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Pricing the CBOE VIX Term Structure and VIX Futures with

Realized Volatility

Zhuo Huang Chen Tong Tianyi Wang

May 2017

Abstract

Using an extended LHARG model proposed by Majewskia et al. (2015), we derive the closed-form

pricing formulas for both the CBOE VIX term structure and VIX futures with different maturity.

Our empirical results suggest that the quarterly and yearly components of lagged realized volatility

should be added into the model to capture long-term volatility dynamics. With the realized volatility

based on high frequency data, the proposed model provide superior pricing performance compared

to the classic Heston-Nandi GARCH model, both in-sample and out-of-sample. The improvement is

more pronounced during high volatility periods.

Keywords: Implied volatility, Volatility term structure, VIX futures, Realized volatility

JEL classification: C19;C22;C80

National School of Development, Peking University, Beijing 100871, P.R. China, Email: zhuohuang@nsd.edu.cn. ZhuoHuang acknowledges financial support from the National Natural Science Foundation of China (71671004).National School of Development, Peking University, Beijing 100871, P.R. China, Email:tongchen@pku.edu.cn.Corresponding author, Department of Financial Engineering, School of Banking and Finance, University of Interna-

tional Business and Economics, Beijing 100029, P.R. China, Email: tianyiwang@uibe.edu.cn. Tianyi Wang acknowledgesfinancial support from the Youth Fund of National Natural Science Foundation of China (71301027), the Ministry ofEducation of China, Humanities and Social Sciences Youth Fund (13YJC790146), and the Fundamental Research Fundfor the Central Universities in UIBE(14YQ05).

1

1 Introduction

The well-known Chicago Board Options Exchange (CBOE) VIX index, computed from a panel of

options prices, is a model-free measure of expected average variance for next 30 days under the risk

neutral measure. The index has become the benchmark for stock market volatility and it is used as

the investor fear gauge for the equity markets. With the launch of VIX futures in 2004 and VIX

options in 2006, volatility derivatives have received increasingly attention in the market as the market

average daily volume of VIX options and VIX futures have expanded to over 137 and 25 times in the

last decade. The VIX-linded products essentials creates a volatility market that enables investors to

trade volatility directly just like to equity or fixed income securities1. In addition to the VIX index that

measures 1-month implied volatility, CBOE has also lunched a series of implied volatility indices across

different maturities in recent years, to reflect volatility term structure under the risk neutral measure.

The CBOE S&P 500 3-Month Volatility Index (Ticker:VXV) was lunched on November 2007. The

CBOE Mid-Term Volatility Index (Ticker: VXMT), a measure of the expected volatility of the S&P

500 Index over a 6-month time horizon, was launched on November 20132.

Zhang and Zhu (2006) gave the first attempt to price VIX futures based on the classic continuous-

time Heston model. Lin (2007) extended the model with simultaneous jumps in both returns and

volatility to price VIX futures with approximation formula. Adding jumps into mean-reverting process

are also investigated by Sepp (2008), Zhang et al. (2010) and Zhu and Lian (2012) etc. Under the

discrete-time volatility framework, Wang et al. (2016) derived the pricing formulas for both VIX and

VIX futures using the discrete-time Heston-Nandi GARCH model. Inspired by Hao and Zhang (2013)

and Kanniainen et al. (2014), the model is estimated in a joint fashion where both the underling

dynamics as well as the VIX futures prices are taken into account in the objective function. Results

show that the model can yield good in-sample and out-of-sample prices when VIX or VIX futures are

involved in estimation.

Since the seminal work of Andersen et al. (2003), the realized volatility computed from high frequency

intra-day returns have proved to be an accurate measurement of the latent volatility process. Models

using the realized measures to model and forecast volatility have attracted great attention in recent years.

Such models include but not limited to the heterogenous autoregressive (HAR) model of Corsi (2009),

the MEM model of Engle and Gallo (2006), the HEAVY model of Shephard and Sheppard (2010) and

the Realized GARCH model of Hansen et al. (2012). Among these models, the HAR model is the most

popular in applied research on volatility modelling due to its estimation simplicity and good forecasting

performance. The model introduces a cascade structure into the linear autoregression framework in

1Luo and Zhang (2014) provide a good discuss of market for volatility derivatives.2CBOE reported historical data of VXMT back to January 2008.

