GARCH pricing and hedging of VIX options

• Published date: 01 June 2022
• Author: Qiang Liu,Yuhan Jiao,Shuxin Guo
• Date: 01 June 2022
• DOI: http://doi.org/10.1002/fut.22318

Received: 12 September 2021

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Accepted: 25 January 2022

DOI: 10.1002/fut.22318

RESEARCH ARTICLE

GARCH pricing and hedging of VIX options

Qiang Liu

1

|Yuhan Jiao

1

|Shuxin Guo

2

1

School of Finance, Southwestern

University of Finance and Economics,

Chengdu, China

2

Department of Finance, School of

Economics and Management, Southwest

Jiaotong University, Chengdu, China

Correspondence

Shuxin Guo, Department of Finance,

School of Economics and Management,

Southwest Jiaotong University, No. 111,

Section 1, North Erhuan Rd, Chengdu,

610031 Sichuan, China.

Email: shuxinguo@home.swjtu.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Numbers: 71701171,

72073109, 72071162

Abstract

We are the first to study the pricing and hedging of VIX options via Monte

Carlo (MC) under GARCH(1,1) and Glosten–Jagannathan–Runkle GARCH

(1,1) models. Our pricing is ab initio and out‐of‐sample and can be im-

plemented in real time. Importantly, we propose the so‐called single‐option

hedge error, a better measure than that of Bakshi et al., and suggest several

techniques to expedite MC by over 1000 times, which rivals the potential

speedup of quantum computers. Empirically, our proposed approach outper-

forms two types of benchmarks; further, the asymmetric Glosten–Ja-

gannathan–Runkle outperforms the symmetric GARCH. Overall, our paper

has both theoretical and practical implications.

KEYWORDS

error measure of delta hedging, GARCH, Glosten–Jagannathan–Runkle, Monte Carlo

speedup, out‐of‐sample pricing, VIX options

JEL CLASSIFICATION

G13, G12

1|INTRODUCTION

An extensive literature investigates the pricing of volatility or volatility index (VIX) options using continuous‐time

volatility models (Grünbichler & Longstaff 1996; Cao et al., 2019; Carr & Madan, 2014; Detemple & Osakwe, 2000;

Goard & Mazur, 2012; Jeon et al., 2021; Jing et al., 2020,2021; Lian & Zhu, 2013; Lin & Chang, 2009; Luo et al., 2019;

Mencia & Sentana, 2013; Pacati et al., 2018; Park, 2016; Wang & Daigler, 2011). Less attention, however, is devoted to

discrete‐time volatility models (such as GARCH models, e.g.). Furthermore, the practically important issue of hedging

VIX options is present in previous studies. These two aspects of research are both worthwhile. On the one hand,

GARCH models are well‐known, straightforward, and widely used for modeling volatilities. Furthermore, the pricing

of other VIX derivatives (i.e., VIX futures) by discrete‐time GARCH models has already been studied (Guo & Liu, 2020;

Huang et al., 2019; Wang et al., 2017). Therefore, there is no reason to ignore the possible application of GARCH

models to VIX options pricing. On the other hand, the hedging of VIX options, to some extent, may be more important

than the pricing itself and can be critical in options trading. Therefore, we study the pricing and hedging of VIX options

under GARCH models in this paper extensively.

Our study is inspired by the recently proposed GARCH pricing of VIX futures of Guo and Liu (2020). The Guo–Liu

approach is appealing in four aspects. First, it is the first to utilize the two most important GARCH models, namely the

symmetric discrete‐time GARCH(1,1) (G11) and the asymmetric Glosten–Jagannathan–Runkle GARCH(1,1) (GJR)

(Glosten et al., 1993), to price VIX futures. Second, the pricing only uses more basic variables, such as S&P 500 index

and the market VIX, and therefore it is essentially ab initio or from first principles. Third, the time‐tfutures price relies

on information up to time‐t, and consequently, the approach is out‐of‐sample in nature. Fourth, it can be utilized in

J Futures Markets. 2022;42:1039–1066. wileyonlinelibrary.com/journal/fut © 2022 Wiley Periodicals LLC

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real‐time pricing and is of practical implications because it depends only on a single VIX, namely the current market

VIX. Critically, these features of the Guo–Liu approach are not easily identifiable from other related works (e.g., Wang

et al., 2017; Yang et al., 2019; Zhu & Lian, 2012). Therefore, it is worthwhile to investigate whether we can utilize G11

or GJR to price VIX options.

