Non-Gaussian Stochastic Models and Their Applications in Econophysics

UNIVERSITÀ DEGLI STUDI DI PAVIA DOTTORATO DI RICERCA IN FISICA – XX CICLO

Enrica Vera Cisana

Tesi per il conseguimento del titolo

Università degli Dipartimento di Fisica Istituto Nazionale di Studi di Pavia Nucleare e Teorica Fisica Nucleare

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 Cover

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.

Non-Gaussian Stochastic Models and Their Applications in Econophysics Enrica Vera Cisana PhD thesis – University of Pavia Printed in Pavia, Italy, November 2007 ISBN 978-88-95767-06-2 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 price differences ...... 11 1.2 One-minute returns of Standard&Poor’s 500 index ...... 12 1.3 Volatility clustering in the Dow Jones index ...... 13 1.4 Dow Jones index daily returns on the period 1900–2000 ...... 14 1.5 Distribution of the empirical volatility fitted by the theoretical Log- Normal and Inverse-Gamma distributions ...... 15 1.6 Prices of real European call options on the German DAX index versus the moneyness ...... 16 1.7 Smile-shape curve characterizing implied volatility ...... 17 1.8 Empirical form of the leverage effect ...... 18 1.9 Empirical form of the volatility autocorrelation ...... 19

2.1 Methodology to compute Value at Risk and Expected Shortfall . . 24 2.2 Convergence of VaR and ES Student-t formulae toward Gaussian limit ...... 26 2.3 Sensitivity of VaR and ES to variation of the tail index value . . . 28 2.4 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 volatility 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 Autostrade SpA and index Mib30 ...... 33

3.1 Characteristic function of returns in the Vasicek model ...... 44 3.2 Probability density function of χ2 variables ...... 49 3.3 Stationary probability distribution of variance and characteristic function of the return pdf in the Heston model ...... 51 3.4 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 of the Vasicek, Heston and expOU models ...... 66 4.3 Numerical returns in comparison with the theory of the expOU model 67 4.4 Numerical returns in comparison with the theory of the Vasicek and Heston models ...... 68 4.5 Numerical leverage eﬀect in comparison with the theory of the Va- sicek and expOU models ...... 70 4.6 Numerical volatility autocorrelation in comparison with the theory of the Vasicek, Heston and expOU models ...... 70 4.7 Historical evolution of Fiat SpA, Brembo and Bulgari SpA shares in both linear and log-linear scales ...... 71 4.8 Fit to historical daily volatility distribution of Bulgari SpA and Fiat SpA ...... 72 4.9 Probability densities of daily returns in comparison with the Nor- mal, the Student-t and the theory of the expOU model for Fiat SpA, Brembo and Bulgari SpA ...... 74 4.10 Cumulative density function of daily returns and negative daily returns for Fiat SpA, Brembo and Bulgari SpA ...... 75 4.11 Probability densities of returns on diﬀerent time lags in comparison with the theory of the expOU model for Fiat SpA, Brembo and Bulgari SpA ...... 76

vi List of Tables

2.1 Crossover values of the tail index for VaR and ES corresponding to different significance levels ...... 27 2.2 Estimated parameters values (mean, volatility and tail index) of the 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 evaluations) for the time series as in Tab. 2.2 ...... 32 2.4 Estimated VaR values with 68% CL intervals (bootstrap evaluations), according to different methodologies, for the time series as in Tab. 2.2 ...... 34 2.5 Estimated ES values with 68% CL intervals (bootstrap evaluations), according to different methodologies, for the time series as in Tab. 2.2 ...... 35

3.1 Models of volatility ...... 40 3.2 Theoretical features of the Vasicek, Heston and expOU stochastic volatility models ...... 59

4.1 Estimated mean values of Fiat SpA, Brembo and Bulgari SpA return series for diﬀerent time lags ...... 73 4.2 Estimated parameter values of the ﬁtted curves shown in Fig. 4.9 for 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 physicists to put physics into Wall Street, or the paper of Fabrizio Lillo, Salvatore Miccichè and Rosario N. Mantegna [2], the latter being one of the first physicists to put finance into academical physics, could certainly help understand how one can justify using the methods of physics and the formalism of mathematics in the frenzied world of finance. Even so, no all the doubts could have been dispelled: if the value of a certain financial good is determined by people, how can the human behaviour be described by equations or predetermined rules? This could sound, perhaps, a more “philosophical” question, arising from the dissimilar nature of knowledge of physics and social sciences. On this matter, we cannot forget the prophetical words of Ettore Majorana, who identified the analogy between physical and social sciences into the statistical character of the laws describing elementary processes, emerging in the framework of quantum mechanics [3]. More recently, Giorgio Parisi has remarked that, besides statistical physics and quantum mechanics quoted by Majorana, another revolution has “shaken” the world of physics during 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 and the positive consequence of this process is that the scope of physics becomes much larger and the constructions of physics find many more applications. The contribution of physicists to the study of economical markets must be interpreted in the light of these words: complexity turns out to be, thus, the reading-key. There are many natural phenomena which exhibit the typical features of complex systems, like earthquakes, the DNA, the traffic flow and, clearly, financial markets. Among the methods developed to study their complex dynamics, the theory of stochastic processes still plays a crucial role in describing and modeling economical systems. Thus, it is not so surprising that the first contact between physics and finance must be traced back to 1900, when Louis Bachelier, attempting to reproduce the price dynamics of some goods traded in the Paris Stock Exchange, developed the model of random walk [5]. This physical phenomenon, which is the mathematical formulation of the Brownian motion [6], is commonly considered by physicists as the archetype of stochastic processes.

1 Introduction

The relevance of Bachelier’s pioneering ideas, forgotten for a long time, has been rediscovered within the economical community in the late ’50s and definitively hallowed by the works of Fisher Black and Myron Scholes [7], and Robert Merton [8] (BSM). In their theory of option pricing, they assumed that the fluctuations in the value of the underlying feature a geometric Brownian motion (GBM), a stochastic process strictly related to the original Brownian motion. Since its publication in 1973, this theory has become a standard in the world of quantitative finance, so that we could state that the history of quantitative finance and econophysics after 1973 is, essentially, the history of the ways in which academics and financial practitioners have refined and extended the BSM model. From this viewpoint, the contribution of physicists has been fundamental, especially if we connect their increasing interest in financial issues to the vast amount of data (prices, orders, volumes, collected over time horizons ranging from a few minutes up to years) available thanks to the informatitation of exchange markets. The empirical studies performed by economists,mathematicians and physicists [9, 10, 11, 12] allowed to prove the existence of some mismatches between the simple statistical properties of the ideal GBM price dynamics and the more complex behaviour observed in real markets, commonly denoted as stylized facts. Among them, we would remind the proof that price returns feature a distribution different from the Gaussian one predicted by the GBM process, where each event happens randomly and independently of all others. We will discuss this point in detail in Chapter 1. To cope with these non-trivial evidences, various theoretical models have been proposed in the literature, in order to go beyond the Gaussian paradigm and progressively refine the statistical characterization of economical dynamics. These models have contributed to the proliferation of different approaches followed by researches in econophysics to interpret and, sometimes, try to forecast the variegate feature in the behaviour of financial market, one of the old systems that the man has built by means of his social organization. The work presented along this thesis is placed into this scenario: its main topics concern, in fact, models based on non-Gaussian stochastic dynamics for the time evolution of price returns, as well as their possible applications in the field of quantitative finance. To understand the necessity of such models for a realistic description of financial markets, in Chapter 1 we will discuss the BSM theory, stressing the main empirical evidences not predicted within this standard framework. The increasing effort to extend models beyond this paradigm is closely related to the possibility of capturing the true statistics of financial phenomena, as well as achieving reliable results in typical financial problems. To prove this latter statement, in Chapter 2 we will illustrate a possible application of the Student-t distribution, a model emerging from the field of statistical physics, to risk management, in order to obtain market risk measures in the presence of non-Gaussian fluctuations. Chapter 3 will be devoted to the family of stochastic volatility models, a natural extension of the GBM paradigm. The models belonging to this class are characterized, in fact, by a random volatility driving the price returns dynamics 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-Uhlenbeck ones), whose theoretical features will be described in detail in Chapter 3 and tested ab initio by means of numerical simulations in Chapter 4. This is indeed a crucial task, especially in view of ﬁnancial applications, like option pricing or risk management, often investigable only through numerical strategies. To conclude, the latter part of Chapter 4 will be dedicated to an empirical analysis of some ﬁnancial time series from the Italian stock market. The aim of this analysis is to critically study the models presented in Chapter 3, evaluating their degree of realism once compared with real data. The results emerged from our works, as well as some possible perspective, will be summarized at the end of the thesis.

The studies discussed in the present work have been developed in collaboration with 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 to the 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 following international meetings:

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

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

(p) A Comparative Study of Stochastic Volatility Models, Applications of • Physics 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 1 Why non-Gaussian models for market dynamics?

The main topic of this work is the application of non-Gaussian stochastic models to the characterization of real financial dynamics. Why is it so necessary to employ such models to describe in a realistic way financial markets? The aim of 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 markets are complex systems, thus investigable within the framework of stochastic processes. Thanks to the last decades’ informatization of exchange markets, there has been an increasing research of models able to capture carefully the statistical features shown by financial variables, like price changes and their corresponding log-returns, volatility, volumes,. . . We agree with Bouchaud and Potter, which point out in Ref. [12] that the word modeling can be interpreted in two different meanings 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 able to 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 the price statistics modeling, belongs to the first class. Within this framework, stock price features a Log-Normal distribution, whereas the difference between the logarithm of prices (log-returns) perform a Brownian motion, turning out to be normally distributed. The GBM plays a key role in the quantitative finance, since it 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 able to provide only a first approximation of the true statistics of price, since it fails in predicting several established phenomena observed in real markets. As we will see 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 the different-time correlations of the price returns or volatility processes itself. To cope with these non-trivial stylized facts, various theoretical models have been proposed in the literature, based on non-Gaussian dynamics for the time evolution 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 paradigm of quantitative finance, and the behaviour exhibited by prices in real markets. This is organized as follows: in Section 1.1 we describe financial markets, their complex nature and, therefore, the possibility of modeling them in a stochastic approach. A brief historical outline, from the pioneering work of Louis Bachelier up to the BSM model, is also given. The latter is shown in Section 1.2. To conclude, in Section 1.3 we illustrate the main mismatches between the GBM theory and the stylized facts which characterize real markets and lead, therefore, to the introduction of non-Gaussian stochastic dynamics for markets modeling.

1.1 Financial markets: complex systems and stochastic dynamics

It is well documented by many studies [2, 11, 12, 13] that financial markets exhibit the typical features of complex systems, since they are open and made up by numerous elements that continuously interplay. One could think about the stock market and its continuous succession of shares sold and bought by investors, strongly influenced by economic, political and social factors. Thus, the complex behaviour of markets can be studied within the framework of the statistical and theoretical physics, like many other natural phenomena (diffusion, chaos, forma- tion of avalanches, earthquakes,. . . ). In particular, the development of the theory of stochastic processes has played a key role in the description of such systems from 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 Théorie de la spéculation at the Sorbonne, University of Paris [5]. In this work, attempting to model the price dynamics of some derivatives traded in the Paris Stock Exchange, he developed the 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 its mathematical formulation, i.e. the random walk, turn out to be the archetypes of the stochastic processes. In the 1908 Paul Langevin, trying to explain the Brownian motion phenomenon, derived the first example of stochastic differential equation [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. Itô has quoted it as one of the main motivations to introduce his stochastic calculus and the geometric Brownian motion [19]. The relevance of Brownian motion was rediscovered in the financial community in the late 50’s, thanks to the studies performed by M. Osborne [20], P. Samuelson and R. Merton [21, 22], as a good candidate in reproducing the dynamics of price changes and their statistical features. Physics is becoming aware of Bachelier’s work only now through the meet between statistical physics and quantitative finance. In 1973 F. Blach and M. Scholes [7] and, independently, R. Merton [8] developed a framework to determinate the fair price of an option assuming that the fluctuations in the value of the underlying feature a geometric Brownian dynamics, thus reducing this problem to a diffusion equation. The model has become a standard in the theory of option pricing and, more generally, within the overall framework of quantitative finance, contributing to the awesome growth of the discipline during the last 30 years. For this reason, in the next section we briefly discuss the BSM model, even if in this work we haven’t dealt with problems of option pricing.

1.2 The Black and Scholes-Merton paradigm

Since its publication, the BSM theory has become the paradigm of the quantitative finance applied to complete markets. As already mentioned, this models starts from the basic assumption that the asset price follows a typical dynamics of the GBM, a stochastic process developed within the framework of finance and, unlike the Bachelier’s Brownian motion, with no particular relevance in physical sciences. It could be interesting, therefore, to briefly discuss this process before passing to BSM model for option pricing. A more exhaustive illustration can be found in Ref. [23], on which what follows is based. In GBM framework, the price S(t) is evolved according to the following stochastic 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 constant instantaneous drift and standard deviation rates. Actually, the process (1.1) is referred as GBM. It’s easy verify that applying the Itˆo lemma [23] to the logarithm of S(t), commonly known as log-return, Eq. (1.1) yields

. σ2 d(ln S(t)) = dX(t) = µ dt + σdW (t) . (1.2) − 2 Eq.(1.2) is the typical sde that characterizes a generalized Wiener process; it tell us that the logarithm of price follows a Brownian motion with an eﬀective drift 2 µ σ and variance σ2dt. Recalling the theory of stochastic processes, we − 2 can conclude that price log-returns are distributed according to a non-zero mean

7 1. Why non-Gaussian models for market dynamics?

Normal, i.e. X (µ σ2 )dt, σ2dt . ∼ N − 2 As a results, the price S(t) features a Log-Normal distribution 2 2 1 [ln(S/S0) (µ σ /2)t] p(S, t S0, t0) = exp − − , (1.3) 2 2 | S 2πσ (t t0) (− 2σ (t t0) ) − − characterized by meanp and variance given by

µ(t t0) E [S(t)] = S0 e − , (1.4) and 2 2 2µ(t t0 ) σ (t t0) Var [S(t)] = S e − e − 1 . (1.5) 0 − It’s worth mentioning that such a result has allowed to go beyond the major problem posed by the original Bachelier’s random walk: the possibility of negative prices, emerging from the predicted Normal distribution featured by the prices. We would also note that both the processes S(t) and X(t) are aﬀected by the same source 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 call option 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 main hypothesis concerns the absence of arbitrage. This assumption is one of the leading argument in the derivation of BSM results, since it is strictly related to the concept of efficient market, namely a market in which all participants are able to obtain all the information necessary for trading, the liquidity is high and there are no transition costs. Besides these two fundamental hypothesis, other ones, more technical, are assumed in the model, as the possibility of continuously trading or the absence of credit risk. The aim of BSM analysis was to solve the problem of finding the fair price of a 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, simply called underlying. A complete description of financial derivatives, in all their innumerable and strange forms and features, can be found in Ref. [24]. However, it’s worth stressing that the derivative price strongly depends on the stochastic dynamics chosen for the price underlying. As mentioned above, in this framework it turns out to be the GBM. In their works, BSM have demonstrate the possibility to remove at all the ∂C risk from Π(S, t) (portfolio hedging) by choosing dynamically ∆ = ∂S , in order to rebalance the fluctuations of Π(S, t) and makes its evolution deterministic. In consequence of the hedging, the portfolio must evolve, as a bank account, according to 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 price becomes a deterministic partial diﬀerential equation, the popular Black-Scholes equation, given by

∂C ∂C 1 ∂2C + rS + σ2S2 = rC . (1.8) ∂t ∂S 2 ∂S2 We would note that Eq. (1.8) contains only the parameters entering the underlying process sde (1.1), namely the constant σ and µ; indeed, the latter is substituted with r because of the hedging. The Black-Scholes equation can be solved by means of the Feynman-Kac formula specifying the proper final condition, given by the option value at the expiry date (maturity) T (S, T ) = max(S(T ) K, 0) . (1.9) C − where K is the strike price, namely the underlying asset price fixed when the option is written, whereas S(T ) is the spot price, namely the effective value of the underlying 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 in Eq. (1.3), the fair call option price reads

r(T t) (S, t) = e− − E [max(S K, 0) S = S] C T − | t r(T t) ∞ . (1.10) = e− − dS0 (S0 K) p (S0, T S, t) − LN | ZK The presence of the extra term leads to the appearance of the discount factor r(T t) e− − . The ﬁnal result is

r(T t) (S, t) = SN(d ) Ke− − N(d ) (1.11) C 1 − 2 where N(x) is the Normal cumulative function and

ln(S/K) + (r σ2/2)(T t) d1,2 = − . (1.12) σ√T t − 1.3 Stylized facts of real market dynamics

Despite the widespread success of the BSM theory, many academics and ﬁnance practitioners believe that this model is too idealized to describe real market phenomena, because of its unrealistic assumptions and the chosen asset price dynamics. Many empirical studies [12, 25, 26, 27] have argued the presence of some mismatches between the simple statistical properties of the ideal GBM prices and the more complex behaviour observed in real market. These stylized facts, whose evidence arose in the past 30 years, concern the leptokurtic nature of price return distribution, the non-constant character of volatility and the existence of correlations of price changes and volatility at diﬀerent times.

