Extremum Options Pricing Under A Non-Affine Stochastic Volatility Model

In recent years,the volume of option is increasing day by day.With the rapid development of financial market,many investors are attracted to option products and take part in.China’s options business was introduced relatively late,but the pace of development is also fast.Constantly,many financial institutions innovate options and design a variety of exotic options to satisfy the needs of different investors.Among them,extremum options are a class of multi-asset portfolio exotic option,whose payoff depends on the maximum or minimum value of multiple underlying assets at maturity.Extremum options are widely traded on organized exchanges and in equity,fixed income,currency and commodity markets.In the enterprise financing and bond pricing models,this kind of options are widely used for speculation,hedging correlation risks,and evaluating real assets.Extremum options can make investors gain more income or minimum loss,so extremum options are favored by investors.It is of practical significance to explore the pricing of extremum option.About options pricing study,Black and Scholes proposed Black-Scholes model(BS model)in the study of option pricing,which provides a theoretical price that both sides of option trading are convinced of.However,BS model also exposes many shortcomings,for example,it assumes that the reverse rate of the underlying asset follows the normal distribution of logarithms.Moreover,BS model assumes that the volatility of the underlying price is constant,which is inconsistent with the “volatility smile or smirk phenomenon”in the actual situation.In order to solve these problems,Heston stochastic volatility model is proposed,which describes the asset volatility process with time-varying volatility.The pricing result is better than BS model.But single factor stochastic volatility model also has defects,the thick tail of asset payoff distribution and the persistence of volatility cannot be fitted at the same time.Multi-factor stochastic volatility model describe the uncertainty factors of volatility better,and the dynamic characteristics of volatility to improve the flexibility of the stochastic volatility process.In recent years,scholars have found that the volatility process of the underlying asset is not affine form,and the characteristics of nonlinear regression is presented.Therefore,It is of great significance to study the pricing of extremum options under the non-affine volatility model.On the basis of previous studies,this paper assumes that the financial market is friction-free and arbitrage-free,and propose a more reasonable option pricing model.Then,the analytical formula of extremum option and geometric Asian extremum option pricing is derived and the option price is calculated by using the relevant option pricing theories and numerical methods.Firstly,in view of time variability of volatility and nonlinear characteristics of volatility process,the non-affine stochastic volatility model of n assets is established,and the extremum option of two assets is studied.the joint characteristic function of logarithms of two asset prices are derived by using the Feynman-Kac theorem and the one-order Taylor approximation expansion.the semi-closed analytical pricing formulas of two asset extremum options are derived by using measure transform technique and Fourier inverse transform method.Then the pricing formulas are extended to the case of extremum options with multiple assets.the pricing results of extremum options under affine model,non-affine model and Black-Scholes model are analyzed,as well as the effects of some model parameters on options by integration method.Numerical results show that the analytical formulas obtained has higher computational efficiency and accuracy than Monte Carlo simulation method.In addition,the results of sensitivity analysis show that the non-affine model is more effective than other existing models in capturing the effects of option pricing volatility.Secondly,the extremum options are extended to the Asian extremum option.By using the model of this paper,the multi-time discrete joint characteristic function and related pricing formula are derived.In numerical examples,Asian extremum options and extremum options are compared,and the effect of model parameters on the Asian extremum option is analyzed.The results show that geometric Asian extremum options have lower price and are more conducive to hedging than standard extremum option.In sensitivity analysis,the change of model parameters has little impact on option prices,which is more conducive to resisting market risks and preventing option products from being manipulated artificially.