國
立
交
通
大
學
管理科學系
博
士
論
文
No. 046
已實現波動度於動態期貨避險之應用
Applications of Realized Volatility for Futures Hedging
研 究 生：賴雨聖
指導教授：許和鈞 教授
國
立
交
通
大
學
管理科學系
博
士
論
文
No. 046
已實現波動度於動態期貨避險之應用
Applications of Realized Volatility for Futures Hedging
研 究 生：賴雨聖
研究指導委員會：沈華榮 教授
謝國文 教授
鍾惠民 教授
指導教授：許和鈞 教授
中 華 民 國 九 十 八 年 十 一 月
已實現波動度於動態期貨避險之應用
Applications of Realized Volatility for Futures Hedging
研 究 生：賴雨聖 Student：Yu-Sheng Lai
指導教授：許和鈞 Advisor：Her-Jiun Sheu
國 立 交 通 大 學
管 理 科 學 系
博 士 論 文
A DissertationSubmitted to Department of Management Science College of Management
National Chiao Tung University in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
in
Management
November 2009
Hsin-Chu, Taiwan, Republic of China
已實現波動度於動態期貨避險之應用
已實現波動度於動態期貨避險之應用
已實現波動度於動態期貨避險之應用
已實現波動度於動態期貨避險之應用
研究生：賴雨聖 指導教授：許和鈞 國立交通大學管理科學系博士班中文摘要
中文摘要
中文摘要
中文摘要
本論文旨在應用已實現波動度(realized volatility)於動態避險議題上。首先我們提出新的 以高頻日內資訊為基礎之多變量波動度模式，並將此模式應用至動態避險比率之估計上。在 實證分析上，我們以美國S&P 500期貨為研究對象，探討此新的避險模式是否可以改進期貨避 險之績效。與傳統以低頻資訊為基礎之避險模式比較後發現，新的避險模式不但可以大幅降 低投資組合在樣本外之風險暴露額，並可為避險者產生正面的經濟價值。探究其主因，主要 是因為已實現波動度可以較報酬率平方(return squares)提供更精確之波動度估計值，以致在波 動度(或是避險比率)的預測上將得以較傳統方式有較佳的表現。 此外我們亦探討如何將此方法應用於事後(ex-post)避險績效之評估上。將Andersen et al. (2005, 2006)之研究成果應用至期貨避險議題上，我們建構之已實現避險比率得以用於評估各 種事前(ex-ante)避險模式之預測能力，同時漸進理論並提供衡量避險比率精確度之方法。另 者，已實現避險效果亦可用於衡量期貨避險之績效。 本論文最後探討已實現避險比率之動態性質。藉由門檻自我迴歸模式之分析，我們得以 研究避險比率是否存在門檻效果並探討其在不同狀態下之動態性質。應用Hansen (1996)提出 之拔靴檢定方法，我們發現門檻效果確實存在於已實現避險比率中。當避險比率高(低)於門 檻變數的情況下，避險比率本身將會有較低(高)的波動度。門檻效果以及正向自我迴歸現象 的存在均顯示避險比率應是與時變動(time-varying)的，因此實證結果支持動態避險比率假說。 藉由日內高頻資料的應用，已實現波動度提供我們一個不需模式設定(model-free)的方法 來估計不可觀測到的波動度指標。本論文的研究成果指出，高頻日內資料的使用將可為許多 財務上的議題，例如資產配置或期貨避險等，開啟另一嶄新的研究題材。 關鍵辭：動態避險比率；多變量波動度模式；已實現波動度；避險績效。Applications of Realized Volatility for Futures Hedging
Student: Yu-Sheng Lai Advisor: Her-Jiun Sheu Department of Management Science
National Chiao Tung University
Abstract
The dissertation applies the realized volatility (RV) approach to the futures hedging problem. Firstly, we propose a new class of multivariate volatility models encompassing RV estimates to estimate the risk-minimizing hedge ratio, and compare the performance of the proposed models with those generated by return-based models. In an out-of-sample context with a daily rebalancing approach, the empirical results show that improvement can be substantial when switching from daily to intraday. This essentially comes from the advantage that the intraday-based RV potentially can provide more accurate daily covariance matrix estimates than RV utilizing daily prices.
Next, we describe ex-post measures for assessing ex-ante hedge ratio estimates. Applying the realized beta framework of Andersen et al. (2005, 2006), the realized hedge ratio, realized hedging effectiveness, and the asymptotic confidence interval are constructed. The realized hedge ratio, which is consistent with the integrated hedge ratio, provides a natural benchmark for assessing the forecasting ability of any ex-ante hedge ratio estimates. Meanwhile, the asymptotic distribution provides insights into the precision of the realized hedge ratio. Furthermore, the realized hedging effectiveness provides an ex-post estimate for the integrated hedging effectiveness.
Then, the dynamics of the realized hedge ratio is investigated via a two-regime Self-Exciting Threshold Autoregressive (SETAR) model. The SETAR is tackled with a linear Autoregressive (AR) model if the threshold effect is not significant. Empirical results conclude the realized daily hedge ratio is characterized as regime-dependent dynamics and is likely to be positively autocorrelated so that the usual assumption of constant hedge ratio seems inappropriate.
The RV approach, which utilizes finer information in intraday high-frequency data, provides a direct and consistent technique for estimating the latent volatility without the need for relying on explicit models. Our investigation may provoke further study on the benefits of utilizing intraday information in volatility modeling, which is relevant to asset allocating and futures hedging.
Keywords: Dynamic hedge ratio; Multivariate volatility model; Realized volatility; Hedging performance.
