Optimal dynamic hedging via copula-threshold-GARCH models

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 79 (2009) 2609–2624

Optimal dynamic hedging via copula-threshold-GARCH models

YiHao Lai a, Cathy W.S. Chen b,∗, Richard Gerlach c

a Institute of Applied Economics, National Taiwan Ocean University and Department of Finance, Da-Yeh University, Taiwanb Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taiwanc Discipline of Econometrics and Business Statistics, University of Sydney, Australia

Available online 24 December 2008

Abstract

The contribution of this paper is twofold. First, we exploit copula methodology, with two threshold GARCH models as marginals,to construct a bivariate copula-threshold-GARCH model, simultaneously capturing asymmetric nonlinear behaviour in univariatestock returns of spot and futures markets and bivariate dependency, in a flexible manner. Two elliptical copulas (Gaussian andStudent’s-t) and three Archimedean copulas (Clayton, Gumbel and the Mixture of Clayton and Gumbel) are utilized. Second, weemploy the presenting models to investigate the hedging performance for five East Asian spot and futures stock markets: HongKong, Japan, Korea, Singapore and Taiwan. Compared with conventional hedging strategies, including Engle’s dynamic conditionalcorrelation GARCH model, the results show that hedge ratios constructed by a Gaussian or Mixture copula are the best-performedin variance reduction for all markets except Japan and Singapore, and provide close to the best returns on a hedging portfolio overthe sample period.© 2008 IMACS. Published by Elsevier B.V. All rights reserved.

JEL classification: C50; D81; G15

Keywords: Hedge ratio; Threshold-GARCH; Copula; Spot and futures market; Stock return

1. Introduction

Stock futures markets provide a channel for stock portfolio holders to potentially transfer risks, through paying apremium to those willing to bear them. However, the effectiveness of such a hedging strategy relies heavily on theaccuracy of the hedge ratio estimation. There is much empirical literature on the calculation of optimal hedge ratios:traditional strategies include the one-to-one ratio and variance-minimization approaches. The one-to-one strategyinvolves adopting the opposite positions in spot and futures with the same magnitude, eliminating price risk only whenspot and future prices move perfectly together. The variance-minimization strategy chooses an optimal position in thefutures market in order to minimize the variance of the spot-futures portfolio, the so-called optimal hedge ratio in theliterature. This is usually estimated as the slope (beta) coefficient in the ordinary least squares (OLS) regression ofspot returns on futures returns, i.e. the optimal hedge ratio is the covariance between spot and futures returns dividedby the variance of futures returns.

The conventional OLS approach assumes constant variance and covariance over time, contradicting the well-knowntime-varying nature of the second moment in asset returns. The extensive framework of GARCH-type models proposed

∗ Corresponding author. Tel.: +886 424517250x4412; fax: +886 424517092.E-mail address: [email protected] (C.W.S. Chen).

0378-4754/$36.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2008.12.010

mailto:[email protected]

dx.doi.org/10.1016/j.matcom.2008.12.010

2610 Y. Lai et al. / Mathematics and Computers in Simulation 79 (2009) 2609–2624

by Engle [7] and Bollerslev [2] were designed to model this time-varying volatility structure. The general consensusis that time-varying hedge ratios built by bivariate GARCH-type models have superior performance in minimizingthe portfolio volatility, e.g. see [16,19]. Sim and Zurbruegg [22] used a bivariate error-corrected GARCH modelto investigate the dynamic hedging effectiveness in the South Korean stock futures, finding that the comparativeperformance of a constant hedge ratio, while being generally inferior, actually improved after the Asian financialcrisis of 1997; Lien et al. [14] found the OLS hedge ratio outperformed the constant-correlation bivariate GARCHmodel; while Choudhry [5] showed that the time-varying hedge ratio performs better than the constant hedge ratio.The previous literature constructs the optimal hedge ratio via the covariance structure under a (dynamic) bivariatenormal assumption, in particular neglecting the impact of a possibly non-normal bivariate dependence structure. Thisstudy considers the non-normality of stock returns and various dependence structures between spot and futures, andinvestigates the corresponding effects on hedge performance.

Symmetric GARCH models do not capture the well-documented financial market trait of asymmetric volatility. Tomodel asymmetric volatility, several specifications have been proposed: such as EGARCH [18] and GJR-GARCH [10].Rabemananjara and Zakoian [21] developed a threshold-GARCH (TGARCH) model, allowing a nonlinear volatilityreaction to positive and negative return shocks. Chan and McAleer [3] discuss estimation of a STAR-GARCH model,with a smooth transition nonlinear mean equation and symmetric GARCH errors. In this paper, we employ a nonlinearTGARCH model similar to that in [25], allowing asymmetric effects from past shocks to appear in the conditionalvariance equation.

An important development in modeling conditional dependence, known as the copula, was first proposed by Sklar[23], stating that any N-dimensional joint distribution function may be decomposed into N marginal distributions anda copula function, that completely describes the dependence between the N variables. Various copula functions allowgreat flexibility in modeling joint distributions, beyond the typical bivariate Gaussian or bivariate t-distributions, sincethe marginals and the joint distribution in copula modeling do not necessarily belong to the same family. Copulas havebecome popular in the literature recently, e.g. see [9,17]. Patton [20] extends the standard copula concept to model time-varying conditional heteroskedasticity, using symmetric GARCH marginals in a copula-GARCH model. In this paper,the joint conditional density function for spot and futures returns is specified as the product of a conditional copulaand the two marginal conditional densities, specified as TGARCH models with potentially different parameters anderror distributions, thus extending Patton [20]. One of the advantages of copula-GARCH models in general is that wecan describe different sorts of conditional dependency by changing the copula function, without altering the marginaldensities of the TGARCH models. Like the dynamic conditional correlation (DCC) model of Engle [8], in this papercopula-TGARCH models are estimated using a two-step maximum likelihood approach, solving the dimensionalityproblem; see [13].

