Estimation of the optimal futures hedge ratio for equity index portfolios using a realized beta generalized autoregressive conditional heteroskedasticity model

• Published date: 01 November 2018
• Date: 01 November 2018
• Author: Yu‐Sheng Lai
• DOI: http://doi.org/10.1002/fut.21937

Received: 8 August 2017

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Revised: 11 May 2018

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Accepted: 13 May 2018

DOI: 10.1002/fut.21937

RESEARCH ARTICLE

Estimation of the optimal futures hedge ratio for equity

index portfolios using a realized beta generalized

autoregressive conditional heteroskedasticity model

Yu‐Sheng Lai

Department of Banking and Finance,

National Chi Nan University, Puli,

Taiwan

Correspondence

Yu‐Sheng Lai, Department of Banking

and Finance, National Chi Nan

University, No. 1, Daxue Rd., Puli

Township, Nantou County 54561, Taiwan.

Email: yushenglai@ncnu.edu.tw

Funding information

Ministry of Science and Technology,

Taiwan, Grant/Award Number:

105‐2410‐H‐260‐008

This paper employs a realized beta generalized autoregressive conditional

heteroskedasticity model for optimal futures hedging. The model has a flexible

structure and is complete because all observed returns and realized measures

are jointly modeled in a system. This enables the incorporation of important

features that may affect the hedge ratio estimation. The model is applied to

equity indices, and substantial dependence between return and volatility

indicates the essential of modeling statistical leverage. Predictive ability testing

confirms the superiority of the model for reducing the hedged portfolio risk.

The predictive ability of the model can translate into pronounced economic

benefits, particularly for short‐term hedges.

KEYWORDS

EGARCH, estimation uncertainty, futures hedge ratio, high‐frequency data, predictive ability

testing

1

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INTRODUCTION

Understanding and predicting temporal dependence in the second‐order moments of spot and futures returns are

crucial for optimal futures hedging. This is because the optimal hedge ratio is determined by the ratio of the conditional

covariance of spot and futures returns to the conditional variance of futures returns (Baillie & Myers, 1991; Kroner &

Sultan, 1993). Several studies have used multivariate generalized autoregressive conditional heteroskedasticity

(GARCH) models for the modeling of returns, forming so‐called GARCH hedging strategies (Baillie & Myers, 1991;

Brooks, Henry, & Persand, 2002; Cecchetti, Cumby, & Figlewski, 1988; Kroner & Sultan, 1993; Lien, Tse, & Tsui, 2002;

Park & Switzer, 1995). Although these GARCH models capture the time‐varying covariance structure of spot and

futures returns, recent studies have documented that a GARCH hedging strategy incorporating realized measures

computed from high‐frequency data (e.g., popular realized covariances) results in higher hedging performance than the

conventional GARCH strategy, which does not incorporate high‐frequency information (Lai, 2016; Lai & Sheu, 2010).

For example, for the modeling of volatility, conventional GARCH models by R. F. Engle (1982) and Bollerslev (1986)

rely exclusively on the daily return. However, the squared (cross‐product) return provides an extremely weak signal as

for the current level of (co)volatility; thus the GARCH models are poorly suited particularly during periods of rapid

changes in the underlying (co)volatility structure (Andersen, Bollerslev, Diebold, & Labys, 2003). To alleviate this

problem, a new class of volatility models that incorporate realized measures are introduced (R. F. Engle, 2002; R. F.

Engle & Gallo, 2006; Hansen, Huang, & Shek, 2012; Shephard & Sheppard, 2010). Incorporating realized measures into

GARCH equations (known as the GARCH‐X model) is found to not only improve the empirical fit but also enable the

forecasts to swiftly adapt to the changing markets.

J Futures Markets. 2018;38:1370–1390.wileyonlinelibrary.com/journal/fut1370

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© 2018 Wiley Periodicals, Inc.