9 1. Why non-Gaussian models for market dynamics?

1.3.1 Non-Gaussian nature of log-return distribution

In the previous section, we have seen that the main theoretical prediction of the GMB model is that the logarithm of prices would feature a Gaussian density broadening in time. Actually, it is well known that empirical price returns, especially in the limit of high frequency, do not follow the Gaussian paradigm and are 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 before Samuelson’s formalization. After having collected a sufficient amount of data on several US cotton exchanges [9], he proposed that the price changes distribution for commodities, goods whose market is much less liquid than the stock one, was not Gaussian but Lévy-like. This stable distribution is known to posses variance and higher-order moments infinite and to verify a generalized Central Theorem Limit, being an attractor in the functional space of pdfs [11]. Mandelbrot’s intuitions have been confirmed not long after also for stock prices, thanks to the investigations of E. Fama [10]. The last decades’ development of information technology in the financial markets and the consequent enormous body of empirical research on this topic, have provide strong confirmations of the non-Gaussian nature of price returns, especially at infra-day and daily time horizons. Contributions by R.N. Mantegna and H.E. Stanley [25], for example, have shown unambiguously that the short time dynamics of prices breaks the Gaussian scaling. In the same work the authors have also pointed out the finiteness of the second moment of price returns probability density, a matter of debate in finance for several year. The discover of this evidence has been peculiar in order to carefully characterize the price process. Moreover, other studies [28] have also proved that the distribution’s tails tend to exhibit features quite different from the central body ones. They show α in fact a typical power-law behaviour, namely P (x) x − with α 4, a value ∼ | | ' in agreement with the requested finiteness of the second moment. However, as clearly explained by H. Stanley in his talk at APFA6 Conference (Lisbon, July 2007), in order to quantify with accuracy the tail behaviour, a very huge amount of data is strongly demanded. To cope with these empirical evidences, many theoretical models, often based on non-Gaussian stochastic dynamics, have been proposed in the literature attempting to capture the leptokurtic nature of log-returns. A first example concern the so-called Truncated Lévy Flights [29], models in which the pdf central body features a stable Lévy distribution, whereas the rare events have a typical power- law behaviour. In this way, the trouble of an infinite variance affecting the original Lévy distribution can be avoid at all, allowing a better characterization of the tails dynamics. Another model often used in the literature is the Student-t distribution [30, 31], because of its ability in reproducing with good accuracy the price returns distribution at short time horizon, especially the power law character of the tails, thanks to its strong leptokurtic nature. In the econophysics literature, the Student-t is also known as Tsallis distribution, emerging within the framework of statistical physics [32]. In Chapter 2 we will illustrate a possible application of this model to risk management, in order to achieve market risk measures in

10 1.3. Stylized facts of real market dynamics the presence of non-Gaussian ﬂuctuations. Finally, the evidence that volatility is non constant, the main topic discussed in the next subsection, but shows itself an 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 empirical distributions respect to the Gaussian, higher-order moments are often used. In particular, they allow to measure the pdf asymmetry and the “fatness” of the tails emerging once compared real data with the Gaussian paradigm. Looking at Fig. 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 in the NYSE [11]. The pdf is clearly more leptokurtic and skewed than the Gaussian one (solid line). in the left tail of the empirical distribution respect to the right one. The figure refers, in particular, to high-frequency price differences of the Xeros stock traded in the New York Stock Exchange; it’s worth mentioning that a more evident asymmetry 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 obtain higher-order moments making use of the cumulants of a distribution, defined as

n n ∂ κn = ( i) ln [ϕ(ω, t)] , (1.13) − ∂wn ω=0   since the relationships between moments and cumulants are known. Note that the function ϕ(ω, t) entering Eq. (1.13) represents the characteristic function of a general pdf p(x, t). The skewness, in particular, can be expressed in term of the second and the third cumulants as

. κ3 γ1 = 3/2 . (1.14) κ2

11 1. Why non-Gaussian models for market dynamics?

Figure 1.2: One-minute returns of Standard&Poor’s 500 index (points) compared with the Gaussian, an inverted parabola in lin-log scale (dotted line), and the L´evy 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 the right one and, conversely, positive values characterize a distribution with the right tail fatter than the left one. Consequently, the skewness reduces to zero in the Gaussian case. It has been observed in particular that, starting from a null return time horizon (at which the pdf is simply a Dirac delta), the skewness grows up to a maximum reached at a time scale of a few tens of days and then decreases very slowly to zero, coherently with the Central Limit Theorem that requests a Gaussian density when t . Looking at Fig. 1.2, w→e could∞ appreciate the presence of tails heavier than the Gaussian ones; this fact indicates that the empirical distribution has an excess of kurtosis, a typical sign of a non-Gaussian nature. To quantify the fatness of the tails respect to the Normal pdf, the fourth moment, called kurtosis γ2, has been introduced. This moment can be computed using the second and the fourth cumulant as . κ4 γ2 = 2 . (1.15) κ2

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 empirical kurtosis decreases monotonically, starting from a maximum and converging to zero a bit faster than skewness. The topics discussed along this section prove that high-frequency returns violate 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 the right one, and a short-time (i.e. infraday-daily data) rare event probability much

12 1.3. Stylized facts of real market dynamics 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, which measures the strength of the price fluctuation, is a constant parameter. Neverthe- less, observing real data leads to quite different conclusions: also the statistical features of volatility are characterized by stylized facts. Fig. 1.3 shows the daily returns of Dow Jones (DJ) Industrial Average index on a period of approximately 40 years (the true temporal range is not carried in the reference [34]). Although the period is relatively quiet and without crashes, the fluctuation 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 the tendency of volatility to come in bursts [34].

fied in correspondence of financial crashes. Fig. 1.4 displays the same DJ returns over the entire twentieth century: besides the periodical ‘pulse’ in volatility, wild oscillations following the 1929 and, on a less catastrophic scale, the 1987 crashes are clearly visible. In general, one can notice that the fluctuation amplitude, and thus the volatility, tends to increase suddenly, stay high for a limited time interval, ranging from few hours to several months, and then go back to a situation of more 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 study its statistics. This implies facing the problem of its estimation, since volatility itself cannot be directly observed or, more commonly, is a hidden process. Measures of volatility have to be derived from ‘proxies’ calculated from the price changes, which conversely can be measured on real markets. Recalling that in the GBM theory log-returns distribution is characterized by V ar[X] = σ2dt, we can imme- diately conclude that Var [X] σ2 , (1.16) ' t where we have used the so-called zero-mean returns X, which we will better describe in Chapter 3 within the framework of stochastic volatility models. Basically

13 1. Why non-Gaussian models for market dynamics?

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

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

N 2 2 1 Sk+1 σhis = ln m , (1.17) (N 1)∆t Sk − − Xk=1 where N is the length of the prices time series Sk (k = 1, .., N) and m is its mean, estimated as empirical moment (see Eq. (2.8) in Chapter 2). However, such proxy can be considered eﬀective only over limited time intervals, otherwise it becomes very inaccurate. When the information on the time at which volatility is estimated must be retained, like in the case of volatility autocorrelation, the so-called instantaneous volatility can be deﬁned by introducing

[X(t + ∆t) X(t)]2 lim − . (1.18) ∆t 0 → s ∆t This equation must be taken as an inﬁnitesimal diﬀerence, leading to the following expression for the instantaneous volatility

. dX(t)2 σist(t) = . (1.19) r dt On the other hand, when high-frequency data are available, we can introduce the high-frequency volatility, deﬁned by Bouchaud and Potters [12] as

1 N σ = ∆S , (1.20) hf N∆t | k| Xk=1 where ∆t is of the order of minutes. It’s worth recalling that for high-frequency data price changes, returns and log-returns can be taken as equivalent variables. In particular, for daily data, namely ∆t = 1 day, volatility can be estimated directly

14 1.3. Stylized facts of real market dynamics from historical daily series as absolute return; this quantity is commonly known as empirical daily volatility. Using the high-frequency proxy of Eq. (1.20), the same authors have found that the empirical volatility distribution function is equally well approximated by a 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 large tail of the measured distribution. Today, there is an almost general consent that the empirical volatility could be modeled through a Log-Normal density in its body, whereas large values could be characterized by a power-law tail. This is indeed a stylized fact about volatility.

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

Another corroboration of the non-constant nature of real volatility comes from the so-called volatility smile. In a BSM scenario, the fair price of a call option given in Eq. (1.11) depends only on the current spot price S, or equivalently on . the 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 K on 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 the official interest rate r is fixed, the only other parameter involved is the volatility, that seems to change with the underlying price. Indeed, another strategy to estimate the unobservable volatility is to invert the BS formulae for a certain option (put or call) taking for granted the real market price at which this is O 15 1. Why non-Gaussian models for market dynamics?

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

exchanged. Namely, one can define an implied volatility σimp imposing the equality . (S, t; r; K, T ) = (S, t; r, σ ; K, T ) , (1.21) Omarket OBS imp and work out its dependence on the moneyness. The result is a smile-shaped curve centered on m = 1, i.e. S = K. This fact indicates that the risk perceived by the option writer, proportional to the price asked to the holder, increases if the spot price deviates from K. The slope of the two branches depends also on the maturity, being stronger for smaller T . Until the 1987 crash, the first relevant one after the institution of the option exchange, the curve was a proper smile, symmetrical with respect to m=1. After the crash the market has settled so as to generate rather smirky curves, like the one displayed in Fig. 1.7, namely with the half corresponding to the unfavorable situation for the holder (m <1 branch for call options, the opposite for puts) raised with respect to the other. This reflects an increased worry against losses, which generate an increase in the effective volatility.

1.3.3 Price-volatility correlations Another well established stylized facts concern the existence of non-trivial correlations between price changes at different times, the so-called leverage effect, as well 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 future volatilities. Its name is due to the jargonistic name for the ratio of a company’s debt to its current capital: when the latter falls, the leverage rises, and so does the investors’ mistrust in the company.

16 1.3. Stylized facts of real market dynamics

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

To quantify this eﬀect, the leverage function (τ) has been introduced [37] and deﬁned through L . E [σ(t + τ)2dX(t)] (τ) = . (1.22) L E [σ(t)2]2 Estimating the (squared) volatility with the instantaneous proxy of Eq. (1.19), the previous equation becomes

. E [dX(t + τ)2dX(t)] (τ) = . (1.23) L E [dX(t)2]2

In the same work [37] the authors, analysing the daily relative changes of some 437 US stocks as long as 7 major international indexes, have found that (τ) is well described by the following empirical law L

bτ Ae− if τ > 0 (τ) = − , (1.24) L ( 0 if τ < 0 with A > 0. Fig. 1.8 proves the very good agreement between empirical data and the derived Eq. (1.24). Hence, there is a negative correlation with an exponential time decay between past price changes and future volatilities, but no correlation is found between past volatilities and future price changes. In other words, a sort of causality exists in the leverage effect dynamics. Another interesting evidence as regards the leverage effect concerns the existence of some “connections” between the leverage itself, the volatility smile and the skewness. The asymmetry observed in the volatility smile effect could be, in fact, controlled by a negative skewness in the price return distribution, which could be generated, in turn, by a negative correlation between price changes and

17 1. Why non-Gaussian models for market dynamics?

Figure 1.8: Comparison between the analytical expression of leverage effect (solid line) given in Eq. (1.24) and the empirical correlations (points), averaged over 7 major stock indexes. The solid line clearly shows an exponential trend [37]. volatility, namely the leverage effect. Thus, these three stylized facts could actually be generated by the same aspect of the underlying dynamics. As for the presence of the smile, it can be explained in term of a non-ability of BSM theory in capturing carefully the rare events of the pdf. In this scenario, the smile could be also related to an excess of kurtosis of the return distribution. This fact would confirm again the results discussed in Section 1.3.1, since the observed kurtosis is stronger at short time horizons as well as the smile is more marked for options with close maturity date. Another established stylized fact regards the time correlation of the estimated volatility, an effect investigated mainly trough the ’90s. Analogously to Eq. (1.23), a convenient empirical quantity introduced to quantify this autocorrelation is . σ(t)2σ(t + τ)2 σ(t)2 2 (τ) = h i − h i C Var [σ(t)2] (1.25) dX(t)2dX(t + τ)2 dX(t)2 2 = h i − h i , dX(t)4 dX(t)2 2 h i − h i where the second equality is obtained substituting the volatility with its instantaneous proxy (1.19). Empirical investigations [38, 39] have proved that the autocorrelation function evaluated from real data turns out to be positive and characterized by a double time scale: a short one of the order of a few tens days and a much longer one amounting to hundreds of (financial) days. Fig. 1.9 demonstrates the presence of both the scales within the empirical volatility autocorrelation function estimated from DJ data. Therefore, the dynamics of volatility cannot be accounted for using a single time-scale: its fluctuation is a multi-time-scales phenomenon. This characteristic double time scale, one of the most difficult feature to be reproduced by current financial stochastic models, seems to reflect the presence of

18 1.3. Stylized facts of real market dynamics two diﬀerent kinds of market’s reaction to the variability of an asset. In the few days following a volatility increase, the volatility itself tends to be higher, because investors 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 cautiousness lasts for a long time.

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

19 1. Why non-Gaussian models for market dynamics?

20 Chapter 2 A non-Gaussian approach to risk measures

In the previous chapter we saw that the diffusion process unanimously accepted as the most universal model for speculative markets, i.e. geometric Brownian motion, fails to predict several established facts observed in the price variations, like the leptokurtic nature of log return probability density. The increasing effort to extend the models beyond the Gaussian paradigm is closely related to the possibility of capturing the true statistical properties of financial markets, i.e. the stylized facts previously described. Moreover, it’s well known that non-Gaussian approach has strong influences also in several financial applications, like option pricing [40], allowing evaluations more reliable and closer to the real nature of financial data. Nevertheless market analysts and operators often seem not to pay particular attention to these topics. Within this scenario, the aim of the present chapter is to present a careful analysis of financial market risk measures in term of a non-Gaussian model for price fluctuations. It’s worth underlying that the true financial risk resides in the rare events of the probability distribution. Thus modeling with a great accuracy the tails 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. The approach we developed leads to estimates for the risk associated with a single asset in good agreement with a full historical evaluation and thus utilizable also in the financial practice. This chapter is based on the work discussed in [41, 42] and is organized as follows. In Section 2.1 a brief introduction to financial risk and the most widely used measure 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 Expected Shortfall are derived as generalizations of the analytical formulae known in the literature under the normality assumption. It is also shown how the standard Gaussian formulae of the parametric approach are recovered, in the appropriate limit, as a special case. In Section 2.3 an empirical analysis of daily returns series from 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 ingredients needed 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 Expected Shortfall are shown in Section 2.4 and compared with the results of the parametric normal method, of its improved version known as RiskMetrics methodology and of the historical simulation. Particular attention is paid to quantify the size of the errors 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 development of reliable methods of measuring and controlling financial risk. One of the main question about it is where risk comes from. The sources of risk are various: it could be human-generated, such as inflation, business cycles, government policies or wars indeed. It can also occur from unforeseen natural events like earthquakes, or arises from the movements in financial markets, the long-term economics growth or technological innovations. The sources of risk are not likely to be eliminated; quoting some words of Walter Wriston, former chairman of Citicorp, one could conclude 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 are identified, measured and controlled. According to the new capital adequacy framework, commonly known as Basel II accord [43], any financial institution has to meet stringent capital requirements in order to cover the various sources of risk that they incur as a result of their normal operation. Basically, three different categories of risk are of interest: credit risk, operational risk and market risk. In particular, market risk concerns the hazard of losing money due to the fluctuations of the prices of those instruments entering a financial portfolio and is, therefore, particularly important for financial risk management. Although in modern parlance the term risk is strictly related to the danger of loss, financial theory defines it as the dispersion of unexpected outcomes due to the financial market movements. In the light of this assumption, traditionally the concept of risk has been related to the volatility, which is known to quantify the fluctuations 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 should be 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 untimely related to the idea that the distribution of price change is Gaussian. In the case of Lévy fluctuation, this definition is straight meaningless since this distribution is characterized by a infinite variance. Moreover, Gaussian models fail to reproduce the 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 determining the probability of extreme losses is to model the distribution functions making use of non-Gaussian models. From this discussion, clearly emerges the necessity of another definition of risk which gets over the concept of volatility. Today, in the financial industry, the most widely used measure to manage market risk is Value-at-Risk [12, 44]. This method was developed in response to the financial disaster of early 1990s, arisen from the poor supervision and management of 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 to VaR, an easy-to-understand method for quantify market risk. In short, VaR refers to the maximum potential loss over a given period at a certain confidence level and can be used to measure the risk of individual assets and portfolios of assets as well. VaR has become a standard component in the methodology of academics and 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 main advantages of this method. To better clarify this point, we can make use of the following example. A bank may say, for instance, that the daily VaR of its trading portfolio is 20 million euro at 99% confidence level. This means that under normal market conditions, the most the portfolio can lose in a day is 20 million euro and it can occur just 1 change in a 100. This statement illustrates how VaR summarizes the various bank’s exposures to the risk in a single number, easily communicable also to a nontechnical audience. In the left panel of Fig. 2.1 the practice way to compute VaR, denoted as Λ?, is shown. Still, as discussed in the literature [12, 45], VaR suffers from some inconsistencies: first, it can violate the sub-additivity rule for portfolio risk, which is a required property for any consistent measure of risk, and, secondly, it doesn’t quantify the typical loss incurred when the risk threshold is exceeded. In other words, the VaR method doesn’t take into account the fact that the losses can accumulate in time, leading to a overall loss which might exceed the VaR value. To overcome the drawbacks of VaR, the Expected Shortfall (or Conditional VaR) is introduced, and sometimes used in financial risk management, as a more coherent measure of risk. Unlike VaR, Expected Shortfall (ES) allows to quantify the average size of the loss when the cutoff value is hit. In the right panel of Fig. 2.1 the methodology to compute ES, denoted as E?, is shown in comparison with the corresponding Λ? value, both calculated at the 5% significance level: the differences in computing, and thus using, VaR and ES as risk measures clearly emerge. Moreover, it’s quite evident the necessity to model the distribution’s left tail, 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 for calculating VaR and Expected Shortfall. The first method consists in assuming some probability distribution function for price changes and calculating the risk measures as closed-form solutions. This approach is called parametric or analytical and is easy to implement since analytical expressions can often be obtained.

