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Nonlinear Economiccs and Financial

Modelling

Essays in Honour of Carl Chiarella

Ð PrmgerS

EditorsRoberto DieciDepaf ment of MathematicsUniversity of BolognaBolognaItz.ly

Ca¡s HommesAmsterdam School of EconomicsUniversity of AmsterdamAmsterdamThe Netherlands

Xue-Zhong HeUTS Business SchoolUniversity of Technology, SydneYSydney, NSWAustralia

ISBN 978-3-319-01469-6 ISBN978-3-319-Ot47O-2 (eBook)

DOI 10. 1007/978-3-319-07 4'7 0-2Springer Cham Heidelberg New York Do¡drecht l,ondon

Library of Congress Control Number: 20149'14539

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While the advice and infomation in tbis bæk ae believed to be true and accumte at the date ofpublication, neither the authors nor the editors nor the publisher can accept ãy legat responsibiliry foràny e*ors or omissions that may be made- The publisher mahes no wffianty, express or implied, withrespect to the material contained herein.

Printed on acid-fræ paper

Preface

has built his entire subsequent c¿ìreer at UTS since then (apart from a three year

Springer is pa¡t of Springer Science+Business Media (www-springer.com)

!

the nonlinear viewpoint in economics. The thesis led to his first book, The Ele-

ments of d Nonlinear Theory of Economic Dynamics, published in the Springer-

Verlag Lecture Note Series in 1990. In 1989, Carl was appointed as a Professor inthe School of Finance and Economics at UTS, a position that he still occupies.

Apart from his early work on nuclear reactor theory, Carl has made numerous

scientific achievements and important contributions to the economics and finance

area, in particular to nonlinear dynamic economic, financial market modeling, and

optron pnclng.As a mathematical economist, Carl has a strong research interest in modeling

key economic adjustment processes as nonlinear dynamical systems. Carl's ea¡lier

work on chaotic economic dynamics in 1980s, in particular The Cobweb Model:

Its Instability and the Onset of Clzaos published in Economic Modeling in 1988

and The þnamics of Speculative Beltnviour, published in Annals of Operations

Resea¡ch in 1992, have been pioneering contributions in this area, which had

profound influence on many resea¡chers in this field. Carl has made a significant

contribution to at least two areas of dynamic economic modeling. The first is on

out-of-equilibrium models of macroeconomic dynamics. It develops a systematic

approach to the disequilibrium tradition of macroeconomic dynamic analysis,

leading to nine jointly authored books (with Peter Flaschel and others) on inte-grated Keynesian dynamic macroeconomic models, including three with Cam-

bridge University Press and three with Springer-Verlag- The other is on financial

market models with heterogeneous boundedly rationa-l economic agents, showing

that price movements of financial assets are the result of nonlinea¡ dynamic

feedback processes driven by the interaction of investors with heterogeneous

beliefs and bounded rationality-Through his many conferences and visits, the University of Urbino and the

University of Bielefeld have become second home for Carl. Carl's visits to Urbino

started in 1998 and have become regular since then. The attraction of Urbino forCarl is not only the glorious history, beautiful palaces, and churches, but also a

group of brilliant researcbers around Laura and Gian ltalo in the theory of non-

linear dynamical systems. Through tbe Vienna Workshops on Economic

Dynamics initiated by Gustav Feichtinger, Carl established his intensive research

collaboration with the research groups around Peter Flaschel, Willi Semmler, and

Volker Böhm in Bielefeld. Carl's collaborations with these groups belong to the

highlights of his career.

As one of the main organizers of the annual Quantitative Methods in Finance

conference at UTS since 1997, Carl has made a significant contribution to

American option pricing, where he has mainly contributed to the development and

numerical implementation of various solution methods- He has also been active in

pricing interest rate derivative securities along two directions. The first is toimplement on market data the va¡ious interest rate terrn structure and interest rate

derivative pricing models that have been developed over the last two decades

using nonlinear filtering and Bayesian updating methods. The second consists in

finding improved computational procedures within the stochastic calculus frame-

work of the tenn structure and option prices by allowing the volatility function of

Preface vll

the Heath-Jarrow-Mofton model to depend on the forward ¡ate, and allowing forforward rate dynamics of this framework.15 books and 200 papers, supervised more than

in more than 30 research projects including the

Australian Resea¡ch Council (ARC) Discovery Grants. He was the Co-Editor of

the Journal of Economíc Dynamics and control from 20&1 to 2072 and has been

Associate Editor of many leading finance and economics journals, including

resea¡ch topics Carl has

e breadth of topics Carl

nspiration his work has

given to all of us over so many years. Indeed his work inspires a new generation to

iurther develop this exciting and challenging resea¡ch agenda'

PrefaceVI

Bologna, APril 2014

SydneY

Amsterdam

Roberto DieciXue-Zhong He

Cars Hommes

Contents

IntroductionRoberto Dieci, Xue-Zhong He and Cars Hommes

Part I Carl Chiarella: An Interview and Some Perspectives

AnlnterviewtoCarlChiarella,anltalo-AustralianGlobeTrotterWho Studies Dynamic Models for Economics and Finance

Gian Italo Bischi

What's Beyond? Some Perspectives on the Future

of Mathematical Economics -

Carl Chiarella

Part II Nonlinear Economic Dynarnics

Expectations, Firms' Indebtedness and Business Fluctuations

in a Structural Keynesian Monetary Growth Framework '

Matthieu Charpe, Peter Flaschel' Christian R. Proaño and Willi Semmler

Mathematical Modelling of Financial Instabilityand Macroeconomic Ståbilisâtion Policies.Toichiro Asada

Bifurcation Structure in a Model of Monetary Dynamicswith Two Kink PointsAnna Agliari, Laura Ga¡dini and Iryna Sushko

Boundedly Rational Monopoly with Single ContinuouslyDistributed Time DelayAkio Matsumoto and Ferenc Szidarovszky

l1

l9

4l

27

6s

83

lx

xContents

. 109

135

Contents

A Multi-factor Structural Model for AustralianElectricity Market Risk. . .

John Breslin, Les Clewlow and Chris Strickland

On an Integral Arising in Mathematical Finânce

Mark Craddock

Change of Numéraire and a Jump-DiffusionOption Pricing FormulaGerald H. L. Cheang and Gim-Aik Teh

xl

Learning and Macro-Economic DynamicsSimone Landini, Mauro Gallegati, Jtseph e. Siigirr,Xihao Li and Corrado Di Guilmi

How Non-normal Is US Output?Reiner Franke

Part III Financial Market Modeiling

Heterogeneous Beliefs and euote Transparencyin an Order-Driven Market. . . . . .-. .Polina Kovaleva and Giulia Iori

lhe.limnlicitf of Optimat Trading in Order Book Markets . . .Daniel Ladley and paolo pellizzan

I:qiry Switching Models in the Foreign Exchange Market . . .Wai-Mun Chia, Mengling Li and Uuunt,i"nZn"ng-'

Time-varying cross-specuration in currency Futures Markets:An Empirical Anatysis. .

