Non-Gaussian Stochastic Models and Their Applications in ...siba.unipv.it/fisica/ScientificaActa/Tesi ottobre 2007/Cisana.pdfNon-Gaussian Stochastic Models and Their Applications in

UNIVERSITÀ DEGLI STUDI DI PAVIA

DOTTORATO DI RICERCA IN FISICA – XX CICLO

Non-Gaussian Stochastic Models and Their Applications in Econophysics

Enrica Vera Cisana

Tesi per il conseguimento del titolo

DOTTORATO DI RICERCA IN FISICA – XX CICLO

Non-Gaussian Stochastic Models and Their Applications in Econophysics

dissertation submitted by

Enrica Vera Cisana

to obtain the degree of

DOTTORE DI RICERCA IN FISICA

Supervisor: Prof. Guido Montagna

Referee: Prof. Rosario N. Mantegna

Università degli Studi di Pavia

Dipartimento di FisicaNucleare e Teorica

Istituto Nazionale di Fisica Nucleare

Cover

Non-Gaussian Stochastic Models and Their Applications in EconophysicsEnrica Vera CisanaPhD thesis – University of PaviaPrinted in Pavia, Italy, November 2007ISBN 978-88-95767-06-2

Left: Traders at work in the New York Stock Exchange (NYSE).Right, top: Simulated paths of the time evolution of volatility process in

the Heston model. More details can be found in Fig. 4.1 of this work.

Right, bottom: Pictorial representation of the frontier of two-dimensional Brownian motion.

To Maria

To Silvana

E intanto il tempo passa e tu non passi mai...

Se potessi far tornare indietro il mondo

farei tornare poi senz’altro te

per un attimo di eterno e di profondo

in cui tutto sembra e niente c’ e

Negramaro, Estate & Immenso

Contents

Introduction 1

1 Why non-Gaussian models for market dynamics? 5

1.1 Financial markets: complex systems and stochastic dynamics . . . 6

1.2 The Black and Scholes-Merton paradigm . . . . . . . . . . . . . . 7

1.3 Stylized facts of real market dynamics . . . . . . . . . . . . . . . 9

1.3.1 Non-Gaussian nature of log-return distribution . . . . . . . 10

1.3.2 Random volatility . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 Price-volatility correlations . . . . . . . . . . . . . . . . . . 16

2 A non-Gaussian approach to risk measures 21

2.1 A brief introduction to financial risk . . . . . . . . . . . . . . . . 22

2.2 Non-Gaussian closed-form expressions for risk measures . . . . . . 25

2.3 Empirical analysis of financial data . . . . . . . . . . . . . . . . . 28

2.4 Risk analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Stochastic volatility models: the theoretical approach 37

3.1 Correlated stochastic volatility models . . . . . . . . . . . . . . . 38

3.2 The Vasicek model . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 The Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 The exponential Ornstein-Uhlenbeck model . . . . . . . . . . . . . 54

4 A comparative analysis of stochastic volatility models 61

4.1 Discretization algorithm . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Volatility and return processes . . . . . . . . . . . . . . . . 64

4.2.2 Leverage effect and volatility autocorrelation . . . . . . . . 69

4.3 Comparison between theory and empirical data . . . . . . . . . . 71

Conclusions and perspectives 77

A The zero-mean return features 81

iii

CONTENTS

B The Ornstein-Uhlenbeck process 85

Bibliography 89

Acknowledgement 95

iv

List of Figures

1.1 The leptokurtic and skewed probability density of Xerox stock pricedifferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 One-minute returns of Standard&Poor’s 500 index . . . . . . . . . 121.3 Volatility clustering in the Dow Jones index . . . . . . . . . . . . 131.4 Dow Jones index daily returns on the period 1900–2000 . . . . . . 141.5 Distribution of the empirical volatility fitted by the theoretical Log-

Normal and Inverse-Gamma distributions . . . . . . . . . . . . . . 151.6 Prices of real European call options on the German DAX index

versus the moneyness . . . . . . . . . . . . . . . . . . . . . . . . . 161.7 Smile-shape curve characterizing implied volatility . . . . . . . . . 171.8 Empirical form of the leverage effect . . . . . . . . . . . . . . . . 181.9 Empirical form of the volatility autocorrelation . . . . . . . . . . 19

2.1 Methodology to compute Value at Risk and Expected Shortfall . . 242.2 Convergence of VaR and ES Student-t formulae toward Gaussian

limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Sensitivity of VaR and ES to variation of the tail index value . . . 282.4 Comparison between empirical complementary cdfs of negative daily

returns and two theoretical fits (Gaussian and Student-t) for Au-tostrade SpA, Telecom Italia, Mibtel and Mib30 time series . . . . 30

2.5 Bootstrap histograms for the tail index and the RiskMetrics volatil-ity proxy for Autostrade SpA . . . . . . . . . . . . . . . . . . . . 32

2.6 VaR and ES values with 68% CL intervals (bootstrap evaluations),according to different methodologies, for the Italian asset AutostradeSpA and index Mib30 . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Characteristic function of returns in the Vasicek model . . . . . . 443.2 Probability density function of χ2 variables . . . . . . . . . . . . . 493.3 Stationary probability distribution of variance and characteristic

function of the return pdf in the Heston model . . . . . . . . . . . 513.4 Volatility autocorrelation in the expOU model . . . . . . . . . . . 58

4.1 Simulated volatility paths for the Heston model . . . . . . . . . . 65

v

LIST OF FIGURES

4.2 Stationary numerical volatility in comparison with the theory ofthe Vasicek, Heston and expOU models . . . . . . . . . . . . . . . 66

4.3 Numerical returns in comparison with the theory of the expOU model 674.4 Numerical returns in comparison with the theory of the Vasicek

and Heston models . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5 Numerical leverage effect in comparison with the theory of the Va-

sicek and expOU models . . . . . . . . . . . . . . . . . . . . . . . 704.6 Numerical volatility autocorrelation in comparison with the theory

of the Vasicek, Heston and expOU models . . . . . . . . . . . . . 704.7 Historical evolution of Fiat SpA, Brembo and Bulgari SpA shares

in both linear and log-linear scales . . . . . . . . . . . . . . . . . . 714.8 Fit to historical daily volatility distribution of Bulgari SpA and

Fiat SpA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.9 Probability densities of daily returns in comparison with the Nor-

mal, the Student-t and the theory of the expOU model for FiatSpA, Brembo and Bulgari SpA . . . . . . . . . . . . . . . . . . . 74

4.10 Cumulative density function of daily returns and negative dailyreturns for Fiat SpA, Brembo and Bulgari SpA . . . . . . . . . . 75

4.11 Probability densities of returns on different time lags in comparisonwith the theory of the expOU model for Fiat SpA, Brembo andBulgari SpA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

vi

List of Tables

2.1 Crossover values of the tail index for VaR and ES corresponding todifferent significance levels . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Estimated parameters values (mean, volatility and tail index) ofthe fitted curves of Fig. 2.4 for Autostrade SpA, Telecom Italia,Mibtel and Mib30 time series . . . . . . . . . . . . . . . . . . . . 31

2.3 Parameters values with the 68% CL intervals (bootstrap evalua-tions) for the time series as in Tab. 2.2 . . . . . . . . . . . . . . . 32

2.4 Estimated VaR values with 68% CL intervals (bootstrap evalua-tions), according to different methodologies, for the time series asin Tab. 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Estimated ES values with 68% CL intervals (bootstrap evalua-tions), according to different methodologies, for the time series asin Tab. 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Models of volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Theoretical features of the Vasicek, Heston and expOU stochastic

volatility models . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 Estimated mean values of Fiat SpA, Brembo and Bulgari SpA re-turn series for different time lags . . . . . . . . . . . . . . . . . . . 73

4.2 Estimated parameter values of the fitted curves shown in Fig. 4.9for Fiat SpA, Brembo and Bulgari SpA time series . . . . . . . . 74

vii

LIST OF TABLES

viii

Introduction

Physics and finance could seem, in principle, two different cultures.Nevertheless, reading the book of Emanuel Derman [1], one of the first physi-

cists to put physics into Wall Street, or the paper of Fabrizio Lillo, SalvatoreMicciche and Rosario N. Mantegna [2], the latter being one of the first physiciststo put finance into academical physics, could certainly help understand how onecan justify using the methods of physics and the formalism of mathematics in thefrenzied world of finance. Even so, no all the doubts could have been dispelled: ifthe value of a certain financial good is determined by people, how can the humanbehaviour be described by equations or predetermined rules? This could sound,perhaps, a more “philosophical” question, arising from the dissimilar nature ofknowledge of physics and social sciences. On this matter, we cannot forget theprophetical words of Ettore Majorana, who identified the analogy between physicaland social sciences into the statistical character of the laws describing elementaryprocesses, emerging in the framework of quantum mechanics [3]. More recently,Giorgio Parisi has remarked that, besides statistical physics and quantum me-chanics quoted by Majorana, another revolution has “shaken” the world of physicsduring the last century, that is the study of complex systems. Parisi writes [4]that these three revolutions...have changed the meaning of the word prediction andthe positive consequence of this process is that the scope of physics becomes much

larger and the constructions of physics find many more applications. The contri-bution of physicists to the study of economical markets must be interpreted in thelight of these words: complexity turns out to be, thus, the reading-key.

There are many natural phenomena which exhibit the typical features of com-plex systems, like earthquakes, the DNA, the traffic flow and, clearly, financialmarkets. Among the methods developed to study their complex dynamics, thetheory of stochastic processes still plays a crucial role in describing and modelingeconomical systems. Thus, it is not so surprising that the first contact betweenphysics and finance must be traced back to 1900, when Louis Bachelier, attempt-ing to reproduce the price dynamics of some goods traded in the Paris StockExchange, developed the model of random walk [5]. This physical phenomenon,which is the mathematical formulation of the Brownian motion [6], is commonlyconsidered by physicists as the archetype of stochastic processes.

1

Introduction

The relevance of Bachelier’s pioneering ideas, forgotten for a long time, hasbeen rediscovered within the economical community in the late ’50s and defini-tively hallowed by the works of Fisher Black and Myron Scholes [7], and RobertMerton [8] (BSM). In their theory of option pricing, they assumed that the fluctu-ations in the value of the underlying feature a geometric Brownian motion (GBM),a stochastic process strictly related to the original Brownian motion. Since its pub-lication in 1973, this theory has become a standard in the world of quantitativefinance, so that we could state that the history of quantitative finance and econo-physics after 1973 is, essentially, the history of the ways in which academics andfinancial practitioners have refined and extended the BSM model. From this view-point, the contribution of physicists has been fundamental, especially if we connecttheir increasing interest in financial issues to the vast amount of data (prices, or-ders, volumes, collected over time horizons ranging from a few minutes up to years)available thanks to the informatitation of exchange markets. The empirical studiesperformed by economists,mathematicians and physicists [9, 10, 11, 12] allowed toprove the existence of some mismatches between the simple statistical propertiesof the ideal GBM price dynamics and the more complex behaviour observed inreal markets, commonly denoted as stylized facts. Among them, we would remindthe proof that price returns feature a distribution different from the Gaussian onepredicted by the GBM process, where each event happens randomly and indepen-dently of all others. We will discuss this point in detail in Chapter 1. To copewith these non-trivial evidences, various theoretical models have been proposedin the literature, in order to go beyond the Gaussian paradigm and progressivelyrefine the statistical characterization of economical dynamics. These models havecontributed to the proliferation of different approaches followed by researches ineconophysics to interpret and, sometimes, try to forecast the variegate feature inthe behaviour of financial market, one of the old systems that the man has builtby means of his social organization.

The work presented along this thesis is placed into this scenario: its maintopics concern, in fact, models based on non-Gaussian stochastic dynamics forthe time evolution of price returns, as well as their possible applications in thefield of quantitative finance. To understand the necessity of such models for arealistic description of financial markets, in Chapter 1 we will discuss the BSMtheory, stressing the main empirical evidences not predicted within this standardframework. The increasing effort to extend models beyond this paradigm is closelyrelated to the possibility of capturing the true statistics of financial phenomena, aswell as achieving reliable results in typical financial problems. To prove this latterstatement, in Chapter 2 we will illustrate a possible application of the Student-tdistribution, a model emerging from the field of statistical physics, to risk man-agement, in order to obtain market risk measures in the presence of non-Gaussianfluctuations. Chapter 3 will be devoted to the family of stochastic volatility mod-els, a natural extension of the GBM paradigm. The models belonging to thisclass are characterized, in fact, by a random volatility driving the price returnsdynamics and driven on its own by a Wiener process. Among them, we will deal

2

Introduction

with three popular models (Vasicek, Heston and exponential Ornstein-Uhlenbeckones), whose theoretical features will be described in detail in Chapter 3 andtested ab initio by means of numerical simulations in Chapter 4. This is indeed acrucial task, especially in view of financial applications, like option pricing or riskmanagement, often investigable only through numerical strategies. To conclude,the latter part of Chapter 4 will be dedicated to an empirical analysis of somefinancial time series from the Italian stock market. The aim of this analysis isto critically study the models presented in Chapter 3, evaluating their degree ofrealism once compared with real data. The results emerged from our works, aswell as some possible perspective, will be summarized at the end of the thesis.

The studies discussed in the present work have been developed in collaborationwith Giacomo Bormetti, Lorenzo Fermi, Guido Montagna and Oreste Nicrosini.They are documented on the following publications:

• G. Bormetti, E. Cisana, G. Montagna and O. Nicrosini, A Non-Gaussian

Approach to Risk Measures, Physica A 376 (2007) 532.

• G. Bormetti, E. Cisana, G. Montagna and O. Nicrosini, Risk Measures with

non-Gaussian Fluctuations, physics/0607247.

• E. Cisana, L. Fermi, G. Montagna and O. Nicrosini, A Comparative Study of

Stochastic Volatility Models, arXiv:0709.0810 [physics.soc-ph], submitted tothe Proceedings of Applications of Physics in Financial Analysis 6 (APFA6)conference (Lisbon July 4 - 7 2007).

They have been also presented in the form of poster(p) and talk(t) in the followinginternational meetings:

• (t) Risk Measures with non-Gaussian Fluctuations, Applications of Physicsin Financial Analysis 5, Torino June 29 - July 1 2006.

• (t) Non-Gaussian Risk Management, International School of Complexityand Socio-Economics Phenomena, Erice September 17 - 23 2007.

• (p) A Comparative Study of Stochastic Volatility Models, Applications ofPhysics in Financial Analysis 6, Lisbon July 4 - 7 2007.

• (p) A Comparative Study of Stochastic Volatility Models, Econophysics Col-loquium, Ancona September 27 - 29 2007.

3

Introduction

4

Chapter 1Why non-Gaussian models for

market dynamics?

The main topic of this work is the application of non-Gaussian stochastic mod-els to the characterization of real financial dynamics. Why is it so necessary toemploy such models to describe in a realistic way financial markets? The aimof the present chapter is to illustrate the motivations behind this question and,therefore, the subjects discussed along this thesis.

It has been shown by many studies [2, 11, 12, 13] that financial marketsare complex systems, thus investigable within the framework of stochastic pro-cesses. Thanks to the last decades’ informatization of exchange markets, therehas been an increasing research of models able to capture carefully the statisticalfeatures shown by financial variables, like price changes and their correspond-ing log-returns, volatility, volumes,. . .We agree with Bouchaud and Potter, whichpoint out in Ref. [12] that the word modeling can be interpreted in two differentmeanings within the scientific community. The first one, developed in this work,consists in describing the observed reality by means of theoretical mathematical-based models. The second one, harder to achieve, means to find a set of causes ableto explain the reality and to justify the chosen mathematical formalism. Namely,these models try to reproduce the market dynamics in term of microscopic (i.e.agent based) mechanisms (see, for example, Ref [14, 15, 16, 17]).

The geometric Brownian motion (GBM), still considered as a standard in theprice statistics modeling, belongs to the first class. Within this framework, stockprice features a Log-Normal distribution, whereas the difference between the log-arithm of prices (log-returns) perform a Brownian motion, turning out to be nor-mally distributed. The GBM plays a key role in the quantitative finance, sinceit is the basis model for the underlying price dynamics in the Black and Scholes-Merton (BSM) theory of option pricing, commonly used in the financial practice.However, many empirical analyses [11, 12] have pointed out that the GBM is ableto provide only a first approximation of the true statistics of price, since it failsin predicting several established phenomena observed in real markets. As we willsee in Section 1.3, they concern the non-Gaussian nature of log-return probability

5

1. Why non-Gaussian models for market dynamics?

distribution function (pdf), the non-constant character of volatility, as well as thedifferent-time correlations of the price returns or volatility processes itself. Tocope with these non-trivial stylized facts, various theoretical models have beenproposed in the literature, based on non-Gaussian dynamics for the time evolu-tion of price log-return.

The aim of this chapter is to deal with the main features of financial markets,stressing the discrepancies between the too idealized GBM, the present paradigmof quantitative finance, and the behaviour exhibited by prices in real markets.This is organized as follows: in Section 1.1 we describe financial markets, theircomplex nature and, therefore, the possibility of modeling them in a stochasticapproach. A brief historical outline, from the pioneering work of Louis Bachelierup to the BSM model, is also given. The latter is shown in Section 1.2. To con-clude, in Section 1.3 we illustrate the main mismatches between the GBM theoryand the stylized facts which characterize real markets and lead, therefore, to theintroduction of non-Gaussian stochastic dynamics for markets modeling.

1.1 Financial markets: complex systems and sto-

chastic dynamics

It is well documented by many studies [2, 11, 12, 13] that financial markets ex-hibit the typical features of complex systems, since they are open and made upby numerous elements that continuously interplay. One could think about thestock market and its continuous succession of shares sold and bought by investors,strongly influenced by economic, political and social factors. Thus, the complexbehaviour of markets can be studied within the framework of the statistical andtheoretical physics, like many other natural phenomena (diffusion, chaos, forma-tion of avalanches, earthquakes,. . . ). In particular, the development of the theoryof stochastic processes has played a key role in the description of such systemsfrom a physical point of view.

The origin of financial dynamics modeling with a stochastic, “statistical - me-chanical” approach can be dated back to 1900, when the young French mathe-matician Louis Bachelier presented his PhD thesis Theorie de la speculation atthe Sorbonne, University of Paris [5]. In this work, attempting to model the pricedynamics of some derivatives traded in the Paris Stock Exchange, he developedthe theory of random walk, five years before the famous Einstein’s interpreta-tion of the Brownian motion [6] 1. As known, Brownian motion, as well as itsmathematical formulation, i.e. the random walk, turn out to be the archetypesof the stochastic processes. In the 1908 Paul Langevin, trying to explain theBrownian motion phenomenon, derived the first example of stochastic differentialequation [18], which is currently the basis of the stochastic calculus.

The pioneering work of Bachelier has been forgotten for at least 60 years, even

1For a brief introduction to the physical phenomenon of Brownian motion and the Ornstein-Uhlenbeck process, the model describing its velocity process, see Appendix B

6

1.2. The Black and Scholes-Merton paradigm

if it’s worth mentioning that K. Ito has quoted it as one of the main motivationsto introduce his stochastic calculus and the geometric Brownian motion [19]. Therelevance of Brownian motion was rediscovered in the financial community in thelate 50’s, thanks to the studies performed by M. Osborne [20], P. Samuelson andR. Merton [21, 22], as a good candidate in reproducing the dynamics of pricechanges and their statistical features. Physics is becoming aware of Bachelier’swork only now through the meet between statistical physics and quantitative fi-nance.

In 1973 F. Blach and M. Scholes [7] and, independently, R. Merton [8] devel-oped a framework to determinate the fair price of an option assuming that thefluctuations in the value of the underlying feature a geometric Brownian dynam-ics, thus reducing this problem to a diffusion equation. The model has becomea standard in the theory of option pricing and, more generally, within the over-all framework of quantitative finance, contributing to the awesome growth of thediscipline during the last 30 years. For this reason, in the next section we brieflydiscuss the BSM model, even if in this work we haven’t dealt with problems ofoption pricing.

1.2 The Black and Scholes-Merton paradigm

Since its publication, the BSM theory has become the paradigm of the quanti-tative finance applied to complete markets. As already mentioned, this modelsstarts from the basic assumption that the asset price follows a typical dynamicsof the GBM, a stochastic process developed within the framework of finance and,unlike the Bachelier’s Brownian motion, with no particular relevance in physicalsciences. It could be interesting, therefore, to briefly discuss this process beforepassing to BSM model for option pricing. A more exhaustive illustration can befound in Ref. [23], on which what follows is based.

In GBM framework, the price S(t) is evolved according to the following sto-chastic differential equation (sde)

dS(t) = µS(t)dt + σS(t)dW (t) , (1.1)

where W (t) is a Wiener process, whereas µ and σ represent respectively the con-stant instantaneous drift and standard deviation rates. Actually, the process (1.1)is referred as GBM. It’s easy verify that applying the Ito lemma [23] to the loga-rithm of S(t), commonly known as log-return, Eq. (1.1) yields

d(lnS(t)).= dX(t) =

(µ − σ2

2

)dt + σdW (t) . (1.2)

Eq.(1.2) is the typical sde that characterizes a generalized Wiener process; it tellus that the logarithm of price follows a Brownian motion with an effective drift(µ − σ2

2

)and variance σ2dt. Recalling the theory of stochastic processes, we

can conclude that price log-returns are distributed according to a non-zero mean

7


Normal, i.e. X ∼ N((µ − σ2

2)dt, σ2dt

).

As a results, the price S(t) features a Log-Normal distribution

p(S, t|S0, t0) =1

S√

2πσ2(t − t0)exp

− [ln(S/S0) − (µ − σ2/2)t]

2

2σ2(t − t0)

, (1.3)

characterized by mean and variance given by

E [S(t)] = S0 eµ(t−t0) , (1.4)

andVar [S(t)] = S2

0 e2µ(t−t0)(eσ2(t−t0) − 1

). (1.5)

It’s worth mentioning that such a result has allowed to go beyond the majorproblem posed by the original Bachelier’s random walk: the possibility of negativeprices, emerging from the predicted Normal distribution featured by the prices. Wewould also note that both the processes S(t) and X(t) are affected by the samesource of uncertainty, that is the stochastic process dW .