致謝辭
致謝辭
致謝辭
致謝辭
在交大管科系博士班的學習過程，隨著博士論文的付梓即將劃上句點。回想剛進入校園 修業時的諸多挑戰，到畢業喜悅收成這段時間的點點滴滴，這一切都要感謝許多人對我的提 攜與幫助。 本論文能順利完成，首先最要感謝的就是我的指導老師－許和鈞教授，一路走來，老師 總是不厭其煩地叮嚀我論文的進度，並隨時教導我正確的研究態度與方式，我的研究成果能 順利發表於學術期刊及研討會上，這都需感謝許老師對於我的辛勤指導跟協助。另外我也要 特別感謝交通大學鍾惠民教授、中研院經濟所周雨田研究員及陳宜廷研究員，多次透過課堂 以及私下的互動機會，給予我學術上的引導與建議，並讓我了解身為研究者應有之積極態度 與堅持。此外我也要感謝我的碩士論文指導老師－盧陽正教授，由於盧老師的鼓勵與推薦， 讓我累積更多的本錢與勇氣能繼續攻讀博士學位，我亦永遠銘記在心。當然我的博士論文能 夠完成，也要感謝交通大學謝國文教授、中央大學周冠男教授以及聯合大學林美貞教授在百 忙中抽空擔任我的論文口試委員，並給予我許多寶貴的建議與方向，讓本篇論文更臻周延。 我也要感謝一群好友：交大財金所陳煒朋、劉炳麟、陳清和；交大管科系徐淑芳、王若 蓮、張玲玲、魏裕珍等同窗；交大經管所劉志良；以及暨南大學陳永泓。感謝你們的鼓勵與 陪伴，讓我苦悶的研究生活添加了許多樂趣及回憶。同時也要感謝管科系林碧梧小姐與葉秀 敏小姐在行政事務上的協助，讓我可以省卻許多繁瑣的業務手續。 最後感謝我的家人，由於你們的支持，使得我在求學的路上，少了許多的擔憂。老婆珮 琪是我這一路上最佳的諮詢者，除了擔起經濟重任外並身負小女瀅如的教養全責。感謝岳父 岳母接手小孩日間的褓母工作，免去我們夫妻倆許多煩惱。最後將本論文獻給我摯愛的父母 親及家人，感謝您無怨無悔的養育與無時無刻的關懷照顧，讓我能專注於課業研究中。從小 到大的求學路上，謝謝您們多年來的支持與鼓勵，我要把我所有的驕傲與你們分享。由衷感 懷之情，溢於言表！ 賴雨聖 民國九十八年十一月中Table of Contents
中文摘要 ……… i Abstract ………... ii 致謝辭 ……… iii Table of Contents ………. iv List of Tables ………... v List of Figures ………... vi Chapter 1. Introduction ………... 1 1.1. Motivation ………. 11.2. Dynamic Futures Hedge and Hedging Effectiveness ………... 3
1.3. Integrated Covariance Estimation using Intraday Data ... 6
1.4. Research Objectives ………... 7
1.5. Organization of the Dissertation ………... 8
Chapter 2. The Incremental Value of a Futures Hedge using Realized Volatility …………... 10
2.1. Research Problem and Objective ………... 10
2.2. The Conventional Hedging Models …………....………... 12
2.3. Alternative Models using Realized Volatility ………... 15
2.4. Empirical Analyses ………... 17
Chapter 3. An Application of Realized Regression to the Hedging Problem ……... 34
3.1. Research Problem and Objective ………... 34
3.2. Realized Hedge Ratio and Hedging Effectiveness ………... 36
3.3. An Illustrated Example ………. 38
Chapter 4. Regime-Dependent Dynamics of a Futures Hedge ……… 44
4.1. Research Problem and Objective ………... 44
4.2. A Two-Regime SETAR Model ………. 45
4.3. Realized Daily Hedge Ratios ...………. 46
4.4. Empirical Results ………... 47
Chapter 5. Conclusive Remarks ……….. 49
Bibliography ……….………... 52
List of Tables
Table 2.1 Data Description: Daily Price Returns ………. 18 Table 2.2 Data Description: Realized Variance, Covariance, and Correlation ……… 20 Table 2.3 Estimation Results of RV-Based and Return-Based Models ………... 21 Table 2.4 Summary Statistics of Out-of-Sample Hedge Ratios ………... 27 Table 2.5 Out-of-Sample Comparisons of Hedging Performance: Statistical
Evaluations ………... 28 Table 2.6 Out-of-Sample Comparisons of Hedging Performance: EV Gains ……….. 29 Table 3.1 Summary Statistics of Realized Hedge Ratios against Sampling
Frequencies ………... 39
Table 3.2 Unconditional Sample Means of Realized Weekly Hedging
Effectiveness ……… 42
Table 4.1 Statistics and Dynamic Dependences of Realized Daily Hedge Ratios …... 47 Table 4.2 Two-Regime SETAR Estimates ………... 48
List of Figures
Figure 1.1 Organization of the Dissertation ………... 9 Figure 2.1 In-Sample Comparisons on Conditional Volatility and Correlation Estimates:
RV-Based vs. Return-Based DCC Methods ... 23 Figure 2.2 Out-of-Sample Hedge Ratios: RV-Based vs. Return-Based Methods ………. 25 Figure 2.3 The Effect of Hedge Horizon on Hedge Ratio and Hedging Effectiveness:
RV-Based (Solid) vs. Return-Based (Dash) Methods ……….. 33 Figure 3.1 Average Values of Realized Weekly Hedge Ratios and 95% Confidence
Intervals drawn against Sampling Frequencies ……… 40 Figure 3.2 (a) Realized Weekly Hedge Ratios with 95% Asymptotic Confidence
Intervals; (b) Upper Bound of Realized Weekly Hedging
Chapter 1. Introduction
1.1. Motivation
Futures contracts are important hedging instruments for hedgers. By taking opposite positions in spot and futures markets, the price risk of a spot position can be reduced. Hence, theoretical and empirical aspects of a futures hedge have been the focus of much academic research. At the theoretical level, the hedging theories have shown that a hedge is the optimal when a hedger uses an optimal hedge ratio, e.g., Johnson (1960). Traditionally, this hedge ratio is de-rived by the portfolio approach via an expected-utility maximization scheme.1 With some ad-ditional restrictions, the minimum-variance hedge ratio, which is equivalent to the covariance of spot and futures over the variance of futures, is generally the optimal, e.g., Benninga et al. (1983). Due to the simplicity, this preference-free hedge ratio is easily to be implemented so that it has been widely adopted in the empirical studies.
At the empirical level, a considerable amount of studies on futures hedging have focused on modeling the joint distribution of spot and futures prices and applying the results to estimate the optimal hedge ratio. While the early studies assume that the hedge ratio is constant over time (e.g., Ederington, 1979), recent studies have documented that the joint distribution, and hence the hedge ratio, should be time dependent due to the time-varying nature of risks (e.g., Kroner & Sultan, 1993).2 Since then, varieties of multivariate volatility techniques have been applied. Baillie and Myers (1991), Myers (1991), Kroner and Sultan (1993), Brooks et al. (2002), Lien et al. (2002), and Lien and Yang (2006) are examples of studies that apply
1 On the other hand, the development of stochastic dominance theory has also facilitated the implementation of a futures hedge. An overview of this application could be found in Lien and Tse (2002).
2 With the aim of reducing risk, this hedge ratio is equivalent to the ratio of the conditional covariance between spot and futures over the conditional variance of futures.
eralized autoregressive conditional heteroscedasticity (GARCH) models.3 As a result, the central issue in the context of a dynamic hedge is to provide the conditional covariance matrix forecast that can characterize the dynamics of the distribution more realistically.
Conventionally, the standard GARCH conditional covariance matrix forecast is simply specified as some functions of its past values as well as outer product of past daily or weekly returns.4 In other words, by only utilizing daily or weekly price information, this category of volatility models provides relatively convenient and traceable method to give the forecast. Nevertheless, even with correctly specified models, it has been shown that the GARCH-type forecasts do not fully mimic the properties of integrated covariance matrix, which serves as an ideal theoretical ex-post benchmark for assessing the quality of ex-ante covariance matrix forecasts (see Andersen et al, 2006). One possible reason of this imperfect forecast may come from that the low frequency data do not fully convey all the relevant information so that the outer product of past returns itself is a nosy proxy of the multivariate volatility.
To judge the success of a futures hedge, the hedging effectiveness (HE) of Ederington (1979) has been extensively adopted in the empirical studies as a benchmark for hedging per-formance and as a measure to select the best hedging method. This traditional measure defines the effectiveness of a hedge by calculating the percentage reduction from the variance of the spot position to the variance of the hedged portfolio. Empirical results on futures hedging us-ing this unconditional HE has shown that the simple ordinary-least-square (OLS) regression method generally has a best performance in sample, and the superiority of this OLS method is also supported by some out-of-sample comparisons.5 For example, Kroner and Sultan (1993)
3
In addition to the GARCH framework, other studies use stochastic volatility models (e.g., Lien & Wilson, 2001).
4 The standard multivariate GARCH models specify the conditional covariance matrix by either generalizing the univariate GARCH models of Bollerslev (1986) or by combining some univariate GARCH models. The former includes the VECH model of Bollerslev et al. (1988) and the BEKK (Baba-Engle-Kraft-Kroner) model of Engle and Kroner (1995). The latter includes the constant conditional correlation (CCC) model of Bollerslev (1990), the dynamic conditional correlation (DCC) model of Engle (2002), and the copula-based GARCH model of Pat-ton (2004, 2006). A review of a wide range of multivariate GARCH models is referred to Bauwens et al. (2006). 5
indicate that the OLS method outperforms all other constant hedge methods in a within-sample context; and, Lien et al. (2002) indicate that the OLS hedge even can beat the constant conditional correlation GARCH (CCC-GARCH) hedge in an out-of-sample context. Lien (2005a,b, 2008, 2009) shows that the OLS hedge tends to outperform others, such as the naïve, error correction (EC), or the GARCH hedges, for within-sample comparisons because the hedging effectiveness of Ederington (1979) only considers the proportional reductions in the unconditional variance. Hence, whether the dynamic hedge can surpass the static OLS is still studied both theoretically and empirically, although the literature has identified with the hedge ratio should be time varying as the new information has arrived to the market (e.g., Kroner & Sultan, 1993).