This paper employs the realized beta GARCH framework proposed by Hansen, Lunde, and Voev (2014) to jointly

model the return vector and the associated realized measures of volatility and covolatility. The use of realized measures

facilitates the construction of precise volatility proxies for quadratic covariation (Andersen, Bollerslev, Diebold, &

Labys, 2001; Barndorff‐Nielsen & Shephard, 2002, 2004; Barndorff‐Nielsen, Hansen, Lunde, & Shephard, 2011). In the

modeling of returns, incorporating precise realized measures into GARCH equations enables the crowding out of noisy

return‐based proxies. The measurement equations provide a direct link between realized measures and the conditional

moments of returns through the measurement errors. The dynamic descriptions of realized measures are crucial for

producing multiperiod‐ahead predictions. In contrast to the realized beta GARCH model, the GARCH‐X models used

by Lai and Sheu (2010) and Lai (2016), which ignore these equations, can only produce one‐period‐ahead forecasts.

Additionally, the presence of leverage functions facilitates the direct modeling of the asymmetric responses of volatility

to return shocks, which has been found to be important for equity futures hedging (Brooks et al., 2002; Lai & Sheu,

2011). In the presence of a spillover parameter, the realized beta GARCH model enables the examination of the

potential impact of spot volatility on futures volatility. Haigh and Holt (2002) discussed the importance of accounting

for volatility spillovers in the estimation of time‐varying hedge ratios in energy futures markets.

This paper contributes a joint modeling of the following financial series features analyzed in previous works that could

affect the optimal hedge ratio and its hedging effectiveness: (i) heteroskedasticity in spot and futures markets

(e.g., Baillie & Myers, 1991; Cecchetti et al., 1988; Lien et al., 2002); (ii) heteroskedasticity with a short‐time response

(e.g., Lai & Lien, 2017; Lai & Sheu, 2010); (iii) the asymmetric response of volatility to positive and negative shocks (e.g.,

Brooks et al., 2002); and (iv) the volatility spillover between spot and futures markets (e.g., Haigh & Holt, 2002). Notably, the

existence of measurement equations allows to produce multiperiod predictions when considering all of these crucial features

in the estimation of hedge ratios. Previous studies such as Figlewski (1985); Chen, Lee, and Shrestha (2004); Lien and

Shrestha (2007); and Lai and Sheu (2010) have reported the optimal hedge ratios for different hedging horizon lengths.

However, none of them has simultaneously considered these features in their multiperiod hedging decisions.

In the empirical analysis, the hedging performance of the realized beta GARCH model is examined using data on equity

index futures contracts with the associated underlying spot indices; six hedging horizons ranging from 1 day to 4 weeks are

used in the analysis. The analysis focuses on comparing out‐of‐sample predictive ability, because forecasting is central to

financial decision‐making. Under the null hypothesis of equally predictive ability, the forecasting performance of the

sophisticated model is compared with that of a simpler benchmark model. To account for the effect of estimation uncertainty

in the forecast evaluation process, the finite‐sample predictive ability tests of Giacomini and White (2006) are used; in

addition, a daily rolling scheme with two estimationwindowsisused.Inthepseudo‐out‐of‐sample environment, pseudo‐

out‐of‐sample comparisons enable estimation uncertainty to be accounted in the testing inference, whereas the commonly

applied inference proposed by Diebold and Mariano (1995) is designed for model‐free forecast comparisons.

The remainder of this paper is structured as follows: Section 2 presents the econometric methodology for a modeling

of the joint density; Section 3 provides the data and preliminary analysis results; Section 4 describes the method for

estimating multiperiod hedge ratios and measuring their hedging performance and provides inferences drawn from the

tests of predictive ability; Section 5 describes the main empirical results of the model; and Section 6 concludes the

paper.

2

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REALIZED BETA GARCH MODEL

2.1

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Model specifications

The realized GARCH model developed by Hansen et al. (2012) provides a new framework for the joint modeling of

returns and realized volatility measures. Considering that realized measures provide much more information on the

current level of volatility than the squared return, incorporating realized measures into the GARCH estimation greatly

improves the empirical fit. As a multivariate extension, the realized beta GARCH model by Hansen et al. (2014) is also

complete because all observed variables (returns and realized measures) are jointly modeled in a system. The

specification used for the modeling of spot (market) and futures (individual) returns is discussed as follows.

Let rt0, and rt1, denote the market and individual returns at time

t

, respectively, and let xt0, and x

t

1, denote the

corresponding realized variance measures.

1

LAI

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1371

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How to use high‐frequency data to construct the returns and realized volatility measures will be presented in Section 3.