Option pricing using path integrals.

Abstract

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable.