Pricing And Hedging Of VIX Options Under The GARCH Models

The Cboe VIX options,traded at the Chicago Board Options Exchange,are straightforward and the most important tool for managing volatility risk in the U.S.market.An extensive literature study the pricing of VIX options using continuoustime volatility models.Of course,the pricing of derivatives is the most fundamental issue in financial engineering.Less attention,however,is devoted to discrete-time volatility models,such as GARCH,for pricing VIX options.GARCH models are wellknown,simple,and widely used for modeling volatilities.In fact,discrete-time GARCH models are used for pricing VIX futures.Therefore,the application of GARCH models to VIX options pricing would be a logic next step and worthwhile.Furthermore,the practically important issue of hedging VIX options is not featured in previous studies,to the best of our knowledge.The hedging of options arguably may be more important to options’ market makers than the pricing of options.Therefore,we investigate the pricing and hedging of VIX options under GARCH models extensively in this work,which no doubt is importance theoretically and practically.Our study is inspired by the recently proposed GARCH pricing of VIX futures by Guo and Liu(2020).The Guo-Liu approach is the first to utilize two of the most important GARCH models,namely the symmetric GARCH(1,1)model(G11)and the asymmetric Glosten-Jagannathan-Runkle GARCH(1,1)model(GJR),to price VIX futures.The Guo-Liu approach uses only basic variables,namely S&P 500 index and the market VIX,and therefore it is essentially ab initio or from first principles.Unlike others approaches,the Guo-Liu pricing is out-of-sample,can be applied in real time,and is highly efficient.In this work,we extend for the first time the Guo-Liu approach to the pricing of VIX options.Our studies are presented in three chapters.In Chapter 3,we propose to extend the Guo-Liu approach to price VIX options.Unlike classical option pricing methods,we do not directly deal with the underlying VIX.Instead,we model the S&P 500 index,the options of which are used to compute the VIX index,by GARCH models.With GARCH parameters,we derive an equation that relates VIX to the variance of S&P 500 index and GARCH parameters and use it to compute VIX as well as VIX options.The key innovation of the Guo-Liu approach is to use VIX to extract riskneutral GARCH parameters and then price VIX futures.We suggest that the riskneutral GARCH parameters from VIX can also be utilized to price VIX options.This will be our first benchmark pricing model.We propose to extract risk-neutral GARCH parameters from VIX futures and then price VIX options.Specifically,we price VIX options in two ways.In the first approach,we use the generalized Fourier transform(FTI method),which is commonly used in continuous-time volatility models.We derive analytical formulas for VIX option price under G11 and GJR as functions of the variance of S&P 500 index and VIX at expiration.In the second approach,we propose to obtain the numerical solution of VIX option prices via Monte Carlo simulation or Monte Carlo integration(MCI).Importantly,we propose several method to speed up Monte Carlo and make it feasible in practice.As the Guo-Liu approach,our porpsed method is out-of-sample,can be applied in real time,and is effiecient.Empirically,we conduct our study of VIX options pricing with all trading data in 2019.The filtered data set includes 25,630 calls and 14,528 puts and is extensive in terms of coverage.The options mature from 10 to 186 calendar days,with moneyness between 0.39 and 3.24 for calls and 6.93 and 0.57 for puts.By comparing the pricing results of the two methods,we find that the FTI method is not only timeconsuming in calculation,but also has numerical errors.About a quarter of the option prices are set as negative values,most of which are deep-out-of-money options.The MCI technique,on the other hand,is very computationally efficient,and the G11 and GJR models(Fut-GJR-MCI/Fut-G11-MCI),which extract parameters from VIX futures market prices,have the fewest out-of-sample pricing errors for VIX call and put options.In Chapter 4,we propose two kinds of benchmark models for VIX option pricing that directly model VIX for further comparative research.The traditional Black-Scholes-Merton(BSM)technique assumes that VIX follows geometric Brownian motion.The innovation of this paper is to propose four kinds of predicted volatility in addition to the basic historical volatility and implied volatility.In the second,it is assumed that VIX is characterized by random volatility.We directly model the return of VIX using GARCH(G11 and GJR)model,estimate the model’s physical parameters using historical VIX data,and