Eq. (1.1) constitutes one of the two main assumptions of the BSM model.Namely, they consider a portfolio Π(S, t) composed of a long position in a calloption C(S, t) and a short position in ∆ shares of the underlying S(t)

Π(S, t) = C(S, t) + ∆(S, t)S(t) , (1.6)

where the price underlying dynamics is governed by a GBM. The other mainhypothesis concerns the absence of arbitrage. This assumption is one of the lead-ing argument in the derivation of BSM results, since it is strictly related to theconcept of efficient market, namely a market in which all participants are able toobtain all the information necessary for trading, the liquidity is high and there areno transition costs. Besides these two fundamental hypothesis, other ones, moretechnical, are assumed in the model, as the possibility of continuously trading orthe absence of credit risk.

The aim of BSM analysis was to solve the problem of finding the fair price ofa plain vanilla option. As known, an option is a derivative financial instrument,whose price depends on the value assumed in time by an underlying asset, sim-ply called underlying. A complete description of financial derivatives, in all theirinnumerable and strange forms and features, can be found in Ref. [24]. However,it’s worth stressing that the derivative price strongly depends on the stochasticdynamics chosen for the price underlying. As mentioned above, in this frameworkit turns out to be the GBM.

In their works, BSM have demonstrate the possibility to remove at all therisk from Π(S, t) (portfolio hedging) by choosing dynamically ∆ = ∂C

∂S, in order to

rebalance the fluctuations of Π(S, t) and makes its evolution deterministic. In con-sequence of the hedging, the portfolio must evolve, as a bank account, accordingto

dΠ(S, t) = rΠ(S, t)dt , (1.7)

8

1.3. Stylized facts of real market dynamics

where r is the risk-free interest rate. Thus, the resulting sde for the option pricebecomes a deterministic partial differential equation, the popular Black-Scholesequation, given by

∂C

∂t+ rS

∂C

∂S+

1

2σ2S2 ∂2C

∂S2= rC . (1.8)

We would note that Eq. (1.8) contains only the parameters entering the underlyingprocess sde (1.1), namely the constant σ and µ; indeed, the latter is substitutedwith r because of the hedging.

The Black-Scholes equation can be solved by means of the Feynman-Kac for-mula specifying the proper final condition, given by the option value at the expirydate (maturity) T

C(S, T ) = max(S(T ) − K, 0) . (1.9)

where K is the strike price, namely the underlying asset price fixed when theoption is written, whereas S(T ) is the spot price, namely the effective value of theunderlying asset at the maturity. Calculating the expectation value of Eq. (1.8)on the stochastic process for S, which is the Log-Normal distribution given inEq. (1.3), the fair call option price reads

C(S, t) = e−r(T−t)E [max(ST − K, 0)|St = S]

= e−r(T−t)

∫ ∞

K

dS ′ (S ′ − K) pLN(S ′, T |S, t). (1.10)

The presence of the extra term leads to the appearance of the discount factore−r(T−t). The final result is

C(S, t) = SN(d1) − Ke−r(T−t)N(d2) (1.11)

where N(x) is the Normal cumulative function and

d1,2 =ln(S/K) + (r ± σ2/2)(T − t)

σ√

T − t. (1.12)

1.3 Stylized facts of real market dynamics

Despite the widespread success of the BSM theory, many academics and financepractitioners believe that this model is too idealized to describe real market phe-nomena, because of its unrealistic assumptions and the chosen asset price dynam-ics. Many empirical studies [12, 25, 26, 27] have argued the presence of somemismatches between the simple statistical properties of the ideal GBM prices andthe more complex behaviour observed in real market. These stylized facts, whoseevidence arose in the past 30 years, concern the leptokurtic nature of price re-turn distribution, the non-constant character of volatility and the existence ofcorrelations of price changes and volatility at different times.

9


1.3.1 Non-Gaussian nature of log-return distribution

In the previous section, we have seen that the main theoretical prediction of theGMB model is that the logarithm of prices would feature a Gaussian densitybroadening in time. Actually, it is well known that empirical price returns, espe-cially in the limit of high frequency, do not follow the Gaussian paradigm andare characterized by heavier tails and a higher peak than a Normal distribution.Such a deviation has been noted for the first time by B. Mandelbrot, even beforeSamuelson’s formalization. After having collected a sufficient amount of data onseveral US cotton exchanges [9], he proposed that the price changes distribution forcommodities, goods whose market is much less liquid than the stock one, was notGaussian but Levy-like. This stable distribution is known to posses variance andhigher-order moments infinite and to verify a generalized Central Theorem Limit,being an attractor in the functional space of pdfs [11]. Mandelbrot’s intuitionshave been confirmed not long after also for stock prices, thanks to the investiga-tions of E. Fama [10]. The last decades’ development of information technology inthe financial markets and the consequent enormous body of empirical research onthis topic, have provide strong confirmations of the non-Gaussian nature of pricereturns, especially at infra-day and daily time horizons. Contributions by R.N.Mantegna and H.E. Stanley [25], for example, have shown unambiguously that theshort time dynamics of prices breaks the Gaussian scaling. In the same work theauthors have also pointed out the finiteness of the second moment of price returnsprobability density, a matter of debate in finance for several year. The discoverof this evidence has been peculiar in order to carefully characterize the price pro-cess. Moreover, other studies [28] have also proved that the distribution’s tailstend to exhibit features quite different from the central body ones. They showin fact a typical power-law behaviour, namely P (x) ∼ |x|−α with α ' 4, a valuein agreement with the requested finiteness of the second moment. However, asclearly explained by H. Stanley in his talk at APFA6 Conference (Lisbon, July2007), in order to quantify with accuracy the tail behaviour, a very huge amountof data is strongly demanded.

To cope with these empirical evidences, many theoretical models, often basedon non-Gaussian stochastic dynamics, have been proposed in the literature at-tempting to capture the leptokurtic nature of log-returns. A first example concernthe so-called Truncated Levy Flights [29], models in which the pdf central bodyfeatures a stable Levy distribution, whereas the rare events have a typical power-law behaviour. In this way, the trouble of an infinite variance affecting the originalLevy distribution can be avoid at all, allowing a better characterization of the tailsdynamics. Another model often used in the literature is the Student-t distribu-tion [30, 31], because of its ability in reproducing with good accuracy the pricereturns distribution at short time horizon, especially the power law character ofthe tails, thanks to its strong leptokurtic nature. In the econophysics literature,the Student-t is also known as Tsallis distribution, emerging within the frameworkof statistical physics [32]. In Chapter 2 we will illustrate a possible applicationof this model to risk management, in order to achieve market risk measures in

10


the presence of non-Gaussian fluctuations. Finally, the evidence that volatility isnon constant, the main topic discussed in the next subsection, but shows itselfan own dynamics, has led to the development of ARCH and GARCH models [33]and stochastic volatility ones, which will be described in detail in Chapter 3.

To conclude, it’s worth recalling that to quantify the deviation of empiricaldistributions respect to the Gaussian, higher-order moments are often used. Inparticular, they allow to measure the pdf asymmetry and the “fatness” of thetails emerging once compared real data with the Gaussian paradigm. Looking atFig. 1.1, one could note in fact a slightly stronger concentration of probability

Figure 1.1: High frequency price differences (points) of the Xerox stock traded inthe NYSE [11]. The pdf is clearly more leptokurtic and skewed than the Gaussianone (solid line).

in the left tail of the empirical distribution respect to the right one. The figurerefers, in particular, to high-frequency price differences of the Xeros stock tradedin the New York Stock Exchange; it’s worth mentioning that a more evident asym-metry can be registered by observing mid-time (i.e. weekly, for example) data.This feature can be quantified through the third cumulant of the distribution,whose normalized version is the so-called skewness γ1. In general, we can obtainhigher-order moments making use of the cumulants of a distribution, defined as

κn = (−i)n ∂n

∂wnln [ϕ(ω, t)]

ω=0

, (1.13)

since the relationships between moments and cumulants are known. Note thatthe function ϕ(ω, t) entering Eq. (1.13) represents the characteristic function of ageneral pdf p(x, t). The skewness, in particular, can be expressed in term of thesecond and the third cumulants as

γ1.=

κ3

κ3/22

. (1.14)

11


Figure 1.2: One-minute returns of Standard&Poor’s 500 index (points) comparedwith the Gaussian, an inverted parabola in lin-log scale (dotted line), and theLevy distribution (solid line) with index α=1.40 [25].

This moment gives information about the asymmetry of the probability density:specifically, negative values of γ1 tell us that the left tail is heavier than theright one and, conversely, positive values characterize a distribution with the righttail fatter than the left one. Consequently, the skewness reduces to zero in theGaussian case. It has been observed in particular that, starting from a null returntime horizon (at which the pdf is simply a Dirac delta), the skewness grows upto a maximum reached at a time scale of a few tens of days and then decreasesvery slowly to zero, coherently with the Central Limit Theorem that requests aGaussian density when t → ∞.

Looking at Fig. 1.2, we could appreciate the presence of tails heavier than theGaussian ones; this fact indicates that the empirical distribution has an excessof kurtosis, a typical sign of a non-Gaussian nature. To quantify the fatness ofthe tails respect to the Normal pdf, the fourth moment, called kurtosis γ2, hasbeen introduced. This moment can be computed using the second and the fourthcumulant as

γ2.=

κ4

κ22

. (1.15)

Specifically, for short time scales the pdf is pronouncedly leptokurtic with γ2 > 0:the tails are initially much heavier respect to the Gaussian case. The empiricalkurtosis decreases monotonically, starting from a maximum and converging tozero a bit faster than skewness.

The topics discussed along this section prove that high-frequency returns vi-olate completely the Gaussian paradigm at least up to scales of ∼1000 minutes.In particular, real log-return distributions exhibit a mid-time (i.e. weekly data)“pessimistic” asymmetry, since the left tail, where losses reside, is fatter than theright one, and a short-time (i.e. infraday-daily data) rare event probability much

12


larger than the one predicted by the Gaussian.

1.3.2 Random volatility

A crucial assumption in the Black and Scholes model is that the volatility σ, whichmeasures the strength of the price fluctuation, is a constant parameter. Neverthe-less, observing real data leads to quite different conclusions: also the statisticalfeatures of volatility are characterized by stylized facts.

Fig. 1.3 shows the daily returns of Dow Jones (DJ) Industrial Average index ona period of approximately 40 years (the true temporal range is not carried in thereference [34]). Although the period is relatively quiet and without crashes, thefluctuation amplitude is quite variable. This behaviour is of course greatly ampli-

Figure 1.3: Dow Jones index daily returns on a 40-years period showing thetendency of volatility to come in bursts [34].

fied in correspondence of financial crashes. Fig. 1.4 displays the same DJ returnsover the entire twentieth century: besides the periodical ‘pulse’ in volatility, wildoscillations following the 1929 and, on a less catastrophic scale, the 1987 crashesare clearly visible. In general, one can notice that the fluctuation amplitude, andthus the volatility, tends to increase suddenly, stay high for a limited time inter-val, ranging from few hours to several months, and then go back to a situation ofmore stable prices. Volatility seems to come in bursts and concentrate in time.This phenomenon, usually referred to as volatility clustering or heteroskedasticity,reflects the multi-scale nature of volatility.

The observed variable character of volatility has encouraged authors to studyits statistics. This implies facing the problem of its estimation, since volatility it-self cannot be directly observed or, more commonly, is a hidden process. Measuresof volatility have to be derived from ‘proxies’ calculated from the price changes,which conversely can be measured on real markets. Recalling that in the GBMtheory log-returns distribution is characterized by V ar[X] = σ2dt, we can imme-diately conclude that

σ2 ' Var [X]

t, (1.16)

where we have used the so-called zero-mean returns X, which we will better de-scribe in Chapter 3 within the framework of stochastic volatility models. Basically

13


Figure 1.4: DJ index daily returns on the period 1900-2000: the 1929 and 1987crashes are clearly visible [12].

Eq. (1.16) implies that the volatility can be estimated using time series of pricescollected on a fixed time horizon ∆t. In such a way, we can evaluate the historicalvolatility, given by

σ2his =

1

(N − 1)∆t

N∑

k=1

[ln

Sk+1

Sk

− m

]2

, (1.17)

where N is the length of the prices time series Sk (k = 1, .., N) and m is itsmean, estimated as empirical moment (see Eq. (2.8) in Chapter 2). However,such proxy can be considered effective only over limited time intervals, otherwiseit becomes very inaccurate. When the information on the time at which volatilityis estimated must be retained, like in the case of volatility autocorrelation, theso-called instantaneous volatility can be defined by introducing

lim∆t→0

√[X(t + ∆t) − X(t)]2

∆t. (1.18)

This equation must be taken as an infinitesimal difference, leading to the followingexpression for the instantaneous volatility

σist(t).=

√dX(t)2

dt. (1.19)

On the other hand, when high-frequency data are available, we can introduce thehigh-frequency volatility, defined by Bouchaud and Potters [12] as

σhf =1

N∆t

N∑

k=1

|∆Sk| , (1.20)

where ∆t is of the order of minutes. It’s worth recalling that for high-frequencydata price changes, returns and log-returns can be taken as equivalent variables. Inparticular, for daily data, namely ∆t = 1 day, volatility can be estimated directly

14


from historical daily series as absolute return; this quantity is commonly knownas empirical daily volatility.

Using the high-frequency proxy of Eq. (1.20), the same authors have foundthat the empirical volatility distribution function is equally well approximated bya Log-Normal and an inverse-Gamma distribution. This result is shown in Fig. 1.5;however, it must be noticed that the Log-Normal tends to underestimate the largetail of the measured distribution. Today, there is an almost general consent thatthe empirical volatility could be modeled through a Log-Normal density in itsbody, whereas large values could be characterized by a power-law tail. This isindeed a stylized fact about volatility.

Figure 1.5: Distribution of the measured σhf of the S&P 500 index fitted by aLog-Normal curve (dotted line) and by an inverse-Gamma (dashed line). Theformer somewhat underestimates the tail, while the latter overestimates it (noticethe lin-log scale) [12].

Another corroboration of the non-constant nature of real volatility comes fromthe so-called volatility smile. In a BSM scenario, the fair price of a call optiongiven in Eq. (1.11) depends only on the current spot price S, or equivalently onthe moneyness m

.= K/S, once fixed the maturity T and the strike price K. The

same is true also for a put option. Therefore, all the options with equal T and Kon different underlyings should fall onto the same moneyness-price curve.

In real markets this is not the case, as demonstrated in Fig. 1.6. Since theofficial interest rate r is fixed, the only other parameter involved is the volatility,that seems to change with the underlying price. Indeed, another strategy toestimate the unobservable volatility is to invert the BS formulae for a certainoption O (put or call) taking for granted the real market price at which this is

15


Figure 1.6: Prices of European call options on the German DAX indexat one month to maturity (expressed in units of the index value) versusmoneyness K/S. The two solid lines represent the BS relation for σ=35%yrs−1/2 (upper) and σ=20% yrs−1/2 (lower) [23].

exchanged. Namely, one can define an implied volatility σimp imposing the equality

Omarket(S, t; r; K, T ).= OBS(S, t; r, σimp; K, T ) , (1.21)

and work out its dependence on the moneyness. The result is a smile-shaped curvecentered on m = 1, i.e. S = K. This fact indicates that the risk perceived bythe option writer, proportional to the price asked to the holder, increases if thespot price deviates from K. The slope of the two branches depends also on thematurity, being stronger for smaller T .

Until the 1987 crash, the first relevant one after the institution of the optionexchange, the curve was a proper smile, symmetrical with respect to m=1. Afterthe crash the market has settled so as to generate rather smirky curves, like theone displayed in Fig. 1.7, namely with the half corresponding to the unfavorablesituation for the holder (m <1 branch for call options, the opposite for puts) raisedwith respect to the other. This reflects an increased worry against losses, whichgenerate an increase in the effective volatility.

1.3.3 Price-volatility correlations

Another well established stylized facts concern the existence of non-trivial corre-lations between price changes at different times, the so-called leverage effect, aswell as the autocorrelation of the volatility process itself.

The leverage effect was notice for the first time by F. Black [36] in the mid’70s. It consists of a non-zero correlation between past price changes and futurevolatilities. Its name is due to the jargonistic name for the ratio of a company’sdebt to its current capital: when the latter falls, the leverage rises, and so doesthe investors’ mistrust in the company.

16


Figure 1.7: Implied volatility versus the quantity K−S for a set of optionswith a maturity of 1 month: the smirk effect is clearly evident [35].

To quantify this effect, the leverage function L(τ) has been introduced [37]and defined through

L(τ).=

E [σ(t + τ)2dX(t)]

E [σ(t)2]2. (1.22)

Estimating the (squared) volatility with the instantaneous proxy of Eq. (1.19),the previous equation becomes

L(τ).=

E [dX(t + τ)2dX(t)]

E [dX(t)2]2. (1.23)

In the same work [37] the authors, analysing the daily relative changes of some437 US stocks as long as 7 major international indexes, have found that L(τ) iswell described by the following empirical law

L(τ) =

−Ae−bτ if τ > 0

0 if τ < 0, (1.24)

with A > 0. Fig. 1.8 proves the very good agreement between empirical data andthe derived Eq. (1.24). Hence, there is a negative correlation with an exponentialtime decay between past price changes and future volatilities, but no correlationis found between past volatilities and future price changes. In other words, a sortof causality exists in the leverage effect dynamics.

Another interesting evidence as regards the leverage effect concerns the exis-tence of some “connections” between the leverage itself, the volatility smile andthe skewness. The asymmetry observed in the volatility smile effect could be,in fact, controlled by a negative skewness in the price return distribution, whichcould be generated, in turn, by a negative correlation between price changes and

17


Figure 1.8: Comparison between the analytical expression of leverage ef-fect (solid line) given in Eq. (1.24) and the empirical correlations (points),averaged over 7 major stock indexes. The solid line clearly shows an ex-ponential trend [37].

volatility, namely the leverage effect. Thus, these three stylized facts could ac-tually be generated by the same aspect of the underlying dynamics. As for thepresence of the smile, it can be explained in term of a non-ability of BSM theoryin capturing carefully the rare events of the pdf. In this scenario, the smile couldbe also related to an excess of kurtosis of the return distribution. This fact wouldconfirm again the results discussed in Section 1.3.1, since the observed kurtosis isstronger at short time horizons as well as the smile is more marked for optionswith close maturity date.

Another established stylized fact regards the time correlation of the estimatedvolatility, an effect investigated mainly trough the ’90s. Analogously to Eq. (1.23),a convenient empirical quantity introduced to quantify this autocorrelation is

C(τ).=

〈σ(t)2σ(t + τ)2〉 − 〈σ(t)2〉2Var [σ(t)2]

=〈dX(t)2dX(t + τ)2〉 − 〈dX(t)2〉2

〈dX(t)4〉 − 〈dX(t)2〉2 ,

(1.25)

where the second equality is obtained substituting the volatility with its instanta-neous proxy (1.19). Empirical investigations [38, 39] have proved that the autocor-relation function evaluated from real data turns out to be positive and character-ized by a double time scale: a short one of the order of a few tens days and a muchlonger one amounting to hundreds of (financial) days. Fig. 1.9 demonstrates thepresence of both the scales within the empirical volatility autocorrelation func-tion estimated from DJ data. Therefore, the dynamics of volatility cannot beaccounted for using a single time-scale: its fluctuation is a multi-time-scales phe-nomenon.

This characteristic double time scale, one of the most difficult feature to bereproduced by current financial stochastic models, seems to reflect the presence of

18


two different kinds of market’s reaction to the variability of an asset. In the fewdays following a volatility increase, the volatility itself tends to be higher, becauseinvestors are more wary of the asset’s behaviour and react faster to new changes.This ‘impulsive’ phase finishes quite soon, but a certain amount of cautiousnesslasts for a long time.

Figure 1.9: The empirical form of the volatility autocorrelation (points) showsclearly the presence of a double exponential time-scales, necessary to model itcarefully [39].

19


20

Chapter 2A non-Gaussian approach to risk

measures

In the previous chapter we saw that the diffusion process unanimously acceptedas the most universal model for speculative markets, i.e. geometric Brownianmotion, fails to predict several established facts observed in the price variations,like the leptokurtic nature of log return probability density. The increasing ef-fort to extend the models beyond the Gaussian paradigm is closely related to thepossibility of capturing the true statistical properties of financial markets, i.e. thestylized facts previously described. Moreover, it’s well known that non-Gaussianapproach has strong influences also in several financial applications, like optionpricing [40], allowing evaluations more reliable and closer to the real nature offinancial data. Nevertheless market analysts and operators often seem not to payparticular attention to these topics.

Within this scenario, the aim of the present chapter is to present a careful anal-ysis of financial market risk measures in term of a non-Gaussian model for pricefluctuations. It’s worth underlying that the true financial risk resides in the rareevents of the probability distribution. Thus modeling with a great accuracy thetails of the probability distribution functions is strongly demanded. To this end,we have made use of the Student-t distribution mentioned in Section 1.3.1. Theapproach we developed leads to estimates for the risk associated with a singleasset in good agreement with a full historical evaluation and thus utilizable alsoin the financial practice.

This chapter is based on the work discussed in [41, 42] and is organized as fol-lows. In Section 2.1 a brief introduction to financial risk and the most widely usedmeasure of market risk -Value at Risk (VaR) and Expected Shortfall- is given. Af-ter it, in Section 2.2 non-Gaussian closed-form expressions for VaR and ExpectedShortfall are derived as generalizations of the analytical formulae known in theliterature under the normality assumption. It is also shown how the standardGaussian formulae of the parametric approach are recovered, in the appropriatelimit, as a special case. In Section 2.3 an empirical analysis of daily returns seriesfrom the Italian stock market is performed, in order to constrain the Student-t

21

2. A non-Gaussian approach to risk measures

parameters entering the formulae of Section 2.2 and to describe the ingredientsneeded for the fore-coming risk analysis. The latter is carried out in Section 2.4.The implications of the parametric non-Gaussian approach for VaR and ExpectedShortfall are shown in Section 2.4 and compared with the results of the parametricnormal method, of its improved version known as RiskMetrics methodology andof the historical simulation. Particular attention is paid to quantify the size of theerrors affecting the various risk measures, by employing a bootstrap technique.

2.1 A brief introduction to financial risk

A topic of increasing importance in modern economy and society is the develop-ment of reliable methods of measuring and controlling financial risk. One of themain question about it is where risk comes from. The sources of risk are various: itcould be human-generated, such as inflation, business cycles, government policiesor wars indeed. It can also occur from unforeseen natural events like earthquakes,or arises from the movements in financial markets, the long-term economics growthor technological innovations. The sources of risk are not likely to be eliminated;quoting some words of Walter Wriston, former chairman of Citicorp, one couldconclude that All of life is the management of risk, not its elimination. Thus,risk management is defined as the process by which different risk exposures areidentified, measured and controlled.

