VIX Option Pricing With Jump Self-exciting Effect

With the continuous deepening of world economic integration and the increasing degree of mutual influence of financial markets in various countries,the trading volume of various derivatives of the whole financial market is huge.Any local changes in the financial market will quickly spread to other area and have an impact.The complexity of investor transactions and hedging risks is increasing,and it is necessary to always pay attention to market sentiment and expected market changes,and adjust investment portfolios in a timely manner.In this context,the volatility index that characterizes the market investors’ expectations of future changes in asset prices and measures future market volatility has entered the investor ’s sight,and has become an important asset for investors to hedge market fluctuations and effectively control risks when constructing investment portfolios.The market transactions of various types of derivatives such as futures and options of VIX index are becoming increasingly active.Volatility index has also started to receive widespread attention from the academia and the industry.The Chicago Board Options Exchange(CBOE)faced the increasingly active over-thecounter volatility derivatives trading in the 1990 s.It first compiled and released the volatility index in 1993 and named it the VIX index.After perfecting the compilation method,the current VIX index has become the primary representative of the volatility index,the most important sentiment indicator in the market,and an important reference benchmark for market participants to judge the volatility of the US and even global financial markets.This paper analyzes and studies the VIX index through five parts.First,it introduces the related concepts and definitions of the VIX index,then summarizes domestic and foreign research trends on the VIX index.Second,in view of the characteristics of mean-reverting,volatility aggregation and jumping shown by the historical data of the VIX index,this paper uses independent modeling methods and selects the affine jumping diffusion structure for modeling with the Hawkes process with self-exciting effect as the jump process and divides the model types into log-mean reverting model without self-exciting jump and log-mean reverting model with self-exciting jump.After that,constructs the log-mean-reverting constant volatility model(MR),the log-mean-reverting stochastic volatility model(SV),log-mean-reverting selfstimulating jump model(HJ)and log-mean-reverting stochastic volatility self-stimulating jump model(SVHJ)and derives the corresponding conditional eigenfunctions of these models and their option price expressions by Fourier transform method.Then,use market data for in-sample correction and out-of-sample prediction,and summarize the correction parameter results of the four models.According to the fitting error,forecast error,parameter term structure and implied volatility tilt of each model’s price,analyze the model’s description performance of VIX features and the modeling effect.Next,select the base date data,and perform numerical analysis on several parameters based on the model correction parameters obtained from the expiration date data in order to compare the model parameter sensitivity and summarize the relevant pricing characteristics.Finally,summarize each part of this article,and the application value of adding Hawkes jump process into VIX option pricing model.The research results in this paper show that compared with the log-mean reverting constant volatility model and the log-mean reverting stochastic volatility model without jumps,adding parameters to construct the model from the perspective of jump self-exciting can greatly reduce the fitting error,improve the model’s ability to predict the price of long-term option contracts and implied volatility surfaces,and observe and extract changes in market expectations from more perspectives,which makes the study of VIX index modeling methods further applicable for risk management and investment portfolio construction.