Option Pricing with the Realized GARCH Model: An Analytical Approximation Approach

• Date: 01 April 2017
• Published date: 01 April 2017
• DOI: http://doi.org/10.1002/fut.21821
• Author: Zhuo Huang,Peter Reinhard Hansen,Tianyi Wang

Option Pricing with the Realized

GARCH Model: An Analytical

Approximation Approach

Zhuo Huang , Tianyi Wang,*and Peter Reinhard Hansen

We derive a pricing formula for European options for the Realized GARCH framework based

on an analytical approximation using an Edgeworth expansion for the density of cumulative

return. Existing approximations in this context are based on a Gram–Charlier expansion while

the proper Edgeworth expansion is more accurate. In relation to existing discrete-time op-

tion pricing models with realized volatility, our model is log-linear, non-affine, with a flexible

leverage effect. We conduct an extensive empirical analysis on S&P500 index options and the

results show that our computationally fast formula outperforms competing methods in terms

of pricing errors, both in-sample and out-of-sample. ©2016 Wiley Periodicals, Inc. Jrl Fut

Mark 37:328–358, 2017

1. INTRODUCTION

It is well known that realized measures of volatility,which are computed from high frequency

data, provide accurate measurements of the latent volatility process. The prime example is the

realized variance, see, for example, Andersen, Bollerslev, Diebold, and Labys (2003). Volatil-

ity is fundamental for option pricing, so it is natural to explore ways to incorporate realized

measures into option pricing issues. Several papers have recently shown that discrete time

models that incorporate the realized variance, can significantly improve the performance of

option pricing. For example, Christoffersen, Feunou, Jacobs, and Meddahi (2014) develop

an affine discrete-time model to provide a closed-form option valuation formula through the

conditional moment-generating function. The volatility dynamic is modeled as a weighted av-

erage between components from daily returns and realized variances, where both components

Zhuo Huang is at the National School of Development, Peking University, Beijing, China. Tianyi Wang

is at the School of Banking and Finance, University of International Business and Economics, Beijing,

China. Peter Reinhard Hansen is at the University of North Carolina at Chapel Hill, North Carolina, USA

and CREATES, Aarhus, Denmark. We are grateful to Bob Webb (editor) and all participates of the 1st

China Derivatives Markets Conference (CDMC) and the 6th IMS-FIPS Workshop for useful comments

that substantially improved the paper. We thank Xiaolin Jiang for his excellent research assistance. Zhuo

Huang acknowledges financial support from the Fund of the National Natural Science Foundation of China

(71201001,71671004). Tianyi Wang(corresponding author) acknowledges financial support from the Youth

Fund of National Natural Science Foundation of China (71301027), the Ministry of Education of China,

Humanities and Social Sciences Youth Fund (13YJC790146), and the Fundamental Research Fund for

the Central Universities in UIBE(14YQ05). Peter Reinhard Hansen acknowledges support from the Center

for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research

Foundation.

*Correspondence author,University of International Business and Economics, School of Banking and Finance,

906 BoXue Building, No.10 Huixin East Street, Chaoyang district, Beijing 100029, China. Tel: 86-10-64492513,

Fax: 86-10-64495059, e-mail:tianyiwang@uibe.edu.cn

Received August 2016; Accepted September 2016

The Journal of Futures Markets, Vol. 37, No.4, 328–358 (2017)

©2016 Wiley Periodicals, Inc.

Published online 18 November 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21821

Option Pricing with the Realized GARCH Model 329

have a Heston–Nandi1structure. They show that including the realized variance results in

a considerable pricing improvement. Their paper also suggests the need for further research

on pricing, using non-affine models and modeling leverage effect separately for both return

and realized measures. A related framework is that in Corsi, Fusari, and Vecchia(2013), who

employ a Heterogeneous Autoregressive Gamma (HARG) model. This model assumes that

the realized variance follows a simple process (with linear long-memory features) and option

pricing can be obtained using Monte Carlo simulation. This model was further developed in

Majewskia, Bormetti, and Corsi (2015), who enhance the HARG model with a Heston–Nandi

type leverage. Their framework includes a class of linear GARCH models with parabolic lever-

age, including those in Heston and Nandi (2000) and Christoffersen, Jacob, Ornthanalai, and

Wang (2008), and the framework conveniently leads to a closed-form option pricing formula.

In this paper, we derive the option pricing formula for the Realized GARCH framework,

which may result in better pricing performance, because the Realized GARCH framework

has proven to be superior to conventional GARCH models for the modeling of returns and

for forecasting volatility. The Realized GARCH model was proposed by Hansen, Huang,

and Shek (2012), and further refined by Hansen and Huang (2016), which is the variant we

adopt for the option pricing in this paper. The model may be labeled as a non-affine log-linear

Realized Exponential GARCH model.

The Realized GARCH framework is attractive for option pricing for several reasons.

First, the realized variance is incorporated in the model and linked to the latent conditional

volatility through a measurement equation. This not only improves the accuracy of the volatil-

ity forecast, but also allows for an additional risk premium that relates to volatility-specific

shocks. Second, the model benefits from having both return and volatility shocks, similar to

stochastic volatility models. Still, the Realized GARCH model is an observation-driven model

that permit straight forward estimation by the maximum likelihood. Third, the measurement

equation in our model does not require the realized measure to be an unbiased estimator of

the daily volatility. Unbiased estimators are difficult to obtain because high-frequency data

is only available for a fraction of the day. Market microstructure noise that is not properly

accounted for, see Hansen and Lunde (2006), can also induce bias in realized measures.

In contrast, the HARG model requires an unbiased estimator, and assumes that a rescaling

of the realized variance (based on the trading hour’s information) achieves this objective.

Fourth, the model operates with distinct leverage functions for returns and realized vari-

ances. The importance of this flexibility is documented in Hansen, Huang, and Wang (2016)

for the pricing of the CBOE volatility index (VIX). Fifth, the log-linear specification avoids

many of the constraints that often must be imposed to guarantee positivity of the volatility

process. Taking the logarithm also serves to reduce the impact of outliers in the realized

measure of volatility. This transformation makes the model more stable, especially during

periods with high volatility of volatility.

A quasi closed-form option pricing formula is very hard to obtain for a non-affine model,

using the standard method based on the moment-generating function and the inverse Fourier

transformation. When a closed-form expression is unavailable one can resort to Monte Carlo

methods, see, for example, Corsi et al. (2013) and Kanniainen, Lin, and Yang (2014). While

this method is straight forward to apply, it can be very time consuming to achieve a desir-

able accuracy. This makes analytical approximation methods an attractive alternative. For

GARCH-type models, including non-affine models, Duan, Gauthier, and Simonato (1999)

1In the Heston–Nandi GARCH model (Heston & Nandi, 2000), volatility is filtered through return data and a

closed-form pricing formula is provided.