23 2. A non-Gaussian approach to risk measures

? P 5% 40 40

35 35

? 30 30 E = 0.021

25 25

PSfrag replacements ? 20 Λ = 0.0233 20 Pdf PSfrag replacements Pdf 15 15 ? 5% ? Λ = 0.016 10 Λ = 0.0164 10

5 1% 5

0 0 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01

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

The parametric approach usually relies on the normality assumption for the returns distribution, although some analytical results using non-Gaussian functional forms are available in the literature [46, 47]. However, in the light of the considerations previous discussed, models beyond the Gaussian paradigm are strongly required to obtain results closest to the true statical features of prices. In order to capture the leptokurtic (fat-tailed) nature of price returns, the historical simulation approach is often used as an alternative to the parametric method. It employs recent historical data and risk measures are derived from the percentiles of the distribution of real data. This method is potentially the most accurate because it accounts for the real statistics of price changes but it is computationally quite demanding (especially when applied to large portfolios) and absolutely depending on the past history of empirical data. A third approach consists in Monte Carlo simulations of the stochastic dynamics of a given model for stock price returns and in calculating risk measures according to Monte Carlo statistics. This method, however, requires very intensive simulations to achieve risk measures predictions with acceptable numerical errors. As a result of the present situation, reliable and possibly fast methods to calculate financial risk are strongly demanded. Inspired by this motivation, we evaluate financial market risk measures with non-Gaussian fluctuations in order to under- line the potentials of this approach, as well as limitations, in comparison with standard procedures used in financial analysis. To capture the excess of kurtosis of empirical data with respect to the normal distribution, the statistics of price changes is modeled in terms of a Student-t distribution, which is known to approximate with good accuracy the distribution derived from market data at a given time horizon [11, 12] and is widely used in the financial literature and in a number 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 study of 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 results for a simple practical implementation. At a variance of the recent calculation in Ref. [47], where analytical results for risk measures using Student-t distributions are presented, we critically investigate the implications of our non-Gaussian analytical solutions on the basis of an empirical analysis of ﬁnancial data and we perform 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 value due 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 a fixed time horizon ∆t. In terms of price changes ∆S, or, equivalently, of returns . R = ∆S/S, VaR can be computed as follows

Λ? Λ?/S . − − ? = d∆S P˜ (∆S) = S dR P (R), (2.1) P ∆t ∆t Z−∞ Z−∞ where P˜∆t(∆S) and P∆t(R) are the probability density functions (pdfs) for price changes and for returns over a time horizon ∆t, respectively. For financial analysts, VaR has become the standard measure used to quantify market risk because it has the great advantage to aggregate several risk component into a single number. As already remarked, in spite of its conceptual simplicity VaR shows two main drawbacks: it is not necessary sub-additive and it does not quantify the size of the potential loss when the threshold Λ? is exceeded. A quantity that does not suffer of these disadvantages is the so-called Expected Shortfall (ES) or Conditional VaR (CVaR), E?. It is defined as

Λ? Λ?/S . 1 − S − E? = d∆S ( ∆S) P˜ (∆S) = dR ( R) P (R), (2.2) ? − ∆t ? − ∆t P Z−∞ P Z−∞ with ? and Λ? as in Eq. (2.1). TheP 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 (m, σ2). In that case, VaR and ES analytical expressions reduce to the follo∼wingN closed- form formulae ? 1 ? Λ = mS + σS √2 erfc− (2 ) (2.3) − 0 0 P and ? σS0 1 1 ? 2 E = mS0 + exp [erfc− (2 )] , (2.4) − ? √2π {− P } P 1 where S0 is the spot price and erfc− is the inverse of the complementary error function [51]. However, we have widely discussed in the previous chapter that the normality hypothesis is often inadequate to reproduce daily returns and, more

25 2. A non-Gaussian approach to risk measures

PSfrag replacements 10 PSfrag replacements 10 Normal 9 1% 5% 9 1% 5% ν = 2.75 8 ν = 3.50 8 ν = 4.50 7 ν = 100 7 Normal 6 6 ? ν = 2.75 5 ? Λ E 5 4 ν = 3.50 4 3 ν = 4.50 2 3

ν = 100 1 2

0 1 0.001 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.001 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 ? ? P P Figure 2.2: Convergence of the VaR (left) and ES (right) Student-t formulae toward Gaussian results when approaching the limit ν + . → ∞

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

ν ν 1 a m,a(R) = ν+1 , (2.5) S B(ν/2, 1/2) [a2 + (R m)2] 2 − where ν (1, + ) is the tail index and B(ν/2, 1/2) is the beta function. It is easy to verify∈ that,∞ for ν > 2, the variance is given by σ2 = a2/(ν 2), while, for − ν > 4, the excess kurtosis reduces to k = 6/(ν 4). Under this assumption, we obtain closed-form generalized expression for VaR− and ES given by

1 λ? ? √ Λ = mS0 + σS0 ν 2 −? (2.6) − − r λ and − ? σS0 √ν 2 ? ν 1 E = mS + − [λ ] 2 , (2.7) − 0 ?B(ν/2, 1/2) ν 1 P − ? . 1 1 where λ = I2− ? (ν/2, 1/2) and I2− ? (ν/2, 1/2) is the inverse of the incomplete beta function evaluatedP in 2 ?, accordingP to the deﬁnition of Ref. [51]. These parametric formulae, that generalizeP standard expressions known in the literature under the normality assumption, are one of the main results of this study. As shown in Fig. 2.2, we have checked numerically the convergence of formulae (2.6) and (2.7) to the Gaussian results (2.3) and (2.4), in the appropriate limit ν + . We chose ν = 2.75, 3.5, 4.5, 100 and m = 0, σS0 = 1, but we checked →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 significance levels ?. P ? 1% 2% 3% 4% 5% P ν (VaR) 2.44 3.21 5.28 32.38 100 cross νcross(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 correctly recover the Gaussian formulae as a special case. It is also worth noting that each line, corresponding to a fixed ν, crosses over the Gaussian one for a certain ?. Analogously, for a fixed ?, there exists a ν value whose line crosses the Gaus-P P cross sian result at that significance level. In the light of this observation, we report in ? Tab. 2.1 the values of νcross corresponding to a given for both VaR and ES. As can be observed, the growth of ν with ? is veryPrapid for VaR, while for ES cross P and 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, since the crossover values for the latter are much more stable than those associated to the first one. This result can be interpreted as a consequence of ES as a more coherent risk measure than VaR. Because the key parameter to capture the fat-tailed nature of returns in terms of 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 differentiation of Eqs. (2.6) and (2.7), with respect to ν variations. Figure 2.3 shows the 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 for two significance levels typically used in financial analysis, i.e. ? = 1%, 5%. The tail index is allowed to vary between 2.5 and 5, in order to elucidateP the sensitivity of the risk measures for typical values of ν derived from our time series analysis discussed in the following (see Tab. 2.3). As expected, the strongest sensitivity is observed for the smallest values of ν, in the range between 2.5 and 3.5, corresponding to a particularly pronounced leptokurtic character of the Student-t distribution. Furthermore, the relative sensitivity 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 ? = 5% (dotted lines) than for ? = 1% (dashed lines). For relatively large valuesP of ν (above 3.5), the relativePsensitivity is below the 10% level, with an almost flat behaviour 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 corresponding error ∆ν 0.2 (see Tab. 2.3), in the worst scenario, i.e. for∼ Λ? at 5% significance ∼ level, 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 2. A non-Gaussian approach to risk measures

1 1 ? P 1% ? 0.8 0.8 P 5% 0 0.6 0.6

0.4 PSfrag replacements 0.4 ? ?

ν 0.2 0.2 ν Λ Λ ∂ ∂ ∂ ∂

0 ? 0 1 Λ PSfrag replacements -0.2 -0.2 -0.4 -0.4

2.5 3 3.5 4 4.5 5 2.5 3 3.5 4 4.5 5

1 1

0.8 0.8

0.6 0.6

0.4 0.4 ? ? ν ν

0.2 E 0.2 E ∂ ∂ ∂ ∂ ?

0 1 0 E PSfrag replacements -0.2 PSfrag replacements -0.2 -0.4 -0.4

2.5 3 3.5 4 4.5 5 2.5 3 3.5 4 4.5 5 ν ν

Figure 2.3: Sensitivity (left panels) and log-sensitivity (right panels) of VaR and ES to a variation of the tail index ν. The solid line represents the zero level, while dashed 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 presence of particularly fat-tailed returns, an accurate determination of the tail index is 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 of N = 1000 daily returns, from the Italian stock market. Two series are collections of 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 indexes Mib30 and Mibtel (from March 27th 2002 to March 13th 2006). The data have been freely downloaded from Yahoo Finance Web site [52]. Figure 2.4 shows a comparison between the historical complementary cumulative density function P> of the negative daily returns and two theoretical fits obtained using Gaussian and Student-t distributions. The parameters values of the fitted curves, as obtained according to the likelihood procedure described below, are displayed in Tab. 2.2. In principle, we could perform the fit according to different methods, but we have to balance between accuracy and computational time. Therefore, we estimate mean

28 2.3. Empirical analysis of ﬁnancial data and variance as empirical moments, i.e.

N 1 . 1 − m = Rt i (2.8) N − i=0 X and N 1 − 2 . 1 2 σ = (Rt i m) , (2.9) N 1 − − − i=0 . X 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 unit . . variance) 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 argumen− t that minimizes the negative log-likelihood, according to the formula

N 1 − ν ν = argmin ν>2 log 0,√ν 2(rt i) , (2.10) − S − − " i=0 # X ν where the constraint ν > 2 prevents the variance to be divergent and 0,√ν 2 is as S − in Eq. (2.5), with m = 0 and a = √ν 2. This apparently simple optimization problem can not be solved analytically−. In fact, the normalization factor in the Eq. (2.5) does depend on the tail index ν in a non-trivial way. Actually, the beta function B(ν/2, 1/2) only admits an integral representation and therefore we implemented a numerical algorithm to search for the minimum. As shown in Section 2.2, the excess kurtosis k depends only on ν and this provides 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 be seen that all the exponents obtained in the likelihood-based approach are smaller than 3.5. For this reason, the implementation of the excess kurtosis method is inadequate for the time series under study here. In order to test the robustness of our results, we also performed a more general three-dimensional minimization procedure over the free parameters (m, σ, ν). The multidimensional optimization problem was solved by using the MINUIT program from CERN library [53]. The obtained numerical results are in full agreement with the previous ones, but the process is more computationally burden and more cumbersome, since it requires a lot care in avoiding troubles related to the appearing of local minima in the minimization strategy. In Fig. 2.4 we show the cumulative distribution P> obtained using the empirical parameters of Tab. 2.2. As expected, we measure daily volatilities of the order of 1% and quite negligible means ( 0.01%). The tail parameters fall in ∼ the range (2.9, 3.5), thus confirming the strong leptokurtic nature of the returns distributions, both for single assets and market indexes. The quality of our fit clearly emerges from Fig. 2.4, where one can see a very good agreement between Student-t and historical complementary cdfs, while the Gaussian distribution fails to reproduce the data.

29 2. A non-Gaussian approach to risk measures

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

1 1 Data Normal Student

0.1 0.1

> PSfrag replacements P

0.01 0.01 PSfrag replacements

0.001 0.001 0.001 0.01 0.1 0.001 0.01 0.1

1 1

0.1 0.1 > P

0.01 0.01 PSfrag replacements PSfrag replacements

0.001 0.001 0.001 0.01 0.1 0.001 0.01 0.1 R R − − Figure 2.4: From top left clockwise: Autostrade SpA, Telecom Italia (from May 15th 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 market risk through VaR and ES according to diﬀerent methodologies. The standard approach 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), that depend 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 parameter is the volatility σ. Several techniques are discussed in the literature to model and forecast 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, Telecom Italia, Mibtel and Mib30 time series. m and σ are estimated from empirical moments, while ν is obtained through a negative log-likelihood minimization as in Eq. (2.10).

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

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

N 1 − 2 . 1 λ i 2 σt+1 t = − λ (Rt i m) , (2.11) | 1 λN+1 − − i=0 − X where λ (0, 1] is a decay factor. The choice of λ depends on the time horizon ∈ and, 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 considers Eq. (2.11) as the defining equation for an autoregressive process followed by σt+1 t | (coupled with Rt = σtt with t i.i.d.(0, 1)), Refs. [60, 61] provide reasons for the claimed good success of the RiskMetrics∼ methodology. In order to relax standard assumption about the return pdf without loosing the advantages coming from a closed-form expression, we have presented in Section 2.2 generalized formulae for VaR and ES based on a Student-t modeling of price returns. In this framework, the tail index ν emerges as a third relevant parameter, which is possible to constrain using a maximum likelihood technique, as previously described. As a benchmark of all our results, we also quote VaR and ES estimates following a historical approach, which is a procedure widely used in the practice. According to this approach, after ordering the N data in increasing order, we consider the ? th [N ] return R([N ?]) as an estimate for VaR and the empirical mean over first P [NP?] returns as an estimate for ES 1. PAt a variance with respect to previous investigations [49, 60], we also provide 68% confidence level (CL) intervals associated to the parameters in order to give

1 th The symbol [ ] stands for integer part, while R(j) is the standard notation for the j term of the order statistic of R. Since N 1 we neglect the fact that the pth entry is a biased estimator E of the p/N-quantile, i.e. [R(p)] = p/(N + 1).

31 2. A non-Gaussian approach to risk measures

F F 0.005 0.01

0 0 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 ∗ ∗ ν σt+1|t

Figure 2.5: Bootstrap histograms for tail index ν (left) and for the RiskMetrics 3 volatility proxy σt+1 t (right) for Autostrade SpA (M = 10 bootstrap copies). | Table 2.3: Parameters values and bootstrap estimates for the 68% CL intervals for the time series as in Tab. 2.2.

m σ σt+1 t ν R(10) | +0.04 +0.08 +0.31 +0.20 +0.31 Autostrade 0.12 0.05% 1.38 0.10% 1.83 0.33% 2.91 0.21 3.51 0.15% −+0.06 −+0.11 −+0.42 −+0.21 − −+0.87 Telecom 0.02 0.07% 2.23 0.11% 1.54 0.47% 3.14 0.22 6.14 1.35% − +0−.02 −+0.03 −+0.19 −+0.18 − −+0.25 Mibtel 0.02 0.04% 1.03 0.04% 0.69 0.20% 3.35 0.19 2.96 0.24% −+0.03 −+0.03 −+0.22 −+0.15 − +0− .30 Mib30 0.02 0.04% 1.16 0.05% 0.72 0.22% 3.22 0.16 3.33 0.25% − − − − − −

a sound statistical meaning to our comparative analysis. In this way we can estimate VaR and ES dispersion. To this extent, we implement a bootstrap technique [62]. Given the N measured returns, we generate M = 1000 synthetic copies of R, R∗ , with j = 1, . . . , M, by random sampling with replacement according to { j } the probability p = (1/N, . . . , 1/N). For each Rj∗ we estimate the quantities of interest and we obtain bootstrap central values and conﬁdence levels. For example, we use for the mean the relations

M N 1 . 1 1 − mb∗ = mj∗ with mj∗ = (Rj∗)t i (2.12) M N − j=1 i=0 X X

and we deﬁne the 1 2α CL interval as [mα∗ , m1∗ α], with ma∗ such that P (m∗ − − ≤ m∗) = a and a = α, 1 α. For 68% CL, α = 16%. In Fig. 2.6 and Tabs. 2.3, a − 2.4, 2.5 we quote results according to m (mb∗ mα∗ ) + (m1∗ α mb∗). In this way, − − − − we use the bootstrap approach in order to estimate the dispersion of the mean around the measured value m. In Fig. 2.5 we show the bootstrap evaluation of the tail index ν for Autostrade SpA: we would note that the bootstrap central

value νb∗ is clearly very close to the estimated value ν. In order to quantify the dispersion around νb∗, we also include the 68% CL intervals, namely ν16%∗ and ν84%∗ . Similar considerations are valid also for the volatility proxy, estimated with the RiskMetrics 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

6 6 ? P 1% ? P 5% 5 PSfrag replacements 5

4 4 PSfrag replacements 3 3 ? Λ

2 2

1 1

0 0 Student−t Normal Historical RiskMetrics Student−t Normal Historical RiskMetrics 7 7

6 6

PSfrag replacements 5 PSfrag replacements 5 4 4 ? 3 3 E

2 2

1 1

0 0 Student−t Normal Historical RiskMetrics Student−t Normal Historical RiskMetrics Autostrade Mib30 Figure 2.6: VaR Λ? (upper panel) and ES E? (lower panel) central values with 68% CL intervals for Autostrade SpA (left) and for Mib30 (right), according to the four different methodologies discussed in the text. The significance level ? is fixed to 1% (circles, solid lines) and 5% (triangles, dashed lines). P

returns series under study. These numerical results come from a straightforward application of the re-sampling technique. It is worth mentioning that it is possible, and sometimes necessary, to use improved versions of the bootstrap. As a rule of thumb, we consider the bootstrap approach accurate when, given a generic parameter, the difference between its empirical value and the bootstrap central value estimate is close to zero and 68% CL interval is symmetric to a good approximation. In our numerical simulation, we measured a systematic non zero bias for σt+1 t and from Tab. 2.3 it is quite evident the asymmetry of R([N ?]) intervals for | P both Autostrade and Telecom data. We can, therefore, consider the corresponding CL intervals as a first approximation of the right ones, since bias and skewness corrections would require sophisticated and ad-hoc techniques [62], which are beyond the scope of the present work. In Fig. 2.6 we show VaR and ES central values and 68% CL bars for Autostrade SpA and Mib30, corresponding to 1% and 5% significance level and according to the four methodologies previously described. In Tabs. 2.4 and 2.5 we detail all the numerical results, including also Telecom Italia and Mibtel data. As already noted in Ref. [60], at 5% significance level Student-t and Normal approaches are substantially equivalent, but here such a statement sounds more statistically robust, thanks to the bootstrap 68% confidence levels and to the comparison with the historical simulation. At this significance level, we

33 2. A non-Gaussian approach to risk measures

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. For each time series, the results of Student-t and Normal ﬁt, historical simulation and RiskMetrics methodology are shown.