The goal of this paper is then to investigate and compare the performance of a range of copula and more conventionalhedge strategies for the spot and futures stock prices from the East Asian markets in Hong Kong, Japan, Korea, Singaporeand Taiwan. Our contribution aims at providing an outperforming and flexibly implemented time-varying hedge ratio,estimated by the copula-TGARCH model, as detailed in Section 2. The paper is organized as follows: Section 2discusses the estimation methods for the models considered; optimal hedge ratios are derived in Section 3; Section 4discusses the data, presents the empirical results and compares the hedge performance across the five markets; finallySection 5 concludes.

2. Copula-threshold-GARCH methodology

2.1. Copula function

A copula function represents a flexible dependence structure between two random variables. Sklar’s theorem provedthat a 2-dimensional joint distribution function G with continuous marginals FX and FY has a unique copula repre-sentation so that G(x,y) = C(FX(x), FY(y)). According to Sklar’s theorem, for a joint distribution function, the marginaldistributions and the dependence structure described by a copula can be separated. Let RS

t and RFt be random variables

denoting spot and futures returns at period t, and let their conditional cumulative distribution functions (CDF) berespectively given by FS

t (RSt |It−1) and FF

t (RFt |It−1); where It−1 denotes all past returns, {Rj

t−i} i = 1, 2, . . . , t − 1and j = S, F. The conditional copula function Ct(ut, vt |It−1) is defined by the time-varying CDF of spot and futures

Y. Lai et al. / Mathematics and Computers in Simulation 79 (2009) 2609–2624 2611

returns in (2.1), where ut = FSt (RS

t |It−1) and vt = FFt (RF

t |It−1) are distributed as continuous uniform variables on(0, 1). Extending Sklar’s theorem, the bivariate conditional CDF of RS

t and RFt can be written as

G(RSt , RF

t |It−1) = Ct(FSt (RS

t |It−1), FFt (RF

t |It−1)|It−1). (2.1)

Assuming all CDFs are differentiable, the conditional joint density is then given by

g(RSt , RF

t |It−1) = ∂Gt(RSt , RF

t |It−1)

∂RSt ∂RF

t

= ct(FSt (RS

t |It−1), FFt (RF

t |It−1)|It−1) × fSt (RS

t |It−1) × fFt (RF

t |It−1),

(2.2)

where ct(ut, vt |It−1) = ∂2Ct(ut, vt |It−1)/∂ut ∂vt is the conditional copula density function. Thus, the bivariate condi-tional density function of RS

t and RFt given above is represented by the product of the copula density and the two

conditional marginal densities fSt (RS

t |It−1) and fFt (RF

t |It−1).We denote the parameters in the models for ct(·), fS

t (·) and fFt (·) as θc, θS and θF, respectively. The log-likelihood

function is then:

log g(RSt , RF

t |It−1, θ) = log ct(ut, vt |It−1, θc) + log fSt (RS

t |It−1, θS) + log fFt (RF

t |It−1, θF ). (2.3)

For a simpler representation, we write (2.3) as

L(θ) = Lc(θc) + LS(θS) + LF (θF ), (2.4)

where Lk(·) is the log-likelihood function of copula (k = C), spot (S) and futures (F) densities, respectively. It can bedifficult to optimize when the dimension (i.e. number of model parameters) of the bivariate conditional density is large.Joe and Xu [13] proposed a two-step estimation, which is called inference for the margins (IFM), to partially resolvethis problem, where one can estimate the marginal densities and the copula density separately; we follow this approachin this paper.

For IFM, firstly the parameters in the marginals are estimated via maximum likelihood as

θ̂S = arg maxT∑

t=1

log fSt (rS

t |It−1, θS), (2.5)

θ̂F = arg maxT∑

t=1

log fFt (rF

t |It−1, θF ). (2.6)

Next, the marginal CDFs are applied to the standardized residuals, using the estimates from (2.5) and (2.6), toprovide estimates of the probabilities ut and vt , t = 1, . . ., n, which are then used to estimate the copula parameters via:

θ̂c = arg maxT∑

t=1

log ct(ut, vt |It−1, θc). (2.7)

Joe [12] shows the high efficiency of the easy-implemented IFM method, compared with the usual maximumlikelihood method.

