Pricing Of Barrier Options And Lookback Options Based On Fractional Brownian Motion

In economics, Options is the core of risk management tool。Since 1973, the famous Black -Scholes model appears, the options pricing theory and applications have developed rapidly , but assumption of the classical Black ? Scholesmodel is too strict and there are some of friction in actual financial markets. Especially with the development of science and technology, methods of transfer information updated constantly and means of describe financial markets also improved rapidly. And with the proposal of the fractal market hypothesis, many studies have proved that financial markets have fractal characteristics . Therefore, fractal market assumptions to be taken seriously, options pricing theory based on fractional brownian motion has gradually become a new hotspot.This paper described the classic barrier options and look options on derivatives markets properly, and analysized the limitations of standard brownian motion in financial market, used the advantage of fractional brownian motion to depict assets movement, studied barrier options and look options which the underlying asset price movement obeys Hurst index range belongs to (1 3,1 2) , had some new results. There are two main innovations.Firstly, this paper used theory of martingale and no arbitrage portfolio method, had partial differential equation of the barrier options under the brownian motion which in special Hurst index , fined its solution through transformed the partial differential equation with its boundary condition which barrier options is satisfied to the standard heated equation , obtained expressions which corresponded the European option.Secondly, this paper treated the maximum price of the asset in the validity period in market, used similar no-arbitrage portfolio to avoid risk, had an equation under the brownian motion which in special Hurst index through many times transformations and extended the boundary condition , got the analytical solution of lookback options.Barrier options and lookback options are two typical exotic options, this research broaden and enrich the theory of options pricing, also provide a reference to investors in the actual operation.