPt−H ,θ

Qt−H)−Cmarket

t

Cmarkett

)2is

calculated, where wCt is the particular call contract trading volume at day-t and V olt denotes the total calltrading volume within the corresponding moneyness/expiration cross-section observed at t. The resultingerrors are summed up and averaged over the number of the trading days considered in the assessment.The VIX call derivative sample period extends from February 24, 2006 up to April 30, 2010, which standsfor 1,052 trading days.

123

MRE Out-of-sample: 1 day Out-of-sample: 5 days Out-of-sample: 20 daysMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion models:Whaley 0.106 1.374 2.167 0.143 1.448 2.362 0.166 1.549 2.344

DOTM GL -0.231 0.026 -0.027 -0.225 0.032 -0.025 -0.214 0.050 -0.047DO -0.136 0.085 -0.040 -0.123 0.096 -0.038 -0.097 0.127 -0.048

Whaley 0.178 0.732 1.123 0.212 0.819 1.275 0.324 0.989 1.370OTM GL 0.100 0.318 0.263 0.110 0.330 0.286 0.123 0.348 0.257

DO 0.127 0.264 0.177 0.140 0.279 0.189 0.164 0.307 0.183Whaley 0.176 0.631 0.965 0.222 0.731 1.161 0.292 0.887 1.383

ATM GL 0.134 0.351 0.322 0.141 0.353 0.331 0.154 0.371 0.341DO 0.159 0.266 0.202 0.171 0.286 0.223 0.193 0.313 0.230

Whaley 0.183 0.548 0.783 0.223 0.621 0.949 0.362 0.969 1.186ITM GL 0.140 0.335 0.318 0.145 0.349 0.307 0.149 0.353 0.274

DO 0.154 0.244 0.176 0.164 0.263 0.175 0.176 0.285 0.151Whaley 0.125 0.337 0.553 0.169 0.416 0.738 0.255 0.615 0.967

DITM GL 0.139 0.293 0.339 0.148 0.310 0.341 0.154 0.313 0.318DO 0.128 0.207 0.187 0.142 0.221 0.204 0.149 0.234 0.206

Jump-diffusion models:Merton -0.189 0.736 1.430 -0.113 0.802 1.599 -0.121 0.909 1.694

DOTM GL-UJ -0.027 -0.175 -0.391 0.010 -0.152 -0.383 0.096 -0.087 -0.363DO-UJ -0.029 -0.047 -0.247 -0.013 -0.038 -0.247 0.027 -0.001 -0.243Merton -0.071 0.441 0.798 0.004 0.522 0.954 0.039 0.653 1.096

OTM GL-UJ 0.197 0.135 -0.047 0.209 0.141 -0.051 0.237 0.176 -0.043DO-UJ 0.140 0.125 -0.017 0.146 0.135 -0.015 0.164 0.160 -0.014Merton -0.018 0.393 0.722 0.047 0.490 0.910 0.082 0.630 1.075

ATM GL-UJ 0.188 0.178 0.035 0.188 0.177 0.020 0.223 0.208 0.040DO-UJ 0.146 0.141 0.031 0.149 0.150 0.033 0.169 0.174 0.047Merton 0.025 0.349 0.607 0.095 0.411 0.714 0.187 0.726 0.961

ITM GL-UJ 0.150 0.184 0.054 0.145 0.171 0.019 0.142 0.177 -0.010DO-UJ 0.125 0.132 0.022 0.125 0.135 0.005 0.123 0.150 -0.016Merton 0.054 0.238 0.452 0.107 0.312 0.626 0.167 0.495 0.861

DITM GL-UJ 0.092 0.162 0.106 0.091 0.157 0.086 0.085 0.134 0.058DO-UJ 0.083 0.103 0.047 0.089 0.110 0.048 0.087 0.108 0.041

Table 7.6: (Cont. of Table 7.5) The VIX call mean-relative-errors (MRE).

124

MSRE Out-of-sample: 1 day Out-of-sample: 5 days Out-of-sample: 20 daysMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion:Whaley 1.355 9.582 14.093 1.786 10.841 18.571 2.210 13.974 17.863

DOTM GL 0.484 0.549 0.641 0.594 0.591 0.610 0.833 0.817 0.675DO 0.466 0.585 0.607 0.514 0.617 0.592 0.687 0.771 0.615

Whaley 0.651 1.713 2.939 0.827 2.667 4.151 3.082 6.921 5.676OTM GL 0.314 0.283 0.280 0.484 0.341 0.272 0.743 0.555 0.406

DO 0.264 0.237 0.250 0.465 0.287 0.262 0.632 0.460 0.353Whaley 0.372 1.299 1.935 0.977 1.977 3.220 1.960 5.440 8.830

ATM GL 0.302 0.239 0.193 0.351 0.316 0.230 0.513 0.468 0.331DO 0.267 0.184 0.165 0.277 0.237 0.211 0.434 0.376 0.289

Whaley 0.506 1.236 1.457 0.725 1.565 2.284 3.562 14.617 4.662ITM GL 0.176 0.205 0.181 0.239 0.261 0.218 0.343 0.416 0.330

DO 0.149 0.146 0.134 0.212 0.201 0.179 0.295 0.342 0.290Whaley 0.172 0.400 0.812 0.300 0.654 1.395 0.807 2.564 2.265

DITM GL 0.084 0.150 0.160 0.102 0.166 0.180 0.177 0.256 0.260DO 0.069 0.108 0.105 0.089 0.125 0.123 0.160 0.214 0.212

Jump-diffusion:Merton 0.237 1.141 3.905 2.339 1.948 5.790 5.860 5.084 8.294

DOTM GL-UJ 0.280 0.301 0.447 0.553 0.358 0.445 1.003 0.565 0.460DO-UJ 0.213 0.218 0.380 0.334 0.249 0.388 0.534 0.361 0.389Merton 0.209 0.367 1.151 1.330 1.023 2.221 2.453 4.888 3.581

OTM GL-UJ 0.192 0.120 0.199 0.379 0.192 0.225 0.706 0.393 0.297DO-UJ 0.136 0.097 0.186 0.228 0.140 0.212 0.412 0.288 0.267Merton 0.103 0.339 1.019 0.923 0.853 1.980 1.643 4.241 7.384

ATM GL-UJ 0.120 0.104 0.141 0.232 0.165 0.182 0.506 0.366 0.268DO-UJ 0.080 0.071 0.130 0.135 0.122 0.170 0.300 0.268 0.237Merton 0.130 0.291 0.824 0.993 0.620 1.507 2.697 10.913 3.573

ITM GL-UJ 0.075 0.082 0.097 0.157 0.127 0.138 0.342 0.313 0.257DO-UJ 0.048 0.061 0.088 0.096 0.097 0.123 0.220 0.244 0.235Merton 0.087 0.165 0.525 0.264 0.386 1.131 0.773 2.248 2.236

DITM GL-UJ 0.036 0.070 0.059 0.077 0.094 0.103 0.185 0.198 0.200DO-UJ 0.024 0.042 0.050 0.054 0.067 0.089 0.140 0.169 0.183

Table 7.7: (Notion similar as in Table 7.5) The out-of-sample mean relative squared errors (MSRE). TheP -process parameter values are the ML estimates over the 252-day overlapping VIX series starting onFebruary 24, 2006, and followed up to April 30, 2010 (1,052 daily parameter estimates).

125

MRE Out-of-sample: 1 day Out-of-sample: 5 days Out-of-sample: 20 daysMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion:Whaley 0.568 1.954 2.422 0.610 2.029 2.627 0.643 2.113 2.557

DOTM GL -0.209 -0.348 -0.608 -0.162 -0.318 -0.590 -0.050 -0.218 -0.530DO -0.140 -0.286 -0.603 -0.113 -0.261 -0.582 -0.034 -0.184 -0.538

Whaley 0.397 0.902 1.158 0.431 0.987 1.282 0.552 1.161 1.395OTM GL 0.194 0.048 -0.207 0.223 0.055 -0.206 0.283 0.119 -0.155

DO 0.186 0.027 -0.244 0.211 0.029 -0.236 0.240 0.074 -0.206Whaley 0.330 0.761 0.929 0.376 0.844 1.127 0.443 1.000 1.358

ATM GL 0.265 0.135 -0.100 0.248 0.140 -0.084 0.262 0.163 -0.072DO 0.248 0.100 -0.138 0.220 0.096 -0.115 0.215 0.104 -0.118

Whaley 0.292 0.640 0.749 0.331 0.702 0.897 0.465 1.051 1.157ITM GL 0.224 0.185 0.027 0.206 0.170 -0.006 0.159 0.156 -0.059

DO 0.196 0.136 -0.037 0.172 0.116 -0.069 0.110 0.096 -0.126Whaley 0.150 0.347 0.480 0.193 0.417 0.651 0.277 0.609 0.884

DITM GL 0.163 0.211 0.162 0.125 0.176 0.145 0.043 0.095 0.012DO 0.127 0.162 0.098 0.088 0.125 0.079 0.000 0.043 -0.056

Jump-diffusion:Merton -0.173 0.601 1.295 -0.036 0.693 1.455 -0.030 0.816 1.527

DOTM GL-UJ 0.024 -0.417 -0.618 0.110 -0.377 -0.601 0.254 -0.277 -0.564DO-UJ 0.060 -0.328 -0.563 0.100 -0.305 -0.541 0.178 -0.228 -0.517Merton -0.124 0.322 0.710 -0.021 0.406 0.848 0.050 0.596 0.987

OTM GL-UJ 0.186 -0.130 -0.357 0.229 -0.117 -0.356 0.313 -0.041 -0.320DO-UJ 0.154 -0.108 -0.344 0.169 -0.103 -0.336 0.210 -0.042 -0.310Merton -0.082 0.288 0.654 0.013 0.390 0.819 0.073 0.591 1.099

ATM GL-UJ 0.166 -0.030 -0.250 0.172 -0.040 -0.258 0.257 0.031 -0.224DO-UJ 0.125 -0.044 -0.255 0.118 -0.042 -0.246 0.165 0.009 -0.220Merton -0.024 0.252 0.539 0.084 0.335 0.686 0.168 0.693 0.944

ITM GL-UJ 0.125 0.024 -0.141 0.121 0.002 -0.183 0.135 0.016 -0.216DO-UJ 0.086 0.009 -0.170 0.072 -0.009 -0.199 0.060 0.000 -0.228Merton 0.022 0.168 0.346 0.092 0.246 0.541 0.153 0.440 0.826

DITM GL-UJ 0.066 0.081 -0.024 0.037 0.050 -0.065 0.003 -0.009 -0.141DO-UJ 0.037 0.056 -0.042 0.005 0.029 -0.075 -0.042 -0.027 -0.153

Table 7.8: (Cont. of Table 7.7) The VIX call mean-relative-errors (MRE).

126

Out-of-sample: 1 day Out-of-sample: 5 days Out-of-sample: 20 daysModel τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

10×MSRE θP over the whole VIX seriesWhaley 0.053 0.205 1.013 0.083 0.292 1.710 0.150 0.448 1.404GL 0.036 0.165 0.391 0.041 0.176 0.395 0.059 0.212 0.445DO 0.024 0.101 0.209 0.033 0.113 0.218 0.048 0.155 0.270

Merton 0.045 0.144 0.796 0.072 0.232 1.180 0.153 0.430 1.298GL-UJ 0.015 0.055 0.107 0.024 0.068 0.119 0.053 0.128 0.184DO-UJ 0.013 0.049 0.099 0.021 0.062 0.111 0.041 0.113 0.171

10×MRE θP over the whole VIX seriesWhaley -0.193 0.087 0.779 -0.174 0.137 0.938 -0.151 0.161 0.918GL 0.296 0.813 1.155 0.290 0.804 1.133 0.260 0.745 1.091DO 0.257 0.668 0.875 0.255 0.667 0.869 0.235 0.625 0.843

Merton -0.141 0.212 0.985 -0.129 0.249 1.095 -0.092 0.297 1.159GL-UJ 0.208 0.460 0.559 0.209 0.458 0.550 0.207 0.440 0.537DO-UJ 0.165 0.374 0.438 0.160 0.369 0.429 0.137 0.333 0.404

10×MSRE θP over the 252-day VIX seriesWhaley 0.069 0.279 1.180 0.099 0.361 1.914 0.166 0.507 1.500GL 0.082 0.122 0.181 0.087 0.131 0.186 0.134 0.184 0.238DO 0.077 0.097 0.138 0.083 0.108 0.142 0.134 0.160 0.195

Merton 0.045 0.124 0.698 0.072 0.211 1.095 0.152 0.410 1.198GL-UJ 0.026 0.071 0.148 0.054 0.088 0.154 0.149 0.161 0.209DO-UJ 0.018 0.052 0.114 0.037 0.067 0.123 0.116 0.138 0.180

10×MRE θP over the 252-day VIX seriesWhaley -0.351 -0.321 0.105 -0.332 -0.273 0.259 -0.307 -0.245 0.247GL 0.564 0.598 0.553 0.530 0.579 0.559 0.429 0.518 0.530DO 0.500 0.485 0.416 0.460 0.461 0.414 0.338 0.392 0.374

Merton -0.184 0.085 0.754 -0.167 0.130 0.881 -0.134 0.178 0.939GL-UJ 0.166 0.081 0.055 0.156 0.070 0.042 0.124 0.052 0.027DO-UJ 0.120 0.080 0.067 0.093 0.065 0.054 0.017 0.029 0.027

Table 7.9: The out-of-sample mean squared relative errors (MSRE) and mean relative errors (MRE) forthe VIX futures pricing within different maturity (τ) sections. The P -dynamics maximum likelihoodestimates are acquired over the whole VIX historical series (January 2, 1990 – April 30, 2010), or reflectthe local VIX characteristics over the 252-day backward series, updated daily starting on February 24,2006 until April 30, 2010. The daily updated Q-measure parameters are implied jointly from the prices ofall the VIX derivative traded at day-t, assuming that both volatility (diffusive) and jump risk are pricedby the market. Each day, the model parameters are backed out by H=1, 5, and 20 days, and the particular

expiration contract pricing error: wFtV olt

(F (θPt−H ,θ

Qt−H)−Fmarket

t

Fmarkett

)2is calculated, where wFt is the particular

futures contract trading volume at day-t and V olt corresponds to the total futures trading volume withinthe corresponding maturity group observed at t. The resulting errors are summed up and averaged overthe number of the trading days considered in the sample. The VIX futures derivative sample periodextends from February 24, 2006 up to April 30, 2010: 1,052 trading days.

