The Research On VIX Option Pricing

The researches on VIX volatility index and its derivative securities pricing have been hot topics in the field of financial mathematics research in recent years.This article uses Hawkes jump diffusion model to describe the dynamic change of VIX index,and studies the option pricing problem of VIX index under this model.The first chapter of this article introduces the development of the VIX index,VIX options,and research status at home and abroad.Chapter 2 first reviews the current research status of the dynamic model of VIX jump diffusion and the defini-tion and application development of Hawkes process.Secondly,the jump diffusion process and Hawkes process are combined to build a dynamic model of Hawkes jump diffusion with VIX index.The model consists of two parts:one is the mean-reverting square-root stochastic process,which is used to describe the change characteristics of the mean return of the VIX index;the other is the Hawkes process,which is used to describe the jump characteristics of the VIX index,and its arrival intensity follows an exponential decay process.Based on this model,using the properties of Ito formula and martingale,through the Fourier transform theorem and inversion formula,the pricing formula of European call options with VIX index as the underlying asset is given.Finally,it is assumed that the size of the jump obeys the exponential distri-bution,and the simulation verification is performed under the given parameters,and numerical examples of option pricing are given.In chapter 3,diffusion disturbance is added to jump arrival intensity of the Hawkes jump diffusion model.The jump arrival intensity of the Hawkes process is affected by both the jump term and the Brownian motion,which makes the model more general and consistent with reality.Using a similar method in Chapter 2,the pricing formula of European call options with the VIX index as the underlying asset under this model is given and and the simulation verification is performed,showing the impact of the Hawkes process on the dynamic change of the VIX index after the Brownian motion term is added.