2.2. Marginal distributions in a threshold-GARCH framework

We specify the conditional marginal densities for spot and futures returns using a TGARCH(1,1) framework, definedby

RSt = φ0 + φ1R

St−1 + φ2R

Ft−1 + εt

hSt =

⎧⎨⎩

α(1)0 + α

(1)1 ε2

t−1 + α(1)2 hS

t−1, εt−1 < 0

α(2)0 + α

(2)1 ε2

t−1 + α(2)2 hS

t−1, εt−1 ≥ 0

εt |It−1 ∼ t(0, hSt ; vS),

(2.8)


and

RFt = ω0 + ω1R

Ft−1 + ω2R

St−1 + ηt

hFt =

⎧⎨⎩

β(1)0 + β

(1)1 η2

t−1 + β(1)2 hF

t−1, ηt−1 < 0

β(2)0 + β

(2)1 η2

t−1 + β(2)2 hF

t−1, ηt−1 ≥ 0

ηt|It−1 ∼ t(0, hFt ; vF ),

(2.9)

where εt and ηt are error terms following Student’s-t distributions with degrees of freedom vS and vF , respectively; hSt

and hFt are the conditional variances for the spot and futures returns; superscript (i), i = 1 or 2, signifies the state in the

nonlinear model; It−1 is the past information set. Since both marginal models must condition on the same informationup to t − 1, we include both lagged returns series in each mean equation. The conditional variance processes hS

t andhF

t are allowed to have potentially asymmetric responses to negative or positive previous random shocks (εt−1 andηt−1), so that when the stock return (either spot or futures) is less (larger) than its conditional expectation, the volatilitybehaviour is determined by state-1 (state-2) parameters. This model extends the TGARCH model in Rabemananjaraand Zakoian [21] by using Student’s-t errors and an AR mean equation and is similar to the model proposed in So etal. [25]. All parameters in the TGARCH models above are estimated separately for spot and futures, as they representmarginal densities. It is usual to enforce sufficient stationarity and positivity constraints on the parameters in GARCH-type models, such as α

(j)0 > 0, β

(j)0 > 0, α

(j)1 > 0, β

(j)1 > 0, α

(j)1 + α

(j)2 > 0 and β

(j)1 + β

(j)2 > 0. However, Medeiros

and Veiga [15] and Wong and Li [26] have argued that multi-regime time series processes may have explosive regimes,yet still remain stationary and ergodic. As such we do not enforce these restrictive conditions in estimation.

2.3. Bivariate copula density

The first-step marginal AR(1)-TGARCH(1,1) parameter estimates provide estimated values of ut = FSt (rS

t |It−1)and vt = FF

t (rFt |It−1). These values are then used in the second-step estimation of the copula dependence structure

between RSt and RF

t .Since the copula function determines the dependence structure, its selection for real data should depend on the

characteristics of dependence observed in the data set. The usual choice is an elliptical copula such as Gaussian orStudent’s-t copula:

cGaussiant (ut, vt ; ρt) = 1√

1 − ρ2t

exp

{− 1

2(1 − ρ2t )

[a2t + b2

t − 2ρtatbt] + 1

2[a2

t + b2t ]

}(2.10)

cStudent’s-tt (ut, vt ; ρt, v

c) = (1 − ρ2t )

1/2 Γ (vc + 2)/2

Γ (vc/2)

[Γ (vc/2)

Γ (vc + 1)/2

]2[1 + a2

t + b2t − 2ρtatbt

vc(1 − ρ2t )

]−(vc+2)/2

×[(

1 + a2t

vc

)(1 + b2

t

vc

)](vc+1)/2

, (2.11)

where Γ (·) is Gamma function and(1) In the Student’s-t copula, at = t−1

vc (ut), bt = t−1vc (vt) and tvc (·) represent the CDF of the Student’s-t distribution

with degree of freedom vc.(2) In the Gaussian copula, at = Φ−1(ut), bt = Φ−1(vt) and Φ(·) represent the CDF of the standard normal distribution.Cherubini et al. [4] refer to the fat tail phenomenon in a multivariate setting as tail dependency, which means the

weight of the joint density in one or both tails is larger than that of a multivariate normal density. Considering theprobability that an event with probability lower than v occurs in X, given that an event has occurred with probabilitylower than v in Y; we have the tail dependency formula:

λL(v) = limv→0

Pr[X ≤ F−1X (v)|Y ≤ F−1

Y (v)] = limv→0

C(v, v)

v, (2.12)


λU (v) = limv→1

Pr[X ≥ F−1X (v)|Y ≥ F−1

Y (v)] = limv→1

1 − 2v + C(v, v)

1 − v, (2.13)

where C(·,·) is the CDF of the copula and FX(·) and FY(·) are the marginal CDFs for X and Y, respectively. Gaussianand Student’s-t copulas feature symmetric tail dependencies, given by λGaussian

U = λGaussianL = 0 and λStudent’s-t

U =λStudent’s-t

L = 2tv+1(−√v + 1

√1 − ρ/

√1 + ρ) > 0, where tv+1(·) is the CDF of the Student’s-t distribution with the

degree of freedom v + 1, and ρ is the linear correlation coefficient between X and Y, respectively.Such symmetric tail dependence implies that extremely favourable or unfavourable market conditions drive the

dependency by the same magnitude in each tail. However, this may not be supported empirically. While markets aremore likely to collapse together during a period of stress (i.e. bear markets) and to boom or rise together during aprosperous period (i.e. bull markets), their dual rates of collapse or boom may not be the same or even similar. Thus, itmakes sense to further consider models allowing an asymmetric dependence between the lower and upper tails of thecopula; e.g. as in Patton [20]. The family of Archimedean copulas, different from the elliptical ones, allows this varietyof tail dependency; see Nelsen [17] for more fundamental properties of Archimedean copulas. This study employsthe Clayton [6], Gumbel [11] and a Mixture of Clayton and Gumbel copulas, to capture the potential asymmetric taildependency. The probability density functions for these copulas are

cClaytont (ut, vt ; δ

C) = (1 + δC)(utvt)−1−δC

(u−δC

t + v−δC

t − 1)−1−2δC/δC

, (2.14)

cGumbelt (ut, vt ; δ

G)