127

Mod

elMerton

GL-U

JDO-U

JParam

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µσ

λγ

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µJ

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mκ

σλ

γη

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1.054

7.106

0.363

-16.84

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0.775

-34.50

18.33

-SE

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4.622

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0.672

0.011

-6.924

2.458

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Mean

Q-

-0.294

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--

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Median

Q-

-0.202

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Max

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2021.2

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P0.790

0.579

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4.452

25.50

0.399

-47.86

58.65

--32.64

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0.828

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15.80

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P-0.882

0.270

-28.2

-0.004

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1.833

11.67

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0.699

-142.43

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1.102

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0.311

1.72

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-20.86

13.93

4.329

1.79

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68.19

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Q-

-0.574

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--3297.6

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Q-

-5.757

-123.1

-14504.3

--

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37.97

--

62.86

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--

17271.2

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128

Chapter 8

Hedging Assessment of the VolatilityDerivative Pricing Models Based onJump-Diffusions

8.1 Introduction

In this Chapter, we investigate how much generalization of the VIX dynamics to the jump-diffusion changes the VIX derivative hedging performance, compared to the one providedby the baseline diffusion models (Whaley, GL, and DO). The analysis starts with thestandard delta hedging setup as a bottom-line. In particular, only one VIX futures isused to hedge a short call option position (Sepp [2008a], Lin and Chang [2010]). In thecase of the futures, a position is covered with the underlying index position, which isthen replicated by the S&P 500 options. In the next step, the option replicating portfoliostrategy is extended by one more futures contract, in order to hedge the expected jumpsize risk. The standard delta hedging for contingent claims fails usually in eliminating thejump risk exposure, even if the time grid is infinitesimal (Naik and Lee [1990]). Typically,the jump size is not known in advance, and the risk associated with the jumps cannot behedged away completely, which yields an incomplete market. Within this setting, hedgingbecomes an approximation rather than a replication issue (Tankov and Voltchkova [2009]).Following the suggestion in Bates [1988], the jump risk can be minimized by increasingthe number of hedging instruments in the portfolio. However, as underlined in He et al.[2006], it is not clear how many of them are required in order to achieve a satisfactoryrisk reduction. For most of the jump-diffusion processes, due to a continuum of expectedjump sizes, an infinite number of derivative instruments must be traded for the market

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to make it complete.1 This strategy, with respect to the VIX options, finds a referencein Sepp [2008a], who hedges a VIX option short position using two VIX futures. One ofthem expires the same day as the hedged option contract, while the other has the next-longer time to maturity. Simulation based evidence suggests that this strategy might bepromising in empirical applications.

The Chapter consists of three sections. The first one provides description of the hedgingsetup, which concerns mainly the delta-jump framework, as for the standard delta one,we rely on the Chapter 6 setup. The next section concentrates on the empirical outcomesfrom the diffusion (delta hedging) and then jump-diffusion (delta-jump hedging) models.The last section concludes the main outcomes.

8.2 Hedging setup

The analysis concentrates on two different hedging setups for the VIX call options: thedelta- and delta-jump hedging, and the delta-type framework for the VIX futures. Thedelta one is designed to hedge against the underlying VIX diffusive volatility risk. However,such an approach might be inappropriate to hedge the VIX call options because of possiblejumps in the volatility index dynamics. Then, the delta-jump strategy is designed todiminish the (expected) jump risk exposure. In both strategies, the VIX call optionhedging is done by trading the VIX futures contracts. The futures strategy is carried outwithin three different ways of constructing the hedging portfolio, which includes a longposition in the underlying VIX.

8.2.1 Delta strategy

The VIX call option and futures delta hedging setups correspond to the ones outlinedin Chapter 6, section 6.3. In particular, for options we consider (6.15) framework, whilefor futures (6.21), (6.27), and (6.34) strategies. See Appendix A.6 for the hedge ratioformulas.

8.2.2 Delta-jump strategy

The portfolio consists of a short position in the contingent claim on the VIX (Vt) withspot price Ct, hedged with bt = [b1 . . . bI ]t units of the derivative (written on the VIX)

1However, as shown in Andersen and Andreasen [2000] for the S&P 500 options, a significant part ofthe jump risk can be eliminated by adding just a few option positions (e.g., two options) to the standarddelta hedge.

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worth Ft = [F1 . . . FI ]′t, and at amount of cash invested in the risk-free bond with constant

interest rate r. At the setup time t, the portfolio has the value

πt = −Ct + at + btFt. (8.1)

It is assumed to be self-financing, hence, the following needs to hold,

dπt = −dCt + dat + btdFt. (8.2)

The underlying asset price Vt is assumed to follow an (affine) jump-diffusion process ingeneral form:

dVt = α(Vt, θP )Vtdt+ β(Vt, θP )dW Pt + ξ(Vt, Jt)dqPt , (8.3)

where Wt denotes a standard Wiener process, qt is a Poisson processes, Vt is the sportunderlying value, and Jt is the jump size. For the particular α, β, and ξ functions, seeChapter 3. Then, the particular elements of (8.2) can be derived using Ito’s rule forjump-diffusion (Cont and Tankov [2004], pp. 275)

dCt = [C ′t + 12β

2(Vt)C′′

V + α(Vt)C′

V ]dt+ β(Vt)C′

V dWPt + ∆CdqPt ,

dFt = [F′t + 12β

2(Vt)F′′

V + α(Vt)F′

V ]dt+ β(Vt)F′

V dWPt + ∆FdqPt ,

dat = ratdt, (8.4)

where C ′t ≡ ∂C∂t, C ′V ≡ ∂C

∂V, C ′′V ≡ ∂2C

∂V 2 , F′t ≡ ∂F∂t, F′V ≡ ∂F

∂V, F′′V ≡ ∂2F

∂V 2 , and ∆C, ∆F denotethe derivative price change due to the occurrence of a jump. Applied to (8.2) yields

dπt =[−C ′t −

12β

2(Vt)C′′

V + Ft

(F′t + 1

2β2(Vt)F

′′

V

)+ rat

]dt

+(−C ′V + btF′

V )[α(Vt)dt+ β(Vt)dW Pt ]

+[−∆C + bt∆F]dqPt . (8.5)

If portfolio (8.1) is delta-neutral ( ∂π∂V

= 0) then

−C ′V + btF′

V = 0, (8.6)

which substituted in (8.5) results in

dπt =[−C ′t −

12β

2(Vt)C′′

V + bt(

F′t + 12β

2(Vt)F′′

V

)+ rat

]dt︸ ︷︷ ︸

deterministic part

+ [−∆C + bt∆F]dqPt︸ ︷︷ ︸stochastic part

, (8.7)

which is a pure jump process with drift. The delta-hedge strategy does not eliminate allrisk sources, but leaves the portfolio exposed to the jump risk in the Poisson process q.A reduction of the remaining jump-risk in the case of jump-diffusion models becomes theissue.

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Jump-risk reduction

The general framework derived above shows that, in the case of delta hedging, only someof the risk is being perfectly hedged. There is a part, which relates to the jumps occurringover dt, and this part cannot be hedged by the standard delta hedging procedure. Thisrisk comes from the expected jump size (Sepp [2008a]). Theoretically, an infinite numberof instruments is needed to hedge against a continuum of jump sizes. Since such a setupis not feasible in practice, the point is to determine optimally bt, given a finite numberof instruments I, such that the risk is eliminated to a significant extent. The frameworkapplied in Andersen and Andreasen [2000] and then followed by He et al. [2006] or Sepp[2008a] relies on choosing the optimal level of bt, by minimizing a mean-squared criterionderived from the stochastic part in (8.7), subject to a delta-neutrality constraint. Followingthis methodology, bt is determined by solving the minimization problem

argminb∗t

∫∞0 [−∆C + b∗t∆F]2 fP (J)dJ

s.t. − ∂C∂V

∣∣∣∣Vt

+ b∗t ∂F∂V

∣∣∣∣Vt

= 0, (8.8)

where fP (J) denotes the P -measure jump size density function. Solution to (8.8) is foundnumerically, using a constraint optimization method implemented in the R nleqslv package(version 1.9.4).2 Andersen and Andreasen [2000] underline that the delta constraint is notthe only condition, which might be applied here, and propose an additional condition:EP (−∆C + b∗t∆F) = 0, however, we do not include this extension. Computation ofthe integral in the expectation part is carried out with the Gauss-Legendre quadraturemethod, implemented in the R statmod package (version 1.4.16).

Application and assessment

In the original futures sample (N=8,895), 14% of the daily records are not traded. Thismeans that for some inception portfolios, the hedging futures contracts do not have atraded counterpart at the termination time. Thus, in the empirical application, we followthe limited approach, where a time t short position in the VIX option worth Ct is hedgedwith the portfolio of only two VIX futures contracts worth F1,t and F2,t. The assumption

2Sepp [2008a] proposed to solve the following system of equations:∫ ∞0

[−∆C + b∗t∆F]2 fP (J)dJ = 0

−∂C∂V

∣∣∣Vt

+ b∗t∂F∂V

∣∣∣Vt

= 0. (8.9)

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is that the considered hedging contracts expire within the same and longer as the hedgedoption period, respectively. After (8.1), such a portfolio has at time t the value

πt = −Ct(V,K, τ1) + at + b1,tF1,t(V, τ1) + b2,tF2,t(V, τ2). (8.10)

Two strategies are followed. In the first one only the delta hedging is applied and thehedge ratios at moment t are derived from b1,t = ∂C

∂V×(∂F1∂V

)−1and b2,t = ∂C

∂V×(∂F2∂V

)−1. In

the second case, the hedge ratios reflect also the expected jump risk and fulfil the followingcondition (see (8.8)):

argminb1,b2

∫∞0 [−∆Ct(V ) + b1,t∆F1,t(V ) + b2t∆F2,t(V )]2 fP (J)dJ,

s.t. − ∂C∂V|Vt + b1,t

∂F1∂V|Vt + b2,t

∂F2∂V|Vt = 0, (8.11)

where fP (J) is the model-relevant jump amplitude density function. The call and futuresdelta values can be determined from the relevant delta hedging formulas. Given the partialderivative values (b1, b2), the portfolio cash position is determined from

at = −Ct + b1,tF1,t + b2,tF2,t. (8.12)

Since in practice, a continuous time rebalancing is not possible, the portfolio is checked atdiscrete time points, t+m∆t, where m=1, 2, . . . , M= τ

∆t , given the weights derived for t.The portfolio at its termination yields the value

πt+∆t = −Ct+∆t + b1,tF1,t+∆t + b2,tF2,t+∆t + at(1 + r∆t), (8.13)

which is indeed the hedging error resulting from the applied hedging strategy. The deltahedging performance is compared with the delta-neutral frameworks, where only one fu-tures contract is used to hedge the short position in the VIX option,

πt+∆t = −Ct+∆t + b1,tF1,t+∆t + at(1 + r∆t). (8.14)

Similar to the delta-neutral strategy, the empirical hedging assessment relies on the meanabsolute/dollar hedging error (MAHE/MDHE) measure, weighted by the minimal optiontrading volume observed between the portfolio setup and its termination (see (6.11) and(6.12)).

8.3 Empirical Results

The VIX call and futures empirical hedging outcomes are provided and discussed in thecurrent section. The jump-diffusion model results are compared with the baseline diffusion

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based outcomes, assuming the different hedging and parameter estimation frameworks. Ashort position in the option is hedged with the VIX futures contracts, following the delta-and delta-jump strategies. The first approach is applied to the diffusion and jump-diffusionbased models. Then, the jump-diffusion results are extended into the delta-jump hedg-ing framework. Theoretically, while constructing the delta portfolio, we assume that thefutures contract will expire on the same date as the hedged call option. Then, in the delta-jump strategy, this portfolio is extended by one more futures contract, expiring within alonger (than the option) time horizon, to cover the jump risk exposure. However in prac-tice, construction of such a portfolio is not always possible, as such futures contracts mightnot be available or traded at the setup date. In such situations, we change the criteria forthe delta-jump strategy first, by choosing the nearest two futures maturing before the op-tion. Then, we adjust the delta framework accordingly by using the one futures contract,which expires later, such that the delta-jump portfolio might be considered as the exten-sion of the delta one. The decision about the futures choice is taken at the portfolio setup,and if at least one of the chosen futures is not traded at the termination, both portfoliosare not considered in the analysis. Given these modifications, the current diffusion resultsmay be different from those provided in Chapter 6.

The hedging outcomes are based on the P -process estimates over the VIX whole his-torical sample (January 2, 1990 – April 30, 2010), as well as on its 252-day backwardseries (starting on February 24, 2006). Given these estimates, the Q-measure parame-ters are calibrated daily to all the VIX derivative prices, under the assumption that boththe volatility and jump risks are prices by the market. There are three different hedg-ing horizons considered: the short- (1-day), mid- (5-day), and long- (10-day) term. Themodel performance is assessed with respect to the mean absolute and dollar hedging errors:MAHE and MDHE, respectively.