1.2. Dynamic Futures Hedge and Hedging Effectiveness
Consider a one-period futures hedging problem. Suppose an investor (hedger) who longs a fixed spot portfolio at beginning would like to reduce the price risk of the spot at the end of the period. To achieve this goal, he may go to a futures market to short a proportion of futures contracts. Usually, nearby contracts are used due to liquidity concerns. Let St and Ft be
the logarithmic prices of spot and futures, respectively, at time t; and, βt be a hedge ratio,
which is defined by the amount of futures per unit spot, at time t. Then the realization on the hedged portfolio return for the hedging period (from time t to t+1) is given by
, 1 , 1 , 1 p t s t t f t
r + =r + −βr + (1.1)
where rs t,+1 =St+1−St is the return for holding the spot, and rf t,+1 =Ft+1−Ft is the
re-turn for holding the futures. Note that the hedge ratio is an unknown decision variable for the
(2006) argue that the chosen (unconditional) risk measures that are used in calculating the hedging effectiveness has important implications in determining the best hedging method when the hedgers have their specific aims. With the use of an extensive set of risk measures, including variance, semi-variance, lower partial moments, value at risk, and conditional value at risk (or called expected shortfall), they find that the best hedging method is sensitive to the specific risk measures.
hedger so that it needs to be estimated via some models.6
Assume that the hedger would like to choose the optimal futures holdings by maximizing the expected (mean-variance) utility function:
, 1 , 1 , 1
(p t | t; , )t ( p t | t) var( p t | t)
EU r + Φ β γ =E r + Φ −γ r + Φ (1.2)
at time t with the degree of risk aversion γ >0, where the expectation and the variance operators are calculated conditional on the set of all available information Φt at time t
(Kroner & Sultan, 1993). With some mathematical derivations, the optimal hedge ratio equals to , 1 , 1 , 1 * , 1 ( | ) 2 cov( , | ) 2 var( | ) f t t s t f t t t f t t E r r r r γ β γ + + + + Φ + Φ = Φ (1.3)
When the expected return for holding futures is zero (i.e. E r( f t,+1 |Φ =t) 0) or the degree of risk aversion is high (γ → ∞), the expected utility-maximizing problem simplifies to the (conditional) variance-minimizing problem:
, 1 min var( | ; ) t p t t t r β + Φ β (1.4)
and the optimal hedge ratio (or minimum-variance hedge ratio) is formulated as
, 1 , 1 * , 1 cov( , | ) var( | ) s t f t t t f t t r r r β + + + Φ = Φ (1.5)
The derivation of this hedge ratio is generally valid for von-Neumann Morgenstern utility functions for U ′( )⋅ > and 0 U ′′()⋅ < (Benninga et al., 1983). This simplified hedge ratio 0 only depends on conditional second moments of spot and futures, and therefore, one central issue in the context of a dynamic hedge is to provide the conditional covariance matrix
6 Traditional approach to commodity futures hedging adopts the naïve strategy, which suggests a hedger who longs a unit of spot position should sell a unit of futures today and then buy the contracts back when he sells the spot. A perfect hedge is achieved when the spot and futures prices both move by the same amount; however, in practice, it is found that the prices usually do not have identical co-movements.
casts that can characterize the dynamics of the second moments more realistically.
Assume that the 2 1× vector of returns Rt =(rs t,,rf t,)′ follows the discrete-time process:
1 2 t t t t
R =M + Ω Z , t =1,2,…,T (1.6) where Mt ≡E R( t |Φt−1) and Ω ≡t var(Rt |Φt−1) represent the 2 1× conditional mean
vector and 2 2× conditional covariance matrix of Rt, respectively; Zt =(ηs t,,ηf t,)′ is an 2 1× vector of serially uncorrelated disturbances with E Z( )t = 0 and var( )Zt =I. Note
that Mt consists of individual conditional means, and Ωt consists of individual conditional variance σ (i t2, i=s f, ) in the diagonal and conditional covariance σsf t, in the off-diagonal. This decomposition is of much popular with the empirical success of multivariate volatility models, e.g. the multivariate GARCH-type models; and, over years, a large body of studies has applied these models to estimate the dynamic hedge ratio, e.g., Kroner and Sultan (1993), and among many others.
To judge the success of these methods, the hedging effectiveness (HE) of Ederington (1979) has been extensively adopted as a benchmark for the hedging performance.7 The measure de-fines the effectiveness of a futures hedge by calculating the percentage reduction from the variance of the spot position to the variance of the hedged portfolio, or
, 1 , 1 var( ) HE 1 var( ) p t s t r r + + ≡ − (1.7)
where the realization on the hedged portfolio return rp t,+1 equals to rs t,+1−βt*rf t,+1; rs t,+1 and rf t,+1 represent realizations of spot and futures returns, respectively; var( )⋅ denotes the variance operator. Based on this performance measure, a hedging method is deemed better than others if it can generate a higher HE or equivalently a smaller var(rp t,+1) for the hedged portfolio.
7 In addition to the statistical measure, alternative economic measures are also discussed, such as the Sharpe-type hedging performance of Howard and DَAntonio (1984) and the economic value of Lence (1995).
1.3. Integrated Covariance Estimation using Intraday Data
The availability of intraday high-frequency data for many financial assets has benefited the measurement of realizations on the unobserved latent volatility process. Without through standard time series techniques, e.g., the GARCH models, the realized volatility (RV) ap-proach has provided a consistent model-free estimate of the price volatility over a given dis-crete-time interval. To illustrate, suppose the 2 1× vector of (spot-futures) returns is arisen from the continuous-time diffusion process:
( ) ( ) ( ) ( )
dP t =M t dt + Σt dW t , t ∈[0, ]T (1.8) where P t( )={ ( ), ( )}S t F t ′ represents the 2 1× vector of logarithmic prices; M t( ) repre-sents the 2 1× instantaneous drifts; Σ( )t represents the 2 2× instantaneous positively definite diffusion matrix that consists of individual instantaneous variance σi2( )t (i =s f, ) in the diagonal and instantaneous covariance σsf( )t in the off-diagonal; W t( ) represents the 2 1× vector of independent standard Brownian motions. Over the [t−, ]t time inter-val, the return for the continuous-time model is defined as
( , ) ( ) ( ) t ( ) t ( ) ( ) t t R t P t P t M s ds s dW s − − = − − =
∫
+∫
Σ (1.9)When =1, it represents the one period return from time t−1 to t. As compared with the discrete-time representation, the conditional mean vector and covariance matrix are replaced by the corresponding integrated mean and covariance process with the innovation driven by the continuously evolving standard Brownian motion. As a result, the integrated covariance matrix for the one-period, which is formulated by
1 ICov( ) t ( ) ( ) t t s s ds − ′ =
∫
Σ Σ (1.10)is closely related to the conditional covariance matrix in the discrete-time framework so that the integrated variance (covariance) has been served as an ideal ex-post benchmark for
as-sessing the quality of ex-ante variance (covariance) forecasts (Andersen et al., 2006).
The availability of intraday high-frequency data has provided a way to estimate the inte-grated covariance matrix. Define the realized covariance matrix for the time interval [t−1, ]t as 1/ 1 RCov( , ) ( 1 , ) ( 1 , ) j t R t j R t j = ′ =
∑
− + ⋅ − + ⋅ (1.11)As the sampling frequency of returns goes to infinity, the bivariate realized covariation con-verges to the corresponding bivariate integrated covariation:
RCov( , )t →ICov( )t , →0 (1.12) Details of the proof are referred to Andersen et al. (2001a, 2001b, 2003), and Barndorff-Nielsen and Shephard (2004). For notation simplicity, hereafter, we denote RVs t,
and RVf t, the realized variance of the spot and the futures, respectively; and RCovt the
realized spot-futures covariance, at time t using the equally sampled discrete-time re-turns.