Student-t Normal Historical RiskMetrics +0.175 +0.197 +0.149 +0.733 Autostrade VaR 1% 3.472 0.185 3.091 0.204 3.516 0.306 4.138 0.764 +0− .071 +0− .139 +0− .175 +0− .520 VaR 5% 1.717 0.071 2.150 0.145 1.810 0.156 2.890 0.540 −+0.279 +0− .275 +1− .348 +0− .990 Telecom VaR 1% 5.900 0.230 5.200 0.277 6.137 0.866 3.595 1.085 +0− .135 +0− .214 +0− .110 +0− .694 VaR 5% 3.121 0.137 3.682 0.202 3.398 0.127 2.548 0.777 −+0.106 +0− .097 +0− .255 +0− .524 Mib30 VaR 1% 3.047 0.105 2.675 0.096 3.331 0.304 1.662 0.516 +0− .066 +0− .073 +0− .090 +0− .375 VaR 5% 1.612 0.067 1.885 0.072 2.010 0.157 1.169 0.358 +0− .097 +0− .088 +0− .240 +0− .453 Mibtel VaR 1% 2.718 0.092 2.378 0.084 2.967 0.255 1.581 0.449 +0− .062 +0− .065 +0− .150 +0− .324 VaR 5% 1.454 0.062 1.674 0.065 1.811 0.173 1.110 0.316 − − − − register for VaR a different behaviour between single assets and indexes. While assets 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 between the Normal and historical methodology. In order to enforce this empirical evidence, it would be necessary to analyse additional time series to see to what extent this difference between assets and indexes holds. From Fig. 2.6, Tab. 2.4 and Tab. 2.5 it can also be seen that Λ? and E? central values calculated according to RiskMetrics methodology are quite fluctuating and characterized by the largest CL bars. The decreasing of ? traduces in a major differentiation of the different approaches. In general, wPe obtain the best agreement between the Student-t approach and the historical simulation, both for Λ? and E?, whereas, as before, the RiskMetrics methodology overestimates or underestimates the results of the historical evaluation and is affected by rather large uncertainties. Moreover, the risk measures obtained through our model show non negligible differences with respect to the widely used Normal and RiskMetrics methodologies, indicating that the approach may have helpful implications for practical applications in the field of financial risk management. It is also worth noting that, from the results shown in Fig. 2.2 and Tab. 2.1, we expect that for a fixed significance level there exists a crossover value, νcross, below which the generalized Student-t VaR and ES formulae underestimate the Gaus- sian predictions. This effect was already mentioned in Ref. [49], but the analytical formulae here derived allow us to better characterize it. Under the hypothesis of a Student-t distribution, the crossover value does not depend on the first and second moments and therefore the knowledge, for a given time series, of the tail exponent only is sufficient to conclude, a priori, whether the fat-tailed results for VaR and ES 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 +0.431 +0.229 +0.607 +0.837 Autostrade ES 1% 5.503 0.421 3.559 0.231 5.076 0.634 4.759 0.876 +0− .153 +0− .175 +0− .248 +0− .653 ES 5% 2.946 0.159 2.727 0.182 3.006 0.235 3.655 0.677 +0− .579 +0− .311 +1− .456 +1− .133 Telecom ES 1% 8.912 0.583 5.954 0.310 9.685 1.475 4.116 1.250 +0− .242 +0− .248 +0− .478 +0− .879 ES 5% 5.035 0.246 4.613 0.246 5.320 0.466 3.190 0.969 −+0.199 +0− .111 +0− .223 +0− .599 Mib30 ES 1% 4.572 0.191 3.068 0.109 3.918 0.234 1.908 0.590 +0− .093 +0− .088 +0− .145 +0− .467 ES 5% 2.596 0.091 2.369 0.086 2.804 0.155 1.471 0.458 +0− .179 +0− .099 +0− .215 +0− .524 Mibtel ES 1% 4.021 0.171 2.728 0.094 3.501 0.224 1.815 0.516 +0− .084 +0− .078 +0− .128 +0− .399 ES 5% 2.314 0.081 2.106 0.077 2.524 0.136 1.399 0.400 − − − −

35 2. A non-Gaussian approach to risk measures

36 Chapter 3 Stochastic 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 volatility, as shown in Section 1.2, is no longer sufficient to capture modern market phenomena. Its natural extension could be therefore modifying the specifications of 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 price volatility is not constant, but shows itself some dynamics (see Chapter 1). Today there is an enormous body of research on volatility models, both in academies and in 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 indeed a key point within this topic. Engle and Patton pointed out in Ref. [63] that a good volatility model should be able to forecast volatility and this is strongly required in almost all financial applications. A portfolio maker, for instance, would sell its portfolio or a stock before it becomes too volatile, and this is true also for a private investor. To forecast volatility, and absolute returns as well, is not the only ingredient required to make these models reliable and thus attractive. A good volatility model must also be able to capture the stylized facts about volatility of financial asset price shown in Section 1.3.2, concerning the volatility clustering, the leverage effect, the multiple time-scale of the volatility autocorrelation and, within the field of derivatives pricing, the smile-shape of the implied volatility. These facts, typical of real market, cannot be captured in the Black-Scholes framework since it 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 the volatility is not a constant, but rather some sort of deterministic function of the underlying price. This hypothesis allows to stay within the boundaries of the Black-Scholes theory for option pricing, even if in most cases it is impossible to derive 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 worth mentioning that in 2003 Robert Engle was awarded the Nobel Price in Economics just for the ARCH model suggestion. A strong point of these models is their ability in reproducing quite well the implied volatility, but they often present the disadvantage of having a large number of parameter with no particular physical or economic significance, needing to be tuned ad hoc and, furthermore, substantially changing with time. An overview of this class can be found in Ref. [64]. Another possible choice of volatility models are the correlated stochastic volatility (SV) models, introduced in the literature in the late 80’s. The models belonging to this class assume the original log-Brownian motion model but, as their name indicates, with random volatility. They constitute thus a natural extension of Black 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 continuous time, showing their ability to capture the commonly held stylized facts about volatility. This is organized as follows: in Section 3.1 an overview of this class is reported, stressing the most important features the various models have in common. At present, several SV models are discussed in the literature, differing in the dynamics attached to the volatility. With respect to the analysis presented in Ref. [38], we have focused our investigation on three popular models: the Vasicek, the Heston and the exponential Ornstein-Uhlenbeck ones. In Sections 3.2, 3.3 and 3.4 we present the main theoretical features of each model (volatility distribution, return distribution, higher-order moments and different-time correlations) in order to emphasize the strong points, as well as their limitations, on reproducing and forecasting the true statistics of volatility and returns. An exhaustive discussion 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 theory and ignored the statistical properties of the market model, although they were indeed able to reproduce the smile effect. The relatively small number of works dealing specifically with price dynamics based on SV models written before 2000 is essentially due to the trouble in deriving analytically their statistical properties. The analysis is even much more involved when there are correlations between volatility and stock, as it seems to be the case. Moreover, it was commonly asserted that empirical data available did not allow for obtaining a reliable estimation of all parameters involved in a SV model. As we shall see this is no longer true, at least for the most popular ones. Indeed, the leverage correlation between stock price and volatility, though increasing the complication of the analysis, is crucial for the complete estimation of the parameters involved. The idea behind SV models is that the volatility σ is not a constant, but itself a stochastic time-dependent variable. Thus, in such approach, its observed random 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 the market dynamic is contained in a two-dimensional stochastic process in which the asset price St and the process Yt obey the stochastic diﬀerential equations (in the Itˆo prescription)

dS(t) = µ S(t)dt + f(Y (t)) S(t) dW1(t) , S(0) = S0 , (3.1) dY (t) = α(m Y (t)) dt + g(Y (t)) dW (t) , Y (0) = Y ( − 2 0 where the parameter µ is the drift coeﬃcient for the price process. The correlation between stock price changes and volatility can be incorporated explicitly by means of the relation between the two Wiener processes W1 and W2 entering Eqs. (3.1)

dW (t) = ρdW (t) + 1 ρ2dZ(t) (3.2) 2 1 − where Z(t) is a Wiener process independentpof W1 and ρ is the correlation coeﬃ- cient. We can convince ourselves of the meaning of the coeﬃcient ρ by evaluating the following expectation value

(3.2) dW (t) dW (t) = dW (t) ρdW (t) + 1 ρ2 dW (t) dZ(t) = h 1 2 i h 1 1 i h − 1 i (3.3) 2 ρdt + 1 ρ dW1p(t) dZ(t) = ρdt . − h i h i We would note that the ultimatepresult is obtained recalling that each Wiener process must satisfy dW = 0 and dW 2 = dt. In general, ρ can assume any value entering the intervh ali[ 1; 1], buth fromi several empirical analysis it has been proved that its sign is often−negative. This fact could account for the skewness and the leverage effect observed in the financial data, as already mentioned in Chapter 1. Namely, Yt is the process driving the volatility and can be in principle formulated 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 be taken as mean reverting process. Mean reversion is a well known stylized fact about asset price volatility and concerns the existence of an asymptotic value m to which σt will eventually return in a typical time 1/α (note that the parameters m and α are the ones entering Eqs. (3.1)). This feature reflects the economic idea of a normal level of volatility toward which an efficient market in health conditions tends, even if practitioners often differ on this level and whether it is really constant in time and through institutional changes. The mean reversion is easily seen (beside being quite trivial) by rewriting the second equation of (3.1) in a properly integral fashion:

t t Y (t) = Y (t0) + α (m Y (t0))dt0 + g(Y (t0)) ξ2(t0)dt0 − (3.4) Zt0 Zt0 dW2(t0) | {z } where ξ2(t) is a Gaussian white noise whose stochastic time derivative is the Wiener process W2(t). Recalling that a Wiener process has null expectation value

39 3. Stochastic volatility models: the theoretical approach the previous statement is proved, since

α(t t0) α(t t0) E[Y (t)] = Y e− − + m(1 e− − ) (3.5) 0 − clearly has the stationary value m as (t t ) . It’s also worth noting that − 0 → ∞ the mean reverting character of σt implies that current information have almost zero eﬀects on the long time forecast. The most popular SV models appearing in the literature during the last two decades are listed in Tab. 3.1. Among them, in the following sections we will examine 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 volatility 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 κY Scott eY Mean-reverting OU k Vasicek Y Mean-reverting OU k Stein-Stein Y Mean-reverting OU k | | Heston √Y CIR κ√Y

Before dealing with the Vasicek model, it’s worth mentioning that in what follows it turns out to be more convenient to work with the so-called zero-mean return, deﬁned as . dS dX = µdt , (3.6) S − 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 this process 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 process of the form dY (t) = α(Y (t) m)dt + kdW (t) . (3.8) − − 2 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

t α(t t0) α(t t0) α(t t0 ) Y (t) = Y (t )e− − + m 1 e− − + k e− − dW (t0) . (3.9) 0 − 2 Zt0 Eq (3.9) is the starting point of the following discussion and constitutes as well the solution of Eq (3.8). Looking at Tab. 3.1, in this framework σ Y and thus ≡ the volatility average value employs the form of Eq. (3.5)

(t t ) α(t t0) α(t t0) 0 E[σ(t)] = σ e− − + m(1 e− − ) − →∞ m (3.10) 0 − −→ where the limit (t t ) indicates that the volatility process is in the stationary − 0 → ∞ state. Here we have assumed that the process started at time t0 when volatility is σ0. Making use of Eq. (3.10), the analytical expressions for the variance and the correlation of σ can be derived, and read

2 2 (t t ) k 2α(t t0 ) 0 k . Var[σ(t)] = 1 e− − − →∞ = β (3.11) 2α − −→ 2α and 2 2 k ατ E [σ(t)σ(t + τ)] = m + e− . (3.12) 2α As known, the “original” Ornstein-Uhlenbeck process [68] is the model describing the velocity process of a physical Brownian motion carried out by a particle in a solution. The solution of the Fokker-Planck equation for such a process is therefore known and reads

α(σ−E[σ(t)])2 1 2 − −2α(t−t ) p(σ; t) = e− k [1 e 0 ] , (3.13) 2 πk (1 e 2α(t t0)) α − − − q i.e. the well known Maxwell velocity distribution. The stationary limits of Eq. (3.13) is given by

1 α(σ−m)2 2 pst(σ) = e− k . (3.14) πk2/α

Since the Ornstein-Uhlenbeck processp turns out to be one of the most interesting and, from a physical viewpoint, peculiar process within the family of stochastic processes, 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), arguing that it does not appear to be a positive-deﬁnite stochastic variable, a fact that can be regarded as troublesome for a SV model. Nevertheless we can convince ourselves that this is not the case. We have already discussed in Section 1.3.2 that the actual evaluation of volatility is very diﬃcult to achieve, since volatility cannot be directly observed. In practice, we can use as a good proxy the instantaneous

41 3. Stochastic volatility models: the theoretical approach volatility, deﬁned in Eq. (1.19). Combining this expression with Eq. (3.6) and recalling that dW 2 = dt, we get

σ (t) = σ(t)2 = σ(t) , (3.15) ist | | which shows that there is no need top attach a sign to the random variable σ(t). Vasicek and Stein-Stein models, as can be read in Tab. 3.1, diﬀer because in the latter one, instead of Y , Y is taken as the driving process for the volatility entering the former Eq. (3.1).| This| assumption allows to avoid at all the trouble of the volatility sign in the Stein-Stein framework. In what follows we will deal with the Vasicek model only. Before addressing to the return process, it’s also worth discussing whether this model is mean reverting. Recall from Section 3.1 that the mean reversion is a necessary feature for a good volatility model. A possible deﬁnition of the “normal level of volatility” is given by [63]

dX(t)2 . lim E σ0 = NLV , (3.16) t →∞ " dt #

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

2 2 2 k O-U NLV = lim E σ (t) σ0 = m + . (3.17) t →∞ | 2α When, on the other hand, the volatility is notyet in the stationary state, the average over σ2 is given by

2 2 αt αt 2 k 2αt E σ (t) σ , 0 = σ e− + m(1 e− ) + 1 e− . (3.18) | 0 0 − 2α − This average tends tothenormal level as αt increases. The magnitude of α allows to classify the SV models into fast mean reverting processes, when 1/α t, or slow mean reverting processes, when 1/α t, where t is the typical time scale we are 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 this framework dX(t) = σ(t)dW (t) 1 , (3.19) dσ(t) = α(m σ(t)) dt + k dW (t) . ( − 2 42 3.2. The Vasicek model

In general, we can perform the analysis in term of return, but we prefer to deal with the zero-mean return since, as can be guessed by looking at Eq. (3.6), the expressions derived are much more handier. The system (3.19) can be seen equivalently as one single equation for the two-dimensional stochastic process (X(t), σ(t)), leading to 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]|

2 2 2 ∂p2 ∂p2 1 2 ∂ p2 ∂ p2 1 2 ∂ p2 = α(σ0 m) σ0 2 ρkσ0 k 2 , (3.20) ∂t0 − ∂σ0 − 2 ∂x0 − ∂σ0∂x0 − 2 ∂σ0 with the ﬁnal condition

p (x, σ, t x , σ , t) = δ(x x ) δ(σ σ ). (3.21) 2 | 0 0 − 0 − 0 Eq. (3.20) is the starting point of the analysis carried out by Masoliver and Perell´o in Ref. [69], on which is based what follows. Making use of the Fourier analysis, they derived a semi-analytical expression for the unconditional characteristic function (cf) of the return process, which reads

