Option Pricing Models Driven By Fractional Brownian Motion And The Numerical Research

Up to now the Black-Scholes model has been paid attentions by many researchers.For the classical Black-Scholes model,the underlying asset is driven by geometric Brownian motion;but a large number of empirical studies show that the changes of underlying asset prices actually have a "fat tail" feature,which is the characteristics of fractional Brownian motion.In the financial markets,the underlying asset prices intermittently jump due to emergencies such as financial crisis,natural disasters,etc.Therefore,the fractional jump-diffusion process is used to characterize the changes of underlying asset prices in this thesis,and then the European option pricing model driven by the fractional jump-diffusion and European digital options driven by the mixed fractional jump-diffusion are studied,respectively.The main results are as follows:Firstly,we study the European pricing options under jump-diffusion process in the fractional Hull-White interest rate model.Using the(35)-hedging principle and the fractional jump-diffusion Ito-formula,an option pricing model is derived.The European call option pricing formula and the European call-put option parity formula are obtained by the partial differential equation method,and then the European put option pricing formula is derived.Using the same method,the pricing formula and parity formula of European digital call and put options are derived,and the explicit solution of the underlying asset price and interest rate is obtained.The impacts of Hurst index H and ? values on European option prices are analyzed by the numerical method,respectively.Secondly,we study the European pricing options under mixed jump-diffusion process in the mixed fractional Hull-White interest rate model.Using(35)-hedging techniques and mixed fractional jump-diffusion Ito-formula,a European option pricing model is derived.By the variable transformation and the heat conduction equation,the European call option pricing formula,the European call-put option parity formula and the European put option pricing formula are obtained.Then the impacts of Hurst index H and ? values on European option prices are analyzed by the numerical method,respectively.