According to the new capital adequacy framework, commonly known as Basel IIaccord [43], any financial institution has to meet stringent capital requirements inorder to cover the various sources of risk that they incur as a result of their normaloperation. Basically, three different categories of risk are of interest: credit risk,operational risk and market risk. In particular, market risk concerns the hazardof losing money due to the fluctuations of the prices of those instruments enter-ing a financial portfolio and is, therefore, particularly important for financial riskmanagement.

Although in modern parlance the term risk is strictly related to the danger ofloss, financial theory defines it as the dispersion of unexpected outcomes due tothe financial market movements. In the light of this assumption, traditionally theconcept of risk has been related to the volatility, which is known to quantify thefluctuations of a financial instrument’s price around its mean value. Also today,the volatility is often chosen as measure of risk associated to a given investment.It’s worth noting that in such way both positive and negative deviations shouldbe viewed as potential source of risk. This is indeed wide of the mark. More-over, the measure of risk in term of volatility suffers from other inconsistencies,as clearly pointed out in Ref. [12]. In particular, this risk indicator is untimelyrelated to the idea that the distribution of price change is Gaussian. In the case ofLevy fluctuation, this definition is straight meaningless since this distribution ischaracterized by a infinite variance. Moreover, Gaussian models fail to reproducethe rare events of the probability distributions when apply to finite data series(which corresponds to the financial reality). Now, it is well known that the ex-

22

2.1. A brief introduction to financial risk

treme events lead to the risk and, thus, the most important factor determiningthe probability of extreme losses is to model the distribution functions makinguse of non-Gaussian models. From this discussion, clearly emerges the necessityof another definition of risk which gets over the concept of volatility.

Today, in the financial industry, the most widely used measure to manage mar-ket risk is Value-at-Risk [12, 44]. This method was developed in response to thefinancial disaster of early 1990s, arisen from the poor supervision and managementof financial risk, that gave rise to losses of billions of dollars in many financial in-stitutions. After this fact, several financial regulators and risk managers turned toVaR, an easy-to-understand method for quantify market risk. In short, VaR refersto the maximum potential loss over a given period at a certain confidence leveland can be used to measure the risk of individual assets and portfolios of assetsas well. VaR has become a standard component in the methodology of academicsand financial practitioners, because it is an easy concept which can provide a rea-sonably accurate estimate of risk at a reasonable computational time. Moreover,VaR provides a summary measure of market risk. This is indeed one of the mainadvantages of this method. To better clarify this point, we can make use of thefollowing example. A bank may say, for instance, that the daily VaR of its tradingportfolio is 20 million euro at 99% confidence level. This means that under normalmarket conditions, the most the portfolio can lose in a day is 20 million euro and itcan occur just 1 change in a 100. This statement illustrates how VaR summarizesthe various bank’s exposures to the risk in a single number, easily communicablealso to a nontechnical audience. In the left panel of Fig. 2.1 the practice way tocompute VaR, denoted as Λ?, is shown.

Still, as discussed in the literature [12, 45], VaR suffers from some inconsis-tencies: first, it can violate the sub-additivity rule for portfolio risk, which is arequired property for any consistent measure of risk, and, secondly, it doesn’tquantify the typical loss incurred when the risk threshold is exceeded. In otherwords, the VaR method doesn’t take into account the fact that the losses can ac-cumulate in time, leading to a overall loss which might exceed the VaR value. Toovercome the drawbacks of VaR, the Expected Shortfall (or Conditional VaR)is introduced, and sometimes used in financial risk management, as a more co-herent measure of risk. Unlike VaR, Expected Shortfall (ES) allows to quantifythe average size of the loss when the cutoff value is hit. In the right panel ofFig. 2.1 the methodology to compute ES, denoted as E?, is shown in comparisonwith the corresponding Λ? value, both calculated at the 5% significance level: thedifferences in computing, and thus using, VaR and ES as risk measures clearlyemerge. Moreover, it’s quite evident the necessity to model the distribution’s lefttail, where the risk of losses resides, in a very accurate and reliable way.

Three main approaches are known in the literature and used in practice forcalculating VaR and Expected Shortfall. The first method consists in assumingsome probability distribution function for price changes and calculating the riskmeasures as closed-form solutions. This approach is called parametric or analyt-ical and is easy to implement since analytical expressions can often be obtained.

23


0

5

10

15

20

25

30

35

40

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02

PSfrag replacements

1%

5%

Λ?= 0.0233

Λ?= 0.0164

Pdf

0

5

10

15

20

25

30

35

40

-0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01

PSfrag replacements

P?5%

Λ?= 0.016

E?= 0.021

Pdf

Figure 2.1: Methodology to compute Value at Risk (Λ?) at 99% and 95% confi-dence level (left) and Expected Shortfall (E?) at 5% significance level P? (right).Right panel clearly shows the difference between the two methods, as well as thecorresponding risk measures, for the same, fixed, confidence level.

The parametric approach usually relies on the normality assumption for the re-turns distribution, although some analytical results using non-Gaussian functionalforms are available in the literature [46, 47]. However, in the light of the consid-erations previous discussed, models beyond the Gaussian paradigm are stronglyrequired to obtain results closest to the true statical features of prices. In order tocapture the leptokurtic (fat-tailed) nature of price returns, the historical simula-tion approach is often used as an alternative to the parametric method. It employsrecent historical data and risk measures are derived from the percentiles of thedistribution of real data. This method is potentially the most accurate becauseit accounts for the real statistics of price changes but it is computationally quitedemanding (especially when applied to large portfolios) and absolutely dependingon the past history of empirical data. A third approach consists in Monte Carlosimulations of the stochastic dynamics of a given model for stock price returns andin calculating risk measures according to Monte Carlo statistics. This method,however, requires very intensive simulations to achieve risk measures predictionswith acceptable numerical errors.

As a result of the present situation, reliable and possibly fast methods to calcu-late financial risk are strongly demanded. Inspired by this motivation, we evaluatefinancial market risk measures with non-Gaussian fluctuations in order to under-line the potentials of this approach, as well as limitations, in comparison withstandard procedures used in financial analysis. To capture the excess of kurtosisof empirical data with respect to the normal distribution, the statistics of pricechanges is modeled in terms of a Student-t distribution, which is known to approx-imate with good accuracy the distribution derived from market data at a giventime horizon [11, 12] and is widely used in the financial literature and in a num-ber of financial applications, ranging from option pricing [48] to risk analysis [49].With respect to the investigation of Ref. [49], we include in our analysis the studyof the Expected Shortfall and we present, in the spirit of a parametric approach,

24

2.2. Non-Gaussian closed-form expressions for risk measures

analytical expressions for the risk measures in order to provide accessible resultsfor a simple practical implementation. At a variance of the recent calculation inRef. [47], where analytical results for risk measures using Student-t distributionsare presented, we critically investigate the implications of our non-Gaussian an-alytical solutions on the basis of an empirical analysis of financial data and weperform detailed comparisons with the results of widely used procedures.

2.2 Non-Gaussian closed-form expressions for risk

measures

Value-at-Risk is referred to the probability of extreme losses in a portfolio valuedue to adverse market movements. In particular, for a given significance level P ?

(typically 1% or 5%), VaR (Λ?) is defined as the maximum potential loss over afixed time horizon ∆t. In terms of price changes ∆S, or, equivalently, of returnsR

.= ∆S/S, VaR can be computed as follows

P? .=

∫ −Λ?

−∞d∆S P∆t(∆S) = S

∫ −Λ?/S

−∞dR P∆t(R), (2.1)

where P∆t(∆S) and P∆t(R) are the probability density functions (pdfs) for pricechanges and for returns over a time horizon ∆t, respectively. For financial analysts,VaR has become the standard measure used to quantify market risk because ithas the great advantage to aggregate several risk component into a single number.As already remarked, in spite of its conceptual simplicity VaR shows two maindrawbacks: it is not necessary sub-additive and it does not quantify the size ofthe potential loss when the threshold Λ? is exceeded.

A quantity that does not suffer of these disadvantages is the so-called ExpectedShortfall (ES) or Conditional VaR (CVaR), E?. It is defined as

E? .=

1

P?

∫ −Λ?

−∞d∆S (−∆S) P∆t(∆S) =

S

P?

∫ −Λ?/S

−∞dR (−R) P∆t(R), (2.2)

with P? and Λ? as in Eq. (2.1).The standard approach in the financial literature [44, 50] is to assume the

returns as normally distributed, with mean m and variance σ2, i.e. R ∼ N (m, σ2).In that case, VaR and ES analytical expressions reduce to the following closed-form formulae

Λ? = −mS0 + σS0

√2 erfc−1(2P?) (2.3)

and

E? = −mS0 +σS0

P?

1√2π

exp−[erfc−1(2P?)]2, (2.4)

where S0 is the spot price and erfc−1 is the inverse of the complementary errorfunction [51]. However, we have widely discussed in the previous chapter thatthe normality hypothesis is often inadequate to reproduce daily returns and, more

25


0

1

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3

4

5

6

7

8

9

10

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.001

PSfrag replacements

1% 5%

Normal

ν = 2.75

ν = 3.50

ν = 4.50

ν = 100

Λ?

P?

1

2

3

4

5

6

7

8

9

10

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.001

PSfrag replacements

1% 5%Normalν = 2.75ν = 3.50ν = 4.50ν = 100

E?

P?

Figure 2.2: Convergence of the VaR (left) and ES (right) Student-t formulaetoward Gaussian results when approaching the limit ν → +∞.

generally, high-frequency stock price variations. A better agreement with data isobtained using leptokurtic distributions, such as truncated Levy distributions orStudent-t ones. Despite this interesting feature, the former family has the maindisadvantage that it is defined only through its characteristic function and wehave no analytic expression for the pdfs [29]. Moreover, in order to compute thecumulative density function (cdf), which is a necessary ingredient of our analysis,we have to resort to numerical approximations. For the reasons above, to modelthe returns, we make use of a Student-t distribution defined as

Sνm,a(R) =

1

B(ν/2, 1/2)

aν

[a2 + (R − m)2]ν+12

, (2.5)

where ν ∈ (1, +∞) is the tail index and B(ν/2, 1/2) is the beta function. It iseasy to verify that, for ν > 2, the variance is given by σ2 = a2/(ν − 2), while, forν > 4, the excess kurtosis reduces to k = 6/(ν − 4). Under this assumption, weobtain closed-form generalized expression for VaR and ES given by

Λ? = −mS0 + σS0

√ν − 2

√1 − λ?

λ?(2.6)

and

E? = −mS0 +σS0

P?B(ν/2, 1/2)

√ν − 2

ν − 1[λ?]

ν−12 , (2.7)

where λ? .= I−1

2P?(ν/2, 1/2) and I−12P?(ν/2, 1/2) is the inverse of the incomplete beta

function evaluated in 2P?, according to the definition of Ref. [51]. These paramet-ric formulae, that generalize standard expressions known in the literature underthe normality assumption, are one of the main results of this study.

As shown in Fig. 2.2, we have checked numerically the convergence of for-mulae (2.6) and (2.7) to the Gaussian results (2.3) and (2.4), in the appropriatelimit ν → +∞. We chose ν = 2.75, 3.5, 4.5, 100 and m = 0, σS0 = 1, but wechecked that the value of these last parameters does not affect the convergence,

26

2.2. Non-Gaussian closed-form expressions for risk measures

Table 2.1: Values of ν crossover for VaR and ES corresponding to different signif-icance levels P?.

P? 1% 2% 3% 4% 5%

νcross(VaR) 2.44 3.21 5.28 32.38 100νcross(ES) 2.09 2.18 2.28 2.38 2.51

as expected. As can be seen, the points corresponding to ν = 100 are almost co-incident with the Gaussian predictions, demonstrating that our results correctlyrecover the Gaussian formulae as a special case. It is also worth noting that eachline, corresponding to a fixed ν, crosses over the Gaussian one for a certain P ?.Analogously, for a fixed P?, there exists a νcross value whose line crosses the Gaus-sian result at that significance level. In the light of this observation, we report inTab. 2.1 the values of νcross corresponding to a given P? for both VaR and ES. Ascan be observed, the growth of νcross with P? is very rapid for VaR, while for ESand for usually adopted significance values, νcross keeps in the interval [2.09, 2.51].From this point of view, VaR and ES are quite different measures of risk, sincethe crossover values for the latter are much more stable than those associated tothe first one. This result can be interpreted as a consequence of ES as a morecoherent risk measure than VaR.

Because the key parameter to capture the fat-tailed nature of returns in termsof a Student-t pdf is the tail index ν, we show in Fig. 2.3 the sensitivity of VaR(upper panels) and ES (lower panels), as obtained through numerical differen-tiation of Eqs. (2.6) and (2.7), with respect to ν variations. Figure 2.3 showsthe behaviour of the derivatives ∂Λ?/∂ν, ∂E?/∂ν (left panels) and of the log-derivatives 1/Λ? ∂Λ?/∂ν, 1/E? ∂E?/∂ν (right panels) as a function of ν and fortwo significance levels typically used in financial analysis, i.e. P? = 1%, 5%. Thetail index is allowed to vary between 2.5 and 5, in order to elucidate the sensitivityof the risk measures for typical values of ν derived from our time series analysisdiscussed in the following (see Tab. 2.3).

As expected, the strongest sensitivity is observed for the smallest values ofν, in the range between 2.5 and 3.5, corresponding to a particularly pronouncedleptokurtic character of the Student-t distribution. Furthermore, the relative sen-sitivity is higher for VaR (up to about 60%) than for ES (up to about 40%),providing further support to ES as a more robust risk indicator than VaR. Also,for a given risk measure, we observe a stronger relative sensitivity for P ? = 5%(dotted lines) than for P? = 1% (dashed lines). For relatively large values of ν(above 3.5), the relative sensitivity is below the 10% level, with an almost flatbehaviour indicating convergence of Eqs. (2.6) and (2.7) to the Gaussian expec-tations in the appropriate limit. For a typical case of ν ∼ 3 and correspondingerror ∆ν ∼ 0.2 (see Tab. 2.3), in the worst scenario, i.e. for Λ? at 5% significancelevel, one can infer a relative variation of VaR, ∆Λ?/Λ?, of about 4%.

As a whole, these results point out that returns distributions with a moderate

27


-0.4

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2.5 3 3.5 4 4.5 5

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?

∂ν

-0.4

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1 Λ?

∂Λ

?

∂ν

P?1%

P?5%

0

-0.4

-0.2

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?

∂ν

ν

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2.5 3 3.5 4 4.5 5

PSfrag replacements1 E?

∂E

?

∂ν

ν

Figure 2.3: Sensitivity (left panels) and log-sensitivity (right panels) of VaR andES to a variation of the tail index ν. The solid line represents the zero level, whiledashed and dotted lines correspond to 1% and 5% significance level, respectively.

leptokurtic nature can lead to sufficiently precise calculations of risk measures,even in the case of a poorly determined tail index. On the contrary, in the pres-ence of particularly fat-tailed returns, an accurate determination of the tail indexis mandatory to derive precise and reliable estimates of Var and ES.

2.3 Empirical analysis of financial data

The data sets used in our analysis consist of four financial time series, composed ofN = 1000 daily returns, from the Italian stock market. Two series are collectionsof data from the Italian assets Autostrade SpA and Telecom Italia (from May 15th

2001 to May 5th 2005), while the other two correspond to the financial indexesMib30 and Mibtel (from March 27th 2002 to March 13th 2006). The data havebeen freely downloaded from Yahoo Finance Web site [52]. Figure 2.4 shows acomparison between the historical complementary cumulative density function P>

of the negative daily returns and two theoretical fits obtained using Gaussian andStudent-t distributions. The parameters values of the fitted curves, as obtainedaccording to the likelihood procedure described below, are displayed in Tab. 2.2. Inprinciple, we could perform the fit according to different methods, but we have tobalance between accuracy and computational time. Therefore, we estimate mean

28

2.3. Empirical analysis of financial data

and variance as empirical moments, i.e.

m.=

1

N

N−1∑

i=0

Rt−i (2.8)

and

σ2 .=

1

N − 1

N−1∑

i=0

(Rt−i − m)2, (2.9)

where R.= (Rt, . . . , Rt−N+1) is the N -dimensional vector of returns. Using the

above m and σ values, we derive a standardized vector (with zero mean and unitvariance) r

.= (rt, . . . , rt−N+1), where rt−i

.= (Rt−i − m)/σ for i = 0, . . . , N − 1.

In order to find the best value for the tail parameter ν, we look for the argumentthat minimizes the negative log-likelihood, according to the formula

ν = argmin ν>2

[−

N−1∑

i=0

logSν0,√

ν−2(rt−i)

], (2.10)

where the constraint ν > 2 prevents the variance to be divergent and Sν0,√

ν−2is as

in Eq. (2.5), with m = 0 and a =√

ν − 2. This apparently simple optimizationproblem can not be solved analytically. In fact, the normalization factor in theEq. (2.5) does depend on the tail index ν in a non-trivial way. Actually, the betafunction B(ν/2, 1/2) only admits an integral representation and therefore we im-plemented a numerical algorithm to search for the minimum.

As shown in Section 2.2, the excess kurtosis k depends only on ν and thisprovides an alternative and more efficient way to estimate the tail parameter [49].However, this approach forces ν to be bigger than 4, while from Tab. 2.2 it can beseen that all the exponents obtained in the likelihood-based approach are smallerthan 3.5. For this reason, the implementation of the excess kurtosis method isinadequate for the time series under study here. In order to test the robustnessof our results, we also performed a more general three-dimensional minimizationprocedure over the free parameters (m, σ, ν). The multidimensional optimizationproblem was solved by using the MINUIT program from CERN library [53]. Theobtained numerical results are in full agreement with the previous ones, but theprocess is more computationally burden and more cumbersome, since it requiresa lot care in avoiding troubles related to the appearing of local minima in theminimization strategy.

In Fig. 2.4 we show the cumulative distribution P> obtained using the em-pirical parameters of Tab. 2.2. As expected, we measure daily volatilities of theorder of 1% and quite negligible means (∼ 0.01%). The tail parameters fall inthe range (2.9, 3.5), thus confirming the strong leptokurtic nature of the returnsdistributions, both for single assets and market indexes. The quality of our fitclearly emerges from Fig. 2.4, where one can see a very good agreement betweenStudent-t and historical complementary cdfs, while the Gaussian distribution failsto reproduce the data.

29


Before addressing a risk analysis in the next section, it is worth mentioning, forcompleteness, that other approaches to model with accuracy the tail exponent ofthe returns cdfs are discussed in the literature. They are based on Extreme ValueTheory [54] and Hill’s estimator [55, 56, 57]. However, since they mainly focus onthe tails, they require very long time series to accumulate sufficient statistics andare not considered in the present study.

0.001

0.01

0.1

1

0.001 0.01 0.1

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P>

0.001

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0.1

1

0.001 0.01 0.1

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DataNormalStudent

0.001

0.01

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1

0.001 0.01 0.1

PSfrag replacements

P>

−R

0.001

0.01

0.1

1

0.001 0.01 0.1

PSfrag replacements

−R

Figure 2.4: From top left clockwise: Autostrade SpA, Telecom Italia (from May15th 2001 to May 5th 2005), Mibtel and Mib30 (from March 27th 2002 to March 13th

2005) P> of negative daily returns. Points represent historical complementary cdf,while dashed and solid lines correspond to Gaussian and Student fits, respectively.The parameters values of the fitted curves are detailed in Tab. 2.2.

2.4 Risk analysis

In this section we present a comparison of the results obtained estimating the mar-ket risk through VaR and ES according to different methodologies. The standardapproach is based on the normality assumption for the distribution of the returns.For this case we are provided of closed-form solutions, Eqs. (2.3) and (2.4), thatdepend on the two parameters m and σ. For the time series under consideration,the effect of the mean, as shown before, is negligible, and the surviving parameteris the volatility σ. Several techniques are discussed in the literature to model andforecast volatility, based on stochastic volatility approaches (topic of Chapter 3),

30

2.4. Risk analysis

Table 2.2: Mean m, volatility σ, and tail exponent ν, for Autostrade SpA, TelecomItalia, Mibtel and Mib30 time series. m and σ are estimated from empiricalmoments, while ν is obtained through a negative log-likelihood minimization asin Eq. (2.10).

m σ ν

Autostrade 0.12% 1.38% 2.91Telecom −0.02% 2.23% 3.14Mibtel 0.02% 1.03% 3.35Mib30 0.02% 1.16% 3.22

GARCH-like [58] and multi-fractal models [59]. They usually require very longtime series (typically 300 high frequency returns per day over ∼ 5 - 10 years)and are quite demanding from a computational point of view. As discussed inSection 2.3, we limit our analysis to 1000 daily data and we estimate the volatilityusing the empirical second moment. In order to avoid the problem of a uniformweight for the returns, RiskMetrics introduces the use of an exponential weightedmoving average of squared returns according to the formula [50]

σ2t+1|t

.=

1 − λ

1 − λN+1

N−1∑

i=0

λi(Rt−i − m)2, (2.11)

where λ ∈ (0, 1] is a decay factor. The choice of λ depends on the time horizonand, for ∆t = 1 day, λ = 0.94 is the usually adopted value [50]. σt+1|t repre-sents volatility estimate at time t conditional on the realized R. If one considersEq. (2.11) as the defining equation for an autoregressive process followed by σt+1|t(coupled with Rt = σtεt with εt ∼ i.i.d.(0, 1)), Refs. [60, 61] provide reasons for theclaimed good success of the RiskMetrics methodology. In order to relax standardassumption about the return pdf without loosing the advantages coming froma closed-form expression, we have presented in Section 2.2 generalized formulaefor VaR and ES based on a Student-t modeling of price returns. In this frame-work, the tail index ν emerges as a third relevant parameter, which is possibleto constrain using a maximum likelihood technique, as previously described. Asa benchmark of all our results, we also quote VaR and ES estimates following ahistorical approach, which is a procedure widely used in the practice. Accordingto this approach, after ordering the N data in increasing order, we consider the[NP?]th return R([NP?]) as an estimate for VaR and the empirical mean over first[NP?] returns as an estimate for ES 1.

