Opton Pricing Of The Generalized Black-Scholes Model

The present dissertation is concerned with the option pricing of the generalized Black-Scholes model.The main aims of the dissertion are:(1)To establish the fractional mode of option pricing with respect to a fractional Brownian motion Bn(t) with Hurst exponent H ∈ [5,1);(2)To establish the generalized mode of option pricing and to give the analytical solution for European option pricing;(3)To establish the mode of option pricing with respect to a stochastic process X(t) - ta g(i, T)dB(r)aiid to show that this mode is equal to the Black-Scholes mode in a sense.This dissertation is divided into four chapters:Chapter 1 is preface.In chapter 2,we establish the fractional mode of European option pricing with respect to a fractional Brownian motion BH(t) with Hurst exponent H E [1/2, 1).First,we define the stochastic integral of random function f(t,u) with respect to a fractional Brownian motion BH(t) with Hurst exponent H ∈ [1/2,1) and get a generalized type ltd differential formula;Second,using the above formula ,the stochastic differential equations and the pricing formulas of the contigent claims are obtained with Hurst exponent H varying in [5,!).The essential difference of our model from Black-Scholes one is that although the variance of stock log-price is not equal to zero in no-trading days, the instaneous variance rate of the stock log-price in our model is zero , and the corresponding equation of option price is one order partial differential equation while in Black-Scholes model,the corresponding equation of option price is two order partial differential equation invariously.In chapter 3,we assume that a stock log-price S(t) satisfies:dS(t) = a(t)dt + f(t)dX(t)where f(t) is a smooth function for t ∈ [a, b] of bounded variation and ,X(t) is mean square continuous for t ∈ [a, b] and not differentiate with respect to t a.s,and E(x(t + Δt) - x(t))2 = c(#i)2H(1/2 < H < 1). A analytical solution for the option pricing is given by us.In chapter 4,we assume that a stock log-price S(t] satisfies:dS(t) = a(t)dt + f(t)dX(t)where f(t)is a bounded variation function and smooth for t 6 [a, b], X(t) =ftag(t,T)dB(r).We prove that X(t) is mean square continuous for t ∈ [a,b] andnot differentiable with respect to t a.s, A analytical solution for the option pricingis given by us.