extract G11 and GJR risk-neutral parameters using VIX futures prices.There are two methods to calculate VIX option prices,based on physical and risk-neutral parameters.One is Monte Carlo method,hereinafter referred to as GARCH-MC method.The other is to obtain the expected volatility of VIX through GARCH model and price VIX options with BSM formula,which is referred to as the GARCH-BSM method.The empirical option data is identical to that in Chapter 3.The best performing BSM benchmark model is implied volatility,which is followed by predicted volatility using GJR or G11 parameters retrieved from VIX futures and greatly outperforming historical volatility.These findings imply that the risk-neutral GARCH parameter may be used to better forecast VIX volatility.In addition,when compared to the model in Chapter 3,the Fut-GJR-MCI/Fut-G11-MCI model exhibits lower price errors in more option groups.In the second type of benchmark model,we find that the pricing performance of the model is relatively poor under the GARCH-MC method based on normal random innovation.Under the GARCHBSM method,we find that the volatility predicted by the G11 model with riskneutral parameters has a relatively small pricing error for VIX call options,and its performance is even better than that of the Fut-GJR-MCI/Fut-G11-MCI model in Chapter 3 in multiple option groups,which may be a problem worth further discussion.Yet,the price error of VIX puts is worse.Overall,the Fut-GJR-MCI/FutG11-MCI model outperforms the others in more option groups.Chapter 5 studies the dynamic hedging of VIX options.We derive the VIX option delta formula based on MCI method,and compare the delta values calculated by pathwise method and finite difference approximation method.The results of the two approaches are relatively comparable,but the pathwise method has the benefit of having an unbiased delta estimator and a high computation efficiency.Since the pathwise technique cannot compute gamma,we develop the finite difference approximation formula for gamma,which is a central difference estimator.The benefit of utilizing the MCI technique to compute the Greek value is that by performing a Monte Carlo simulation,options prices on five underlying asset values can be calculated and the delta and gamma may be obtained at the same time.Most notably,we offer a more acceptable and precise measure of delta hedging error,known as the single option hedging error,which is more accurate than the delta hedging error described by Bakshi,Cao,and Chen(1997).It is worth noting that our dynamic hedging scheme can in principle be utilized in real-time hedging of VIX options,too.To improve the empirical analysis of dynamic hedging of VIX options,we suggested determining the options price sequence based on maturity and exercise price.The options all expire in 2019,and 11,433 call options and 10,380 put options are ultimately obtained.For the hedging research,five models with small price errors in Chapters 3 and 4 were chosen,and the hedging position was adjusted every7 calendar days.We show that,assuming the VIX is a traded underlying asset,the Fut-GJR-MCI model outperforms the top three benchmark BSM models in hedging calls and puts,and that this finding is robust for alternative positioning intervals.Yet,when VIX futures are used as hedging assets,the Fut-GJR-MCI model outperforms the benchmark BSM model on call options while the benchmark BSM model outperforms on put options.This finding may deserve additional investigation.We make six contributions to the derivatives literature in this paper.First,we are the first to propose out-of-sample pricing of VIX options using two of the most important discrete GARCH models,namely GARCH(1,1)and GJR GARCH(1,1).Our approach is easy to understand and straightforward.Second,we propose to extract risk neutral GARCH parameters from the current market price of VIX futures.As a result,our pricing of VIX options is by definition out-of-sample and can be utilized in real time.Third,we propose a suite of techniques to speedup Monte Carlo simulation,which has wide implications for the application of Monte Carlo in other cases.Fourth,we suggest a better error measure than those of Bakshi,Cao and Chen(1997)to gauge dynamic delta hedges.Fifth,we propose to forecast the volatility of VIX via GARCH models and use BSM formulas to price VIX options.Sixth,we suggest to model VIX directly with GARCH models and use them as a second benchmark.Our study contributes to the literature of VIX options,Monte Carlo,and derivatives theory,provides traders with efficient and real-time tools for pricing and hedging VIX options,and will serve as a foundation for design and trading of volatility index and volatility derivatives in China in the future.