VIX term structure and VIX futures pricing with realized volatility

• DOI: http://doi.org/10.1002/fut.21955
• Author: Tianyi Wang,Zhuo Huang,Chen Tong
• Published date: 01 January 2019
• Date: 01 January 2019

Received: 2 January 2018

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Revised: 11 June 2018

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Accepted: 13 June 2018

DOI: 10.1002/fut.21955

RESEARCH ARTICLE

VIX term structure and VIX futures pricing with realized

volatility

Zhuo Huang

1

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Chen Tong

1

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Tianyi Wang

2

1

National School of Development, Peking

University, Beijing, China

2

Department of Financial Engineering,

School of Banking and Finance,

University of International Business and

Economics, Beijing, China

Correspondence

Tianyi Wang, Department of Financial

Engineering, School of Banking and

Finance, University of International

Business and Economics, Beijing 100029,

China.

Email: tianyiwang@uibe.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Numbers: 71301027,

71671004

Using an extended LHARG model proposed by Majewski et al. (2015, JEcon,

187, 521–531), we derive the closed‐form pricing formulas for both the

Chicago Board Options Exchange VIX term structure and VIX futures with

different maturities. Our empirical results suggest that the quarterly and

yearly components of lagged realized volatility should be added into the

model to capture the long‐term volatility dynamics. By using the realized

volatility based on high‐frequency data, the proposed model provides

superior pricing performance compared with the classic Heston–Nandi

GARCH model under a variance‐dependent pricing kernel, both in‐sample

and out‐of‐sample. The improvement is more pronounced during high

volatility periods.

KEYWORDS

implied volatility, realized volatility, VIX futures, volatility term structure

JEL CLASSIFICATION

C19, C22, C80

1

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INTRODUCTION

The well‐known Chicago Board Options Exchange (CBOE) VIX index, computed from a panel of options prices, is a

model‐free measure of expected average variance for the next 30 days under the risk‐neutral measure. The index has

become the benchmark for stock market volatility and is used as the “investor fear gauge”for financial market

practitioners. With the launch of VIX futures in 2004 and VIX options in 2006, volatility derivatives have received

increasing attention from the market, as the average daily trading volume of VIX futures and VIX options have

increased by over 25 and 137 times, respectively, in the last decade. VIX‐linked products essentially create a volatility

market that enables investors to trade volatility directly, as with equity or fixed income securities.

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In addition to the

VIX index that measures 1‐month implied volatility, the CBOE has also launched a series of implied volatility indices

across different maturities in recent years to reflect the volatility term structure under the risk‐neutral measure. The

CBOE S&P 500 3‐month Volatility Index (ticker: VXV) was launched in November 2007. The CBOE Mid‐Term

Volatility Index (ticker: VXMT), a measure of the expected volatility of the S&P 500 index over a 6‐month time horizon,

was launched in November 2013.

2

The whole family of CBOE VIX indices and VIX futures provides rich information for

the implied volatility term structure of the stock market.

Zhang and Zhu (2006) were the first to attempt to price VIX futures based on the classic continuous‐time Heston

model. The importance of the volatility term structure in VIX futures pricing was illustrated in Zhu and Zhang (2007).

J Futures Markets. 2019;39:72–93.wileyonlinelibrary.com/journal/fut72

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© 2018 Wiley Periodicals, Inc.

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Luo & Zhang (2014) provide a good discussion of the market for volatility derivatives.

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CBOE reported historical data of VXMT back to January 2008.

Lin (2007) extended the model with simultaneous jumps in both returns and volatility to price VIX futures with

approximation formulas. Adding jumps to the mean‐reverting process was also investigated by Sepp (2008), Zhang, Shu,

and Brenner (2010), and Zhu and Lian (2012), and so forth. As a generalization, Mencia and Sentana (2013) provided a

model with a time‐varying central tendency, jumps, and stochastic volatility to price VIX derivatives. Under the

discrete‐time Heston–Nandi generalized autoregressive conditional heteroskedasticity (GARCH) framework, Wang,

Shen, Jiang, and Huang (2017) derived the pricing formulas for both VIX and VIX futures. Following the ideas of Hao

and Zhang (2013) and Kanniainen, Lin, and Yang (2014), the model parameters were jointly estimated, as the log‐

likelihoods of both the stock returns (realized volatility if required) and the risk‐neutral information (VIX panel

3

and/or

VIX futures) were included in the objective function. The empirical results suggested that including the risk‐neutral

information in the objective function improves parameter estimation and yields better pricing performance. Other VIX‐

related products such as the VIX, ETNs, and VXX are also discussed by Eraker and Wu (2017), Gehricke and Zhang

(2018), and so forth.