= exp{−[[− ln(ut)]δG + [− ln(vt)]δ

G]1/δG

}[ln(ut) ln(vt)]δG−1{[[− ln(ut)]δ

G + [− ln(vt)]δG

]1/δG

+ δG − 1}utvt[[− ln(ut)]δ

G + [− ln(vt)]δG

]2−(1/δG)

,

(2.15)

cMixturet (ut, vt ; q, δC, δG) = qc

Claytont (ut, vt ; δ

C) + (1 − q)cGumbelt (ut, vt ; δ

G), (2.16)

where δc ∈ [−1, ∞)\{0} and δG ∈ [1, ∞) are parameters for the Clayton and Gumbel copulas respectively, and q ∈ [0, 1]is the weighting parameter for the Mixture copula. The upper and lower tail dependencies are: λ

ClaytonU (v) = 0 and

λClaytonL (v) = 2−1/δC

for the Clayton copula and λGumbelU (v) = 2 − 21/δG

and λGumbelL (v) = 0 for Gumbel copula.

For the Gaussian and Student’s-t copulas, we specify the linear dependence parameter ρt so that it depends on theprevious dependence ρt−1 and the past absolute difference of the cumulative probabilities, |ut−1 − vt−1|, as suggestedby Patton [20]:

ρt = κ + λ1ρt−1 + γζt, (2.17)

where an exponentially weighted moving average of all past absolute differences of the cumulated probabilities isassumed

ζ = λ2ζt−1 + (1 − λ2) |ut−1 − vt−1| , (2.18)

giving higher weights to more recent observations. Combining (2.17) and (2.18), we get

ρt = ϕ + (λ1 + λ2)ρt−1 − λ1λ2ρt−2 + π |ut−1 − vt−1| , (2.19)

where ϕ = (1 − λ2)κ, π = (1 − λ2)γ , ut = FSt (rS

t |It−1) and vt = FFt (rF

t |It−1). As suggested by Patton [20] and Bartramet al. [1], a smaller absolute difference of two cumulative probabilities implies that spot and futures move more together(higher dependence), i.e. π is expected to be negative. Eq. (2.19) is an AR(2)-like model, hence the required parameterrestrictions for stationarity are:

(−λ1λ2) + (λ1 + λ2) < 1, − (λ1λ2) − (λ1 + λ2) < 1, and | − λ1λ2| < 1. (2.20)

Archimedean copulas have no linear dependence parameter ρt in their density functions. As such, we specify arelation between the parameter δz in copula z and the scale invariant dependence measure Kendall’s τ, which can berepresented as δC = 2τ/(1 − τ) for the Clayton and δG = 1/(1 − τ) for the Gumbel copula. Further, we specify a relation


between the linear dependence ρt and τt for the Archimedean copulas, which is τt = (2/π)sin−1(ρt) (see [17] for moredetails).

Thus, we incorporate the linear dependence process into the Archimedean copula density estimation as

cClaytont (ut, vt ; δ

C) = cClaytont

(ut, vt ;

2τ

1 − τ

)= c

Claytont

(ut, vt ;

2[(2/π) sin−1(ρt)]

1 − (2/π) sin−1(ρt)

)(2.21)

cGumbelt (ut, vt ; δ

G) = cGumbelt

(ut, vt ;

1

1 − τ

)= cGumbel

t

(ut, vt ;

1

1 − (2/π) sin−1(ρt)

)(2.22)

We examine the five copula dependence structures for this data in detail in the empirical results given in Section 4.

3. Estimation of optimal hedge ratio

The logic of the variance minimization hedging strategy is to invest in the amount of futures, β, that minimizes thevariance of the returns of a portfolio, consisting of the spot and futures position. Let RP

t represent the return on theportfolio given by

RPt = RS

t − βRFt . (3.1)

The variance of the portfolio is thus:

Var(RPt ) = Var(RS

t ) + β2 Var(RFt ) − 2β Cov(RS

t , RFt ). (3.2)

Minimizing Var(RPt ) with respect to β gives

β = Cov(RSt , RF

t )

Var(RFt )

. (3.3)

The estimate of the constant hedge ratio βOLS via the conventional OLS regression can be obtained by

RSt = α + βOLSRF

t + εt. (3.4)

As mentioned in Section 1, the OLS method assumes constant variation over time, which is not supported by mostempirical literature in finance. To account for the time-varying feature, we also use Engle’s DCC bivariate GARCHmodel as a benchmark. The conditional correlation is then estimated by Qt = ρ̄(1 − �1 − �2) + �1ξt−1ξ

Tt−1 +

�2Qt−1 where Qt is the covariance matrix of the vector of first-step standardized residuals (ξt), and ρ̄ is the unconditionalcorrelation coefficient. The linear time-varying correlation coefficients, ρDCC

t , can be derived from the off-diagonalelement of the dynamic conditional correlation matrix, Rt = diag{Qt}−1Qt diag{Qt}−1.

Once the time-varying correlation coefficients of the DCC and each Copula-TGARCH model (ρDCCt and ρ

copulat )

are estimated, as above and in Section 2, the corresponding optimal hedge ratio can be calculated by

βit = ρi

t

√hS

t hFt

hFt

, i = DCC and copula, (3.5)

where hSt , hF

t represent the conditional variance for spot and futures, estimated in the first step, respectively.