And¡eas notnig anaïnì."" nãr¡,g

Computational fssues in fhe Stochastic Discount FactorFramework for Equity Risk premium . . .Ramaprasad Bhar and A. G. Mallians

Part IV Quantitative Finance

On the Risk Evaluation Method Based on the Market ModelMasaaki Kijima and yukio Muromachi

On Multicurve Models for the Term Structure. . .

Lau¡a Morino and Wolfgang J. Runggaldier

Pricing an American Call Under Stochastic Volatitityand Interest Rates. . . . .

Boda Kang and Gunter lf. V"y.,

!n the Volatility of Commodity Futures prices .

Les Clewlow, Boda Kang and Christina Sklibosios Nikitopoulos

335

355

371

163

183

201

225

235

253

275

291

315

-l

l.'

Contributors

Anna Agliari DISES, Catholic University Piacenza, Piacenza, ltaly

Toichiro Asada Faculty of Economics, Chuo University, Hachioji, Tokyo, Japan

Ramaprasad Bhar School of Risk and Actua¡ial Studies, The University of New

South Wales, SYdneY, NSW, Australia

Gian ltalo Bischi Università di Urbino Ca¡lo Bo, Urbino, Italy

John Breslin Lacima, Sydney, NSW, Australia

Matthieu Charpe International Labor Organization, Geneva, Switzerland

Gerald H. L. Cheang School of Information Technology and Mathematical

Sciences, Centre for Industrial and Applied Mathematics, University of South

Australia, Adelaide, SA, Australia

Wai-Mun Chia Division of Economics, Nanyang Technological University,

Singapore, Singapore

Carl Chiarella UTS Business School, University of Technology, Sydney, NSW,Australia

Les Clewlow Lacima Group, Sydney, NSW, Australia

Mark Craddock School of Mathematical Sciences, University of Technology,Sydney, Broadway, NSrù/, Australia

Corrado Di Guilmi UTS Business School, University of Technology, Sydney,Sydney, NSW, Australia

Roberto Dieci Dipartimento di Marematica, Università di Bologna, Bologna,Italy

Peter Flaschel Bieìefeld University, Bielefeld, Germany

Reiner Franke University of Kiel, Kiel, Germany

Mauro Gallegati DiSES, Università Politecnica delle Marche, Ancona, Italy

Laura Gardini DESP, Università di Urbino Carlo Bo, Urbino, Italy

xllt

xlv Contributors

Xue-Zhong He UTS Business School, University of Technology, Sydney, Syd-ney, NSW, Australia

Cars Hommes CeNDEF, Amsterdam School of Economics, University ofAmsterdam, Amsterdam, The Netherlands

Giulia Iori Department of Economics, City University, London, UK

Boda Kang Department of Mathematics, University of York, Heslington, york,UK

Masaaki Kijima Tokyo Metropoliran Universiry, Hachiohji, Tokyo, Japan

Polina Kovaleva Department of Economics, City University, London, UK

Daniel Ladley Department of Economics, University of L,eicester, Leicester, UK

Simone Landini I.R.E.S. Piemonre, Turin, Iraly

Mengling Li Division of Mathematical Sciences, Nanyang Technological Uni-versity, Singapore, Singapore

Xihao Li DiSES, Università Politecnica delle Ma¡che, Ancona, Italy

A. G. Malliaris Department of Economics and Finance, euinlan School ofBusiness, Loyola University, Chicago, IL, USA

Akio Matsumoto Department of Economics, Chuo University, Hachioji, Tokyo,Japan

Gunter H. Meyer School of Mathematics, Georgia Instiilte of Technology,Atlanta, GA, USA

Laura Morino Dipartimento di Matematica Pura ed Applicata, Università diPadova, Padua, Italy; Deloitte Consulting Sr1., Milan, Italy

Yukio Muromachi Tokyo Metropolitan University, Hachiohji, Tokyo, Japan

Christina Sklibosios Nikitopoulos UTS Business School, University of Tech-nology, Sydney, Sydney, NSW, Australia

Paolo Pellizzari Depafment of Econornics, Ca'Foscari University, Venice, Italy

Christian R. Proaño The New School for Social Research, New York, NY, USA

Andreea Röthig Institute for Automatic Control and Mechatronics, TechnischeUniversität Dâ¡mstadt, Darmstadt, Germany

Andreas Röthig Deutsche Bundesbank, Frankfurt am Main, Germany

Wolfgang J. Runggaldier Dipartimento di Matematica Pura ed Applicata, Uni-versità di Padova, Padua, Italy

Willi Semmler The New School for Social Research, New York, NY, USA

Contributoß xv

Joseph E. Stiglitz columbia Business School, columbia university, New York,

NY, USA

' Chris Strickland Lacima, Sydney, NSVy', Australia

Iryna Sushko Institute of Mathematics NASU, Kiev, Ukraine

Ferenc Szidarovszky Department of Applied Mathematics, University of Pécs,

: Pé"t, Hungary

: Gim-Aik Teh BNP Paribas' Hong Kong SAR, China

. Huanhuan zheng Institute of Global Economics and Finance, The chinese

., uniu"rrity of Hong Kong, Shatin N. T, Hong Kong; Department of Economics and

. Related Studies, The University of York, Heslington' York' UK3:

Í

-ii

i¡

of Mathematical Economics and of the possible impact of the recent economrc cnsls

on economic theorising.ThesecondpartofthebookcontainingsixchaptersdealswithNonlinearEco-

nomic Dynamics, the area of Economic Theory that mostly attracted Carl's interest

Introduction

Roberto Dieci, Xue-Zhong He and Cars Hommes

R. DieciDipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato' 5'

40126 Bologna, Italye-mail: robeno-dieci @ unibo-it

X.-2. He (E)UTS Business School, University ofTechnology, Sydney,Sydney, NSW, Austnliae-maiì: tony.he I @uts.edu-au

C. HommesCeNDEF, Amsterdam School of Economics, University of Amsterdam. Amsterdam,The Netherlandse-mail: C.H.Hommes@uva_nl

R. Dieci et al- (eds-), Nonlinaar Ecotto¡tic Drnqmics and Fircncial Modelling,DOI: 10.1007/978-3-3lg-01470-Z-1. @ Springer Inremationaì Publishing Switzerland 2014

2 R' Dieci et al'

since the b e frrst two chapters stand in the tradition

of the join ors in the field of disequilibrium macro-

economicmodels inand Williinto a larger scale disequilibrium macroet

Chiarella et al. (2@5). tn u i"tty interdependent m¿cro-model they investigate both

analytically and through "";;;";i;;"iation the feedback impact ofendogenously

sional fixed price model without active stabilisation policy and considers' as exten-

sions of this core version, " iått-¿ì-*ti"nal model òf monetary stabilisation policy

with flexible Prices and a

mix. Besides Providing a

the steady state of the eco

and central bank's credibi

main feedback mechanisms operating in

deal more closely with the bifurcations

may emerge from traditional economlc

infármatioì assumptions a¡e abandoned

the use of simple rules of thumb in maki

earities. This is a resea¡ch fietd in which

As a result, the price dynamics of the phy

one-dimensional map having two "kink

and even chaotic dynamics may appear I

of economic time series.

