Research On VIX Option Pricing Under Non-affine Heston Stochastic Volatility Double-jump-diffusions Model

stochastic volatility jump-diffusions(SVJ)model is attracting more and more attention from financial academic and industrial world in recent years,but most of them consider jump items with the jump amplitude of normal distribution,on the basis of affine stochastic volatility model.There are two problems with such a model:(1)Affine stochastic volatility model can't accurately describe the dynamics of underlying asset price fluctuations;(2)most of the literature assumes that the return jump follows a normal distribution;whether this is necessary and whether there is a better distribution assumption remains to be investigated.Existing empirical studies show that the non-affine stochastic volatility model surpasses the traditional affine stochastic volatility model and can more fully describe the dynamics of underlying asset price fluctuations.In addition,not only the underlying price trend there is a sudden and large change,its volatility also has similar jump situation.Therefore,this paper selects the nonaffine stochastic volatility model with double jumps as the underlying asset price base model,and studies the influence of different jump amplitude distribution combinations as well as the synchronization and asynchronization of jumps on the pricing performance of the model in the options market.First,this paper explores the statistical performance of non-affine stochastic volatility model with double-jump.Three important distributions(normal distribution,exponential distribution,and double exponential distribution)widely appeared in the existing literature were selected to study the influence of various possible combinations of jump amplitude distribution on the market performance.Secondly,the options pricing formulas under the non-affine stochastic volatility model with synchronous and asynchronous double-jump are derived.Since non-affine stochastic volatility model has no analytical expression of characteristic function,this paper deals with the partial differential equation of logarithmic price characteristic function of underlying assets by perturbation method,and derives the analytical approximation of characteristic function.Then,the Fourier-cosine option pricing method is used to derive the option pricing formula under the non-affine random volatility model with jump.Finally,the above research work is applied to the option pricing of VIX index.The results show that:(1)In terms of option pricing performance,the non-affine stochastic volatility model with double jump is superior to the non-affine stochastic volatility model without jump.(2)The stochastic volatility model with asynchronous double jump is superior to the stochastic volatility model with synchronous double jump.(3)The Heston nonaffine stochastic volatility model with double jump can obtain the optimal performance of option pricing under the assumption of the combination of exponential distribution and double exponential distribution.The innovations of this article are as follows:(1)The return jump and variance jump are added to the non-affine Heston stochastic volatility model,which can not only capture the dynamic variability of continuous fluctuations,but also describe the discontinuous jump performance when major events occur;thereby further improving the performance of the model.(2)Combined with the perturbation method and the Fourier-cosine method,this paper solves the problem that it is difficult to get the analytic formula of option pricing under the non-affine random volatility model with jump.