Pricing VXX options by modeling VIX directly

• Published date: 01 May 2022
• Author: Wei Lin,Jin E. Zhang
• Date: 01 May 2022
• DOI: http://doi.org/10.1002/fut.22313

Received: 6 April 2021

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Accepted: 8 January 2022

DOI: 10.1002/fut.22313

RESEARCH ARTICLE

Pricing VXX options by modeling VIX directly

Wei Lin

1

|Jin E. Zhang

2

1

School of Mathematics, Hangzhou

Normal University, NO. 2318,

Yuhangtang Road, Yuhang District,

Hangzhou, Zhejiang Province, China

2

Department of Accountancy and

Finance, Otago Business School,

University of Otago, Dunedin,

New Zealand

Correspondence

Wei Lin, School of Mathematics,

Hangzhou Normal University, Hangzhou,

Zhejiang Province, 31121, China.

Email: wei.lin@postgrad.otago.ac.nz,

weilin1991@zju.edu.cn

Jin E. Zhang, Department of Accountancy

and Finance, Otago Bussiness School,

University of Otago, Dunedin, 9054,

New Zealand.

Email: jin.zhang@otago.ac.nz

Abstract

In this paper, we firstdevelop a theoretical and model‐free VXX formula in

terms of Volatility Index (VIX) futures in both discrete and continuous forms.

The discrete form ofVXX can quantify theroll yield of VXX, which can be used

to explain VXX's underperformance. Using the log‐normal Ornstein–Uhlenbeck

(LOU) diffusion model, we show how the number of rolls of VIX futures affects

the VXX option pricing formula and its implied volatility (IV). To further verify

the nonflat IV of VXX, the VXX option pricing formula under the LOU with

stochastic volatility model is also derived. Finally, we analyze their pricing

performance, and the ability to forecast implied volatilities.

KEYWORDS

stochastic volatility, VIX option pricing, VXX option pricing

JEL CLASSIFICATION

G12, G13

1|INTRODUCTION

In this paper, we first theoretically develop a formula using a series of Volatility Index (VIX) futures to represent VXX

which is the iPath SPX VIX Short‐Term Futures exchange‐traded note, and support this by subsequent deduction and

prove its reliability via two models which price VIX and VXX options together. The paper evaluates these models' pricing

ability and their implied volatility (IV)'s forecasting ability. In parallel with the boom and prosperity observed duringthe

last decade within the VIX and VXX market, both researchers (e.g., Grasselli et al., 2020; Grasselli & Wagalath, 2020)in

option pricing and practitioners are eager to find a joint analytical framework that explores the intermarket connections

and prices both options consistently. In the small body of existing literature on VXX, one main focus has been VXX option

pricing which model VXX directly but fail to explore the intermarket connections, and the other main focus has been

consistent‐pricing approach models, which cannot deduce an explicit VXX formula for its option pricing. Thus, referring to

the official “S&P VIX Futures Indices Methodology,”

1

the authors of this paper appear to be the first to establish a model‐

free relationship between VXX and VIX futures which succeeds in using VIX futures to explain the VXX formula, making it

possible to price VIX and VXX options consistently.

Originally introduced in 1993 and redefined in 2003, the Chicago Board Options Exchange (CBOE) VIX has become a

standard measure of volatility risk in the financial market. VIX is designed to reflect the market's expectation of volatility in

the next 30days and is commonly regarded as the “fear gauge.”Later, VIX futures contracts were launched by CBOE in

March 2004, and European‐style VIX options in February 2006. A common way to trade implied and realized market

volatility is to use exchange‐traded instruments, such as VIX futures contracts and related exchange‐traded notes (ETNs).

J Futures Markets. 2022;42:888–922.wileyonlinelibrary.com/journal/fut888

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© 2022 Wiley Periodicals LLC

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Link: https://www.spglobal.com/spdji/en/documents/methodologies/methodology-sp-vix-futures-indices.pdf.

Traders who adopt this approach long or short VIX index futures, depending on their volatility expectations. Some traders

use the actual VIX futures, but a simpler and more common way is to use ETNs that replicate VIX futures strategies.

Trading VIX futures has been extremely popular in recent years, mainly by using exchange‐traded products (ETPs), such as

the iPath S&P 500 index (SPX) VIX Short‐Term Futures ETN (VXX). 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 are rebalanced to achieve an almost

constant 1‐month maturity. Due to the mutual promotion between VXX and VIX futures, the VIX futures ETP market has

been dramatically growing in popularity since the VXX was launched. The daily dollar trading volume has remained above

$1.4 billion, with an average daily market cap of about $1 billion. CBOE introduced the VXX options in May 2010. Each

VXX option contract includes 100 VXX ETNs. VXX options, as a new volatility trading vehicle, have become more and more

popular. In Figure 1, we can see that the VXX option market has grown to an average trading volume of about 0.25 million

(3.48 billion in dollars) and an average daily open interest of 2.7 million (35.9 billion in dollars) contracts over the past

10 years. The open interest in VXX options has grown consistently from 2010to 2019, reaching about 5.8 million contracts

on August 18, 2017, while the trading volume has hovered around 0.37 million contracts per day since 2018.