ThethirdpartofthebookfocussesonFinancialMarketModelling'oneofthemain areas of Carl's work. The first two chapters Present agent-based models of

financial ma¡kets with limit order books, extending some of carl's earlier work on

this topic in Chia¡ella et al.of an order-driven marketthe interrelations betweenorder-driven markets. Theiskewness of stock retums and clustered volatility when book depth is visible to

traders. Full quote transparency contributes to convergence in traders' actions' while

pa¡tial tra¡sparency res;ictions may lead to long-range dependencies- DanielLadleyand, Paolo Pellizzañstudy optimal trading strategies in order book-based continuousdouble auction markets. Their framework is still analytically tractable and optimal

Introduction

i::

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4 R. Diæi et al

trading strategies can be identified using numerical techniques. They find that theoptimal strategies are well approximated by linear strategies using only the bestquotes. This study illustrates that, in complex ma¡kets, optimal behaviour may bewell approximated by simple (linear) heuristics, a major theme in Carl's work. Thefollowing chapter by Wai-Mun Chia, Mengling Li and Huanhuan Zteng, studiesregime switching models in foreign exchange markets and fits in a rapidly growingliterature on estimating heterogeneous agent models (HAMs), an a¡ea where Carlhas made major contributions, e.g. recently in Chia¡ella et al- (2012)- Three differentempirical models are compared, endogenous switching with fractions determined byrelative performance, endogenous switching based on macroeconomic fundamentalsand models with heterogeneous beliefs based on a Ma¡kov-switching process. In-sample and out-of-sample forecasting are compared across these different HAMsusing monthly AUDruSD exchange rate data. The last two chapters in this part ofthe book are concemed with time series modelling and empirical analysis of financialma¡kets- Andreas Röthig and Andreea Röfåig study the time-varying cross-markettrading activities of speculators in US cunency futures markets, extending earlierwork in Röthig and Chiarella (201 l). They investigate linkages between speculativeactivities in different currency futures markets. The results show positive responsesof total,4ong/short speculative activities in the GBP, CAD and JPY futures marketsto an increase in total/long/short speculation in the CHF futures ma¡kets, indicatingthe presence ofcross-market herding activities. These cross-market linkages betweenspeculative activities a¡e relatively stable over time from 1994-2013 and therefore donot suggest that changes in regulation or new market participants or trading strategieshad a significant and lasting impact on cross-market speculative activities- The finalchapter in this part of the book,by Ranaprasad Bhar and A.G. Malliaris, deals withthe Stochastic Discount Factor (SDF) methodology as a general empirical frameworkfor asset pricing. In particular, the authors suggest a multifactor model for the SDFtaking both macroeconomic fundamental factors, such as the yield curve, the VIXindex and a measure fortrading liquidity, as well as behavioural factors into account toidentify significant determinants of the daily equity premium. The chapter proposesto include momentum return as a behavioural factor in the SDF and offers a way toaddress this issue empirically. The chapter also shows how copula methods can beused in this context to overcome analytical complexities for software implementationin defining the dependence between asset returns and the SDF. These five chaptersillustrate the broad contributions of Carl to Financial Market Modelling.

The fourth part of the book consisting of seven chapters focusses on Quanti-tative Finance, another main area of Carl's work. The first two chapters developa new framework for risk management of interest-rate products, extending someof Carl's ea¡lier wo¡k on this topic in Chiarella and Kwon (2003), Chiarellaet al. (2007) and Chiarella et al. (2010). To develop a new methodology forrisk management of interest-rate sensitive products, Masazki Kijima and Y*ioMuromachi present a risk evaluation model for interest-rate sensitive productswithin the no-arbitrage framework. They first consider a yield-curve model underthe observed probability measure, based on the results of the principa.l compo-nent analysis (PCA), to generate future scenarios of interest rates, and then iden-

Introduction

tif}' market prices of risk for the pricing of interest-rate derivatives under the

risk-neutral measure at any future time- Thus risk measures such as Value-at-

Risk (VaR) of portfolios with interest-rate sensitive products can be evaluated

through simple Monte Carlo simulation. They also show, howeve¡ that some

ma¡ket models often used in practice are not consistent with the no-arbitrage

paradigm. Motivated by a significant increase in the spread between LIBORs of

ãiff"."nt tenors as well as the spread between LIBOR and the discount curve during

the financial cnsis, Laura Morino and Wolfgang l. Runggaldier extend Carl's work

(chiarella et al. 2oo7,2ol0) beyond a pure credit risk setting to a more general post-

crisis multicurve set-up. while carl's work follows an HJM-based approach, here the

authors use a short rate modelling with a short rate spread and consider a two-curve

model with one curve for discounting (OIS swap curve) and one for generating future

cash flows (LIBOR for a give tenor). The clean-valuation approach of pricing FRAs

and CAPS without counterparty risk exhibits an "adjustment factor" when passing

from the one-curve to the two-curve setting. The bottom-up short rate modelling

where the short rate and a short rate spread are driven by affine factors allows for

correlation between short rate and short rate spread as well as to exploit the convenient

afflne structure methodology- The next two chapters contlibute to price American

call option and futures price volatility with the framework of stochastic volatility, two

areas Carl has made a significant contribution, see for example, Chiarella and Kwon

compulational domain with appropriate boundary conditions. Comparing with finite

difference approximation, they find that the time discrete method of lines is accurate

and efficient in producing option prices, early exercise boundaries and option hedge

rarios such as delta and gamma. Using a continuous time forward price model with

srochastic volatility, Les Clewlow, Boda Kang and Christina Sklibosios Nikitopou-

Ios introduce three distinct volatility structures to capture the impact of long-term,

medium-term and short-term futures price volatility in commodity futures markets.

