Pricing the CBOE VIX Futures with the Heston–Nandi GARCH Model

• Date: 01 July 2017
• DOI: http://doi.org/10.1002/fut.21820
• Author: Yiwen Shen,Tianyi Wang,Yueting Jiang,Zhuo Huang
• Published date: 01 July 2017

Pricing the CBOE VIX Futures with the

Heston–Nandi GARCH Model

Tianyi Wang, Yiwen Shen, Yueting Jiang, and Zhuo Huang *

We propose a closed-form pricing formula for the Chicago Board Options Exchange Volatility

Index (CBOE VIX) futures based on the classic discrete-time Heston–Nandi GARCH model.

The parameters are estimated using several sets of data, including the S&P 500 returns, the

CBOE VIX, VIX futures prices and combinations of these data sources. Based on the resulting

empirical pricing performances, we recommend the use of both VIX and VIX futures prices for

a joint estimation of model parameters. Such estimation method can effectively capture the

variations of the market VIX and the VIX futures prices simultaneously for both in-sample and

out-of-sample analysis. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:641–659, 2017

1. INTRODUCTION

The idea of using derivatives o f market volatility to manage fina ncial risk can be traced

back to long before the Chicago Board Options Exchange (CBOE) developed its Volatility

Index (VIX). Brenner and Ga lai (1989) introduced a vola tility index (the Sigma inde x) and

discussed derivatives suc h as options and futures in relat ion to this index. Following thi s

idea, Whaley (1993) introdu ced the old version of the VIX, wh ich depended on the

inversion of the Black–Scholes formula. However, standardized derivative contracts on the

VIX were not available until the CBO E calculated the VIX on a model-fre e basis in 2003.

Since the introduction o f VIX futures in 2004 and of VI X options in 2006, volatil ity

derivatives have become a popul ar set of derivatives in the marke t, especially after the

subprime crisis.

Several models have been proposed for pricing VIX futures and other volatility

derivatives. Zhang and Zhu (2006) first studied VIX futures with the Heston model.

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Zhu and

Tianyi Wang is at the Department of Financial Engineering, School of Banking and Finance, University of

International Business and Economics, Beijing, China. Yiwen Shen is at the Department of Industrial

Engineering & Operations Research, Columbia University, New York. Yueting Jiang is at the HSBC Business

School, Peking University, Shenzhen, China. Zhuo Huang is at the National School of Development, Peking

University, Beijing, China. We are grateful to Bob Webb (editor) and an anonymous referee whose comments

substantially improved the paper. The authors acknowledge financial support from the National Natural

Science Foundation of China (71301027, 71201001, 71671004), the Ministry of Education of China,

Humanities and Social Sciences Youth Fund (13YJC790146), and the Fundamental Research Fund for the

Central Universities in UIBE (14YQ05).

JEL Classification: C19, C22, C80

*Correspondence author, National School of Development, Peking University, Beijing, China. Tel: 86-10-

62751424, Fax: 86-10-62751424, e-mail: zhuohuang@nsd.pku.edu.cn

Received December 2015; Accepted August 2016

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Proposed by Heston (1993) for option pricing.

The Journal of Futures Markets, Vol. 37, No. 7, 641–659 (2017)

© 2016 Wiley Periodicals, Inc.

Published online 9 November 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21820

Zhang (2007) also proposed a non-arbitrage model for VIX futures, based on VIX term

structures. In considering the possible jumps in log-returns, Duffie, Pan, and Singleton

(2000) proposed an affine jump-diffusion process for log-returns, which soon became a new

benchmark process in the asset pricing literature. On the basis of this process and its

modifications, Lin (2007) proposed a stochastic volatility model with simultaneous jumps in

both returns and volatility to price VIX futures, which yielded an approximation formula.

Sepp (2008) added jumps into the square root mean-reverting process to price VIX options.

Zhang, Shu, and Brenner (2010) included additional stochastic long-run variance into the

square root mean-reverting process that linked the VIX and VIX futures prices. By adding

jumps to the classical Heston model, Zhu and Lian (2012) found an analytical pricing

formula for VIX futures.

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Using intraday data, Frijns, Tourani-Rad, and Webb (2016) found

strong evidence for bi-directional Granger causality between the VIX and the VIX futures.

Despite the development of a significant literature on the stochastic process for volatility

derivatives, little attention has been paid to discrete-time GARCH family models. Studies on

volatilityderivativesunder the GARCHframework havemainly focused onequity option pricing

(e.g., Christoffersen, Jacobs, Ornthanalai, & Wang, 2008; Christoffersen, Feunou, Jacobs, &

Meddahi, 2014; Duan, 1995, 1999; Heston & Nandi, 2000; Duan, Ritchken, & Sun, 2005).

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To

the best of ourknowledge, little (if any)literature exists on thepricing of VIX derivatives under

the GARCH framework. One possible reason is that the conventional local risk-neutral

valuationrelationship(LRNVR) onlycompensates for the equityrisk premium,and thus there is

no roomfor an independentvariance risk premiumwithin a singleshock in the GARCH models.

To overcome this problem, recent studies have estimated parameters with information from

both the underlyings and the risk-neutral measures (such as option prices and the VIX). For

example, Hao and Zhang (2013) suggested that such a joint estimation can significantly improve

the GARCH model’s ability in fitting the market VIX. Kanniainen, Lin, and Yang, (2014)

showed that joint estimation with VIX data can greatly improve the GARCH model’s option

pricingperformance.These resultssuggest that the GARCHmodels couldpotentially be usedin

volatility derivatives pricing, when an appropriate estimation method is adopted.

To fill this gap in the literature on volatility derivatives pricing, we investigate the

pricing of VIX futures with discrete-time GARCH-type models. One appealing advantage

of GARCH-type models is their convenience in conducting parameter estimations. Unlike

stochastic volatility models with their unobservable volatility shocks, the underlying

volatility process in GARCH models is recursively observable, and thus the estimation is

straightforward using the maximum likelihood estimation (MLE). For a large sample of

futures prices with a substantial cross-sectional dimension over a long period, it is

important to use a less computationally demanding model to implement the estimation

procedure. In this paper, we discuss VIX futures pricing under the classic discrete-time

Heston–Nandi GARCH model of Heston and Nandi (2000), which is very popular in the

option pricing literature. We derive an explicit pricing formula for VIX futures via an

integration of a transformed moment-generation function of the conditional volatility.

Several estimation methods are provided that differ in terms of the data used, and their

pricing performances are investigated. Among these methods, the model estimated by

jointly using the VIX and VIX futures prices yields a satisfying pricing performance and a

good balance in fitting both the VIX and VIX futures. We also conduct a rolling window

out-of-sample analysis and find similar empirical results. Such facts indicate the model is

free from in-sample overfitting when proper joint estimation is used.

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Luo and Zhang (2014) provide a good discussion on the literature of VIX derivatives pricing.

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Christoffersen, Jacobs, and Ornthanalai (2012) provided an extensive review of equity option pricing with GARCH

family models.

642 Wang et al.