4. Data and empirical results

We estimate the models using the daily stock returns from the spot and futures markets of Hong Kong, Japan, Korea,Singapore and Taiwan. All data come from DATASTREAM. The spot price indices are the Hang Seng Price Index ofHong Kong (HNGKNGI), Nikkei 225 Index of Japan (JAPDOWA), Korea Stock Exchange Composite Price Index(KORCOMP), Straits Times Price Index of Singapore (SNGPORI) and the Taiwan Stock Exchange Capitalizationweighted Stock Index (TAIEX). The corresponding futures price indices are the Hong Kong Futures Exchange HangSeng Index (HSI), Osaka Stock Exchange Nikkei 225 Index (ONA), Korea Stock Exchange KOSPI 200 Index (KKX),Singapore International Monetary Exchange Straits Times Index (SST) and Taiwan Futures Exchange Index (TAIFEX).All futures price indices are continuous series, as defined by DATASTREAM. Stock returns of spot and futures are the


log-difference of the corresponding index multiplied by 100, i.e. rit = 100 × (log(Pi

t ) − log(Pit−1)), i = S, F. The sample

period is from January 1, 1998 to June 10, 2005, except for Singapore and Taiwan, whose starting dates for futuresmarket indices are June 28, 2000 and July 21, 1998, respectively. This period excludes most effects from the Asianfinancial crisis of 1997.

Table 1 shows some preliminary descriptive statistics. During the sample period, the average return on the spot andfutures are both positive for Hong Kong, Korea and Singapore, while they are both negative for Japan and Taiwan. Theunconditional sample standard deviation indicates that the Korean market was the most and Singapore was the leastvolatile during this period. The kurtosis statistics illustrate significant leptokurtosis for all markets. Consequently, theJarque-Bera test statistics confirm returns on spot and futures for all markets are significantly non-normal. The two-stage estimation method and the dynamic copula models require the marginal models to have the same information set,as is specified in (2.8) and (2.9). For parsimony, the first-order cross-correlation matrix is used to check if lagged spotand futures returns series can be excluded from each mean equation. The cross-correlation matrices indicate significantauto- and cross-correlation in spot returns, but none in futures returns, for Hong Kong, Korea and Taiwan. Significantauto- and cross-correlation coefficients were found in the futures returns, but not the spot returns, in Japan. Finally, allauto- and cross-correlation coefficients were insignificant in Singapore. The mean equations in (2.8) and (2.9) wereadjusted to include only significant lagged terms.

Table 2 shows the results from the marginal TGARCH(1,1) models, as in (2.8) and (2.9) but adjusted as above,applied to the spot and futures returns in each market. After estimation, we further eliminated the insignificant varianceparameters, at the 5% level, motivated again by model parsimony.

The remaining estimates are significant, except for the mean intercept terms (excepting Korea), the lagged meanterms for Japan and Taiwan and β

(2)1 in the Hong Kong futures. We leave these coefficients in the model since we found

significant 1st order auto- and cross-correlation in Taiwan’s spot and Japan’s futures return series (see Table 1). Wedid not delete β

(2)1 in the Hong Kong futures so as to maintain a consistent representation across markets (if we did the

results change only slightly). We note the negative influence of the lagged spot return, and the positive influence ofthe lagged futures return, on spot returns in Hong Kong and Korea. The converse pattern exists for futures returns inJapan.

The TGARCH estimates indicate a clear asymmetric volatility response to positive and negative shocks in both spotand futures markets in all countries. When the past shock is positive (e.g. εt−1 ≥ 0), we see that all countries’ spot andfutures market estimates satisfy the usual constraints: α(2)

i > 0, β(2)i > 0, α(2)

1 + α(2)2 < 1 and β

(2)1 + β

(2)2 < 1, except the

Korean spot market which has an explosive volatility process. However, when the past shock is negative, the volatilityof returns on spot and futures seem to follow an explosive GARCH process (α(1)

1 + α(1)2 > 1 and β

(1)1 + β

(1)2 > 1) for

all markets (except Korean spot), and also without a drift term (i.e. α(1)0 = 0, β

(1)0 = 0), except Korea and Singapore.

Though the spot model of Korea has a negative drift when the shock is positive and the future model of Singaporehas an explosive GARCH process with a negative drift, the unconditional variance is still positive. For example, theunconditional variance of Korea spot in regime 2 is −0.233/(1 − 0.031 − 1.017) = 4.85. Medeiros and Veiga [15] andWong and Li [26] have argued that a multi-regime GARCH process may have an explosive regime, and yet still bestationary and ergodic. This implies that large negative shocks may cause extreme instability for short periods of time,after which the process reverts back to stationarity.

The degrees of freedom for the Student’s-t distributions for the returns on spot and futures are all reasonably low,ranging from 4 to 11, and indicate the use of a Student’s-t error distribution is appropriate compared with the normal.They also imply the existence of the fourth moment in each market return series (since they exceed 4). Since theerrors are assumed as Student’s-t distributed and the Ljung-Box Q statistics are based on a normality assumption,Smith [24] suggests using the CDF of the Student’s-t, with the estimated degree of freedom, to transform the originalstandardized residuals into cumulative probabilities and then map these probabilities using the inverse Gaussian CDFfunction into a (possibly) standard normal variable. The resulting series will be standard normal if the model fits thedata (i.e. the original standardized residuals follow an independent Student’s-t). The Ljung-Box Q statistics on theprobability transformed standardized residuals and squared transformed standardized residuals suggest the absence ofserial correlation up to 10 lags at the 5% level of significance. The TGARCH Student’s-t error marginal models seemwell specified.