1.4. Research Objectives
The objective of this dissertation is to apply the model-free RV approach to the one-period hedging problem. To do so, there are three topics to be investigated in the following article.
Firstly, a large number of empirical studies on futures hedge have concerned with the con-ventional multivariate GARCH-type models, which only utilize daily (weekly) price informa-tion, in estimating the dynamic hedge ratio. With the use of intraday informainforma-tion, Chapter 2 presents alternative discrete-time multivariate volatility models encompassing the elements of realized covariance in estimating the dynamic hedge ratio. In addition, the benefits on the dy-namic hedge over those conventional methods using return-based volatility models are
com-pared based on an extensive set of statistical and economical performance measures.
Next, applying the realized beta framework of Andersen et al. (2005, 2006), Chapter 3 ex-tends the framework to analyze the hedging problem. With the availability of intraday data, it is shown that the realized hedge ratio and realized hedging effectiveness provide alternative ex-post benchmarks for evaluating the performance of ex-ante hedging methods. Furthermore, an empirical example is also exhibited in this chapter.
Finally, with the construction of realized daily hedge ratio series, the dynamics of the hedge ratio is analyzed in Chapter 4. The importance of this analysis is that the insight into the hedge ratio behavior may further the development of dynamic hedge ratio models. Moreover, a two-regime threshold autoregressive model is also applied to detect the regime-switching feather of the ratio. If the feature is appeared in the hedge ratio, the argument of time-varying hedge ratio is further supported by the evidence.
1.5. Organization of the Dissertation
The organization of the dissertation is depicted in Figure 1.1 and briefly introduced as follows. Chapter 1 introduces the dissertation with the organization. Chapter 2 proposes a new class of discrete-time multivariate volatility models encompassing the elements of realized covariance matrix to estimate the risk-minimizing hedge ratio, and compare the performance with those generated by return-based volatility models. Chapter 3 presents a RV-based method for ana-lyzing the one-period hedging problem with an illustrated example. Chapter 4 assesses the dynamics of realized daily hedge ratios. Chapter 5 concludes the dissertation with suggestions for the future research.
Chapter 2. The Incremental Value of a Futures Hedge
using Realized Volatility
This chapter proposes a new class of multivariate volatility models encompassing RV esti-mates to estimate the risk-minimizing hedge ratio, and compare the hedging performance of the proposed models with those generated by return-based models. In an out-of-sample con-text with a daily rebalancing approach, based on an extensive set of statistical and economic performance measures, the empirical results show that improvement can be substantial when switching from daily to intraday. This essentially comes from the advantage that the intra-day-based RV potentially can provide more accurate daily covariance matrix estimates than RV utilizing daily prices. Finally, this study also analyzes the effect of hedge horizon on hedge ratio and hedging effectiveness for both the in-sample and the out-of-sample data.
2.1. Research Problem and Objective
It has been shown that most of the latent GARCH models fail to satisfactorily describe the high kurtosis, small first-order autocorrelation of squared returns, and slow decay of the autocorrelation of squared returns toward zero that have been observed in many daily or weekly financial returns (e.g., Carnero et al., 2004).9 The search for alternative volatility techniques has motivated scholars to exploit information in intraday high-frequency data (e.g., Anderson & Bollerslev, 1998; Andersen et al., 2003; Barndorff-Nielsen & Shephard, 2004; Hayashi & Yoshida, 2005; Voev & Lunde, 2007). This so-called RV approach provides a di-rect and consistent technique to estimate the latent volatility process without the need for
9 To capture the high kurtosis of returns and low first-order autocorrelation of return squares simultaneously, GARCH models often require a high persistence and/or leptokurtic conditional distributions when they are fitted to the financial time series (Carnero et al., 2004). The restrictions on GARCH then severely restrict the allowed dynamic dependence of the volatility.
lying on explicit models.10 Therefore, it has served as an ex-post benchmark for assessing the quality of any ex-ante volatility forecasts (e.g., Andersen et al, 2006). In forecasting future volatility, studies have shown that RV can provide more accurate forecasts than methods using daily squared returns (e.g., Blair et al., 2001). Recently, RV has been employed to explore the economic value (EV) of investment by Fleming et al. (2003), Bandi et al. (2008), and De Pooter et al. (2008).11 They find that the EV is substantial when switching from daily to in-traday returns, even without applying bias-correction techniques in constructing realized es-timates. As a result, the superiority of RV-based investment may result because it takes into account finer intraday information and thus potentially provides more accurate daily covari-ance matrix estimates or forecasts than methods utilizing daily prices.
This study attempts to improve the performance of a risk-minimizing futures hedge when the intraday-based RV approach is incorporated into forecasting the relevant covariance ma-trix. It is not clear, however, whether the accurate RV is used or not may differentiate the performance of a futures hedge. To address this issue, a new RV-based method is demon-strated in this study. The proposed method builds on the bivariate error correction model by employing the flexible CCC-GARCH error structure of Kroner and Sultan (1993). The ad-vantage of extending their model is that it can capture the long-run cointegration relationship and the time-varying second moments simultaneously when the dynamics of the joint distri-bution is specified. Moreover, when the leverage effect and/or the dynamic correlation are re-vealed, the asymmetric volatility and/or the dynamic conditional correlation (DCC) models can also be incorporated into this method. These desirable properties thus provide a more
10 Generally, in the absence of market microstructure and non-synchronous trading (Epps effect), a daily meas-ure of variance is computed as the sum of the squared intraday equidistant returns, and a daily measmeas-ure of co-variance is obtained by summing the products of intraday equidistant returns, for the given trading day. For a re-view of the RV refers to McAleer and Medeiros (2008).
11 For example, Fleming et al. (2003) show how a risk-averse investor would be willing to pay 50 to 200 basis points per year to capture the multivariate volatility forecasts based on intraday returns instead of daily returns in the context of investment decisions on three actively traded futures contracts (S&P 500 index, Treasury bonds, and gold). It is assumed that the investor follows a volatility-timing strategy, which rebalances his portfolio only when the estimated conditional covariance matrix of the daily returns changes. This case treats the expected daily returns as time-invariant.
convenient way to lodge the realized covariance matrix in the GARCH error structures as compared with the rolling estimators of Fleming et al. (2003), Bandi et al. (2008), and De Pooter et al. (2008). The RV-based method, which uses finer volatility proxies in estimating and forecasting the conditional covariance matrix, is expected to provide better descriptions on the spot-futures dynamics and the resulting hedge ratios than methods using squared return shocks.
In the empirical analyses, the RV-based hedge ratios are calculated in an out-of-sample context spanning the period of December 19, 2003 and March 31, 2009 for the highly traded S&P 500 index futures contracts. we compare the performance of the RV-based hedge with the return-based GARCH and the OLS hedges using an extensive set of statistical and eco-nomic measures. The comparisons are conducted for both short and long hedgers. To antici-pate the results, this study finds that the RV-based hedge can substantially outperform the re-turn-based GARCH and/or the static OLS hedges especially during the surge in volatility pe-riod. Then the RV-based method is applied to examine the effect of hedging horizon (ranges from one week to three month) on hedge ratio and hedging effectiveness. The results show that hedge ratio tends to increase and to approach unity (i.e., naïve hedge ratio) with the length of hedging horizon; and, hedging effectiveness tends to increase as the length of hedg-ing horizon increases. The rest of the chapter is organized as follows. First, we present the conventional hedging method and demonstrate the RV-based method. Next, we present the data and their properties with the empirical results. Finally, the last section concludes the study.
