Pricing VIX options with volatility clustering

• Author: Yong Ma,Bo Jing,Shenghong Li
• DOI: http://doi.org/10.1002/fut.22092
• Published date: 01 June 2020
• Date: 01 June 2020

J Futures Markets. 2020;40:928–944.wileyonlinelibrary.com/journal/fut928

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© 2020 Wiley Periodicals, Inc.

Received: 24 May 2019

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Accepted: 31 December 2019

DOI: 10.1002/fut.22092

RESEARCH ARTICLE

Pricing VIX options with volatility clustering

Bo Jing

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Shenghong Li

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Yong Ma

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Department of Mathematics, School of

Mathematical Sciences, Zhejiang

University, Hangzhou, China

2

Department of Financial Engineering,

College of Finance and Statistics, Hunan

University, Changsha, China

Correspondence

Yong Ma, Department of Financial

Engineering, College of Finance and

Statistics, Hunan University, Yuelu

District, Changsha 410006, China.

Email: yma@hnu.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Numbers: 11571310,

71601075, 71971077, 71871087

Abstract

We investigate the valuation of volatility index (VIX) options by developing a

model with a self‐exciting Hawkes process that allows for clustering in the VIX.

In the proposed framework, we find semianalytical expressions for the

characteristic function and forward characteristic function, and then we solve

the pricing problem of standard‐start and forward‐start options via the fast

Fourier transform. The empirical results provide evidence to support the

significance of accounting for volatility clustering when pricing VIX options.

KEYWORDS

fast Fourier transform, self‐exciting Hawkes process, VIX options, volatility clustering

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INTRODUCTION

Since the 2008 financial crisis, the demand for volatility‐related products has grown markedly as a result of economic

uncertainty. Among these diversified volatility products, volatility index (VIX) options, launched by the Chicago Board

Options Exchange (CBOE) as the tradable assets in 2006, have gained wide popularity. Unlike the equity options, the

underlying asset of VIX options, that is, the CBOE VIX, cannot be traded directly. The trading of VIX options enables

market players to diversify stock market risk and manage volatility effectively, due to the fact that the relationship

between stock returns and market volatilities is basically negative. With significant growth in the trading volume of

volatility products, much academic attention has been given to the valuation of VIX options and other volatility

derivatives. Abundant literature has emerged on this study topic. For example, Detemple and Osakwe (2000) valued

volatility options when volatility follows an exponential Ornstein‐Uhlenbeck (OU) process. Mencía and Sentana (2013)

extended the model of Detemple and Osakwe (2000) by allowing for a time‐varying central tendency and unspanned

stochastic volatility. Park (2016) proposed a class of models with asymmetric volatility and jumps for the valuation

of VIX derivatives.

So far, the previous studies on the pricing of VIX derivatives are classified into two main categories based on

different modeling ideas. The first one is to model the joint dynamics of the S&P 500 index (SPX) and its instantaneous

variance (e.g., Baldeaux & Badran, 2014; Lin & Chang, 2010; Lo, Shih, Wang, & Yu, 2019; Zhang & Zhu, 2006). The VIX,

as a measure of the implied 30‐day volatility of the SPX options, can be derived from the joint dynamics. The second one

is to specify the VIX dynamics or the logarithmic VIX dynamics directly (e.g., Drimus & Farkas, 2013; Goard & Mazur,

2013; Grünbichler & Longstaff, 1996; Park, 2016), which is the route we take in this paper. In accordance with market

practice, the hedging of options on the VIX usually makes use of corresponding futures, the pricing of VIX options,

therefore, can be addressed by directly assuming the VIX dynamics, analogous to the valuation of stock index options.

The fact that a standard Brownian‐driven model is incapable of generating large price movements, even if it is

extended to allow for stochastic volatility, is widely recognized. Eraker, Johannes, and Polson (2003) and Wagner and

Szimayer (2004) provided empirical evidence of significant upward jumps in implied volatility and concluded that

jumps have an impact on option values. Table 1 presents the descriptive statistics of the VIX log‐returns. The VIX log‐

returns exhibit a high kurtosis suggesting the presence of large movements. To capture these large movements in

volatility, it is typical to add a jump component, which is characterized by the compound Poisson processes, to the

diffusion model in a study of pricing volatility options (e.g., Kaeck & Alexander, 2013; Lo et al., 2019; Psychoyios,

Dotsis, & Markellos, 2010; Sepp, 2008). Unfortunately, to our best knowledge, existing models for the valuation of VIX

options fail to replicate the pattern that extreme price movements are empirically highly clustered. This pattern is

observed extensively in financial markets and cannot be neglected (e.g., Aït‐Sahalia, Cacho‐Diaz, & Laeven, 2015; Lux &

Marchesi, 2000). We also find that extreme log‐returns of the VIX often exhibit the behavior of clustering, especially

during the periods of financial turmoil. Accordingly, the main objective of this paper is to develop a novel pricing model

that is capable of characterizing the clustering behavior in VIX markets.