1 B2k2/α 4mB 4m2A ϕX (ω, t) = exp C + − − . (3.22) 2 − 4(1 + k2A/α) 1 + k A/α where the functionsp A, B, C are given by ω2 sinh ηt A(ω, t) = , (3.23) 2 η cosh ηt + ζ sinh ηt ω2αm cosh ηt 1 B(ω, t) = − , (3.24) η η cosh ηt + ζ sinh ηt (ωαm)2 1 ζ C(ω, t) = + iωρk α t/2 + ln cosh ηt + sinh ηt η2 − 2 η (ωαm)2 2ζ(cosh ηt 1) + η sinh ηt − , (3.25) − 2η3 η cosh ηt + ζ sinh ηt with

η = α2 2iρkαω + (1 ρ2)k2ω2, ζ = α iωρk. (3.26) − − − The Vasicek mopdel succeeds in predicting the semi-closed form solution (3.22) for the return process. Nevertheless it has to be inverse-transformed to compare it with real ﬁnancial data. It’s worth stressing that in order to derive the unconditional 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 returns in 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

ω2m2t/2 ϕX (ω, t) = e− , (3.27)

43 3. Stochastic volatility models: the theoretical approach

rho = 0 0.8 rho = 0.99

0.6 phi

0.4

0.2

0 -300 -200 -100 0 100 200 300 p_x

Figure 3.1: Characteristic function of the return in the Vasicek model for two values of the correlation coefficient ρ. which is the characteristic function of the zero-mean return when X(t) follows a one-dimensional diffusive process with constant volatility σ = m. Hence, Eq. (3.22) correctly embodies the geometric Brownian motion model as a particular case. In Fig. 3.1 the characteristic function of returns Eq. (3.22) is shown. Moreover, assuming that k 1, (3.28) α one can also prove that Eq. (3.22) converges to the Gaussian distribution when approaching the limit t → ∞ k t ϕ (ω, t) exp ω2 1 + ν2 + O m2 , (t ), (3.29) X ∼ − α 2 → ∞ . 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 model must provide. The Gaussian density (3.29) assures that the Vasicek model is able to reproduce this feature. It’s also worth noting that to assume k α means that the volatility is weakly random in comparison with its deterministic drift, since k is the strength of the volatility driving noise (the so-called volatility of volatility) whereas α gives information about its deterministic drift. Thus the ratio k/α measures in some way the degree of volatility randomness. Once obtained the semi-closed form solution carried in Eq. (3.22) for the return process, we are able to evaluate analytical expression for the higher-order moments introduced in Section 1.3.1, i.e. skewness and kurtosis. Making use of the definition given in Eq. (1.15), we can derive the expressions for the asymptotic limits of the kurtosis in this framework, given by 2 2 . κ4 6ν (ν + 2) γ2 = 2 2 2 (αt 1), (3.30) κ2 ∼ (ν + 1)

44 3.2. The Vasicek model and 2 2 2 2 . κ4 6ν [ν (1 + 4ρ ) + 4(1 + ρ )] 1 γ2 = 2 2 2 (αt 1). (3.31) κ2 ∼ (ν + 1) αt As known, the kurtosis allows to quantify the “fatness” of the tail of the distribution compared to the Gaussian case. Thus, it is a crucial quantity in measuring financial market risk. From Eq. (3.30) we observe that the model produces a never 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 convergence is as slow as 1/t, in agreement with the Gaussian limit of the pdf. In addition, we observe that for short times kurtosis does not contain the correlation coefficient 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

κ3 ν γ1 3ρ √2αt (αt 1), (3.32) ≡ 3/2 ∼ √ 2 κ2 ν + 1 and κ ν(ν2 + 2) 1 γ 3 6ρ (αt 1). (3.33) 1 ≡ 3/2 ∼ (ν2 + 1)3/2 √ κ2 2αt Unlike kurtosis, this moment is known to give information about the asymmetry of the pdf, reducing to zero in the Gaussian case. From Eqs. (3.32) and (3.33) it emerges that the skewness goes to zero at both short and long times, but if the kurtosis 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 ρ determines the skewness sign. Many empirical observations suggest that ρ must be negative and, consequently, we have to expect that the the left tail is fatter then the right one at any times. Before addressing to different-time correlations, it’s worth focusing on the tails of the distribution. It is well establish that the return distribution exhibits an excess 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 small andmoderate time, when the Central Limit Theorem has no effect. Tails are derived from Eq. (3.22) by keeping the first orders in ω, which means large values of zero-mean returns x. The analytical computation shown in Ref. [69] yields

1 1 2 pX (x, t) exp a(t) + b(t) a(t) x (x + ), ∼ 2 −b(t) − → ∞ a(t) + b(t) p (3.34) and p

1 1 2 pX (x, t) exp a(t) + b(t) + a(t) x , (x ). ∼ 2 b(t) → −∞ a(t) + b(t) p (3.35) with p . ρkt . k2 a(t) = , and b(t) = t(2 ρ2αt). (3.36) 4 8α − 45 3. Stochastic volatility models: the theoretical approach

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 is in disagreement with the power-law nature of the tails observed at infraday-daily time horizons, pointing out that the Vasicek model is unable to reproduce this key feature. To conclude, we would note that the sign of ρ allows to determine which tail is heavier since a(t) = ρkt/4. In particular, when ρ < 0 the fattest tail is the one representing losses, while when ρ > 0 the fattest tail corresponds to proﬁts. If ρ = 0 there is no diﬀerence between the two tails and the distribution becomes asymmetric, 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 reproduce the leverage effect and the volatility autocorrelation introduced in Section 1.3.3. To derive analytically the leverage function we can use the definition given in Eq. (1.23), which allows to derive [69] the following expression in the Vasicek framework 2 ατ ν√2α (1 + ν e− ) ατ (τ) = 2ρ 2 2 e− H(τ) , (3.37) L " (1 + ν ) m # where H(τ) is the Heaviside step function

1, if τ > 0 H(τ) = (3.38) (0, if τ < 0

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

2 ατ ν√2α (1 + ν e− ) A = 2ρ 2 2 . (3.39) " (1 + ν ) m #

Note that, as for the skewness and the tails, the sign of the leverage function is completely determined by the sign of ρ, since Eq. (3.37) is linear with the correlation coefficient. The evaluation of the volatility autocorrelation (τ) is quite more involved than the leverage effect, recalling that its analyticalC definition is given by the complex Eq. (1.25). To compute all the quantities entering the Eq. (1.25) we have to use Eqs. (A.3), (A.5) and (A.15) and, after some lengthy calculations, we obtain

2 4 2 ατ 2 ατ σ(t)σ(t + τ) σ(t) ν e− (ν e− + 1) (τ) = h i − h i = . (3.40) C 4 σ(t)2 2 3 σ(t) 4 4ν2(ν2 + 2) h i − h i 46 3.3. The Heston model

From Eq. (3.40) emerges that the volatility autocorrelation has one single char- 1 acteristic time scale coinciding with the leverage relaxation time α− . This facts sounds unrealistic, since (τ) is characterized by a double time-scale instead that C a single one, as shown in Section 1.3.3. The main theoretical features of Vasicek model are summarized in Tab. 3.2 at the end of the present chapter. To conclude, we would stress that what we have illustrated proves that the Vasicek model is paradigmatic for the simplicity of the assumptions on the processes, which allows a full calculation development. In this framework, the leverage eﬀect and the higher-order moments are correctly predicted. Therefore, it fails in measuring two important stylized facts about volatility: its non-Gaussian probability distribution and the double time-scale of the 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 of interest rates, the so-called CIR process (also known as Feller process or square root process) was later chosen by Heston [72] to drive the volatility fluctuations in a SV model capable to obtain option prices in a semi-closed form, i.e. the characteristic functions of prices are in closed form. The Heston model was also able to explain the asymmetry of the volatility smile with the introduction of the correlation (3.2) between the two Wiener processes W1 and W2. As known, such asymmetry is indeed caused by a negative correlation between past prices and future 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:p as the volatility is usually defined as the standard deviation of the price distribution, we are actually declaring that the driving process Yt has significance of variance. In what follows, we shall use v instead of Y to recall this fact, as it has become customary to do since Heston’s original work. The starting point of our analysis is to assume that the volatility σ obeys a zero-reverting Ornstein-Uhlenbeck process, i.e. Eq. (3.8) with m = 0

dσ(t) = ασ(t)dt + kdW (t) . (3.42) − The Itˆo 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 p k2 γ = 2α , κ = 2k , θ = . (3.44) 2α 47 3. Stochastic volatility models: the theoretical approach

The factor √v as coeﬃcient of the random term of dv has appeared, as requested by Eq. (3.41). It’s also worth stressing that the process which has to be taken as primary in the evolution of volatility is v, which obeys the mean reverting equation (3.43). Reminding the general outline given in Section 3.2 for the Vasicek model, we can ﬁrstly describe the main statistical features of such a process. To this end, we have to specify its integral form; from Eq. (3.4) we get

t t v(t) = v0 + γ (θ v(t0))dt0 + κ v(t0)dW2(t0) , (3.45) t0 − t0 Z Z p where v0 = v(t0) is supposed to be known. Taking the expectation value of Eq. (3.45), we can therefore obtain the mean value

(t t ) γ(t t0) γ(t t0) 0 E [v(t) v ] = v e− − + θ 1 e− − − →∞ θ . (3.46) | 0 0 − −→ Note that the expression (3.46) states that the normal level of volatility in the Heston 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 in the 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

2 2 2 θκ κ γ(t t0) κ θ 2γ(t t0 ) Var[v(t) v ] = + (v θ)e− − + v e− − . (3.47) | 0 2γ γ 0 − γ 2 − 0 which represents the conditional variance for the v process. It is well known that to completely characterize a statistical variable the analytical 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 a non-central χ2 distribution. Now, from the statistics [73] we know that a variable given by the sum of n squared Normals follows a central χ2 density with n degrees of freedom. Instead the attribute non-central refers to a variable defined by the sum 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 squared Ornstein-Uhlenbeck processes, leading to the following expression for its sde nκ2 dv(t) = γ v(t) dt + κ v(t)dZ(t) . (3.48) − − 4γ p From the comparison between Eq. (3.48) and the simpler Eq. (3.43) emerges that the only affected parameter in the process seems to be the long-time mean θ, passing from k2/2α in Eq. (3.44) to nk2 θ = , (3.49) 4γ 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 3.3. The Heston model 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 preserve the process v(t) from becoming negative, which would be troublesome since in dv there happens to appear a √v. Thus to correctly deﬁne the variance process, we must require that 4γθ n = > 2 (3.50) κ2 where Eq. (3.49) has been used. Eq. (3.50) is an important constraint to take into account in calculating the form of pv(v).

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

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

(t t ) (t t ) 4γ 0 4γ γ(t t0) 0 ct = − →∞ , λt = ctv0e− − − →∞ 0 (3.51) κ2 (1 e γ(t t0)) −→ κ2 −→ − − − where λt is deﬁned as the non-centrality parameter, linear with respect to the variance starting value v0. In term of ct and λt the variance probability distribution is given by [74] λ/2 i ∞ e− (λ/2) pt(v) = ct pχ2 (ctv) , (3.52) i! n+2i i=0 X 2 where p 2 (c v) is a central χ distribution with n + 2i degrees of freedom. χn+2i t Once obtained Eq. (3.52), one can derive the stationary distribution of v. Observing the stationary limits in Eqs. (3.51), we get

(α 1) αv/θ v − e− . 2γθ n p (v) = , α = , (3.53) st (θ/α)α Γ(α) κ2 ≡ 2 which is the well-known Gamma distribution. The parameter α is the ratio between the average variance θ and its characteristic ﬂuctuation κ2/2γ during the

49 3. Stochastic volatility models: the theoretical approach relaxation time 1/γ; note that when α , p (v) δ(v θ). → ∞ st → − The corresponding probability density for the volatility process σ = √v is easily derivable and reads α 2 t 2α (2α 1) ασ2/θ p (σ; t) = 2σ c p 2 4γθ (c σ ) →∞ p (σ) = σ − e− . σ t χ ( ,λt) t st α κ2 −→ θ Γ(α) (3.54) In Fig. 3.3 (left panel) the stationary distributions for the variance and the volatility processes are shown. Finally, we would note that the Heston model has the notable advantage of providing a closed form expression for the volatility distribution, which exhibits a strongly required non-Gaussian character, unlike the Vasicek model. However, as mentioned in Chapter 1, it has been empirically proved [12] that the volatility could feature an inverse-Gamma distribution instead that a Gamma one, as found in this framework.

The return process Before dealing to the probability distribution of the return process, it’s worth mentioning that, with respect to the investigation of Ref. [75], for the Heston model the convenient definition of zero-mean return differs from the one given in Eq. (3.6). Indeed we will use the more original . S(t) x(t) = ln µt , (3.55) S0 − where, instead of returns, we make use of log-returns. Eq. (3.55) gives the slightly more complicated sde v(t) dx(t) = dt + v(t) dW (t) . (3.56) − 2 1 Following the same reasoning of Section 3.2,p we define the transition probability P (x, v v ) for the two-dimensional stochastic process (x , v ) t | i t t v(t) dx(t) = dt + v(t) dW (t) − 2 1 , (3.57)  dv(t) = γ(v(t) θp) dt + κ v(t)dW (t) .  − − Pt(x, v vi) represen ts the probability density tophave log-return x and variance v | at time t, given the initial log-return x = 0 and variance vi at t = 0. In order to find an analytical formula for the time dependent probability distribution of the return, the Fokker-Planck equation, which governs the time evolution of Pt, has to be solved. It has been done exactly by Dr˘agulescu and Yakovenko in Ref. [75] making use of the Fourier analysis, leading to the following semi-closed unconditional solution for the return process in term of a Fourier integral + 1 ∞ P (x) = dp eipxx+Ft(px) (3.58) t 2π x Z−∞ 50 3.3. The Heston model

0.8 1 0.7 0.8 ) ) θ 0.6 1/2 0.6 θ / (v/ * σ ( Π )

σ 0.4 0.5 ( * Π 0.2 0.4 0 0 1 2 3 0.3 σ/θ1/2

0.2 Stationary distribution

0.1

0 0 1 2 3 4 5 v/θ

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 of v. Inset: The corresponding stationary probability distribution pst(σ) of volatility (Eq. (3.54)) [75]. Right panel: Characteristic function of the return pdf in the Heston model for two values of the correlation coefficient ρ. with γθ 2γθ Ωt Ω2 Γ2 + 2γΓ Ωt F (p ) = Γt ln cosh + − sinh . (3.59) t x κ2 − κ2 2 2γΩ 2 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 hypothesis, the Pt(x) obtained is real (ReF is an even function of px and ImF is an odd one), and that Ft(0) = 0, which implies that Pt(px = 0) is correctly normalized at all times. The integral in Eq. (3.58) can be calculated numerically, as we will see in the next 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. Finding special conditions in which Pt(x) or at least its chief features, e.g. the slopes of its tails, are exactly calculable is hence fairly appealing. Hence, in what follows we will illustrate the asymptotic behaviours of the returns at both short and long times, stressing the possibility to characterize them analytically. This is a strong point of this model, since we know that the log-returns can be directly evaluated from financial time series and thus compared with theoretical predictions. Con- versely, variance is a hidden stochastic variable that has to be measured by means of estimators based on the same time series, although its probability density can be 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 equilibrium value θ within the characteristic relaxation time 1/γ. Thus, the main difference between the regimes of Pt(x) must lie in the amount of time elapsed since the initial condition (0, vi) in comparison with 1/γ. We will therefore draw our attention

51 3. Stochastic volatility models: the theoretical approach to two main cases: short times (γt 1) or long times (γt 1). Both the regimes have been studied in detail in Ref. [75]. Here, the main results found, without the entire 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 regime Pt(x) acquires, in fact, meaning of short-time transition probability. As t is small the conditional probability distribution reads

2 1 (x+vit/2) 2v t Pt(x vi) = e− i . (3.60) | √2πvit Eq. (3.60) shows that at short time P (x v ) evolves in a Gaussian manner, because t | i variance has little time to change. Again assuming that the variance is in the stationary state allows to pass from the conditional probability density of Eq. (3.60) to the unconditional one for the log-return process, given by

1 α x/2 2 − e− α α 1/2 Pt(x) = y − Kα 1/2(y), (3.61) Γ(α) πθt − r where Ka is the modified Bessel function of (real) order a and we introduced the scaling variable 2αx2 2√γ x y = = | | . (3.62) r θt κ √t In the limit y 1, namely very large x , using the approximation Kν(y) y | | ≈ e− π/2y we find 1/2 α 2 − α α 1 y p P (x) y − e− . (3.63) t ≈ Γ(α) θt r Eqs. (3.62) and (3.63) prove that the tails of the distribution are exponential in x. Thus the Heston model fails to predict the power-law shape of the return tails at short time, a well known stylized fact about returns. The asymptotic behaviour at long time (γt 1) is strictly related with risk management. As t tends to infinity, Eq. (3.59) reduces to γθt F (p ) (Γ Ω) (3.64) t x ≈ κ2 − and, after laborious computation, we are able to obtain the scaling form

p0x Pt(x) = Nt e− P (z) , P (z) = K1(z)/z , (3.65) ∗ ∗ with

2 2 2 . ω0 (x + ργθt/κ) γθt . ω0γθt Λt z = + , Nt = e (3.66) κ s 1 ρ2 κ πκ3 1 ρ2 − − where Nt is the time-dependent normalization factor. Notep that the dependence of Pt(x) on the two arguments x and t is given by the function P (z) of the single ∗ 52 3.3. The Heston model scaling argument z in Eq. (3.66). z As for the tails, we can use the asymptotic expression K1(z) e− π/2z in Eq. (3.65) and take the logarithm of P . For large x , x γθt/κ≈and so we get | | | | p Pt(x) ω0 ln p0x x . (3.67) Nt ≈ − − κ 1 ρ2 | | − Thus, Pt(x) has exponential tails also for largep time horizon. It is also worth mentioning that in the considered limit the slopes d ln P/dx of the exponential tails do not depend on time t. In particular, the slopes (3.67) for positive and negative x are not equal, leading to an asymmetric distribution Pt(x) with respect to positive and negative price changes. Finally, we would note that in the region x 0 of small log-return Eq. (3.67) becomes ∼ P (x) ω (x + ργθt/κ)2 ln t p x 0 , (3.68) N ≈ − 0 − 2(1 ρ2)γθt t0 − 2 where Nt0 = Nt exp( ω0γθt/κ ). 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 agreemen| | t with the Central Limit Theorem.

