跳自刺激效应下的VIX期权定价研究

针对VIX指数的均值回复、波动率聚集特征以及实证研究最近发现的跳跃自刺激性,本文采用具有自刺激性的Hawkes过程对VIX指数的跳跃进行建模,进而构建仿射跳跃扩散模型用于VIX期权定价,得到Hawkes跳跃扩散过程的条件特征函数,然后在风险中性定价框架内采用傅里叶变换方法推导出VIX期权的价值表达式。实证结果表明,本文模型不仅能克服一般均值回复模型拟合误差大的缺点,且能产生正的隐含波动率倾斜和隐含波动率微笑;另一方面,由于考虑了跳跃的自刺激,本文模型在泊松跳跃均值回复模型基础上进一步改善了VIX期权价格的预测效果。

The valuation of VIX options with self-exciting jumps

The VIX index is launched by the Chicago Board Options Exchange(CBOE)based on the implied volatilities of the S&P 500 stock index options.The VIX index,also known as the“fear index”,reflects market participants′expectation of volatilities in the US stock market over the next 30 days.CBOE launched VIX futures in 2004 and VIX options in 2006.As important financial derivatives in the world,VIX futures and VIX options are effective tools for financial market participants to hedge market volatility and manage market risks,and provide more investment opportunities for asset managers.Since 2015,VIX has been the second largest underlying in CBOE following the S&P 500 index.Currently,the VIX option market is very active in trading and has a large market scale,e.g.,the trading volume is nearly 700,000 per day in 2018.While the exchange-traded VIX options are booming,the over-thecounter VIX options is also growing rapidly.Therefore,in order to maintain the healthy and stable development of the VIX option market,proposing a reasonable model for fairly pricing VIX options is not only an important academic issue,but also widely concerned by investors,market makers and regulators.There are two main methods for pricing VIX options:consistent modeling and standalone modeling.The consistent modeling method refers to setting up the joint dynamic processes of the S&P 500 index and its volatility,and then obtaining the dynamics of VIX index and the value of VIX derivatives based on the underlying prices.However,the standalone method directly gives the dynamics of VIX index.Because of the tractability of the standalone method,this study will adopt it to price VIX options.It has been widely found in the existing literature that the VIX index has the mean-reverting feature and it jumps from time to time.Recent empirical studies have further found that the jumps of the VIX index come in cluster.In literature,Poisson processes in which events occur independently are applied to model the arrivals of jumps so that it cannot describe the characteristics of jump clustering.From the perspective of jump,this study uses the Hawkes process with self-exciting effect to model the jump clustering of VIX index in the standalone modeling framework,and then constructs the Hawkes jump diffusion process with mean-reverting property to study VIX option pricing.Since the proposed Hawkes jump diffusion process belongs to a class of affine processes,we are able to obtain the analytical expression for its conditional characteristic function.Then,based on the classic risk-neutral pricing(also known as martingale pricing)method,we evaluate VIX options by the Fourier transform and its inverse transform techniques which are commonly used in option pricing.In the end,we derive the analytical expression for the theoretical value of the VIX option.In order to evaluate the pricing performance of the proposed model(HJD),we compare it to the mean-reverting model without taking into account the jump(LR)and the mean-reverting model with Poisson jump(LRJ)in terms of the in-sample fitting accuracy and the out-of-sample forecasting ability.The empirical results show that,firstly,the fitting error and prediction error of the HJD model are much lower than the LR model,which to a large extent overcomes the shortcomings of the LR model with large fitting error and low prediction accuracy.Secondly,compared to the LRJ model,although the HJD model does not significantly bring down the fitting error,its prediction error,especially the mean absolute error,is greatly reduced up to 26.14%.Finally,given multiple expiration dates and strike prices,the HJD model can produce positive implied volatility skew and implied volatility smile that LR model fails to do,and it can produce the volatility surface matching market data.In summary,from the perspective of pricing performance,the model considering the jump characteristics of the VIX index can substantially reduce the fitting error;from the perspective of the out-of-sample prediction,the model considering the jump clustering characteristics can improve the forecasting ability of the VIX option price.As a result,compared to the traditional models,the proposed VIX option pricing model based on self-exciting jumps in this paper has stronger price fitting ability and prediction ability because it incorporates the clustering jumps into the traditional mean-reverting models.The proposed model can be also applied to the pricing of other volatility derivatives.