Numerical Calculation And Empirical Analysis Of American Options Pricing Based On Fractional Brownian Motion

In the financial markets, stochastic model driven by fractional Brownian motion canbe better explain the more and more phenomenon-Economics of scale, Seasonal effects,Leptokurtic and so on. This paper studies numerical solution of American options pricingbased on fractional Brownian motion. It has great significance in theory and practice.This article is divided into five sections. In the first section, we briefly discuss thedefinition and classification of the options, and introduce the option pricing theory in thestandard Brownian motion and fractional Brownian motion. It also reviews the research ofthe American options pricing. In the second section, we discuss the concept and nature ofthe fractional Brownian motion. Based on a fundamental result on theL~2-approximationof the noise by semi-martingales, then we obtain the determining solution of American putoptions when Hurst index H∈(1/3,1/2) and it is expanded H∈(1/n,1/(n-1)). Then weintroduce the stochastic simulation of fractional Brown motion and numerical solution ofAmerican options. In the third section, firstly we explain the basic idea of the Monte Carlosimulation, and then simulate the increments of fractional Brownian motion and theprocesses of the underlying assets with extended Maruyama notion. Secondly, we get theprice of American put options with the finite difference method when Hurst indexH∈(1/3,1/2). Then we give a numerical example. Thirdly, this method is used toAmerican options pricing when Hurst index H∈(1/4,1/3). In the fourth section, we usethe R/S analysis method to select two stocks which has the similar Hurst index in theChinese stock markets, and give the empirical analysis. In the fifth section, we summarizethe advantages and inadequacies of this article.