Flexible covariance dynamics, high‐frequency data, and optimal futures hedging

• DOI: http://doi.org/10.1002/fut.22054
• Author: Yu‐Sheng Lai
• Published date: 01 December 2019
• Date: 01 December 2019

J Futures Markets. 2019;39:1529–1548. wileyonlinelibrary.com/journal/fut © 2019 Wiley Periodicals, Inc.

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1529

Received: 21 August 2018

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Accepted: 17 August 2019

DOI: 10.1002/fut.22054

RESEARCH ARTICLE

Flexible covariance dynamics, high‐frequency data, and

optimal futures hedging

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., 54561 Puli,

Nantou, Taiwan.

Email: yushenglai@ncnu.edu.tw

Funding information

Ministry of Science and Technology,

Taiwan, Grant/Award Number:

106‐2410‐H‐260‐013

Abstract

This paper investigates the out‐of‐sample performance of hedged portfolios

constructed using a novel rotated ARCH (RARCH) model class, which enables

flexible covariance dynamics for spot and futures returns. The model’s

empirical fit can be significantly improved when it incorporates rotated realized

covariance matrix measures. The empirical results suggest that a highly risk‐

averse hedger implementing the restricted RARCH model would be willing to

pay substantial switching fees to capture the incremental gains generated by the

flexible and informative alternative; this thus supports the economic

importance of incorporating high‐frequency data into flexible RARCH modeling

processes for the construction of optimal hedged portfolios.

KEYWORDS

flexible covariance dynamics, futures hedge ratio, high‐frequency data, predictive ability testing,

rotated ARCH model

1

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INTRODUCTION

Modeling and forecasting temporal dependencies in the conditional covariance matrix of spot and futures returns

constitute crucial processes for estimating the optimal futures hedge ratio, which is formulated as the conditional

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

Henry, & Persand, 2002; Kroner & Sultan, 1993). Studies have adopted various bivariate generalized autoregressive

conditional heteroskedasticity (GARCH) models to specify the conditional covariance matrix, thus establishing

dynamic hedging strategies. For example, the Baba–Engle–Kraft–Kroner (BEKK) model proposed by Engle and Kroner

(1995) is widely employed in the literature. Although highly general, the empirical application of this model is rather

limited because it requires a high number of parameters and necessitates ensuring the positive definiteness and

stationarity of the conditional covariance matrix. To overcome this difficulty, feasible parameterizations, such as

restricting the parameter matrices to be scalar or diagonal, are usually adopted in practice (Alexander & Barbosa, 2008;

Haigh & Holt, 2002). Some studies, however, have reported that the restricted BEKK strategy fails to outperform

conventional strategies out of sample (Alexander & Barbosa, 2008; Kavussanos & Visvikis, 2008). The poor hedging

performance might be because the covariance dynamics described by the restricted models can be less flexible and thus

deteriorate the accuracy of the forecasted covariance matrix and the estimated hedge ratio (Kroner & Ng, 1998).

This paper employs a new class of multivariate volatility models provided by Noureldin, Shephard, and Sheppard

(2014) to investigate the behavior of spot and futures returns and the empirical performance of the associated futures

hedge ratios. Compared with the conventional GARCH models, this new model class, which entails a transformation of

raw returns in the modeling process, is not only attractive in terms of estimation and inference but also affords richer

dynamics in covariance process modeling. The model class involves feasible parameterization processes, signifying that

the covariance stationarity and positive definiteness can be easily ensured under covariance targeting. These attractive

features enable us to analyze the empirical implications of the imposed restriction when estimating the futures hedge

ratio.

Recent studies have demonstrated that standard GARCH models that apply information from high‐frequency data—

called GARCHX models—provide considerable improvements in terms of model fitting and volatility forecasting

(Engle, 2002b; Hansen, Huang, & Shek, 2012; Shephard & Sheppard, 2010). Standard GARCH models typically project

conditional volatility dynamics using lagged squared return; such a focus obviously ignores the information contained

within price changes. Currently, financial data often contain intraday (e.g., 5 min) prices; realized measures

(e.g., realized variance) constructed from high‐frequency data provide accurate estimates of the current volatility levels

(Andersen, Bollerslev, Diebold, & Labys, 2001; Barndorff‐Nielsen & Shephard, 2004). Accordingly, a few studies have

shown that the use of GARCHX models leads to appropriate futures hedging decisions compared with the use of

GARCH models (Lai, 2016; Lai & Sheu, 2010). This understanding motivates this paper to propose augmenting a

flexible rotated ARCH (RARCH) model with high‐frequency data, called the RARCHX model, for modeling spot and

futures returns, thus improving the efficiency in the construction of the hedged portfolios.

A typical GARCHX model encompasses standard realized covariance measures to project the time‐varying

conditional covariance matrix. By contrast, the RARCHX model parameterizes the conditional covariance dynamics

using rotated realized covariance measures. This not only preserves attractive features obtained from RARCH modeling

but also enables the rapid adjustment of the covariance dynamics in a changing market. To the author’s knowledge, this

is the first study to examine the statistical and economic importance of modeling flexible covariance dynamics with

rotated realized covariance estimators to establish futures hedging decisions.

1

Empirical assessments are conducted

using data on equity index markets; the empirical results indicate that the gains obtained from incorporating a rotated

realized covariance estimator in the RARCH model class for futures hedging can be substantial, in terms of both

variance reduction and economic value perspectives.

The remainder of this paper is organized as follows. Section 2 describes RARCH‐type models for estimating the

optimal futures hedge ratio. Section 3 discusses the data and provides preliminary empirical results regarding model

fitting. Section 4 provides the main empirical results. Finally, Section 5 concludes the paper.

2

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RARCH MODELS AND HEDGING

2.1

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Optimal hedge ratio

The basic concept of hedging involves reducing the risk of price fluctuation in a spot position by including futures

contracts in a portfolio. Let

rrr=( , )

tstft,,

be the log‐return vector at time

t

that consists of spot and futures assets.

Assume that a hedger would like to take a short position in the futures market for protection against the risk of a

declining spot price between times

t

and

t

+

1

. Under a two‐period framework, the return on the hedged portfolio is

given by

rrδr=−

pt st t ft,+1 ,+1 ,+

1

, where

δt

represents the hedge ratio determined at time

t

. The optimal hedge ratio,

δσ

σρσ

σ

==

,

*

t

sf t

ft

sf t

st

ft

,+1

,+1

2,+1

,+1

,+1

(1)

is derived by minimizing the variance of hedged portfolio returns at time

t

+

1

, given by

σ

σ=−

pt st,+1

2,+1

2

δσ δ σ

2

+

tsft tft

,+1 2,+

1

2. In addition,

σ

σ()

st ft,+1 ,+1 denotes the conditional standard deviation of the spot (futures) return,

and

σ

ρ()

sf t sf t

,+1 ,+1 denotes the conditional covariance (correlation) between spot and futures returns (Baillie & Myers,

1991; Brooks et al., 2002; Kroner & Sultan, 1993). Clearly, empirical hedge ratio estimation requires modeling the

conditional moments for the bivariate joint distribution of rt.

2.2

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RARCH model class

The RARCH model introduced by Noureldin et al. (2014) enables the estimation of the multivariate volatility of asset

returns with rich dynamics; this can be achieved by fitting the model to the rotated returns. Let reH=

t

t

1/2 , assuming

1

High‐frequency‐based volatility models, such as the model proposed by Noureldin, Shephard, and Sheppard (2012), rely on unrotated realized measures in modeling processes. These models have

been demonstrated to provide pronounced improvements in log‐likelihoods compared with standard GARCH models that solely use daily returns.

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