Leverage eﬀect and volatility autocorrelation 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 autocorrelation can be rewritten . E [v(t + τ)dx(t)] (τ) = (3.69) L E [v(t)]2 and . v(t)v(t + τ) v(t) 2 (τ) = h i − h i . (3.70) C Var [v(t)] Nonetheless, the Heston model does not provide a closed equation for the volatility, since Eq. (3.45) is not exactly solvable. Thus it seems impossible to derive an analytical expression for the Heston leverage. Masoliver, Perell´o and Anento reported in Ref. [38] the approximated form

γ γτ (τ) 4ρ e− , (3.71) L ∼ κ which is exact only when θ = κ2/2α, but Ref. [75] shows that it fails in measuring leverage eﬀect for the Dow Jones index data. Thus, the only way to compute the Heston leverage is to derive it by means of numerical simulations. Unlike leverage, volatility autocorrelation can be exactly computable assuming that at time t the variance is in the stationary regime. It reads [77]

γτ (τ) = e− . (3.72) C 53 3. Stochastic volatility models: the theoretical approach

As in the Vasicek model (Eq. (3.40)), the Heston model is unable to reproduce the multiple time-scale empirically observed in (τ). In particular, it fails in reproducing its slowest time decay. C To conclude, we would note that the Heston model has the notable advantage of 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 computable in the principal regimes of interest. Nevertheless, it fails to predict some major stylized facts, as the power-law shape of the return tails at short time or the double time scales in the volatility autocorrelation. The main theoretical features of the model, illustrated in this section, are listed in Tab. 3.2 at the end of the present chapter.

3.4 The exponential Ornstein-Uhlenbeck model

The exponential Ornstein-Uhlenbeck model was introduced by Scott no later than 1987 [78] as a possible stochastic volatility modification of the Black-Scholes model for option pricing. Nevertheless, the former version of the model yields a Gaus- sian pdf for the stationary distribution of the volatility. This feature is indeed wide of the mark, since it has been already discussed in Section 1.3.1 that the volatility could feature a Log-Normal or an inverse Gamma distribution. One of the main motivation which has led to the formulation of the so-called correlated exponential Ornstein-Uhlenbeck model, topic of this section, is indeed to find a model whose volatility is Log-normally distributed. As we shall see, the other major capability of the model is to predict two distinct time-scales in the decay of volatility autocorrelation. It’s worth mentioning that also the Vasicek and the Heston models could be sophisticated ad hoc in order to account for the double relaxation time-scale, but this method has neither physical nor financial assumptions. 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 + kdW (t) . (3.73) − 2 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. Since Y (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 value ' 1/2 m, which obviously has dimension t− . With respect to the investigation of Ref. [39], in the following we will assume that at time t the process Y (t) (hence the 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

t t 0 (t t ) 0 α(t t0) α(t t ) 0 α(t t ) Y (t) = Y e− − + k e− − dW (t0) − →∞ k e− − dW (t0) , 0 2 −→ 2 Z−∞ Z−∞ (3.75) with the ﬁrst and second order moments given by

E [Yst] = 0 , (3.76) and 2 2 (t t ) k 2α(t t0 ) 0 k . Var[Y (t)] = 1 e− − − →∞ = β . (3.77) 2α − −→ 2α Eqs. (3.74) and (3.75) are thestarting point to derive the main features of the volatility process. In the expOU model, the average value and the variance assume the following expressions

(t t ) α(t t0) β 2α(t t0 ) 0 β/2 E [σ(t)] = m exp Y e− − + 1 e− − − →∞ me , (3.78) 0 2 − −→ 2 α(t t0) 2α(t t0 ) Var [σ(t)] = m exp 2Y e− − + β 1 e− − 0 − (t t ) 2α(t t0 ) 0 2 β β exp β 1 e− − 1 − →∞ m e e 1 , − − −→ − (3.79) while the autocorrelation ofthe stationary volatilit y reads

2 ατ E [σ(t)σ(t + τ)] = m exp β(1 + e− ) . (3.80) Since Y (t) is an Ornstein-Uhlenbeck process, it follows aGaussian density with 2α(t t0) zero mean and variance β 1 e− − . This feature gives us the form of the − distribution for the volatility process α(t t0) 2 1 [ln(σ/m) e− − ln(σ0/m)] p(σ, t σ0, t0) = exp − , 2α(t t0 ) | σ 2πβ(1 e 2α(t t0 )) − 2β(1 e− − ) − − − − (3.81) whose stationaryplimit is 1 ln2(σ/m) p (σ) = exp . (3.82) st σ√2πβ − 2β Eq. (3.81) conﬁrms the correct prediction of the Log-Normal distribution of the volatility.

The return process In order to ﬁnd the return probability density, we are dealing with the following system of two coupled sdes

dX(t) = meY (t)dW (t) 1 , (3.83) dY (t) = αY (t)dt + kdW (t) . ( − 2 55 3. Stochastic volatility models: the theoretical approach where the returns are deﬁned as in the Vasicek model. The calculation of the return pdf was performed by Masoliver and Perell´o in Ref. [39] with a technique analogous to the one followed in Sections 3.2 and 3.3. Unlike the Vasicek and Heston models, the characteristic function found in this framework can be analytically inverse-transformed, leading to the following closed-form expression for the entire pdf

1 x2/2m2t ρka(αt) x − pX (x, t) e 1 1/2 3/2 H3 ' √2πm2t " − α (2αt) −√2m2t (3.84) k2b(αt) x + H4 , 8α(αt)2 −√ 2 2m t# where Hn(x) are Hermite polynomials and the functions a(αt) and b(αt) are de- ﬁned by z a(z) = z 1 e− , (3.85) − − and

2 2 z z z b(z) = (1 + 2ρ )z + 2ρ ze− 2 1 e− 1 e− . (3.86) − − − − . with z = αt. It’s worth noting that Eq. (3.84) is an approximate expression for the return distribution, since it has been obtained under the assumption of stationary volatility. Moreover, we have also assumed that k 1 , (3.87) m i.e. we have dealt with the case in which the “volatility of volatility” k is much greater than the normal level of volatility represented by m. This situation is actually close to reality, in that typical λ values lie around 102 103. Before addressing with the diﬀerent-time correlations, we ÷would stress that Eq. (3.84) is a correction for the Gaussian density

1 x2/2m2t p(x, t) = e− , (3.88) √2πm2t which would correspond to the return pdf if the volatility was a deterministic quantity, i.e. if k = 0 in Eq. (3.73). The deviation from the Gaussian character of the return pdf in Eq. (3.84) is also evidenced by the existence of non-zero cumulants of order higher than two, implying non-zero skewness and kurtosis. Recalling Eqs. (1.14) and (1.15), we get [39] a(αt) b(αt) γ 6ρ 2β , γ 24β . (3.89) 1 ' (αt)3/2 2 ' (αt)2 p The return distribution, thus, is designed to capture the kurtosis and the skewness eﬀect.

56 3.4. The exponential Ornstein-Uhlenbeck model

Leverage eﬀect and volatility autocorrelation 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 reliable proxies to evaluate it from the data. This is an important task, especially when we want compare empirical data with theoretical predictions. For this reason, it seems to be more convenient to evaluate the volatility autocorrelation (τ) by means of C E [σ(t)2σ(t + τ)2] E [σ(t)2]2 (τ) = − , (3.90) C 3E [σ4] E [σ(t)2]2 − instead of Eq. (1.25), which allows to calculate the following expression

ατ exp[4βe− ] 1 (τ) = − . (3.91) C 3e4β 1 − 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

∞ n 1 (4β) nατ (τ) = e− (3.92) C 3e4β 1 n! n=1 − X a discrete inﬁnite multiplicity of relaxation times appears. Nevertheless the relevant ones are just two, clearly visible in Fig. 3.4 which is obtained plotting ln (τ) C in function of the dimensionless ατ for several values of the β parameter. In detail, we see that all the curves assume approximately the same value for τ = 0, in agreement with the fact

4β e 1 β&1 1 C(0) = − . (3.93) 3e4β 1 −→ 3 − Moreover, we observe that the initial slope of each curve, i.e. the short-time decay rate, namely the ﬁrst time scale, depends on the value of β, while the asymptotic slope is the same for every curve and hence it must be a function of α. We can characterize this double time-scale also analytically, as follows. For long times, i.e. ατ 1, recalling the deﬁnition of β we can rewrite Eq. (3.91) as 1 4β 2k2τ 2 2 (τ) = e − 1 + O α τ . (3.94) C 3e4β 1 − − h i In this case the short time behaviour of (τ) has the characteristic time 1/2k2. In the opposite case ατ 1 we can neglect,C in Eq. (3.92), terms of the order nατ of e− with n 2 and get ≥

4β ατ (τ) e− . (3.95) C ≈ 3e4β 1 − 57 3. Stochastic volatility models: the theoretical approach

10-2

10-4 β=5

10-6 β=10 10-8

10-10 β=20 volatility autocorrelation 10-12

10-14

β=30 10-16

0 2 4 6 8 10 normalized time ατ

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

Thus, the long time behaviour of the volatility autocorrelation is governed by the 1 characteristic time α− . Therefore, within the expOU framework the autocorrelation of the volatility presents, at least, two time scales: the longest one governed by τlong = 1/α and 2 a shorter one governed by τshort = 1/2k . We observe that the dimensionless parameter β can be rewritten in terms of τlong and τshort as τ β = long , (3.96) 4τshort which shows that β is also a measure of the distance between the long and the short time scales (see Fig. 3.4). To derive the expression of the expOU leverage we can use Eq. (1.23), which allows to obtain the following formula [39]

2ρk ατ 3 (τ) = exp ατ + 2β e− H(τ), (3.97) L m − − 4 which reproduces the form given by Eq. (1.24), although the implied deﬁnition of A and b will depend on the diﬀerent scales involved. As for (τ), we can study the asymptotic behaviours of (τ) at both short C L and long times. For long time, i.e. ατ 1, Eq. (3.97) reduces to (τ > 0)

2 3β/2 ατ 2ατ (τ) = ρke− e− + O e− (ατ 1), (3.98) L m which has the form (1.24) if we assume b = α and

2 3β/2 . A = ρke− = A . (3.99) m long 58 3.4. The exponential Ornstein-Uhlenbeck model

Since the value of β is usually quite large, Along results exponentially small and the long-time behaviour of (τ) turns out to be undetectable in the practice of empirical observations. ConsequenL tly, the eﬀect of the leverage correlation can be sought only in the short-time regime ατ 1. In this limit expanding Eq. (3.97) yields (τ > 0) 2 β/2 k2τ (τ) ρke e− (ατ 1), (3.100) L ' m 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 2 A = ρkeβ/2 A . (3.101) m ≡ short Hence, the leverage correlation is determined by Eq. (3.100) which features one 2 single time scale given by τlev = 1/k . Note that this time scale is of the same order 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 expOU models: the Log-Normal distribution of the volatility and the presence of the double time-scale in the volatility autocorrelation, stylized facts measured in many empirical analyses of real ﬁnancial time series. The main theoretical features derived along this section are detailed in Tab. 3.2.

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

Vasicek Heston expOU Volatility pdf Normal Gamma Log-Normal Log-return pdf ϕx(ω, t) ϕx(ω, t) ϕx(ω, t) non-inv non-inv approx. invertible ατ k2τ (τ) ρe− H(τ) ? ρe− H(τ) L −ατ ατ ατ exp[4β e ] 1 (τ) e− e− − C ∼ 3e4β 1 1 time scale 1 time scale 2 time −scales

59 3. Stochastic volatility models: the theoretical approach

60 Chapter 4 A comparative analysis of stochastic volatility models

In the previous chapter we have described in detail the main theoretical features of the Vasicek, Heston and expOU SV models, stressing their ability in reproducing some aspects characteristic of the market dynamics. This is true especially for the expOU model, since the predicted Log-Normal volatility distribution and the double time-scale in the autocorrelation of the volatility process make it the most realistic among the models considered. Now, a natural prosecution of such analysis could be to implement numerically the models, paying particular attention to their predictive effectiveness on the outcomes of numerical simulations. This is an important task, especially in view of financial applications like risk management or option pricing, which can be often treated only by means of numerical methods and algorithms. Thus, in this scenario, a good volatility model must not only meet the Engle and Patton’s demands illustrated in Chapter 3, but also maintain all its features when it is simulated. This is actually a basic request once dealing with numerical issues. The numerical implementation of the models is the main topic of the former part of the present chapter. We illustrate in detail the comparison between the theoretical predictions of Chapter 3 and the numerical results obtained by means of original simulations of the models. We were particularly interested in confirming the closed or semi-closed solution for volatility and returns distributions, as well as the different-time correlation functions. As we will see, we proved an almost perfect convergence between analytical and numerical results. The latter part of the chapter is devoted to an empirical analysis of real financial time series. Unlike the numerical approach, this could directly provide confirmations of the ability of the 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 be successful in predicting the stylized facts of real market. This chapter is based on the work presented in Ref. [79] and is organized as follows: in Section 4.1 we briefly discuss the discretization algorithm used to approximate 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 its stability, are also pointed out. In Section 4.2 the results achieved from the numerical simulations are shown and compared with the theoretical predictions of Chap- ter 3 for both volatility and return probability distributions and diﬀerent-time correlations. 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 Fourier Transform technique used. Finally, in Section 4.3 the empirical analysis of three time series from the Italian stock market is presented, in order to emphasize the realism, as well as some limitations, of the expOU model once comparing with real data.

4.1 Discretization algorithm

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

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 , (4.1)  −  dW (t) = ρ dW (t) + 1 ρ2 dZ(t) 2 1 −  p whereW1(t), W2(t) and Z(t) are Wiener processes generated by Gaussian white noises. In principle, we could implement numerically such a system according to several methods, variously sophisticated (Euler, Runge-Kutta, predictor-corrector . . . ), but their suitability depends on the nature of the problem treated. To individuate the most correct, we must identify the ordinary diﬀerential problem closest to the stochastic one we are dealing with. Looking at Eqs. (4.1), it is evident that the two model-dependent equations, if stripped of the stochastic terms, are elementary ﬁrst-order coupled equations with initial conditions. Therefore we can discretize them following the straightforward Euler method. Namely, if we apply the method to y˙ = f(t, y(t)) at step i it yields simply

(∆y) = f(t , y )∆t = y = y + f (t , y ) ∆t . (4.2) i i i ⇒ i+1 i i i The Euler method assures, for a sufficiently small step size, a good approximation of the underlying differential equation. It’s also worth mentioning that it has a rough first-order accuracy. The Euler discretization algorithm continues to be substantially valid also when the stochastic terms are added, but the meaning of accuracy must be modified. To better understand it, we have to recall that one of the key points in the Itô formalism is that the stochastic quantities written using the differential notation, like dW , have the meaning of infinitesimal standard deviation corresponding to infinitesimal time increments dt [23]. In other words, dW 2(t) = dt must be interpreted in “mean-square sense”: the variance of the Wiener process realizations (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 any sense. Therefore the accuracy of the numerical simulation can only be understood and verified in terms of averages (mean, variance, correlations) calculated on an ideally infinite set of realizations. These preliminary remarks insure that the so- called Euler-Maruyama algorithm, which is the extension of the Euler method to stochastic differential equations, correctly approximates the paths described by the sde given in the models. Following its prescriptions, at step i the volatility driving process Y is incremented to

Y = Y + α (m Y ) ∆t + g (Y ) ξ √∆t , (4.3) i+1 i − i i i where ξ(t) is a Gaussian white noise whose stochastic time derivative is the Wiener process W2(t), as in Eq. (3.4). It’s worth mentioning that to better estimate the accuracy of such a method more sophisticated techniques could be used, falling in the scope of pure numerical analysis and, thus, out of interest for the present work. 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 the real process ones at ﬁnite time horizons, and its stability, i.e. its correct behaviour at long times. We can consider for instance the stability of the Euler-Maruyama method in a Heston model simulation, limiting to the behaviour of the average value [80]. Recalling Eq. (3.43), the discretization yields

v = v + γ (θ v ) ∆t + κ√vξ √∆t . (4.4) i+1 i − i i

An iterated substitution of vi in the right side of Eq. (4.4). followed by the application of the average, gives directly the mean value of the variance v at the last step n E [v ] = (1 γ∆t)n (E [v ] θ) + θ . (4.5) n − 0 − The ﬁnal result depends on the value of ∆t, given a certain γ. For

n ∆t < 2/γ, E [v ] →∞ θ ; • n −→ ∆t = 2/γ, E [v ] = ( 1)nE [v ] + (( 1)n+1 + 1) θ ; • n − 0 − n ∆t > 2/γ, E [v ] →∞ . • | n | −→ ∞ In order to have the simulated paths tending (in average) to the correct stationary value θ the time step size ∆t must not be too larger than the mean-reverting time 1/γ. Note that the average cancels out the contribution from the stochastic term in Eq. (4.4): the previous conclusion can therefore be regarded as a general result for all the mean-reverting SV models. Thus, we can take twice the relaxation time as the upper bound for the choice of the step size of our simulations. Much more specialized calculations give the expectation values of the maximum discrepancies between the simulation output and the real process, which, in agreement with the above reasoning, can be used to estimate the simulation accuracy. As a rule of thumb, they give for E[ Y sim Y (t ) ] a supreme value which is linear with | n − n | 63 4. A comparative analysis of stochastic volatility models

√∆t and a sup E[(Y sim Y (t ))2] proportional to ∆t. We can conclude that, n − n the smaller the time step ∆t, the more accurate simulations will be. This could clearly cause some limitations in the computation time availability and also in the ﬁnite machine precision.