8.3.1 VIX option hedging

The analysis concerns the VIX call options exclusively and starts with the results based onthe fixed (‘whole sample’) P -parameter estimates, Table 8.1. Within the diffusion baseddelta hedging outcomes, none of the models clearly outperforms the other specificationswithin all the moneyness/expiration cross-sections, as well as the portfolio rebalancingfrequencies. The DO and Whaley remain complementary in hedging the VIX calls. Thefirst model performs usually best for the DOTM calls, regardless of the hedging period.The only exception concerns the short-term contracts for the daily frequency, where bothWhaley and DO provide nearly comparable MAHE outcomes over the GL result: 0.073and 0.072 vs. 0.083, respectively. The lower the portfolio revision frequency (5, 10 days),

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the more precise DO’s performance within the DOTM group. This is an important result,bearing in mind that 62% of the total VIX call options trade concentrates within thismoneyness group. Among the other sections, the leading role switches between the DOand Whaley models. Namely, the best outcomes for the short-term hedging horizon belongto the DO specification, with some exceptions in favor to Whaley, however, the differencesare not that large. The longer time between the portfolio setup and its termination, themore credit is given to the Whaley model. The GL performance is worse than the Whaleyand DO in majority of the cross-sections.

The jump extension provided to the underlying VIX dynamics translates into thehedging performance. The mean-reverting models improve in almost all sections, exceptthe one with short-term (D)ITM calls. Within the cheapest short-term DOTM calls,given that the portfolio is rebalanced daily, DO-UJ improves by 9% over the DO outcome(MAHE=0.072). In the same group, the Merton model is inferior to the Whaley one(MAHE=0.079 vs. 0.073) and does not work better within the other short-term moneynesssections. It works slightly better for the mid- and long-term DOTM calls. Overall, theDO-UJ model performs best in almost all sections.

The delta hedging for the jump-diffusion models is extended into the delta-jump frame-work to hedge against the sudden upward VIX changes (the expected jump risk). Sucha modification yields further improvements in some cross-sections. The DO-UJ modelremains the best performing specification over the short-term calls, regardless of the port-folio rebalancing frequency. The improvements over the delta hedging outcomes are mainlyencountered for the mid- and long-term rebalanced portfolios with the short-term calls,which account for around 85% of the total VIX call option trade. The magnitude in theMAHE change stays usually between 0.02 and 0.1 over the jump-diffusion delta outcomes.The observed improvements for these particular rebalancing rates may be explained withthe ‘whole sample’ jump intensity parameter estimates (Table 7.10). Out of the DO-UJand GL-UJ models, it takes on average 7 and 15 days for a jump of the average size tooccur, respectively. Thus, during the 1-day interval the model expectation for the jumpincidence becomes small, and the delta-jump results remain almost the same as for thedelta ones, within the jump-diffusion frameworks. The delta-jump strategy for the Mer-ton model improves mainly over its delta results (e.g., (D)ITM section), but does notoutperform Whaley’s results.

Relaxing the assumption about the constant P -estimates provides usually higher, thanfor the ‘whole sample’ counterparts, absolute hedging error magnitudes (see Table 8.2).Within the diffusion based framework, only the GL works better in the short-term maturitygroup, but despite these improvements, the DO gets most credit in all the cross-sections.The jump-extension improves the Whaley performance in the mid- and long-term DOTM

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sections only. Among the jump-diffusion models, the GL-UJ improvements do not have anysystematic character and concern some long-maturity sections. The DO-UJ outperformsits diffusion based version in almost all the cross-sections. In the short-term daily rebal-anced DOTM calls, the improvement rate reaches 13% over the baseline MAHE=0.077,and remains very similar within the other hedging horizons. The DO model works betterin the DITM short-term calls, clearly improving in the lower portfolio rebalancing fre-quencies. The delta-jump extension improves only the DO-UJ performance for the dailyrebalanced portfolios with long-maturing call positions. Despite this effect dies out for thelonger hedging periods, it remains the best among all the frameworks considered.

Tables 8.3 and 8.4 complement the absolute hedging error results with ones in thedollar terms. Their changes between different estimation, the underlying dynamics, andhedging frameworks do not happen in any systematic way. In all the cases, the outcomesremain usually positive and comparable between the diffusion and jump-diffusion sections.The values in short-term hedging horizon oscillate strongly around zero and grow as theportfolio rebalancing happens less often.

8.3.2 VIX futures hedging

Within the framework based on the fixed P -estimates, the GL-UJ and DO-UJ modelsperform clearly better than the Merton model, over all considered strategies (see Table8.5). This becomes a particularly apparent for the long-term contract portfolios rebalancedon the 10-day basis, where the Merton MAHE magnitude remains three times higher. Inthe reference VIX-based hedging framework, the GL-UJ model works better for the short-term contracts improving, the DO-UJ performance (MDHE=0.437) by 11% and the bestDO’s outcome (MDHE=0.489, see Table 6.6) by 20.4%, as the portfolios get rebalanceddaily. As the futures time to maturing or hedging horizon become longer, it is loosingmomentum with respect to DO-UJ’s performance. Within the same hedging setup, theMerton and Whaley do not provide much different results in the absolute terms, withpreference for the latter specification. The more practically relevant approach, based onthe S&P 500 market information, increases the MDHE magnitude and confirms the GL-UJleading role in hedging the short-term futures portfolios. The DO-UJ model works usuallybetter for the long-term contracts. The jump extension improvements with respect tothe diffusion based outcomes are not that evident in the baseline VIX framework. Theybecome more apparent as the portfolio construction relies on the S&P 500 options and itsrebalancing period becomes longer.

Switch to more flexible P -parameter estimates yields worse in terms of the MAHEoutcomes within the reference VIX based framework, when compared to the fixed estimates

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based results (see Table 8.6). The DO-UJ model performs best in this framework for allthe maturity and hedging horizon sections. However, it is not able to improve DO’soutcomes (see Tables 6.6 and 6.7). Using the S&P 500 based information pays off in abetter DO-UJ’s performance for the short-term futures, over the relevant (best) GL-UJ,given the fixed P -estimates, results. The improvement magnitude, in the 10-day hedgingperiods stays around 14% over the baseline GL-UJ MAHE=2.254 and 2.218 outcomes.The short-maturity performance of the DO-UJ is not extended into the mid- and long-maturity contracts, where the outcomes based on the fixed P -estimates (Table 8.5) remainbest in both diffusion and jump-diffusion model categories.

The hedging errors in the dollar terms (MDHE) stay negative for all the models inmajority of the maturity sections, and decrease as the period between the portfolio setupand its termination becomes longer. Switch from the VIX to S&P 500 market basedinformation results in even lower dollar errors, which reflects an increasing variation of themodel outcomes.

8.4 Summary

The Chapter concerns the empirical hedging performance for the VIX call and futuresderivatives, within one-dimensional jump-diffusion pricing models. For the call options,there are two hedging strategies evaluated. First, the delta hedging, where the short VIXcall option position is covered with a long one in the VIX futures, such that both contractsexpire on the same date. The primary objective was an assessment of the jump-diffusionbased hedging performance, and its comparison with the baseline diffusion outcomes. Inthe next step, the hedging framework was adjusted to hedge the risk of the (expected)jumps in the underlying: the delta-jump strategy. For the futures, we consider the deltahedging framework using the underlying spot VIX position, or its replicated with the S&P500 options version. All models were assessed within two different information frameworks.The model estimates reflecting the long-term underlying tendencies, which stayed constantover the whole analyzed period. Alternatively, this condition was relaxed and the estimateswere allowed to change every day, reflecting the local VIX characteristics. The portfoliorevision frequencies were carried out within three different time horizons: short- (1-day),mid- (5-day), and long- (10-day) term.

The mean absolute hedging error outcomes do not favor any particular pricing frame-work. In the case of the VIX call option contracts, the more flexible parameter estimatesbased on the 252-day VIX series do not provide hedging improvements over the constant‘whole sample’ ones. Within the latter estimation framework and the short-maturing or

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DOTM sections, the Detemple-Osakwe (DO) model performs usually best, regardless thehedging horizon. The Whaley model is slightly better than DO for the mid- and long-term expiring portfolios, outside the DOTM moneyness section. Worth mentioning is thefact that both models provide very similar error values in the short-term DOTM sec-tion, which indicates on their complementarity in the VIX call option hedging. The DOjump extension (the DO-UJ model) improves its diffusion based outcomes within all thecross-sections, regardless of the portfolio rebalancing frequency considered in the analysis.The improvement in the short-term DOTM section stands for 8.3% in the MAHE terms.Switch to the delta-jump hedging framework provides further improvements in the short-term call option cross-sections, given that the portfolio is rebalanced on a lower than dailyfrequency. The improvement magnitude encountered for this section stays between 0.02and 0.1 over the delta based jump-diffusion framework.

The outcomes for the futures give more credit to the mean-reverting models, than forthe GBM based specifications. The GL-UJ performs better in the short- and mid-termportfolios, while the DO-UJ gets more recognition for the long-term cases. The jumpamendment provided to the diffusion based models pays-off with a better hedging perfor-mance within the short- and mid-term contracts, given that the portfolio is rebalanceddaily or weekly.

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MAHE 1-day (∆t) revision 5-day revision 10-day revisionMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion based: delta hedging with one VIX futuresWhaley 0.073 0.078 0.082 0.162 0.147 0.133 0.259 0.222 0.183

DOTM GL 0.083 0.066 0.069 0.175 0.125 0.118 0.263 0.187 0.156DO 0.072 0.062 0.065 0.158 0.116 0.111 0.250 0.178 0.145

Whaley 0.094 0.080 0.079 0.203 0.148 0.128 0.332 0.231 0.159OTM GL 0.107 0.086 0.085 0.228 0.167 0.131 0.366 0.255 0.172

DO 0.093 0.080 0.080 0.207 0.149 0.124 0.335 0.227 0.159Whaley 0.102 0.087 0.101 0.220 0.149 0.118 0.377 0.231 0.160

ATM GL 0.114 0.097 0.108 0.248 0.171 0.135 0.432 0.261 0.180DO 0.098 0.091 0.103 0.216 0.156 0.125 0.389 0.238 0.168

Whaley 0.111 0.093 0.098 0.231 0.148 0.133 0.374 0.234 0.193ITM GL 0.118 0.102 0.106 0.274 0.171 0.159 0.447 0.281 0.197

DO 0.103 0.097 0.101 0.229 0.160 0.151 0.387 0.253 0.191Whaley 0.126 0.122 0.120 0.226 0.188 0.163 0.355 0.303 0.212

DITM GL 0.124 0.137 0.124 0.246 0.215 0.173 0.395 0.376 0.193DO 0.114 0.126 0.122 0.209 0.191 0.166 0.351 0.320 0.188

Jump-diffusion based: delta hedging with one VIX futuresMerton 0.079 0.066 0.076 0.172 0.123 0.129 0.260 0.187 0.182

DOTM GL-UJ 0.077 0.061 0.065 0.163 0.114 0.116 0.249 0.171 0.153DO-UJ 0.066 0.058 0.062 0.151 0.106 0.106 0.240 0.165 0.142Merton 0.105 0.080 0.084 0.225 0.151 0.141 0.351 0.232 0.192

OTM GL-UJ 0.105 0.076 0.077 0.220 0.142 0.121 0.347 0.218 0.161DO-UJ 0.089 0.075 0.076 0.197 0.137 0.118 0.329 0.216 0.154Merton 0.110 0.089 0.117 0.238 0.149 0.129 0.401 0.240 0.190

ATM GL-UJ 0.120 0.085 0.100 0.251 0.145 0.110 0.412 0.223 0.160DO-UJ 0.095 0.085 0.100 0.206 0.145 0.109 0.380 0.222 0.160Merton 0.111 0.096 0.107 0.247 0.158 0.145 0.414 0.260 0.215

ITM GL-UJ 0.130 0.096 0.098 0.279 0.159 0.142 0.439 0.253 0.187DO-UJ 0.102 0.093 0.099 0.221 0.151 0.142 0.384 0.244 0.185Merton 0.128 0.136 0.131 0.249 0.233 0.190 0.430 0.380 0.267

DITM GL-UJ 0.144 0.126 0.118 0.284 0.203 0.151 0.443 0.325 0.193DO-UJ 0.118 0.120 0.118 0.208 0.186 0.154 0.356 0.304 0.192

Jump-diffusion based: delta-jump hedging with two VIX futuresMerton 0.079 0.066 0.077 0.172 0.125 0.129 0.261 0.190 0.180

DOTM GL-UJ 0.077 0.061 0.066 0.162 0.115 0.117 0.247 0.172 0.153DO-UJ 0.066 0.058 0.063 0.151 0.106 0.107 0.238 0.165 0.142Merton 0.104 0.080 0.084 0.224 0.150 0.142 0.351 0.233 0.183

OTM GL-UJ 0.106 0.076 0.078 0.219 0.143 0.124 0.341 0.220 0.162DO-UJ 0.090 0.075 0.076 0.195 0.137 0.120 0.323 0.216 0.154Merton 0.109 0.090 0.114 0.237 0.148 0.127 0.403 0.239 0.184

ATM GL-UJ 0.121 0.087 0.099 0.247 0.149 0.114 0.403 0.227 0.163DO-UJ 0.095 0.086 0.099 0.202 0.147 0.112 0.370 0.224 0.161Merton 0.110 0.095 0.104 0.242 0.150 0.138 0.406 0.246 0.198

ITM GL-UJ 0.130 0.097 0.097 0.277 0.161 0.148 0.432 0.257 0.194DO-UJ 0.102 0.093 0.098 0.218 0.152 0.146 0.377 0.246 0.188Merton 0.127 0.125 0.124 0.239 0.200 0.166 0.401 0.337 0.209

DITM GL-UJ 0.145 0.126 0.117 0.283 0.208 0.150 0.437 0.329 0.194DO-UJ 0.118 0.120 0.118 0.204 0.188 0.152 0.347 0.305 0.191

Table 8.1: The mean absolute hedging errors (MAHE) for the VIX call options implied by using different hedging frameworks,over the moneyness (Mness) and expiration (τ) cross-sections. A particular option hedging error is weighted by its minimaltrading volume between the one recorded at the portfolio setup and its termination. The P -process parameters θP areestimated with the maximum-likelihood method, using the underlying VIX sample spanning the February 24, 2006 up toApril 30, 2010 period. The market volatility and jump size risk premiums are jointly implied from all VIX derivatives(options and futures) prices listed at the portfolio setup date.

