Dynamic Hedging Performance of the CSI 300 Index Futures - The ...

Dynamic Hedging Performance of the CSI 300 Index Futures - The Realized Minimum-Variance Hedge Ratio Approach Hui Qu School of Management and Engineering Nanjing University, China Tel: 86-13951604813; Email: [email protected]

Tianyang Wang (Corresponding Author) Finance and Real Estate Department Colorado State University, USA Tel: 1-9704912381; Email: [email protected]

Yi Zhang School of Management and Engineering Nanjing University, China Tel: 86-13770680103; Email: [email protected]

Pengfei Sun School of Management and Engineering Nanjing University, China Tel: 86-15298386501; Email: [email protected]

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Dynamic Hedging Performance of the CSI 300 Index Futures

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- The Realized Minimum-Variance Hedge Ratio Approach

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Abstract: This paper comprehensively investigates the dynamic hedging performance

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of China’s CSI 300 index futures by using the realized minimum-variance hedge ratio

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(RMVHR) as an efficient way to utilize the high-frequency intraday information. We

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thoroughly examine a number of RMVHR-based time-series models for CSI 300 index

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futures, and evaluate the out-of-sample dynamic hedging performance in comparison

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to the conventional hedging models using both lower frequency and high frequency

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data. Our results show that the dynamic hedging performance of the RMVHR-based

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methods robustly dominates that of the conventional methods in terms of major

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performance measures including the hedge ratio, the hedging effectiveness, the

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portfolio return and the Sharp ratio in the out-of-sample forecast period. Furthermore,

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the superiority of the RMVHR-based methods is consistent during different volatility

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regimes of China’s financial markets, including the market turbulence in 2015.

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Keywords: Realized Minimum-Variance Hedge Ratio; Dynamic Hedging Performance,

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High-Frequency Data; Hedging Effectiveness; Volatility Regime

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1

Introduction

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The launch of the CSI 300 equity index futures on April 16, 2010 marked a

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milestone development in the evolution of China’s financial market. For the first time,

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China’s financial market provides investors with an essential tool to hedge the

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systematic risk of holding the market, proxied by the underlying CSI 300 equity index,

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a free-float weighted index comprises 300 of the largest and actively traded A-share

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stocks on the Shanghai and Shenzhen Stock exchanges.

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While its inception was widely hailed as an effective hedging tool and even a

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stabilizing force in China’s financial markets among investors and regulators 1 , the

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effectiveness of the hedging strategy using the CSI 300 index futures contracts in the

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dynamic settings has been a subject of very limited academic research. Despite the

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growing importance of stock index futures and almost eight years since its introduction

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in China, the existing literature on its hedging effectiveness mainly focuses on the

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conventional constant hedge ratios with low frequencies of data (mostly daily) over

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relatively short hedging horizons (c.f. Zhang and Shen 2011; Wen, Wei, and Huang

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2011; He and Yang 2012; Hou and Li 2013; Pan and Sun 2014), and hence is subject to

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important weaknesses. First, many criticisms have been suggested against the

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conventional hedge ratios methods used in the early literature because they can’t fully

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incorporate the dynamic hedging relation and the past available information (e.g.,

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Baillie and Myers, 1991; Yang and Allen, 2004; Park and Jei, 2010). Second, empirical

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study has suggested valuable informational content of high-frequency intraday data in

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estimating hedge ratios over the low frequency data (e.g. Lai and Sheu 2010; Han and

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Ren 2012; Markopoulou et al. 2016; Miao et al. 2017). In addition, the China’s stock

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market turbulence in 2015 provides an interesting window to examine the hedging

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performance of index futures in China’s financial markets, but none of the earlier

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studies evaluates the hedging performance of the CSI 300 index futures during the

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turbulent market fluctuations. This is especially important as the China Securities

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Regulatory Commission (CSRC) openly blamed the stock market collapse on

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“malicious short-selling” of index futures as “weapons of mass destruction” by

On December 5, 2014, Xiao Gang, chairman of the China Securities Regulatory Commission (CSRC) remarked, stock index futures are “sophisticated risk management tools for improving the stock market operation mechanism, providing hedging instruments, improving the investment product market system and promoting stable development of great significance.” 1

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speculators and questioning its conventional role as a hedging instrument. Therefore, a

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clear picture of the dynamic hedging performance of the CSI 300 index futures is

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essential for investors and calls for close attention from the academics.

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The new technical development in hedging ratio measures, the increasing

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availability of high-frequency intraday data, and the tumultuous market environment in

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China’s recent financial market motivate us to assess and examine the hedging

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performance of the new proposed realized minimum-variance hedge ratio (RMVHR)

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(Markopouloua, et al. 2016), constructed from high frequency data (five-minute

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frequency) of China’s CSI 300 index and index futures. Our study shows that the

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dynamic hedging performance of the RMVHR-based methods dominates that of the

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conventional methods in terms of major performance measures including the hedge

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ratio, the hedging effectiveness, the portfolio return and the Sharp ratio in the out-of-

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sample forecast period. Furthermore, the superiority of the RMVHR-based methods is

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consistent during different volatility regimes of China’s financial markets, including

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China’s market turbulence in 2015.

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This research contributes to the existing literature in the following four dimensions.