To achieve our aim, we introduce a Hawkes process, first described in the seminal work by Hawkes (1971), to

model the dynamics of the jump component. In a Hawkes process, the occurrence of an event can affect the

intensity of arrivals of future events, which is self‐exciting property. Owing to this property, Hawkes processes

are suitable for dealing with the clustering of events. Originally, they were introduced to predict the occurrences

ofaftershocksgiventhefactthattheoccurrenceofashockislikelytoincreasetheprobabilityoffutureshocks

occurring (Ogata, 1988). They were also applied in genome analysis (Reynaud‐Bouret & Schbath, 2010) and in

the modeling of crime rates (Mohler, Short, Brantingham, Schoenberg, & Tita, 2011). Over recent years, Hawkes

processes are increasingly popular in the financial field. For example, Grothe, Korniichuk, and Manner (2014)

used self‐exciting Hawkes processes to study the extreme negative returns in the European and U.S. financial

markets. Ma, Shrestha, and Xu (2017) adopted Hawkes jump‐diffusion processes to value vulnerable options and

emphasized the role of self‐exciting jumps in option prices. The applications of Hawkes processes are especially

successful in the context of high‐frequency finance. Chavez‐Demoulin and McGill (2012) introduced a model

with Hawkes processes for the excesses of the time‐series over a high threshold in the framework of high‐

frequency financial data. With mutually exciting Hawkes processes, Bacry, Delattre, Hoffmann, and Muzy (2013)

proposed a price model that allows one to recover major high‐frequency stylized facts. Other relevant literature

includes Bacry and Muzy (2014), Da Fonseca and Zaatour (2014), Filimonov and Sornette (2015), Omi, Hirata,

and Aihara (2017), and among others. To the best of our knowledge, Hawkes processes have never been

employed in the context of pricing volatility options. Inspired by the growing literature on the application of

Hawkes processes in the area of finance, this paper aims to fill this gap.

This paper extends the existing literature on the valuation of volatility options in several ways. First, we

propose a model that takes the clustering pattern of volatility into account for pricing VIX options. This critical

pattern is modeled by introducing a self‐exciting Hawkes process with stochastic intensity in the dynamics of the

logarithm of the VIX. Previous empirical studies have indicated that the introduction of stochastic volatility is

necessary when pricing VIX options (e.g., Kaeck & Alexander, 2013; Mencía & Sentana, 2013; Park, 2016).

Therefore, we also incorporate a stochastic volatility component into the underlying dynamics. Second, in the

model framework, we derive the characteristic function and forward characteristic function, and then solve the

pricing problem of standard‐start, as well as forward‐start options on the VIX via the fast Fourier transform

(FFT) method. Forward‐start options are one of the simple exotic derivatives, which are very popular in the field

of equity derivatives. With the development of the VIX product markets, we expect them to be increasingly

popular. Finally, to illustrate the ability of the proposed valuation model to match the VIX option surface,

empirical studies are conducted in which we compare the results to the benchmark models in the aspect of fitting

market data. We adopt three commonly used performance criteria to assess the in‐and out‐of‐sample pricing

performance of different models. Our empirical results provide evidence of the significance of accounting for

clustering behavior when pricing VIX options.

The remainder of the paper is structured as follows. Section 2 introduces a self‐exciting Hawkes process and outlines

the model specification. Section 3 solves the pricing problem of standard‐start, as well as forward‐start options on the

TABLE 1 Descriptive statistics for VIX log‐returns

Observations Mean Median Max Min Std Skew Kurt

VIX log‐returns 3,524 −0.0002 −0.0054 0.4960 −0.3506 0.0703 0.7138 4.2474

Note: The table provides descriptive statistics for VIX log‐returns from January 2, 2004 to January 2, 2018.

Abbreviations: Kurt, excess kurtosis; Std, standard deviation.

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