Pricing VIX Futures and Options With Good and Bad Volatility of Volatility

• Published date: 01 November 2024
• Author: Zhiyu Guo,Zhuo Huang,Chen Tong
• Date: 01 November 2024
• DOI: http://doi.org/10.1002/fut.22545

Journal of Futures Markets

RESEARCH ARTICLE

Pricing VIX Futures and Options With Good and Bad

Volatility of Volatility

Zhiyu Guo

1,2

| Zhuo Huang

2,3

| Chen Tong

4,5

1

School of International Economics, China Foreign AffairsUniversity, Beijing, China |

2

Institute of Digital Finance, Peking University, Beijing, China |

3

China

Center for Economic Research, National School of Development, Peking University, Beijing, China |

4

Department of Finance,School of Economics& WangYanan

Institute for Studies in Economics (WISE), Xiamen University, Xiamen, China |

5

Laboratory of Digital Finance, Xiamen University, Xiamen, China

Correspondence: Chen Tong (tongchen@xmu.edu.cn)

Received: 19 January 2024 | Revised: 3 April 2024 | Accepted: 13 July 2024

Funding: This research was supported by the Fund of the National Natural Science Foundation of China (72271010, 72241418), Youth Fund of the National

Natural Science Foundation of China (72301227), and the Ministry of Education of China, Humanities and Social Sciences Youth Fund (22YJC790117).

Keywords: high‐frequency VIX | realized semivariance| VIX futures and option pricing

ABSTRACT

This article studies the pricing of VIX futures and options by directly modeling the dynamics of VIX, based on realized

semivariances computed from high‐frequency data of VIX. We derive the closed‐form pricing formula for both the VIX futures

and options. The empirical results show that the new model provides superior pricing performance compared with the model

based on conventional unsigned realized variance and the classic Heston‐Nandi GARCH model, both in sample and out of

sample. Our study confirms that the decomposition of realized variance into upside and downside components helps to improve

the pricing performance for VIX futures and options.

JEL Classification: C51, C52, G12, G13

1 | Introduction

The CBOE VIX, as a measure of market implied volatility of

S&P 500 index, has been widely studied over the past two

decades. The launch of futures and options written on VIX in

2004 and 2006 enables investors to trade volatility directly and

hedge their exposure of portfolio to volatility risk. Since then,

VIX derivatives have become an important part of financial

markets, which prompt researchers to devote more attention to

study the valuation of VIX derivatives.

The literature on pricing VIX derivatives can be mainly divided

into two categories. One strandistomodelthedynamicsofS&P

500 index (SPX) under the physical measure and then derive the

pricing formulas for VIX and VIX derivatives after risk

neutralization. Under the continuous‐time framework, Zhang

and Zhu (2006) explored the pricing of VIX futures by utilizing the

Heston stochastic model of SPX. Sepp (2008)consideredtheVIX

option pricing and assumed that the variance of SPX returns is

driven by the Heston stochastic model with variance jumps. Other

studies under the continuous‐time framework include Lin (2007),

Lin and Chang (2009), Lian and Zhu (2013), and Luo, Zhang, and

Zhang (2019). Under the discrete‐time framework, Wang et al.

(2017)andTong(2024) priced the CBOE VIX under affine and

non‐affine GARCH models, respectively.

1

Huang, Tong, and Wang

(2019) and Wang and Wang (2020)adoptedtheHAR‐gamma

process with flexible leverage components (LHARG) and the

GARV model with two hidden components and jump (GARV‐2C‐

J) to depict the dynamics of SPX and price VIX futures,

respectively.ForVIXoptions,Caoetal.(2 020)providedsemi

closed‐form solutions under affine GARCH models, and Tong and

Huang (2021) derived analytical pricing formulas when the SPX

return follows the GARV and Realized GARCH models.

The second strand of the literature is to directly model the dynamics

of VIX under the risk‐neutral measure. For example, Grünbichler

© 2024 Wiley Periodicals LLC.

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https://doi.org/10.1002/fut.22545

and Longstaff (1996)usedthemean‐reverting square‐root process to

model the implied volatility index and price options, Goard and

Mazur (2013) developed analytic solutions for VIX option prices

when the VIX are modeled by the 3/2‐model, and Mencia and

Sentana (2013) compared the empirical performance of several

models on pricing VIX derivatives. Other work under the

continuous‐time framework included Park (2016) and Jing, Li,

and Ma (2020). For discrete‐time models, the literature is more

about pricing VIX futures: Yin, Bian, and Wang (2021)applied the

HAR model with asymmetric jumps to model the logarithm of VIX

and priced VIX futures. Based on a similar process, Tong, Huang,

and Wang (2022) investigated the information role of VIX futures in

pricing VIX options. Wang et al. (2022) extended the model in Yin,

Bian, and Wang (2021) by incorporating dynamic volatility with

short‐and long‐run components and dynamic jump intensity. The

above‐mentioned two studies only use daily series of VIX, not the

information contained in the high‐frequency data of VIX. Jiang

et al. (2022)appliedVIX5‐min high frequency data to identify the

interday and intraday jumps of VIX and incorporated these jumps

when modeling the logarithmic VIX to price VIX futures.