2.2. The Conventional Hedging Models
Kroner and Sultan (1993) have proposed a bivariate GARCH error correction model for mod-eling the joint distribution of spot and futures. As such, the most prominent application of this
model is to estimate time-varying hedge ratios. The econometric model for the daily returns conditioning on the set of all relevant information Φt−1 at time t−1 can be described as
, 0 1 1 1 , , 0 1 1 1 , ( ) ( ) s t s s t t s t f t f f t t f t r S F r S F α α δ ε α α δ ε − − − − = + − + = + − + (2.1) , 1 , | (0, ) s t t t f t N H ε ε − Φ ∼ (2.2)
where (St−1−δFt−1) is the error correction term (ECT). The incorporation of ECT in the conditional mean equations is essential, especially in currency and equity markets (see Park & Switzer, 1995; Brooks et al., 2002; Choudhry, 2003). The residual vector (εs t,,εf t,)′ is postu-lated as a bivariate normal distribution with a 2 1× zero mean vector and a 2 2× time-varying covariance matrix
1/ 2 1/ 2 , , , , 1/ 2 1/ 2 , , , , 0 1 0 1 0 0 s t sf t s t s t t t t sf t f t f t f t h h h h H D RD h h h h ρ ρ = = = (2.3)
where hsf t, is a covariance, and hs t, and hf t, are conditional variances for the spot and
fu-tures returns, respectively, and ρ is the time-invariant correlation coefficient between them. That is, the model applies the CCC estimator of Bollerslev (1990) to model and forecast the bivariate conditional covariance matrix.
The expressions of the conditional variances hs t, and hf t, in Equation (2.3) are typically
thought of as univariate GARCH-type models. For example, the GARCH(1,1) structure in-troduced by Bollerslev (1986) for the returns can be specified as
2 , 0 1 , 1 2 , 1 2 , 0 1 , 1 2 , 1 s t s s s t s s t f t f f f t f f t h h h h β β β ε β β β ε − − − − = + + = + + (2.4)
When the leverage effect is revealed, asymmetric volatility models are commonly formulized, such as the GJR (Glosten-Jagannathan-Runkle) model of Glosten et al. (1993). This simply modifies the standard GARCH(1,1) model with an additional ARCH (autoregressive
condi-tional heteroscedasticity) term condicondi-tional on the sign of the past innovation. The condicondi-tional volatilities of Equation (2.4) are then reformulated as
2 2 , 0 1 , 1 2 , 1 3 , 1 , 1 2 2 , 0 1 , 1 2 , 1 3 , 1 , 1 ( 0) ( 0) s t s s s t s s t s s t s t f t f f f t f f t f f t f t h h I h h I β β β ε β ε ε β β β ε β ε ε − − − − − − − − = + + + < = + + + < (2.5)
where I( )⋅ denotes an indicator function. For estimating the constant correlation coefficient, the ever-popular rolling estimator uses equal weight to all past T return innovations:
, , 1 2 2 , , 1 1 ˆ [ 1,1] ( )( ) T s t f t t T T s t f t t t ε ε ρ ε ε = = = =
∑
∈ −∑
∑
(2.6)To fit in with reality, however, the conditional correlation coefficient can be relaxed to vary with time (Engle, 2002; Tse & Tsui, 2002). That is, the conditional covariance matrix of Equation (2.3) is generalized to the DCC formulation:
1/ 2 1/ 2 , , , , 1/ 2 1/ 2 , , , , 0 1 0 1 0 0 s t sf t s t t s t t t t t t sf t f t f t f t h h h h H D R D h h h h ρ ρ = = = (2.7)
The evolution of ρt in this paper is analogous to the univariate GARCH equation:
, , 1 1 2 1 1 2 2 2 , , 1 1 (1 ) ( )( ) M s t h f t h h t t M M s t h f t h h h η η ρ θ θ ρ θ ρ θ η η − − = − − − = = = − − + +
∑
∑
∑
(2.8)where θ1 and θ2 are non-negative with θ1+θ2 ≤1; the sample size M =2 for
estimat-ing the sample correlation coefficient with ηi t h,− =εi t h,− /hi t h1/ 2,− (i =s f, ) follows Tse and Tsui (2002); ρ is the unconditional correlation between spot and futures.
Given the null hypothesis that the set of all relevant information Φt−1 is observed with correctly specified models, these CCC or DCC models with the ECT describe the dynamic nature of spot-futures distributions. In particular, this GARCH class of volatility (correlation) models provides simple ways to forecast the bivariate volatility by using actual return innova-tions and the estimated hedge ratio by minimizing the risk of the hedged portfolio return at time t is given by
*
ˆ
t
β =hˆsf t,+1/hˆf t,+1 (2.9)
where hˆsf t,+1 and hˆf t,+1 are the covariance and variance forecasts using Equations (2.3) or (2.7). In brief, Equations (2.1) through (2.9) then construct the conventional hedge ratio mod-els, namely, the ECT-GARCH-CCC model (Equations (2.1)-(2.4), (2.6), and (2.9)), the ECT-GARCH-DCC model (Equations (2.1)-(2.2), (2.4), and (2.7)-(2.9)), the ECT-GJR-CCC model (Equations (2.1)-(2.3), (2.5)-(2.6), and (2.9)), and the ECT-GJR-DCC model (Equa-tions (2.1)-(2.2), (2.5), and (2.7)-(2.9)).
2.3. Alternative Models using Realized Volatility
To explore the incremental value of a RV-based hedge, the realized variance and/or correla-tion are encompassed within the convencorrela-tional CCC and DCC models with the ECT specifica-tion. Following the previous specifications, the RV-based GARCH(1,1) volatility can be ex-pressed as , 0 1 , 1 2 , 1 , 0 1 , 1 2 , 1 RV RV s t s s s t s s t f t f f f t f f t h h h h β β β β β β − − − − = + + = + + (2.10)
where each RVi t−, 1 (i=s f, ) is defined by summing up -minute squared returns at time
1
t− . Analogously, the RV-based GJR(1,1) model can be reformulated as
, 0 1 , 1 2 , 1 3 , 1 , 1 , 0 1 , 1 2 , 1 3 , 1 , 1 RV RV ( 0) RV RV ( 0) s t s s s t s s t s s t s t f t f f f t f f t f f t f t h h I h h I β β β β ε β β β β ε − − − − − − − − = + + + < = + + + < (2.11)
That is, the right-hand side squared residuals in the conventional GARCH or GJR models are replaced by realized variances. To encompass the realized correlation
1/ 2 1 1 , 1 , 1
RCorrt− =RCovt− (RVs t−RVf t−) within the conditional correlation dynamics, Equa-tion (2.8) can be modified as
1 2 1 1 2 1
(1 ) RCorr
t t t
where the realized covariance RCovt−1 is defined by cumulating the cross-products of the
intraday spot and futures returns at time t−1. For estimating the constant correlation in the RV-based CCC model, however, the sample mean of the realized correlations should be a bi-ased estimate due to non-synchronous trading (Epps effect) and/or market microstructure noise (see, e.g., Hayashi & Yoshida, 2005; Voev & Lunde, 2007). To simplify the estimation process, the rolling estimator of Equation (2.6) is still adopted. Then, these modifications construct a new class of RV-based hedge ratio models, namely, the ECT-RV-GARCH-CCC model (Equations (2.1)-(2.3), (2.6), (2.9), and (2.10)), the ECT-RV-GARCH-DCC model (Equations (2.1)-(2.2), (2.7), (2.9), (2.10), and (2.12)), the ECT-RV-GJR-CCC model (Equa-tions (2.1)-(2.3), (2.6), (2.9), and (2.11)), and the ECT-RV-GJR-DCC model (Equa(Equa-tions (2.1)-(2.2), (2.7), (2.9), and (2.11)-(2.12)).