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First, to our best knowledge, this is the first study to comprehensively investigate

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the dynamic hedging performance of CSI 300 index futures by applying and extending

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the new proposed realized minimum-variance hedge ratio. In contrast to earlier studies

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that use relatively low frequency and short examination windows, we use the intraday

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five-minute data of CSI 300 index and index futures, and employ a variety of time-

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series models to forecast the RMVHR ratio from a much longer sample period. Our

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study shows that our proposed method is more efficient than conventional models in

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terms of major performance measures including the hedge ratio, the hedging

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effectiveness, the portfolio return and the Sharp ratio criteria. 2

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Second, we propose a method to directly measure the marginal benefits of using

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RMVHR in China’s financial markets and show that constructing the RMVHR and

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directly forecasting it is a more efficient way to utilize the high-frequency intraday

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information. More specifically, in addition to the conventional low-frequency models

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in the comparison group, we also assess the hedging performance of the dynamic

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conditional correlation (DCC)-RV model (Lai and Sheu 2010) and the Vector

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Heterogeneous Autoregressive (VHAR) model (Busch et al., 2011) that also utilize

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high-frequency data. Because the VHAR model of the realized covariance (RCov)

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matrix and the heterogeneous autoregressive (HAR) model (Corsi, 2009) of RMVHR

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utilize exactly the same information set (intraday five-minute returns of spot and futures)

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and have similar structures, the comparison provides direct measure of the marginal

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benefit of RMVHR and illustrates its superiority in utilizing the high-frequency

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intraday information.

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Third, while none of the earlier studies examine the crisis period in 2015, we

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carefully test the robustness of our results to different market conditions including

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China’s turbulent market fluctuations in the out-of-sample forecast period. Using the

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nonparametric change point model (Ross et al. 2011), different volatility regimes of the

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underlying CSI 300 index were detected in the forecast period. We show that the

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superiority of the hedging performance of CSI 300 index futures using RMVHR is

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robust to different volatility regimes.

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Last but not the least, observing that the residuals for the HAR model of RMVHR

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have significant ARCH effects and non-zero skewness, we further extend the HAR

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model by modeling the variance of the dynamic hedge ratio with the GARCH

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specification and skewed-t innovations. We show that the resulting HAR-GARCH 3

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model of RMVHR has lower hedge ratio, competitive hedging effectiveness, much

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higher portfolio return and Sharp ratio than the basic HAR model, and its superiority is

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robust to different market conditions.

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The remainder of the paper is organized as follows. Section 2 provides a brief

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overview of the literature and draws comparison with our study. Section 3 presents

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methodology of the RMVHR models and its comparison models. Section 4 explains

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the data and the results are discussed in Section 5. This is followed by a discussion of

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robustness tests in Section 6. Section 7 concludes the paper.

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2. Literature Review

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The effectiveness of a hedging strategy using futures contracts depends heavily on

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the accuracy of the optimal hedge ratio estimation. The view of determining hedge ratio

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that prevails today originates from Johnson (1960) and Stein (1961), who apply the

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Markowitz modern portfolio theory and propose the minimum-variance method with

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the objective to minimize portfolio risk. Following them, Ederington (1979) proposes

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to calculate constant hedge ratio using ordinary least squares (OLS) model. In addition,

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Ghosh (1993) proposes the error correction (ECM) model for determining hedge ratio

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based on the cointegration theory, which considers both the long-term equilibrium and

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the short-term dynamics between spot and futures. However, both models are static

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hedging methods since they assume constant variance and covariance between spot and

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futures returns over time and do not consider the time-varying (co)variance of spot and

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futures.

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With

the

development

of

the

generalized

autoregressive

conditional

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heteroscedastic (GARCH) models, dynamic hedging based on the bivariate GARCH

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models have gained rising attentions in the literature. The bivariate GARCH-ECM 4

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model (Kroner and Sultan 1993) considers the cointegration relationship and

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characterizes the time-varying covariance of spot and futures. It has been shown to have

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better dynamic hedging performance than the static hedging methods, and thus gained

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wide applications. Alternative common choices of the bivariate GARCH model include

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the BEKK-GARCH model (Engle and Kroner 1995), the constant conditional

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correlation (CCC)-GARCH model (Bollerslev 1990), the DCC-GARCH model (Engle

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2002) and copula-GARCH model (Hsu et al. 2008; Lai et al. 2009). However, these

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GARCH-class models are likely to overestimate the persistence in volatility since

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relevant sudden changes in variance are ignored (Wei et al. 2011). In addition, the early

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studies mainly use relatively low frequency (daily in most cases) data to latently

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characterize the time-varying covariance of spot and futures, therefore they cannot

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capture the intraday variation of prices and are slow in catching up the covariance

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changes.

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The harnessing of high-frequency information and the new development in

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financial econometrics have enabled significant progress in direct measuring and

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modeling of covariance. The realized volatility (RV) calculated as the sum of squared

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intraday returns provides an unbiased estimator of the quadratic variation (Andersen

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and Bollerslev, 1998). As a natural extension of RV into the multivariate case, the

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realized covariance (RCov) matrix calculated as the sum of the cross products of high-

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frequency intraday return vectors provides an unbiased estimator of the quadratic

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covariation (Barndorff-Nielsen and Shephard, 2004). Unfortunately, the calculation of

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the RCov matrix suffers from the problems of market microstructure noise and

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nonsynchronous trading. While more complicated estimators have been proposed (e.g.,

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the multivariate realized kernel (Barndorff-Nielsen et al., 2011), the incorporation of

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investor sentiment and weather effects (Yang, Jhang and Chang 2016), and the two5

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time scale covariance (Zhang, 2011), their computation complexities impede wide

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applications in practice.

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As a more practical alternative, the easily implemented sparse sampling method

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using high frequencies of data has been employed in empirical applications. For

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instance, Lai and Sheu (2010) proposed the DCC-RV model using 15-minutes

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frequency of data, which encompasses the realized volatility (correlation) in the

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conditional variance (correlation) functions for spot and futures and shows substantial

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improvement in hedging performance for the S&P 500 index futures. Han and Ren

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(2012) proposed the vector heterogeneous autoregressive (VHAR) type of model

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(Busch et al., 2011) which directly models the realized covariance matrix, and achieves

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better hedging performance compared with the bivariate GARCH-ECM type models.