They then use an extensive 21-year database of commodity futures prices to estimate

the model for six key commodities: gold, crude oil, natural gas' soybean, sugar and

corn. They identify the shape and the persistence of each volatility factor, their con-

tribution to the total variance, the extent to which commodity futures volatility can

be spanned and the nature of the return-volatility relation. In the next chapter, based

on the structural relationships in the electricity market, John Breslin, I'es Clewlowand Cfuis Strickland develop a general framework for the modelling of Australianelectricity market risk. The framework is consistent with temperature and load meanforecasts, market forward price quotes, the dependence of load on temperature andthe dependence of price on load. The model uses basic building blocks of a¡ HJMform for which Carl has contributed important results. The model can be used notonly for accurate evaluation of the market risk of an electricity generation and retailcompany, but also for the valuation ofelectricity market derivatives and assets. Theydemonstrate the application of the framework to the Australian National Electricity

76 R. Dieci et al.

Market- In the followingchapfer, Mark Craddockpresents a tractable solution to theYakubovich parabolic PDE u¡ : x2uxx * xu, - ¡2r¡, which arises in a number offina¡cial mathematical problems such as Asian option pricing, the Hartman-Watsonlaw and pricing zero coupon bonds in the Dothan model. After deriving the heat ker-nel for the PDE, Mark uses the Fourier sine transform to reduce the kernel to a simpleform, which may be explicitly evaluated as a series of error functions. Some finan-cial applications are then discussed. In the final chapter, by applying the approachof charging numeraire, Gerald Cheang and Gim-Aik Teå extend the European calloption pricing formula in the literature to the case when both stock prices and inter-est rates are driven by jump-diffusion processes. The pricing model is an extendedMerton jump-diffusion stock price model with a stochastic interest rate term sruc-ture that is an HJM-type model with jumps. The approach does not require Fouriertransforms as used in the existing literature. It allows us to price the option when the

bond price dynamics is also discontinuous. When the jump-sizes are fixed instead,then they get the special case of the HJM model with fixed jumps as in Chia¡ella andNikitopoulos Sklibosios (2003).

References

Adolfsson, T., Chiarella, C., Ziogas, A., &Ziveyí,1. (201 3). Representation and numerical approx-imation of American option prices under Heston stochastic volatility dynamics, QuantitativeFinance Research Centre, University of Technology Sydney. Working paper No. 327.

Agliari, 4., Chiarella. C., & Gardini, L- (2004). A stability analysis ofthe perfect foresight map innonlinea¡ models of monetary dynamics. Chaos, Solitons and Fractals, 2 I ,37 1386-

Buchanan, M. (2007)- The social alom: why the rich get riche4 cheaters get cought, and your

neighbor usmlly looks like you- New York: Bloomsbury.Chiarella, C. (l99O)-Thc elements of a nonlineartheory of economic dyrunics- New York: Springer-

Chiarella, C., Flaschel, P., & Franke, R. (2005). Foundatíorc Jor a disequilibrium theory of the

business cycle- Cambridge, UK: Cambridge University Press-

Chiarella, C-, He, X.-2., Huang, W., & Zheng, H. (2012). Estimating behaviouml heterogeneityunder regime switching. Jourml of Economic Behavior and. Organization,83(3),44646O-

Chiarella, C., Iori, G., & Perello, J. (2009). The impact of heterogeneous trading rules on the limitorder book and order fr,ows- Journal of Economic Dynamics and Contol, 33(3),525-537 .

Chiarella, C., Kang,8., Meyer, G. H.,&Ziogas, A.(2009). The evaluationof American option pricesunder stochaslic volatiliry and jump-diffusion dynamics using the method of lines. IntematiorulJourml of Theoretical and Applied Fimnce, 12(3),393425.

Chiarella, C., Kang, B., Nikitopoulos, C. S., & Tô, T- (2013). Humps in the volatility structure ofthe crude oil futures market: New evidence. Ene rgy Economics, 40, 989-lOCo.

Chiarella, C., & Kwon, O- (2001)- Foruard rate dependent Markovian Íansformations of the Heath-Janow-Morton tem structure model. Fircnce and Stochastics, 5(2),237-257.

Chiareìla, C., & Kwon, O. (2003). Finite dimensional affine realisations of HJM models in termsof forua¡d rates and yields. RavÍew of Derivatives Research, 6, 129-155.

Chiarella, C., Maina, S. C., & Nikitopoulos, C. S. (2010). Ma¡kovian defaultâble HJM term structuremodels with unspanned stochastic volatility- Quantitative Finance Research Centre Research

Paper no.283 (2010), University ofTechnology, Sydney.Chiarella, C., & Nikitopouìos. C. S. (2003). A class ofjump-diffusion bond pricing models within

the HJM fmmework. Asø-Pacifc Fimncíal Markets, 10,87-127.

chiarella, c., Nikitopoulos, c. S., & Schloegl, E. (2007). A Markovian defaultable term structure

model with state dependent volatilities- Internatiorul Journal ofTheoretical andApplied Finance,

t0.155-202.Chiarella'C.'Ziogas,A.,&Ziveyi,J.(2010).RepresentationofAmericanoptionpricesunder

Heston stochastic volatility dynamics using integÉl transfoms' ln Contemporary quqntitative

fnance: essays in honour of Eckhard Platen (pp'281-315)' New Yo¡k: Springer'

náttrig, n., & ðhirella, C. (201l). Small traders in cunency futures markets. Joumal of Futures

Markzts,3 I ,898-913-Sarg"nl f. J., & Wallace' W. (1973)- The stability of models of money and growth with perf€ct

foresighl. Econometrica. 4l , lM3-8.

Introduction

314

References

B. Kang and G. H. Meyer

Adolfsson, T., Chiaella, C.imation of American opFinance Research Centre

Black, F., & Scholes, M. (l8t,637459.

Kjellin' R'' & Lovgren' G. (2006). option pricing under srochastic volatility, Master's rhesis, u. ofGothenburg-

On the Votatility of Commodity Futures prices

l,es Clewlow, Boda Kang and Christina Sklibosios Nikitopoulos

L. Clewlow

Sl1a.Croun, Sydney. NSVr', AusrmtiaE-marl: les @ lacimagroup.com

1 Introduction

' commodity futures markets arepraying a reading ro.re in the current financial arena.commercial parricipanrs rr tir ptrvrluffi ;;;i;;.-"di ties have tradi tionar rybeen the main traders of commodity fr,;; _;;i;;rl"o*"u"., a rapidty increasinqnumber of financia'y motivaæd^i:_;g..,, ¡ü;:'i"å*" a"or, institutional invesro¡iand insurance companies hav-e entered the ma¡kets å'nd have been using commodi_ties derivatives for porrfolio diu"..in"utr*, u"o iå. iåo*,", infladon and rhe weakU.S. dolla¡ (Tokic (201l)). co__oaiÇ .nuä;;;"

"swings and significant ,ál^rlnyespecialry over rhe orl:,":i"n:"d^noreworrhy price

analysi s and,h. n'un u g"n.," n, .r,i,i'.]-å*# u,y,. ;'r"rïL."Xil:#ff ::::* r"

""In this chaprer. a forwardprice ..ã"i *ììi¡i inJú"n et at. (1992) frameworkfor rhe enrire rerrn srrucrure of rurr., p.J", ;;;;;;ä with a murti-facror srochas-tic volatiliry model. The proposed ,n.|"_ruoo.roã.ïl:shorr-term, medium-term, as weil as rong+erm ,u,u.", olÏt

to:allur: the impacr of

,'Ëilä:j;:îî:ïfr "ï;'i:i**'f {i'ilöö:ïil''':ï?Jl:i.i::'î"ffi fl :;,in". ¡" ,t" r1"ä;;:":¡#'* model admits finite-dimensio",l .."1;;,i";:';;;:¿"u,",orì*"ì"-,ä,-,;::,"iäi[H'åiä,",'.iäî"iJä:Ëfr

,i¡."ff îf I2lvears from lst ianuary r990 to å1., ói""-iï. ioìî. ,",""r,n, rhe most liquiã

of York. York. UK

Sydney, Sydney, NSW, Ausrralia

et al. (eds.), Nonlinear Economic Dyrlamics and Financíal Mode lling.