At a variance with respect to previous investigations [49, 60], we also provide68% confidence level (CL) intervals associated to the parameters in order to give

1The symbol [ ] stands for integer part, while R(j) is the standard notation for the jth term of

the order statistic of R. Since N 1 we neglect the fact that the pth entry is a biased estimatorof the p/N -quantile, i.e. E[R(p)] = p/(N + 1).

31


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2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4

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ν∗

Fre

quency

ν∗

ν∗16%

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ν∗84%

0

0.005

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0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

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σ∗t+1|t

Fre

quency

σt+1|t

σt+1|t∗16%

σ∗t+1|t

σt+1|t∗84%

Figure 2.5: Bootstrap histograms for tail index ν (left) and for the RiskMetricsvolatility proxy σt+1|t (right) for Autostrade SpA (M = 103 bootstrap copies).

Table 2.3: Parameters values and bootstrap estimates for the 68% CL intervalsfor the time series as in Tab. 2.2.

m σ σt+1|t ν R(10)

Autostrade 0.12+0.04−0.05% 1.38+0.08

−0.10% 1.83+0.31−0.33% 2.91+0.20

−0.21 −3.51+0.31−0.15%

Telecom −0.02+0.06−0.07% 2.23+0.11

−0.11% 1.54+0.42−0.47% 3.14+0.21

−0.22 −6.14+0.87−1.35%

Mibtel 0.02+0.02−0.04% 1.03+0.03

−0.04% 0.69+0.19−0.20% 3.35+0.18

−0.19 −2.96+0.25−0.24%

Mib30 0.02+0.03−0.04% 1.16+0.03

−0.05% 0.72+0.22−0.22% 3.22+0.15

−0.16 −3.33+0.30−0.25%

a sound statistical meaning to our comparative analysis. In this way we can esti-mate VaR and ES dispersion. To this extent, we implement a bootstrap technique[62]. Given the N measured returns, we generate M = 1000 synthetic copies ofR, R∗

j, with j = 1, . . . , M , by random sampling with replacement according tothe probability p = (1/N, . . . , 1/N). For each R∗

j we estimate the quantities of in-terest and we obtain bootstrap central values and confidence levels. For example,we use for the mean the relations

m∗b

.=

1

M

M∑

j=1

m∗j with m∗

j =1

N

N−1∑

i=0

(R∗j )t−i (2.12)

and we define the 1 − 2α CL interval as [m∗α, m∗

1−α], with m∗a such that P (m∗ ≤

m∗a) = a and a = α, 1 − α. For 68% CL, α = 16%. In Fig. 2.6 and Tabs. 2.3,

2.4, 2.5 we quote results according to m− (m∗b −m∗

α) + (m∗1−α −m∗

b). In this way,we use the bootstrap approach in order to estimate the dispersion of the meanaround the measured value m. In Fig. 2.5 we show the bootstrap evaluation ofthe tail index ν for Autostrade SpA: we would note that the bootstrap centralvalue ν∗

b is clearly very close to the estimated value ν. In order to quantify thedispersion around ν∗

b , we also include the 68% CL intervals, namely ν∗16% and ν∗

84%.Similar considerations are valid also for the volatility proxy, estimated with theRiskMetrics methodology, σt+1|t displayed on the right panel of Fig. 2.5.

Table 2.3 shows central values and estimated 68% CL intervals for the daily

32

2.4. Risk analysis

0

1

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5

6

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?

Student−t Normal Historical RiskMetrics

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6

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5

6

7

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E?


Autostrade

0

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2

3

4

5

6

7

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Mib30

Figure 2.6: VaR Λ? (upper panel) and ES E? (lower panel) central values with68% CL intervals for Autostrade SpA (left) and for Mib30 (right), according tothe four different methodologies discussed in the text. The significance level P?

is fixed to 1% (circles, solid lines) and 5% (triangles, dashed lines).

returns series under study. These numerical results come from a straightforwardapplication of the re-sampling technique. It is worth mentioning that it is possi-ble, and sometimes necessary, to use improved versions of the bootstrap. As arule of thumb, we consider the bootstrap approach accurate when, given a genericparameter, the difference between its empirical value and the bootstrap centralvalue estimate is close to zero and 68% CL interval is symmetric to a good approx-imation. In our numerical simulation, we measured a systematic non zero bias forσt+1|t and from Tab. 2.3 it is quite evident the asymmetry of R([NP?]) intervals forboth Autostrade and Telecom data. We can, therefore, consider the correspondingCL intervals as a first approximation of the right ones, since bias and skewnesscorrections would require sophisticated and ad-hoc techniques [62], which are be-yond the scope of the present work. In Fig. 2.6 we show VaR and ES central valuesand 68% CL bars for Autostrade SpA and Mib30, corresponding to 1% and 5%significance level and according to the four methodologies previously described. InTabs. 2.4 and 2.5 we detail all the numerical results, including also Telecom Italiaand Mibtel data. As already noted in Ref. [60], at 5% significance level Student-tand Normal approaches are substantially equivalent, but here such a statementsounds more statistically robust, thanks to the bootstrap 68% confidence levelsand to the comparison with the historical simulation. At this significance level, we

33


Table 2.4: Estimated VaR values (mean and 68% CL interval) for 1% and 5%significance levels from Autostrade SpA, Telecom Italia, Mib30 and Mibtel. Foreach time series, the results of Student-t and Normal fit, historical simulation andRiskMetrics methodology are shown.

Student-t Normal Historical RiskMetrics

Autostrade VaR 1% 3.472+0.175−0.185 3.091+0.197

−0.204 3.516+0.149−0.306 4.138+0.733

−0.764

VaR 5% 1.717+0.071−0.071 2.150+0.139

−0.145 1.810+0.175−0.156 2.890+0.520

−0.540

Telecom VaR 1% 5.900+0.279−0.230 5.200+0.275

−0.277 6.137+1.348−0.866 3.595+0.990

−1.085

VaR 5% 3.121+0.135−0.137 3.682+0.214

−0.202 3.398+0.110−0.127 2.548+0.694

−0.777

Mib30 VaR 1% 3.047+0.106−0.105 2.675+0.097

−0.096 3.331+0.255−0.304 1.662+0.524

−0.516

VaR 5% 1.612+0.066−0.067 1.885+0.073

−0.072 2.010+0.090−0.157 1.169+0.375

−0.358

Mibtel VaR 1% 2.718+0.097−0.092 2.378+0.088

−0.084 2.967+0.240−0.255 1.581+0.453

−0.449

VaR 5% 1.454+0.062−0.062 1.674+0.065

−0.065 1.811+0.150−0.173 1.110+0.324

−0.316

register for VaR a different behaviour between single assets and indexes. Whileassets show the best agreement between the Student-t and historical approaches(see also Tab. 2.4), for Mib30 and Mibtel data we observe the best agreement be-tween the Normal and historical methodology. In order to enforce this empiricalevidence, it would be necessary to analyse additional time series to see to whatextent this difference between assets and indexes holds. From Fig. 2.6, Tab. 2.4and Tab. 2.5 it can also be seen that Λ? and E? central values calculated accordingto RiskMetrics methodology are quite fluctuating and characterized by the largestCL bars. The decreasing of P? traduces in a major differentiation of the differ-ent approaches. In general, we obtain the best agreement between the Student-tapproach and the historical simulation, both for Λ? and E?, whereas, as before,the RiskMetrics methodology overestimates or underestimates the results of thehistorical evaluation and is affected by rather large uncertainties. Moreover, therisk measures obtained through our model show non negligible differences with re-spect to the widely used Normal and RiskMetrics methodologies, indicating thatthe approach may have helpful implications for practical applications in the fieldof financial risk management.

It is also worth noting that, from the results shown in Fig. 2.2 and Tab. 2.1, weexpect that for a fixed significance level there exists a crossover value, νcross, belowwhich the generalized Student-t VaR and ES formulae underestimate the Gaus-sian predictions. This effect was already mentioned in Ref. [49], but the analyticalformulae here derived allow us to better characterize it. Under the hypothesis of aStudent-t distribution, the crossover value does not depend on the first and secondmoments and therefore the knowledge, for a given time series, of the tail exponentonly is sufficient to conclude, a priori, whether the fat-tailed results for VaR andES will underestimate or not the corresponding Gaussian estimates.

34

2.4. Risk analysis

Table 2.5: Estimated ES values (mean and 68% CL interval) for 1% and 5%significance levels. Time series and methodologies as in Tab. 2.4.

Student-t Normal Historical RiskMetrics

Autostrade ES 1% 5.503+0.431−0.421 3.559+0.229

−0.231 5.076+0.607−0.634 4.759+0.837

−0.876

ES 5% 2.946+0.153−0.159 2.727+0.175

−0.182 3.006+0.248−0.235 3.655+0.653

−0.677

Telecom ES 1% 8.912+0.579−0.583 5.954+0.311

−0.310 9.685+1.456−1.475 4.116+1.133

−1.250

ES 5% 5.035+0.242−0.246 4.613+0.248

−0.246 5.320+0.478−0.466 3.190+0.879

−0.969

Mib30 ES 1% 4.572+0.199−0.191 3.068+0.111

−0.109 3.918+0.223−0.234 1.908+0.599

−0.590

ES 5% 2.596+0.093−0.091 2.369+0.088

−0.086 2.804+0.145−0.155 1.471+0.467

−0.458

Mibtel ES 1% 4.021+0.179−0.171 2.728+0.099

−0.094 3.501+0.215−0.224 1.815+0.524

−0.516

ES 5% 2.314+0.084−0.081 2.106+0.078

−0.077 2.524+0.128−0.136 1.399+0.399

−0.400

35


36

Chapter 3Stochastic volatility models: the

theoretical approach

It is widely recognized that the simplicity of the Black and Scholes-Merton model,which relates derivatives prices to current stock price through a constant volatil-ity, as shown in Section 1.2, is no longer sufficient to capture modern marketphenomena. Its natural extension could be therefore modifying the specificationsof volatility to make it a time-dependent process. This assumption leads to the so-called volatility models, arisen from the evidence that the empirical market pricevolatility is not constant, but shows itself some dynamics (see Chapter 1). Todaythere is an enormous body of research on volatility models, both in academies andin financial practice, and it seems to be very fruitful also for future studies.

What features must characterize a volatility model to make it a good one is in-deed a key point within this topic. Engle and Patton pointed out in Ref. [63] thata good volatility model should be able to forecast volatility and this is strongly re-quired in almost all financial applications. A portfolio maker, for instance, wouldsell its portfolio or a stock before it becomes too volatile, and this is true also fora private investor.

To forecast volatility, and absolute returns as well, is not the only ingredientrequired to make these models reliable and thus attractive. A good volatilitymodel must also be able to capture the stylized facts about volatility of financialasset price shown in Section 1.3.2, concerning the volatility clustering, the lever-age effect, the multiple time-scale of the volatility autocorrelation and, within thefield of derivatives pricing, the smile-shape of the implied volatility. These facts,typical of real market, cannot be captured in the Black-Scholes framework sinceit untimely employs the hypothesis of log-normality of underlying distribution.

Today there are two general classes of volatility models in widespread use.The first type tries to remedy the inconsistencies described assuming that thevolatility is not a constant, but rather some sort of deterministic function of theunderlying price. This hypothesis allows to stay within the boundaries of theBlack-Scholes theory for option pricing, even if in most cases it is impossible toderive a closed-form option price. Based on this approach, there exist the au-

37

3. Stochastic volatility models: the theoretical approach

toregressive conditional heteroskedastic models (known as ARCH and GARCH)introduced by Engle [33] and then extended by many other authors. It’s worthmentioning that in 2003 Robert Engle was awarded the Nobel Price in Economicsjust for the ARCH model suggestion. A strong point of these models is theirability in reproducing quite well the implied volatility, but they often present thedisadvantage of having a large number of parameter with no particular physical oreconomic significance, needing to be tuned ad hoc and, furthermore, substantiallychanging with time. An overview of this class can be found in Ref. [64].

Another possible choice of volatility models are the correlated stochastic volatil-ity (SV) models, introduced in the literature in the late 80’s. The models belongingto this class assume the original log-Brownian motion model but, as their nameindicates, with random volatility. They constitute thus a natural extension ofBlack and Scholes-Merton framework and, here, the topic of the present chapter.The aim of this chapter is to deal with the stochastic volatility models with con-tinuous time, showing their ability to capture the commonly held stylized factsabout volatility. This is organized as follows: in Section 3.1 an overview of thisclass is reported, stressing the most important features the various models have incommon. At present, several SV models are discussed in the literature, differing inthe dynamics attached to the volatility. With respect to the analysis presented inRef. [38], we have focused our investigation on three popular models: the Vasicek,the Heston and the exponential Ornstein-Uhlenbeck ones. In Sections 3.2, 3.3and 3.4 we present the main theoretical features of each model (volatility distri-bution, return distribution, higher-order moments and different-time correlations)in order to emphasize the strong points, as well as their limitations, on repro-ducing and forecasting the true statistics of volatility and returns. An exhaustivediscussion on the overall family of SV models is presented in Ref. [65].

3.1 Correlated stochastic volatility models

The first works on SV models were basically interested on option pricing theoryand ignored the statistical properties of the market model, although they wereindeed able to reproduce the smile effect. The relatively small number of worksdealing specifically with price dynamics based on SV models written before 2000 isessentially due to the trouble in deriving analytically their statistical properties.The analysis is even much more involved when there are correlations betweenvolatility and stock, as it seems to be the case. Moreover, it was commonly assertedthat empirical data available did not allow for obtaining a reliable estimation ofall parameters involved in a SV model. As we shall see this is no longer true, atleast for the most popular ones. Indeed, the leverage correlation between stockprice and volatility, though increasing the complication of the analysis, is crucialfor the complete estimation of the parameters involved.

The idea behind SV models is that the volatility σ is not a constant, butitself a stochastic time-dependent variable. Thus, in such approach, its observedrandom behaviour is directly modeled. In general, it is assumed that volatility is

38

3.1. Correlated stochastic volatility models

a function σt = f(Yt) of a stochastic process Yt. According to these models themarket dynamic is contained in a two-dimensional stochastic process in which theasset price St and the process Yt obey the stochastic differential equations (in theIto prescription)

dS(t) = µ S(t)dt + f(Y (t)) S(t) dW1(t) , S(0) = S0

dY (t) = α(m − Y (t)) dt + g(Y (t)) dW2(t) , Y (0) = Y0

, (3.1)

where the parameter µ is the drift coefficient for the price process. The correlationbetween stock price changes and volatility can be incorporated explicitly by meansof the relation between the two Wiener processes W1 and W2 entering Eqs. (3.1)

dW2(t) = ρdW1(t) +√

1 − ρ2dZ(t) (3.2)

where Z(t) is a Wiener process independent of W1 and ρ is the correlation coeffi-cient. We can convince ourselves of the meaning of the coefficient ρ by evaluatingthe following expectation value

〈dW1(t) dW2(t)〉(3.2)= 〈dW1(t) ρdW1(t)〉 + 〈

√1 − ρ2 dW1(t) dZ(t)〉 =

ρdt +√

1 − ρ2 〈dW1(t)〉〈dZ(t)〉 = ρdt .(3.3)

We would note that the ultimate result is obtained recalling that each Wienerprocess must satisfy 〈dW 〉 = 0 and 〈dW 2〉 = dt. In general, ρ can assume anyvalue entering the interval [−1; 1], but from several empirical analysis it has beenproved that its sign is often negative. This fact could account for the skewnessand the leverage effect observed in the financial data, as already mentioned inChapter 1.

Namely, Yt is the process driving the volatility and can be in principle formu-lated in different forms, leading to a specific SV model, as we will see in detail.Nevertheless all SV models have to account for one common feature: Yt must betaken as mean reverting process. Mean reversion is a well known stylized factabout asset price volatility and concerns the existence of an asymptotic value mto which σt will eventually return in a typical time 1/α (note that the parametersm and α are the ones entering Eqs. (3.1)). This feature reflects the economic ideaof a normal level of volatility toward which an efficient market in health condi-tions tends, even if practitioners often differ on this level and whether it is reallyconstant in time and through institutional changes. The mean reversion is easilyseen (beside being quite trivial) by rewriting the second equation of (3.1) in aproperly integral fashion:

Y (t) = Y (t0) + α

∫ t

t0

(m − Y (t′))dt′ +

∫ t

t0

g(Y (t′)) ξ2(t′)dt′︸︷︷︸

dW2(t′)

(3.4)

where ξ2(t) is a Gaussian white noise whose stochastic time derivative is theWiener process W2(t). Recalling that a Wiener process has null expectation value

39


the previous statement is proved, since

E[Y (t)] = Y0 e−α(t−t0) + m(1 − e−α(t−t0)) (3.5)

clearly has the stationary value m as (t − t0) → ∞. It’s also worth noting thatthe mean reverting character of σt implies that current information have almostzero effects on the long time forecast.

The most popular SV models appearing in the literature during the last twodecades are listed in Tab. 3.1. Among them, in the following sections we willexamine in detail the main theoretical features of the Vasicek, Heston and Scott(also known as exponential-OU) models.

Table 3.1: Most popular functions of volatility appearing in the stochastic volatil-ity models. For each model the author’s name, the expression of the volatility,the driving process Y and the function g entering the stochastic term of Eq. (3.1)are given.

Authors f(Y ) Y process g(Y )

Hull-White√

Y Log-Normal κYScott eY Mean-reverting OU kVasicek Y Mean-reverting OU kStein-Stein |Y | Mean-reverting OU k

Heston√

Y CIR κ√

Y

Before dealing with the Vasicek model, it’s worth mentioning that in whatfollows it turns out to be more convenient to work with the so-called zero-meanreturn, defined as

dX.=

dS

S− µdt , (3.6)

and whose stochastic differential equation (sde) reads

dX(t) = σ(t)dW1(t) . (3.7)

The zero-mean return X(t) has a fairly simpler dynamics than the price S(t)because it only contains the random term σ(t)dW1(t). On the other hand thisprocess still retains the interesting features of the whole dynamics. In Appendix A,an explicit expression for X(t) and some key features are given.

3.2 The Vasicek model

The Vasicek model [66], and its variant formulated by Stein and Stein [67], pos-tulates that volatility is driven by an Ornstein-Uhlenbeck mean reverting processof the form

dY (t) = −α(Y (t) − m)dt + kdW2(t) . (3.8)

40

3.2. The Vasicek model

As g(Y ) is a simple multiplication by a constant, the second equation in (3.1)becomes linear in Y and Eq. (3.4) can be rewritten as

Y (t) = Y (t0)e−α(t−t0) + m

(1 − e−α(t−t0)

)+ k

∫ t

t0

e−α(t−t0)dW2(t′) . (3.9)

Eq (3.9) is the starting point of the following discussion and constitutes as wellthe solution of Eq (3.8). Looking at Tab. 3.1, in this framework σ ≡ Y and thusthe volatility average value employs the form of Eq. (3.5)

E[σ(t)] = σ0 e−α(t−t0) + m(1 − e−α(t−t0))(t−t0)→∞−→ m (3.10)

where the limit (t−t0) → ∞ indicates that the volatility process is in the stationarystate. Here we have assumed that the process started at time t0 when volatility isσ0. Making use of Eq. (3.10), the analytical expressions for the variance and thecorrelation of σ can be derived, and read

Var[σ(t)] =k2

2α

(1 − e−2α(t−t0)

) (t−t0)→∞−→ k2

2α

.= β (3.11)

and

E [σ(t)σ(t + τ)] = m2 +k2

2αe−ατ . (3.12)

As known, the “original” Ornstein-Uhlenbeck process [68] is the model describingthe velocity process of a physical Brownian motion carried out by a particle in asolution. The solution of the Fokker-Planck equation for such a process is thereforeknown and reads

p(σ; t) =1√

πk2

α(1 − e−2α(t−t0))

e− α(σ−E[σ(t)])2

k2[1−e−2α(t−t0)] , (3.13)

i.e. the well known Maxwell velocity distribution. The stationary limits ofEq. (3.13) is given by

pst(σ) =1√

πk2/αe−

α(σ−m)2

k2 . (3.14)

Since the Ornstein-Uhlenbeck process turns out to be one of the most interestingand, from a physical viewpoint, peculiar process within the family of stochasticprocesses, in Appendix B we give a brief description of it.

Looking at Eq. (3.13) one may be concerned about the sign of σ(t), arguingthat it does not appear to be a positive-definite stochastic variable, a fact thatcan be regarded as troublesome for a SV model. Nevertheless we can convinceourselves that this is not the case. We have already discussed in Section 1.3.2 thatthe actual evaluation of volatility is very difficult to achieve, since volatility cannotbe directly observed. In practice, we can use as a good proxy the instantaneous

41


volatility, defined in Eq. (1.19). Combining this expression with Eq. (3.6) andrecalling that dW 2 = dt, we get

σist(t) =√

σ(t)2 = |σ(t)| , (3.15)

which shows that there is no need to attach a sign to the random variable σ(t).Vasicek and Stein-Stein models, as can be read in Tab. 3.1, differ because inthe latter one, instead of Y , |Y | is taken as the driving process for the volatilityentering the former Eq. (3.1). This assumption allows to avoid at all the troubleof the volatility sign in the Stein-Stein framework. In what follows we will dealwith the Vasicek model only.

Before addressing to the return process, it’s also worth discussing whether thismodel is mean reverting. Recall from Section 3.1 that the mean reversion is anecessary feature for a good volatility model. A possible definition of the “normallevel of volatility” is given by [63]

limt→∞

E

[dX(t)2

dt

∣∣∣∣∣σ0

].= NLV , (3.16)

i.e. the stationary average of the square of instantaneous volatility (1.19). Thelimit over time t indicates that the process has begun infinitely back in the pastand, therefore, the volatility process is in the stationary state. The Ornstein-Uhlenbeck volatility process, with its non-zero stationary expectation values, seemsto be a good candidate for describing this effect, since Eq. (3.16) can be writtenin this framework [69]

O-U NLV = limt→∞

E[σ2(t)|σ0

]= m2 +

k2

2α. (3.17)

When, on the other hand, the volatility is not yet in the stationary state, theaverage over σ2 is given by

E[σ2(t)|σ0, 0

]=

[σ0e

−αt + m(1 − e−αt)]2

+k2

2α

(1 − e−2αt

). (3.18)

This average tends to the normal level as αt increases. The magnitude of α allowsto classify the SV models into fast mean reverting processes, when 1/α t, orslow mean reverting processes, when 1/α t, where t is the typical time scale weare interested in.

