Instantaneous squared VIX and VIX derivatives

• Published date: 01 October 2019
• DOI: http://doi.org/10.1002/fut.22037
• Author: Xingguo Luo,Jin E. Zhang,Wenjun Zhang
• Date: 01 October 2019

J Futures Markets. 2019;39:1193–1213. wileyonlinelibrary.com/journal/fut © 2019 Wiley Periodicals, Inc.

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1193

Received: 19 February 2019

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Revised: 29 May 2019

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Accepted: 30 May 2019

DOI: 10.1002/fut.22037

RESEARCH ARTICLE

Instantaneous squared VIX and VIX derivatives

Xingguo Luo

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Jin E. Zhang

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Wenjun Zhang

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1

Department of Finance, School of

Economics and Academy of Financial

Research, Zhejiang University, Hangzhou,

P.R. China

2

Department of Accountancy and

Finance, Otago Business School,

University of Otago, Dunedin,

New Zealand

3

Department of Mathematical Sciences,

School of Engineering, Computer and

Mathematical Sciences, Auckland

University of Technology, Auckland,

New Zealand

Correspondence

Xingguo Luo, School of Economics and

Academy of Financial Research, Zhejiang

University, Hangzhou 310027, PR China.

Email: xgluo@zju.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Number: Project no.

71771199; Fundamental Research Funds

for the Central Universities;

Establishment grant from the University

of Otago

Abstract

In this paper, we propose a parsimonious and efficient model to price

derivatives written on VIXs with different horizons. Our model is built on Luo

and Zhang’s (2012, J Futures Markets, 32, 1092–1123) concept of the

instantaneous squared VIX (ISVIX) that is the sum of instantaneous diffusive

and jump variances of the SPX return. Modeling the ISVIX as a mean‐reverting

jump‐diffusion process with a stochastic long‐term mean, we obtain analytical

formulas for VIX options and futures. Estimation with VIX term structure and

calibration with VIX options data show that our model performs well in

matching both time series and cross‐sectional VIX derivatives market prices.

KEYWORDS

instantaneous squared VIX, VIX, VIX futures, VIX option

JEL CLASSIFICATION

C13; G13

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INTRODUCTION

The VIX options and futures market has grown extremely rapidly. For example, the market size of VIX futures and

options has expanded since their introduction in 2004 and 2006 to an average daily volume of around 295,000 and

667,000, respectively, in 2018 compared with 1,731 and 23,491, respectively, in 2006.

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On October 1, 2013, the Chicago

Board Options Exchange (CBOE) introduced the Short‐Term Volatility Index (VXST), which is a 9‐day VIX.

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Subsequently, the CBOE Futures Exchange (CFE) launched VXST futures on February 13, 2014 and VXST options on

April 10, 2014. This rapid development of VIX/VXST derivatives not only generates rich information beyond traditional

equity index/stock derivatives but also creates an urgent need for cutting‐edge academic research on consistent pricing

between VIX and VXST options. However, there is as yet no commonly agreed framework within which we can both

theoretically price VIX derivatives and SPX options simultaneously and empirically calibrate model parameters in an

efficient way, not to mention study VXST derivatives.

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Generally, VIX refers to 30‐day future volatility. In this paper, the VIX includes a volatility index with any arbitrary time‐to‐maturity. Unless otherwise noted, VIX refers to the 30‐day VIX. Data

source: http://www.cboe.com/micro/vix/pricecharts.aspx. For an average VIX value of 20, these volumes correspond to market values of 5.9 and 1.3 billion US dollars. Carr and Lee (2009) provide an

overview of the market for volatility derivatives, including variance swaps and VIX futures and options.

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For more information, please refer to http://www.cboe.com/VXST. Note that the S&P 500 3‐month volatility index under the ticker “VXV”was launched on November 12, 2007 by the CBOE.

There is a fast‐growing literature on VIX derivatives pricing. Up to now, these studies can be divided into four

groups. The first group starts from classical option pricing theories in specifying underlying dynamics and derive

VIX from the underlying. Due to the extensive development of option pricing literature, this approach is widely

used. In fact, Zhang and Zhu (2006) is the first study on the VIX futures by considering Heston (1993) model. Zhu

and Zhang (2007) extend Zhang and Zhu (2006) to a model with time‐varying long‐term mean of variance. Later

on, Lin (2007), Lu and Zhu (2010), Dupoyet, Daigler and Chen (2011), and Zhu and Lian (2012) examine more

complicated models for VIX futures. Meanwhile, Sepp (2008a, 2008b), Albanese, Lo, and Mijatović(2009), Lin

and Chang (2009, 2010), Li (2010), Chung, Tsai, Wang, and Weng (2011), Wang and Daigler (2011), Chen and

Poon (2013), Lian and Zhu (2013), Papanicolaou and Sircar (2014), Branger, Kraftschik, and Volkert (2016), Song

and Xiu (2016), Lin, Li, Luo, and Chern (2017), Romo (2017), Bardgett, Gourier, and Leippold (2019), and Lo,

Shih, Wang, and Yu (2019) investigate various specifications for pricing VIX options. For example, Branger et al.