Èmail:

1007/978_3_ 3t9-07470t t8.a Springer Intemational Publishing Swiøe rtand, 20 I 4315

316

commodity futures mafkets, the following commodity futures prices are included in

the study; gold, crude oil, natural gas, soybean, sugar and corn'

The moãel is well suited to identify the shape, the persistence and the intensity

of mean reversion and the

also assessed and indicate

ty factor. Fufhermore, ú¡s

s volatility can be spanned

by futures contfacts (i.e. hedged by futures contracts only), and the nature of the

retum-volatility re s markets'

Forward price and Schwafz (1998)' Clewlow and

Strickland (2000) well as convenience yield models of

Gibson and Schwa¡rz (1990) and Korn (2005) have studied commodity futures ma¡-

kets but they are restricted to deterministic volatility. Stochastic volaúlity models

have been proposed by Schwartz a¡d Trolle (2@9a) and chiarella et al. (2013) and

have analysed crude oil futures ma¡ket volatility, under exponenúal decaying and

hump-shaped volatility specifrcations, respectively. The contribution of this paper

rests on using three distinct volatility structures and aiming to analyze their nature

and assess their contribution-The paper is organized as follows. Section2 presents a Ma¡kovian affine forwa¡d

price model with stochastic volatility. Section

ity futures price data used in the analysis and

estimate the model. Section4 presents and dis

2 A Markovian Commodity Futures Price Model

arguments in commodity futures ma¡kets imply that the futures price process ls

equal to the expected future commodity spot price under an equivalent risk-neutral

probabitity measure O (see Dufñe (2001)), namely

F (t, T, V ù : Bø[S(r, VùlØ,]-

On the Volatility of Commodity Futures prices 3n

risk_neutral

;ìä:":::;

dF(r,T,V.) 3FGJ,vr) : à"'"'T'Y¡)dw¡(t)' (l)

where, w(t) : lwt(t), wz?), w3eÐ is a three_dimensional wiener process- TheØs-adaptedfirturespricevolatilityprocessesø¡ (t,T,y¡)havethefuncúonarforms,forallT > t,

o1Q,T,v,): orrÆt.

oz(,T,Y¡) - n2¿-n('-')F?, e)q(t, T,Vt) : rc¡(i. - t¡e-mT -t) ¡lyt.

with n¡, (t : I, 2,3) and n¡, G : 2,3) consranrs. The volatility process V1 :{Vl, V3,Vf¡ is a three-dimensional Hestón (l993ttype process such that

dYi: ¡1,¡(v¡ -vi)dt + r¡$I¿w! <O, (3)

,"rti::llr]li-,le following correlarion srructure of innovarions between volatilitydrq rutures pnce retums is assumed

EaÍdw¡(t) . dwl Ø) - p¡dt,for i : j; and. o, for i f j. (4)

where p¡ are constants for i - 1,2,3- The correlation structure of innovationsbetween volatility and futures prices determines tÌ¡e extent to which the volatility risk

L- Clewlow et al.

I The usual conditions satisfied by a fittered complete probabitity space ae: (a) .Øo contains all.the

p-null sers of.-ø and (b) rhe filÍ;tion is right continuóus- See Protrer (2004) for technical detarts'

318 L. Clewlow et al-

can be hedged (spanned) by futures contracts. When the Wiener processes W¡ (r) and

W,v {t) are uncorrelated then the volatility risk is unhedgeable by futures contracts.

When the ìùy'iener process es W¡ (t) and, W ìv (t) are correlated, then the volatility riskcan be partially hedged by futures contracts- Consequently, futures derivatives thatare sensitive to volatility, such as options on futures, cannot be completely hedged

by using only futures contracts. Furthermore, this modelling framework allows us toassess the dynamic relationship between futures price returns and volatility changes.

A negative (positive) correlation implies a negative (positive) relation between priceretums and their volatility, a well-known empirical phenomenon termed as asym-metric (inverted asymmetric) volatility. Yet again, the proposed model can capturethe volatility reaction of each of the volatility factors, as these factors a¡e modelledas separate identities.

For modelling convenience, we express the system ofEqs. (1) and (3) in terms ofindependent Wiener process, such that

dFft.T.v,\ ¿ffi :loie.r,v.)dwite), (s)

dYi: ¡.t¡(u' -vitat + ,,rfurþdryt ç¡ + J-- fiawirr)) , rol

where l4zr(r) : W(t) a¡¡d Wz(t¡ a¡e three-dimensional independent Wienerprocesses. Accordingly, the volatility factors Vl with p¡ : 0 carries a risk that

cannot be spanned by futures contracts alone and when p¡ . 0 (p¡ > 0) then the

volatility lactor V! has an asymmetric (inverted asymmetric) reacúon.It is well known that for general volatility specifications, the forward price model

for pricing the commodity futures (5) is Ma¡kovian in an infinite dimensional state

space. However, the volatility specifications (2) produce finite dimensional realisa-

tions of the forward price model, see Chiarella and Kwon (2001) and Björk et al.

(2N4).

Theorem I Under the volntility specifications of (2),1¡ F(t,T,Y¡) is affine in nine

state varinbles, as described below:

On the Volatility of Commodity Futures prices 3Ig

þzQ - t) : nz(T - t)e-,B?-t¡, þce _ t) : 63¿-m(r-t),^ß(T -t): fu(T - t)2. u(T _t):2ljze _t)1c(T _t),.ys: p4(T _ t)2.

The state variables ó^(t), n _ l,stochas tic dife rential eq uation.s

,4 and x¡(t), j : 1,...,5 satisfy the

dù(t) : ,f¡awrç.¡,d þ2Q) : -q2g2e)d r + rffl aw21t¡,

dh Q) : -rßó3ç)dt + {-v! aw, ç¡,dþa(t) : (-nzóqe) + ôzGDdr,dx¡(fl:\!¿¡,dx2Q¡ - (-rrrrr, +vl) at,

dh(t) : (-2r64e) +v!¡at.dx4q) : (-2ryne) * xtTDdt,dx5t) - (-2rpx5e) +2n7Ddt,

subject to ó'(0) : x¡ (0) : 0, for a, n anà j - The associated stochastic voratilityprocess y ¡ : {vl, vi, v! . ¡ ¡oítows ,n" iy""|àir'io¡.Proof The technical details are summa¡ized in the Appendix. tr

The proposed model admits finite dim. ofDuffie and Kan (1996) and ir is cons

observed futures price curve; consequ': estimation purposes, it is necessary to, aspresentedinSect.3.2.