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Most recently, Markopoulou et al. (2016) proposed a new concept of the realized

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minimum-variance hedge ratio (RMVHR) employing high-frequency data. RMVHR is

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defined as the ratio of the realized covariance between futures and spot returns divided

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by the realized volatility of futures. In contrast, all former mentioned dynamic hedging

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methods determine the dynamic hedge ratio as the ratio of the forecasted covariance of

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futures and spot returns divided by the forecasted futures variance. Markopoulou et al.

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(2016) show that directly modeling and forecasting the RMVHR can result in

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substantial hedging performance improvements when compared with conventional

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low-frequency models for the S&P 500 and the FTSE 100 indices as well as the

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EUR/USD and the GBP/USD foreign exchange rates. However, it is unclear if the

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RMVHR would also be beneficial in China’s market given several distinguishing

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elements in the Chinese stock market, including trading restrictions such as price-limit

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rules, margin trading, short-selling restrictions and T+1 trading constraints, closed

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nature of the market to foreign shareholders, and a large portion of retail investors in 6

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contrast to developed markets.

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It is important to highlight how our study differs from the previous research on

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hedging performance of CSI 300 index futures in China’s market. In China’s market,

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preliminary research has been conducted to assess the hedging performance of CSI 300

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index futures using constant hedge ratios (cf. Hou and Li 2013), relatively low

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frequency (mostly daily) (c.f. Zhang and Shen 2011; Wen, Wei, and Huang 2011) and

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short examination windows surrounding the introduction of equity futures in China in

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2010 (c.f. Wei et al. 2011; Hou and Li 2013; Pan and Sun 2014). A notable downside

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of these prior studies is their reliance on lower frequency data, relatively short sample

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periods and examination of periods that does not fully reflect the broader evolution of

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the equity index and futures market. More importantly, none of the earlier studies

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examine the crisis period in 2015, when the Chinese regulatory authority officially

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undertook a series of regulatory interventions and launched a formal investigation into

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the role played by the futures market in exacerbating the crisis in response to the stock

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market crash in June 2015, which as a result raises the question of the conventional

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hedging role of the index futures in China’s market.

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We believe that our study offers the most comprehensive investigation of the

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dynamic hedging performance of CSI 300 index futures in the Chinese stock market.

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Our analysis benefits from a longer sample period that spans January 2012 through

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December 2017, which captures the overall evolution of the market as evinced by its

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steep rise and dramatic decline. We applied and further extended the newly proposed

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RMVHR approach to examine the dynamic hedging performance of CSI 300 index

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futures throughout the long horizon of up and down in Chinese stock market.

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Additionally, our study employs intraday data measured at 5-minutes intervals thus

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providing an added level of granularity as well as robustness to the analysis. 7

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3. Methodology

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3.1 Realized Measures

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Let the discretely sampled Δ-period log return be denoted by rt+j·Δ,Δ = lnpt+j·Δ -

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lnpt+(j-1)·Δ, j = 1, 2, …, M, t = 0, 1, 2, …, where pt+j·Δ is the high-frequency price observed

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at time j·Δ within day t+1 and M = 1/Δ is the number of sampling intervals per day. The

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daily realized volatility is defined by the summation of the squared intraday returns as

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RVt ( Δ) ≡ ∑ j =1 rt 2-1+ j ⋅ Δ, Δ (Andersen and Bollerslev, 1998), which converges uniformly

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in probability to the quadratic variation as Δ→0.

1/ Δ

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Let rt+j·Δ,Δ = [rSt+j·Δ,Δ, rFt+j·Δ,Δ]’ be the column vector of returns, where rSt+j·Δ,Δ is the

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day (t+1) Δ-period log return of the CSI 300 index and rFt+j·Δ,Δ is the day (t+1) Δ-period

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log return of the CSI 300 index futures. The daily realized covariance matrix is defined

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by the summation of the cross products of intraday return vectors as

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RCov tS , F ( Δ) ≡ ∑ j =1 rt -1+ j ⋅ Δ, Δ ⋅ rt′-1+ j ⋅ Δ, Δ (Barndorff-Nielsen and Shephard, 2004), which

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converges uniformly in probability to the quadratic covariation as Δ→0.

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1/ Δ

The minimum-variance hedge ratio of day t can be calculated as H tS Cov( RtS , RtF ) S ,F , where RtS and RtF are the day t log returns of the spot = ρ × t F F Var ( Rt ) Ht

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= HR t

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and the futures, respectively. ρtS,F is the day t correlation between the spot and the

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futures returns, and HSt and HFt are the day t variances of the spot and the futures,

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respectively. According to this, the day t realized minimum-variance hedge ratio

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(RMVHR) is defined as RMVHRt ( Δ) ≡

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F RCovtS , F ( Δ) is the sub-diagonal element of RCov tS , F ( Δ) , and RVt ( Δ) is the day t

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realized variance of the futures. For notational simplicity, we omit the notation ( Δ) in

RCovtS , F ( Δ) (Markopulou et al., 2016), where RVt F ( Δ)

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the realized measures when presenting the forecasting models.

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3.2 Forecasting models

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As for the RMVHR forecasting models, we consider the following time-series

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models:

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1) The ARMA model: = . RMVHRt = c + ∑ i 1ϕ i RMVHRt −i + ∑ j 1ϑ j ε t − j + ε t =

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2) The ARMA-GARCH model:

p

q

RMVHR = c + ∑ ϕ RMVHR + ∑ ϑ j ε t − j + ε t , p

q

t i t −i = i 1= j 1

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ε t =σ t et , 2 2 2 σ= ω + ∑ k 1α t = k ε t − k + ∑ l 1 β lσ t − l . = m

n

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cst + ϕ st RMVHRt −1 + ε t , 3) The Regime-switching (RS) model: RMVHRt =

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where st is the state variable that takes the values 1 and 2. The state transitions are given

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pi , j P= st −1 i ) , i, j = 1,2. by a Markov chain with transition probabilities = ( st j |=

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(

)

(

)

4) The ARFIMA model: 1 − ∑ ϕ L (1 − L ) ( RMVHR − µ ) = 1 + ∑ ϑ j Lj ε t ,

p q d i i t =i 1 = j 1

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where d is the differencing order and L is the lag operator.

