Pricing VIX derivatives with infinite‐activity jumps

• Published date: 01 March 2020
• Author: Shu Su,Xinfeng Ruan,Jiling Cao,Wenjun Zhang
• Date: 01 March 2020
• DOI: http://doi.org/10.1002/fut.22074

J Futures Markets. 2020;40:329–354. wileyonlinelibrary.com/journal/fut © 2019 Wiley Periodicals, Inc.

|

329

Received: 23 May 2019

|

Accepted: 11 October 2019

DOI: 10.1002/fut.22074

RESEARCH ARTICLE

Pricing VIX derivatives with infinite‐activity jumps

Jiling Cao

1

|

Xinfeng Ruan

2

|

Shu Su

1

|

Wenjun Zhang

1

1

Department of Mathematical Sciences,

School of Engineering, Computer and

Mathematical Sciences, Auckland

University of Technology, Auckland,

New Zealand

2

Department of Accountancy and

Finance, Otago Business School,

University of Otago, Dunedin,

New Zealand

Correspondence

Wenjun Zhang, Department of

Mathematical Sciences, School of

Engineering, Computer and Mathematical

Sciences, Auckland University of

Technology, Private Bag 92006, Auckland

1142, New Zealand.

Email: wenjun.zhang@aut.ac.nz

Abstract

In this paper, we investigate a two‐factor VIX model with infinite‐activity

jumps, which is a more realistic way to reduce errors in pricing VIX derivatives,

compared with Mencía and Sentana (2013), J Financ Econ,108, 367–391. Our

two‐factor model features central tendency, stochastic volatility and infinite‐

activity pure jump Lévy processes which include the variance gamma (VG) and

the normal inverse Gaussian (NIG) processes as special cases. We find empirical

evidence that the model with infinite‐activity jumps is superior to the models

with finite‐activity jumps, particularly in pricing VIX options. As a result,

infinite‐activity jumps should not be ignored in pricing VIX derivatives.

KEYWORDS

infinite‐activity jumps, maximum log‐likelihood estimation, unscented Kalman filter, VIX

derivatives

JEL CLASSIFICATION

G12, G13

1

|

INTRODUCTION

Time‐varying volatility is a key risk factor in financial markets; thus, understanding the dynamics of volatility variation

is crucial for market practitioners and academic researchers. The Chicago Board Options Exchange (CBOE) Volatility

Index, known by its ticker symbol VIX, is a risk‐neutral gauge of the future implied volatility in the U.S. stock market. It

is calculated by the S&P 500 index options market over the next 30‐day period and indicates overall stock market fear.

1

VIX can be traded through the VIX derivatives. Because of a negative correlation between the return of VIX and the

return of the S&P 500 index, market participants treat the VIX derivatives as an important trading vehicle for reducing

exposure to risk. Reported by CBOE, the VIX derivatives include VIX futures and VIX options. In 2004, CBOE

introduced VIX futures. According to the CBOE website, the average daily volume of VIX futures rose more than 135‐

fold from 2004 to 2017, with a significant growth after 2010.

2

After the successful launch of VIX futures, CBOE

introduced VIX options in 2006, which can provide an alternative way for hedging the market risk with relatively low

costs. Since then, the VIX options have become popular financial tools in risk management. Not surprisingly, the

average daily volume of VIX options expanded over 28 times between 2006 and 2017.

3

Therefore, it is important to find

an accurate valuation model of the VIX derivatives.

The methodologies for pricing VIX derivatives can be classified into two directions. In the first direction, the VIX

derivatives pricing formulae are derived from the instantaneous volatilities of the S&P 500 index, which can provide a

clear picture of the relationship among the S&P 500 index, VIX, and VIX derivatives. Researchers in this direction

1

See https://www.cboe.com/micro/vix/vixwhite.pdf.

2

See http://www.cboe.com/blogs/options‐hub/2017/03/01/eight‐charts‐highlighting‐growth‐in‐options‐and‐vix‐futures.

3

See http://www.cboe.com/blogs/options‐hub/2017/08/25/record‐days‐for‐vix‐futures‐and‐options‐volume‐and‐open‐interest‐this‐month.

believe that the instantaneous volatility process has a mean‐reverting property. In addition, some of them add finite‐

activity jumps into the S&P 500 return process, instantaneous volatility process, or both of them. However, it is

challenging to derive an analytical VIX derivative pricing formula when more and more notable features are added to a

model. For more information about research work in this direction, we refer the reader to (Goutte, Ismail, & Pham,

2017; Lin, 2007; Lin & Chang, 2009; Zhang & Zhu, 2006; S. P. Zhu & Lian, 2012; Y. Zhu & Zhang, 2007); and others.

