The Pricing Problem Of Butterfly Option In Stochastic Volatility Models

Option which is a quite important financial derivative in modern financial markets has a very long history.Since 1973,when Black and Scholes put forward the classical B-S option pricing formula,the problem of pricing financial derivatives has increasingly become one of the most widely studied work in Mathematical Finance.At the same time,as people show different requests for risk and payoff,there appear a large number of exotic options,such as Butterfly option,Binary option,Barrier option……,and the sort of exotic options is still increasing.Naturally,the pricing problem of these exotic options has become one of the key topics in modern financial fields,because they are more in tune with the risk or payoff demand of the investors,and it is well-known that the pricing problem of these exotic options has academic value and financial significance.Although our B-S model is very popular for its simple formula and easy calculation,the price of option in this model often has a big difference from that in the market,because the assumption of this model is too ideal,and a lot of researches have shown that the market price of option implies the "smile" effect of volatility.Therefore,people are continuously trying to modify the assumptions of the model in order to make the model better agree with the real world.One of the significant improvements is that people take the volatility of the underlying asset as a stochastic process correlated with the underlying asset instead of taking it as a constant,and this kind of model is so-called a Stochastic Volatility Model.In this article,we will consider the pricing problem of butterfly option in Stochastic Volatility Model,and we will assume the price of the underlying asset is in the Hull-White Stochastic Volatility Model and the mean-reverting Ornstein-Uhlenbeck Stochastic Volatility Model respectively.For the former one,when the Brown motion that drives the underlying asset is uncorrelated with the one that drives the volatility of the underlying asset,we will apply the martingale method and the Taylor expansions to obtain the pricing formula;when they are correlated with each other,we will use the slochastic partial differential equation pricing method and the Antithetic Monte Carlo Simulation method to get some numerical results of this model.For the latter one,apart from applying the stochastic partial differential equation pricing method as the former one,we will also take the martingale method combining with the characteristic function method and the Inverse Fourier transform to get a closed form solution of the price of the butterfly option in this model.