2

which the current daily realized variance is regressed on the lagged realized variance over the past

day, past week and past month. Empirical studies show the HAR model provide a parsimonious but

good approximation for the long memory process of volatility. For option pricing purpose, Corsi et al.

(2013) extended the HAR model with an Gamma innovation (namely the Heterogenous Autoregressive

Gamma, HARG) and specified an exponentially affine pricing. Based on that, Majewskia et al. (2015)

further developed the model by allowing more flexible leverage components (LHARG) and derived the

analytical pricing formula for European options.

While most studies focus on the performances of the HAR model and its extensions in forecasting

volatility or realized volatility under the physical measure, Corsi et al. (2013) and Majewskia et al. (2015)

have shown the HAR framework is also capable to match the volatility information implied by option

prices, i.e., under the risk neutral measure. In this paper, we derive the analytical formulas for VIX term

structure and VIX futures based on an modified version of LHARG model. The empirical results suggest

we should add the quarterly and yearly average of lagged realized volatility into the LHARG model

to capture volatility dynamics in longer horizons in order to price volatility index and its derivatives.

Compared with the pricing formula under the classic Heston-Nandi GARCH model in Wang et al. (2016),

our proposed model provides superior performance in pricing VIX term structures and VIX futures. The

improvement is more pronounced during high volatility periods, when the realized volatility contains

relatively more accurate information about underlying volatility. Our empirical findings are robust in

out-of-sample analysis.

The remainder of the paper is organized as follows. Section 2 discusses the model setup and derive

the pricing formula for VIX term structure and VIX futures. Section 3 discusses model estimation using

different datasets. Section 4 presents the empirical results and Section 5 concludes.

2 THE MODEL

2.1 LHARG Model and Risk Neutralization

In this paper, we denote the original LHARG model as LHARG-M since it contains volatility components

up to monthly average. It is extended with quarterly average (63 trading days) to LHARG-Q and with

both quarterly and yearly average (252 trading days) to LHARG-QY. If properties are applied to all

three models, we use LHARG to save space. The LHARG-QY model is given by:

Rt+1 = r + RVt+1 1

2RVt+1 +

RVt+1t+1, t+1 i.i.dN(0, 1) (2.1)

RVt+1|Ft (,(RVt,Lt), )

3

(RVt,Lt) =1

(d+dRV

(d)t + wRV

(w)t + mRV

(m)t + qRV

(q)t + yRV

(y)t +

d`(d)t + w`

(w)t + m`

(m)t + q`

(q)t + y`

(y)t )

We define components as follows:

RV(d)t = RVt `

(d)t =

2t 1 2t

RVt

RV(w)t =

1

4

4i=1

RVti `(w)t =

1

4

4i=1

(2ti 1 2tiRVti)

RV(m)t =

1

17

21i=5

RVti `(m)t =

1

17

21i=5

(2ti 1 2tiRVti)

RV(q)t =

1

41

62i=22

RVti `(q)t =

1

41

62i=22

(2ti 1 2tiRVti)

RV(y)t =

1

189

251i=63

RVti `(y)t =

1

189

251i=63

(2ti 1 2tiRVti)

The Rt, RVt, r denote the log-return of underlying index, the realzied volatility and the risk free rate

respectively. captures the equity risk premium. The coditional distribution of RVt+1 features a

noncentral gamma distribution (denoted as ()) with shape and scale parameters equals and . Thelocation parameter is given by (RVt,Lt). It is easy to see that LHARG-Q is nested in LHARG-QY

and the original LHARG-M can be recovered by setting q, y, q and y equal to zero.

The specification of the leverage function has been inspired by Christoffersen et al. (2008) and

enriched by a heterogeneous structure3. Unlike the the Heston-Nandi type leverage function, (RVt,Lt)

is no longer guaranteed to be positive. Nevertheless, Majewskia et al. (2015) provided numerical evidence

of the effectiveness of the analytical results in describing a regularized version of the model.

According to the properties of noncentral gamma distribution, the conditional expectation of RVt+1

in physical (P ) measure is

EPt [RVt+1] = + (RVt,Lt)

= + d+ dRV(d)t + wRV

(w)t + mRV

(m)t + qRV

(q)t + yRV

(y)t +

d`(d)t + w`

(w)t + m`

(m)t + q`

(q)t + y`

(y)t

3Majewskia et al

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