^ In^addition, the ma¡ket price of voratility risk is mode,ed with ..complete,,

affinespecifications (see Doran and Ronn (200giand oai anJ singteton (2000)) and morespecifically as

aw!çt¡: dw¡(t) + ^,tR¿r,

awlv 1t¡ : dw¡v (t) + ¡! rR ¿t.

(8)

r5ln F(t,T,yr¡ :ln F(O,T,vù+>P"Q -t)ô^(t)- ¡Z^ttt, -t)x¡(t), (7)

(e)

where

þtC - 1) : nt, 'n(T - r¡ : "1.

7zQ - t) - n2¿-'tzÎ-t\, n(T - ) : xl¿-2nz[-t)

320 L. Clewlow et al-

for i : 1, 2,3, where W! tÐ zrira Wlv Ø are Wiener processes under the physicalmeasure P- Next, the proposed model is estimated by fitting the model to commodityfutures prices of six key commodities: gold, crude oil, natural gas, soybean, sugarand corn.

3 Data and the Estimation Method

3.1 Dafg

://www.barchan.com, æ well as by computing the

example, on Sept 9, 201 3 volumes werei 261.394I 12,208 for Nov 13 soybeans, 1N,027 for Oc¡ 13

for Oct l3 com.

An extended dataset of six comrnodity futures prices provided by cME is used thatspans 2lyears from lst January 1990 to 3lst December 2010. The selected com-

metal (including gold), energy (including crude oil and natural gas) and agricultural(including soybeans, sugar and com) commodities products.

over this period, all comrnodity ma¡kets experienced extreme price movemenßand volatility due to noteworthy financial market and macroeconomic events such asthe oil price crisis in I 990, the financial crisis in 2008, the crude oil bubble in 200g andthe ongoing food crisiof available futures cohas increased significwith positive open interest. For example, for crude oil, the open interest per day hasincreased from 17 on the I st of January l99o to 6i on the 3 lst of December i0l0and the maximum maturity of crude oil futures contracts with positive open interesthas increased from 499 (calendar) days to 3,128 days.

contracts. Thus, the available contract months and the liquidity for each commod-ity are invesúgated, and the following selection per commodity has been made. Forcrude oil, the first seven monthly contracts are used, near to the trade date, followedby the three cont¡acts which have either March, June, September or December expi-ration months and then we include the next five December contracts. Therefore, the

On the Volatility of Commodity Futures prices

32t

tracts used on a daily basis varies betw

maturities. The com futures more Iiand December thus all available mof contracts per day being 15- In a

pnce retums for I month and l2monthterest.

3.1.1 Number of Stochastic Factors

cting the evolution ofthe futures curve isanalysis (pCA) of rhe furures pricebutions for different .o._odi,i",volatility functions. Three factors

3.1.2 The Discount Function

n and Sieget (1997)3,6,9 and l2 month

aneous rorward,",.3,'Ji:Íiïnl,)^", .

forwa¡d interest rate curve as

322

Future prÈ6 - Gold

L- Clewlow et al.

Future pri6 - Crudeoil

Time t Mafurity T¡me t Maturity

Future pn-æs - Naturalcæ Future prics - Soybeans

Time t Matur¡ty Time t Matuûty

Future pr¡ces - Sugar FutuE priæs - Corn

Maturity Time t MaturityTime t

Fig. I Commodity futures prices- The figure plots lhe prices of selected commodity futures con-

tracts from January 2, I 990 to Dæember 31, 2010

On the Volatility of Commodity Futures prices

Table I Descriprive crude oil andMaturity Gold Crude oil

IM 12M IMÈ

L

.9È

L

-9c

Id

-9ù

ù

r

Mean

St. Dev.

Kurtosis

Skewness

0.000220

o.ott97239-837774

-0.t24339

0.000200

o.ot0799

28.t33742

-0-162282

o.00029t

0.026æ4

20.164654

-0.893782

0.000306

0-01 7870

8.156276

natuml gas

t2MNatural gas

IM0-000 I 85

o.o37743

l 6.00890 r

t¿5

t2M0.000 r 79

o.022392

t9.721668

-0.069086-0.298393 0.301666The table displays the descriptive statistics for daily log returns2.1990 and Dæember 31. 2010 for goìd, crude oil and natuml gas

Table 2 Descriptive statistics-soybeans, sugar and comMaturity Soybeans Sugar

IM l2M IM t2M

of futures prices between January

Mean

St. Dev.

Kurtosis

Skewness

0.000090

0.015746

12.0250t3

-0.932488

0.000084

0.013324

7.275205

-0-34t833

0.000061

0.022sU

12.437959

-0.345857

o.000225

0.014234

7.495627

-0.243873

Corn

IM0.000077

0.016618

2t.tæ643

-0.847868

t2M0.000096

0.0t2512

7.647 190

-o.147756The table displays rhe descriprive statistics for daily log2, 1990 and December3l, 201 0 for soybeans, sugar and com

Table 3 Accumulated percenCage of factor contribut¡onCommodity One factor Two factorsGold

retums of futures prices between January

Crude oil

Natural gas

Soybeans

Sugar

Corn

o.973s

0.9503

0.7636

0.8973

0.8788

o.8764

0.98 r 0

0.9825

o.8625

o.9413

0.9560

0.9236

Three factors

0.988 I

0.99t90.9308

0.9639

0.9832

0.9487

Four factors

0.9926

0.9957

0.9666

o.9798

o.99s6

0.9712

f (t,T) : oto* ate-q(r-tJ + az\(T - t)¿-q(-t¡ (10)

from which LIBOR and swap rates can be priced. This also yields for zero-couPon

bond prices the expression

IP(t.r)- "*p I -c,o(r - o - !@t+oz) (r - e-e(r-ù) + c,z(r- rl|. rrrl

The table displays the accumulated percentage ofpCA factor contribution towards each commodityfutures retum variation- We found that three factors are able to explain most of the variations of thefutures ¡etums for the commodities of interest

and 0 arc recalibrated, byeen the model implied for-and swap curve consistingswap rate on that date.

3.2 Estímøtion Method

The estimation approach is quasi-maximum likelihood in combination with rhefilter- The model is cast into a state-space form, which consists of the

equaÍons and the observation equations- For estimation purposes, a time-system

version of the model (7) is considered, by assuming for all I,

324 L- Clewlow et al- On the Volatility of Commodity Futures prices 325

Xt+t : <Þo * <ÞxXt * wt+t, wt+t - iíd N(O, e), ç2)d in closed form.inea¡ functions ofthe state va¡iables, thus thedescribe the relationship between observed

EDdrqdtu dtuUl@FEÉoE

I

õ

9I

à

9

9!