The return process

In order to find an analytical expression for the return probability distribution,we are dealing with the system of two coupled sdes (3.1), which becomes in thisframework

dX(t) = σ(t)dW1(t)

dσ(t) = α(m − σ(t)) dt + k dW2(t) ., (3.19)

42


In general, we can perform the analysis in term of return, but we prefer to deal withthe zero-mean return since, as can be guessed by looking at Eq. (3.6), the expres-sions derived are much more handier. The system (3.19) can be seen equivalently asone single equation for the two-dimensional stochastic process (X(t), σ(t)), leadingto the introduction of the conditional pdf for such a process p2(x, σ, t|x0, σ0, t0).This density obeys the following backward Fokker-Planck equation [70]

∂p2

∂t0= α(σ0 − m)

∂p2

∂σ0

− 1

2σ2

0

∂2p2

∂x20

− ρkσ0∂2p2

∂σ0∂x0

− 1

2k2 ∂2p2

∂σ20

, (3.20)

with the final condition

p2(x, σ, t|x0, σ0, t) = δ(x − x0) δ(σ − σ0). (3.21)

Eq. (3.20) is the starting point of the analysis carried out by Masoliver and Perelloin Ref. [69], on which is based what follows. Making use of the Fourier analysis,they derived a semi-analytical expression for the unconditional characteristic func-tion (cf) of the return process, which reads

ϕX(ω, t) =1√

1 + k2A/αexp

[−C +

B2k2/α − 4mB − 4m2A

4(1 + k2A/α)

]. (3.22)

where the functions A, B, C are given by

A(ω, t) =ω2

2

(sinh ηt

η cosh ηt + ζ sinh ηt

), (3.23)

B(ω, t) =ω2αm

η

(cosh ηt − 1


), (3.24)

C(ω, t) =

[(ωαm)2

η2+ iωρk − α

]t/2 +

1

2ln

(cosh ηt +

ζ

ηsinh ηt

)

−(ωαm)2

2η3

[2ζ(cosh ηt − 1) + η sinh ηt


], (3.25)

with

η =√

α2 − 2iρkαω + (1 − ρ2)k2ω2, ζ = α − iωρk. (3.26)

The Vasicek model succeeds in predicting the semi-closed form solution (3.22)for the return process. Nevertheless it has to be inverse-transformed to compareit with real financial data. It’s worth stressing that in order to derive the un-conditional solution (3.22) the volatility is assumed to be in stationary regime.Equivalently, one may state that the pX found is intended to predict the returnsin situation of initial volatility at a normal level.

Note that Eq. (3.22) has the right limit when volatility is constant, i.e. non-random. Indeed, in such a case k = 0, Eqs. (3.22)-(3.26) yield

ϕX(ω, t) = e−ω2m2t/2, (3.27)

43


p_x-300 -200 -100 0 100 200 300

ph

i

0

0.2

0.4

0.6

0.8

1

rho = 0

rho = 0.99

Figure 3.1: Characteristic function of the return in the Vasicek model for twovalues of the correlation coefficient ρ.

which is the characteristic function of the zero-mean return when X(t) follows aone-dimensional diffusive process with constant volatility σ = m. Hence, Eq. (3.22)correctly embodies the geometric Brownian motion model as a particular case. InFig. 3.1 the characteristic function of returns Eq. (3.22) is shown.Moreover, assuming that

k

α 1, (3.28)

one can also prove that Eq. (3.22) converges to the Gaussian distribution whenapproaching the limit t → ∞

ϕX(ω, t) ∼ exp

−ω2

[1 + ν2 + O

(k

α

)]m2 t

2

, (t → ∞), (3.29)

with ν2 .= k2/2αm2. The Gaussian character of the long time returns in financial

market, requested by Central Limit Theorem, is a stylized fact that any SV modelmust provide. The Gaussian density (3.29) assures that the Vasicek model is ableto reproduce this feature.

It’s also worth noting that to assume k α means that the volatility is weaklyrandom in comparison with its deterministic drift, since k is the strength of thevolatility driving noise (the so-called volatility of volatility) whereas α gives infor-mation about its deterministic drift. Thus the ratio k/α measures in some waythe degree of volatility randomness.

Once obtained the semi-closed form solution carried in Eq. (3.22) for the re-turn process, we are able to evaluate analytical expression for the higher-ordermoments introduced in Section 1.3.1, i.e. skewness and kurtosis. Making use ofthe definition given in Eq. (1.15), we can derive the expressions for the asymptoticlimits of the kurtosis in this framework, given by

γ2.=

κ4

κ22

∼ 6ν2(ν2 + 2)

(ν2 + 1)2(αt 1), (3.30)

44


and

γ2.=

κ4

κ22

∼ 6ν2[ν2(1 + 4ρ2) + 4(1 + ρ2)]

(ν2 + 1)2

1

αt(αt 1). (3.31)

As known, the kurtosis allows to quantify the “fatness” of the tail of the distribu-tion compared to the Gaussian case. Thus, it is a crucial quantity in measuringfinancial market risk. From Eq. (3.30) we observe that the model produces anever negligible kurtosis, even after an infinitesimal time. On the other hand,from Eq. (3.31) we see that kurtosis goes to zero as time increases and the con-vergence is as slow as 1/t, in agreement with the Gaussian limit of the pdf. Inaddition, we observe that for short times kurtosis does not contain the correlationcoefficient ρ, but in the long run a non zero ρ magnifies the distribution kurtosis.

Eq. (1.14) leads to the following asymptotic limits for the skewness

γ1 ≡κ3

κ3/22

∼ 3ρν√

ν2 + 1

√2αt (αt 1), (3.32)

and

γ1 ≡κ3

κ3/22

∼ 6ρν(ν2 + 2)

(ν2 + 1)3/2

1√2αt

(αt 1). (3.33)

Unlike kurtosis, this moment is known to give information about the asymmetryof the pdf, reducing to zero in the Gaussian case. From Eqs. (3.32) and (3.33) itemerges that the skewness goes to zero at both short and long times, but if thekurtosis is proportional to 1/t, the skewness vanishes as slowly as 1/

√t. Finally,

we note that skewness is proportional to ρ and, in consequence, the sign of ρdetermines the skewness sign. Many empirical observations suggest that ρ mustbe negative and, consequently, we have to expect that the the left tail is fatterthen the right one at any times.

Before addressing to different-time correlations, it’s worth focusing on the tailsof the distribution. It is well establish that the return distribution exhibits anexcess of kurtosis (see also Section 1.3.1) which is going to vanish for long times(αt 1) when it reaches the Gaussian limit. Thus we limit our analysis at smalland moderate time, when the Central Limit Theorem has no effect. Tails arederived from Eq. (3.22) by keeping the first orders in ω, which means large valuesof zero-mean returns x. The analytical computation shown in Ref. [69] yields

pX(x, t) ∼ 1√a(t)2 + b(t)

exp

[− 1

b(t)

(√a(t)2 + b(t) − a(t)

)x

](x → +∞),

(3.34)and

pX(x, t) ∼ 1√a(t)2 + b(t)

exp

[1

b(t)

(√a(t)2 + b(t) + a(t)

)x

], (x → −∞).

(3.35)with

a(t).=

ρkt

4, and b(t)

.=

k2

8αt(2 − ρ2αt). (3.36)

45


In particular, Eq. (3.34) describes the exponential decay of the right tail, i.e.large gains, whereas Eq. (3.35) reproduces an asymmetric decay of the left tail,i.e. large losses. Therefore both the tails are exponential at any time. This fact isin disagreement with the power-law nature of the tails observed at infraday-dailytime horizons, pointing out that the Vasicek model is unable to reproduce thiskey feature.

To conclude, we would note that the sign of ρ allows to determine which tailis heavier since a(t) = ρkt/4. In particular, when ρ < 0 the fattest tail is the onerepresenting losses, while when ρ > 0 the fattest tail corresponds to profits. Ifρ = 0 there is no difference between the two tails and the distribution becomesasymmetric, as in the Gaussian case.

Leverage effect and volatility autocorrelation

The last issue about the Vasicek model is to verify whether it is able to reproducethe leverage effect and the volatility autocorrelation introduced in Section 1.3.3.

To derive analytically the leverage function we can use the definition givenin Eq. (1.23), which allows to derive [69] the following expression in the Vasicekframework

L(τ) = 2ρ

[ν√

2α (1 + ν2e−ατ )

(1 + ν2)2m

]e−ατH(τ) , (3.37)

where H(τ) is the Heaviside step function

H(τ) =

1, if τ > 0

0, if τ < 0(3.38)

From Eq. (3.37) emerges that the Vasicek model is able to reproduce the empiricallaw of Eq. (1.24), which captures carefully the measured leverage function (seeFig. 1.8 in Chapter 1), by choosing b = α and

A = 2ρ

[ν√

2α (1 + ν2e−ατ )

(1 + ν2)2m

]. (3.39)

Note that, as for the skewness and the tails, the sign of the leverage functionis completely determined by the sign of ρ, since Eq. (3.37) is linear with thecorrelation coefficient.

The evaluation of the volatility autocorrelation C(τ) is quite more involvedthan the leverage effect, recalling that its analytical definition is given by thecomplex Eq. (1.25). To compute all the quantities entering the Eq. (1.25) wehave to use Eqs. (A.3), (A.5) and (A.15) and, after some lengthy calculations, weobtain

C(τ) =〈σ(t)σ(t + τ)〉2 − 〈σ(t)〉4

4〈σ(t)2〉2 − 3〈σ(t)〉4 =ν2e−ατ (ν2e−ατ + 1)

4ν2(ν2 + 2). (3.40)

46

3.3. The Heston model

From Eq. (3.40) emerges that the volatility autocorrelation has one single char-acteristic time scale coinciding with the leverage relaxation time α−1. This factssounds unrealistic, since C(τ) is characterized by a double time-scale instead thata single one, as shown in Section 1.3.3.

The main theoretical features of Vasicek model are summarized in Tab. 3.2at the end of the present chapter. To conclude, we would stress that what wehave illustrated proves that the Vasicek model is paradigmatic for the simplicityof the assumptions on the processes, which allows a full calculation development.In this framework, the leverage effect and the higher-order moments are correctlypredicted. Therefore, it fails in measuring two important stylized facts aboutvolatility: its non-Gaussian probability distribution and the double time-scale ofthe volatility autocorrelation. This is indeed a big limitation of this model.

3.3 The Heston model

Introduced by Cox, Ingersoll and Ross in 1985 [71] to describe the dynamics ofinterest rates, the so-called CIR process (also known as Feller process or squareroot process) was later chosen by Heston [72] to drive the volatility fluctuationsin a SV model capable to obtain option prices in a semi-closed form, i.e. thecharacteristic functions of prices are in closed form. The Heston model was alsoable to explain the asymmetry of the volatility smile with the introduction of thecorrelation (3.2) between the two Wiener processes W1 and W2. As known, suchasymmetry is indeed caused by a negative correlation between past prices and fu-ture volatility, which is in its turn given by the coefficient ρ appearing in Eq. (3.2).

In the Heston model the assumption for the functions f and g entering Eqs. (3.1)are

f(Yt) = g(Yt) =√

Yt . (3.41)

The meaning of the choice of f is quite plain: as the volatility is usually definedas the standard deviation of the price distribution, we are actually declaring thatthe driving process Yt has significance of variance. In what follows, we shall use vinstead of Y to recall this fact, as it has become customary to do since Heston’soriginal work.

The starting point of our analysis is to assume that the volatility σ obeys azero-reverting Ornstein-Uhlenbeck process, i.e. Eq. (3.8) with m = 0

dσ(t) = −ασ(t)dt + kdW (t) . (3.42)

The Ito lemma applied to Eq. (3.42) yields to the sde for the variance process v,which reads

dv(t) = dσ(t)2 = −γ(v(t) − θ) dt + κ√

v(t)dW (t) , (3.43)

where we defined

γ = 2α , κ = 2k , θ =k2

2α. (3.44)

47


The factor√

v as coefficient of the random term of dv has appeared, as requestedby Eq. (3.41). It’s also worth stressing that the process which has to be takenas primary in the evolution of volatility is v, which obeys the mean revertingequation (3.43).

Reminding the general outline given in Section 3.2 for the Vasicek model, wecan firstly describe the main statistical features of such a process. To this end, wehave to specify its integral form; from Eq. (3.4) we get

v(t) = v0 + γ

∫ t

t0

(θ − v(t′))dt′ + κ

∫ t

t0

√v(t′)dW2(t

′) , (3.45)

where v0 = v(t0) is supposed to be known. Taking the expectation value ofEq. (3.45), we can therefore obtain the mean value

E [v(t)|v0] = v0e−γ(t−t0) + θ

(1 − e−γ(t−t0)

) (t−t0)→∞−→ θ . (3.46)

Note that the expression (3.46) states that the normal level of volatility in theHeston model, as defined in Eq. (3.16), is given by the value of θ.

Calculating the analytical expression for the variance is more involved than inthe Vasicek model, because we have to manipulate in a non-trivial way Eq. (3.43)in order to obtain the second central moment of v. The ultimate result yields

Var[v(t)|v0] =θκ2

2γ+

κ2

γ(v0 − θ)e−γ(t−t0) +

κ2

γ

(θ

2− v0

)e−2γ(t−t0) . (3.47)

which represents the conditional variance for the v process.It is well known that to completely characterize a statistical variable the ana-

lytical expression of its probability distribution is strongly requested. For a CIR-like process, this is known and, hence, we can state that the variance features anon-central χ2 distribution. Now, from the statistics [73] we know that a variablegiven by the sum of n squared Normals follows a central χ2 density with n degreesof freedom. Instead the attribute non-central refers to a variable defined by thesum of n squared Gaussian variables with non-zero mean and unitary variance.In the light of this statement, we can rewrite the variance v as a sum of n squaredOrnstein-Uhlenbeck processes, leading to the following expression for its sde

dv(t) = −γ

(v(t) − nκ2

4γ

)dt + κ

√v(t)dZ(t) . (3.48)

From the comparison between Eq. (3.48) and the simpler Eq. (3.43) emerges thatthe only affected parameter in the process seems to be the long-time mean θ,passing from k2/2α in Eq. (3.44) to

θ =nk2

4γ, (3.49)

where n represents the number of degrees of freedom of the distribution of v(t).Although this statement could be seen as a mere definition, it is indeed a key

48


point for the well-definiteness of process (3.43).In Fig. 3.2 we show a central χ2 distribution; it’s quite clear that it has null

pdf in zero if n > 2. This circumstance is precisely what we need to preservethe process v(t) from becoming negative, which would be troublesome since in dvthere happens to appear a

√v. Thus to correctly define the variance process, we

must require that

n =4γθ

κ2> 2 (3.50)

where Eq. (3.49) has been used. Eq. (3.50) is an important constraint to take intoaccount in calculating the form of pv(v).

Figure 3.2: Probability density function of χ2 variables with various numbers ofdegrees of freedom.

Before writing explicitly the analytical expression of the non-central χ2 func-tion, it is convenient to introduce the following parameters

ct =4γ

κ2 (1 − e−γ(t−t0))

(t−t0)→∞−→ 4γ

κ2, λt = ctv0e

−γ(t−t0) (t−t0)→∞−→ 0 (3.51)

where λt is defined as the non-centrality parameter, linear with respect to thevariance starting value v0. In term of ct and λt the variance probability distributionis given by [74]

pt(v) = ct

∞∑

i=0

e−λ/2(λ/2)i

i!pχ2

n+2i(ctv) , (3.52)

where pχ2n+2i

(ctv) is a central χ2 distribution with n + 2i degrees of freedom.

Once obtained Eq. (3.52), one can derive the stationary distribution of v.Observing the stationary limits in Eqs. (3.51), we get

pst(v) =v(α−1)e−αv/θ

(θ/α)α Γ(α), α

.=

2γθ

κ2≡ n

2, (3.53)

which is the well-known Gamma distribution. The parameter α is the ratio be-tween the average variance θ and its characteristic fluctuation κ2/2γ during the

49


relaxation time 1/γ; note that when α → ∞, pst(v) → δ(v − θ).The corresponding probability density for the volatility process σ =

√v is

easily derivable and reads

pσ(σ; t) = 2σ ct pχ2( 4γθ

κ2 ,λt)(ctσ

2)t→∞−→ pst(σ) =

2αα

θα Γ(α)σ(2α−1)e−ασ2/θ .

(3.54)In Fig. 3.3 (left panel) the stationary distributions for the variance and the volatil-ity processes are shown.

Finally, we would note that the Heston model has the notable advantage ofproviding a closed form expression for the volatility distribution, which exhibitsa strongly required non-Gaussian character, unlike the Vasicek model. However,as mentioned in Chapter 1, it has been empirically proved [12] that the volatilitycould feature an inverse-Gamma distribution instead that a Gamma one, as foundin this framework.

The return process

Before dealing to the probability distribution of the return process, it’s worthmentioning that, with respect to the investigation of Ref. [75], for the Hestonmodel the convenient definition of zero-mean return differs from the one given inEq. (3.6). Indeed we will use the more original

x(t).= ln

S(t)

S0− µt , (3.55)

where, instead of returns, we make use of log-returns. Eq. (3.55) gives the slightlymore complicated sde

dx(t) = −v(t)

2dt +

√v(t) dW1(t) . (3.56)

Following the same reasoning of Section 3.2, we define the transition probabilityPt(x, v | vi) for the two-dimensional stochastic process (xt, vt)

dx(t) = −v(t)

2dt +

√v(t) dW1(t)

dv(t) = −γ(v(t) − θ) dt + κ√

v(t)dW (t) .

, (3.57)

Pt(x, v | vi) represents the probability density to have log-return x and variance vat time t, given the initial log-return x = 0 and variance vi at t = 0. In orderto find an analytical formula for the time dependent probability distribution ofthe return, the Fokker-Planck equation, which governs the time evolution of Pt,has to be solved. It has been done exactly by Dragulescu and Yakovenko inRef. [75] making use of the Fourier analysis, leading to the following semi-closedunconditional solution for the return process in term of a Fourier integral

Pt(x) =1

2π

∫ +∞

−∞dpx eipxx+Ft(px) (3.58)

50


0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

v/θ

Sta

tiona

ry d

istr

ibut

ion

Π*(v

/θ)

0 1 2 30

0.2

0.4

0.6

0.8

1

σ/θ1/2

Π*(σ

) (σ/θ

1/2 )

Figure 3.3: Left panel: The stationary probability distribution pst(v) of variance(Eq. (3.53)), shown for α = 1.3. The vertical line indicates the average value ofv. Inset: The corresponding stationary probability distribution pst(σ) of volatility(Eq. (3.54)) [75]. Right panel: Characteristic function of the return pdf in theHeston model for two values of the correlation coefficient ρ.

with

Ft(px) =γθ

κ2Γt − 2γθ

κ2ln

[cosh

Ωt

2+

Ω2 − Γ2 + 2γΓ

2γΩsinh

Ωt

2

]. (3.59)

Analogously to what assumed for the Vasicek model, we have conjectured that vi

is in its stationary regime. Moreover, it is quite easy to check, under this hypoth-esis, the Pt(x) obtained is real (ReF is an even function of px and ImF is an oddone), and that Ft(0) = 0, which implies that Pt(px = 0) is correctly normalized atall times.

The integral in Eq. (3.58) can be calculated numerically, as we will see in thenext chapter, or analytically, but only in certain regimes. In fact, despite the in-tricate appearance of eFt in Eq. (3.58), the function has actually a smooth shape(Fig. 3.3, right panel), very similar to the characteristic function of Vasicek model(Fig. 3.1). Its inverse transform will be carried out in the next chapter. Never-theless, computing a Fast Fourier Transform is often a non-trivial issue. Findingspecial conditions in which Pt(x) or at least its chief features, e.g. the slopes ofits tails, are exactly calculable is hence fairly appealing. Hence, in what followswe will illustrate the asymptotic behaviours of the returns at both short and longtimes, stressing the possibility to characterize them analytically. This is a strongpoint of this model, since we know that the log-returns can be directly evaluatedfrom financial time series and thus compared with theoretical predictions. Con-versely, variance is a hidden stochastic variable that has to be measured by meansof estimators based on the same time series, although its probability density canbe entirely characterized from an analytical point of view (3.52).

By looking at Eq.(3.43), it is clear that the variance revers to the equilibriumvalue θ within the characteristic relaxation time 1/γ. Thus, the main differencebetween the regimes of Pt(x) must lie in the amount of time elapsed since the ini-tial condition (0, vi) in comparison with 1/γ. We will therefore draw our attention

51


to two main cases: short times (γt 1) or long times (γt 1). Both the regimeshave been studied in detail in Ref. [75]. Here, the main results found, without theentire calculation, are reported.

The asymptotic behaviour of Pt(x) at short times (γt 1) is of great interest,because it can account for derivative pricing via path-integral [76]. In this regimePt(x) acquires, in fact, meaning of short-time transition probability. As t is smallthe conditional probability distribution reads

Pt(x | vi) =1√

2πvite− (x+vit/2)2

2vit . (3.60)

Eq. (3.60) shows that at short time Pt(x|vi) evolves in a Gaussian manner, becausevariance has little time to change. Again assuming that the variance is in the sta-tionary state allows to pass from the conditional probability density of Eq. (3.60)to the unconditional one for the log-return process, given by

Pt(x) =21−αe−x/2

Γ(α)

√α

πθtyα−1/2Kα−1/2(y), (3.61)

where Ka is the modified Bessel function of (real) order a and we introduced thescaling variable

y =

√2αx2

θt=

2√

γ

κ

|x|√t

. (3.62)

In the limit y 1, namely very large |x|, using the approximation Kν(y) ≈e−y

√π/2y we find

Pt(x) ≈ 21/2−α

Γ(α)

√α

θtyα−1e−y . (3.63)

Eqs. (3.62) and (3.63) prove that the tails of the distribution are exponential inx. Thus the Heston model fails to predict the power-law shape of the return tailsat short time, a well known stylized fact about returns.