To estimate the parameters in the RV-based or the conventional CCC or DCC models, we follow the two-step estimation procedure of Bollerslev (1990) and Engle (2002). Since this class of multivariate volatility models has separate parameters, it can be estimated easily and consistently in two steps.12 With the normality assumption of Equation (2.2), we can maxi-mize each volatility term with the conditional mean in the first step:
ˆ arg max{ ( )} i
i LV i
ϑ = ϑ (2.13)
where ϑi =( , , )α β δi i and LVi denote the Gaussian quasi-likelihood function for asset i, and then maximize the correlation term:
ˆ
θ =arg max{L θ ϑC( | )}ˆ (2.14)
in the second step, LC represents the Gaussian quasi-likelihood function of the correlation
part, and θ =ρ for the CCC and θ =( , )θ θ1 2 for the DCC. Without the normality
12 Engle and Granger (1987) indicate that a cointegration system could consistently be estimated via a two-step estimator, where both steps require only single equation least squares, so that we can estimate the DCC-type models with the ECT by using the two-step procedure of Bollerslev (1990) and Engle (2002).
tion, these estimators still have the quasi-maximum likelihood (QML) interpretation.
2.4. Empirical Analyses
Data Descriptions
The performance of the RV-based hedging method is examined empirically on the S&P 500 index futures contracts traded on Chicago Mercantile Exchange (CME). The sample period is from January 1, 1998 to March 31, 2009, which covers the period of subprime mortgage crisis. We obtain the daily closing (settlement) prices for the spot (futures) from the Datastream and the intraday transaction prices for them from Tick Data Inc.13 Note that this study rolls the nearest month contract to the next month when the daily volume of the current contract is ex-ceeded. Specifically, daily and intraday prices of all days corresponding to U.S. public holi-days are removed. Hence, there are 2828 trading holi-days for the period examined.
Table 2.1 reports the diagnostic checks on the distribution properties of the daily spot (fu-tures) returns, which are calculated as differenced natural logarithmic daily closing (settle-ment) prices. The results of unit root and cointegration tests are shown in Panel A, and sum-mary statistics on the returns are reported in Panel B. The augmented Dickey-Fuller tests show that the spot and futures log-prices have a unit root, but their first-differenced series are stationary. The Johansen trace statistic indicates that the spot and futures prices are cointe-grated with the cointegrating parameter δˆ≈1. The unconditional distributions of the uni-variate returns reveal non-normality, as evidenced by the non-zero skewness, high kurtosis, and significant Jarque-Bera statistics. Panel C provides the autocorrelation functions (ACF) as well as the Ljung-Box statistics of the squared (cross-product) daily returns. The first-order ACF ranges from 0.1970 to 0.4093 but decays slowly toward zero. Based on the empirical
13 The intraday transaction observations consist of open, high, low, and close prices at the one-minute sampling frequency.
evidences and the findings of Carnero et al. (2004), the conventional GARCH models may be inadequate to describe the spot-futures volatility dynamics, hence, it is not clear whether the inadequacy may damage the performance of a futures hedge.
TABLE 2.1 Data Description: Daily Price Returns
Indices Statistics
Panel A: Unit root and cointegration tests ADF (price) ADF (return) Trace δˆ Spot -1.36* -42.15* Futures -1.36* -42.40* 84.97 * 0.9841*
Panel B: Summary statistics of returns
Mean Std. Dev. Skewness Kurtosis Correlation JB
Spot -0.0001* 0.0138* -0.1280* 10.5955* 6798.74*
Futures -0.0001* 0.0141* 0.0369* 12.4322* 0.9760
*
10480.13* Panel C: Autocorrelation functions of squared (cross-product) returns
ACF(1) ACF(2) ACF(3) ACF(4) ACF(5) Q2(10)
Spot 0.3661* 0.1785* 0.2935* 0.3307* 0.2982* 2230.80*
Futures 0.4093* 0.1692* 0.2526* 0.3000* 0.2516* 2032.04*
Spot-Futures 0.1970* 0.4310* 0.1727* 0.3139* 0.3488* 1279.50*
Notes: The daily spot (futures) returns are calculated as differenced natural logarithmic closing (settlement) prices where public holidays are removed. The sample period for the prices runs from January 1, 1998 to March 31, 2009 and the sample size for each is 2828. The values in rows ADF, Trace, JB, ACF(k), and Q2(10) are sta-tistics of the augmented Dickey-Fuller unit root test, the Johansen cointegration test, the Jarque-Bera normality test, the order k autocorrelation of squared returns, and the Ljung-Box test for the serial correlations in the squared returns. δˆ is the estimated cointegrating parameter. * indicate significance at the 5% level.
To construct the realized (co-)variance for the alternative method, the intraday futures prices after 3:00 p.m. Chicago time for each day t are dropped since the futures market closes fifteen minutes later than the spot market.14 we divide the contemporaneous time sec-tion across the markets, which runs from 8:30 a.m. until 3:00 p.m. (390 minutes), into m (non-overlapping) intervals of equal lengths ≡390 / m such that the times
1 j j
t =t− + for j = …1, ,m with t0 =8:30 a.m. Chicago time. The log close transaction price at time
j
t is denoted as p t( )j , then the equidistant intraday returns on day t, ( ) ( 1) j
t j j
r ≡p t −p t− ,
14 The (floor) trading section for the S&P 500 index futures on the CME runs from 8:30 a.m. Chicago time until 3:15 p.m. Chicago time.
for j =2,…,m; and, the first period (j =1) intraday return is defined as the difference be-tween the log close and open transaction prices during that time interval. With these mathe-matical definitions, the realized variance is then defined as RV( ),m m1 2,j
j
i t ≡ ∑ = ri t for i=s f, ,
and the realized covariance is defined as RCov( ) 1 ,j ,j
m m j
t ≡ ∑ = r rs t f t , for each day t.
This study uses fifteen-minute intraday prices (m =26) to construct the realized (co-)variance estimates, and summarizes their descriptive statistics in Table 2.2. Since only observations during the floor trading section are sampled, the average realized variances are smaller than the corresponding unconditional variances obtained from daily returns. For ex-ample, the average value of the realized variance estimates for the S&P 500 cash index is 1.11e-4, which is about 58% of the unconditional variance 1.90e-4 calculated from Table 2.1. To measure the realized variance (covariance) for the whole day, the squared (cross-product) overnight returns can further be incorporated into the realized estimators (Martens, 2002; Fleming et al., 2003, Hansen & Lunde, 2005; De Pooter et al., 2008). For the realized correla-tions, the average level for the S&P 500 (about 0.95) shows slight bias toward zero as com-pared with the corresponding unconditional correlation estimates (about 0.98) using Equation (2.6). According to Hayashi and Yoshida (2005) and Voev and Lunde (2007), the biasness may come from the non-synchronous trading and/or the market microstructure noise, but it can be corrected using some bias-correction techniques.15 The ACF of these realized esti-mates are also reported in Table 2.2. It shows that these realized second moments reveal con-siderable persistency. However, the ACF of the realized variance (covariance) is higher than the corresponding ACF of squared (cross-product) returns in Table 2.1. For example, in the spot market, the ACF(1) of the realized variance is about 0.66, which is higher than the ACF(1) of the squared returns in Table 2.1 by about 0.37. It is expected that the behavior dif-ference between the realized variance (covariance) and squared (cross-product) returns should
15 We do not adjust for the biases in the empirical analyses because the bias-correction procedures do not guar-antee a positive definite realized covariance matrix estimates; see, De Pooter et al. (2008) for discussions.
produce different volatility (covariance) estimates and forecasts based on the alternative and the conventional methods.
