Asymptotic Properties For Stochastic Volatility Models And Applications

In the financial market,stochastic volatility models are extensively used to price the financial products by traders and risk managers,especially to price the options.In 1973,Black and Scholes used stochastic model to describe the stock price process and proposed the formula for pricing European option,which laid the foundation of option pricing theory.However,in the practical applications,there are some great limitations in the stochastic model proposed by Black and Scholes.The main reason is that the assumption of constant volatility are not consistent with the real market.Therefore,in order to remedy the above mentioned drawbacks,researchers have proposed many other models.One approach,known as stochastic volatility model,is to assume that the volatility is also characterized by stochastic process.These models fit the market volatility very well,and are widespread studied.The Hull-White and Heston stochastic volatility model are the most popular in this field.In this thesis,we focus on the asymptotic properties for four kind of Euler-Maruyama discretization of a class of Hull-White stochastic volatility model.On the other hand,we also study the asymptotic properties of Heston model with mod-? method,and then apply the obtained results to the option pricing.The thesis is organized as follows:In the first chapter,we introduce some basic concepts and methods used later:large deviation principle,mod-? method,Euler-Maruyama discretization,option,implied volatility and risk-neutral measure.Then,we state the background and our motivation.In the second chapter,we give our main results and applications.In the third chapter,we concentrate on the moderate deviations principle for the four kind of Euler-Maruyama discretization of a class of Hull-White model.In the fourth chapter,we apply mod-? method to study the sharp large deviation principle and moderate deviation principle of logarithmic asset pricesX_t in Heston model under the small and large time,and then apply the results to the European option pricing.In the fifth chapter,we recall the main ideas of the thesis,and postulate the further research work.