4.2 Numerical results

The theoretical results derived in Chapter 3 are tested ab initio by means of original numerical simulations of the models, whose sde are discretized following the technique introduced in the previous section. The aim of this study consists in verifying the theory, regarding the time evolution of volatility and return processes of the generated populations. We are particularly interested in evaluating the shape and the main statistical features (average values and higher-order moments) of their probability distributions at various time, including the stationary limit. In order to perform an exhaustive numerical investigation, we check also the capability of our simulations to correctly reproduce the time series correlation effects (leverage effect and volatility autocorrelation). This is indeed a crucial point, since these aspects are among the most realistic features of SV models and it would be disappointing to find that the discretization procedure does not con- serve them. Afterwards we display the most relevant results emerged from the comparative study we carried out. The entire analysis, as well as several technical details, 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. The theoretical 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 times the Vasicek and the expOU volatilities feature the same stationary distributions of Tab. 3.2, that is the Normal and the Log-Normal ones respectively. On the contrary, the Heston volatility follows at short times the non-central χ2 density given in Eq. (3.52), clearly different from the Gamma pdf characterizing its long times limit. Moreover, Eq. (3.52) cannot be simplified and we have to rewrite it an a properly series-expansion fashion in order to implement it numerically. Namely, Eq. (3.52) yields

(c v+λ )/2 n/2 1 (3.52) e t t (c v) p (v; t) = c − t − v t 2n/2 Γ(n/2) i (4.6) ∞ 1 1 λ c v 1 + t t , i! n(n + 2) . . . (n + 2i 2) 2 " i=1 # X − where n is deﬁned as the eﬀective number of degrees of freedom 4γθ/σ2. The latter expression has been derived starting from Ref. [81].

64 4.2. Numerical results

In order to test the compatibility between theoretical predictions and numerical simulations we have to produce a suﬃciently large Monte Carlo population 4 6 of volatility paths ( 10 ÷ ). This is indeed one of the messages come out from ∼ what discussed in Section 4.1. To generate them, the sde for the volatility process is 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 relaxation time, corresponds to some years. In Fig. 4.1 we show for instance ten simulated paths 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 make use of model parameters quoted in the literature [39, 69, 75].

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

0.5

0.45

0.4

0.35

0.3 Volatility

0.25

0.2

0.15

0 50 100 150 200 250 Time

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

This comparative analysis has led to an almost perfect coincidence between theory 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 its stationary limit in comparison with the predictions reported in Tab. 3.2. The numerical evaluation substantially confirms the theoretical relations, except for the most extreme events in the expOU model, not captured by the Log-Normal distribution. 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 curves are not fitted, but plotted over the numerical results. In this way we can check directly the correctness of our simulations, as well as the exactness of the theoretical models. As for the volatility, we perform a comparative study of the returns too. It’s worth 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 implementation 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 the return increment. In this way σ can drive the evolution of the return process step by step. To comply with what discuss in Section 3.1, we generate and compare

65 4. A comparative analysis of stochastic volatility models with the theory only zero-mean log-return series.

Vasicek model Heston model expOU model

10 Simulated Normal 1 Gamma 1 Log-Normal

10-1

10-1 10-2

10-2

10-2 -3 Probability density Probability density Probability density 10

10-3

10-3 10-4

10-4

10-4 10-5

-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6

σ σ σ

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

We are particularly interested in confirming the theoretical predictions derived in Chapter 3 for the entire shape of the return distribution. This task is quite simple for the expOU model, because the predicted solution of Eq. (3.84) is a closed form, directly comparable with the simulation output. This is shown in Fig. 4.3 on different time lags, from 1 up to 250 days. Like what has been done for the volatility, we did not fit the theoretical curves looking for the parameters which assure the best agreement, but we have only plotted them in comparison with the numerical data: the overall agreement is almost perfect. Fig. 4.3 allows to appreciate also the skewness of the distribution, more evident as the time increases. We would recall that Eq. (3.84) has been derived under the assumption of the stationarity of volatility. In order to account for it, we set the initial values of the parameters entering the formula (3.84) very closed to their mean reversion values. This allows to sensibly compare the numerical results with the theory, also at short time horizons. Unlike the expOU model, the predicted Vasicek and Heston solutions for the return process are semi-closed formulae, not analytically invertible. Namely, they are expressed analytically through their characteristic functions, which can be switched to the corresponding probability densities only by means of numerical techniques. In order to solve this problem and obtain theoretical expressions comparable with our results, a Fast Fourier Transform (FFT) algorithm was implemented making use of the built-in functions appearing in the software ROOT [82] and MATHEMATICAr. Namely, we have to numerically compute the following Fourier integral + 1 ∞ iωx P (x) = dω e P (ω) , (4.7) t 2π t Z−∞ e 66 4.2. Numerical results

expOU model

1 day

6 5 days 10 20 days 40 days 250 days 5 10 Theory

104 p(X) 103

102

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

Figure 4.3: Numerical returns (points) on different time lags in comparison with the theory of the expOU model (line). The curves are shifted each other by one decade for greater clarity. where the sign of the phase factor exponent in the integrand is positive for the Heston 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 the characteristic function we would inverse-transform, which assumes respectively the expression of Eqs. (3.22) and (3.59) in the Vasiceke and Heston framework. Since the latter expressions cannot be inverse-transformed analytically, we must take into account the numerical formulation of the FFT algorithm. This consists in the discretization of the integral entering Eq. (4.7), where the sum can be restrict to a finite amount of N + 1 discrete values of x in which Pt(x) is computed. More exactly, the Fourier integral can be discretized on an finite interval ( xmax, xmax) making use of routines of the form −

N 1 − 1 2πi nk P (x ) = e− N P (ω ) . (4.8) t n N t k Xk=0 e 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 πN πN e = , . (4.9) Iω −2 x 2 x max max The resulting P are transformed in the desired P (x ) by multiplicand them { n} { t n } by N/I, where I is the width of the interval . To check the validity of the result, I Pt(x) must be properly normalized. A more exhaustive description of the FFT technique can be found in Ref. [51]. Fig. 4.4 displays the time evolution of the Vasicek and Heston return distributions, together with the theoretical curves achieved by means of the FFT technique above illustrated. As for the expOU model, the analytical solutions are plotted in comparison with the simulated returns in order to directly evaluate their convergence. We note that in the Vasicek framework the agreement is poor at infra-week time 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 4. A comparative analysis of stochastic volatility models expOU model, to account for the stationary volatility assumption, we set the initial value of volatility equal to its mean reverting one. Looking at Figs. 4.4, it also emerges that the skewness increases for increasing time horizon, being quite negligible at short times. Recalling what we have discussed in Section 1.3.1, one could note that such a behaviour correctly reproduces the skewness’s trend empirically observed in real markets. Moreover it proves the non-Gaussian character of the simulated distributions, visible as an asymmetry arising for non zero ρ when the evolution time is large enough. A skewned distribution results, in fact, from a negative correlation between volatility and return processes (see Section 1.3.1); to simulate this feature we have implemented the processes setting a negative value for the parameter ρ. In order to quantitatively characterize it, we have also checked the simulations capability to correctly reproduce the higher-order moments, comparing them with the theory. As saw in Section 3.2, the Vasicek framework predicts the behaviour of skewness and kurtosis at both short and long times. We note that the simulation outputs agree very well with the predicted Eqs. (3.30-3.33), showing the slow convergence to zero of both the moments, coherently 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 at least for several relaxation→ ∞ times. Conversely, no analytical expressions are predicted in the Heston model for higher-order moments; thus, we can analyse them only qualitatively.

Vasicek model Heston model

104 104

103 103

102 102 p(X) p(X)

10 10 1 day 5 days 20 days 40 days 1 1 250 days Theory 10-1 10-1 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Return,X Return,X

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

Before addressing to the correlation functions, it’s worth reminding that in Chapter 3 we have reported several approximated results, valid in the Heston and Vasicek frameworks for the complete pdf as well as just for the tails. Such results are obviously eﬃcient only in particular regimes, but they have the remarkable advantage of being all expressed in closed-form. We check the convergence between these expressions and numerical results, ﬁnding out in almost all the cases a good agreement [77]. To conclude, the numerical results demonstrate that, for what concerns the volatility and return processes, the models can be implemented maintaining all

68 4.2. Numerical results their features. This has great importance in view of ﬁnancial applications, like the above-illustrated risk management, in which the shape of the return distribution plays 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 to test whether they are able to correctly capture the volatility correlation functions, whose analytical expressions are listed in Tab. 3.2. From the numerical point of view this problem is quite different from the ones we previously faced, because the leverage effect of Eq. (1.23) and the autocorrelation function of Eq. (1.25) depend on the time delay τ and not on the absolute time t. Thus, to numerically simulate these quantities a whole time series must be considered: namely, for a fixed τ the averages appearing in the analytical formulae are calculated upon all the existing instants t. This procedure is widely used to compute leverages and correlations on real financial data, since for a certain index or share clearly only one time series exists. We have followed the same method to obtain these statistical functions for the outcomes of our simulations: a very long series of increments, corresponding to up to a century of exchanges, has been simulated. In Figs. 4.5 and 4.6 we present, respectively, the resulting numerical leverages and volatility autocorrelations, compared with the theoretical predictions displayed in Tab. 3.2. For the Vasicek model, it’s quite evident that the numerical output superposes almost perfectly the prediction of Eq. (3.37) for the leverage effect, as well as Eq. (3.40) for the volatility autocorrelation. Conversely, in the Heston model there is no theoretical formula for leverage effect to look at and we can just limit ourselves to qualitative considerations on the simulation output, as in the case of higher-order moments. For what concerns the volatility autocorrelation, the approach which led to Eq. (3.72) must be kept in mind: it has been obtained directly from the squared volatility process v(t) rather than from the squared increment dX(t)2. Therefore it is natural to implement Eq. (3.70) taking as input the realizations of the variance process instead that the volatility one. The numerical outputs recovers perfectly the theoretical expectation, as shown in Fig. 4.6. Finally, the expOU leverage and volatility autocorrelation show a clear, but quite 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 predicted presence of two typical time scales in the volatility autocorrelation, namely Eqs. (3.94) and (3.95). The parameter values fixed along the simulation has lead the following time scales 1 1 τ = = 4 days and τ = = 25 days . (4.10) short 2k2 long α Such values yield a very fast damped autocorrelation function, which allows to appreciate in Fig. 4.6 (bottom panel) a sharp distinction between the two characteristic time-scales.

69 4. A comparative analysis of stochastic volatility models

Vasicek model expOU model 10 4

2 0

-10 -2

Leverage Leverage -4 -20

-6

-30

-8 Simulated Theory

-40 -10 -20 0 20 40 60 80 100 -20 0 20 40 60 80 Delay (days) Delay (days)

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 can be found in Tab. 3.2.

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

Vasicek model Heston model

0.25 1

0.2

0.8

0.15

0.6

0.1

0.4

0.05 Volatility autocorrelation Volatility autocorrelation

0.2 0

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

expOU model

Simulated Theory 10-1

10-2 Volatility autocorrelation

10-3

0 10 20 30 40 50 60 Delay (days)

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 theoretical curves can be found in Tab. 3.2. In the case of expOU model, the double characteristic 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 an empirical analysis of financial data from the Italian stock market. Unlike what previously produced, the aim of this analysis is to critically compare the models and to establish whether they turn out to be realistic once comparing with empirical data. The time series used, freely downloaded from Yahoo Web Site [52], are collections of daily closing prices of the Italian assets Bulgari SpA, Brembo and Fiat SpA from January 2000 to May 2007. Thus we are dealing with N = 1920 data. Fig. 4.7 displays the historical evolution of the three stocks in the time interval considered: the price of each share clearly features a negative growth rate until the 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 then a steady rise in its price, due to the change occurred in the company’s management and its consequent new throw in the world-wide market. Besides these interesting matters about market dynamics, knowing the state of a share has great importance when we are dealing with fit of the data share on theoretical curves. The value of the parameters entering the fitted curves could, in fact, change in time passing from a period of positive to one of negative growth. Silva and Yakovenko in 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 returns derived in the Heston model. In Fig. 4.8 we show a fit to empirical volatility on the theoretical predictions of the SV models considered, displayed in Tab. 3.2, for Bulgari SpA and Fiat SpA.

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

20 10

Price 15 Price

10 Fiat Brembo 5 Bulgari

2000 2001 2002 2003 2004 2005 2006 2007 2000 2001 2002 2003 2004 2005 2006 2007 Year Year

Figure 4.7: Historical evolution of Fiat SpA (red), Brembo (green) and Bulgari SpA (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 proxy introduced in Section 1.3.2, we estimate empirical daily volatility as absolute daily

71 4. A comparative analysis of stochastic volatility models returns. Namely, σ = R , (4.11) i | i| . where R = (R1, . . . , RN ) is the N-dimensional vector of daily returns. The parameter values of the fitted curves are obtained according to a multidimensional minimization procedure based on the maximum likelihood approach. As in Sec- tion 2.3, in order to solve the optimization problem and find the best parameter values 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 agreement between data and the Log-Normal distribution predicted in the expOU framework, whereas the Normal and the Gamma densities tend to underestimate the large distribution’s tail. This conclusion sounds not so surprising, since we have already discussed the ability of Log-Normal to reproduce empirical volatility for both individual share and indexes. It’s worth mentioning that a similar analysis was performed also by Miccichè et al in Ref. [84], where the theoretical prediction of 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 equity market. Also this study proves that historical volatility features a Log-Normal distribution, especially in the region of low values of volatility, coherently with the conclusions 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 volatility, as expected, especially in the body of the distribution.

Bulgari data, 03/01/2000-29/05/2007 Fiat data, 03/01/2000-29/05/2007 102 Vasicek Heston ExpOU

10 10 Probability density Probability density 1 1

0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 0.12 σ σ

Figure 4.8: Fit to empirical daily volatility distribution on the theoretical prediction of Vasicek (green), Heston (blue) and expOU (pink) models for Bulgari SpA and 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 returns and the analytical expression of Eq. (3.84) derived by Masoliver and Perell´o (MP) in the expOU framework, in order to test its ability in modeling the return features. The same curve is shown in Fig. 4.3, in fully agreement with the numerical outcomes. With respect to what discussed in Section 3.4, we dealt, like in the

72 4.3. Comparison between theory and empirical data numerical analysis, only with zero-mean log-returns Xτ , estimated as in Ref. [39]

N 1 1 − Xτ = Rτ Rτ = Rτ Rτ , (4.12) i i − h i i − N i i=0 X where Rτ is the vector of returns and the parameter τ stands for time lag (see below). In Tab. 4.1 we report the mean values Rτ of the time series considered, evaluated from empirical moment as clearly emergesh i from Eq. (4.12). We note that for daily returns, i.e. τ = 1, the means are quite negligible, involving no difference between log-returns and zero-mean log-returns. Nevertheless, to comply τ with the literature, we analyse only Xi 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 % h i h i h i h i h i 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 Normal and the fat-tailed Student-t introduced in Section 2.2. We would recall that the MP function derived in Section 3.4 is an approximate solution, valid only in the stationary limit of volatility. This hypothesis is well verified for our stocks, since they have been trading in the Italian stock market (and also in the NYSE in the case of Fiat SpA) from the last century (Fiat SpA) and the 90’s (Brembo and Bulgari SpA). In order to constrain the parameter entering the theoretical formulae, we implement 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 is very ticklish to manage, due to the appearance of local minima in the optimization function. Nevertheless, the parameter correlation matrix proves that m, which describes the normal level of volatility, has quite negligible correlation with other parameters. 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∼) which are, unlike m, strongly correlated. In spite of the trouble of such procedure, our fits turn out to be always stable. The parameters of the fitted curves, displayed in Tab. 4.2, are similar to the ones quoted in the literature [39]. Note also that κ > 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 function are in good agreement with empirical distributions, both in the central body and in the tails, while Normal fails to reproduce the data. This is true especially for Fiat SpA and Bulgari SpA, whereas the tails of Brembo data tend to be widely

73 4. A comparative analysis of stochastic volatility models

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

Normal Student MP func

10 10 10 p(X) p(X) p(X)

1 1 1

10-1 10-1 10-1 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 Daily return, X Daily return, X Daily return, X

Figure 4.9: Fit of daily returns (histogram) on the theoretical prediction of the expOU 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 mean is indicated with the symbol µ to not generate confusion with the parameter m entering the MP formula.