In the other direction, research directly model VIX dynamics and then derive the derivatives pricing formulae from

the proposed models. In this direction, they acknowledge that volatility varies stochastically and reverts toward long‐

term mean (Grünbichler & Longstaff, 1996; Mencía & Sentana, 2013; Psychoyios & Dotsis, 2010, etc.). Thus, different

types of mean‐reverting processes, such as the squared‐root mean‐reverting process, the arithmetic mean‐reverting

process, and the geometric mean‐reverting process, are applied for modeling VIX. In addition, Goard and Mazur

(2013), Kaeck and Alexander (2013), Mencía and Sentana (2013), and Psychoyios and Dotsis (2010), find clear evidence

of jumps in the VIX return process, in which the compound Poisson process is employed to characterize finite‐activity

jumps triggered by influential financial events. As an advantage, analytical pricing formulae for VIX derivatives can be

obtained based on even more complex VIX models. For example, Mencía and Sentana (2013) extend previous models

into a two‐factor model with three major features of VIX dynamics: central tendency, stochastic volatility of VIX and

finite‐activity jumps, and they document that their two‐factor model works better than other existing models, in terms

of pricing VIX derivatives. Our research is along this direction and proposes a new two‐factor model with infinite‐

activity jumps, which can significantly improve the pricing performance of VIX derivatives.

There has been a growing interest in pricing stock options with infinite‐activity jump processes (Carr, Geman,

Madan, & Yor, 2003; Carr & Wu, 2003, 2004; Clark, 1973; Geman, 2002; Lian, Zhu, Elliott, & Cui, 2017, etc.). Some

pieces of credible evidence show that infinite‐activity jumps (a.k.a. high‐frequency jumps) occur in the dynamics of

financial asset return. For instance, Wu (2011) figures out that small jumps frequently appear in the volatility dynamics

based on the analysis of high‐frequency S&P 500 returns. As another example, Yang and Kanniainen (2017) reveal that

the models with infinite‐activity jump processes can fit the S&P 500 option data better. The major advantage of infinite‐

activity jump processes over finite‐activity jump processes is that they are able to capture not only finite activities but

also infinite activities within a finite time interval. Through non‐parametric analysis of high‐frequency VIX data,

Todorov and Tauchen (2011) document that VIX dynamics have infinite‐activity jumps with infinite variation.

However, there is little literature on pricing VIX derivatives with infinite‐activity jump processes. To fill this gap, we

propose a new two‐factor model with infinite‐activity jumps to price VIX derivatives.

Several infinite‐activity jump Lévy processes can be applied in modeling. For example, the generalized hyperbolic

Lévy process with five parameters is one of the most fundamental Lévy processes (Barndorff‐Nielsen, 1978). It has two

impressive subclasses, the hyperbolic Lévy process (Eberlein, Keller, & Prause, 1988) and the normal inverse Gaussian

(NIG) process (Barndorff‐Nielsen, 1997). The main difference between these subclasses is the tail behavior: the tails of

the NIG process are thicker than those of hyperbolic Lévy process. Another Lévy jump process with infinite divisibility,

popularly applied in finance, is the CGMY introduced by Carr, Geman, Madan, and Yor (2002)). This is a generalization

of the variance gamma (VG) process by adding a parameter

Y

permitting finite or infinite activity and finite or infinite

variation. When Y<0

, the process permits finite‐activity like compound Poisson process. On the other hand, if

Y

is

between

0

and

2

, the process permits infinite activity. The process has finite variation when

Y

0

<<

1

and infinite

variation when

Y

1

<<

2

(Carr et al., 2002). The VG process (Madan, Carr, & Chang, 1999) is a special case of the

CGMY, which also belongs to the generalized hyperbolic Lévy process class.

We select the two most representative Lévy processes as our jump process: the NIG process and the VG process. The

VG process has a finite variation with a moderately low activity rate of small jumps, whereas the NIG process has an

infinite variation with a stable arrival rate of small jumps (Cont & Tankov, 2004). From the comparison, we can see how

different types of Lévy processes perform in the VIX derivatives pricing. Todorov and Tauchen (2011) find that VIX

dynamics have infinite‐activity jumps. Our paper is different to Li, Li, and Zhang (2017), in which a pure jump

semimartingale (one type of time‐changed Lévy process in Carr & Wu, 2004) is generated by an additive subordination.

Our model considers the time‐varying mean level of the log VIX and is estimated using a much longer data sample.

To this end, we apply an appropriate estimation procedure to calibrate models and various measurement criteria to

test the performance of our model. In this paper, we apply a combined estimation approach of the unscented Kalman

filter (UKF) and the maximum log‐likelihood estimation method (MLE) to our model. Christoffersen, Dorion, Jacobs,

and Karoui (2014) find that the UKF is a good method to estimate affine factor models. Following the literature, we use

the root mean squared errors (RMSEs) as our major performance measurement criterion (Mencía & Sentana, 2013;

Park, 2015; Yang & Kanniainen, 2017, etc.).

330

|

CAO ET AL.