The seminal work of Andersen, Bollerslev, Diebold, and Labys (2003) proved the realized volatility computed from

high‐frequency intraday returns to be an accurate measure of the latent volatility process. Volatility models with the

realized measures have attracted great attention in recent years. Leading models include the heterogeneous

autoregressive (HAR) model (Corsi, 2009), the MEM model (Engle & Gallo, 2006), the HEAVY model (Shephard &

Sheppard, 2010), and the Realized GARCH model (Hansen, Huang, & Shek, 2012). Among these models, the HAR

model is receiving increasing attention in volatility modeling and financial applications because of its estimation

simplicity and good forecasting performance. The model introduces a cascade structure into the linear autoregression

framework, in which the current daily realized variance (RV) is regressed on the lagged realized variance over the

previous day, week, and month. Empirical studies show that the HAR model provides a parsimonious but effective

approximation of the long memory process of volatility.

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Most studies focus on the performance of the HAR model and its extensions into forecasting volatility or realized

volatility under the physical measure, but Corsi, Fusari, and Vecchia (2013) and Majewski, Bormetti, and Corsi (2015)

have shown that the HAR framework is also capable of matching the volatility information implied by option prices,

that is, under the risk‐neutral measure. Corsi et al. (2013) extended the HAR model with a gamma innovation (the

heterogeneous autoregressive gamma, HARG) and specified an exponentially affine pricing kernel. Majewski et al.

(2015) further developed the model by allowing more flexible leverage components (LHARG) and derived the analytical

pricing formula for European options. In this study, we extend the LHARG model by including lagged quarterly and

yearly realized variance into the RV dynamics and derive the analytical formulas for the VIX term structure in addition

to the VIX futures. We find that adding these two terms enhances the model’s ability to capture volatility dynamics over

longer horizons, and it is also empirically important for the purpose of pricing VIX term structures and VIX futures.

Compared with the classic Heston–Nandi GARCH model under a variance‐dependent pricing kernel in Christoffersen,

Heston, and Jacobs (2013)

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, our proposed model provides superior performance in pricing VIX term structures and VIX

futures. The improvement is more pronounced during high volatility periods when the realized volatility provides more

accurate information about underlying volatility than the squared daily returns. A rolling window out‐of‐sample pricing

analysis is also conducted and the main empirical findings are robust.

The remainder of the paper is organized as follows: Section 2 introduces the model setup and derives the pricing

formula for VIX term structures and VIX futures; Section 3 discusses the model estimation using different datasets;

Section 4 presents the empirical results; and Section 5 concludes.

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THE MODEL

2.1

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LHARG model and risk neutralization

In this study, we denote the original LHARG model of Majewski et al. (2015) as LHARG‐M, as it contains volatility

components up to the monthly average. We extend the model to LHARG‐Q by including the quarterly average

3

“VIX panel”in this paper refers to the collection of time series of volatility indices. Each cross section of this panel is term structure of the implied volatility for a given day.

4

See Corsi and Audrino (2012) for a review of this model. Some latest developments of HAR‐type models are discussed by Gong and Lin (2018).

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Due to the similarity of the risk‐neutral Heston–Nandi GARCH model under a variance‐dependent pricing kernel and locally risk‐neutralization valuation relationship (LRNVR), the pricing formula

can be adapted from Wang et al. (2017). The pricing performance of the LRNVR‐based (Duan, 1995) Heston–Nandi GARCH model underperforms compared with the one used here. Results are

available upon request.

HUANG ET AL.

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