Equivalent Martingale measures and option pricing in jump-diffusion markets.

Abstract

One of the key questions in financial mathematics is the choice of an appropriate model for the financial market. There are a number of models available, such as Geometrical Brownian motion and different types of Levy processes, that are not general enough to reflect all the characteristics of fluctuations in stock price but for which the parameters can be estimated with relative ease. There are more general semimartingale models for which parameter estimation and numerical calculation become very difficult questions. The goal of this thesis is to present a tractable model for which we can carry out computations, and it seems that by varying the parameters this model can be related to real market data. We will use the equivalent measure approach to obtain estimates of the price of European call options for our model. Since our market is incomplete, a consequence of the inclusion of jump processes in the model, we will choose the "best" equivalent martingale measure by applying various techniques and compare the results for different choices. We will also illustrate how this theory works on particular examples. We consider applications not only to the cases of continuous and Levy process markets but also to cases that reflect the main advantages of our jump diffusion model. Finally we numerically illustrate option pricing in our setting.