2

The other VIX

futures ETP options VXZ (iPath Series B S&P 500 VIX Mid‐Term Futures ETN) which is similar to VXX but the only

difference is the mid term rather than the short term in VXX, for example, have also grown in popularity, but their trading

volume and open interest are still quite low, relative to the VXX options. This is one reason why our study focuses on the

VXX option pricing and its IV. Another reason is that SPX, VIX, and VXX can be seen as a “family”of options since VXX is

a kind of VIX futures derivative while VIX is derived from SPX. Thus, it appears to be a natural choice to establish a joint

analytical framework for modeling these three markets consistently. Besides, a united framework could be more convenient

for traders to trade in SPX, VIX, and VXX markets when managing their portfolios and hedging.

Before offering our solution to this problem, here we first present a brief overview of the previous literature on VXX

modeling. There are roughly three categories of approaches in the field of VXX modeling. In one line of research, authors

directly model VXX dynamics and focuson the issue of VXX options pricing. Bao et al. (2012) make a comparative study

of alternative models for VXX option pricing, taking into consideration factors, such as mean‐reversion, jumps, default

risk, and positivevolatility skew. The paper suggests that a jump‐to‐default extended logarithmic model with jumps(LRJ)

and positive correlated stochastic volatility serves as the best model in all the required aspects. Tan et al. (2021)proposea

comprehensivejump‐to‐default e xtended two‐factor stochastic volatilityplus asymmetry jumps model for thevaluation of

VXX derivatives. Cao et al. (2021) conduct a comprehensive study on the specifications of VXX option pricing models

under Lévy processes. However, a direct modeling approach in VXX cannot explain the systematical loss of VXX, whichis

obviously observable in the VXX market. In the second line of research, Grasselli and Wagalath (2020)proposea

framework for modeling in a consistent manner the VIX index and the VXX. This study makes it possible to linkthe

properties of VXX to those of the VIX in a tractable way. It is capable of quantifying the systematic loss observed

FIGURE 1Market dollor volume and

open interest of VXX. This figure shows the

daily trading volume and open interest of

VXX from May 28, 2010 to January 25, 2019

2

The VXX volume data are directly extracted from OptionMetrics (2020b).

LIN AND ZHANG

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empirically for VXX when the VIX futures term structureis in contango. However, this approach cannot provide exact

VXX options pricing. In the last line of research, the joint dynamics of SPX and its stochastic volatility are specified, and

expression for spot VIX is then derived as forward realized volatility of SPX. Given themethodology in Gehricke and

Zhang (2018), VXX also can be developed by using Zhang's formula in Zhang and Zhu (2006) for VIX futures prices.

Later, Gehricke and Zhang (2020b) generalize his model to a modified Heston framework, where the volatility process

includes jumps and a stochastic long‐term mean. But this approach does not provide VXX options pricing either.

In this paper, we first establish a joint and model‐free analytical framework for consistently modeling the VIX and

VXX. We develop a new model‐free VXX formula and present it in terms of VIX futures in both discrete and

continuous forms. In the discrete situation, we consider several different naive VXXs, depending on the number of

rolls: never roll, roll once, roll twice, and daily roll. Different cases of VXX can be represented as numbers of VIX

futures ratios. This VXX representation is the main contribution to the financial‐economic literature on VXX since this

representation makes it possible for traders and researchers to investigate VIX and VXX option markets together and

explore their connection. Taking the log‐normal Ornstein–Uhlenbeck (LOU) model as an example, we derive a VIX

and VXX options pricing formula consistently. To further create nonflat IV and verify the forecasting ability perfor-

mance in VIX and VXX markets, we then consider the log‐normal Ornstein–Uhlenbeck with stochastic volatility

(LOUSV) model to create the nonflat IV smirk. Although the LOUSV model dynamics are nonaffine, we contribute two

mathematical methods for deducing the characteristic function ofV

(

lnVIX ,

)

TT

by transferring it to affine processes.

Figure 2shows the constant 30‐day daily IV level in VIX and VXX options from January 1, 2018 to December 26, 2018.

To create a constant maturity IV curve, we interpolate it to the closest and second maturity.

3

The constant 30‐day daily

IV level of VXX is shown as the blue line, while the purple line shows the corresponding VXX's IV level in the market.

Roughly speaking, two IVs have the same trends, and the VXX's IV level can be explained approximately by 70%–80%

of the VIX's IV level, which provides the possibility that the VXX's IV can be covered/forecasted by the VIX's IV.

Finally, numerical experiments study the performance based on a pricing test and IV forecasting test, and further

clarify the VXX formula and conclude that the LOUSV model is a well‐performing model.

The structure of this paper is as follows. The VXX is first defined in Section 2. In Section 3, we establish a model‐

free VXX formula in terms of VIX futures both in continuous and discrete forms. Section 4gives an example of the

LOU model to derive the VIX and VXX options pricing formula and its IV level in several different scenarios based on

the number of rolls. In Section 5, the formula for VIX and VXX options under LOUSV is derived. All numerical

FIGURE 2Constant 30‐day daily IV

level in VIX and VXX options. This figure

shows the constant 30‐day daily IV level of

VIX and VXX option from January 1, 2018 to

December 26, 2018. To create constant

maturity IV curve, we interpolate it to the

closest and second maturity. The constant

30‐day daily IV level of VXX is shown as the

blue line while the purple one is the

corresponding VXX's IV level in the

market. IV, implied volatility; VIX, Volatility

Index

3

The VIX and VXX option data are directly extracted from OptionMetrics (2020a,2020b).

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