Fractional Brownian Motion And Its Application In Option Pircing

1973, Fisher Black and Myron Scholes used the principle of no-arbitrage as abasic premise and put forward a series of assumptions, derived the European optionprices to meet the partial differential equations, the partial differe ntial equations inessence is a heat conduction equation. They plus exercise price as the boundaryconditions, solving the partial differential equations of the European option priceformula, obtained the famous Black-Scholes formula.The Black-Scholes formula has a very important hypothesis that the asset pricesare subject to the geometric Brown movement so that these prices are independentrandom walk, the price of the underlying assets has no memory. But whether practiceor scholars theory research found that in the securities market, the prices of all sorts ofassets exist long memory, namely the price of a time in long time also may be on theprice of the later produces an effect, the price does not follow the geometric Brownmotion, but instead of the more general geometric fractional Brown motion.This article contains five parts:The first chapter is the introduction, this paper introduces the background andsignificance, the innovation and the insufficiency.The second chapter introduces the definition of the Brownian Motion and itsnature, and then discuss that why primary asset prices follows the Brown movementare unreasonable, and then puts forward the improvement, it is concluded that theunderlying asset price is subject to the geometric Brown motion, with this basisassumption, we sum up the Black-Scholes formula’s derivation.The third chapter discusses the Fractional Brownian Motion and its nature, thenthe fractional difference, fractional differentiation, basic knowledge of the fractionalderivative. The most important is, this paper introduces the basic definition of theactuarial options pricing method and its advantage compared with traditional methods.Use the option actuarial pricing method, we sum up the fractional Black-Scholes formula.The fourth chapter use computer program (Matlab) simulation and comparison thedifferent H (ie, the Hurst exponent) of the option price and its various hedgecoefficient.The fifth part conclude different Hurst exponent H’s effect on fractionalBlack-Scholes formula.