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5) The HAR model:, RMVHRt = α 0 + α d RMVHRt −1 + α w RMVHRt(−w1) + α m RMVHRt(−m1) + ε t

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where

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weekly and monthly RMVHRs.

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6) The HAR-GARCH model:

RMVHRt(−w1) =

1 5 1 22 RMVHRt −i , RMVHRt(−m1) = ∑ i =1 RMVHRt −i are the past ∑ i =1 5 22

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RMVHRt = α 0 + α d RMVHRt −1 + α w RMVHRt(−w1) + α m RMVHRt(−m1) + ε t ,

ε t =σ t et , 2 2 2 σ= ω + ∑ k 1α t = k ε t − k + ∑ l 1 β lσ t − l , = m

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where et follows skewed-t distribution.

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As for the conventional hedging approaches, we include the static OLS and ECM

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models, the dynamic DCC model, DCC-RV model and the VHAR model. The former

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three models completely rely on the daily log returns of the spot (RtS) and the futures

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(RtF). The DCC-RV model incorporates high-frequency based realized covariance

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matrix (volatilities and correlation) in the DCC framework; while the VHAR model

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directly models the high-frequency based realized covariance matrix.

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α β RtF + ε t . 7) The OLS model: RtS =+

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α β RtF + γ RtS-1 − θ RtF−1 + ε t , 8) The ECM model: RtS =+

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where

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equilibrium between the spot and the futures.

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)

(

(R

S t -1

− θ RtF−1 ) is the error correction term that characterizes the long-term

The OLS model and the ECM model are all static models, and the estimated

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parameter β is the (constant) hedge ratio.

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9) The DCC model: RtS = µ S + γ S ( RtS-1 − θ RtF−1 ) + ε tS ,

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RtF = µ F + γ F ( RtS-1 − θ RtF−1 ) + ε tF ,  ε tS   F  | ψt −1 ~ N (0, H t ),  εt  10

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where ψt-1 is the information set up to day (t-1) and Ht is the conditional covariance

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matrix modeled as: S H tS , F   H t =  H tF   0  S S S S 2 S β 0 + β1 ε t −1 +β 2 H tS−1 , H= t

0   1 × , S F H tF   ρt

 HS H t =  St, F  Ht 253

× 1   0 

0   = Dt R t Dt , H tF 

(*)

F β 0F + β1F ε tF−12 +β 2F H tF−1 , H= t 1

S ρtS , F   H t

(*) 1

R t = diag ( Qt ) 2 Qt diag ( Qt ) 2 , −

−

Qt = (1 − α − β )Q + α z t −1z t −1′ + β Qt −1 ,

εS

(*)

H tS   is the standardized residual vector, and Q is the unconditional H tF 

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where z t = 

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correlation matrix of the spot and the futures returns. α and β are nonnegative scalars

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with α + β ≤ 1 .

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10) The DCC-RV model has similar formulation compared to the DCC model, with

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t

ε F  t

modifications in the three equations of 9) that are marked with (*): S H= β 0S + β1S RVt −S1 +β 2S H tS−1 , t

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(*)

F H= β 0F + β1F RVt −F1 +β 2F H tF−1 , t

(*)

Qt = (1 − α − β )Q + α RCorrtS−1, F + β Qt −1 , 260

where RCorrtS , F is the realized correlation matrix whose sub-diagonal element is S ,F

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calculated as RCorrt

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11) The VHAR model:

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(*)

=

RCovtS , F RVt S ⋅ RVt F

.

The matrix logarithm transformation method is adopted to guarantee the positive definiteness

of

the

forecasted

covariance 11

matrix.

Specifically,

define

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= X t vech( = At ) A t = logm(RCOVtS , F ) and define

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is constructed as:

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 X tS   S ,F  Xt  =  X tF   

 α S   β11( d )  S ,F   (d )  α  +  β 21  α F   β31( d )   

 β11( w)  +  β 21( w)  β31( w) 

β12( w) β 22( w) β32( w)

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where

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X tF−1( m ) =

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X tS−1( w) =

β12( d ) β 22( d ) β32( d )

1 22 X tF−i ∑ i =1 22

X tS−,1F ( w) =

S t

, X tS , F , X tF ) . The VHAR model '

β13( d )   X tS−1    β 23( d )   X tS−,1F  β33( d )   X tF−1 

β13( w)   X tS−1( w)   β11( m )    β 23( w)   X tS−,1F ( w)  +  β 21( m ) β33( w)   X tF−1( w)   β31( m )

1 5 S ∑ X t −i 5 i =1

(X

,

β12( m ) β 22( m ) β32( m )

X tF−1( w) =

β13( m )   X tS−1( m )   ε tS      β 23( m )   X tS−,1F ( m )  +  ε tS , F  , β33( m )   X tF−1( m )   ε tF 

1 5 X tF−i ∑ i =1 5

,

X tS−1( m ) =

22 1 X tS−i , ∑ i =1 22

1 5 S ,F 1 22 X t −i , X tS−,1F ( m ) = ∑ i =1 X tS−,iF . ∑ i =1 22 5

The inverse of the vech() function and the matrix exponential transformation is then

271

applied to get the prediction of the covariance matrix.

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4.