The asymptotic behaviour at long time (γt 1) is strictly related with riskmanagement. As t tends to infinity, Eq. (3.59) reduces to

Ft(px) ≈γθt

κ2(Γ − Ω) (3.64)

and, after laborious computation, we are able to obtain the scaling form

Pt(x) = Nt e−p0xP∗(z) , P∗(z) = K1(z)/z , (3.65)

with

z.=

ω0

κ

√(x + ργθt/κ)2

1 − ρ2+

(γθt

κ

)2

, Nt.=

ω20γθt

πκ3√

1 − ρ2eΛt (3.66)

where Nt is the time-dependent normalization factor. Note that the dependenceof Pt(x) on the two arguments x and t is given by the function P∗(z) of the single

52


scaling argument z in Eq. (3.66).As for the tails, we can use the asymptotic expression K1(z) ≈ e−z

√π/2z in

Eq. (3.65) and take the logarithm of P . For large |x|, |x| γθt/κ and so we get

lnPt(x)

Nt≈ −p0x − ω0

κ√

1 − ρ2|x|. (3.67)

Thus, Pt(x) has exponential tails also for large time horizon. It is also worthmentioning that in the considered limit the slopes d lnP/dx of the exponentialtails do not depend on time t. In particular, the slopes (3.67) for positive andnegative x are not equal, leading to an asymmetric distribution Pt(x) with respectto positive and negative price changes.

Finally, we would note that in the region x ∼ 0 of small log-return Eq. (3.67)becomes

lnPt(x)

N ′t

≈ −p0x − ω0(x + ργθt/κ)2

2(1 − ρ2)γθt, (3.68)

where N ′t = Nt exp(−ω0γθt/κ2). Thus, for small log-returns |x|, Pt(x) is Gaussian

with the width increasing linearly in time.In summary, ln Pt(x) is linear in x for large |x| and quadratic for small |x|. As

time passes, the distribution broadens and gets closer to a Gaussian, in agreementwith the Central Limit Theorem.


Since in the Heston model the primary volatility process is the variance v(t),Eq. (1.23) for the leverage function and Eq. (1.25) for the volatility autocorrelationcan be rewritten

L(τ).=

E [v(t + τ)dx(t)]

E [v(t)]2(3.69)

and

C(τ).=

〈v(t)v(t + τ)〉 − 〈v(t)〉2Var [v(t)]

. (3.70)

Nonetheless, the Heston model does not provide a closed equation for the volatil-ity, since Eq. (3.45) is not exactly solvable. Thus it seems impossible to derivean analytical expression for the Heston leverage. Masoliver, Perello and Anentoreported in Ref. [38] the approximated form

L(τ) ∼ 4ργ

κe−γτ , (3.71)

which is exact only when θ = κ2/2α, but Ref. [75] shows that it fails in measuringleverage effect for the Dow Jones index data. Thus, the only way to compute theHeston leverage is to derive it by means of numerical simulations.

Unlike leverage, volatility autocorrelation can be exactly computable assumingthat at time t the variance is in the stationary regime. It reads [77]

C(τ) = e−γτ . (3.72)

53


As in the Vasicek model (Eq. (3.40)), the Heston model is unable to reproducethe multiple time-scale empirically observed in C(τ). In particular, it fails inreproducing its slowest time decay.

To conclude, we would note that the Heston model has the notable advantageof being able to provide, besides a closed form for the volatility distribution, a semi-closed form for the complete Pt(x). This density is also analytically computablein the principal regimes of interest. Nevertheless, it fails to predict some majorstylized facts, as the power-law shape of the return tails at short time or thedouble time scales in the volatility autocorrelation. The main theoretical featuresof the model, illustrated in this section, are listed in Tab. 3.2 at the end of thepresent chapter.

3.4 The exponential Ornstein-Uhlenbeck model

The exponential Ornstein-Uhlenbeck model was introduced by Scott no later than1987 [78] as a possible stochastic volatility modification of the Black-Scholes modelfor option pricing. Nevertheless, the former version of the model yields a Gaus-sian pdf for the stationary distribution of the volatility. This feature is indeedwide of the mark, since it has been already discussed in Section 1.3.1 that thevolatility could feature a Log-Normal or an inverse Gamma distribution. One ofthe main motivation which has led to the formulation of the so-called correlatedexponential Ornstein-Uhlenbeck model, topic of this section, is indeed to find amodel whose volatility is Log-normally distributed. As we shall see, the othermajor capability of the model is to predict two distinct time-scales in the decayof volatility autocorrelation. It’s worth mentioning that also the Vasicek and theHeston models could be sophisticated ad hoc in order to account for the doublerelaxation time-scale, but this method has neither physical nor financial assump-tions.

As can be viewed in Tab. 3.1 in the expOU framework the driving process Y (t)is taken as a zero-reverting Ornstein-Uhlenbeck process, which obeys Eq. (3.8)with the long time mean term equal to zero

dY (t) = −αY (t)dt + kdW2(t) . (3.73)

In term of the Y (t) process the volatility is given by

σ(t) = meY (t) (3.74)

Eq. (3.74) clears the meaning of the attribute exponential for such a model. SinceY (t) is zero-reverting, eY (t) fluctuates around 1 (with a symmetric pdf if we sup-pose small oscillations then eY ' 1 + Y ) and σ(t) varies around the scale valuem, which obviously has dimension t−1/2. With respect to the investigation ofRef. [39], in the following we will assume that at time t the process Y (t) (hencethe volatility) has reached its stationary state.

54

3.4. The exponential Ornstein-Uhlenbeck model

In the stationary limit, the integral form of Eq. (3.9) reads

Y (t) = Y0e−α(t−t0) + k

∫ t

−∞e−α(t−t′)dW2(t

′)(t−t0)→∞−→ k

∫ t

−∞e−α(t−t′)dW2(t

′) ,

(3.75)with the first and second order moments given by

E [Yst] = 0 , (3.76)

and

Var[Y (t)] =k2

2α

(1 − e−2α(t−t0)

) (t−t0)→∞−→ k2

2α

.= β . (3.77)

Eqs. (3.74) and (3.75) are the starting point to derive the main features of thevolatility process. In the expOU model, the average value and the variance assumethe following expressions

E [σ(t)] = m exp

[Y0e

−α(t−t0) +β

2

(1 − e−2α(t−t0)

)] (t−t0)→∞−→ meβ/2 , (3.78)

Var [σ(t)] = m2 exp[2Y0e

−α(t−t0) + β(1 − e−2α(t−t0)

)]

exp

[β

(1 − e−2α(t−t0)

)]− 1

(t−t0)→∞−→ m2eβ(eβ − 1

),

(3.79)while the autocorrelation of the stationary volatility reads

E [σ(t)σ(t + τ)] = m2 exp[β(1 + e−ατ )

]. (3.80)

Since Y (t) is an Ornstein-Uhlenbeck process, it follows a Gaussian density withzero mean and variance β

(1 − e−2α(t−t0)

). This feature gives us the form of the

distribution for the volatility process

p(σ, t|σ0, t0) =1

σ√

2πβ(1 − e−2α(t−t0))exp

− [ln(σ/m) − e−α(t−t0) ln(σ0/m)]2

2β(1 − e−2α(t−t0))

,

(3.81)whose stationary limit is

pst(σ) =1

σ√

2πβexp

− ln2(σ/m)

2β

. (3.82)

Eq. (3.81) confirms the correct prediction of the Log-Normal distribution of thevolatility.

The return process

In order to find the return probability density, we are dealing with the followingsystem of two coupled sdes

dX(t) = meY (t)dW1(t)

dY (t) = −αY (t)dt + kdW2(t) ., (3.83)

55


where the returns are defined as in the Vasicek model. The calculation of thereturn pdf was performed by Masoliver and Perello in Ref. [39] with a techniqueanalogous to the one followed in Sections 3.2 and 3.3. Unlike the Vasicek andHeston models, the characteristic function found in this framework can be ana-lytically inverse-transformed, leading to the following closed-form expression forthe entire pdf

pX(x, t) ' 1√2πm2t

e−x2/2m2t

[1 − ρka(αt)

α1/2(2αt)3/2H3

(− x√

2m2t

)

+k2b(αt)

8α(αt)2H4

(− x√

2m2t

)],

(3.84)

where Hn(x) are Hermite polynomials and the functions a(αt) and b(αt) are de-fined by

a(z) = z −(1 − e−z

), (3.85)

and

b(z) = (1 + 2ρ2)z + 2ρ2[ze−z − 2

(1 − e−z

)]−

(1 − e−z

). (3.86)

with z.= αt.

It’s worth noting that Eq. (3.84) is an approximate expression for the returndistribution, since it has been obtained under the assumption of stationary volatil-ity. Moreover, we have also assumed that

k

m 1 , (3.87)

i.e. we have dealt with the case in which the “volatility of volatility” k is muchgreater than the normal level of volatility represented by m. This situation isactually close to reality, in that typical λ values lie around 102 ÷ 103.

Before addressing with the different-time correlations, we would stress thatEq. (3.84) is a correction for the Gaussian density

p(x, t) =1√

2πm2te−x2/2m2t , (3.88)

which would correspond to the return pdf if the volatility was a deterministicquantity, i.e. if k = 0 in Eq. (3.73). The deviation from the Gaussian characterof the return pdf in Eq. (3.84) is also evidenced by the existence of non-zerocumulants of order higher than two, implying non-zero skewness and kurtosis.Recalling Eqs. (1.14) and (1.15), we get [39]

γ1 ' 6ρ√

2βa(αt)

(αt)3/2, γ2 ' 24β

b(αt)

(αt)2. (3.89)

The return distribution, thus, is designed to capture the kurtosis and the skewnesseffect.

56



The autocorrelation of volatility has been already given in Eq. (3.80). However,as already argued, the volatility is a hidden variable and thus we need reliableproxies to evaluate it from the data. This is an important task, especially whenwe want compare empirical data with theoretical predictions. For this reason, itseems to be more convenient to evaluate the volatility autocorrelation C(τ) bymeans of

C(τ) =E [σ(t)2σ(t + τ)2] − E [σ(t)2]

2

3E [σ4] − E [σ(t)2]2, (3.90)

instead of Eq. (1.25), which allows to calculate the following expression

C(τ) =exp[4βe−ατ ] − 1

3e4β − 1. (3.91)

The presence of the nested exponents makes this formula particularly interesting,because, in contrast with the corresponding Vasicek (3.40) and Heston (3.72)expressions, it displays the presence of multiple time scales in the autocorrelation.Indeed, writing Eq. (3.91) as

C(τ) =1

3e4β − 1

∞∑

n=1

(4β)n

n!e−nατ (3.92)

a discrete infinite multiplicity of relaxation times appears. Nevertheless the rele-vant ones are just two, clearly visible in Fig. 3.4 which is obtained plotting ln C(τ)in function of the dimensionless ατ for several values of the β parameter. In de-tail, we see that all the curves assume approximately the same value for τ = 0, inagreement with the fact

C(0) =e4β − 1

3e4β − 1

β&1−→ 1

3. (3.93)

Moreover, we observe that the initial slope of each curve, i.e. the short-time decayrate, namely the first time scale, depends on the value of β, while the asymptoticslope is the same for every curve and hence it must be a function of α. We cancharacterize this double time-scale also analytically, as follows.

For long times, i.e. ατ 1, recalling the definition of β we can rewriteEq. (3.91) as

C(τ) =1

3e4β − 1

[e4β−2k2τ − 1

]+ O

(α2τ 2

). (3.94)

In this case the short time behaviour of C(τ) has the characteristic time 1/2k2.In the opposite case ατ 1 we can neglect, in Eq. (3.92), terms of the order

of e−nατ with n ≥ 2 and get

C(τ) ≈ 4β

3e4β − 1e−ατ . (3.95)

57


10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

0 2 4 6 8 10

vola

tility

aut

ocor

rela

tion

normalized time ατ

β=30

β=20

β=10

β=5

Figure 3.4: Plots of C(τ) given by Eq. (3.92) as a function of ατ for differentβ = k2/2α in a semi log-scale. The figure clearly shows the existence of twoasymptotic time scales separated by a sum of multiple time scales. This effect isenhanced in bigger values of β [39].

Thus, the long time behaviour of the volatility autocorrelation is governed by thecharacteristic time α−1.

Therefore, within the expOU framework the autocorrelation of the volatilitypresents, at least, two time scales: the longest one governed by τlong = 1/α anda shorter one governed by τshort = 1/2k2. We observe that the dimensionlessparameter β can be rewritten in terms of τlong and τshort as

β =τlong

4τshort, (3.96)

which shows that β is also a measure of the distance between the long and theshort time scales (see Fig. 3.4).

To derive the expression of the expOU leverage we can use Eq. (1.23), whichallows to obtain the following formula [39]

L(τ) =2ρk

mexp

[−ατ + 2β

(e−ατ − 3

4

)]H(τ), (3.97)

which reproduces the form given by Eq. (1.24), although the implied definition ofA and b will depend on the different scales involved.

As for C(τ), we can study the asymptotic behaviours of L(τ) at both shortand long times. For long time, i.e. ατ 1, Eq. (3.97) reduces to (τ > 0)

L(τ) =2

mρke−3β/2

[e−ατ + O

(e−2ατ

)](ατ 1), (3.98)

which has the form (1.24) if we assume b = α and

A =2

mρke−3β/2 .

= Along. (3.99)

58


Since the value of β is usually quite large, Along results exponentially small andthe long-time behaviour of L(τ) turns out to be undetectable in the practice ofempirical observations. Consequently, the effect of the leverage correlation can besought only in the short-time regime ατ 1. In this limit expanding Eq. (3.97)yields (τ > 0)

L(τ) ' 2

mρkeβ/2e−k2τ (ατ 1), (3.100)

where we have taken into account that α k2 in the exponent. Again Eq. (3.100)can be reduced to Eq. (1.24) with b = k2 and

A =2

mρkeβ/2 ≡ Ashort. (3.101)

Hence, the leverage correlation is determined by Eq. (3.100) which features onesingle time scale given by τlev = 1/k2. Note that this time scale is of the sameorder than the short time scale of the autocorrelation of the volatility Eq. (3.94),in accordance with empirical observations.

To conclude, we would stress again the two major capabilities of the expOUmodels: the Log-Normal distribution of the volatility and the presence of thedouble time-scale in the volatility autocorrelation, stylized facts measured in manyempirical analyses of real financial time series. The main theoretical featuresderived along this section are detailed in Tab. 3.2.

Table 3.2: Main theoretical features of the Vasicek, Heston and expOU SV mod-els. For each one, we display the volatility and the return distribution, togetherwith the analytical expressions, where exist, for leverage and volatility autocorre-lation. We remind that ϕx(ω, t) has the meaning of characteristic function.

Vasicek Heston expOU

Volatility pdf Normal Gamma Log-NormalLog-return pdf ϕx(ω, t) ϕx(ω, t) ϕx(ω, t)

non-inv non-inv approx. invertible

L(τ) ρe−ατ H(τ) ? ρe−k2τ H(τ)

C(τ) ∼ e−ατ e−ατ exp[4β e−ατ ]−13e4β−1

1 time scale 1 time scale 2 time scales

59


60

Chapter 4A comparative analysis of

stochastic volatility models

In the previous chapter we have described in detail the main theoretical features ofthe Vasicek, Heston and expOU SV models, stressing their ability in reproducingsome aspects characteristic of the market dynamics. This is true especially forthe expOU model, since the predicted Log-Normal volatility distribution and thedouble time-scale in the autocorrelation of the volatility process make it the mostrealistic among the models considered. Now, a natural prosecution of such analy-sis could be to implement numerically the models, paying particular attention totheir predictive effectiveness on the outcomes of numerical simulations. This is animportant task, especially in view of financial applications like risk managementor option pricing, which can be often treated only by means of numerical methodsand algorithms. Thus, in this scenario, a good volatility model must not onlymeet the Engle and Patton’s demands illustrated in Chapter 3, but also maintainall its features when it is simulated. This is actually a basic request once dealingwith numerical issues.

The numerical implementation of the models is the main topic of the formerpart of the present chapter. We illustrate in detail the comparison between the the-oretical predictions of Chapter 3 and the numerical results obtained by means oforiginal simulations of the models. We were particularly interested in confirmingthe closed or semi-closed solution for volatility and returns distributions, as wellas the different-time correlation functions. As we will see, we proved an almostperfect convergence between analytical and numerical results. The latter part ofthe chapter is devoted to an empirical analysis of real financial time series. Unlikethe numerical approach, this could directly provide confirmations of the ability ofthe considered SV models to correctly reproduce the features of real data. Thus,it allows us to critically compare the models and to establish whether they can besuccessful in predicting the stylized facts of real market.

This chapter is based on the work presented in Ref. [79] and is organized asfollows: in Section 4.1 we briefly discuss the discretization algorithm used to ap-proximate the stochastic paths described by the stochastic differential equations

61

4. A comparative analysis of stochastic volatility models

given in the models. The accuracy of the Euler-Maruyama method, as well as itsstability, are also pointed out. In Section 4.2 the results achieved from the numeri-cal simulations are shown and compared with the theoretical predictions of Chap-ter 3 for both volatility and return probability distributions and different-timecorrelations. The most ticklish point of this analysis is actually the numerical in-version of the characteristic functions given in the theory in Eqs. (3.22) and (3.59).To justify completely our technical choices, we briefly described the Fast FourierTransform technique used. Finally, in Section 4.3 the empirical analysis of threetime series from the Italian stock market is presented, in order to emphasize therealism, as well as some limitations, of the expOU model once comparing withreal data.

4.1 Discretization algorithm

As seen in Section 3.1, the analysis of SV models starts from sde systems of theform

dX(t) = f(Y (t)) dW1(t) , X(0) = 0

dY (t) = α(m − Y (t)) dt + g(Y (t)) dW2(t) , Y (0) = Y0

dW2(t) = ρ dW1(t) +√

1 − ρ2 dZ(t)

, (4.1)

where W1(t), W2(t) and Z(t) are Wiener processes generated by Gaussian whitenoises. In principle, we could implement numerically such a system according toseveral methods, variously sophisticated (Euler, Runge-Kutta, predictor-corrector. . . ), but their suitability depends on the nature of the problem treated. Toindividuate the most correct, we must identify the ordinary differential problemclosest to the stochastic one we are dealing with. Looking at Eqs. (4.1), it is evidentthat the two model-dependent equations, if stripped of the stochastic terms, areelementary first-order coupled equations with initial conditions. Therefore we candiscretize them following the straightforward Euler method. Namely, if we applythe method to y = f(t, y(t)) at step i it yields simply

(∆y)i = f(ti, yi)∆t =⇒ yi+1 = yi + f (ti, yi)∆t . (4.2)

The Euler method assures, for a sufficiently small step size, a good approximationof the underlying differential equation. It’s also worth mentioning that it has arough first-order accuracy.

The Euler discretization algorithm continues to be substantially valid alsowhen the stochastic terms are added, but the meaning of accuracy must be mod-ified. To better understand it, we have to recall that one of the key points inthe Ito formalism is that the stochastic quantities written using the differentialnotation, like dW , have the meaning of infinitesimal standard deviation corre-sponding to infinitesimal time increments dt [23]. In other words, dW 2(t) = dtmust be interpreted in “mean-square sense”: the variance of the Wiener processrealizations (ideally, computed on an infinitely large population of them) grows

62

4.1. Discretization algorithm

linearly with time. Considering a single realization of a process does not make anysense. Therefore the accuracy of the numerical simulation can only be understoodand verified in terms of averages (mean, variance, correlations) calculated on anideally infinite set of realizations. These preliminary remarks insure that the so-called Euler-Maruyama algorithm, which is the extension of the Euler method tostochastic differential equations, correctly approximates the paths described bythe sde given in the models. Following its prescriptions, at step i the volatilitydriving process Y is incremented to

Yi+1 = Yi + α (m − Yi)∆t + g (Yi) ξi

√∆t , (4.3)

where ξ(t) is a Gaussian white noise whose stochastic time derivative is the Wienerprocess W2(t), as in Eq. (3.4). It’s worth mentioning that to better estimate theaccuracy of such a method more sophisticated techniques could be used, fallingin the scope of pure numerical analysis and, thus, out of interest for the presentwork. The most important topics to this regard are the simulation’s convergence,i.e. its degree of ability to generate populations of paths whose average follows thereal process ones at finite time horizons, and its stability, i.e. its correct behaviourat long times. We can consider for instance the stability of the Euler-Maruyamamethod in a Heston model simulation, limiting to the behaviour of the averagevalue [80]. Recalling Eq. (3.43), the discretization yields

vi+1 = vi + γ (θ − vi) ∆t + κ√

vξi

√∆t . (4.4)

An iterated substitution of vi in the right side of Eq. (4.4). followed by theapplication of the average, gives directly the mean value of the variance v at thelast step n

E [vn] = (1 − γ∆t)n (E [v0] − θ) + θ . (4.5)

The final result depends on the value of ∆t, given a certain γ. For

• ∆t < 2/γ, E [vn]n→∞−→ θ ;

• ∆t = 2/γ, E [vn] = (−1)nE [v0] + ((−1)n+1 + 1) θ ;

• ∆t > 2/γ, |E [vn]| n→∞−→ ∞ .

In order to have the simulated paths tending (in average) to the correct stationaryvalue θ the time step size ∆t must not be too larger than the mean-reverting time1/γ. Note that the average cancels out the contribution from the stochastic termin Eq. (4.4): the previous conclusion can therefore be regarded as a general resultfor all the mean-reverting SV models. Thus, we can take twice the relaxation timeas the upper bound for the choice of the step size of our simulations. Much morespecialized calculations give the expectation values of the maximum discrepanciesbetween the simulation output and the real process, which, in agreement withthe above reasoning, can be used to estimate the simulation accuracy. As a ruleof thumb, they give for E[|Y sim

n − Y (tn)|] a supreme value which is linear with

63


√∆t and a sup E[(Y sim

n − Y (tn))2] proportional to ∆t. We can conclude that,the smaller the time step ∆t, the more accurate simulations will be. This couldclearly cause some limitations in the computation time availability and also in thefinite machine precision.

4.2 Numerical results

The theoretical results derived in Chapter 3 are tested ab initio by means of orig-inal numerical simulations of the models, whose sde are discretized following thetechnique introduced in the previous section. The aim of this study consists inverifying the theory, regarding the time evolution of volatility and return pro-cesses of the generated populations. We are particularly interested in evaluatingthe shape and the main statistical features (average values and higher-order mo-ments) of their probability distributions at various time, including the stationarylimit. In order to perform an exhaustive numerical investigation, we check alsothe capability of our simulations to correctly reproduce the time series correlationeffects (leverage effect and volatility autocorrelation). This is indeed a crucialpoint, since these aspects are among the most realistic features of SV models andit would be disappointing to find that the discretization procedure does not con-serve them. Afterwards we display the most relevant results emerged from thecomparative study we carried out. The entire analysis, as well as several technicaldetails, can be found in Ref. [77].