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MAHE 1-day (∆t) revision 5-day revision 10-day revisionMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion based: delta hedging with one VIX futuresWhaley 0.081 0.088 0.084 0.191 0.169 0.135 0.298 0.256 0.185

DOTM GL 0.081 0.067 0.071 0.181 0.137 0.130 0.282 0.209 0.170DO 0.077 0.066 0.070 0.181 0.133 0.127 0.282 0.203 0.169

Whaley 0.097 0.084 0.078 0.214 0.154 0.126 0.347 0.243 0.158OTM GL 0.104 0.088 0.085 0.233 0.166 0.129 0.377 0.254 0.173

DO 0.100 0.084 0.082 0.227 0.156 0.123 0.371 0.239 0.165Whaley 0.106 0.091 0.099 0.233 0.157 0.122 0.387 0.242 0.159

ATM GL 0.106 0.100 0.107 0.238 0.178 0.142 0.431 0.266 0.182DO 0.102 0.095 0.104 0.232 0.165 0.135 0.420 0.250 0.172

Whaley 0.120 0.097 0.099 0.247 0.155 0.134 0.391 0.244 0.198ITM GL 0.111 0.105 0.112 0.247 0.176 0.164 0.413 0.270 0.218

DO 0.108 0.101 0.106 0.238 0.162 0.155 0.402 0.253 0.208Whaley 0.139 0.126 0.121 0.260 0.205 0.171 0.404 0.313 0.229

DITM GL 0.124 0.130 0.130 0.222 0.188 0.185 0.345 0.294 0.186DO 0.122 0.128 0.129 0.215 0.179 0.178 0.343 0.275 0.182

Jump-diffusion based: delta hedging with one VIX futuresMerton 0.082 0.063 0.073 0.182 0.116 0.122 0.271 0.173 0.172

DOTM GL-UJ 0.078 0.067 0.061 0.175 0.142 0.129 0.280 0.214 0.169DO-UJ 0.067 0.060 0.064 0.158 0.118 0.121 0.252 0.181 0.163Merton 0.118 0.084 0.084 0.253 0.160 0.135 0.376 0.235 0.196

OTM GL-UJ 0.106 0.081 0.074 0.236 0.166 0.128 0.386 0.258 0.169DO-UJ 0.091 0.075 0.075 0.206 0.145 0.128 0.345 0.228 0.170Merton 0.127 0.098 0.116 0.267 0.176 0.137 0.424 0.251 0.211

ATM GL-UJ 0.117 0.088 0.085 0.261 0.168 0.110 0.445 0.261 0.156DO-UJ 0.096 0.085 0.092 0.218 0.152 0.107 0.397 0.231 0.155Merton 0.129 0.111 0.112 0.279 0.206 0.178 0.455 0.306 0.249

ITM GL-UJ 0.128 0.097 0.093 0.284 0.167 0.141 0.458 0.272 0.175DO-UJ 0.105 0.092 0.094 0.233 0.152 0.138 0.402 0.246 0.186Merton 0.152 0.164 0.151 0.303 0.314 0.261 0.522 0.464 0.376

DITM GL-UJ 0.149 0.118 0.118 0.287 0.181 0.161 0.444 0.294 0.191DO-UJ 0.123 0.118 0.114 0.225 0.168 0.155 0.377 0.279 0.184

Jump-diffusion based: delta-jump hedging with two VIX futuresMerton 0.081 0.063 0.075 0.182 0.118 0.120 0.272 0.181 0.166

DOTM GL-UJ 0.084 0.072 0.063 0.195 0.151 0.131 0.313 0.233 0.176DO-UJ 0.068 0.060 0.062 0.159 0.118 0.114 0.253 0.183 0.156Merton 0.115 0.077 0.081 0.248 0.144 0.127 0.372 0.225 0.160

OTM GL-UJ 0.123 0.094 0.079 0.286 0.191 0.145 0.469 0.290 0.195DO-UJ 0.092 0.076 0.072 0.207 0.148 0.124 0.344 0.231 0.151Merton 0.119 0.089 0.106 0.258 0.146 0.113 0.413 0.232 0.168

ATM GL-UJ 0.143 0.109 0.092 0.330 0.209 0.134 0.590 0.305 0.192DO-UJ 0.098 0.088 0.087 0.218 0.159 0.106 0.396 0.238 0.146Merton 0.118 0.099 0.105 0.258 0.159 0.145 0.420 0.255 0.207

ITM GL-UJ 0.155 0.109 0.098 0.370 0.203 0.163 0.588 0.317 0.218DO-UJ 0.106 0.094 0.089 0.234 0.157 0.131 0.402 0.253 0.158Merton 0.132 0.130 0.128 0.245 0.205 0.173 0.427 0.350 0.219

DITM GL-UJ 0.190 0.130 0.121 0.380 0.224 0.179 0.662 0.347 0.204DO-UJ 0.125 0.119 0.108 0.227 0.177 0.146 0.381 0.287 0.160

Table 8.2: (Cont. of Table 8.1) The P -process parameters θP are estimated using the maximum-likelihood method over thebacked 252-day rolling VIX series starting on February 24, 2006.

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MDHE 1-day (∆t) revision 5-day revision 10-day revisionMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion based: delta hedging with one VIX futuresWhaley 0.004 0.003 0.007 0.026 0.012 0.019 0.047 0.007 0.016

DOTM GL 0.005 0.002 0.005 0.028 0.012 0.008 0.050 0.016 0.009DO 0.003 0.002 0.005 0.021 0.012 0.007 0.034 0.010 0.008

Whaley 0.011 0.006 -0.002 0.048 0.023 0.023 0.085 0.030 0.037OTM GL 0.008 0.003 0.000 0.033 0.021 0.022 0.047 0.018 0.037

DO 0.006 0.007 0.000 0.032 0.021 0.020 0.046 0.019 0.031Whaley 0.014 0.003 0.013 0.068 0.026 -0.002 0.105 0.029 0.032

ATM GL 0.012 -0.001 0.014 0.051 0.020 -0.001 0.040 0.007 0.020DO 0.008 0.001 0.013 0.048 0.021 0.006 0.044 0.004 0.026

Whaley 0.014 0.003 0.004 0.068 0.021 0.004 0.118 0.026 0.021ITM GL 0.012 0.003 0.006 0.032 0.020 0.014 0.029 0.012 0.024

DO 0.008 0.002 0.007 0.043 0.020 0.013 0.049 0.007 0.026Whaley 0.014 -0.009 -0.006 0.070 -0.001 -0.022 0.118 -0.005 0.008

DITM GL 0.014 -0.008 -0.004 0.043 -0.017 -0.020 0.056 -0.070 -0.016DO 0.010 -0.009 -0.005 0.051 -0.009 -0.019 0.051 -0.039 -0.009

Jump-diffusion based: delta hedging with one VIX futuresMerton 0.004 0.000 0.004 0.034 0.007 0.013 0.063 0.003 0.006

DOTM GL-UJ 0.005 0.002 0.002 0.029 0.010 -0.006 0.060 0.015 -0.005DO-UJ 0.004 0.002 0.004 0.026 0.012 0.002 0.052 0.014 0.004Merton 0.006 0.003 -0.003 0.038 0.009 0.016 0.064 0.010 0.019

OTM GL-UJ 0.007 0.006 0.001 0.034 0.015 0.013 0.058 0.019 0.016DO-UJ 0.006 0.006 0.001 0.030 0.017 0.017 0.052 0.017 0.022Merton 0.008 0.001 0.012 0.050 0.020 0.009 0.057 0.005 0.011

ATM GL-UJ 0.008 0.003 0.011 0.048 0.016 0.008 0.043 0.012 0.017DO-UJ 0.008 0.002 0.012 0.042 0.016 0.007 0.043 0.006 0.018Merton 0.007 0.001 0.006 0.027 0.006 0.009 0.034 0.001 0.008

ITM GL-UJ 0.008 0.001 0.001 0.030 0.008 0.002 0.023 0.003 0.013DO-UJ 0.007 0.001 0.002 0.030 0.007 0.004 0.032 0.000 0.014Merton 0.009 -0.012 -0.008 -0.001 -0.042 -0.013 -0.014 -0.093 0.006

DITM GL-UJ 0.002 -0.010 -0.003 0.005 -0.029 -0.014 -0.029 -0.067 0.001DO-UJ 0.006 -0.008 -0.004 0.025 -0.026 -0.017 0.009 -0.062 -0.006

Jump-diffusion based: delta-jump hedging with two VIX futuresMerton 0.004 0.000 0.003 0.035 0.008 0.012 0.066 0.005 0.007

DOTM GL-UJ 0.005 0.002 0.002 0.026 0.011 -0.005 0.054 0.017 -0.002DO-UJ 0.004 0.002 0.004 0.024 0.013 0.003 0.047 0.015 0.007Merton 0.007 0.003 -0.005 0.041 0.012 0.017 0.072 0.015 0.024

OTM GL-UJ 0.006 0.006 0.000 0.028 0.016 0.015 0.045 0.021 0.020DO-UJ 0.005 0.006 0.000 0.026 0.018 0.019 0.043 0.019 0.026Merton 0.008 0.000 0.010 0.054 0.020 0.002 0.069 0.010 0.014

ATM GL-UJ 0.006 0.005 0.007 0.039 0.018 0.011 0.021 0.017 0.027DO-UJ 0.006 0.004 0.010 0.036 0.017 0.010 0.028 0.009 0.027Merton 0.008 0.001 0.004 0.032 0.008 0.008 0.051 0.007 0.015

ITM GL-UJ 0.006 0.002 0.000 0.019 0.013 0.007 0.006 0.005 0.019DO-UJ 0.006 0.002 0.002 0.023 0.010 0.008 0.021 0.002 0.020Merton 0.012 -0.009 -0.007 0.017 -0.033 -0.015 0.025 -0.067 0.010

DITM GL-UJ -0.001 -0.011 -0.003 -0.006 -0.027 -0.012 -0.058 -0.067 0.009DO-UJ 0.003 -0.008 -0.004 0.017 -0.024 -0.015 -0.011 -0.062 0.000

Table 8.3: The mean dollar hedging errors (MDHE) for the VIX call options implied by using different hedging frameworks,over the moneyness (Mness) and expiration (τ) cross-sections. A particular option hedging error is weighted by its minimaltrading volume between the one recorded at the portfolio setup and its termination. The P -process parameters θP areestimated with the maximum-likelihood (ML) method using the underlying VIX sample spanning the February 24, 2006 upto April 30, 2010 period. The market volatility and jump size risk premiums are jointly implied from all VIX derivatives(options and futures) prices listed at the portfolio setup date.

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MDHE 1-day (∆t) revision 5-day revision 10-day revisionMness Model τ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Diffusion based: delta hedging with one VIX futuresWhaley 0.004 0.003 0.008 0.029 0.014 0.018 0.052 0.009 0.012

DOTM GL 0.001 0.001 0.002 0.009 0.006 -0.002 0.008 -0.001 0.000DO 0.001 0.000 0.003 0.013 0.006 -0.004 0.016 -0.003 -0.007

Whaley 0.011 0.006 -0.002 0.052 0.025 0.019 0.096 0.035 0.026OTM GL 0.001 0.007 -0.002 0.020 0.021 0.030 0.019 0.016 0.038

DO 0.001 0.006 -0.001 0.021 0.020 0.025 0.027 0.018 0.030Whaley 0.015 0.005 0.012 0.074 0.028 -0.007 0.125 0.040 0.028

ATM GL 0.003 0.001 0.008 0.034 0.021 0.009 0.013 0.006 0.033DO 0.003 0.001 0.008 0.032 0.022 0.013 0.023 0.015 0.031

Whaley 0.016 0.002 0.003 0.078 0.024 -0.001 0.143 0.036 0.017ITM GL 0.006 0.003 0.005 0.036 0.020 0.040 0.037 0.014 0.057

DO 0.006 0.002 0.006 0.035 0.017 0.036 0.045 0.020 0.056Whaley 0.015 -0.010 -0.006 0.075 0.007 -0.024 0.128 0.018 0.015

DITM GL 0.009 -0.005 0.000 0.054 0.004 -0.005 0.060 -0.005 0.026DO 0.008 -0.004 0.000 0.047 0.005 -0.005 0.056 0.004 0.037

Jump-diffusion based: delta hedging with one VIX futuresMerton 0.007 0.001 0.007 0.043 0.013 0.011 0.091 0.021 0.006

DOTM GL-UJ 0.005 0.002 0.001 0.025 0.010 -0.011 0.057 0.013 0.010DO-UJ 0.004 0.002 0.001 0.023 0.008 -0.008 0.047 0.008 0.002Merton 0.010 0.004 0.003 0.057 0.011 0.006 0.109 0.029 -0.005

OTM GL-UJ 0.004 0.005 0.001 0.020 0.009 0.006 0.039 0.004 0.024DO-UJ 0.005 0.005 0.002 0.022 0.012 0.005 0.040 0.004 0.007Merton 0.010 0.001 0.012 0.070 0.028 0.015 0.097 0.025 -0.017

ATM GL-UJ 0.003 0.001 0.003 0.022 0.002 -0.003 0.010 -0.011 0.020DO-UJ 0.005 0.001 0.004 0.026 0.007 -0.004 0.024 -0.005 0.002Merton 0.007 0.003 0.005 0.038 0.012 -0.011 0.059 0.015 -0.018

ITM GL-UJ 0.001 -0.002 -0.003 0.006 -0.001 -0.016 -0.015 -0.015 -0.005DO-UJ 0.003 0.000 -0.001 0.016 0.005 -0.008 0.012 -0.003 0.003Merton 0.006 -0.017 -0.021 0.000 -0.021 -0.014 -0.003 -0.053 0.016

DITM GL-UJ -0.004 -0.008 -0.018 -0.003 -0.016 -0.022 -0.049 -0.033 -0.029DO-UJ 0.001 -0.006 -0.005 0.012 -0.007 -0.022 -0.013 -0.016 -0.009

Jump-diffusion based: delta-jump hedging with two VIX futuresMerton 0.007 0.002 0.005 0.044 0.015 0.014 0.091 0.023 0.018

DOTM GL-UJ 0.001 0.003 0.003 0.005 0.015 -0.009 0.013 0.027 0.017DO-UJ 0.003 0.002 0.002 0.019 0.010 -0.005 0.039 0.011 0.012Merton 0.010 0.005 0.000 0.060 0.015 0.022 0.115 0.033 0.029

OTM GL-UJ -0.006 0.007 -0.003 -0.032 0.015 0.008 -0.066 0.013 0.036DO-UJ 0.003 0.006 0.002 0.014 0.013 0.013 0.023 0.007 0.032Merton 0.010 0.001 0.013 0.073 0.024 0.009 0.106 0.021 0.014

ATM GL-UJ -0.012 0.013 -0.008 -0.045 0.018 0.006 -0.168 0.022 0.059DO-UJ 0.002 0.005 0.001 0.016 0.011 0.001 -0.005 0.002 0.030Merton 0.008 0.002 0.000 0.043 0.010 -0.002 0.069 0.015 0.002

ITM GL-UJ -0.017 -0.003 -0.005 -0.082 0.032 0.004 -0.166 -0.007 0.017DO-UJ 0.000 0.000 -0.002 0.003 0.013 0.005 -0.009 0.000 0.025Merton 0.010 -0.011 -0.013 0.014 -0.025 -0.007 0.017 -0.057 0.017

DITM GL-UJ -0.036 -0.017 -0.012 -0.092 -0.001 -0.003 -0.294 -0.037 0.010DO-UJ -0.003 -0.008 -0.003 -0.002 -0.003 -0.012 -0.049 -0.017 0.022

Table 8.4: (Cont. of Table 8.3) The P -process parameters θP are estimated using the maximum-likelihood method over thebacked 252-day rolling VIX series starting on February 24, 2006.