TABLE 2.2 Data Description: Realized Variance, Covariance, and Correlation
Summary Statistics Autocorrelation Functions
Realized
Estimates Mean Median Min Max ACF(1) ACF(2) ACF(3) ACF(4) ACF(5)
,
RVs t 1.11e-4 5.40e-5 3.00e-6 0.0053 0.6550 0.6632 0.5424 0.5995 0.5357 ,
RVf t 1.20e-4 5.90e-5 3.00e-6 0.0055 0.6558 0.6677 0.5442 0.6050 0.5467 RCovt 1.11e-4 5.40e-5 3.00e-6 0.0054 0.6573 0.6710 0.5460 0.6080 0.5451
RCorrt 0.9506 0.9595 0.2855 0.9986 0.3993 0.3823 0.3999 0.3863 0.4036
Notes: The sample period for these (daily) realized estimates, which are constructed from fifteen-minute equi-distant intraday returns, spans the period of January 1, 1998 to March 31, 2009, and the sample size is 2828. It is noted that the futures returns after 3:00 p.m. Chicago time for each day are dropped since the futures markets close fifteen minutes later than the spot markets. The ACF(k) indicates the sample autocorrelation function of the realized estimates corresponding to lags k=1,2,…5. The upper and lower confidence bounds of the ACF with the 5% confidence level are 0.0375 and -0.0375, respectively.
Estimation Results
Table 2.3 presents the estimation results of the return-based and the RV-based GARCH mod-els. Panel A shows the conditional mean and variance estimates, and Panel B shows the con-ditional correlation estimates. Given the evidence of the cointegration relationship between the spot and futures (in Table 2.1), the restricted ECT, (St−1−Ft−1), is parameterized in the conditional mean equations to avoid the loss of long-run information; though, all α1f are in-significant different from zero (Park & Switzer, 1995; Brooks et al., 2002). The insignificance of α0 f and α1f coefficients show the expected returns of futures should be zero, meaning the minimum variance hedge is generally the expected utility maximization hedge (Baillie & Myers, 1991; Kroner & Sultan, 1993).
TABLE 2.3 Estimation Results of RV-Based and Return-Based Models
Return-Based RV-Based
ECT-GARCH ECT-GJR ECT-RV-GARCH ECT-RV-GJR
Parameters / Statistics
CCC DCC CCC DCC CCC DCC CCC DCC
Panel A: Estimates of conditional mean and conditional variance equations
0s α -1.88e-5 (-0.07) -0.0004 (-1.72) -0.0004 (-1.64) -0.0004 (-1.95) 1s α -0.1035 (-2.54) -0.0880 (-2.76) -0.0947 (-2.40) -0.1010 (-2.58) 0s β 1.13e-6 (5.91) 1.31e-6 (7.02) 6.76e-6 (7.57) 4.79e-6 (7.09) 1s β 0.9173 (121.28) 0.9258 (113.79) 0.7702 (33.59) 0.8261 (45.93) 2s β 0.0774 (10.68) 0.0001 (0.01) 0.2297 (9.10) 0.0773 (4.22) 3s β - 0.1311 (10.52) - 0.1929 (7.01) 0f α 0.0005 (2.23) 1.12e-5 (0.05) 0.0002 (0.73) 0.0001 (0.36) 1f α 0.0440 (1.03) 0.0065 (0.19) 0.0613 (1.52) 0.0486 (1.23) 0f β 1.36e-6 (6.99) 1.45e-6 (7.67) 7.33e-6 (8.24) 5.30e-6 (7.65) 1f β 0.9121 (117.37) 0.9217 (114.13) 0.7325 (28.70) 0.7958 (39.87) 2f β 0.0812 (10.98) 0.0001 (0.01) 0.2674 (9.44) 0.0915 (4.29) 3f β - 0.1379 (10.86) - 0.2251 (7.51) Panel B: Estimates of constant (dynamic) conditional correlation processes
1 θ - 0.2537 (3.24) - 0.1416 (2.50) - 0.9929 (660.87) - 0.9932 (687.19) 2 θ - 0.0374 (7.59) - 0.0515 (8.61) - 0.0018 (4.43) - 0.0018 (4.49) sf ρ (ρsf) 0.9732 (2821.10) 0.9716 0.9719 (2728.50) 0.9700 0.9736 (3036.90) 0.9675 0.9728 (2932.00) 0.9667 Notes: The entries (in the parentheses) are the Gaussian QML estimates (and their asymptotic t-statistics) of return-based and RV-based models, where parameters are estimated via a two-step estimation method. This method first estimates ϑi : ( , )= α βi i for i=s f, by maximizing the Gaussian quasi-likelihood function
1 2 1 1 1 2 2 1 , 2 1 , , 0 1 1 1 ( ) ln 2 ln [ ( )] i T T T T V i t i t t i t i t i i t t L ϑ π h h− r α α S F − − = =
= − −
∑
−∑
− − − , then estimates θ bymaxi-mizing the ˆϑ -based quasi-likelihood function i 1 1 ˆ ˆ ( | ) T T ln ( ; ) C t t t L θ ϑ f ε θ =
=
∑
, where ft( )θ represents theconditional probability density function of the standard bivariate normal distribution. **indicates significance at the 5% level.
While the estimates of the conditional mean equations are similar, the results in the condi-tional variance and/or correlation equations are quite different. Concerning the result of the conventional models, the insignificance of the β2i shows the symmetric GARCH specifica-tion seems more suitable for the data. The persistence of the GARCH for the spot and futures
are about 0.9947 and 0.9933, respectively, which suggests the conditional volatilities reveal high persistence. For the correlation equations, the inferences conclude that the DCC is held. Hence, the empirical evidence using daily information indicates the ECT-GARCH-DCC model fits reasonably well to the S&P 500 market. Then turn to the estimation results of the RV-based models. The significance of β3i suggests that the asymmetry in volatility are re-vealed. A positive β3i shows that the impact of a positive return shock on the current volatil-ity is smaller than that of a negative return shock of the same magnitude. Particularly, the per-sistence of volatility implied by the RV-based models is about 0.9999, which is higher than the persistence implied by the return-based models. Besides the persistence, the weight on the persistence parameters between the two methods also differs. Taking the spot volatility as an example, the return-based GARCH estimates of β1i (β2i) is about 0.91 (0.08), whereas the RV-based GARCH estimates of β1i (β2i) is about 0.77 (0.23). The higher weight on the β2i using RV-based GARCH indicates that past RV may provide more information in predicting the current volatility than those using lagged daily return squares. For the conditional correla-tion equacorrela-tions, the significance of θ1 and θ2 for the S&P 500 indicates the null of mean-reverting DCC hypothesis is held. In addition, the higher persistence of the RV-based correlation than the return-based one is also observed form the empirical evidence. It shows that the persistence of correlation implied by the RV-based models is about 0.99, which is much higher than the persistence (ranges from 0.19 to 0.29) implied by the return-based mod-els. Thus, the empirical evidence using RV indicates the ECT-RV-GJR-DCC model seems suitable to the S&P 500 data. Figure 2.1 compares the conditional volatility and correlation estimates using the two methods. We report the best-fitted models among them to save space. It is evident that the RV-based second moment estimates are not equal to those of the re-turn-based models. Essentially, the difference should come from the dynamics difference be-tween the RV and the return squares.
FIGURE 2.1 In-Sample Comparisons on Volatility and Correlation Estimates: RV-Based vs. Return-Based Methods
RV-based Hedge Ratios and their Hedging Performances
We now turn to analyze the performance of a futures hedge using the RV approach. Since the hedging decision has to be made ex-ante, the evaluation is conducted in an out-of-sample context using a rollover method.16 To do so, this study splits the full sample period into two: the in-sample period (from January 1, 1998 to December 18, 2003; 1500 observations), and the out-of-sample period (from December 19, 2003 to March 31, 2009; 1328 observations). Each model is estimated with the use of the in-sample data and then re-estimated with a daily rollover in the out-of-sample period, keeping the estimation sample size of 1500 (fixed). This
16
rollover method is continued for all the 1328 out-of-samples. The estimated hedge ratios as indicated in Equation (2.9) for each model are subsequently constructed, and the correspond-ing realized portfolio returns for both the short (rp t,+1 =rs t,+1−βt f tr,+1) and the long
(rp t,+1 = −rs t,+1 +βt f tr,+1) hedges are calculated. In addition to the dynamic models specified
in the previous sections, we also evaluate the performance of the static OLS method based on this rollover method. Particularly, to see whether the RV-based models can provide a superior hedging performance during the crisis, the results before (Period I: December 19, 2003 through September 28, 2007, 950 observations) and during (Period II: October 1, 2007 through March 31, 2009, 378 observations) that period are separately reported.