4.2.1 Volatility and return processes

We firstly deal with the volatility distribution, at both short and long times. Thetheoretical predictions of each model in the stationary regime (i.e. long times)are listed in Tab. 3.2 of Chapter 3. It’s worth reminding that at short timesthe Vasicek and the expOU volatilities feature the same stationary distributionsof Tab. 3.2, that is the Normal and the Log-Normal ones respectively. On thecontrary, the Heston volatility follows at short times the non-central χ2 densitygiven in Eq. (3.52), clearly different from the Gamma pdf characterizing its longtimes limit. Moreover, Eq. (3.52) cannot be simplified and we have to rewrite it ana properly series-expansion fashion in order to implement it numerically. Namely,Eq. (3.52) yields

pv(v; t)(3.52)= ct

e−(ctv+λt)/2 (ctv)n/2−1

2n/2 Γ(n/2)[1 +

∞∑

i=1

1

i!

1

n(n + 2) . . . (n + 2i − 2)

(λtctv

2

)i]

,

(4.6)

where n is defined as the effective number of degrees of freedom 4γθ/σ2. Thelatter expression has been derived starting from Ref. [81].

64

4.2. Numerical results

In order to test the compatibility between theoretical predictions and numer-ical simulations we have to produce a sufficiently large Monte Carlo populationof volatility paths (∼ 104÷6). This is indeed one of the messages come out fromwhat discussed in Section 4.1. To generate them, the sde for the volatility processis discretized and then evolved according to time steps of one day circa. The to-tal evolution time, usually taken to be much larger than the volatility relaxationtime, corresponds to some years. In Fig. 4.1 we show for instance ten simulatedpaths produced from Eq. (3.43), which is on the grounds of the Heston model.It’s worth mentioning that, to numerically implement the three models, we makeuse of model parameters quoted in the literature [39, 69, 75].

Time

0 50 100 150 200 250

Vol

atili

ty

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Processo CIR per vol=sqrt(v) (10 simulazioni MC)Processo CIR per vol=sqrt(v) (10 simulazioni MC)

Figure 4.1: Simulated volatility paths for the Heston model. Each path is obtainedby means of Monte Carlo numerical implementation of the stochastic differentialequation given in Eq. (3.43).

This comparative analysis has led to an almost perfect coincidence betweentheory and numerical results for all the three models, at both short and long times.As an example, in Fig. 4.2 we present the shape of the numerical volatility in itsstationary limit in comparison with the predictions reported in Tab. 3.2. Thenumerical evaluation substantially confirms the theoretical relations, except forthe most extreme events in the expOU model, not captured by the Log-Normaldistribution. We guess that, in spite of the large number (106) of simulated paths,the statistics is still not sufficient. We would stress that the theoretical curvesare not fitted, but plotted over the numerical results. In this way we can checkdirectly the correctness of our simulations, as well as the exactness of the theoret-ical models.

As for the volatility, we perform a comparative study of the returns too. It’sworth mentioning that, in the literature, such an analysis is not much investigated.In order to numerically simulate the return process, we must deal with the imple-mentation of the complete system of the coupled equations of Eqs. (4.1). Namely,at each instant we compute the volatility value and, then, use it to evaluate thereturn increment. In this way σ can drive the evolution of the return process stepby step. To comply with what discuss in Section 3.1, we generate and compare

65


with the theory only zero-mean log-return series.

σ

-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1

Prob

abilit

y de

nsity

-410

-310

-210

-110

1

Vasicek model

σ

0 0.1 0.2 0.3 0.4 0.5

Prob

abilit

y de

nsity

-410

-310

-210

-110

1

10

Heston model

σ

0 1 2 3 4 5 6

Prob

abilit

y de

nsity

-510

-410

-310

-210

-110

1

Simulated

Normal

Gamma

Log-Normal

expOU model

Figure 4.2: Stationary numerical volatility (points) in comparison with the theoryof the Vasicek (green), Heston (blue) and expOU (pink) models.

We are particularly interested in confirming the theoretical predictions derivedin Chapter 3 for the entire shape of the return distribution. This task is quitesimple for the expOU model, because the predicted solution of Eq. (3.84) is aclosed form, directly comparable with the simulation output. This is shown inFig. 4.3 on different time lags, from 1 up to 250 days. Like what has been donefor the volatility, we did not fit the theoretical curves looking for the parameterswhich assure the best agreement, but we have only plotted them in comparisonwith the numerical data: the overall agreement is almost perfect. Fig. 4.3 allowsto appreciate also the skewness of the distribution, more evident as the time in-creases. We would recall that Eq. (3.84) has been derived under the assumptionof the stationarity of volatility. In order to account for it, we set the initial valuesof the parameters entering the formula (3.84) very closed to their mean reversionvalues. This allows to sensibly compare the numerical results with the theory, alsoat short time horizons.

Unlike the expOU model, the predicted Vasicek and Heston solutions for thereturn process are semi-closed formulae, not analytically invertible. Namely, theyare expressed analytically through their characteristic functions, which can beswitched to the corresponding probability densities only by means of numericaltechniques. In order to solve this problem and obtain theoretical expressions com-parable with our results, a Fast Fourier Transform (FFT) algorithm was imple-mented making use of the built-in functions appearing in the software ROOT [82]and MATHEMATICA

r. Namely, we have to numerically compute the followingFourier integral

Pt(x) =1

2π

∫ +∞

−∞dω e±iωx Pt(ω) , (4.7)

66


Return, X-0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

p(X

)

10

210

310

410

510

610

expOU model1 day

5 days20 days40 days

250 days Theory

Figure 4.3: Numerical returns (points) on different time lags in comparison withthe theory of the expOU model (line). The curves are shifted each other by onedecade for greater clarity.

where the sign of the phase factor exponent in the integrand is positive for theHeston model and negative in Vasicek’s case. One can note that Eq. (4.7) is

similar to Eq. (3.58) of Section 3.3. Thus, the term Pt(ω) turns out to be thecharacteristic function we would inverse-transform, which assumes respectivelythe expression of Eqs. (3.22) and (3.59) in the Vasicek and Heston framework.Since the latter expressions cannot be inverse-transformed analytically, we musttake into account the numerical formulation of the FFT algorithm. This consists inthe discretization of the integral entering Eq. (4.7), where the sum can be restrictto a finite amount of N +1 discrete values of x in which Pt(x) is computed. Moreexactly, the Fourier integral can be discretized on an finite interval (−xmax, xmax)making use of routines of the form

Pt(xn) =1

N

N−1∑

k=0

e−2πiN

nk Pt(ωk) . (4.8)

In order to compute the summation, the analytical values of the function Pt(ω)must be calculated in N points ωk equally spaced on an interval

Iω =

(− πN

2 xmax,

πN

2 xmax

). (4.9)

The resulting Pn are transformed in the desired Pt(xn) by multiplicand themby N/I, where I is the width of the interval I. To check the validity of the result,Pt(x) must be properly normalized. A more exhaustive description of the FFTtechnique can be found in Ref. [51].

Fig. 4.4 displays the time evolution of the Vasicek and Heston return distribu-tions, together with the theoretical curves achieved by means of the FFT techniqueabove illustrated. As for the expOU model, the analytical solutions are plotted incomparison with the simulated returns in order to directly evaluate their conver-gence. We note that in the Vasicek framework the agreement is poor at infra-weektime horizons and gets better as the pdf becomes less singular, whereas in the He-ston model the agreement is very good at both short and long times. As for the

67


expOU model, to account for the stationary volatility assumption, we set the ini-tial value of volatility equal to its mean reverting one. Looking at Figs. 4.4, italso emerges that the skewness increases for increasing time horizon, being quitenegligible at short times. Recalling what we have discussed in Section 1.3.1, onecould note that such a behaviour correctly reproduces the skewness’s trend empir-ically observed in real markets. Moreover it proves the non-Gaussian character ofthe simulated distributions, visible as an asymmetry arising for non zero ρ whenthe evolution time is large enough. A skewned distribution results, in fact, froma negative correlation between volatility and return processes (see Section 1.3.1);to simulate this feature we have implemented the processes setting a negativevalue for the parameter ρ. In order to quantitatively characterize it, we havealso checked the simulations capability to correctly reproduce the higher-ordermoments, comparing them with the theory. As saw in Section 3.2, the Vasicekframework predicts the behaviour of skewness and kurtosis at both short and longtimes. We note that the simulation outputs agree very well with the predictedEqs. (3.30-3.33), showing the slow convergence to zero of both the moments, co-herently with the Central Limit Theorem which requires a Gaussian density (i.e.γ1 = γ2 = 0) when t → ∞. This clearly implies that the pdfs are non-Gaussian atleast for several relaxation times. Conversely, no analytical expressions are pre-dicted in the Heston model for higher-order moments; thus, we can analyse themonly qualitatively.

Return,X-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

p(X

)

-110

1

10

210

310

410

Return,X-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

p(X

)

-110

1

10

210

310

410

Vasicek model

Return,X-0.3 -0.2 -0.1 0 0.1 0.2 0.3

p(X

)

-110

1

10

210

310

410

Return,X-0.3 -0.2 -0.1 0 0.1 0.2 0.3

p(X

)

-110

1

10

210

310

410

1 day

5 days

20 days

40 days

250 days

Theory

Heston model

Figure 4.4: Numerical returns (points) on different time lags in comparison withthe theory of the Vasicek (left) and Heston (right) models (line). The curves areshifted each other by one decade for greater clarity.

Before addressing to the correlation functions, it’s worth reminding that inChapter 3 we have reported several approximated results, valid in the Heston andVasicek frameworks for the complete pdf as well as just for the tails. Such resultsare obviously efficient only in particular regimes, but they have the remarkableadvantage of being all expressed in closed-form. We check the convergence be-tween these expressions and numerical results, finding out in almost all the casesa good agreement [77].

To conclude, the numerical results demonstrate that, for what concerns thevolatility and return processes, the models can be implemented maintaining all

68


their features. This has great importance in view of financial applications, like theabove-illustrated risk management, in which the shape of the return distributionplays a fundamental role.

4.2.2 Leverage effect and volatility autocorrelation

In order to prove the effective correctness of our simulations, a crucial point is totest whether they are able to correctly capture the volatility correlation functions,whose analytical expressions are listed in Tab. 3.2. From the numerical point ofview this problem is quite different from the ones we previously faced, because theleverage effect of Eq. (1.23) and the autocorrelation function of Eq. (1.25) dependon the time delay τ and not on the absolute time t. Thus, to numerically simulatethese quantities a whole time series must be considered: namely, for a fixed τ theaverages appearing in the analytical formulae are calculated upon all the existinginstants t. This procedure is widely used to compute leverages and correlations onreal financial data, since for a certain index or share clearly only one time seriesexists. We have followed the same method to obtain these statistical functions forthe outcomes of our simulations: a very long series of increments, correspondingto up to a century of exchanges, has been simulated.

In Figs. 4.5 and 4.6 we present, respectively, the resulting numerical lever-ages and volatility autocorrelations, compared with the theoretical predictionsdisplayed in Tab. 3.2. For the Vasicek model, it’s quite evident that the numericaloutput superposes almost perfectly the prediction of Eq. (3.37) for the leverageeffect, as well as Eq. (3.40) for the volatility autocorrelation. Conversely, in theHeston model there is no theoretical formula for leverage effect to look at and wecan just limit ourselves to qualitative considerations on the simulation output, asin the case of higher-order moments. For what concerns the volatility autocorre-lation, the approach which led to Eq. (3.72) must be kept in mind: it has beenobtained directly from the squared volatility process v(t) rather than from thesquared increment dX(t)2. Therefore it is natural to implement Eq. (3.70) takingas input the realizations of the variance process instead that the volatility one.The numerical outputs recovers perfectly the theoretical expectation, as shown inFig. 4.6.

Finally, the expOU leverage and volatility autocorrelation show a clear, butquite noisy for the leverage, agreement with the formulae given in Eqs. (3.97)and (3.91). We would recall that one of the strong points of this model is the pre-dicted presence of two typical time scales in the volatility autocorrelation, namelyEqs. (3.94) and (3.95). The parameter values fixed along the simulation has leadthe following time scales

τshort =1

2k2= 4 days and τlong =

1

α= 25 days . (4.10)

Such values yield a very fast damped autocorrelation function, which allows toappreciate in Fig. 4.6 (bottom panel) a sharp distinction between the two charac-teristic time-scales.

69


Delay (days)-20 0 20 40 60 80 100

Leve

rage

-40

-30

-20

-10

0

10

Vasicek model

Delay (days)-20 0 20 40 60 80

Leve

rage

-10

-8

-6

-4

-2

0

2

4

SimulatedTheory

expOU model

Figure 4.5: Numerical leverage effect in comparison with the theory of the Vasicek(left) and expOU (right) models. More details about the theoretical curves canbe found in Tab. 3.2.

We can conclude that the numerical analysis has proved a full convergence be-tween numerical outcomes and theoretical predictions, showing in the same timethat these are correct and, on the other hand, that the models can be implementedmaintaining all their features.

Delay (days)0 20 40 60 80 100

Vol

atili

ty a

utoc

orre

latio

n

0

0.05

0.1

0.15

0.2

0.25

Vasicek model

Delay (days)0 20 40 60 80 100

Vol

atili

ty a

utoc

orre

latio

n

0

0.2

0.4

0.6

0.8

1

Heston model

Delay (days)0 10 20 30 40 50 60

Vol

atili

ty a

utoc

orre

latio

n

-310

-210

-110

Simulated

Theory

expOU model

Figure 4.6: Numerical volatility autocorrelation, calculated on a simulated 100-years time series, in comparison with the theory of the Vasicek (top, left), Heston(top, right) and expOU (bottom) models. More details about the theoreticalcurves can be found in Tab. 3.2. In the case of expOU model, the double charac-teristic time-scales is clearly visible in the log-linear scale.

70

4.3. Comparison between theory and empirical data

4.3 Comparison between theory and empirical

data

As in Section 2.3 within the framework of risk measures, we present again anempirical analysis of financial data from the Italian stock market. Unlike whatpreviously produced, the aim of this analysis is to critically compare the modelsand to establish whether they turn out to be realistic once comparing with empir-ical data. The time series used, freely downloaded from Yahoo Web Site [52], arecollections of daily closing prices of the Italian assets Bulgari SpA, Brembo andFiat SpA from January 2000 to May 2007. Thus we are dealing with N = 1920data.

Fig. 4.7 displays the historical evolution of the three stocks in the time intervalconsidered: the price of each share clearly features a negative growth rate untilthe 2004, followed by a positive exponential growth, perceivable in log-linear scale.This is true especially for Fiat SpA, which shows a very quickly recovery and thena steady rise in its price, due to the change occurred in the company’s managementand its consequent new throw in the world-wide market. Besides these interestingmatters about market dynamics, knowing the state of a share has great impor-tance when we are dealing with fit of the data share on theoretical curves. Thevalue of the parameters entering the fitted curves could, in fact, change in timepassing from a period of positive to one of negative growth. Silva and Yakovenkoin Ref. [83] pointed out clearly such a behaviour for the parameters of Eq. (3.58),emerging from the comparison between US indexes and the distribution of returnsderived in the Heston model.

In Fig. 4.8 we show a fit to empirical volatility on the theoretical predictions ofthe SV models considered, displayed in Tab. 3.2, for Bulgari SpA and Fiat SpA.

Year2000 2001 2002 2003 2004 2005 2006 2007

Pric

e

5

10

15

20

25

30

Stocks data, 03/01/2000-29/05/2007

Year2000 2001 2002 2003 2004 2005 2006 2007

Pric

e

10

Stocks data, 03/01/2000-29/05/2007

FiatBremboBulgari

Figure 4.7: Historical evolution of Fiat SpA (red), Brembo (green) and BulgariSpA (blue) shares between 2000-2007 in both linear-linear (left) and log-linear(right) scales.

A very similar shape characterizes also the volatility of Brembo. Using the proxyintroduced in Section 1.3.2, we estimate empirical daily volatility as absolute daily

71


returns. Namely,

σi = |Ri| , (4.11)

where R.= (R1, . . . , RN ) is the N -dimensional vector of daily returns. The pa-

rameter values of the fitted curves are obtained according to a multidimensionalminimization procedure based on the maximum likelihood approach. As in Sec-tion 2.3, in order to solve the optimization problem and find the best parametervalues necessary for the fit, we implement a numerical algorithm based on the MI-NUIT program of CERN library [53]. Fig. 4.8 displays clearly a better agreementbetween data and the Log-Normal distribution predicted in the expOU frame-work, whereas the Normal and the Gamma densities tend to underestimate thelarge distribution’s tail. This conclusion sounds not so surprising, since we havealready discussed the ability of Log-Normal to reproduce empirical volatility forboth individual share and indexes. It’s worth mentioning that a similar analysiswas performed also by Micciche et al in Ref. [84], where the theoretical predictionof Log-Normal was compared with the theory of the Hull and White SV model [85](see Tab. 3.1 in Chapter 3) for the most capitalized stocks traded in the US equitymarket. Also this study proves that historical volatility features a Log-Normal dis-tribution, especially in the region of low values of volatility, coherently with theconclusions achieved by Bouchaud and Potters [12] (see Section 1.3.2). Conversely,the Hull and White model is a good candidate in reproducing the tail of the pdf.Thus, the expOU model seems able to capture the non-Gaussian nature of volatil-ity, as expected, especially in the body of the distribution.

σ0 0.02 0.04 0.06 0.08 0.1 0.12

Pro

babi

lity

dens

ity

1

10

Bulgari data, 03/01/2000-29/05/2007

σ0 0.02 0.04 0.06 0.08 0.1 0.12

Pro

babi

lity

dens

ity

1

10

210

Fiat data, 03/01/2000-29/05/2007

Vasicek

HestonExpOU

Figure 4.8: Fit to empirical daily volatility distribution on the theoretical predic-tion of Vasicek (green), Heston (blue) and expOU (pink) models for Bulgari SpAand Fiat SpA. More detail about the predicted curves can be found in Tab 3.2.

In the light of this result we propose a comparison between empirical returnsand the analytical expression of Eq. (3.84) derived by Masoliver and Perello (MP)in the expOU framework, in order to test its ability in modeling the return fea-tures. The same curve is shown in Fig. 4.3, in fully agreement with the numericaloutcomes. With respect to what discussed in Section 3.4, we dealt, like in the

72


numerical analysis, only with zero-mean log-returns Xτ , estimated as in Ref. [39]

Xτi = Rτ

i − 〈Rτ〉 = Rτi −

1

N

N−1∑

i=0

Rτi , (4.12)

where Rτ is the vector of returns and the parameter τ stands for time lag (seebelow). In Tab. 4.1 we report the mean values 〈Rτ 〉 of the time series considered,evaluated from empirical moment as clearly emerges from Eq. (4.12). We notethat for daily returns, i.e. τ = 1, the means are quite negligible, involving no dif-ference between log-returns and zero-mean log-returns. Nevertheless, to complywith the literature, we analyse only Xτ

i series also in the case of daily data.

Table 4.1: Mean values of Fiat SpA, Brembo and Bulgari SpA return series Rτ ,estimated as empirical moment, for different time horizons τ (days).

〈R1〉% 〈R5〉% 〈R7〉% 〈R20〉% 〈R40〉%Fiat SpA 0.01% -0.06% -0.09% -0.22% -0.55%Brembo 0.02% 0.05% 0.07% -0.20% 0.33%Bulgari SpA 0.05% 0.10% 0.13% -0.29% -0.52%

Fig. 4.9 compares daily returns with the MP theory, together with the Normaland the fat-tailed Student-t introduced in Section 2.2. We would recall that theMP function derived in Section 3.4 is an approximate solution, valid only in thestationary limit of volatility. This hypothesis is well verified for our stocks, sincethey have been trading in the Italian stock market (and also in the NYSE in thecase of Fiat SpA) from the last century (Fiat SpA) and the 90’s (Brembo andBulgari SpA).

In order to constrain the parameter entering the theoretical formulae, we im-plement a negative log-likelihood algorithm, as for the volatility fit. In particular,for the MP function we perform a multidimensional fit over the free parameters(α,κ,m,ρ) appearing in Eq. (3.84). In general a four-dimensional minimization isvery ticklish to manage, due to the appearance of local minima in the optimizationfunction. Nevertheless, the parameter correlation matrix proves that m, which de-scribes the normal level of volatility, has quite negligible correlation with otherparameters. Moreover, its value is very similar for all the three shares (∼ 0.02).This allows to fix this parameter and perform a more careful fit over (α,κ,ρ) whichare, unlike m, strongly correlated. In spite of the trouble of such procedure, ourfits turn out to be always stable. The parameters of the fitted curves, displayedin Tab. 4.2, are similar to the ones quoted in the literature [39]. Note also thatκ > m, as requested by the theory illustrated in Section 3.4.Looking at Fig. 4.9, we could convince ourselves that Student-t and MP functionare in good agreement with empirical distributions, both in the central body andin the tails, while Normal fails to reproduce the data. This is true especially forFiat SpA and Bulgari SpA, whereas the tails of Brembo data tend to be widely

73


Daily return, X-0.1 -0.05 0 0.05 0.1

p(X

)

-110

1

10

Fiat data, 03/01/2000-29/05/2007

Daily return, X-0.1 -0.05 0 0.05 0.1

p(X

)

-110

1

10

Brembo data, 03/01/2000-29/05/2007

Daily return, X-0.1 -0.05 0 0.05 0.1

p(X

)

-110

1

10

Bulgari data, 03/01/2000-29/05/2007

Normal

Student

MP func

Figure 4.9: Fit of daily returns (histogram) on the theoretical prediction of theexpOU model (green), in comparison with Normal (blue) and Student-t (red)distributions for Fiat SpA, Brembo and Bulgari SpA.

Table 4.2: Estimated parameter values of the fitted curves shown in Fig. 4.9:Normal (µ,σ), Student-t (µ,σ,ν) and MP function (α,κ,m,ρ). Note that the meanis indicated with the symbol µ to not generate confusion with the parameter mentering the MP formula.