õ

i

!9E

Þã9

z¡ : h(X¡) * u¡, u¡ - iid NQ, A). (13)

.is maximised by using the constrained optimization

ibrary. Several different initial hypotheti""i pu*rn.*imonthly data, then on weekly dara and finally on dailyoptima.

tè

9

æÞhtu 6tul¡@tud

F(0,7-) - /o, where ;fo is a constant representing the long-term futures price (at

infinite matuúty). This constant is an additional parameter that is also to be estimated.

The system equations describe the evolution of the underlying state variables- In

our case, the state vector is X¡ : {Xi, m : 1,2, - . . ,121, where X¡ consists of the

12 state va¡iables; x; (/), j : l, . . ., 5, ó,(t), n : l, .. ., 4, and Vi, i : l, 2, 3-The

continuous time dynamics (under the physical probability measure) of these state

va¡iables a¡e defined by Eqs.(6), (8) and (9). The corresponding discrete evolution

4 Empiricat Results

4. 7 Pørameter Estímafíon

4.2 GoldVolafílity

Ë

!

:g

õ

Fdor ku.tiy(t|d)

Fig.2 PCA results (Eigenvalues and volatiliry functions)

3 In absolute terms, V! is the variance p.o"".. und nfi is tbe volatil¡ty process.

326

Table 4 Parameær Estimates

MetalsGold

6.78 r

2

3

J" Factor Ki Tlì lri Y¡ e¡

Stochastic volat¡tity - Gotd

J@ ¡s7 .m ù&Time t

L- Clewlow et al. On the Volatility of Commodity Futures prices327

0.05

o

-o6{.f

ât42þ 425

funct¡on - Gold

Pi À¡ o7

-0.4324 0.0000 -1.9822 0.0100 4.01l0 0.8996 -3.0000 -2.1974

-0.0109 4.9979 -1.ø38 0.0305 l-3018-0.9127 2.9762 -2.97400.0308 4.870s 4-9083 0.0404 1.8998 0-9490 2.8698 -2.s7û

0-0364 0.0000 3.4978 0.1015 0.0100 0.5411 3-0000 -0.69570.3735 0.3914 -2-0000 0.2624 5.Om -0.8337 -3.0000 -l -1797

-0.0804 r.661l 5-0000 5.0000 0.0619 0.9500 -3.0000 -1.6399

0.6

0.5

o-4

0.3

o.2

01

Energy

Cnrde oil

4.50

Natural gas

t.49

AgriculturalSoybeans

6.30

{s44

{-Æ

o.4

03s

0-3

o25

o0.51 1522533s¿Time to matur¡ty t

Volatility function - CrudeOil Stochastic - Crudeo¡t

45 5 Jù7 bto JuXz

-o.2904o.&632.4848

0-5595

0.0448

o.428'l

0.0000

4-2571

5.0000

0-0000

4-9300

2.8152

-2.0000 0.9012 5.0000 0.3116 -3.0000 -l.l33r5.0000 0.1914 5.0000 -0.9500 -3.0000 -2.8280

-2.0000 s.0000 5.0000 -0.5886 -3.0000 -0.9792

5.0000 0.0356 0.964 0.7338 l-9856 1.4988

4.8135 0.0130 0.9412-o.o197 0.268s -2.49830-n85 4.8349 0.0100 -0.2922 0-2036 -0.8500

45

3.5

î02 3

>- 25

2

1.5

I

0.5

0

I

þ ors

01

005

0

{05

Sugar

0.2228 0.0000 -2.0000 0.s134 5.0000 -0.9155 -3.0000 -3.00000.0000 -1.6488 -2.0000 0.t219 1.7671 -0.9221 -2.9525 -2.9494r.0191 3.3670 0.0828 5.0000 5.0000 -0-9s00 0.0642 -o.0t99

Corn

-0.4r88 0.0000 -0.0762 0.1527 5-0000 0.9500 3.0000 -2-5545

-0.0143 3.8123 -1.8526 0.1028 4.6859 -0.8463 0.1100 -0.86520.9395 l-3926 1.3400 0.0100 0-4731 0.9500 -3.woo-2.7719

The table displaysthequasi maximumlikelihoodestimates forthe three-factormodel specifications.

F is the homogenous futures price at time 0, namely F(O' t) : Í"'Yt

rhe long-rerm volatility is the dominant volatility factor in the gold futures market.

Furthermore, the innovations of this volatility factor has a correlation of 0.9 with the

innovations ofthe gold futures price returns, implying that gold futures price volatility

can be mostly hedged by gold futures contracts. Moreover' the positive sign of úe

correlation coemcient confi tms the well-documented Positive gold return-volatility

relation that in the spot gold market has predominantly been explained by the safe

haven effect, see Baur (2012).

Fig. 3 Estimated volatility factors for metals and energy futures prices

4.3 Crude OítVolatílþ

t.94 I

2

J

0.5

o.4

0.3

o.2

0 0-5 1 1.52 25 3 3.5 4 4.5 5

25 3 3.5 4 45 5

T¡me to maturity rtunc-t¡on - NaturalGas

.Jæ.@.8J¡dJú.& .@.H.6.h51o

T¡me t

Stochast¡c volat¡lity - Naturalcas

.l.tffi.6bþbÞ

T¡me t

6.47 I

2

5

7

6

>-4

3

þ 0

0

-01

0 05 I 1.5 2

Time to maturity r

328

t)

Volatil¡ty function - Soybeans

0 05 r 15 2 25 3 35 4 4.5

T¡me to maturity rVolat¡lity function - Sugar

0 05 1 1.5 2 25 3 35 4 4.5 5

Time to maturity r

funct¡on - Com

L. Clewlow et al-

Stochast¡c

]]me t

Stæhaslic

T¡me t

Stochastic - Corn

On the Volatility of Commodity Futures prices

efficientry medium-term crude o' futures vorat'ity rather than shorr-term crude oircorrelation coefificients in these two dom_turn_volatility relation, a positive return_

ory effect, see Ng and pirrong (1994)Iained by the volatility feedback effect.

??o

þ

o.7

06

05

o.4

03

o2

o1

0

4

3.5

3

>-2

t.5

1

0-5

0

30

25

20

>- f5

10

5

03

o.25

o2

015

0l

005

0

-o 05

03

o_2

01

0

-9 ¡.rþ

1.4

1.6

1.4

12

1

>-0.8

0.6

o.4

o.2

42

-03

4.4

Time to maturity r

Fi& 4 Estimated volatility factors for agricultural futures prices

Time t

volatility state factor Vl is more persistent (reverts slower to the mean) and farmore volatile compared to the hump-shaped state factor Vl, see also the middlepanels ofFig-3. Therefore, the short-term volatility is the more influential volatilityfacto¡ followed by the medium-term and the long-term volatility factor- The corre-lation coefûcient between volatility and futurcs price returns are -0.8337 and0.95,respectively. These correlations imply that crude oil futures contracts can hedge more

0 0.5 1 15 2 25 3 3.5 4 45 5

330 L- Clewlow et al.

4.5 Soybean Volnrilify

From Table 5, two driving volatility factors are present, with the long-term volatilityfactor contributingT9-5{Øo of total variance and the hump-shaped volatility factorcontributing 20.36Vo of ¡hetotal variance. The hump-shaped state factorVf is highlypersistent (reverts very slowly to the mean) and very volatile compared to the long-term volatility state factor Vf which is not as persistent and less volatile, see the toppanels of Fig.4. The correlation coefficients a¡e 0.7338 for the long-term volatilityfactor and -0.2922 for the hump-shaped. Thus, soybean futures contracts alone

cannot hedge medium-term volatility, yet they can hedge more efûciently the long-term volatility. Moreover, the more contributing long{erm volatility factor involves

a positive retum-volatility relation, while the hump-shaped volatility factor entails a

negative return-volatility relation.

