Large Deviation Theory And Option Pricing

The area of large deviations is a set of asymptotic results on rare events probabilities and a set of methods to derive such results. In this paper, we present some methods and applications of large deviations to finance. Large deviations theory is a very active field in applied probability, and finds important applications in finance, where questions related to extremal events play an increasingly major role. Large deviations methods are largely used in rare events simulation and so appear naturally in the approximation of option pricing,, in particular for barrier options and far from the money options. More recently, there has been a growing literature on small time asymptotics for stochastic volatility models. First we give some basic tools and results on large deviations, in particular the most classical result on large deviations,such as Cramer’s theorem,Freidlin-Wentzell theory on sample path large deviations, and Varadhan’s integral principle. We then describe how large deviation approximation and importance sampling with Monte-Carlo methods are used in rare event simulation for option pricing. We know Monte-Carlo method is used for estimating the expectations arising in option pricing,and the basic principle of importance sampling is to reduce variance by changing probability measure from which paths are generated.Here,we briefly describe the importance sampling variance reduction technique for diffusion via Girsanov’s theorem,which consists in determining a process φ.We will present two approaches leading to the construction of such processes φ. In the first approach the process φ is stochastic, and requires an approximation of the expectation of interest. In the second approach, the process φ is deterministic and derived through a simple optimization problem. Both approaches rely on asymptotic results from the theory of large deviations.