Data Description

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Our empirical data consist of five-minute (1/Δ = 48) prices of China’s CSI 300

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index and index futures from January 4, 2012 to December 29, 2017, covering a total of

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1456 trading days. 2 We chose the five-minute sparse sampling approach following the

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majority of previous studies as it provides a good trade-off between accuracy and market

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microstructure noise (nonsynchronous trading). The trading time of the CSI 300 index

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futures used to be 9:15am – 11:30am for morning session and 13:00pm – 15:15pm for

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afternoon session. Since January 1, 2016, China Financial Futures Exchange has

2 There are 1458 trading days from January 4, 2012 to December 29, 2017. However, trading on January 4, 2016 and January 7, 2016 closed much earlier, due to the circuit breaker mechanism being triggered. Thus these two days are deleted from our sample. 12

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adjusted the opening and closing times for the CSI 300 index futures to 9:30am and

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15:00pm, respectively, to match those of the CSI 300 index. Thus in this empirical

282

research, we use the five-minutes prices between 9:30am – 11:30am and 13:00pm –

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15:00pm for both the CSI 300 index and the CSI 300 index futures, deleting all price

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records in the non-overlapping periods for consistency and comparison purpose.

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[Insert Figure 1 Here]

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Figure 1 displays the time series plots of the log daily prices for the CSI 300 index

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and the CSI 300 index futures in the whole sample period. It shows that the log daily

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prices of the CSI 300 index futures are very close to those of the CSI 300 index in most

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of the trading days, and that the Chinese stock market has gone through relative tranquil

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and extremely volatile periods during our sample period. This observation inspires us to

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test the robustness of our results to different market conditions, which will be explained

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later.

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[Insert Figure 2 Here]

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Figure 2 displays the time series plots of the realized volatilities for the CSI 300

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index (RVtS) and the CSI 300 index futures (RVtF) as well as the realized covariance

296

between the spot and the futures (RCovtS,F) in the whole sample period. It shows that the

297

realized volatility of the CSI 300 index futures has a similar pattern as that of the CSI

298

300 index, although it is more volatile. Both the realized volatility series and the realized

299

covariance series are relatively tranquil during the period from January 4, 2012 to the

300

end of 2014, but are very turbulent around the year of 2015. Such pattern necessitates

301

our robustness check in different volatility regimes.

302

[Insert Table 1 Here] 13

303

Table 1 reports descriptive statistics for the realized volatilities (RVtS and RVtF), the

304

realized covariance (RCovtS,F), and the realized minimum-variance hedge ratio

305

(RMVHRt) of the CSI 300 index and index futures over the entire sample period. We can

306

see that the realized volatility of the CSI 300 index futures has higher standard deviation

307

than that of the CSI 300 index, indicating that the CSI 300 index futures is more volatile.

308

The ADF and PP test statistics show that these four realized measures are all stationary,

309

and thus can all be directly modeled. The Ljung-Box test statistics show that these four

310

realized measures all exhibit up to 20th order serial correlation, and thus the long-

311

memory models may be appropriate choices to model the RMVHR and the RCov matrix.

312

[Insert Table 2 Here]

313

Table 2 reports the diagnostic statistics for the regression residuals of the HAR

314

model of RMVHR in the whole sample period from January 4, 2012 to December 29,

315

2017. The ARCH-LM tests indicate strong autoregressive conditional heteroscedasticity

316

effects, which calls for GARCH model for the residual variance. The residuals have

317

positive skewness and kurtosis higher than 3, and are significantly non-normal as

318

confirmed by the Jarque-Bera normality test statistics. Thus, we assume skewed-t

319

innovations.

320

5.

Hedging Performance Comparison

321

We set the period from January 2, 2014 to December 29, 2017 (975 trading days) as

322

the out-of-sample forecast period, and perform one-step-ahead rolling window forecast.

323

That is, we use the period from January 4, 2012 to December 31, 2013 (2 years, 481

324

trading days) as the first estimation window, to make forecasts for January 2, 2014. The

325

estimation window is then rolled forward, and we use the period from January 5, 2012 14

326

to January 2, 2014 as the second estimation window, to make forecasts for January 3,

327

2014. The estimation window keeps rolling forward, until we have made forecasts for

328

all the 975 out-of-sample trading days.

329 330

Based on these forecasts, we perform dynamic hedging of the CSI 300 index futures, and calculate the following four hedging performance indicators:

331

(1) Hedge Ratio (HR): HR = E ( HRt ) , where E() means taking expectation. For the

332

RMVHR-based models, HRt is the predicted day t hedge ratio; for the OLS model

333

and the ECM model, HRt is the predicted coefficient βˆ ; for the DCC model

334

H tS Cov( RtS , RtF ) S ,F and the DCC-RV model, HRt is calculated as HR ; = = × ρ t t F F Var ( Rt )

Ht

RCovtS , F ; for the NAÏVE RVt F

335

for the VHAR model, HRt is calculated as HRt =

336

method , HRt is always equal to 1.

337

The hedge ratio indicates the cost and leverage of the hedging, and thus lower HR

338

is preferred.

339

(2) Hedging Effectiveness (HE): HE = E ( HEt ) , HEt = 1 −

2 σ HP 2 ,t , where σ UP ,t is 2 σ UP ,t

340

the day t variance of the un-hedged portfolio, and is calculated as the realized

341

2 variance of the CSI 300 index ( RVt S ), σ HP ,t is the day t variance of the hedged

342

portfolio, and is calculated using the realized variances of the CSI 300 index and

343

index futures ( RVt F ), and the realized covariance of the spot and the futures (

344

2 RCovtS , F ): σ HP RVt S − 2 βˆt RCovtS ,F + βˆt 2 RVt F , where βˆt is the forecasted ,t =

345

minimum-variance hedge ratio for day t. 15

346

The hedging effectiveness indicator assesses the variance reduction rate of the

347

hedged portfolio. Thus, higher HE is preferred since it means that the portfolio

348

risk has been largely reduced.

349 350 351

= RtS − βˆt RtF , where RtHP is the (3) Portfolio Return (PR): PR = E ( RtHP ) , RtHP day t log return of the hedged portfolio. Naturally, higher PR is preferred. (4) Sharpe Ratio (SR): SR =

PR − R f

σ HP

, where σ HP is the standard deviation of the

352

hedged portfolio returns, R f is the risk-free return which we assume 0 here. This

353

indicator takes into consideration the returns and risks at the same time. Similar

354

to the evaluation of PR, higher SR is preferred.

355

[Insert Table 3 Here]

356

Table 3 reports the hedging performance of all the models in the out-of-sample

357

forecast period from January 2, 2014 to December 29, 2017. It is divided into two

358

panels. Panel I displays results for those models that directly model the RMVHR. Panel

359

II displays results for those models that model the daily returns (covariance matrix).

360

The performance of the NAÏVE method that uses a hedge ratio equal to 1 is also

361

reported in Panel II. In each panel, the hedging performance indicators are listed in the

362

first column, while the models are specified in the second row. For each specified

363

hedging performance indicator, the results of all the twelve methods are compared, and

364

the best result is in bold. For example, the number 62.7216% in Panel I is in bold,

365

because the RS using RMVHR model has the lowest HR indicator value of 62.7216%

366

among all these twelve methods.

367

From Table 3 we can see that when the hedge ratio indicator (HR) is considered, 16

368

the magnitude of the numeric numbers in Panel I using RMVHR are all much smaller

369

than the corresponding numbers in Panel II using alternative methods. When the

370

hedging effectiveness indicator (HE) is considered, the numeric numbers in Panel I are

371

mostly larger than those numbers in Panel II with only one exception, i.e., the RS model

372

in Panel I has slightly lower HE than the VHAR model in panel II. When the portfolio

373

return indicator (PR) or the Sharp ratio indicator (SR) is considered, the numeric

374

numbers in Panel I are all much larger than the corresponding numbers in Panel II.

375

Therefore, our empirical test shows that the dynamic hedging performance of CSI 300

376

index futures using RMVHR dominates that of the conventional methods in terms of

377

all these four hedging performance indicators in the out-of-sample forecast period.

378

These six RMVHR models all have quite close hedging effectiveness, among which the

379

RS model has the lowest hedge ratio, the highest portfolio return and the highest Sharp

380

ratio, and thus is superior in terms of these three economic criteria.

381

Additionally, in Panel II, the DCC-RV model has lower HR, higher HE, higher PR

382

and higher SR than the DCC model, which means that incorporating the information in

383

the realized covariance matrix can effectively improve the dynamic hedging

384

performance. The VHAR model has the lowest HR, the highest HE, the highest PR and

385

the highest SR in Panel II, which indicates that directly modeling the realized

386

covariance matrix can better utilize the intraday information and further improve the

387

hedging performance. The VHAR model of RCov in Panel II and the HAR model of

388

RMVHR in Panel I utilize exactly the same information set (intraday five-minute

389

returns of spot and futures) and have similar structures. Since the HAR model in Panel

390

I leads to better hedging performance in terms of all these four hedging performance

391

indicators, we conclude that constructing the RMVHR and directly forecasting it is a

392

more efficient way to utilize intraday information. Last but not the least, the HAR17

393

GARCH model in Panel I utilizes the same length of past RMVHR series as that of the

394

HAR model, using GARCH-skewed-t innovations instead of normal innovations. Since

395

these two models have similar HE, while the HAR-GARCH model has lower HR,

396

higher PR and higher SR than the HAR model, we conclude that modeling the variance

397

of the dynamic hedge ratio is valuable for improving the dynamic hedging performance.

398

6. Robustness Checks

399

To test the robustness of the above results to different market conditions, we use

400

the nonparametric change point model (NPCPM) (Ross et al. 2011) to detect the

401

different volatility regimes of the CSI 300 index in the forecast period. The NPCPM

402

detects the shifts in the volatility by sequential application of Mood’s test (Mood, 1954),

403

which is a nonparametric test for comparing the variances of two samples. Since the

404

Mood’s test assumes the independence of observations, we filter the original return

405

series using a GARCH(1,1) model with student-t innovations following Ross (2013),

406

and use the standardized residuals for the sequential Mood’s tests.

407 408

Assume the two samples for variance comparison are

(r

2,1

1,1

, r1,2 ,..., r1,a )

and

, r2,2 ,..., r2,b ) , where a+b=T. The Mood’s test statistic can be calculated as:

T + 1  ∑ i=1 rank ( r1,i ) − 2  , where rank r1,i 2

409 = M

(r

a

( )

is the rank of r1,i in the combined

410

sample of length T. By comparing the standardized Mood’s test statistic with the

411

simulated thresholds reported in Ross et al. (2011), we can decide whether the null

412

hypothesis of equal variance is rejected. The NPCPM applies sequential Mood’s tests

413

in the following manner to detect the volatility change points:

414

1) Divide the out-of-sample period into two contiguous samples. The first sample 18

415

contains the initial 22 (a month) observations, and the second sample contains

416

the remaining 953 (975-22=953) observations.

417

2) Perform the Mood’s test on these two samples.

418

3) If the null hypothesis of equal variance is not rejected, prolong the first sample

419

by 1 observation, and thus the second sample contains the remaining 952

420

observations. Perform the Mood’s test on these two updated samples.

421

4) Repeat procedure 3) until the null hypothesis is rejected, which means a

422

volatility change point has been detected. Flag this change point and repeat

423

procedures 1)-3) starting from the first observation after the change point.

