Martingale Analysis Of Option Pricing In A Semi-markov Modulated Market

Option pricing is one of the key topics of the financial mathematics. Since Black and Scholes proposed the famous Black-Scholes option pricing model, it has not only been obtained a great number of results in theory, but also been applied in practice broadly. The Black-Scholes pricing model, however, is based on the asset price following geometric Brownian motion in which the relative returns of stock price follow lognormal distribution. In recent, some empirical studies show that the dynamics of stock price do not satisfy the normal distribution.This dissertation is intended to study risk minimizing option pricing in a semi-Markov modulated market. Using martingale methods we find the locally risk minimizing price for European options and the corresponding hedging strategy. Our main contributions are as follows:Firstly briefly introduces the concepts of options, including positions of options, participants, classification of options, some common methods of options pricing, and the international development is reviewed, the origin and significance of the subject is pointed out.Secondly, we explain the common theory and principles of martingale, including the definition of martingale, the definition of stopping time, some research results around the world, the definition of options, general theory of barrier options, and basic theorems of semi-Markov market.Finally,using martingale methods we find the locally risk minimizing price for European options and the corresponding hedging strategy, and the locally risk minimizing price for European barrier options and the corresponding hedging strategy.