1

Equally important, we also study the hedging of VIX options. It is well‐known that hedging is at least as valuable as

pricing with respect to options both in theory and practice. To the best of our knowledge, however, previous research

has not addressed the hedging of VIX options. In this paper, we initiate the study of the hedging issue and report the

empirical results of hedging VIX options.

We present our pricing and hedging analyses in three steps. To begin with, we reproduce closed‐form formulas for

variance and VIX at the maturity of VIX options under both G11 and GJR. Then, we propose to extract risk‐neutral

GARCH parameters from the current market price of VIX futures. Finally, we obtain the prices of VIX options via

Monte Carlo (MC) simulations. Our approach has economic implications as those of the Guo–Liu approach: the

proposed pricing, which is based on the two most important GARCH models, is ab initio or from first principles, is out‐

of‐sample, and can be utilized in real‐time trading.

Second, we focus on the dynamic hedging of VIX options. We derive “formulas”for deltas of VIX calls and puts.

Then, we propose an approximate formula for gamma. To reduce the estimation bias of gamma, we apply the

extrapolation technique to a central‐difference estimator. Further, we suggest obtaining five option prices necessary for

both delta and gamma estimators within one single MC run. This leads to two advantages: computational time‐saving

and much lower numerical errors inherent from random paths in the simulations of prices (Glasserman, 2003). Most

importantly, we propose a more reasonable and accurate delta hedging error measure for empirically gauging hedging

performance. We call it the single‐option hedging error, which is better than that of delta hedging defined in Bakshi

et al. (1997). It is worth noting that our dynamic hedging scheme can in principle be utilized in real‐time hedging of

VIX options, too.

Third, we propose new techniques to speed up the MC simulation that we employed in the pricing and hedging of

VIX options. It is known that traditional MC simulations, though very flexible, can be inefficient computationally. To

make MC applicable in real time, we propose a suite of significant speedup techniques. The two most consequential of

these are sampling from a precomputed pool of independent normal random variates and thread‐based parallel

computing. With a pool of 100,000 normal variates, daily time steps, and 100,000 paths for the options' average maturity

of 91 days in the data set, we estimate that our proposed speedup amounts to over 1000 times.

2

Incidentally, we find

that this speedup is comparable with the potential speedup of 1000 times theorized for MC on quantum computers

available 10 to 20 years in the future (according to Goldman Sachs and QC Ware

3

).

Empirically, we conduct our study of VIX options pricing and hedging 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. The out‐of‐sample pricing

performance of our proposed approach is quite good, compared to two types of benchmarks, namely G11 or GJR

pricing with risk‐neutral parameters extracted from VIX and the Black–Scholes–Merton (BSM) formulas.

In addition to using the historical “volatility”of VIX with BSM, we propose a mechanism to forecast volatilities of

VIX with either G11 or GJR parameters extracted from either VIX or VIX futures and using them to price VIX options

via BSM. With the BSM formulas, the volatility forecasted by futures‐GJR or futures‐G11 (i.e., GJR or G11 parameters

that are extracted from VIX futures) significantly outperforms the historical volatility. It is intriguing that in a few

cases, futures‐GJR‐BSM matches the pricing performance of futures‐G11‐MC (i.e., our proposed MC pricing with G11

parameters extracted from VIX futures). These results imply that we can utilize risk‐neutral GARCH parameters to

forecast the volatility of VIX better.

To empirically study the dynamic delta hedging of VIX options, we propose explicitly identifying an option series by

maturity and strike price. We obtain 11,433 call series and 10,380 put series that end at 1 of the 12 months in 2019. Our

hedge is rebalanced in intervals of 7 calendar days, provided that data is available. We find that futures‐GJR‐MC (i.e.,

our proposed MC pricing with GJR parameters extracted from VIX futures) outperforms the best benchmark BSM (i.e.,

1

In Section 2.1, we provide three reasons for the choice of G11 and GJR.

2

On a low‐end laptop running Windows OS, the average execution time for price and delta is as short as 64 ms per option for GJR with the data set

used in the empirical study.

3

Goldman Sachs and QC Ware Collaboration Brings New Way to Price Risky Assets within Reach of Quantum Computers (https://qcware.com/

news/press-release-april-29/).

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