The results for the five copula dependence models and for the contrasting OLS and DCC models are reported inTable 3, where we have restricted the dependence terms given by (2.19) to be less than 1. All estimates are significant

2616Y.L

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Simulation

79(2009)

2609–2624

Table 1Descriptive statistics of stock returns for futures and spot.

Hong Kong Japan Korea Singapore Taiwan

Futures Spot Futures Spot Futures Spot Futures Spot Futures Spot

Mean 0.015 0.014 −0.015 −0.015 0.056 0.050 0.005 0.005 −0.015 −0.014Max 14.153 13.395 8.004 7.222 14.654 10.024 5.414 4.905 8.492 6.172Min −8.712 −9.285 −7.599 −7.234 −10.531 −12.805 −7.426 −7.714 −8.144 −6.912SD 1.865 1.647 1.477 1.437 2.640 2.225 1.192 1.101 1.928 1.692Skewness 0.300** 0.216** 0.006 −0.011 0.268** −0.057 −0.318** −0.297** −0.032 0.017Kurtosis 7.309** 8.720** 5.017** 4.898** 5.701** 5.594** 6.651** 6.750** 5.292** 4.358**Jarque−Bera 1539.37** 2676.36** 330.88** 292.97** 616.88** 548.41** 745.63** 782.56** 385.04** 135.12**N 1952 1952 1952 1952 1952 1952 1303 1303 1758 1758

Cross-correlationFutures(t − 1) Spot(t − 1) Futures(t − 1) Spot(t − 1) Futures(t − 1) Spot(t − 1) Futures(t − 1) Spot(t − 1) Futures(t − 1) Spot(t − 1)

Futures (t) −0.027 −0.012 −0.066# −0.055# −0.042 −0.011 −0.045 0.025 −0.041 −0.013Spot (t) 0.076# 0.049# −0.007 −0.032 0.070# 0.046# 0.024 0.022 0.082# 0.065#

**,* Significance at 5% and 10% level, respectively. # Absolute value of cross-correlation coefficient greater than 2/√

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Table 2Marginal threshold-GARCH(1,1) from multivariate copula model.


Variable Est. Std. Est. Std. Est. Std. Est. Std. Est. Std.

Spotφ0 0.005 (0.027) −0.009 (0.028) 0.071* (0.037) 0.016 (0.024) −0.003 (0.034)φ1 −0.216** (0.064) −0.122** (0.023) −0.024 (0.065)φ2 0.242** (0.056) 0.152** (0.020) 0.087 (0.057)α

(1)0 0.237** (0.007)

α(1)1 0.060** (0.013) 0.075** (0.018) 0.046** (0.003) 0.066** (0.022) 0.107** (0.027)

α(1)2 0.995** (0.016) 0.973** (0.020) 0.905** (0.003) 0.997** (0.021) 0.979** (0.021)

α(2)0 0.017** (0.008) 0.047** (0.021) −0.233** (0.006) 0.020* (0.011) 0.094** (0.039)

α(2)1 0.017* (0.010) 0.029** (0.011) 0.031** (0.003) 0.030* (0.017) 0.035** (0.015)

α(2)2 0.922** (0.020) 0.905** (0.023) 1.017** (0.002) 0.887** (0.033) 0.848** (0.039)

vS 5.804** (0.800) 7.393** (1.145) 5.815** (0.576) 6.891** (1.076) 10.660** (2.586)Q (10) 3.593 [0.96] 3.541 [0.97] 3.204 [0.98] 11.041 [0.35] 5.585 [0.85]Q2 (10) 12.680 [0.24] 5.026 [0.89] 13.581 [0.19] 4.946 [0.90] 16.142 [0.10]

Futuresω0 0.012 (0.030) −0.005** (0.029) 0.094** (0.004) 0.029 (0.024) 0.005 (0.037)ω1 −0.146** (0.070)ω2 0.097 (0.072)β

(1)0 −0.043** (0.001) −0.034* (0.020)

β(1)1 0.060** (0.011) 0.079** (0.013) 0.051** (0.004) 0.086** (0.020) 0.110** (0.022)

β(1)2 1.001** (0.014) 0.981** (0.019) 1.006** (0.003) 1.014** (0.029) 0.965** (0.022)

β(2)0 0.017** (0.008) 0.056** (0.018) 0.082** (0.004) 0.049* (0.026) 0.095** (0.033)

β(2)1 0.010 (0.008) 0.028** (0.010) 0.050** (0.004) 0.045** (0.017) 0.051** (0.017)

β(2)2 0.922** (0.018) 0.891** (0.013) 0.886** (0.003) 0.854** (0.033) 0.860** (0.031)

vF 5.689** (0.848) 6.958** (1.016) 6.259** (0.679) 4.338** (0.573) 5.262** (0.752)Q (10) 5.377 [0.87] 2.360 [0.99] 2.592 [0.99] 13.182 [0.21] 10.054 [0.44]Q2 (10) 16.049 [0.10] 15.440 [0.112] 13.027 [0.22] 3.071 [0.98] 15.473 [0.12]

The numbers in parentheses are standard errors. Q (10) and Q2 (10) are Ljung-Box autocorrelation test statistic for transformed standardized residuals and squared standardized residuals up to 10lags. The numbers in brackets beside LB Q statistics are p-values. **,* Significance at 5% and 10% levels, respectively.

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Table 3Estimates for dependence and copula structures.

