A dimension‐invariant cascade model for VIX futures

• DOI: http://doi.org/10.1002/fut.22042
• Published date: 01 October 2019
• Author: Brice Dupoyet,Zhiguang Wang
• Date: 01 October 2019

J Futures Markets. 2019;39:1214–1227.wileyonlinelibrary.com/journal/fut1214

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© 2019 Wiley Periodicals, Inc.

Received: 18 April 2018

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Revised: 12 June 2019

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Accepted: 12 June 2019

DOI: 10.1002/fut.22042

RESEARCH ARTICLE

A dimension‐invariant cascade model for VIX futures

Zhiguang Wang

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Brice Dupoyet

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1

Ness School of Management and

Economics, South Dakota State

University, Brookings, South Dakota

2

Department of Finance, Florida

International University, Miami, Florida

Correspondence

Zhiguang Wang, Ness School of

Management and Economics, South

Dakota State University, Box 2220,

Brookings, SD 57007.

Email: zhiguang.wang@sdstate.edu

Abstract

We propose a new stochastic volatility model by allowing for a cascading

structure of volatility components. The model, under a minor assumption,

allows us to add as many components as desired with no additional parameters,

effectively defeating the curse of dimensionality often encountered in

traditional models. We derive a semi‐closed‐form solution to the VIX futures

price, and find that our six‐factor model with only six parameters can closely fit

spot VIX and VIX futures prices from 2004 to 2015 and produce out‐of‐sample

pricing errors of magnitudes similar to those of in‐sample errors.

KEYWORDS

cascade, dimension‐invariant, term structure, VIX futures

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INTRODUCTION

VIX futures and options contracts have now become the second most actively traded contracts on the Chicago Board of

Exchange (CBOE). The number of VIX futures contracts’months has increased from four in March 2004 to ten in 2009,

and eventually stabilized at nine in 2016. VIX futures pricing has always been a focal point of academic research. Along

with the expansion of VIX futures contracts, the literature on stochastic volatility models has evolved from a single

factor (Lin, 2007; Zhang & Huang, 2010; Zhang & Zhu, 2006; Zhu & Zhang, 2007) to two factors (Christoffersen, Jacobs,

Ornthanalai, & Wang, 2008; Egloff, Leippold, & Wu, 2010; Luo & Zhang, 2012; Zhang, Shu, & Brenner, 2010), and more

recently to three factors as in Lu and Zhu (2010). In particular, Lu and Zhu (2010) find that the third factor is

statistically significant for explaining the variance term structure.

Lu and Zhu’s (2010) three‐factor model largely represents the state of the art, or at least the most comprehensive

model, in VIX futures pricing. Their framework offers a rich structure able to accommodate five strips of VIX futures

contracts in sample. The multifactor model improves significantly on the pricing of short‐term contracts (30 and 60

days). However, their results reveal some weaknesses: (a) Their three‐factor model generates large errors for 90‐and

270‐day contracts; (b) the empirical results are not based on original VIX futures data but on interpolated (smoothed)

prices, with the interpolation potentially hiding the actual pricing errors of the model; (c) their out‐of‐sample test is

limited to 8 days, making it difficult for one to judge the merit of the model.

Another shortcoming of current VIX futures factor‐based term structure models is that they suffer from the curse of

dimensionality. One generally needs three extra parameters for each additional factor. Lu and Zhu’s (2010) two‐factor

model has between seven and eight parameters, while their three‐factor model contains between 10 and 12 parameters.

The likely overfitting of the five strips of VIX futures contracts is a distinctive concern.

To address the above issues, we propose a new model of volatility by allowing for a cascading structure of volatility

components, motivated by the interest rate model of Calvet, Fisher, and Wu (2018). The cascading volatility model

essentially has one governing factor with multiple layers or commonly referenced as factors. Each layer, identified by its

speed of mean reversion, mean‐reverts to the next layer until it converges to a constant long‐run mean. A faster (slower)

moving layer represents economic information occurring at a higher (lower) frequency, for example intraday (vs.

quarterly) news arrival. Since such a structure allows one to add as many layers as desired without any additional