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1-day (∆t) revision 5-day revision 10-day revisionτ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Model MAHEB: VX vs. VIX

Merton 0.749 0.922 1.013 1.345 1.779 1.973 1.807 2.310 2.521GL-UJ 0.389 0.297 0.316 0.825 0.728 0.641 1.319 1.086 0.902DO-UJ 0.437 0.258 0.290 0.846 0.644 0.594 1.277 0.985 0.849

C: VX vs. SPX optionsMerton 0.866 1.016 1.105 2.450 2.824 3.048 4.595 4.879 5.029GL-UJ 0.467 0.306 0.317 1.321 0.779 0.645 2.254 1.203 0.927DO-UJ 0.537 0.277 0.289 1.510 0.782 0.611 2.711 1.394 0.949

D: VX vs. SPX options and SPXMerton 0.937 1.130 1.234 2.534 2.989 3.253 4.645 5.010 5.200GL-UJ 0.468 0.291 0.312 1.293 0.759 0.638 2.218 1.184 0.921DO-UJ 0.555 0.269 0.283 1.505 0.763 0.599 2.690 1.377 0.939

MDHEB: VX vs. VIX

Merton -0.010 -0.088 -0.118 0.044 -0.033 0.004 0.067 -0.098 0.002GL-UJ -0.005 -0.028 0.002 0.037 -0.074 -0.027 0.058 -0.131 -0.027DO-UJ 0.016 -0.012 0.009 0.073 -0.044 -0.014 0.108 -0.092 -0.012

C: VX vs. SPX optionsMerton -0.077 -0.212 -0.280 -0.770 -1.034 -1.161 -0.742 -1.124 -0.915GL-UJ -0.044 -0.044 -0.004 -0.384 -0.199 -0.064 -0.486 -0.274 -0.067DO-UJ -0.032 -0.036 0.000 -0.440 -0.236 -0.075 -0.454 -0.230 -0.052

D: VX vs. SPX options and SPXMerton -0.070 -0.212 -0.288 -0.749 -1.017 -1.154 -0.714 -1.097 -0.882GL-UJ -0.041 -0.044 -0.005 -0.375 -0.198 -0.065 -0.476 -0.275 -0.068DO-UJ -0.026 -0.034 0.001 -0.425 -0.231 -0.074 -0.436 -0.225 -0.051

Table 8.5: The VIX futures mean absolute/dollar hedging errors (MAHE/MDHE) weighted by the minimal futures tradingvolume, between the one observed at the portfolio setup and its liquidation. The P -process parameters estimated over theVIX sample spanning January 2, 1990 – April 30, 2010 (5,125 trading days) period. The Q-measure parameters are impliedfrom all traded VIX derivative prices, under the assumption that both the volatility (diffusive) and jump risks are pricedby the market. In the ‘VX vs. VIX’ hedging framework, the VIX futures contract is hedged with a position in the VIXasset. The ‘VX vs. SPX’ framework considers the VIX futures position hedged with the VIX replicating S&P 500 optionsportfolio. The derivative sample spans the period from February 24, 2006 up to April 30, 2010.

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1-day (∆t) revision 5-day revision 10-day revisionτ < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120 < 60 [60, 120) ≥ 120

Model MAHEB: VX vs. VIX

Merton 0.747 0.908 0.990 1.341 1.754 1.926 1.801 2.283 2.477GL-UJ 0.510 0.396 0.337 1.242 0.897 0.685 1.901 1.282 0.955DO-UJ 0.430 0.368 0.333 1.059 0.852 0.678 1.702 1.240 0.947

C: VX vs. SPX optionsMerton 0.863 1.000 1.079 2.447 2.803 3.004 4.576 4.820 4.939GL-UJ 0.524 0.398 0.337 1.311 0.897 0.685 1.999 1.288 0.955DO-UJ 0.466 0.370 0.332 1.210 0.861 0.678 1.952 1.266 0.953

D: VX vs. SPX options and SPXMerton 0.932 1.111 1.204 2.530 2.963 3.200 4.625 4.950 5.107GL-UJ 0.502 0.396 0.337 1.276 0.895 0.685 1.970 1.286 0.955DO-UJ 0.453 0.365 0.331 1.166 0.852 0.676 1.905 1.259 0.951

MDHEB: VX vs. VIX

Merton -0.009 -0.085 -0.113 0.049 -0.022 0.026 0.074 -0.080 0.038GL-UJ -0.002 -0.010 0.008 -0.012 -0.057 -0.019 -0.014 -0.111 -0.015DO-UJ -0.001 -0.013 0.007 -0.029 -0.070 -0.021 -0.064 -0.129 -0.018

C: VX vs. SPX optionsMerton -0.075 -0.206 -0.267 -0.754 -0.990 -1.088 -0.731 -1.080 -0.851GL-UJ -0.018 -0.012 0.008 -0.154 -0.066 -0.021 -0.207 -0.117 -0.015DO-UJ -0.032 -0.019 0.006 -0.304 -0.110 -0.029 -0.442 -0.169 -0.024

D: VX vs. SPX options and SPXMerton -0.069 -0.205 -0.275 -0.732 -0.970 -1.075 -0.701 -1.047 -0.810GL-UJ -0.016 -0.012 0.008 -0.152 -0.066 -0.021 -0.209 -0.118 -0.016DO-UJ -0.031 -0.020 0.006 -0.303 -0.110 -0.030 -0.446 -0.170 -0.025

Table 8.6: (Cont. of Table 8.5) The P -process parameters θP are estimated using the maximum-likelihood method over thebacked 252-day rolling VIX series starting on February 24, 2006.

Chapter 9

Summary and Conclusions

On September 22, 2003, the Chicago Board of Option Exchange (CBOE) disseminated theS&P 500 volatility index VIX, which replaced its ‘old’ version from 1993. The ‘new’ VIXreflects the expected 30-day market volatility, calculated using a kernel-smooth estimator,which takes the current S&P 500 market prices for all out-of-the-money calls and putsfor the front month and second month expirations. The VIX serves as a forward lookingbenchmark of the expected short-term market volatility, and it is often considered as theinvestor ‘fear gauge’. Shortly after the amended VIX version was introduced, the CBOEdisseminated the first volatility derivatives: the VIX futures in March 2004 and VIX optioncontracts in February 2006. Since its inception, the VIX derivative market grew rapidlywith the combined trading activity in VIX options and futures of more than 100,000contracts per day. Primarily designed to provide an instrument allowing the hedging ofthe market volatility risk, it became an important part of the portfolio revenue.

The first pricing models were proposed before these contracts entered the market andreflected rather academic interests in these derivative types. Introduction of the VIXcontracts provided an empirical evidence, necessary to investigate the existing pricingframeworks and gave an impulse for further development of the more efficient and precisemodels.

The VIX derivative pricing methodology develops in two directions. First, the struc-tural approach, where VIX derivative pricing relies on the dynamics of the stock marketrepresented by the S&P 500 index. The spot VIX itself is derived from the index dy-namics, and such an outcome is used as the underlying for the VIX derivative pricing.The alternative reduced-form approach recognizes the current VIX level as the ‘sufficientunderlying’ for its derivatives, and relies on modeling its dynamics directly without in-corporating the S&P 500 information. In the study, we follow the reduced-form pricingframework. The main advantage of the first approach is its completeness, as it links the

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VIX derivative segment with the overall stock market. However, excluding completely theempirical VIX input and replacing it with the S&P 500 model-dependent outcome doesnot seem to be the most efficient strategy. Developing the VIX derivative pricing approachon such an underlying implies another level of complexity and yields a risk of the modelmisspecification. The resulting models remain often difficult to be handled numerically,which seriously undermines their practical utility. These problems are not commonly en-countered in the reduced-form approach, where the available derivative pricing solutionsare less complex than those emerging from the structural approach, and often available inthe closed forms. This usually simplifies the numerical procedures involved in the param-eter estimation process, which has a great value in the empirical applications. The maincritique of the reduced approach concerns the fact, that it tends to ignore the S&P 500dynamics, which remains strongly correlated with the VIX. This relation, despite beingan important aspect of derivative market modeling, does not preclude from relying onlyon the VIX as a ‘sufficient underlying’ statistic. The VIX is a well-recognized marketvolatility measure, and its calculation formula derives from the information included inthe S&P 500 option market prices. In addition, a direct modeling of the VIX dynamicsdiminishes the risk of its misspecification, as it might happen in the structural approach.Given such a reasoning, the reduced-form methodology can be considered as a reliableframework for the VIX derivative pricing and hedging strategies assessment.

The main body of the thesis consists of nine Chapters, which provide the literaturereview, description of the theoretical models applied in the analysis and their empiricalperformance. In particular, we focus on pricing and hedging outcomes from the modelsbased on one-dimensional affine diffusions. Our derivative sample consists of the VIXderivative daily trading records observed between February 24, 2006 and April 30, 2010(1,052 trading days). The model parameters under the physical measure are estimated us-ing the conditional maximum likelihood over the end-of-day VIX historical series recordedbetween January 2, 1990 and April 30, 2010 (5,125 trading days). Alternatively, these pa-rameters are estimated over the 252-day VIX subseries, to reflect scenarios when investorsin their decisions rely on local, rather than global VIX characteristics. Given the physi-cal process estimates, the risk-neutral distribution parameters are implied from the VIXderivative data, by minimizing the mean-squared relative distance between the theoreticaland empirical contract prices.

The first three Chapters cover the literature review, theoretical models, and the datasample descriptions, respectively. The subsequent Chapters focus on the model pricingand hedging assessment aspects.

In Chapter 5, we focus on the pricing performance of the three baseline models: theWhaley [1993]) (Whaley), Grünbichler and Longstaff [1996] (GL), and Detemple and Os-

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akwe [2000] (DO) models. In particular, we consider the role and influence of the availablemarket information on the model performance. It turns out, that without changing themodel assumptions we can influence/improve the pricing results by combining differentmarket information sets. This is an important outcome with respect to the common prac-tice, where the choice of data used in the model estimation has an arbitrary character. Wefound that using the joint derivative information in the risk-neutral distribution param-eters calibration is superior to the particular derivative load, with respect to the in- andout-of-sample model performance. In addition, within the same information set the mod-els with the mean-reverting factor (GL, DO) outperform the one based on the GeometricBrownian Motion (GBM) VIX dynamics (Whaley). Out of these two best specifications,the DO model usually outperforms the GL. Within all these setups, the derivative record-ings are weighted by their daily relative trading volumes. Such a practice discriminates themore representative contract contribution in the aggregated pricing outcomes and remainsa standard practice for the whole thesis.

In Chapter 6, we continue with the baseline model assessment and concentrate on thedelta-hedging performance. The VIX calls are hedged with the VIX futures position, bothexpiring on the same date. The futures are hedged using the underlying VIX position, orwith the portfolio of the S&P 500 options used in the spot VIX calculation. The empiricalhedging outcomes for the call options do not confirm the model hierarchy established inChapter 5. The DO model gets a credit in the portfolios, which include the short-maturingor the deep-of-the-money (DOTM) calls. This is an important outcome from the economicpoint of view, as these contracts accumulated around 95% of the total VIX call optiontrading volume, and 89% of all their transactions value, over the analyzed period. Theremaining mid- and long-term sections (outside DOTM group) are mainly dominated bythe Whaley specification, regardless the hedging horizon length. The role of both models iscomplementary and excludes in some sense the GL model from its application in the VIXcall options hedging. The situation changes for the futures contracts, where the mean-reverting models provide a better performance over the Whaley specification, and remaincomplementary to each other. Interestingly, the P -process model estimates reflecting thelocal VIX characteristics over the 252-day series, are more relevant for hedging the VIXfutures than the dynamics over the VIX whole historical sample. The analysis based on theempirical data, is complemented with the assessment carried out on the model-simulateddata scenarios.

The empirical studies recognize jumps as one of the important characteristics of theVIX. Following this, we extend the baseline dynamics with a jump component and checkhow it influences the model performance for the VIX derivative contracts. The pricingassessment is provided in Chapter 7. One of the more apparent outcomes concerns the

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short-term DOTM VIX calls pricing improvement. For these contracts, the best perform-ing DO with the upward jump extension (DO-UJ) model provides 42% improvement inthe mean-squared relative error (MSRE) terms over its DO counterpart, within the 1-dayforecast horizon. A similar rate applies for the less intensively traded but more expen-sive deep-in-the-money (DITM) contracts. Using the estimates, which reflect the localVIX characteristics, yields inferior results with respect to the ‘whole sample’ based MSREoutcomes. This is also confirmed by higher errors in the dollar terms.