µ% σ% ν α κ m ρ

Fiat SpA 0.01% 2.06% 3.32 0.01 0.22 0.02 -0.7Brembo 0.02% 1.86% 3.08 0.02 0.31 0.02 -0.2Bulgari SpA 0.05% 2.45% 3.60 0.02 0.17 0.02 -0.3

overestimated by the MP theory. This behaviour could be related to the lowercapitalization of Brembo with respect to companies like Bulgari and, above all,Fiat. A low capitalization implies, in fact, a smaller market, lower liquidity of theshare and, consequently, less exchanges. In such a scenario, it can happen that afew, anomalous, transitions (i.e big losses or gains) would assume great relevancein the company’s history. From a statistical viewpoint, these extreme events, thatenrich the distribution tails, tend to assume a statistical weight too relevant sincethe overall number of exchanges is relatively low. This fact could be the cause ofthe overestimation in the tail regions exhibited by the MP fit.

To better evaluate the ability of the three curves in capturing the featuresof empirical distributions, especially in the tails region, we propose in Fig. 4.10the corresponding cumulative density functions P>, numerically evaluated fromthe curves of Fig. 4.9. The figures displayed on the left panel show that the MPfunction fits data better than Student-t density. This is due to the presence inthe Eq. (3.84) of the Hermite polynomials, which allow to model correctly thefunction shape over the data. Similar results emerge also focusing only on the lefttail of the distribution, i.e. negative returns, which has known to be the regionwhere risk is located. As already noted, this conclusion is not completely correct

74


Daily return, X-0.15 -0.1 -0.05 0 0.05 0.1

P>

(X)

-310

-210

-110

1

Fiat data, 03/01/2000-29/05/2007

Daily return, -X-210 -110

P>

(X)

-310

-210

-110

1

Fiat data, 03/01/2000-29/05/2007

DataNormalStudentMP func

Daily return, X-0.15 -0.1 -0.05 0 0.05 0.1

P>

(X)

-310

-210

-110

1

Brembo data, 03/01/2000-29/05/2007


P>

(X)

-310

-210

-110

1

Brembo data, 03/01/2000-29/05/2007

Daily return, X-0.15 -0.1 -0.05 0 0.05 0.1

P>

(X)

-310

-210

-110

1

Bulgari data, 03/01/2000-29/05/2007


P>

(X)

-310

-210

-110

1

Bulgari data, 03/01/2000-29/05/2007

Figure 4.10: From top: Fiat SpA, Brembo and Bulgari SpA cumulative densityfunction of daily returns (left) and negative daily returns (right). Points representempirical data, whereas lines correspond to Normal (blue), Student-t (red) andMP function (green) as in the corresponding Fig. 4.9.

for Brembo, even if the MP function is able to capture the most extreme eventsof its distribution.

Finally, in Fig. 4.11 we present the return distributions for several time hori-zons τ (1,5,7,20,40 days) in comparison with the MP theoretical function, as inFig. 4.3. With respect to the procedure described in Ref. [75], for each time lagwe extract the return series Xτ from Eq. (4.12) fixing the time lag τ . Tab 4.1proves that the larger the value of τ , the closer to zero the means are. Thus,when the lag increases, Rτ and Xτ tend to have a different behaviour. It’s worthmentioning that all the curves appearing in Fig. 4.11 are generated changing inEq. (3.84) only the value of the temporal parameter t according to τ , while forthe other parameters (α,κ,m,ρ) we always use the values listed in Tab. 4.2. Thisis the same procedure adopted to produce the corresponding Fig. 4.3: namely, wedid not fit the model parameters trying to find the best ones, but we limited to

75


make a sharp prediction of the theory plotting the curves in comparison with theempirical data. The overall agreement is very good. This is indeed the meaning ofthe system of Eqs. (4.1) on the grounds of the SV models studied: once fixed theparameters, the equations must reproduce correctly the time evolution of the twocorrelated processes. The quality of the MP theory emerges also noting that, whenthe time lag increases, the left tail of the empirical distributions becomes fatterand the analytical solution tends to decrease its skewness (recall Section 1.3.1),becoming more asymmetric and similar to the data shape.

Thus, expOU model seems to be able to reproduce carefully both volatilityand return distribution, the latter over different time horizons too.

Return, X

-0.6 -0.4 -0.2 0 0.2 0.4

p(X

)

-110

1

10

210

310

410

Fiat data, 03/01/2000-29/05/2007

Return, X

-0.4 -0.2 0 0.2 0.4

p(X

)

-110

1

10

210

310

410

Brembo data, 03/01/2000-29/05/2007

Return, X

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

p(X

)

-110

1

10

210

310

410

Bulgari data, 03/01/2000-29/05/20071 day5 days7 days20 days40 days

MP func

Figure 4.11: Comparison between the probability distribution of returns in theexpOU model (solid line) and empirical data (points) on different time horizonsfor Fiat SpA, Brembo and Bulgari SpA. For clarity, each line is shifted from theother by one decade. The parameter values of the fitted curves are displayed inTab. 4.2.

76

Conclusions and perspectives

The geometric Brownian motion (GBM) is still accepted as a paradigm of thequantitative finance applied to financial complex systems. Nevertheless, as wehave pointed out, within this framework many facts typical of the real markets dy-namics cannot be captured. They concern the non-Gaussian nature of log-returndistribution, the non-constant character of volatility, which features, conversely,its own dynamics, the leverage effect and the volatility autocorrelation described inChapter 1. To resolve the contradictions between the well-known stylized facts andthe widely accepted theory of Black and Scholes-Merton (BSM), models beyondthe Gaussian paradigm are strongly demanded. Inspired by these motivations,the studies presented in this thesis have been devoted to non-Gaussian stochasticmodels, to their often different nature and their possible applications in the fieldof quantitative finance.

In this scenario, we have shown that within the framework of statistical physicsthe Student-t distribution, often used in the econophysics literature, emerges aspeculiar distribution able to capture the non-Gaussian nature of log-returns. Inparticular, in Chapter 2 we have illustrated a possible application of this model torisk management, in order to obtain market risk measures in the presence of non-Gaussian fluctuations. It is known in fact that Student-t allows to model carefullythe power-law tails of return distribution, where the risk resides. We have shownthat the obtained non-Gaussian parametric formulae given in Eqs. (2.6) and (2.7)are able to capture accurately the leptokurtic nature of price changes. Moreover,we have verified that they are also in good agreement with a full historical evalu-ation, once specified in terms of the models parameters optimized by means of anempirical analysis of real daily return series. With respect to similar investigationspresent in the literature [47, 60], our results sound more statistically robust thanksto the confidence levels, estimated with a bootstrap-based technique, attached tothe performed measures. As far as possible perspectives are concerned, it wouldbe interesting to investigate to what extent our conclusions, drawn from an anal-ysis of a sample of Italian financial data, apply also to other financial markets.In particular, one could check whether, at a given significance level, statisticallyrelevant differences are present between the results valid for a single asset andthose relative to a portfolio of assets, as our analysis seems to indicate, at least

77


for 5% VaR. Another interesting development concerns the comparison betweenthe predictions for VaR and ES of our model with the corresponding ones de-rived by means of other statistical procedures to measure tail exponents known inthe literature [54, 56, 57], as well as with the results from simulations of advancedmodels of the financial market dynamics, such as GARCH-like, multi-fractal mod-els [58, 59] and stochastic volatility models.

Taking this latter perspective as a starting point, we have investigated thefamily of stochastic volatility models, widely described in Chapter 3, as a naturalextension of the GBM paradigm. The common denominator of the models belong-ing to this family is in fact that the volatility, a constant in the BSM framework,is itself a stochastic quantity driving the log-returns dynamics and driven on itsown by a Wiener process. Among this class, we have dealt with the Vasicek,Heston and expOU models, stressing their ability, as well as their limitations, incapturing the commonly held stylized facts about volatility and returns. From thetheoretical viewpoint, the expOU model has turned out to be the most realistic,since it displays the capability to correctly reproduce the Log-Normal character ofthe volatility distribution, as well as the typical double-time scales of the volatilityautocorrelation.

The theoretical features derived in Chapter 3 have been tested ab initio onthe outcomes of original numerical simulations, paying particular attention to thepredictive effectiveness of each model. This is indeed a necessary task, especiallyin view of financial applications like the above-mentioned risk management oroption pricing, which can be often treated only by means of numerical methods.For each model we have proved an almost perfect convergence between the the-oretical predictions and the numerical results, which assures that the theory iscorrect and, conversely, the numerical implementations of the models maintain alltheir features. Moreover, we have also performed and presented in Chapter 4 anempirical analysis of Italian time series, whose aim has been to critically studythe models valuating for each one its degree of realism once compared with realdata. Our analysis has indicated that the expOU model is the most successful incapturing the non-Gaussian character of empirical volatility. It also reproducesquite well the return distribution, even if a better agreement with high-frequencydata, especially in the tails, could be achieved by substituting in the theory theWiener noise with a non-Gaussian one [86], therefore leaving definitively the BSMframework. A natural prosecution of such analysis could concern extending theempirical investigation to a wider sample of financial series, in order to check thevalidity of our results with a deeper statistical meaning. On the other hand, itcould also be interesting to include in both the theoretical and empirical analysisother SV models, like the Hull and White [85], and then compare them with theones already taken into account. From our analysis we would conclude that oneof most relevant open questions about the SV model is to establish whether theexpOU model, in its original formulation or in a more complex one (i.e. non-Gaussian noise), could really be a good candidate to capture the real marketphenomena. Moreover, we guess that a key point is also understanding if the

78


model could describe only the dynamics of daily and low-frequency data or couldbe successfully applied also to infra-day time horizons.

To conclude, we would note that the most natural “bridge” between the twomain topics faced in this work (risk measures and SV models) could obviously beto perform risk measures modeling the return distribution in term of the (semi)-analytical expressions derived in the SV frameworks, leading to non-GaussianSV market risk measures, easily comparable with the methodologies described inChapter 2.

The non-Gaussian stochastic models would seem the natural way to go beyondthe GBM paradigm and the inconsistencies emerged from this model. This isan important issue, for both the academic investigations and the finance-practiceapplications still based on BSM. Despite the theoretical difficulties related to thesemodels, clearly harder to overcome when compared with the Gaussian standard,they allow to perform more reliable evaluations, getting closer to the real natureof financial data.

79


80

Appendix AThe zero-mean return features

The zero-mean returns have been defined through their differential dX of Eq. (3.6).The process whose increment has such an expression is given by

X(t) = ln

[S(t)

S0

]− µt +

1

2

∫ t

t0

σ2(t′)dt′ . (A.1)

We now note that, despite the correlation between dσ(t) and dW1(t), σ(t) anddW1(t) are independent random variables. This is a consequence of Ito prescriptionfor stochastic integrals [19], because the process σ(t) is independent of its drivingnoise. Hence we have

E[dX(t)] = E[σ(t)dW1(t)] = E[σ(t)]E[dW1(t)] = 0 , (A.2)

in accordance with the name “zero-mean” given to X(t).To better characterize the zero-mean returns, we will derive several expectation

values concerning dX, in order to construct also two correlation functions enteringthe definition of leverage effect Eq. (1.23) and volatility autocorrelation Eq. (1.25)given in Section 1.3.3. Taking into account the independence of σ(t) and dW1(t)we obtain

E[dX2] = E[σ2]E[dW 21 (t)] = E[σ2]dt , (A.3)

where we used the Ito identity. Combining Eqs. (A.1) and (A.3) we can derivethe second central moment (i.e. variance) given by

Var[dX(t)] = E[dX2] − E[dX]2 = E[σ2]dt . (A.4)

For the fourth moment, in much the same way, we can demonstrate

E[dX4] = E[σ4]E[dW 4] = E[σ4]3E[dW 2]2 = 3E[σ4]dt2 . (A.5)

It must be noted that for the second equality we have made use of Novikov’stheorem [87], a result we will count on several times in the following. Hence

Var[dX2(t)

]= E[dX4] − E[dX2]2 = E

[σ(t)4

]E

[dW (t)4

]

− E[σ(t)2

]2E

[dW (t)2

]2= 3E

[σ(t)4

]dt2 − E

[σ(t)2

]2dt2 .

(A.6)

81

A. The zero-mean return features

Finally, we derive two important averages involving different times. The firstone is

E[dX(t + τ)2dX(t)] = E[σ(t + τ)2dW1(t + τ)2σ(t)dW1(t)] , (A.7)

which gives us information on the correlation between squared X fluctuation(a possible proxy for volatility) and fluctuation themselves. Indeed, observingEq. (1.23), Eq. (A.7) has great importance in calculating the leverage function.Following the Ito convention, if τ > 0 then dW1(t + τ) is uncorrelated with theother factors; moreover we know that E [dW1(t + τ)2] = dt and thus

E[dX(t)dX(t + τ)2

]= E

[σ(t)dW1(t)σ(t + τ)2

]dt (τ > 0). (A.8)

On the other hand, if τ ≤ 0 then dW1(t) is uncorrelated with the remainingvariables; taking into account that E [dW1(t)] = 0, we get

E[dX(t)dX(t + τ)2

]= 0 (τ ≤ 0). (A.9)

The two results are merged in

E[dX(t)dX(t + τ)2

]= E

[σ(t)σ(t + τ)2dW1(t)

]H(τ)dt , (A.10)

where H(τ) is the Heaviside step function defined in Eq. (3.38). Writing explicitlydW1(t) = ξ1(t)dt, the Novikov theorem allows to rewrite Eq. (A.10) as

E[dX(t)dX(t + τ)2

]= 〈σ(t)σ(t + τ)2ξ1(t)〉H(τ)dt2

=

⟨δ [σ(t)σ(t + τ)2]

δξ1(t)

⟩H(τ)dt2 = ρ

⟨δ [σ(t)σ(t + τ)2]

δξ2(t)

⟩H(τ)dt2

= ρ

⟨σ(t + τ)2 δσ(t)

δξ2(t)+ 2σ(t)σ(t + τ)

δσ(t + τ)

δξ2(t)

⟩H(τ)dt2 .

(A.11)

Any ensuing development of the calculation requests the evaluation of the func-tional derivatives of σ, which of course are model-dependent.

In analogous fashion, we can calculate the autocorrelation function

E[dX(t)2dX(t + τ)2

]= 〈σ(t)2σ(t + τ)2dW1(t)ξ1(t)〉dt2

= 2ρ

[⟨δσ(t)

δξ2(t)σ(t)σ(t + τ)2dW1(t)

⟩+

⟨σ(t)2σ(t + τ)

δσ(t + τ)

δξ2(t)dW1(t)

⟩]dt2

+

⟨σ(t)2σ(t + τ)2 δ[dW1(t)]

δξ1(t)

⟩dt2 ,

(A.12)which is needed for the calculations of volatility autocorrelation given in Eq. (1.25).We are presently only able to obtain an expression for the functional derivativeδ[dW1(t)]/δξ1(t): to this aim we write dW1(t) in a somewhat involved form,

dW1(t) =

∫ t+dt

t

ξ1(t′)dt′ . (A.13)

82

Thusδ[dW1(t)]

δξ1(t)=

∫ t+dt

t

δ(t − t′)dt′ = 1 . (A.14)

This result, even not knowing the exact forms of the functional derivatives ofσ, ensures us that the first term in Eq. (A.12), as it contains dW (t) factors, isnegligible in front of the second one. Namely

E[dX(t)2dX(t + τ)2

]=

⟨σ(t)2σ(t + τ)2

⟩dt2 + O(dt3) (A.15)

Of course, the previous formulae are useless if the dynamics of σ, which generateits averages, is not specified through a choice of the functions f and g appearingin Eqs. (3.1).

83

A. The zero-mean return features

84

Appendix BThe Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process [68] is the model describing the velocity processof a Brownian motion. It’s worth recalling that the physical phenomenon knownas Brownian motion, observed for the first time by Robert Brown in 1827 withhis microscope, concerns the small random movement of fine particles suspendedin a fluid solution. Several years later, its dynamics was interpreted by Einsteinin his famous paper dated 1905 [6], in which he illustrated how the statisticaltheory of heat would require the motion of particle in suspension and, thereby,the diffusion. Such a phenomenon would not be allowed by classical thermody-namics: Brownian motion belongs in fact to the class of stochastic process and,more precisely, to the family of Wiener processes.

The quantity better described by the Wiener process is the momentum trans-fer from the molecules to the particle, namely the force acting on the particle.Considering that the latter is immersed in the solute, there will be an additionalfriction force opposing the motion that, considering the small velocities involved,will be well described by a Stokes-like formula in which a γ friction coefficientappears. The stochastic differential equation for the velocity process will thushave a deterministic and a stochastic terms

dv(t) = −γv(t)dt +Γ

mdW (t) , (B.1)

where m is the particle’s mass and Γ is the intensity of the stochastic force.Following the same analysis performed in Section 3.2 for the Vasicek model, we

can compute the average (on the particle ensemble) velocity and squared velocity,given by

E [v(t)] = v0e−γ(t−t0) (t−t0)→∞−→ 0 , (B.2)

where v0 is the initial velocity, common to all the ensemble, and

E[v2(t)

]=

Γ2

2γm2+

(v20 −

Γ2

2γm2

)e−2γ(t−t0) (t−t0)→∞−→ Γ2

2γm2. (B.3)

We would note that the particle velocity has a null stationary mean value, whichmeans that the particles spread over uniformly the entire available solvent volume

85

B. The Ornstein-Uhlenbeck process

so that their velocities cancel out each other in average. Moreover, it is charac-terized by a non-zero stationary mean square, indicating that the particles tendto maintain the velocity associated with Brownian motion. Making use of theequipartition theorem, we can derive the relation that the coefficients γ and Γmust satisfy in order to correctly describe the Brownian motion from a thermo-dynamical viewpoint (like in the Einstein’s approach). It reads

1

2m

⟨v2

⟩∣∣t→∞ =

1

2kBT =⇒ Γ2 = 2γmkBT . (B.4)

This is a case of the fluctuation-dissipation theorem, which links the strength ofthe equilibrium fluctuations Γ to the dissipation intensity γ.

From a statistical point of view, to completely characterize the system one hasto compute the probability distribution of v at time t by solving the Fokker-Planckequation. For such a process, it assumes the following expression [35]

∂

∂tp(v, t) =

∂

∂v(γ vp(v, t)) +

1

2D

∂2

∂v2p(v, t) . (B.5)

Switching to the characteristic function

ϕv(ω, t) =

∫ ∞

−∞dv eiωvp(v, t)

the equation becomes

∂

∂tϕv(ω, t) = −γω

∂

∂ωϕv(ω, t) − 1

2Dω2ϕv(ω, t) . (B.6)

The general solution is computable with the method of characteristics and reads

ϕv(ω, t) = e−Dω2

4γ g(ωe−γt) , (B.7)

with g an arbitrary (well-behaved) function. Since the pdf initial condition

p(v, 0) = δ(v − v0) (B.8)

is translated intoϕv(ω, 0) = eiv0ω , (B.9)

we can rewrite g(ω) as follows

g(ω) = exp

(Dω2

4γ+ iv0ω

). (B.10)

By substituting the latter expression into Eq. (B.7), we can hence derive theultimate solution

ϕv(ω, t) = exp

[−Dω2

4γ(1 − e−2γt) + iωv0e

−γt

]. (B.11)

86

It’s worth noting that Eq. (B.11) is the characteristic function of a Gaussiandistribution with mean coinciding with Eq. (B.2) and variance given by

Var [V (t)] =D

2γ

(1 − e−2γ(t−t0)

) (t−t0)→∞−→ D

2γ, (B.12)

namely in agreement with Eq. (B.3). Recalling Eq. (B.4) we obtain the velocitypdf in terms of the solution temperature:

p(v, t) =

√m

2πkBT (1 − e−2γt)exp

[− m

2kBT

(v − v0e−γt)

2

1 − e−2γt

]

t→∞−→√

m

2πkBTexp

[− mv2

2kBT

],

(B.13)

which is the well-known Maxwell-Boltzmann distribution of velocities in one di-mension. Doob showed in Ref. [88] that the Ornstein-Uhlenbeck process is theonly Gaussian Markov stochastic process having stationary solution. This is an-other proof of the universality of the Maxwell-Boltzmann distribution.

To conclude, we would mention that defining the position stochastic processin term of the velocity process through

x(t) =

∫ t

t0

dt′ v(t′) (B.14)

we are able to calculate its variance, which reads

Var [X(t)] =2kBT

mγt

.= Dxt . (B.15)

Eq. (B.15) is equal to the expression of a Brownian motion variance, if x(t) istaken as an independent Wiener process.

87

B. The Ornstein-Uhlenbeck process

88

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96

Acknowledgement

Oh we can be Heroes, just for one day

David Bowie, Heros

Desidero ringraziare Guido, di cuore, per tutto quello che mi ha insegnato in questianni, per la fiducia, i consigli (scientifici e non), l’allegria che e in grado di trasmet-tere. Oreste, per le tante discussioni scientifiche e la perenne disponibilita. Unringraziamento particolare va a Giacomo e Lorenzo, ho imparato tanto lavorandoe trascorrendo il tempo con voi. Grazie, perche se non mi sono mai pentita dellascelta che ho fatto e soprattutto per merito di tutti voi!Grazie Gio, sei stato un compagno d’ufficio fantastico!Vorrei ringraziare inoltre il prof. Rosario Mantegna, per aver fatto da lettore aquesto lavoro.

Grazie a tutta la mia famiglia. Grazie ai miei genitori, per avermi regalato annibellissimi, pieni di affetto e pure di qualche critica, in fondo non fanno mai male.Grazie Gigi, sei il miglior fratello che si possa avere, nonostante i tuoi silenzi!Grazie Pia, per avermi sempre tenuto per mano, anche nei momenti piu difficili.Grazie al piccolo Davide, che riesce sempre a strapparmi un sorriso. Grazie Laura.Nonna, zia, sono sicura che, in fondo, anche voi in questo momento siete felici edemozionate, quasi piu di me, come lo siete sempre state per ogni mio “successo”.Mi mancate.

Grazie a tutti gli amici e le amiche, fisici e non, pavesi e bergamaschi, che riem-piono le mie giornate e le mie serate. Un ringraziamento speciale va a Franci perl’amicizia che mi regala ogni giorno, le risate, le parole di conforto, il suo affetto.Andrea, se sono arrivata fino a qui e grazie a te, alla tua vicinanza, alla tua ca-pacita (unica) di capirmi sempre...avrei mille motivi per ringraziarti e mille altriancora da farmi perdonare...Voglio dirti solo grazie.

Enrica

97

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