Exotic Option Pricing And Application Based On Stochastic Volatility Model

Options and futures both are basic derivatives,they are very important financial tools in the financial market.Options endow the option buyer the right to buy or sell certain amount of underlying in specific term.Inorder to obtain this right,option buyer must pay the option seller certain amount of fee which is called option premium or option price.As a basic dertivative in the foreign financial market,the main reasons for the rapid development of option market is investors have strong demand for the high efficient risk management and pricing tools.Options are not only important in the function of asset allocation and risk management for the exchange,but also important to function of liquidity,pricing and fund rasing for the exchange.In recent years,the capital market of China has experienced rapid development,the size of market value is the second largest in the whole world.But the structure of the financial market is simple.On this back ground,shanghai stock exchange lauched 50 ETF option on Feburary 2015.The 50 ETF option market signals the advent of the option epoch in the capital market in China,meanwhile it also a new era of of diversified investment and risk management.Black-Scholes model is an important model for option pricng.This model avoids the assumptions on the risk preference of investors and market equibrium,but only based on the arbitrage-free assumption.Black-Scholes model has been applied to many areas,lots of researcher have been studying the accuracy and robusty of Black-Scholes model.Meanwhile,some experts raised different opionions on the defects of Balck-Scholes model,they also extent the model inorder to rectify its defects.Concerning the problem of Black-Scholes model,lots of theories are based on the discussion of the reliability of the assumptions of the Black-Scholes model.Some researchers find that the assumptions on the Black-Scholes model are too strict,especially the assumption on the constant volatility which can not explain the data from the option market.Hence,lots of researchers attempt to extend the volatility assumption on the Black-Scholes model by replacing the constant volatility with stochastic volatility.There are several problems on this research method.Firstly,the process of volatility is different of the process of the underlying asset.The volatility is a process which is unobservable.Secondly,volatility tends to fluctuate at a high level for a while and then fluctuate at low level for a similar period.It will mean revert many times during the life of a derivative contract.Because of the fat tail characteristic of the return of the underlying asset price and the clustering of the volatility of the underlying asset price,this thesis studies the pricing formulae and parameter estimation of several typical options under stochastic volatility.The main results of this thesis are shown as below:Under the framework of fast mean reverting stochastic volatility model,this thesis studies the pricing method of collar option,chooser option,geometric average Asian option and lookback option and improve the pricing method.Financial market data shows that the volatility of the underlying asset price will fluctuate around its mean level randomly,hence using stochastic volatility model will be more appropriate for modeling the price of the risk asset.Stochastic volatility reflects the complex process of the volatility as time passes.Empirical studies show that the volatility of risky asset price is not only changes randomly but also reverts in high frequencies.Based on this fact,some researchers propose the fast mean reverting stochastic volatility model.This kind of model assumes that in a relative short time scale compare to the maturity of the option contract,volatility is a function of a fast mean reverting diffusion process.Compare to the constant volatility model,the partial differential equation is more complicated in the stochastic volatility framework,so it will be more difficult to obtain an analytical formula.This thesis applies singular perturbation analysis method on the pricing problem of collar option,chooser option,geometric average Asian option and lookback option,and derive the approximate analytical formulae for these options.Based on the analytical formulae of the collar and chooser option,greek letters of these two options are also discussed.Estimating the parameters of the fast mean reverting stochastic volatility model by implied volatility data and apply this result to option pricing.Implied volatility is the value of the volatility parameter that must go into the Black-Scholes model to match real option price obtained from financial market.The moneyness of the option refers to the extent of in-the-money or out-of-money,it is the ratio of strike price to the spot price of the underlying asset.The variation of implied volatility is reflected by the option moneyness and the term structure.This thesis studies the linear relationship between the implied volatility,option moneyness and the maturity of the option.By using this linear relationship the parameters of the fast mean reverting stochastic volatility can be more easily estimated.In this study,SPDR & S&P 500 ETF option implied volatility and uderying price data was used to calibrate the model pramameters.The advantage of this method is that the number of parameters are reduced and the process of parameters estimated is simplified.The application of importance sampling Monte Carlo simulation in option pricing under stochastic volatility.One of the main advantages of the Monte Carlo method in the application is that it can improve the efficiency of financial derivatives with higher dimensions,such as derivative pricing under stochastic volatility model.Because the number of state variables are greater than two.The method of importance sampling is one of the widely used variance reduction approaches.Unlike the other variance reduction methods,importance sampling is based on the idea of changing the underlying probability measure from which paths are generated.The importance sampling technique in this thesis is based on the approximation of the option price which derived from the pricing partial differential equation by the singular perturbation.The numerical experiment for the collar option and chooser option demonstrates the significant reduction of the option price variance from the basic Monte Carlo simulation to the importance sampling Monte Carlo simulation.