Gaussian Student’s-t Clayton Gumbel Mixture OLS DCC

Hong Kongλ1 0.412 (0.001)** 0.489 (0.001)** λ1 0.601 (0.001)** 0.942 (0.003)** 0.555 (0.002)** α 0.002 (0.012) �1 0.041 (0.003)**λ2 −0.050 (0.002)** −0.078 (0.002)** λ2 −0.200 (0.003)** −0.808 (0.003)** −0.096 (0.003)** βOLS 0.835 (0.007)** �2 0.947 (0.005)**φ 0.598 (0.001)** 0.536 (0.001)** φ 0.454 (0.001) 0.104 (0.004)** 0.474 (0.002)**π −0.113 (0.033)** −0.125 (0.020)** π −0.279 (0.052)** −0.075 (0.008)** −0.117 (0.027)**vc 4.814 (0.555)** q 0.185 (0.027)**

Japanλ1 0.900 (0.027)** 0.692 (0.107)** λ1 0.678 (0.060)** 0.728 (0.065)** 0.669 (0.137)** α −0.001 (0.010) �1 0.053 (0.014)**λ2 −0.477 (0.188)** −0.540 (0.163)** λ2 −0.479 (0.138)** −0.526 (0.098)** −0.525 (0.190)** βOLS 0.926 (0.007)** �2 0.525 (0.167)**φ 0.145 (0.039)** 0.462 (0.139)** φ 0.451 (0.072)** 0.404 (0.099)** 0.490 (0.173)**π −0.068 (0.014)** −0.125 (0.031)** π −0.268 (0.041)** −0.145 (0.029)** −0.132 (0.035)**vc 5.976 (0.908)** q 0.224 (0.027)**

Koreaλ1 0.926 (0.008)** 0.973 (0.007)** λ1 0.972 (0.005)** 0.937 (0.010)** 0.973 (0.007)** α 0.005 (0.021) �1 0.032 (0.004)**λ2 −0.636 (0.120)** −0.841 (0.130)** λ2 −0.792 (0.138)** −0.773 (0.093)** −0.843 (0.128)** βOLS 0.766 (0.008)** �2 0.958 (0.004)φ 0.123 (0.013) 0.050 (0.013)** φ 0.051 (0.008)** 0.113 (0.017)** 0.052 (0.012)**π −0.126 (0.011)** −0.125 (0.014)** π −0.103 (0.015)** −0.131 (0.017)** −0.060 (0.013)**vc 4.491 (0.565)** q 0.237 (0.027)**

Singaporeλ1 0.921 (0.011)** 0.873 (0.033)** λ1 0.906 (0.018)** 0.906 (0.020)** 0.858 (0.034)** α 0.001 (0.013) �1 0.046 (0.009)**λ2 −0.731 (0.108)** −0.606 (0.211)** λ2 −0.842 (0.064)** −0.659 (0.177)** −0.506 (0.251)** βOLS 0.840 (0.011)** �2 0.920 (0.017)**φ 0.141 (0.017)** 0.203 (0.048)** φ 0.167 (0.028)** 0.158 (0.031)** 0.212 (0.046)**π −0.199 (0.021)** −0.125 (0.049)** π −0.280 (0.039)** −0.215 (0.038)** −0.239 (0.048)**vc 4.040 (0.637)** q 0.262 (0.040)**

Taiwanλ1 0.857 (0.003)** 0.895 (0.004)** λ1 0.066 (0.022)** 0.921 (0.001)** 0.932 (0.005)** α 0.003 (0.015) �1 0.095 (0.018)**λ2 −0.394 (0.049)** −0.454 (0.122)** λ2 0.067 (0.010)** −0.359 (0.012)** −0.548 (0.121)** βOLS 0.816 (0.008)** �2 0.588 (0.038)**φ 0.196 (0.003)** 0.150 (0.007)** φ 0.800 (0.011)** 0.106 (0.001)** 0.105 (0.001)**π −0.112 (0.015)** −0.125 (0.015)** π −0.380 (0.082) −0.085 (0.010)** −0.079 (0.010)**vc 7.883 (1.601)** q 0.204 (0.026)**

The numbers in parentheses are standard errors. ** Significance at 5% level.


Table 4Preliminary statistics for conditional correlation (CC) and hedge ratio (HR).


UnconditionalCC 0.946 0.952 0.906 0.909 0.930HR 0.835 0.926 0.764 0.840 0.816

Gaussian copulaCC 0.957 (0.008) 0.956 (0.008) 0.934 (0.027) 0.872 (0.052) 0.944 (0.015)HR 0.841 (0.059) 0.936 (0.061) 0.823 (0.124) 0.787 (0.089) 0.835 (0.086)

Student’s-t copulaCC 0.959 (0.009) 0.957 (0.010) 0.938 (0.026) 0.874 (0.049) 0.945 (0.013)HR 0.843 (0.060) 0.937 (0.061) 0.827 (0.126) 0.789 (0.090) 0.836 (0.086)

Clayton copulaCC 0.912 (0.022) 0.912 (0.020) 0.877 (0.043) 0.794 (0.065) 0.886 (0.028)HR 0.801 (0.060) 0.892 (0.060) 0.774 (0.126) 0.716 (0.088) 0.784 (0.084)

Gumbel copulaCC 0.955 (0.011) 0.951 (0.011) 0.930 (0.029) 0.864 (0.053) 0.936 (0.017)HR 0.839 (0.060) 0.931 (0.060) 0.820 (0.125) 0.780 (0.089) 0.829 (0.086)

