Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona AN INTRODUCTION TO STOCHASTIC VOLATILITY MODELS Autor: Matias Puig Director: Dr. Jose Manuel Corcuera Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, 29 de juny de 2017 Contents Introduction i 1 Historical Background on Stock Price Models 3 2 Stochastic Integration 7 2.1 Martingales and Brownian motion . 7 2.2 Integral of mean square integrable processes . 9 2.3 Extension of the integral . 12 3 Fundamental Theorems 17 3.1 Itô’s formula . 17 3.2 Stochastic differential equations . 18 3.3 Girsanov’s theorem . 19 3.4 Martingale representation theorem . 21 4 The Black-Scholes Model 25 4.1 Risk-neutral measure . 26 4.2 Arbitrage and admissibility . 28 4.3 Completeness . 30 4.4 Pricing and hedging . 32 4.4.1 Pricing a put option . 33 4.4.2 Pricing a call option . 34 4.4.3 Hedging . 35 5 Stochastic Volatility 39 5.1 Empirical motivations . 39 5.2 A general approach for pricing . 39 5.3 Heston’s model . 42 6 Conclusions 47 7 Bibliography 49 Bibliography: Articles . 49 Bibliography: Books . 49 Bibliography: Online . 50 i Introduction 1 The main goal of this work is to introduce the stochastic volatility models in mathematical finance and to develop a closed-form solution to option pricing in Heston’s stochastic volatiltiy model, following the arguments in Heston 1993. No background in mathematical finance will be assumed, so another main goal of this work is to develop the theory of stochastic integration and to introduce the Black-Scholes market model, the benchmark model in mathe- matical finance. Standard topics in the framework of market models, such as trading strategies, completeness and replication, and the notion of arbitrage, will also be reviewed. 2 Introduction Chapter 1 Historical Background on Stock Price Models Louis Bachelier, in his thesis "Théorie de la Spéculation", made the first contri- bution of advanced mathematics to the study of finance in 1900. This thesis was well received by academics, including his supervisor Henry Poincaré, and was published in the prestigious journal Annales Scientifiques de l’École Normale Supérieure. In this pioneering work, the Brownian motion is used for the modelling of movements in stock prices. In the words of Louis Bachelier, in Bachelier 1900: La détermination des mouvements de la Bourse se subordonne à un nombre infini de facteurs: il est dès lors impossible d’en espérer la prévision mathématique, [...] et la dynamique de la Bourse ne sera jamais une science exacte. Mais il est possible d’étudier mathématiquement l’état statique du marché à un instant donné, c’est-à-dire d’établir la loi de proba- bilité des variations de cours qu’admet à cet instant le marché. Si le marché, en effet, ne prévoit pas les mouvements, il les consid- ère comme étant plus ou moins probables, et cette probabilité peut s’évaluer mathématiquement. Bachelier argued that, over a short time period, fluctuations in price are inde- pendent of the current price and the past values of the price, and that these fluctuations follow a zero mean normal distribution with variance propor- tional to the time difference. He also assumed that the prices are continuous, therefore modelled as a Brownian motion (see Bachelier 2011). Many years later, in the famous article by Black and Scholes, Black and Sc- holes 1973, prices are modeled as a geometric Brownian motion, whose fluctua- tions have a lognormal distribution. This model is based on the assumption that the log returns of a stock price are independent and normally distributed, with variance proportional to the time difference. The log returns are defined as log(p ) log(p ), where p and p denote the prices at times i and j, respec- i − j i j tively, with i > j. The returns of a stock price are defined as the increments pi pj ri = − pj Hence, the approxamation 3 4 Historical Background on Stock Price Models Figure 1.1: Sample path of a standard Brownian motion, used by Bachelier to model fluctuations in stock prices log(1 + r) r, when r 1 (1.1) ⇡ ⌧ Gives the approximation pi pj − = ri log(1 + ri)=log(pi/pj)=log(pi) log(pj) (1.2) pj ⇡ − So that the lognormal distribution for the stock price increments proposed by Black and Scholes obeys to the intuitive idea that the price returns are inde- pendent and normaly distributed. Explicitely, as stated in Black and Scholes 1973, the model proposed by Black and Scholes relies on the following assumptions of an "ideal" market: (a) The interest rate is known and constant through time (b) The distribution of stock prices at the end of any finite interval is lognor- mal (c) The stock pays no dividends (d) The variance rate of the return on the stock is constant (e) The stock price is continuous over time Empirical observations of stock price distributions have motivated numerous extensions of the Black-Scholes model in which one or more of the previ- ous assumptions are relaxed. One of the main criticisms of the Black-Scholes model is that the normal distribution of stock price returns does not explain the significant presence of outliers in the distribution of returns (see, for ex- ample, Mandelbrot 1963). Conveniently, relaxing some of the previous as- sumptions results in distributions with a higher presence of outliers, or fatter 5 Figure 1.2: Sample path of a Geometric Brownian motion (with µ = s = 1), used by Black and Scholes to model stock prices tails, which better adjust to the reality of observed prices. Examples of such extensions include models with dividend payments, stock prices with jumps (non-continuous over time), and models in which the distri- bution of returns is non-Gaussian, among others. These extensions are widely used by practitioners and are described in most text books on mathematical finance (see, for example, Musiela and Rutkowski 2006). In this project we will concentrate on the stochastic volatility models, in which the assumption that the variance rate of the return is constant is relaxed. 6 Historical Background on Stock Price Models Chapter 2 Stochastic Integration Let (W, , P) be a probability space, and T an index set. Consider a func- F tion X(t, w) having two arguments in T and W. By fixing t, X(t, ) is · t T a family of random variables defined in W. On the other hand, by fixing2 w, X( , w) can be seen as a family of ’random maps’. We will define a · w W stochastic process2 following the first point of view and show the equivalence between the two. Definition 2.1 (Stochastic Process). Let E be a metric space with the borel s-field. A stochastic process is a family of E-valued random variables (Xt)t T defined in a 2 probability space (W, , P) F In order to define a random map as a function-valued random variable, we must specify the function space and a s-field in this space. The most natural function space to consider is the space ET of all maps from T to E. The s-field will be the product s-field, that is the one generated by cylinder sets of G 1 1 the form p− (B1) ... p− (Bn), with B1,...,Bn (E) and pt being the t1 \ \ tn 2 B natural projection defined by the rule f f (t). 7! Let (Xt)t T be a stochastic process and define Y(w)(t)=Xt(w). Is the 2 map Y : (W, , P) (ET, ) -measurable? Indeed, for any cylinder set 1 F ! 1 G G 1 1 1 C = p− (B1) ... p− (Bn) , Y− (C)=X− (B1) ... X− (Bn) . t1 \ \ tn 2 G t1 \ \ tn 2 F Since is generated by the cylinder sets, we conclude that Y is -measurable. G G On the other hand, if Y : (W, , P) (ET, ) is -measurable, for any F ! G G t T and w W define Xt(w) := Y(w)(t). Then, for any B (E), 21 2 1 1 2 B X− (B)= w X (w) B = w Y(w)(t) B = Y (p− (B)) . t { | t 2 } { | 2 } − t 2 F Given the equivalence between both definitions, we will consider E-valued stochastic processes on T as any of the definitions above. A sample path of X is a function x(t)=X(t, w). We say that a E-valued stochastic process on T has paths in U ET if its sample paths are included in U. A stochastic process X ⇢ is said to be continuous if its paths are included in (T), the set of continuous C functions. The process X is continuous at t T if its sample paths x(t) are 0 2 continuous at t0 almost surely. Two stochastic processes X and Y are said to be versions of each other if P(X = Y )=1 for all t T. The finite dimensional t t 2 distributions of a stochastic process X are the joint distributions of the random vectors (X ,...,X ), t ,...,t T. t1 tn 1 n 2 2.1 Martingales and Brownian motion We will now consider stochastic processes with the index set T = R+, and with values on R. 7 8 Stochastic Integration Definition 2.2 (Filtration, adapted process). A filtration ( t)t 0 is an increasing F ≥ family of s-algebras included in the s-algebra . A process is said to be adapted to F the filtration ( t)t 0 if for each t, Xt is t measurable. F ≥ F Given a process (Xt)t 0, we can define the natural filtration t = s( Xs, s ≥ F { t ), for which the process X is adapted to.