Pricing Of Exchange Options Based On The Fractional Brownian Motion

In financial analysis, option pricing has been the focus of attention. Since the classic B - S formula since birth, around B - S formula made by various extension and improvement has been emerging. Classic B - S formula is on the assumption that the underlying assets follow standard Brown motion .European option to discuss the option pricing formula. Subsequently, the underlying assets to satisfy the coefficient of SDE has assumed a modified, is no longer a constant coefficient of related with the time that is rather function of time, have assumed coefficient is random, and the process of joining jump, etc. In recent years, have author postulates that satisfy the underlying assets in is the standard of SDE of time sport, but the score sport. Time See, scores from empirical closer to the time the actual sports market, but therefore bring difficulty is: the market is a arbitrage. For option pricing has brought the fundamental difficulties. This paper considers the fractional Brownian motion environment, then exchange options pricing equation and pricing formula when Hurst index H∈(1/ 2,1/ 3), and exchange options pricing equation and discussed the numerical solution when Hurst indexH =1/ 3.This paper first chapter presents the concept and pricing formula of options , the concepts and pricing formula of exchange options; The second chapter presents the fractional Brown motion and the approximation theory of fractional Brown motion in financial problems, and introduced the theory of partial differential equation numerical solution method; The third chapter is the core content of this article, firstly introduces submartingale process based on approximation theory of fractional Brown motion that is not a submartingale process, and then when Hurst index H∈(1/ 2,1/ 3) and when Hurst index H = 1/3two circumstances is derived using construction portfolio of exchange options, and the pricing equation is obtained the pricing equation pricing formula. Due to situation pricing equation is relatively complex, not directly ask out pricing formula, so this text only give two types of solution analysis and numerical solution analysis.