(2016) compare consistent and log VIX models by focusing on both the first and the second moments of the VIX

risk‐neutral distribution in addition to pricing errors. Bardgett et al. (2019) conduct a comprehensive analysis by

combining time series of SPX, VIX, and their options data and employing an accurate approximation and a

powerful filter method to get an estimation of a total of around 25 parameters and three latent variables at one

time. They find that a stochastic central tendency of volatility and jumps in volatility are important in capturing

volatility smiles in both the SPX and VIX markets and the tail of variance risk‐neutral distribution, respectively.

However, these papers do not use the VIX term structure data. More important, the SPX options data must be

used in estimating their models including volatility process.

The second group directly models volatility or variance. For example, Mencia and Sentana (2013), Carr and

Madan (2014), Park (2016), and Yan and Zhao (2019) start from the VIX or the log VIX, while Madan and

Pistorius (2014) consider the squared VIX.

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One drawback of this approach is that the tight link between VIX and

its underlying is not considered, and inconsistency may arise in pricing SPX options and replicating the VIX

index simultaneously. Bergomi (2008) and Cont and Kokholm (2013) overcome this drawback by modeling the

dynamics of the forward variance swap rate, which is similar to the squared VIX. In addition, some studies focus

on modeling the dynamics of the underlying volatility/variance (see, e.g., Detemple & Osakwe, 2000; Goard &

Mazur, 2013; Grunbichler & Longstaff, 1996). The third group consists of Huskaj and Nossman (2013) and Lin

(2013), who specify exogenous dynamics for the VIX futures. In particular, Huskaj and Nossman (2013)

investigate the normal inverse Gaussian process for the term structure of VIX futures, while Lin (2013) defines a

proxy of the future VIX as a forward VIX squared normalized by the VIX futures and studies VIX options by

considering various volatility functions for the proxy. Nevertheless, these two groups do not consider the

relationship between VIX and SPX. The fourth group considers VIX futures pricing with different discrete‐time

GARCH‐type models, including Wang, Shen, Jiang, and Huang (2017) and Huang, Tong, and Wang (2019), while

VIX options are not investigated. These models also rely on SPX data to estimate volatility parameters, which

suffers from the same drawbacks as we mentioned in the first group.

In this paper, we deal with previous concerns by using Duffie, Pan, and Singleton (2000) affine jump‐diffusion

technique and employing Luo and Zhang's (2012) concept of instantaneous squared VIX (ISVIX), which is the

sum of instantaneous diffusive and jump variances of the SPX return.

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The ISVIX is analogous to the

(instantaneous) short rate in interest rate models. Assuming that the ISVIX follows a mean‐reverting jump‐

diffusion process with a stochastic long‐term mean, we are able to derive analytical formulas for the VIX, VIX

futures and options. Further, we use the informative VIX term structure data to determine the mean‐reverting

speed and sequentially calibrate the volatility and jump parameters in the ISVIX process to market prices of VIX

options. As noted by Duan and Yeh (2010), Duan and Yeh (2012), and Bardgett et al. (2019), the 30‐day VIX is not

sufficient for estimating risk‐neutral stochastic volatility models. In fact, the two‐factor stochastic volatility

model of ISVIX requires the use of VIX term structure data. The importance of VIX term structure data in

obtaining risk‐neutral stochastic volatility models with mean‐reverting property is alsomentionedbyDuanand

Yeh (2012). However, Duan and Yeh (2012) do not consider VIX derivatives pricing. Although Huang et al.

(2019) use the VIX term structure data as well, they focus on VIX futures and do not study VIX options. We are

among the first to use the VIX term structure data to estimate the two‐factor stochastic volatility model and

pricing VIX derivatives with different horizons of VIX. The purpose of this paper is to propose a unified

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Kaeck and Alexander (2013) find that it is better to model the log VIX than to model the VIX level directly in terms of several statistical and operational metrics.

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The jump variance of the SPX return is the second term of Equation (3), which is different from variance swap rate and will be further clarified when our model is introduced.

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