Optimal futures hedging by using realized semicovariances: The information contained in signed high‐frequency returns

• Published date: 01 May 2023
• Author: Yu‐Sheng Lai
• Date: 01 May 2023
• DOI: http://doi.org/10.1002/fut.22406

Received: 12 July 2022

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Accepted: 21 February 2023

DOI: 10.1002/fut.22406

RESEARCH ARTICLE

Optimal futures hedging by using realized

semicovariances: The information contained

in signed high‐frequency returns

Yu‐Sheng Lai

Department of Banking and Finance,

National Chi Nan University, Puli,

Nantou, Taiwan

Correspondence

Yu‐Sheng Lai, Department of Banking

and Finance, National Chi Nan

University, No. 1, Daxue Rd, Puli

Township 545301, Nantou County,

Taiwan.

Email: yushenglai@ncnu.edu.tw

Funding information

Ministry of Science and Technology,

Taiwan, Grant/Award Number: 109‐2410‐

H‐260‐007

Abstract

This paper proposes a realized semicovariance‐based generalized autoregres-

sive conditional heteroskedasticity (GARCH) model for optimal futures

hedging in which realized semicovariances are computed from signed high‐

frequency returns. The model enables flexible, continuous leverage for equity

indices and exhibits stronger responses to jointly negative return shocks than

do traditional threshold‐based asymmetric GARCH models. Our results

indicate that the proposed model outperforms simpler models in model fit,

covariance prediction, and portfolio variance reduction and can help achieve

pronounced economic gains for hedgers. The findings demonstrate that signed

high‐frequency returns contain valuable information for explaining covariance

asymmetries and provide managerial implications for market participants to

improve risk management.

KEYWORDS

asymmetric covariance, dynamic hedge ratio, hedging effectiveness, high‐frequency data,

realized semicovariance

JEL CLASSIFICATION

C32, C53, G11

1|INTRODUCTION

Multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models are commonly used to

estimate optimal futures hedge ratios (Alexander & Barbosa, 2008; Baillie & Myers, 1991; Cecchetti et al., 1988;

Kavussanos & Visvikis, 2008; Kroner & Sultan, 1993; Lien et al., 2002; Myers, 1991; T. H. Park & Switzer, 1995). Some

studies have examined asymmetric behavior in the covariance structure of spot and futures returns by using

asymmetric multivariate GARCH models (e.g., Brooks et al., 2002; Cotter & Hanly, 2012; Kroner & Ng, 1998; Lai &

Sheu, 2011; Meneu & Torró, 2003; S. Y. Park & Jei, 2010). Asymmetric GARCH models enable covariances to respond

asymmetrically to positive and negative return shocks, which is known as the “leverage effect”(Black, 1976). Empirical

evidence indicates the presence of asymmetries in spot–futures covariances; however, the associated hedge ratios are

insensitive to asymmetric return shocks. Thus, the improved model fit due to covariance asymmetries does not

necessarily imply lower hedged portfolio variance, especially in an out‐of‐sample context.

The availability of high‐frequency intraday data has driven the development of realized covariance estimators used to

determine the increments to quadratic covariation, which is the standard ex post measure of the covariation of asset prices

J Futures Markets. 2023;43:677–701. wileyonlinelibrary.com/journal/fut © 2023 Wiley Periodicals LLC.

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(e.g., Andersen et al., 2001; Barndorff‐Nielsen & Shephard, 2004). A new class of realized covariance‐based GARCH models

has emerged, such as the multivariate high‐frequency‐based volatility model of Noureldin et al. (2012). In hedging applications,

several studies have demonstrated that incorporating accurately realized covariances into standard GARCH models facilitates

the determination of covariance levels and their dynamic behavior, thereby increasing the accuracy of covariance forecasts and

hedge ratio estimations (e.g., Lai & Lien, 2017;Lai&Sheu,2010). Thus, lower hedged portfolio variances can be achieved by

switching from standard GARCH models to realized covariance‐based GARCH models.