The hedging assessment of the jump extended models is provided in Chapter 8. Thereare two sources of the risk to be hedged: the diffusion (volatility) and jump related risks.The latter splits into the risk associated with the jump size and its timing. We start witha similar as in Chapter 6 setup, where we hedge the VIX options using the position in theVIX futures with (assumed) non-systematic jump risk: delta hedging. In such a frame-work, the jump-extension translates into a better call and futures hedging performance.Within the first derivative group, the mean-reverting DO-UJ model provides the best per-formance within majority of the cross-sections, over the best diffusion based DO or Whaleyoutcomes. However, the differences are not that large and the magnitude of the improve-ment stays below 10% in the absolute errors terms. For the futures, both mean-revertingjump-extended models share the leading roles, with the best short-maturity performanceaccounted by the DO-UJ specification, given the 252-day based P -process estimates. Inthe next step, the assumption about the zero jump risk is relaxed, and as suggested in theliterature, one more futures contract is introduced to the portfolio to hedge against thejump risk: the delta-jump hedging. This extension pays off within the portfolios includingthe short-maturing calls and rebalanced within the mid- (5-day) and long- (10-day) termhorizons. The improvements concern mainly the DO-UJ model performance, and theirmagnitude stays between 2 and 5% of the baseline mean absolute hedging error (MAHE)value.

The thesis results provide some important implications for the assessment of the modelsused for pricing and hedging in the VIX derivative market. It matters for the model pricingand hedging performance which derivative and underlying information is used in the modelestimation process. We can influence/improve model’s abilities in- and out-of-sample,using the information implied jointly from different derivative types, instead of only one(e.g., calls), which happens to be a common practice in the empirical studies. Then,against the common belief, the more rich and empirically relevant parametric structuredoes not always improve the model performance. Our results confirm that the modelsincluding the mean-reverting element, crucial for modeling the market volatility, does notseem to be always relevant in hedging the VIX call contracts. The model of Whaley,based on the GBM assumption of the underlying dynamics, works almost equally well

149

as the DO model in hedging one of the biggest segments of the VIX market: the short-maturing DOTM call options. On the other side, the jump extension, which increases themodel complexity, is not always the best strategy in hedging the VIX futures contracts,and can be efficiently replaced by the ‘simpler’ diffusion based models. The pursuit ofmore empirically relevant models pays-off is a more precise pricing. However, insteadof developing new models, pricing improvements may also be acquired by combining theavailable market information. Hedging is complementary to the model pricing performanceassessment criterion. Here, the outcomes do not improve much when the underlyingdynamics is extended by additional empirically sound factors, and seems to imply thebelief that the simpler model does not necessarily mean the worse model.

The work on the VIX derivative market in not complete yet. The intensive developmentof the new pricing models, together with increasing empirical evidence from the VIXderivative market, makes it one of the most promising theoretical and empirical researchvenues to be followed in the future. The outcomes of our study provide a small steptowards better understanding the possible pricing and hedging implications of the one-dimensional diffusion based pricing frameworks, when different market information andmodel assumptions are considered.

150

Appendix A

Appendix

A.1 Bakshi-Madan pricing approach

After Bakshi and Madan [2000], let Xt be a function of Vt under the physical measure Pis described by a one-dimensional Markov process

dXt = a(Xt, θ)dt+ b(Xt, θ)dW Pt + dZt, (A.1)

where a and b represent drift and volatility components, W PT is a standard Brownian

motion, Zt is a jump component. The European call price given strike K and time toexpiration τ can be derived from

C(Xt, K, τ) = G(Xt, τ)Π1(τ)−KB(Xt, τ)Π2(τ), (A.2)

where B(.) is a time-t price of discount bond with τ periods to expiration, G(.) is a time-tprice of a commitment to deliver at t+ τ the quantity Xt+τ (scaled-forward price). Bothof these obey

B(Xt, τ) = f(Xt, τ, φ = 0) (A.3)

G(Xt, τ) = 1i× ∂f(Xt, τ, φ)

∂φ

∣∣∣∣φ=0

, (A.4)

where i2 = −1 and f denotes time-scaled conditional characteristic function under therisk-neutral Q-measure,

f(Xt, τ, φ) = e−rτϕ(Xt, τ, φ) (A.5)ϕ(Xt, τ, φ) = EQ

t (eiφXt+τ ). (A.6)

The ϕ(.) satisfies the Fokker-Planck backward equation (suppressed notation)

0 = ∂ϕ

∂Xt

at(Xt) + 12∂2ϕ

∂X2t

b2t (Xt)−

∂ϕ

∂τ+ γE[ϕ(Xt + J)− ϕ(Xt)], (A.7)

151

152

subject to the boundary condition ϕt(τ = 0, φ) = eiφXt . The probability Πj(τ), j = 1, 2in (A.2) is determined by

Πj(τ) = 12 + 1

π

∫ ∞0

Re[e−iφK × fj(Xt, τ, φ)

iφ

], (A.8)

where Re denotes the real part of complex number and

f1(τ, φ) = 1iGt(τ) ×

∂f(Xt, τ, φ)∂φ

, (A.9)

f2(τ, φ) = 1Bt(τ) × f(Xt, τ, φ). (A.10)

It is understood that the conditional characteristic function ϕ under dynamics (A.1) isavailable in closed form. Duffie et al. [2000] and Singleton [2001] prove that for affinediffusion or jump-diffusion processes, under some regularity conditions,

ϕ(Xt, τ, φ, θ) = exp{α(τ, φ, θ) + β(τ, φ, θ)Xt}, (A.11)

where α(.) and β(.) satisfy complex-valued Ricatti ordinary differential equations withboundary conditions α(0, φ) = 0 and β(0, φ) = iφ. Derivation of these functions is a crucialpoint of the pricing framework. For some affine process specifications (A.1) these solutionsexist in closed forms. In other cases the solution would have to be found numerically.

A.2 Characteristic function for the mean-revertingmodels with jumps

GL-UJ

The characteristics function (A.11) for Xt ≡ Vt for

ϕ(Vt, τ, φ, θ) = exp{α(τ, φ, θ) + β(τ, φ, θ)Vt}, (A.12)

where

α(τ, φ, θ) = γ

κ− 12η+σ2 ln

(κ− 1

2iφσ2 + iφ(1

2σ2 − κ/η+)e−κτ

κ− iφκ/η+

)

− 2mσ2 ln

(2κ− iφσ2(1− e−κτ )

2κ

),

β(τ, φ, θ) = 2iφκe−κτ2κ− iφσ2(1− e−κτ ) .

For derivations, see (Bakshi and Cao [2002]).

153

DO-UJ

The characteristics function (A.11) for Xt ≡ ln Vt for

ϕ(Vt, τ, φ, θ) = exp{α(τ, φ, θ) + β(τ, φ, θ) ln(Vt)}, (A.13)

where

α(τ, φ, θ) = iφm

κ(1− e−κτ )− φ2σ2 1− e−2κτ

4κ + γ

κln(η+ − iφe−κτ

η+ − iφ

),

β(τ, φ, θ) = iφe−κτ .

For derivations, see Jaimungal and Surkov [2008].

A.3 VIX calculation methodology

The spot value of the VIX is being calculated by the CBOE on an intra-day minute basis,and reflects the minute-by-minute market view on the forward 30-day market volatilityimplied by the S&P 500 European call and put option prices. In practice, the VIX iscalculated as the square root of the expected annualized 30-day S&P 500 rate of returnvariance, understood as the sum of squared standard deviations of the S&P 500 rate ofreturns at every point in time during the 30 days period. For any period, the value ofthe index rate-of-return variance can be approximated by the forward price of a strip ofall available out-of-the-money (OTM) S&P 500 index options expiring within this period,which represents the market risk-neutral expectation of that variance. The general formulafor the calculation of the spot VIX is

σ2t (τ) = 2erτ

τ

It∑i=1

∆Ki

K2i

OSt,i(τ)− 1

τ

[FtKmint

− 1]2

, (A.14)

where:

- τ : the option common time to expiration,

- Ft : the time-t S&P 500 forward index level, derived from the following put-callparity:

Ft = Kmint + erτ [Ct(Kmin

t , τ)− Pt(Kmint , τ)], (A.15)

where Kmint is the strike price for which the differnce between the corresponding call

and out prices (bid-ask average) is minimal (Kmint ≤ Ft),

154

- Ki : the ith OTM option strike price: a call if Ki > Ft and put Ki < Ft

- ∆Ki : the half-distance between the (ordered ascendantly) adjacent strikes,

∆Ki = (Ki+1 −Ki−1)/2,

which is modified at the extremes in the following ways

- for the lowest strike: the difference between the lowest strike and the nexthigher strike

- for the highest strike is the difference between the highest strike and the nextlower strike,

- OSt,i(τ) : the price (bid-ask average) for each options with strike Ki and strictly

positive bids; at Kmint the price is the average between the in-the-money (ITM)

(recall, that Kmint ≤ Ft) call and the OTM put.

The option strips whose prices are used to calculate VIX from formula A.14 are portfoliosof the OTM S&P 500 calls and puts, however, one ITM S&P 500 call at Kmin

t is used inthe index calculation. Hence, the second term in the formula (A.14) represents so calledadjustment term, which converts this call into the relevant OTM put. If the S&P 500option strip at time t expires exactly within 30 days, then the spot value of the VIX indecimal terms is calculated from the annualized formula (A.14),

V IXt =√

36530 τσ

2t (τ), (A.16)

where τ = 30. However, in practice, we are not always able to find options expiring exactlywithin 30 days and some approximation needs to be applied. Therefore, we select two stripsof calls and puts with near and next to 30-day expiration, with the corresponding numberof minutes to expiration n1 and n2, respectively. If the number of days to expiration isless than nine then, the second or third near- and next-term group is considered to avoidmicrostructure effects. The spot VIX value in decimal terms is calculated by weighting thecorresponding values of the annualized formula (A.14). If the SPX option strip consistsof near- (τ1) and next-term (τ2) options, then the following interpolation is applied using(A.14),

V IXt =√

36530 [τ1wσ2

t (τ1) + τ2(1− w)σ2t (τ2)], (A.17)

where w = n2−30n2−n1

for n1 and n2 corresponding to the number of minutes to the the near-and next-term option expiration, respectively.

155

Given (A.16) and (A.17), the spot VIX is considered as a value of the following functions

V IXt ≡ V30(OSt,i(τ), Ft, Kmin

t , r), (A.18)V IXt ≡ V (OS

t,ij(τ1, τ2), Ft, Kmin

t,j , r), (A.19)

suppressed to V30(OSt , τ) and V (OS

t , τ1, τ2), respectively. The OSt denotes the price vector of

the S&P 500 calls and puts for a given range of strikes {Ki}Iti=1 and common expiration(s)at t + 30 or t + τj for j=1(near-term), 2(next-term), selected to calculate the spot VIXvalue (see (A.14)). For further details, see Carr and Lee [2007] or CBOE [2003].

A.4 Hedge ratio derivations: VIX options

In all derivations, the put option prices featured by strike K and time to expiration τ arederived from the put-call parity. For a non-tradable VIX asset with spot value Vt, we useformula (3.7)

P (Vt, K, τ) = C(Vt, K, τ)− e−rτ [F (Vt, τ)−K]. (A.20)

Hence, all the derivations start first with the call and futures contract and then the cor-responding put option price follows from (A.20) formula.

Whaley

The pricing framework is provided in section 3.2.1. Using the closed form pricing solutionsfor a European call option (3.11) and futures contract (3.10), we derive the delta hedgeratio formulas for these contracts with respect to the underlying volatility index VIX (Vt),

∂CWhaley(Vt, K, τ)∂Vt

= eτ(µ−σλ−r)N (d) , (A.21)

∂FWhaley(Vt, τ)∂Vt

= eτ(µ−σλ), (A.22)

where d = ln(Vt/K)+(µ−σλ+σ22 )τ

σ√τ

and N(.) is standard normal cumulative distribution func-tion.The hedge ratio for a European put option is determined from relation (A.20), using theformula for futures contract expiring at the same date as the option. Then, taking thepartial derivative with respect to the underlying spot value Vt yields

∂PWhaley(Vt, K, τ)∂Vt

= eτ(µ−σλ−r)[N(d)− 1]. (A.23)

156

Grünbichler-Longstaff (GL)

The pricing framework is provided in section 3.2.2. Since there is no explicit formulato calculate the hedge call and put ratio, we determine it numerically. Approximationof the partial derivative from a European call closed pricing formula (Psychoyios andSkiadopoulos [2006]):

limh→0

CGL(Vt + h,K, τ)− CGL(Vt − h,K, τ)2h = ∂CGL(Vt, τ)

∂Vt. (A.24)

For puts the same approach applies. The futures hedge ratio follows from the explicitfutures pricing formula,

∂FGL(Vt, τ)∂Vt

= e−τ(κ+σλ). (A.25)

Detemple-Osakwe (DO)

The pricing framework is provided in section 3.2.3. A European call option delta hedgeratio is provided in closed-form (Detemple and Osakwe [2000]),

∂CDO(Vt, τ)∂Vt

= e−rτφV φ−1t e

m−σλκ

(1−φ)+ 12a

2N(d+ a), (A.26)

where N(.) is standard normal CDF and φ = e−κτ , a = σ [(1− φ2)/2κ]1/2, and d =1a

[φ ln(Vt)− ln(K) + m−σλ

κ(1− φ)

]. We derive the delta hedge formula for a futures con-

tract (see also Psychoyios et al. [2003]),

∂FDO(Vt, τ)∂Vt

= e−κτ

Vtexp

[e−κτ ln Vt + m− σλ

κ(1− e−κτ ) + σ2

4κ(1− e−2κτ )].