424

[Insert Figure 3 Here]

425

Figure 3 displays the volatility regimes detected by the NPCPM in the out-of-

426

sample period from January 2, 2014 to December 29, 2017. There are three detected

427

volatility regimes. The first regime is from January 2, 2014 to November 3, 2014,

428

altogether 203 trading days. We refer to it as the low volatility regime (L) since the CSI

429

300 index is very tranquil during this period. The second regime is from November 4,

430

2014 to August 31, 2016 (448 trading days). We refer to it as the high volatility regime

431

(H) since the CSI 300 index is extremely volatile during this period. The last regime is

432

from September 1, 2016 to December 29, 2017 (324 trading days). We again refer to it

433

as the low volatility regime (L) due to its similarity with the first regime.

434

[Insert Table 4 - 5 Here]

435

We perform hedging performance comparison on each of these three volatility

436

regimes, and report the results in Tables 4-5. Comparing these two tables, we can see

437

that the hedging effectiveness is always lower during the low volatility regime than 19

438

during the high volatility regime, with the only exception of the NAÏVE method. This

439

observation confirms the appropriateness of our partition of volatility regimes.

440

Our insights from Table 4 for the low volatility regimes are completely consistent

441

with those from Table 3, and our insights from Table 5 for the high volatility regime are

442

also consistent with those from Table 3, which we summarize as follows:

443

1) The RMVHR-based models (Panel I) have much lower HR, higher HE, PR and

444

SR than those of the daily return based models (Pane II excluding the VHAR model) in

445

both the low volatility regimes and the high volatility regime.

446

2) The lowest hedge ratio, the highest hedging effectiveness, portfolio return and

447

Sharp ratio always come from the RMVHR-based models, regardless of the volatility

448

regime considered.

449

3) Incorporating the information in the realized covariance matrix into the DCC

450

model can effectively improve the dynamic hedging performance, regardless of the

451

volatility regime considered.

452

4) Directly modeling the realized covariance matrix with the VHAR model can

453

better utilize the intraday information than the DCC-RV model and further improve the

454

hedging performance, regardless of the volatility regime considered.

455

5) Directly forecasting the RMVHR is a more efficient way to utilize intraday

456

information during the low volatility regimes, since in Table 4 the HAR model of

457

RMVHR in Panel I has lower HR, higher HE, PR and SR than the VHAR model of

458

RCov in Panel II. However, this conclusion does not hold in the high volatility regime.

459

Nevertheless, by replacing the normal innovations in the HAR model with the GARCH20

460

skewed-t innovations, the HAR-GARCH model in Panel I has lower HR, higher PR

461

and SR than the VHAR model, regardless of the volatility regime considered.

462

7.

Concluding Remarks

463

This paper examines the incremental value of stock futures hedges in China’s

464

market by directly forecasting the realized minimum-variance hedge ratio calculated

465

from high-frequency prices of CSI 300 spot and futures over the conventional hedging

466

models.

467

Using five-minute data of China’s CSI 300 index and index futures, we made out-

468

of-sample forecasts of the RMVHR with a number of popular time-series models, and

469

used the forecasts to perform dynamic hedging of the CSI 300 index futures. We also

470

include the static OLS and ECM models, the VHAR model, the dynamic DCC model

471

based on daily returns, and the DCC-RV model using five-minute prices for comparison.

472

In addition, we detected three volatility regimes in the forecast period using the

473

nonparametric change point model (Ross et al. 2011). Using the hedge ratio, the

474

hedging effectiveness, the portfolio return and the Sharp ratio as criteria, we conducted

475

hedging performance comparison in the out-of-sample forecast period as well as in each

476

detected volatility regime.

477

Our results show that the dynamic hedging performance of CSI 300 index futures

478

using RMVHR dominates that of the conventional methods in terms of all these four

479

hedging performance indicators in the out-of-sample forecast period. Furthermore, the

480

superiority of the RMVHR-based methods is consistent during different volatility

481

regimes of China’s financial markets, including China’s abnormal market fluctuations

482

in 2015. Our research also shed some new lights on the conventional hedging models. 21

483

For instance, incorporating information in the realized measures from high-frequency

484

data improves the dynamic hedging performance. In addition, the VHAR model that

485

directly models the realized covariance matrix better utilizes the intraday information

486

and outperforms the DCC-RV model.

487

Our research is instructive for futures hedgers, risk managers and academics.

488

Future work would involve exploring forecast combination techniques to improve the

489

forecasting capability of RMVHR and the dynamic hedging performance.

490

Acknowledgements

491

This work was supported by the National Natural Science Foundation of China

492

(Research Grant No. 71671084); and the Specialized Research Fund for the Doctoral

493

Program of Higher Education of China (Research Grant No. 20120091120003).

494 495

22

496

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Figure 1. Time series plots of the log daily prices for the CSI 300 index and the CSI 300 index futures from January 4, 2012 to December 29, 2017.

588

26

589 590 591 592 593

Figure 2. Time series plots of the realized volatilities for the CSI 300 index and the CSI 300 index futures, and the realized covariance between the spot and the futures from January 4, 2012 to December 29, 2017.

27

594 595 596

Figure 3. Volatility regimes detected by the NPCPM in the out-of-sample period from January 2, 2014 to December 29, 2017.