µ% σ% ν α κ m ρ Fiat SpA 0.01% 2.06% 3.32 0.01 0.22 0.02 -0.7 Brembo 0.02% 1.86% 3.08 0.02 0.31 0.02 -0.2 Bulgari 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 lower capitalization of Brembo with respect to companies like Bulgari and, above all, Fiat. A low capitalization implies, in fact, a smaller market, lower liquidity of the share and, consequently, less exchanges. In such a scenario, it can happen that a few, anomalous, transitions (i.e big losses or gains) would assume great relevance in the company’s history. From a statistical viewpoint, these extreme events, that enrich the distribution tails, tend to assume a statistical weight too relevant since the overall number of exchanges is relatively low. This fact could be the cause of the overestimation in the tail regions exhibited by the MP fit. To better evaluate the ability of the three curves in capturing the features of empirical distributions, especially in the tails region, we propose in Fig. 4.10 the corresponding cumulative density functions P>, numerically evaluated from the curves of Fig. 4.9. The figures displayed on the left panel show that the MP function fits data better than Student-t density. This is due to the presence in the Eq. (3.84) of the Hermite polynomials, which allow to model correctly the function shape over the data. Similar results emerge also focusing only on the left tail of the distribution, i.e. negative returns, which has known to be the region where risk is located. As already noted, this conclusion is not completely correct

74 4.3. Comparison between theory and empirical data

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

Data 1 1 Normal Student MP func

10-1 10-1 P>(X) P>(X) 10-2 10-2

10-3 10-3

-0.15 -0.1 -0.05 0 0.05 0.1 10-2 10-1 Daily return, X Daily return, -X

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

1 1

10-1 10-1 P>(X) P>(X) 10-2 10-2

10-3 10-3

-0.15 -0.1 -0.05 0 0.05 0.1 10-2 10-1 Daily return, X Daily return, -X

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

1 1

10-1 10-1 P>(X) P>(X) 10-2 10-2

10-3 10-3

-0.15 -0.1 -0.05 0 0.05 0.1 10-2 10-1 Daily return, X Daily return, -X

Figure 4.10: From top: Fiat SpA, Brembo and Bulgari SpA cumulative density function of daily returns (left) and negative daily returns (right). Points represent empirical data, whereas lines correspond to Normal (blue), Student-t (red) and MP function (green) as in the corresponding Fig. 4.9. for Brembo, even if the MP function is able to capture the most extreme events of its distribution. Finally, in Fig. 4.11 we present the return distributions for several time horizons τ (1,5,7,20,40 days) in comparison with the MP theoretical function, as in Fig. 4.3. With respect to the procedure described in Ref. [75], for each time lag we extract the return series Xτ from Eq. (4.12) fixing the time lag τ. Tab 4.1 proves 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 worth mentioning that all the curves appearing in Fig. 4.11 are generated changing in Eq. (3.84) only the value of the temporal parameter t according to τ, while for the other parameters (α,κ,m,ρ) we always use the values listed in Tab. 4.2. This is the same procedure adopted to produce the corresponding Fig. 4.3: namely, we did not fit the model parameters trying to find the best ones, but we limited to

75 4. A comparative analysis of stochastic volatility models make a sharp prediction of the theory plotting the curves in comparison with the empirical data. The overall agreement is very good. This is indeed the meaning of the system of Eqs. (4.1) on the grounds of the SV models studied: once ﬁxed the parameters, the equations must reproduce correctly the time evolution of the two correlated processes. The quality of the MP theory emerges also noting that, when the time lag increases, the left tail of the empirical distributions becomes fatter and 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 volatility and return distribution, the latter over diﬀerent time horizons too.

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

104 104

103 103

102 102 p(X) p(X)

10 10

1 1

10-1 10-1 -0.6 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Return, X Return, X

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

1 day 5 days 7 days 20 days 4 10 40 days MP func

103

102 p(X)

10-1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Return, X

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

76 Conclusions and perspectives

The geometric Brownian motion (GBM) is still accepted as a paradigm of the quantitative finance applied to financial complex systems. Nevertheless, as we have pointed out, within this framework many facts typical of the real markets dynamics cannot be captured. They concern the non-Gaussian nature of log-return distribution, the non-constant character of volatility, which features, conversely, its own dynamics, the leverage effect and the volatility autocorrelation described in Chapter 1. To resolve the contradictions between the well-known stylized facts and the widely accepted theory of Black and Scholes-Merton (BSM), models beyond the Gaussian paradigm are strongly demanded. Inspired by these motivations, the studies presented in this thesis have been devoted to non-Gaussian stochastic models, to their often different nature and their possible applications in the field of quantitative finance. In this scenario, we have shown that within the framework of statistical physics the Student-t distribution, often used in the econophysics literature, emerges as peculiar distribution able to capture the non-Gaussian nature of log-returns. In particular, in Chapter 2 we have illustrated a possible application of this model to risk 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 carefully the power-law tails of return distribution, where the risk resides. We have shown that 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 evaluation, once specified in terms of the models parameters optimized by means of an empirical analysis of real daily return series. With respect to similar investigations present in the literature [47, 60], our results sound more statistically robust thanks to the confidence levels, estimated with a bootstrap-based technique, attached to the performed measures. As far as possible perspectives are concerned, it would be interesting to investigate to what extent our conclusions, drawn from an analysis of a sample of Italian financial data, apply also to other financial markets. In particular, one could check whether, at a given significance level, statistically relevant differences are present between the results valid for a single asset and those relative to a portfolio of assets, as our analysis seems to indicate, at least

77 Conclusions and perspectives for 5% VaR. Another interesting development concerns the comparison between the predictions for VaR and ES of our model with the corresponding ones derived by means of other statistical procedures to measure tail exponents known in the literature [54, 56, 57], as well as with the results from simulations of advanced models of the financial market dynamics, such as GARCH-like, multi-fractal models [58, 59] and stochastic volatility models. Taking this latter perspective as a starting point, we have investigated the family of stochastic volatility models, widely described in Chapter 3, as a natural extension of the GBM paradigm. The common denominator of the models belonging 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 its own 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, in capturing the commonly held stylized facts about volatility and returns. From the theoretical viewpoint, the expOU model has turned out to be the most realistic, since it displays the capability to correctly reproduce the Log-Normal character of the volatility distribution, as well as the typical double-time scales of the volatility autocorrelation. The theoretical features derived in Chapter 3 have been tested ab initio on the outcomes of original numerical simulations, paying particular attention to the predictive effectiveness of each model. This is indeed a necessary task, especially in view of financial applications like the above-mentioned risk management or option pricing, which can be often treated only by means of numerical methods. For each model we have proved an almost perfect convergence between the theoretical predictions and the numerical results, which assures that the theory is correct and, conversely, the numerical implementations of the models maintain all their features. Moreover, we have also performed and presented in Chapter 4 an empirical analysis of Italian time series, whose aim has been to critically study the models valuating for each one its degree of realism once compared with real data. Our analysis has indicated that the expOU model is the most successful in capturing the non-Gaussian character of empirical volatility. It also reproduces quite well the return distribution, even if a better agreement with high-frequency data, especially in the tails, could be achieved by substituting in the theory the Wiener noise with a non-Gaussian one [86], therefore leaving definitively the BSM framework. A natural prosecution of such analysis could concern extending the empirical investigation to a wider sample of financial series, in order to check the validity of our results with a deeper statistical meaning. On the other hand, it could also be interesting to include in both the theoretical and empirical analysis other SV models, like the Hull and White [85], and then compare them with the ones already taken into account. From our analysis we would conclude that one of most relevant open questions about the SV model is to establish whether the expOU 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 market phenomena. Moreover, we guess that a key point is also understanding if the

78 Conclusions and perspectives model could describe only the dynamics of daily and low-frequency data or could be successfully applied also to infra-day time horizons. To conclude, we would note that the most natural “bridge” between the two main topics faced in this work (risk measures and SV models) could obviously be to perform risk measures modeling the return distribution in term of the (semi)- analytical expressions derived in the SV frameworks, leading to non-Gaussian SV market risk measures, easily comparable with the methodologies described in Chapter 2.

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

79 Conclusions and perspectives

80 Appendix A The 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 t S(t) 1 2 X(t) = ln µt + σ (t0)dt0 . (A.1) S − 2 0 Zt0 We now note that, despite the correlation between dσ(t) and dW1(t), σ(t) and dW1(t) are independent random variables. This is a consequence of Itô prescription for stochastic integrals [19], because the process σ(t) is independent of its driving noise. 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 entering the 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 2 2 2 2 E[dX ] = E[σ ]E[dW1 (t)] = E[σ ]dt , (A.3) where we used the Itô identity. Combining Eqs. (A.1) and (A.3) we can derive the second central moment (i.e. variance) given by Var[dX(t)] = E[dX 2] 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’s theorem [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 − 2 2 2 2 4 2 2 2 2 E σ(t) E dW (t) = 3E σ(t) dt E σ(t) dt . − − (A.6) 81 A. The zero-mean return features

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

2 2 2 E[dX(t + τ) dX(t)] = E[σ(t + τ) dW1(t + τ) σ(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, observing Eq. (1.23), Eq. (A.7) has great importance in calculating the leverage function. Following the Itô convention, if τ > 0 then dW1(t + τ) is uncorrelated with the 2 other factors; moreover we know that E [dW1(t + τ) ] = dt and thus

2 2 E dX(t)dX(t + τ) = E σ(t)dW1(t)σ(t + τ) dt (τ > 0). (A.8) On the other hand, if τ 0 then dW (t) is uncorrelated with the remaining ≤ 1 variables; 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

2 2 E dX(t)dX(t + τ) = E σ(t)σ(t + τ) dW1(t) H(τ)dt , (A.10) where H(τ) is the Heaviside step function deﬁned in Eq.(3.38). Writing explicitly dW1(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)2ξ (t) H(τ)dt2 h 1 i δ [σ(t)σ(t + τ)2] δ [σ(t)σ(t + τ)2] = H(τ)dt2 = ρ H(τ)dt2 δξ (t) δξ (t) (A.11) 1 2 δσ(t) δσ(t + τ) = ρ σ(t + τ)2 + 2σ(t)σ(t + τ) H(τ)dt2 . δξ (t) δξ (t) 2 2 Any ensuing development of the calculation requests the evaluation of the functional 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)2dW (t)ξ (t) dt2 h 1 1 i δσ(t) δσ(t + τ) = 2ρ σ(t)σ(t + τ)2dW (t) + σ(t)2σ(t + τ) dW (t) dt2 δξ (t) 1 δξ (t) 1 2 2 δ[dW (t)] + σ(t)2σ(t + τ)2 1 dt2 , δξ1(t) (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 derivative δ[dW1(t)]/δξ1(t): to this aim we write dW1(t) in a somewhat involved form,

t+dt dW1(t) = ξ1(t0)dt0 . (A.13) Zt 82 Thus t+dt δ[dW1(t)] = δ(t t0)dt0 = 1 . (A.14) δξ (t) − 1 Zt This result, even not knowing the exact forms of the functional derivatives of σ, ensures us that the ﬁrst term in Eq. (A.12), as it contains dW (t) factors, is negligible 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 generate its averages, is not speciﬁed through a choice of the functions f and g appearing in Eqs. (3.1).

83 A. The zero-mean return features

84 Appendix B The Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process [68] is the model describing the velocity process of a Brownian motion. It’s worth recalling that the physical phenomenon known as Brownian motion, observed for the first time by Robert Brown in 1827 with his microscope, concerns the small random movement of fine particles suspended in a fluid solution. Several years later, its dynamics was interpreted by Einstein in his famous paper dated 1905 [6], in which he illustrated how the statistical theory 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 additional friction force opposing the motion that, considering the small velocities involved, will be well described by a Stokes-like formula in which a γ friction coefficient appears. The stochastic differential equation for the velocity process will thus have a deterministic and a stochastic terms Γ dv(t) = γv(t)dt + dW (t) , (B.1) − m 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 (t t ) γ(t t0 ) 0 E [v(t)] = v e− − − →∞ 0 , (B.2) 0 −→ where v0 is the initial velocity, common to all the ensemble, and 2 2 2 (t t ) 2 Γ 2 Γ 2γ(t t0 ) 0 Γ E v (t) = + v e− − − →∞ . (B.3) 2γm2 0 − 2γm2 −→ 2γm2 We would notethat the particle velocity has a null stationary mean value, which means 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 characterized by a non-zero stationary mean square, indicating that the particles tend to maintain the velocity associated with Brownian motion. Making use of the equipartition theorem, we can derive the relation that the coeﬃcients γ and Γ must satisfy in order to correctly describe the Brownian motion from a thermo- dynamical viewpoint (like in the Einstein’s approach). It reads

1 2 1 2 m v t = kBT = Γ = 2γmkBT . (B.4) 2 →∞ 2 ⇒

This is a case of the ﬂuctuation-dissipation theorem, which links the strength of the equilibrium ﬂuctuations Γ to the dissipation intensity γ. From a statistical point of view, to completely characterize the system one has to compute the probability distribution of v at time t by solving the Fokker-Planck equation. For such a process, it assumes the following expression [35]

∂ ∂ 1 ∂2 p(v, t) = (γ vp(v, t)) + D p(v, t) . (B.5) ∂t ∂v 2 ∂v2 Switching to the characteristic function

∞ iωv ϕv(ω, t) = dv e p(v, t) Z−∞ the equation becomes ∂ ∂ 1 ϕ (ω, t) = γω ϕ (ω, t) Dω2ϕ (ω, t) . (B.6) ∂t v − ∂ω v − 2 v The general solution is computable with the method of characteristics and reads

2 Dω γt ϕv(ω, t) = e− 4γ g(ωe− ) , (B.7) with g an arbitrary (well-behaved) function. Since the pdf initial condition

p(v, 0) = δ(v v ) (B.8) − 0 is translated into iv0ω ϕv(ω, 0) = e , (B.9) we can rewrite g(ω) as follows

Dω2 g(ω) = exp + iv ω . (B.10) 4γ 0 By substituting the latter expression into Eq. (B.7), we can hence derive the ultimate solution

2 Dω 2γt γt ϕ (ω, t) = exp (1 e− ) + iωv e− . (B.11) v − 4γ − 0 86 It’s worth noting that Eq. (B.11) is the characteristic function of a Gaussian distribution with mean coinciding with Eq. (B.2) and variance given by

(t t ) D 2γ(t t0 ) 0 D Var [V (t)] = 1 e− − − →∞ , (B.12) 2γ − −→ 2γ namely in agreement with Eq. (B.3). Recalling Eq. (B.4) we obtain the velocity pdf in terms of the solution temperature:

γt 2 m m (v v e− ) p(v, t) = exp − 0 2πk T (1 e 2γt) −2k T 1 e 2γt r B − − " B − − # (B.13) 2 t m mv →∞ exp , −→ 2πk T −2k T r B B which is the well-known Maxwell-Boltzmann distribution of velocities in one dimension. Doob showed in Ref. [88] that the Ornstein-Uhlenbeck process is the only Gaussian Markov stochastic process having stationary solution. This is another proof of the universality of the Maxwell-Boltzmann distribution. To conclude, we would mention that deﬁning the position stochastic process in term of the velocity process through

t x(t) = dt0 v(t0) (B.14) Zt0 we are able to calculate its variance, which reads

2kBT . Var [X(t)] = t = D t . (B.15) mγ x

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

87 B. The Ornstein-Uhlenbeck process

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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 questi anni, per la fiducia, i consigli (scientifici e non), l’allegria che è in grado di trasmet- tere. Oreste, per le tante discussioni scientifiche e la perenne disponibilità. Un ringraziamento particolare va a Giacomo e Lorenzo, ho imparato tanto lavorando e trascorrendo il tempo con voi. Grazie, perché se non mi sono mai pentita della scelta che ho fatto è soprattutto per merito di tutti voi! Grazie Giò, sei stato un compagno d’ufficio fantastico! Vorrei ringraziare inoltre il prof. Rosario Mantegna, per aver fatto da lettore a questo lavoro. Grazie a tutta la mia famiglia. Grazie ai miei genitori, per avermi regalato anni bellissimi, 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 ed emozionate, 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 per l’amicizia che mi regala ogni giorno, le risate, le parole di conforto, il suo affetto. Andrea, se sono arrivata fino a qui è grazie a te, alla tua vicinanza, alla tua ca- pacità (unica) di capirmi sempre...avrei mille motivi per ringraziarti e mille altri ancora da farmi perdonare...Voglio dirti solo grazie.

Enrica