Modeling the dynamics of S&P 500 to price VIX derivatives can

provide a consistent framework for the SPX, VIX, and VIX

derivatives, but it can also induce inaccurate integration problems,

and the computation is time consuming, while directly modeling

the dynamics of VIX can overcome these problems. However, the

majority of the literature on pricing VIX derivatives belongs to the

first category, and the exploration of the second modeling method

is still at the initial stage. Therefore, this paper studies VIX option

pricing by directly modeling the dynamics of VIX.

It is well known that the realized variance computed from high‐

frequency data provides accurate measurement of latent volatility

and thus improves the volatility forecasting and derivatives pricing.

Further, Barndorff‐Nielsen, Kinnebrock, and Shephard (2010)

proposed to distinguish upside and downside components from

realized variance, which are calculated by summing up intraday

positive and negative returns, respectively. Existing studies showed

that such decomposition of realized variance of SPX returns can

improve volatility prediction (Patton and Sheppard 2015), VIX

futures pricing (Tong and Huang 2023), and stock option pricing

(Feunou and Okou 2019). However, the studies on the role of

realized semivariance of VIX in derivatives pricing remain limited.

As a volatility index, the dynamics of VIX exhibit characteristics

distinct from those of stock prices. The asymmetric leverage effect

2

dictates that volatility tends to escalate more significantly during

stock price declines than it decreases during market upswings.

This asymmetry implies that the realized upside volatility of VIX

may manifest more pronounced peaks than its downside volatility.

It also reveals the importance of separately modeling the upside

and downside components. These insights, supported by the

summary statistics presented in Table 1, underscore the potential

impact of these unique characteristics on VIX options pricing.

Therefore, this paper constructs the realized semivariance of VIX

based on high‐frequency VIX data and check if the distinction

between realized upside and downside variances can improve the

pricing performance for VIX futures and option. Considering the

long‐memory feature of volatility process,wetaketheheteroge-

neous autoregressive (HAR) framework and combine it with a

realized semivariance‐based model.

Our contributions are threefold. First, we contribute to the

literature on pricing VIX futures and options by directly

modeling VIX with realized semivariances. Our paper is closely

related to Jiang et al. (2022) and Qiao and Jiang (2023), who

priced VIX derivatives using high‐frequency data of VIX.

However, both of these two studies only focused on the pricing

of VIX futures. The former investigated the impact of jumps on

futures prices, and the latter combined realized the

semivariance‐based model with support vector regression

(SVR) to price VIX futures. Our work is different from theirs

since we provide a unified framework to jointly price VIX

futures and options. And to the best of our knowledge (if any),

this paper is the first to investigate the role of realized

semivariance of VIX in pricing VIX options. Second, we drive

the closed‐form expressions of VIX futures price and option

price under the proposed model, in which the moment

generating function is expressed as a exponential linear

function of contemporaneous and lagged logarithm VIX, as

well as the conditional upside and downside variances. This

makes the computation of futures and option prices much

easier and time saving. Third, we conduct an extensive

empirical analysis to investigate the pricing performance of

the proposed realized semivariance‐based model. We compare

its in‐and out‐of‐sample performance with the model based on

conventional unsigned realized variance and the classic Heston‐

Nandi GARCH model. The empirical results show that the

realized semivariance‐based model performs best both in and

out of sample, indicating that separating upside and downside

realized variances of VIX is helpful to obtain more accurate

prices of VIX futures and options.

The remainder of the paper is organized as follows: Section 2

introduces the realized semivariance‐based model, HAR‐RSV.

Section 3derives the pricing formulas of VIX futures and

options under the HAR‐RSV model. In Section 4, we introduce

two competing models and derive their pricing formulas.

Section 5presents empirical analysis, where we describe the

data, estimate the parameters, and investigate the pricing

performance of the models, both in and out of sample. Finally,

Section 6concludes the paper.

2 | The Realized Semivariance‐Based Model

(HAR‐RSV)

According to Barndorff‐Nielsen, Kinnebrock, and Shephard

(2010), for a given day

t

, the upside realized semivariance

R

Vtu

and the downside realized semivariance

R

V

td

for the logarithm

of the

V

IX

t, denoted as y=logVIX

tt, are defined as









RV r r

RV r r

={>0},

={<0},

tu

j

M

tj tj

td

j

M

tj tj

=1

,

2,

=1

,

2,

t

t

where

ryy=−

tj tj tj

,,,−1

is the

j

th intraday return during day

tM,tis the number of returns on that day, and



⋅{} is the

indicator function. By construction, the total realized variance,

R

V

t

, is given by

R

VRV RV=+

ttut

d

.

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