(A.27)

Given the closed form hedge ratio formulas for a European call option and futures contract,(A.26) and (A.27), respectively, the hedge ratio for a European put option is derived fromrelation (A.20),

∂PDO(Vt, K, τ)∂Vt

= ∂CDO(Vt, τ)∂Vt

− e−rτ ∂FDO(Vt, τ)∂Vt

. (A.28)

A.5 Hedge rates derivations: VIX futures

A.5.1 Hedging with the S&P 500 options

The hedging portfolio corresponds to (6.27). The partial derivative (6.29) can be derivedanalytically using the chain rule and a model explicit formulas for the VIX futures price

157

and the spot value of the VIX. In particular,

∂Fmodel(V (OSt ), τ)

∂OSt,i

= ∂Fmodel(V (OSt , τ), τ)

∂V (OSt , τ) × ∂V (OS

t , τ)∂OS

t,i

, (A.29)

where the first right-hand side derivative is specified in (A.22), (A.25), and (A.27). Thesecond derivative can be derived from (A.19) with respect to the ith SPX option priceOSt,i, following the formula (A.16) or (A.17).

If all the hedging options expire within 30 days, formula (A.16) applies,

∂V30(OSt , τ)

∂OSt,i

= 36530

erτ∆Ki

V30(OSt , τ)K2

i

, (A.30)

where for the Ki = Kmint case, OS

t,i = (CSt,i + P S

t,i)/2.Otherwise, the formula (A.17) applies,

∂V (OSt , τ1, τ2)

∂OSt,ij

= 36530

werτ1∆Kij

V (OSt , τ1, τ2)K2

ij

, (A.31)

where for the near- (j=1) and near-term (j=2) expiration group, the OSt,ij

= (CSt,ij

+P St,ij

)/2given Kij = Kmin

t,j .

A.5.2 Hedging with the S&P 500 options and the S&P 500 asset

The hedging portfolio corresponds to (6.34). In this framework, the S&P 500 index forwardvalue Ft is replaced with the spot index position St, using the following put-call parityrelation (compare with (A.15)):

Ct(Kt, τ)− Pt(Kt, τ) = St − e−rτKt, (A.32)

where Kt is the first strike below the Ft = erτSt, with the corresponding call (C) and put(P ) contract prices. In such a framework, the VIX formulas (A.18) and (A.19) change to:

V IXt = V30(OSt,i(τ), St, Kt, r), (A.33)

V IXt = V (OSt,i(τ1, τ2), St, Kt, r), (A.34)

abbreviated to V30(OSt , St, τ) and V (OS

t , St, τ1, τ2), respectively. Similar as in section A.5.1,the partial derivative (6.36) and (6.37) can be derived analytically using the chain ruleand a model explicit formulas for the VIX futures price and the spot value of the VIX. Inparticular,

∂Fmodel(V (OSt , St), τ)

∂OSt,i

= ∂Fmodel(V (OSt , τ), τ)

∂V (OSt , St, τ) × ∂V (OS

t , St, τ)∂OS

t,i

,

∂Fmodel(V (OSt , St), τ)

∂St= ∂Fmodel(V (OS

t , τ), τ)∂V (OS

t , St, τ) × ∂V (OSt , St, τ)∂St

, .

158

For the first hedge ratio, the methodology similar to (A.30) – (A.31) applies, given thenew threshold strike values Kt or Kt,j, respectively. For the second hedge ratio, if all thehedging options expire within 30 days, formula (A.16) applies,

∂V30(OSt , St, τ)∂St

= −36530

erτ (erτSt −Ki)V30(OS

t , St, τ)K2i

, (A.35)

where Ki = Kt. Otherwise, if the formula (A.17) applies,

∂V (OSt , St, τ1, τ2)∂St

= −36530

[werτ1(erτ1St −Ki,1)V (Ot, St, τ1)K2

i,1+ (1− w)e

rτ2(erτ2St −Ki,2)V (OS

t , St, τ2)K2i,2

], (A.36)

whereKij corresponds to the first below Ft = erτSt strike in the near- (j=1) and next-term(j=2) S&P 500 option strips, respectively.

A.6 Delta hedging ratios: jump-diffusion models

Merton

The call delta hedge ratio can be derived from the pricing formula (3.26) using appropriatesolution for the Black-Scholes model (3.27),

∂C(V )∂V

=∑j≥0

e−γQτ (γQτ)jj! e(bj−r)τN(dj), (A.37)

where bj = µQ − γQkQ + jµQJ +σ2

J/2τ

, σj =√σ2 + j

σ2J

τ, and dj = ln(Vt/K)+(bj+σ2

J/2)τσj√τ

. For thefutures pricing formula (3.29)

∂F (V )∂V

= eµQτ . (A.38)

GL-UJ

The delta call and futures contracts formulas are expressed in compact forms using thefutures (3.39) and probability (3.40) formulas

∂C(V )∂V

= e−(κQ+r)τF (Vt, τ)Π1(τ), (A.39)

∂F (V )∂V

= e−κQτF (Vt, τ). (A.40)

159

DO-UJ

Similar to the GL-UJ model, the call and futures delta ratios are provided in compactformulas using (3.41) and (3.42),

∂C(V )∂V

= e−(κQ+r)τ

VtF (Vt, τ)Π1(τ), (A.41)

∂F (V )∂V

= e−κQτ

VtF (Vt, τ). (A.42)

160

Bibliography

Y. Aït-Sahalia. Disentangling Diffusion from Jumps. Journal of Financial Economics, 55:487–528, 2004.

Y. Aït-Sahalia and R. Kimmel. Maximum Likelihood Estimation of Stochastic VolatilityModels. Journal of Financial Economics, 83:413–452, 2007.

C. Alexander and D. Korovilas. The Hazards of Volatility Diversification. ICMA CentreDiscussion Papers in Finance (DP2011-04), 2011.

M. Ammann and S. Süss. Valuation of Volatility Options. working paper, University ofSt. Gallen, 2008.

Y. An, A. Assaf, and J. Jang. Hedging Volatility Risk: The effectiveness of VolatilityOptions. International Journal of Theoretical and Applied Finance, 10/3:517–534, 2007.

L. Andersen and J. Andreasen. Jump-Diffusion Processes: Volatility Smile Fitting andNumerical Methods for Option Pricing. Review of Derivative Research, 4:231–262, 2000.

T. G. Andersen, T. Bollerslev, and F.X. Diebold. Parametric and Nonparametric VolatilityMeasurement. Handbook of Financial Econometrics., pages 67–138, 2010.

D.W.K. Andrews. Heteroskedasticity and Autocorrelation Consistent Covariance MatrixEstimation. Econometrica, 59/3:817–858, 1991.

M. Avellaneda, B. Robert, C. Friedman, N. Grandchamp, and L. Kruk adn J. Newman.Weighted Monte Carlo: A New Technique for Calibrating Asset-pricing Models. Inter-national Journal of Theoretical and Applied Finance, 4/1:91–119, 2000.

G. Bakshi and Ch. Cao. Risk-Neutral Kurtosis, Jumps, and Option Pricing: Evidencefrom 100 Most Actively Traded Firms on the CBOE. working paper, 2002.

G. Bakshi and D. Madan. Spanning and Derivative Security Valuation. Journal of Finan-cial Economics, 55:205–238, 2000.

161

162

G. Bakshi, Ch. Cao, and Z. Chen. Empirical Performance of Alternative Option PricingModels. Journal of Finance, 52/5:2003–2049, 1997.

G. Bakshi, N. Ju, and H. Ou-Yang. Estmation of Continuous-Time Models with anApplication to Equity Volatility Dynamics. Journal of Financial Economics, 82:227–249, 2006.

C. A. Ball and A. Roma. Stochastic Volatility Option Pricing. Journal of Financial andQuantitative Analysis, 29/4:589–607, 1994.

C. A. Ball and W. N. Torous. On Jumps in Common Stock Process and Their Impact onCall Option Pricing. Journal of Finance, 40/1:155–173, 1985.

P.S. Banerjee, J. S. Doran, and D. R. Peterson. Implied Volatility and Future PortfolioReturns. Journal of Banking and Finance, 31:3183–3199, 2007.

D. Bates. Pricing Options under Jump-Diffusion Processes. working paper, The WhartonSchool, University of Pennsylvania, 1988.

D. Bates. The Crash of ’87: Was It Expected? The Evidence from Options Markets.Journal of Finance, 46:1009–1044, 1991.

D. Bates. Testing Option Pricing Models. Statistical Methods in Finance (Ch. 20), pages567–611, 1995.

D. Bates. Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in GermanMark Options. The Review of Financial Studies, 9/1:181–238, 1996.

D. Bates. Post-’87 Crash Fears in the S&P 500 Futures Option Market. Journal ofEconometrics, 94:181–238, 2000.

D. Bates. Empirical Option Pricing: A Retrospection. Journal of Econometrics, 116:387–404, 2003.

M. Beinert and S. Trautmann. Jump-Diffusion Models of German Stock Returns. AStatistical Investigation. Statistical Papers, 32:269–280, 1991.

F. Black. The Pricing of Commodity Contracts. Journal of Financial Economics, 3:167–179, 1976a.

F. Black. Studies in Stock Price Volatility Changes. Proceedings of the 1976 BusinessMeeting of the Business and Economics Statistics Section, American Statistical Associ-ation, pages 177–181, 1976b.

163

F. Black and M. Scholes. The Pricing of Options and Corporate Liabilities. Journal ofPolitical Economy, 81:637–654, 1973.

N. P. B. Bollen. Derivatives and the Price of Risk. The Journal of Futures Markets, 17/7:839–854, 1997.

T. Bollerslev. Generalized Autoregressive Conditional Heteroscedasticity. Journal ofEconometrics, 31:307–327, 1986.

H.P. Boswijk. Volatility Mean Reversion and the Market Price of Volatility Risk. Proceed-ings of the International Conference on Modelling and Forecasting Financial Volatility,pages 159–170, 2001.

M. Bremer and T. Hiraki. Volume and Individual Security Returns on the Tokyo StockExchange. Pacific-Basin Finance Journal, 7 (3-4):351–370, 1999.

M. Brenner and D. Galai. New Financial Instrument for Hedging Changes in Volatility.Financial Analysts Journal, 4:61–65, 1989.

M. Brenner, E. Y. Ou, and J. E. Zhang. Hedging Volatility Risk. Journal of Banking &Finance, 30:811–821, 2006.

M. Broadie and A. Jain. Pricing and Hedging Volatility Derivatives. The Journal ofDerivatives, 2008.

M. Broadie, M. Chernov, and M. Johannes. Model Specification and Risk Premia: Evi-dence from Futures Options. Journal of Finance, LXII/3:1453–1488, 2007.

J. Y. Campbell, A. W. Lo, and A. C. Mac Kinley. The Econometrics of Financial Markets.Princeton University Press, 1997.

P. Carr and R. Lee. Pricing and Hedging VIX Options under Stochastic Volatility andCorrelated Jumps in S&P 500 Price and Variance Processes. Risk, 5, 2007.

P. Carr and R. Lee. Volatility Derivatives. Annual Review of Financial Economics, pages319–339, 2009.

P. Carr and D. Madan. Towards the Theory of Volatility Trading. In R. Jarrow (Ed.)Volatility New Estimation Techniques for Pricing Derivatives, pp.417-427. PrincetonUniversity Press, 1998.

P. Carr and L. Wu. Time-Changed Levy Processes and Option Pricing. Journal of Fi-nancial Economics, 71:113–141, 2004.

164

P. Carr and L. Wu. A Tale of Two Indices. The Journal of Derivatives, 13/3:13–29, 2006.

CBOE. The CBOE Volatility Index - VIX. The VIX White Paper, 2003.

G. Chacko and L. M. Viceira. Spectral GMM Estimation of the Continuous-time Processes.Journal of Econometrics, 116:259–292, 2003.

K. C. Chan, G. A. Karolyi, F. A. Longstaff, and A. B. Sanders. An Empirical Comparisonof Alternative Models of the Short-Term Interest Rate. Journal of Finance, 47:1209–1227, 1992.

J. Cheng, M. Ibraimi, M. Leippold, and J. E. Zhang. A remark on Lin and Chang’spaper ‘Consistent Modelling of S&P 500 and VIX Derivatives’. Journal of EconomicDynamics & Control, in press, 2012.

M. Chernov and E. Ghysels. Estimation of Stochastic Volatility Models for the Purposeof Option Pricing. Computational Finance, 1999.

M. Chernov and E. Ghysels. A Study Toward a Unified Approach to Joint Estimation ofObjective and Risk Neutral Measures for the Purpose of Option Valuation. Journal ofFinancial Economics, 56:407–458, 2000.

P. Christoffersen and K. Jacobs. The Importance of the Loss Function in Option Valuation.Journal of Financial Economics, 72:291–318, 2004a.

P. Christoffersen and K. Jacobs. Which GARCHModel for Option Valuation. ManagementScience, 50/9:1204–1221, 2004b.

P. Christoffersen, K. Jacobs, and K. Mimouni. Volatility Dynamics for Models for the S&P500 Dynamics: Evidence from Realized Volatility, Daily Returns, and Option Prices.Review of Financial Studies, forthcoming, 2011.

R. Cont and P. Tankov. Non-parametric Calibration of Jump-diffusion Option PricingModels. Journal of Computational Finance, 7:1–49, 2004.

Ch. J. Corrado and T. W. Miller. The Forecast Quality of CBOE Implied VolatilityIndexes. Journal of Future Markets, 25/4:339–373, 2005.

D. R. Cox and D. V. Hinkley. Theoretical Statistics. Chapman & Hall, 1974.

J. C. Cox and Jr. S. A. Ross. The Valuation of Options for Alternative Stochastic Pro-cesses. Journal of Financial Economics, 3:145–166, 1976.

165

J. C. Cox, J. E. Ingersoll, and Jr. S. A. Ross. Theory of the Term Structure of InterestRates. Econometrica, 53/2:1029–1054, 1985.

J.C. Cox. Notes on Option Pricing I: Constant Elasticity Variance Diffusions. workingpaper, Stanford University, 22, 1975.

Q. Dai and K. J. Singleton. Specification Analysis of Affine Term Structure Models.Journal of Finance, LV/5:1943–1978, 2000.