597

598

28

599 600 601 602

603 604 605 606 607 608

Table 1. Descriptive statistics for the realized volatilities (RVtS and RVtF), the realized covariance (RCovtS,F), and the realized minimum-variance hedge ratio (RMVHRt) of the CSI 300 index and index futures from January 4, 2012 to December 29, 2017 RMVHRt RVtS RVtF RCovtS,F Mean 0.6600 1.6101 2.1515 1.3713 Standard Deviation 0.1724 3.3081 5.6002 3.4555 Skewness 0.1303 7.2779 10.7185 10.0274 Kurtosis 3.4574 76.4803 172.1786 146.6007 *** *** *** ADF -3.1386 -10.7050 -12.7407 -12.4644*** PP -30.4818*** -16.8493*** -17.3488*** -16.5660*** LB(5) 1007.8*** 2719.2*** 2627.6*** 2695.7*** LB(10) 1698.6*** 3847.8*** 3708.5*** 3668.4*** LB(20) 2891.2*** 5575.6*** 5015.1*** 5007.5*** Note: JB represents the Jarque-Bera normality test statistics, ADF represents the Augmented-Dickey-Fuller test statistics, PP represents the Phillips-Perron test statistics, LB(k) represents the Ljung-Box Q-statistics for kth order serial correlation, *** represents the significance level of 1%. The orders of magnitude for the mean and the standard deviation of RVtS, RVtF and RCovtS,F are 10-4.

29

609 610

Table 2. Diagnostic statistics for the regression residuals of the HAR model of RMVHR in the whole sample period from January 4, 2012 to December 29, 2017. ARCH(5) ARCH(10) ARCH(20) Skew Kurt JB *** *** ** 3.9680 0.2185 3.4579 23.9404*** 2.3323 1.6147

611

Note: ARCH(·) represents the ARCH-LM test statistics. JB represents the Jarque-Bera

612

normality test statistics. ** indicates significance at the 5% level, significance at the 1% level.

613 614 615 616 617 618 619 620 621 622 623 624

30

***

indicates

625 626 627 628

Table 3. Hedging performance comparison in the out-of-sample forecast period from January 2, 2014 to December 29, 2017.

Panel I: modeling the RMVHR RS ARMA ARMA-GARCH ARFIMA HAR HAR-GARCH HR 62.7216% 65.8045% 65.8096% 70.3667% 70.5335% 69.3108% HE 58.3634% 58.7287% 58.6469% 58.5888% 58.5556% 58.7288% PR 5.0476% 3.8319% 3.8302% 3.0697% 2.7218% 3.2610% SR 2.7654% 2.1591% 2.1583% 1.7720% 1.5711% 1.8568% Panel II: modeling the daily returns (covariance matrix) OLS ECM DCC DCC-RV VHAR NAÏVE 91.8041% 91.1134% 71.0610% 100.0000% HR 81.2190% 80.8292% 45.0022% 49.5666% 58.5016% 43.4252% HE 55.2131% 55.2960% 1.8086% -2.8180% -0.3087% 2.7077% 0.1082% PR 1.7645% 1.0355% -1.4666% -0.1835% 1.5693% 0.0521% SR 1.0111% 629 Note: HR represents the hedge ratio, HE represents the hedging effectiveness, PR 630 represents the portfolio return, SR represents the Sharp ratio. The lowest HR, the 631 highest HE, the highest PR and the highest SR are in bold. 632 633 634

31

635 636 637 638

639 640 641 642 643 644

Table 4. Hedging performance comparison in the low volatility regime from January 2, 2014 to November 3, 2014 (203 trading days), and from September 1, 2016 to December 29, 2017(324 trading days), total is 527 trading days. Panel I: modeling the RMVHR RS ARMA ARMAARFIMA HAR HAR-GARCH GARCH HR 62.7947% 65.8457% 65.8505% 69.8919% 69.7844% 68.3916% HE 58.0378% 58.3540% 58.3542% 58.3191% 58.3797% 58.3610% PR 4.5457% 4.5333% 4.5324% 3.6916% 3.8158% 4.2150% SR 5.5696% 5.8338% 5.8331% 4.9322% 5.1236% 5.5303% Panel II: modeling the daily returns (covariance matrix) OLS ECM DCC DCC-RV VHAR NAÏVE HR 83.1962% 82.9233% 107.3345% 99.5945% 71.2981% 100.0000% HE 54.1327% 54.1596% 37.4661% 44.9733% 58.1988% 44.7205% PR 2.6491% 2.6513% -1.5033% -0.0839% 3.2563% -0.0217% SR 3.8532% 3.8483% -1.8078% -0.1130% 4.4796% -0.0283% Note: HR represents the hedge ratio, HE represents the hedging effectiveness, PR represents the portfolio return, SR represents the Sharp ratio. The lowest HR, the highest HE, the highest PR and the highest SR are in bold.

32

645 646 647

Table 5. Hedging performance comparison in the high volatility regime from November 4, 2014 to August 31, 2016 (448 trading days)

Panel I: modeling the RMVHR ARMARS ARMA ARFIMA HAR HAR-GARCH GARCH HR 62.6356% 65.7562% 65.7615% 70.9253% 71.4147% 70.3921% HE 58.7464% 59.1696% 59.1696% 58.9614% 58.8569% 58.8339% PR 5.6381% 3.0067% 3.0041% 2.3382% 1.4349% 2.1387% SR 2.2156% 1.2122% 1.2112% 0.9642% 0.5914% 0.8704% Panel II: modeling the daily returns (covariance matrix) OLS ECM DCC DCC-RV VHAR NAÏVE 73.5352% 81.1366% 70.7822% 100.0000% HR 78.8931% 78.3659% 53.8673% 54.9698% 58.8577% 41.9014% HE 56.4839% 56.6327% 0.8172% -4.3645% -0.5731% 2.0624% 0.2611% PR 0.7240% 0.3312% -1.6232% -0.2440% 0.8516% 0.0885% SR 0.2936% 648 Note: HR represents the hedge ratio, HE represents the hedging effectiveness, PR 649 represents the portfolio return, SR represents the Sharp ratio. The lowest HR, the 650 highest HE, the highest PR and the highest SR are in bold.

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