Table 2.4 exhibits some diagnostic statistics of the hedge ratios; and, Figure 2.2 plots the dynamic hedge ratios.17 On average, the RV-based hedge ratios are larger (smaller) than the conventional return-based hedge ratios in Period I (II); however, they have smaller variation in both the periods. The ADF tests on the hedge ratios illustrate the unit-root hypothesis is re-jected at the 5% level except for the results based on the OLS (in both periods) and the ECT-GARCH-CCC (in period II) models.18 The study also reports the ACF values of these out-of-sample hedge ratios up to lag five. It is apparent that, the ACF of the RV-based hedge ratios is smaller but decays more quickly than the ACF of the return-based hedge ratios.
17 Since the CCC-based hedge ratios are similar to the DCC-based ones, we do not plot them to save space. 18 This agrees with the finding of Lien et al. (2002), who report that the out-of-sample GARCH hedge ratios are stationary.
FIGURE 2.2 Out-of-Sample Hedge Ratios: RV-Based vs. Return-Based Methods
Table 2.5 presents several statistics on the realizations of the hedged portfolio returns. Con-sidering the hedging performance using standard deviation (Std. Dev.), the results show that the RV-based method yields an average sample volatility of 0.1730% (0.4284%) for the Pe-riod I (II), which is smaller than the 0.1745% (0.4404%) of the conventional method. That is, the improvement of the RV-based method over the return-based method is about 0.90% and 2.74% for the Period I and Period II, respectively. Particularly, the RV-based method even can surpass the simple OLS method during the crisis while the return-based GARCH method does not.19 Besides the Std. Dev., two alternatives, namely the value-at-risk (VaR) and the
19 Lien et al. (2002) indicate that the out-of-sample CCC-GARCH hedge does not outperform the OLS hedge in the S&P 500 market, where their data were extracted for the period of January 1988 through June 1998. However,
pected shortfall (ES), are also included in the comparisons since the evaluation results may change if performance criteria other than traditional measures are applied (Cotter & Hanly, 2006).20 The result shows that using the RV-based method generally can provide a better performance in managing portfolio VaR and ES especially during the crisis period. In addition, in Period II, the improvement of RV-based method over the conventional method tends to enlarge when the percentile has moved toward 99%. For example, on average, the improve-ment of long hedges in VaR(0.95), ES(0.95), VaR(0.99), and ES(0.99) is about 2.53%, 8.33%, 20.82%, and 11.43% (the percentage change in VaR and ES reduction), respectively; and they all can surpass the simple OLS method whereas the return-based models cannot. For the per-formance in Period I, it seems that the RV-based method is inferior to the conventional method. In this period, it is observed that the OLS hedge has the best performance in most of the cases. Hence, the empirical evidences conclude that the RV-based hedges are more useful than the return-based hedges in managing portfolio risk especially during the surge of volatil-ity period.
Cotter and Hanly (2006) show that the CCC-GARCH hedge can beat the OLS hedge in the S&P 500 market in terms of variance reduction, where their data were extracted for the period of January 1998 through December 2003.
20 The VaR is defined by the negative of the th
α empirical percentile of the realizations on hedged portfolio returns, i.e., VaR( )α Fˆn1(rp t, 1; )α
− +
= − , where Fˆn denotes the empirical distribution of the hedged portfolio returns using the n realized observations. A major shortcoming with the VaR is that it is not a coherent risk measure (Artzner et al., 1999). Hence, the ES measure has received some attention recently. The ES summarizes the negative of the average returns on the portfolio given that the hedged portfolio return has exceeded its α th empirical percentile, or ES( )α = −E rˆn(p t,+1|rp t,+1≤ −VaR(rp t,+1; ))α , where Eˆn represents the sample
aver-age operator. This gives the hedger additional information about both the probability of losses and possible mag-nitude of losses beyond the α percentile. th
TABLE 2.4 Summary Statistics of Out-of-Sample Hedge Ratios
ECT-GARCH ECT-RV-GARCH ECT-GJR ECT-RV-GJR Statistics
CCC DCC CCC DCC CCC DCC CCC DCC OLS
Panel A: Period I (December 19, 2003 ~ September 28, 2007, 950 observations)
Mean 0.9820 0.9761 0.9908 0.9864 0.9906 0.9828 0.9890 0.9840 0.9602 Std. Dev. 0.0415 0.0427 0.0246 0.0242 0.0438 0.0474 0.0442 0.0438 0.0108 Min 0.8314 0.8037 0.8795 0.8749 0.8321 0.7523 0.8028 0.7979 0.9393 Max 1.0859 1.0778 1.0771 1.0707 1.1562 1.1494 1.1692 1.1613 0.9789 ADF -7.1189 -9.5666 -15.5897 -15.6399 -7.1890 -12.7961 -16.1559 -16.1489 -1.8493 ACF(1) 0.9036 0.8353 0.6509 0.6453 0.8990 0.7332 0.6529 0.6503 0.9962 ACF(2) 0.8272 0.7486 0.4817 0.4735 0.8457 0.6382 0.5309 0.5287 0.9924 ACF(3) 0.7585 0.6850 0.3343 0.3264 0.7859 0.6064 0.4783 0.4756 0.9986 ACF(4) 0.6996 0.6335 0.2570 0.2487 0.7344 0.6017 0.4358 0.4333 0.9848 ACF(5) 0.6500 0.5886 0.2206 0.2104 0.6798 0.5334 0.4030 0.3985 0.9810
Panel B: Period II (October 1, 2007 ~ March 31, 2009, 378 observations)
Mean 1.0123 1.0065 0.9761 0.9714 1.0329 1.0264 0.9704 0.9651 0.9807 Std. Dev. 0.0405 0.0401 0.0319 0.0317 0.0542 0.0532 0.0480 0.0477 0.0081 Min 0.8965 0.8923 0.8923 0.8888 0.9168 0.9120 0.8228 0.8181 0.9648 Max 1.1236 1.1176 1.0793 1.0723 1.1992 1.1785 1.1308 1.1204 1.0077 ADF -3.1989 -3.4508 -6.4422 -6.4631 -3.9150 -4.0537 -7.4427 -7.4055 -1.8333 ACF(1) 0.9493 0.9411 0.8037 0.8026 0.9263 0.9211 0.7518 0.7536 0.9858 ACF(2) 0.8988 0.8878 0.6263 0.6239 0.8623 0.8549 0.6081 0.6090 0.9704 ACF(3) 0.8464 0.8347 0.4938 0.4908 0.8067 0.7973 0.4836 0.4839 0.9510 ACF(4) 0.8004 0.7897 0.3729 0.3690 0.7504 0.7419 0.4093 0.4091 0.9303 ACF(5) 0.7619 0.7509 0.2658 0.2617 0.6902 0.6789 0.2968 0.2963 0.9123
Notes: The ADF indicates the augmented Dickey-Fuller unit root test based on trend stationary AR model, where the 5% critical values for Period I and II are -3.4150 and
-3.4234, respectively. The ACF(k) indicates the sample autocorrelation function of the hedge ratios corresponding to lags k. Their 95% confidence bonds for Period I and II are [-0.0649, 0.0649] and [-0.1029, 0.1029], respectively.