4.6 Sugar Volatiliry

The analysis reveals two main driving volatility factors. The most contributing (with70.68Vo oftotal variance) is the long-term volatility factor and the second contributingvolatility factor (with 29-327a of the total variance) is the hump-shaped volatilityfacto¡ see Table5. The long-term volatility state factor Vf is not as persistent (reverts

quickly to the mean) but equally volatile compared to the hump-shaped state factorVf, see also the middle panels of Fig.4. The correlation coefficient between the

associated volatility factors and futures price retums are both negative and close to

- 1 , thus sugar futures contracts can hedge most of the sugar futures volatility, and a

negative return-volatility relation is implied.

4.7 Corn Volatílity

The parameter estimates for corn from Table4 demonstrate that the long-term volatil-

ity factor dominates in the corn futures ma¡ket. More specifically, from Table 5, the

long-term volatility factor contnbufes93.34%o to the total variance. The second con-

tributing factor is the hump-shaped votatility factor with 6.65 70 to the total variance-

The leading volatility state factor V,r is highly persistent (revefs slowly to the mean)

and very volatile, see also the bottom panels of Fig.4. Therefore, the long-term

volatility is the more influential volatility factor in the corn commodity futures mar-

ket. The correlation coefficient between the associated volatility factors and fuuresprice returns are positive and high, thus corn futures contracts can hedge the com

futures volatility. The positive sign of the correlation coefÊcient of the two domi-

nant volatility factors implies a positive retum-volatility relation, see Ng and Pirrong

(1994) and relates to the inventory effect.

.l.¡me I

Fþ.5 Model goodness of fit

On the Volatility of Commodity Futures prices

RMSE - Gold RMSE - CrudeOit

J.ñ.@.JdffiJd

ïme t

RMSE -

ïme tRMSE - Com

33t

o.1

o.09

o.G

o.o7

0_ffi

o.60.ø

0.æ

o.æ

0.01

0

E

E

o12

01

0@

Eo*0g

oe

oJe90 Ja95 Jam JanOS Janto Jills.l-¡me

t

RMSE - Naturalcas

Time t

RMSE _

01

o-

008

0

0.1

0.09

0.æ

o.o7

0.60.05

0.ø0-æ

o.e0.01

0.1

0.æ

0.æ

o.o7

0.06

0.60.üo.03

o.æ

o.01

o

E

€

oE

001

T¡me t

4.8 Model Fit

332 L- Clewlow et al.

are subject to strong seasonal effects, alternate model specifrcations---€.g- higherdimensions of volatility factors, seasonal dynamics or regime switching volatiliry5çþsms5-ws¡ld have the potential to better capture the natural gas futures pricedynamics. This also applies for the agricultural commodities, where the RMSEs are

relatively low because ofpersistent seasonality effects. For crude oil and gold, the

model fit is reasonable, with exceptional cases being associated with some majorsocio-economic effects such as the Gulf war and Global Financial Crisis (GFC).

5 Conclusion

In this paper, a tractable forward price model with stochastic volatility is proposed

and an empirical study is ca¡ried out to analyze the volatility of the most liquidcommodity futures markets, including the gold, crude oil, natural gas, soybeans,

sugar and corn market. The model allows distinct volatility structures, includingexponential decaying, hump-shaped and infinite maturity and can potentially gauge

the impact of short-term, medium-term and long-term va¡iation.The study shows that for most of the commodities futures ma¡kets, at least two of

the volatility structures are present (with varying levels of persistence and intensity).The long-term volatility is the dominant stochastic volatility factor for most com-modities, except crude oil where exponenúal and hump-shaped volatility factors are

contributing more. The extent to which the volatility can be hedged by futures con-tracts varies across commodities, with futures contracts being least capable to hedge

volatility in the crude oil futures market, natural gas futures market and soybeans

futures ma¡ket.Overall the proposed model provides a relatively good fit for these six com-

modities, even though the distinctive cha¡acteristics of each market is not properlyaccounted for. A comparative investigation between models with different specifi-cations and their fit performance will better reflect on the quality of the model fit, a

study that has been left for further resea¡ch-Under the proposed model, option prices can be obtained quasi-analytically, con-

sequently the model can also be estimated by fitting to commodity futures options.Since options are volatility sensitive derivative instruments, this estimation study has

the potential to provide constnrctive and insightfuì findings about the volatility incommodity futures ma¡kets, as well as volatility hedging effectiveness-

Acknowledgments We thank m anonymous referee for helpful suggestions and detailed feedback.The authors also wish to thank the Australian Resea¡ch Council for 6nancial suppon in relation to

the datapurchase (DP 1095177, The Modelling and Estimation'of Volatility in Energy Markets).

On the Volati¡ity of Commodity Futures prices 333

Ap¡rcndix: Proof of Theoreml

We consider the process X (r, Z) : ln F (t, T, y ¡), where the forward price dynamicsare given by (l) with the vorariliry specification, iz). Then an appricarion ofthe lto,sformula derives

F (t. r, v t): F(0, r) exp :|ii, - i ",

- r, y )d w ¡ (s) - :Z i n -, ., o.,r,]

(t4)

By introducing the state variables

xr(¿) -

e-2ry2e-ÐVZdc

(t - s¡¿-m?-"rrF?o*U¡

f <, - rl"-'*o-")v3¿",0

and performing some basic manipulations, Eq. (la) can be expressed as (7).

References

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A Mutti-factor Structural Modetfor Australian Electricity Market Risk

John Breslin, Les Clewlow and Chris Strickland

I Introduction

J. B¡eslin . L. Clewlow - C. StricklandI¿cima, Lævel 32, I Market Street, Sydney, NSW 2000, Ausrmliae-mail: John.Breslin @ Iacimagroup.com

L. Clewlow (E)e-mail: les@ lacimagroup-com

C. Stricklande-mail: chris@lacimagrcup-com

l;?:: :l:! f"q ), Nonlinear Economic Dynamics and Fircnciat Morteiling. 335u\)t: 10.10071978-3-319-01470-2-1 9, @ Springer Inremarionar pubiishing Swirzerrand 20r 4