Correlated Stochastic Dynamics in Financial Markets

Abstract

Thesis investigates the dynamics of financial markets. Nowadays, this is one of the emergent fields in physics and requires a multidisciplinary approach. The thesis studies the first work made by the financial mathematicians and presents those in a more comprehensible form for a physicist. Option pricing is perhaps most complete problem. Until very recently, stochastic differential equations theory was solely applied to finance by mathematicians. The thesis reviews the theory of Black-Scholes and pays attention to questions that had not interested too much to the mathematicians but that are of importance from a physicist point of view. Among other things, thesis derives the so-called Black-Scholes option price following the rules used by physicists (Stratonovich). Mathematicians have been using Itô convention for deriving this price and thesis founds that both approaches are equivalent. Thesis also focus on the martingale option pricing which directly relates the stock probability density to the option price. The thesis optimizes the martingale method to implement it in cases where only the characteristic function is known. The study of the correlations observed in markets conform the second block of the thesis. Good knowledge of correlations is essential to perform predictions. In this sense, two diffusive models are presented. First model proposes a market described by a singular two-dimensional process driven by an Ornstein-Uhlenbeck process where noise source is Gaussian and white. The model correctly describes the volatility as a function of time by considering the memory effects in the stock price changes. This model gives reason of the market inefficiencies due to the absence of liquidity or any other type of market interties. These correlations appear to have a a long range persistence in the option price and entails a remarkable influence in the risk due to holding an option. The second model is a stochastic volatility model. In this case, prices are described by a two-dimensional process with two Gaussian white noise sources and where volatility follows an Ornstein-Uhlenbeck process. Their statistical properties are studied and these describe most of the empirical market properties such as the leverage effect.