Pricing Options Study

Black and Scholes forlnulated the famous B-S equation for option pricing in 1973, and the most of later researches have been carried out in B-S circumstance,which are also in linear circumstance.However,traditional theories can not explain many abnormal phenomenons on financial markets.In this situation,some researchers turn to nonlinear methodology.Based on Fractal Markets Theory,this dissertation studies lookback option pricing from the point of the character of fractal capital markets,and compares the solutions to the lookback option pricing formulas which were deduced by Goldman from the perspective of theory and practice.This dissertation is based on Fractal Market Theory,and two traditional hypotheses have been adjusted in it,including that stock price follows Geometric Brownian Motion(GBM) and investors are homogenous.For highlighting the impact of relaxation the hypothesis,only one hypothesis has been adjusted every time.So this research concentrates on two aspects:1.Adjusting traditional hypothesis of B-S economy by introducing Fractional Brownian Motion(FBM).Under the hypothesis of stock price following FBM process,pricing models for float striking price lookback Call and Put options,which are solved by Monte Carlo simulation,have been established.2.Introducing heterogeneity.By comparing simulation prices to formula prices,it illustrates how to estimate the parameter J to make simulation prices equal to formula prices.Then the heterogeneous investors trading with lookback options have been simulated.The main conclusions are the following:1.The lookback option prices are affected significantly by H,which shows that the characters,which are not depicted by Brownian Motion,are not negligible.2.Under the B-S hypotheses,simulation prices equal to formula prices when using the current stock price as J.3.The investors,which invest according to Goldman's formula with the current stock price as J,lose money and retreat from markets eventually.4.Lookback option prices are concerning to underlying properties' return,so riskneutral is not true.