R. T. Daigler and L. Rossi. A Portfolio of Stocks and Volatility. The Journal of Investing,pages 99–106, 2006.

H. Daouk and J. Q. Guo. Switching Asymmetric GARCH and Options on a VolatilityIndex. The Journal of Future Markets, 24/3:251–282, 2004.

S. R. Das. The Surprise Element: Jumps in Interest Rates. Journal of Econometrics, 106:27–65, 2002.

K. Demeterfi, E. Derman, M. Kamal, and J. Zou. A Guide to Volatility and VarianceSwaps. Journal of Derivatives, 6/4:9–32, 1999.

G. Deskalakis, D. Psychoyios, and R. N. Markellos. Modeling CO2 Emission AllowancePrices and Derivatives: Evidence from the European Trading Scheme. Journal of Bank-ing and Finance, 33:1230–1241, 2009.

J. Detemple and C. Osakwe. The Valuation of Volatility Options. European FinanceReview, 4:21–50, 2000.

G. Donaldson and M. Kamstra. Volatility Forecasts, Trading Volume, and the ARCHversus Option-Implied Volatility Trade-off. working paper, Federal Reserve Bank ofAtlanta, 2004-06, 2004.

G. Dotsis, D. Psychoyios, and G. Skiadopoulos. An Empirical Comparison of Continuous-Time Models of Implied Volatility Indices. Journal of Banking and Finance, 31:3584–3603, 2007.

J.Ch. Duan and Ch. Y. Yeh. Jump and Volatility Risk Premiums Implied by VIX. Journalof Economic Dynamics and Control, 34:2232–2244, 2010.

D. Duffie, J. Pan, and K. Singleton. Transform Analysis and Asset Pricing for AffineJump-Diffusions. Econometrica, 68:1343–1376, 2000.

166

B. J. Dumas, J. Fleming, and R. Whaley. Implied Volatility Functions: Empirical Tests.Journal of Finance, 53:2059–2106, 1998.

D. Y. Dupont. Extracting Risk-neutral Probability Distributions from Option Prices UsingTrading Volume as a Filter. working paper, Institute for Advanced Studies, Vienna, 2001.

B. Dupoyet, R. T. Daigler, and Z. Chen. A Simplified Pricing Model for Volatility Futures.The Journal of Futures Markets, 31/4:307–339, 2011.

R. Engle and G. M. Gallo. A Multiple Indicators Model for Volatility Using Intra-DailyData. Journal of Econometrics, 131:3–27, 2006.

B. Eraker. MCMC Analysis of Diffusion Models With Application in Finance. Journal ofBusiness and Economic Statistics, 19:177–191, 2001.

B. Eraker, M. Johannes, and N. Polson. The Impact of Jumps in Volatility and Returns.Journal of Finance, 74:1269–1299, 2004.

W. Feller. Two Singular Diffusion Problems. Annals of Mathematic, 54:173–182, 1951.

S. Figlewski. Forecasting Volatility. Financial Markets, Institutions, and Instruments(monograph), 6/1, 1997.

J. Fleming, B. Ostdiek, and R. E. Whaley. Predicting Stock Market Volatility: A NewMeasure. The Journal of Future Markets, 15/3:265–302, 1995.

R. L. Fountain, J. R. Herman, and D. J. Rustvold. An Application of Kendall Distributionand Alternative Dependence Measures: SPX vs. VIX. Insurance Mathematics andEconomics, 42/2:469–472, 2008.

A. R. Gallant and G. Tauchen. Which Moments to Match. Econometric Theory, 12:657–681, 1996.

T. J. George and F. A. Longstaff. Bid-Ask Spread, Trading Activity in the S&P 100 IndexOption Market. Journal of Financial and Quantitative Analysis, 28/3:381–397, 1993.

L. Glosten and L. Harris. Estimating the Components of the Bid-Ask Spread. Journal ofFinancial Economics, 21:123–142, 1988.

Ch. Gourieroux and J. Jasiak. Financial Econometrics. Princeton University Press, 2001.

A. Grünbichler and F. Longstaff. Valuing Futures and Options on Volatility. Journal ofBanking and Finance, 20:985–1001, 1996.

167

L. P. Hansen. Large Sample Properties of Generalized Method of Moments Estimators.Econometrica, 50:1029–1054, 1982.

C. R. Harvey and R. E. Whaley. Market Volatility Predictions and the Efficiency of theS&P 100 Index Option Market. Journal of Financial Economics, 31:43–74, 1992.

C. He, J. S. Kennedy, T. F. Coleman, P. A. Forsyth, Y. Li, and K.R. Vetzal. Calibrationand Hedging under Jump Diffusion. Review of Derivative Research, 9:1–35, 2006.

S. L. Heston. A Closed-Form Solution for Option with Stochastic Volatility with Appli-cation to Bond and Currency Options. The Review of Financial Studies, 6/2:327–343,1993.

S. L. Heston and S. Nandi. Derivatives on Volatility: Some Simple Solutions Based onObservables. working paper, Federal Reserve Bank of Atlanta, 2000-20, 2000a.

S. L. Heston and S. Nandi. A Closed-Form GARCH Option Valuation Model. The Reviewof Financial Studies, 13/3:585–625, 2000b.

J. E. Hillard and J. Reis. Valuation of Commodity Futures and Options Under Stochas-tic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot. Journal ofFinancial and Quantitative Analysis, 33:61–86, 1998.

P. Honoré. Maximum Likelihood Estimation of Non-Linear Continuous Time Term-Structure Models. working paper, Aarhus School of Business, 1997.

P. Honoré. Pitfalls in Estimating Jump-Diffusion Models. working paper, Aarhus Schoolof Business, 1998.

K. C. Hsieh and P. Ritchken. An Empirical Comparison of GARCHOption Pricing Models.Review of Derivative Research, 8:128–150, 2005.

J. Hull and A. White. The Pricing of Assets with Stochastic Volatility. Journal of Finance,42:281–300, 1987.

A.S. Hurn, J.I. Jeisman, and K.A. Lindsay. Seeing the Wood for the Trees: A CriticalEvaluation of Methods to Estimate the Parameters of Stochastic Differential equations.Journal of Fiancial Econometrics, 5/3, 2007.

S. Jaimungal and V. Surkov. FFT-Based Option Pricing under Mean-Reverting LevyProcesses. 5th World Congress of Bachelier Finance Society, 2008.

168

Ch. S. Jones. The Dynamics of Stochastic Volatility: Evidence from Underlying andOption Prices. Journal of Econometrics, 116:181–224, 2003.

A. Kaeck and C. Alexander. VIX Dynamics with Stochastic Volatility of Volatility. ICMACentre Discussion Paper in Finance, 2010.

N. M. Kiefer. Discrete Parameter Variation: Efficient Estimation of a Switching RegressionModel. Econometrica, 46:427–434, 1978.

S. G. Kou. A Jump Diffusion Model for Option Pricing. Management Science, 48/8:1086–1101, 2002.

G-H. Lian and S-P. Zhu. Pricing VIX Options with Stochastic Volatility and RandomJumps. Decisions in Economics and Finance, 2011.

Y. N. Lin. Pricing VIX Futures: Evidence from Integrated Physical and Risk-NeutralProbability Measure. The Journal of Furtures Markets, 27/12:1175–1217, 2007.

Y-N. Lin and Ch-H. Chang. Pricing and Hedging VIX Options under Stochastic Volatilityand Correlated Jumps in S&P 500 Price and Variance Processes. working paper, NationalChung Hsing University, 2008.

Y-N. Lin and Ch-H. Chang. VIX Option Pricing. Journal of Futures Markets, 29/6:523–543, 2009.

Y-N. Lin and Ch-H. Chang. Consistent Modeling of S&P 500 and VIX Derivatives. Journalof Economic Dynamics and Control, 34:2302–2319, 2010.

Z. Lu and Y. Zhu. Volatility Components: Evidence from VIX Futures Market. Journalof Futures Markets, 30/3:230–256, 2010.

K. Matsuda. Introduction to Merton Jump Diffusion Model. working paper, Departmentof Economics, The City University of New York, 2004.

K. Matsuda. Parametric Regularized Calibration of Merton Jump-Diffusion Model withRelative Entropy: What Difference Does It Make? working paper, City University ofNew York, 2005.

A. Melino and S. M. Turnbull. Pricing Foreign Currency Options with Stochastic Volatil-ity. Journal of Econometrics, 45:239–265, 1990.

J. Mencía and E. Sentana. Valuation of VIX Derivatives. working paper, Banco de Espana,No. 1232, 2012.

169

R. C. Merton. Option Pricing when Underlying Stock Returns are Discontinuous. Journalof Financial Economics, 3:125–144, 1976.

S. Mikhailov and U. Nogel. Heston’s Stochastic Volatility Model Implementation, Cali-bration and Some Extensions. Wilmott Magazine, pages 74–79, 2003.

F. Moraux, P. Navatte, and Ch. Villa. The Predictive Power of the French Market Volatil-ity Index: A Multi Horizons Study. European Finance Review, 2:303–320, 1999.

V. Naik and M. Lee. General Equilibrium Pricing of Options on the Market Portfolio withDiscontinuous Returns. Review of Financial Studies, 3:493–521, 1990.

S. Nandi. How important is the Correlation Between Returns and Volatility in a StochasticVolatility Model? Empirical Evidence from Pricing and Hedging SP500 Index OptionMarket. Journal of Banking and Finance, 22:589–610, 1998.

D. Nelson. Conditional Heteroscedasticity in Asset Returns. Econometrica, 59/2:347–370,1991.

J. Pan. The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study. Journal of Financial Economics, 63:3–50, 2002.

S.-H. Poon and C. W. J. Granger. Forecasting Volatility in Financial Markets: A Review.Journal of Economic Literature, XLI:478–539, 2003.

D. Psychoyios and G. Skiadopoulos. Volatility Options: Hedging Effectiveness, Pricing,and Model Error. The Journal of Future Markets, 26:1–31, 2006.

D. Psychoyios, G. Skiadopoulos, and P. Alexakis. A Review of Stochastic Volatility Pro-cesses: Properties and Implications. The Journal of Risk Finance, pages 43–59, 2003.

D. Psychoyios, G. Dotsis, and R. N.Markellos. A Jump Diffusion Model for VIX VolatilityOptions and Futures. Review of Quantitative Finance and Accounting, pages 245–269,2009.

C. A. Ramezani and Y. Zeng. Maximum Likelihood Estimation of Asymmetric Jump-Diffusion Processes: Application to Security Prices. working paper, University of Mis-souri, pages 74–79, 1998.

L. O. Scott. Option Pricing when Variance Changes Randomly: Theory, Estimation andApplication. Journal of Financial and Quantitative Analysis, 22:419–438, 1987.

A. Sepp. VIX Options Pricing in Jump-Diffusion Model. Risk, 4:84–89, 2008a.

170

A. Sepp. Pricing Options on Realized Variance and Heston Model with Jumps in Returnsand Volatility. Journal of Computational Finance, page forthcoming, 2008b.

J. Shu and J. E. Zhang. Causality on the VIX Futures Market. Journal of Futures Markets,32/1:24–46, 2012.

K. Singleton. Estimation of Affine Asset Pricing Models Using the Empirical CharacteristicFunction. Journal of Econometrics, 68:111–141, 2001.

E. M. Stein and J. C. Stein. Stock Price Distribution with Stochastic Volatility: AnAnalytic Approach. The Review of Financial Studies, 4/4:727–752, 1991.

E. Szado. VIX Futures and Options - a Case Study of Portfolio Diversification During the2008 Financial Crisis. Journal of Alternative Investments, 12:68–85, 2009.

P. Tankov and E. Voltchkova. Jump-Diffusion Models: A Practitioner’s Guide. Banqueen Marches, 99, 2009.

V. Todorov and G. Tauchen. The Information Content of the S&P 500 Index and VIXOptions on the Dynamics of the S&P 500 Index. Journal of Business and EconomicStatistics, 29:356–371, 2011.

A.B. Trolle and E.S. Schwartz. Unspanned Stochastic Volatility and the Pricing of Com-modity Derivatives. Review of Financial Studies, 22:4423–4461, 2009.

G. E. Uhlenbeck and L. S. Ornstein. On the Theory of the Brownian Motion. PhysicsReview, 36:823–841, 1930.

O. Vasicek. An Equilibrium Characterization of the Term Structure. Journal of FinancialEconomics, 5:177–188, 1977.

N. Wagner and A. Szimayer. Local and Spillover Shocks in Implied Market Volatility:Evidence for the U.S. and Germany. Research in International Business and Finance,18:237–251, 2004.

Z. Wang and R. T. Daigler. The Performance of VIX Options Pricing Models: EmpiricalEvidence Beyond Simulation. The Journal of Futures Markets, 3:251–281, 2011.

R. E. Whaley. Derivatives on Market Volatility: Hedging Tools Long Overdue. TheJournal of Derivatives, 1, 1993.

R. E. Whaley. The Investor Fear Gauge. Journal of Portfolio Management, 26:12–17,2000.

171

R. E. Whaley. Understanding VIX. Journal of Portfolio Management, 35/3:98–105, 2009.

J. B. Wiggins. Option Values under Stochastic Volatility. Theory and Empirical Estimates.Journal of Financial Economics, 19:351–372, 1987.

J. E. Zhang and Y. Zhu. VIX Futures. The Journal of Future Markets, 6:512–531, 2006.

J. E. Zhang and Y. Zhu. Variance Term Structure and VIX Futures Pricing. InternationalJournal of Theoretical and Applied Finance, 10:111–127, 2007.

J. E. Zhang, J. Shu, and M. Brenner. The New Market for Volatility Trading. The Journalof Futures Markets, 30/9:809–833, 2010.

J.E. Zhang and Y. Huang. The CBOE S&P 500 Three Month Variance Futures. Journalof Futures Markets, 38:48–70, 2010.

S. P. Zhu and G. H. Lian. An Analytical Formula for VIX Futures and Its Applications.The Journal of Futures Markets, 0/0:1–25, 2011.