Modeling VXX under jump diffusion with stochastic long‐term mean

• Author: Sebastian A. Gehricke,Jin E. Zhang
• Date: 01 October 2020
• Published date: 01 October 2020
• DOI: http://doi.org/10.1002/fut.22145

J Futures Markets. 2020;40:1508–1534.wileyonlinelibrary.com/journal/fut1508

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© 2020 Wiley Periodicals LLC

Received: 9 January 2020

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Accepted: 18 May 2020

DOI: 10.1002/fut.22145

RESEARCH ARTICLE

Modeling VXX under jump diffusion with stochastic

long‐term mean

Sebastian A. Gehricke |Jin E. Zhang

Department of Accountancy and Finance,

Otago Business School, University of

Otago, Dunedin, New Zealand

Correspondence

Sebastian A. Gehricke, Department of

Accountancy and Finance, Otago Business

School, University of Otago, Dunedin

9054, New Zealand.

Email: sebastian.gehricke@otago.ac.nz

Abstract

We develop a model for the VXX, the most actively traded VIX futures

exchange‐traded note, using Duffie, Pan, and Singleton's affine jump diffusion

framework, where the volatility process has jumps and a stochastic long‐term

mean. We calibrate the model parameters using the VIX term structure data

and show that our model provides the theoretical link between the VIX, VIX

futures, and the VXX. Our model can be used for pricing VIX futures, the VXX

and other short‐term VIX futures exchange‐traded products (ETPs). Our model

could be extended to price options on the VXX and other short‐term VIX

futures ETPs.

KEYWORDS

VIX futures, VIX futures ETP options, VIX futures ETPs, VXX

JEL CLASSIFICATION

G13

1|INTRODUCTION

In this paper, we extend the model of Gehricke and Zhang (2018), by including jumps and letting the long‐term mean of

volatility be mean‐reverting. This results in a better fit to the VIX term structure and the VXX, the most actively traded

VIX futures exchange‐traded note (ETN), compared with the nested models.

1

We calibrate to the VIX term structure

and then explore the fit of the model for the VXX. The model also performs well for other short‐term VIX futures

exchange‐traded products (ETPs), which dominate the market, and could be extended for VIX futures ETP option

pricing.

VIX index exposure first became accessible to investors in 2004, when VIX futures contracts were launched by the

Chicago Board Options Exchange (CBOE), followed in 2006 by VIX options. More recently, since 2009, VIX futures

ETPs have been heavily traded. The VXX was the first ETP tracking the short‐term VIX futures index (SPVXSTR),

which represents the return on a portfolio of VIX futures that is rebalanced to achieve an almost constant 1‐month

maturity. Since 2009 the number of other VIX futures ETPs has been rapidly growing, but with first mover advantage

the VXX has been the largest and most heavily traded VIX futures ETP throughout this period. In this paper, we model

and fit the short‐term VIX futures ETPs, while calibrating to the VIX term structure.

1

The first nested model is the Heston (1993) model, as used by Gehricke and Zhang (2018). The second nested model is the floating θmodel, as

presented in Appendix A.6 Equations (A28), (A29), and (A30). The last nested model is equivalent to the full model, as presented in Equations (3), (4),

and (5), but where

κ

κκ==

Vθ.

In Figure 1, we can see that the market capitalization has grown to around $4 Billion and the average daily dollar

trading volume is around $2 billion. On some days the ETPs are traded so heavily that the dollar trading volume is

several multiples of the market capitalization, meaning the market can turn over several times in a day.

The long‐exposure ETPs were initially marketed as diversification tools for equity portfolios, due to the negative

correlation between the VIX and the S&P 500; however, several studies have shown that they are not useful for

diversification (Alexander, Korovilas, & Kapraun, 2016; Deng, McCann, & Wang, 2012; Hancock, 2013). The reason

why these products are not good for diversification is due to their underperformance relative to the VIX index, which is

an empirical fact in contrast to the common misconception that investing in the VXX is like investing in the VIX index.

In October of 2017 Wells Fargo was ordered to pay remunerations of $3.4 million to investors because they were

advising them to invest in VIX futures ETPs as hedging tools (Banerji, 2017).

The under (out) performance of the short‐term long‐(short‐) exposure VIX futures ETPs is well documented in the

literature, and can be seen in Table 1. Alexander and Korovilas (2013), Liu and Dash (2012), and Whaley (2013) suggest

that the usually contango (upward‐sloping) VIX futures term structure is the driver of the underperformance of the

VXX. Gehricke and Zhang (2018) are the first to model the VXXs price while accounting for the dynamic relationships

between the SPX, VIX index, VIX futures and the ETN price. They show that the underperformance of the VXX relative

to the VIX index is mainly due to the roll yield, which measures the effect of rebalancing from the nearest to the second

nearest futures contract. We confirm this finding with our extended model. The roll yield will be negative (positive)

when the VIX futures term structure is in contango (backwardation). They show that the negative roll yield is driven by

the market price of variance risk, on aggregate. Their result is consistent with that of Eraker and Wu (2017), who show

that the underperformance of the SPVXSTR index is driven by the variance risk premium. The market price of variance

risk and the variance risk premium are two closely related concepts, which Zhang and Huang (2010) show are almost

proportional to each other. Eraker and Wu (2017) show, in a consumption‐based equilibrium setting, that the

underlying driver of the negative variance risk premium is investor risk aversion.

Recently, there have been several news releases (Burger, 2018; Jakab, 2018; Zuckermann & Fletcher, 2018) on the

sudden collapse of the XIV ETN, February 2018, which tracked the inverse performance of the SPVXSTR index.

FIGURE 1 Market share by maturity target. This figure has four panels. The top left panel shows the total market capitalization

of all the VIX futures ETPs grouped by their target maturity. The bottom left panel shows the daily proportion of total market

capitalization for each group. The top right panel shows the 5 day moving average of the daily dollar trading volume for each group.

The last panel (bottom right) shows the 5 day moving average of the daily proportion of dollar trading volume for each group. ST

represents the ETPs that track the short‐term VIX futures indices, MT represents the ETPs that track the midterm VIX futures

indices, WEEK represents ETPs that provide exposure to shorter term VIX futures (weekly futures) and dynamic/hybrid represents

those ETPs that are not linearly tracking one of the indices. ETP, exchange‐traded product

GEHRICKE AND ZHANG

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