Option Pricing By Esscher Transforms

The thesis is focused on the applications of Esscher transform in option pricing. It is a method of martingale pricing, but better than it because of its good properties. It can solve lots of problem which can't be solved in traditional ways such as the differential equations method. The important methods include risk-neutral Esscher transform and the equivalent martingale measure in classical risk model. We present specific financial models--European call option-pricing formula, barrier call option-pricing formula and reload option-pricing formula in different circumstances. At the same time, some numerical illustrations are also presented in some cases.The First Chapter:We outline the definition, strong points, traditional pricing methods and main work of our study. The Second Chapter:Risk-neutral Esscher transform is discussed. Furthermore, European call option-pricing formulas are obtained in common hypothesis. The Third Chapter:Based on the Esscher transform, we introduce the compound Poisson process and Meixner process and give the pricing formulas. The Fourth Chapter:Motivated by the model in Gerber and Shiu, we consider Down-and-Out Power Calls, Partial-time-start barrier options and Partial-time-end barrier options. Two-Asset Barrier Options are also considered. The Fifth Chapter:We consider the pricing of the reload options with a single reload time. The logarithm of the price of the underlying asset is modeled as a Gamma process as well as Brownian process. Here we present some numerical illustrations .