On the basis of the research of Bollerslev, Patton et al. (2020), this paper proposes a realized semicovariance‐based

GARCH model for asymmetric futures hedging; the realized semicovariances use information from signed high‐

frequency returns. Barndorff‐Nielsen et al. (2010), Bollerslev, Li et al. (2020) (henceforth BLPQ), and Patton and

Sheppard (2015) have demonstrated that the signs of high‐frequency returns contain valuable information for

covariance estimation and have indicated that negative realized semicovariance has a stronger effect on future

covariance than does positively realized semicovariance.

1

Accordingly, decomposing the realized covariance matrix

into four separate realized semicovariance matrices for GARCH modeling can provide refined covariance matrix

dynamics, thereby strengthening the explanatory power of the model fit.

As an illustration of spot and futures assets, Figure 1displays a plot of the daily concordant (

P

N+

tt

)andmixed(



t

)

realized semicovariance components (first panel) with the positive (

P

t)andnegative(

N

t) realized semicovariances (second

panel) for the S&P 500 data for 2007–2009.

2

As presented in the first panel in Figure 1, the concordant and mixed

components are typically similar in magnitude during calm periods characterized by relatively low and slow‐moving

volatility. Conversely, during periods of market stress, the concordant component increases markedly more than the mixed

component decreases, which indicates that the daily realized covariance for spot and futures returns is mainly determined

by the concordant realized semicovariance under such circumstances. The second panel in Figure 1reveals different

dynamic dependencies between the positive and negative realized semicovariances. Consequently, a model that can use

estimates of covariances with signs of high‐frequency returns may be crucial to hedgers because the different realized

semicovariance matrices contain distinct information that is used to predict spot–futures covariances.

The conventional approach for multivariate asymmetric GARCH modeling relies on extending the univariate

threshold‐based model of Glosten, Jagannathan, and Runkle (GJR, 1993), which depends on the signs of daily returns

arising from “good”or “bad”news. By contrast, the realized semicovariance‐based GARCH model uses all positive and

negative intraday returns in a day, thereby creating a continuous rather than threshold leverage effect (Bollerslev,

Patton, et al., 2020). On the basis of a comparison of the news impact surfaces for equity spot–futures covariances, the

empirical evidence indicates that the realized semicovariance‐based GARCH model demonstrates stronger responses to

jointly negative return shocks than do threshold‐based asymmetric GARCH models. Because our results provide clear

evidence of improved explanatory power and covariance forecasts for the realized semicovariance‐based GARCH

model, the informational advantages of using signed intraday returns for the prediction of daily spot–futures

covariances are supported. To the best of our knowledge, this study is the first to incorporate realized semicovariances

into the modeling of spot–futures covariances and the estimation of byproduct hedge ratios.

Another contribution of this study is that it demonstrates that hedgers who use the realized semicovariance‐based

GARCH model to predict the one‐step‐ahead covariance matrix can construct out‐of‐sample hedged portfolios more

effectively. Specifically, the realized hedged portfolio variances obtained through the realized semicovariance‐based GARCH

model (by using signed high‐frequency returns) are smaller on average than those obtained through the asymmetric

threshold‐based GARCH model (by using signed daily returns) and symmetric realized covariance‐based GARCH model (by

using unsigned high‐frequency returns); these two models are the main competitors to the proposed model. The model

confidence set (MCS) procedure of Hansen et al. (2011) reveals that the model encompassing signed intraday returns is

always included in the MCS for covariance matrix prediction and hedge ratio estimation. To understand the economic

importance of the realized semicovariance‐based GARCH model relative to simpler models, we compute the switching fees

and break‐even transaction costs for a hedger with quadratic utility. The results indicate that switching from the simpler

models to the realized semicovariance‐based GARCH model can be economically beneficial for hedgers. Overall, our results

1

By examining 500 randomly selected pairs of the Standard and Poor's (S&P) 500 stocks, BLPQ demonstrated that the daily realized covariance is

mainly determined by the concordant realized semicovariance rather than the mixed realized semicovariance during periods of financial market

stress.

2

We use the S&P 500 and the E‐mini S&P 500 to represent spot and futures data, respectively. Section 3provides a detailed description of the data.

For the daily realized semicovariances, we use equidistant 15‐min returns in the computation; their mathematical definitions are provided in

Section 2.4.

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