Mixture copulaCC 0.958 (0.009) 0.956 (0.010) 0.938 (0.025) 0.871 (0.050) 0.941 (0.015)HR 0.842 (0.060) 0.935 (0.060) 0.827 (0.125) 0.786 (0.090) 0.833 (0.086)

DCCCC 0.954 (0.021) 0.952 (0.007) 0.932 (0.030) 0.889 (0.058) 0.932 (0.018)HR 0.838 (0.065) 0.932 (0.060) 0.822 (0.125) 0.802 (0.094) 0.825 (0.084)

The numbers in parentheses are standard deviations.

with estimates for λ1 and λ2 satisfying stationarity, as in (2.20). The estimates of dependence persistence for theGaussian, Student’s-t, Clayton, Gumbel and the Mixture copulas, as measured by λ1 + λ2 − λ1λ2, are 0.38, 0.45, 0.52,0.90 and 0.51 for Hong Kong, 0.85, 0.53, 0.52, 0.58 and 0.50 for Japan, 0.88, 0.95, 0.95, 0.89 and 0.95 for Korea,0.86, 0.80, 0.83, 0.84 and 0.79 for Singapore, and 0.80, 0.85, 0.13, 0.89 and 0.89 for Taiwan, respectively. This impliesa much higher persistence in conditional dependence between the spot and futures markets in Korea, Singapore andTaiwan, but only moderate persistence in dependence for the Hong Kong and Japanese markets. The parameter π issignificantly negative in each case, as expected, indicating the lagged absolute difference of the cumulated probabilitiesut and vt is relevant when modeling the dependence process. The weight parameter q in the mixture copula ranges from0.185 for Hong Kong to 0.262 for Singapore; with an average of 0.22. This estimate means that the Gumbel copulacaptures the greater part of the dependence between the spot and futures markets in the mixture copula, and that thetail dependencies are stronger in the upper tail. The parameter βOLS in the OLS model is significant for each market,and �1 and �2 in the DCC model are also significant and satisfy the stationarity condition, �1 + �2 < 1.

Table 4 shows properties of the estimated conditional dependence between spot and futures (CC) and the corre-sponding hedge ratio (HR) in each market, as well as the unconditional ones, labeled “unconditional”. The magnitudeof the estimated linear dependence ρt is similar (in mean) across the models considered, although the estimates aremarginally higher for the Student’s-t copula and lowest for the Clayton copula in each market. Furthermore, the depen-dence pattern is the most volatile for the Clayton copula and DCC models in each market, while it is most stable forthe Student’s-t (and Gaussian) copula, as shown by the standard deviations (S.D.). Interestingly, Singapore, with thelowest S.D. in returns on spot and futures, has by far the highest volatility in dependency in all five markets. One canalso calculate the upper and lower tail dependencies for the Clayton and Gumbel copulas, as mentioned previously.For example, the mean value of ρt of Clayton copula is 0.912 for Hong Kong, implying that the Kendall’s τ is 0.731(=2/π sin−1(0.912)), the average of δc is thus 5.435 (=20.731/(1 − 0.731)) and then the lower tail dependency is 0.880(=2−1/5.435). In the same manner, the mean value of ρt of Gumbel copula is 0.955 for Hong Kong, the Kendall’s τ is0.808, the average of δG is 5.208 (=1/(1 − 0.808)) and the upper tail dependency is 0.858 (=2 − 21/5.208).

The constant hedge ratio βOLS, labeled “unconditional” is estimated via (3.4), and the time-varying hedge ratio,β

copulat and βDCC

t , is calculated by (3.5), using the estimated time-varying TGARCH variances hSt and hF

t and depen-


dence structure ρcopulat and ρDCC

t , respectively. For each market, the average hedge ratios are very similar among thedependence models, including the DCC, except the Clayton copula which has the lowest value in each market. Themean value of the hedge ratios for Gaussian, Student’s-t, Clayton, Gumbel, the Mixture Copula and DCC is about0.83 for Hong Kong, 0.93 for Japan, 0.82 for Korea, 0.78 for Singapore and 0.82 for Taiwan. The S.D.s for the hedgeratios are almost identical across models. Compared with the OLS estimates, Singapore has uniformly lower and Koreahas consistently higher average hedge ratios for the copula and DCC strategies. The other markets have more closelysimilar results between the copula, DCC and OLS estimates. However, if the Clayton copula is excluded, all otheraverage hedge ratios are marginally higher than the OLS ratio for Hong Kong, Japan and Taiwan. Since the spot andfutures markets are the most stable in Singapore and the most volatile in Korea, the corresponding estimated hedgeratios seem reasonable.

The hedging performance of the five Copula-TGARCH strategies is reported in Table 5, where the un-hedged, one-to-one method, OLS and DCC hedge ratios are used as benchmarks. Boldfaced numbers denote the best-performed resultsamong copula strategies and traditional strategies, respectively. Similar to Choudhry [5], we define two measures: (i)the difference in mean portfolio returns and (ii) the percentage of variance reduction; to show how effective the copulahedge strategies have been. These are calculated using: (i) R

copulap − Rtraditional

p and (ii) (V copulap − V traditional

p )/V traditionalp

where Rip and V i

p are portfolio return means and variances for i = copula and traditional, respectively.The averages for the mean and variance of the portfolio returns for the copula strategies are 0.00634 and 0.28160

(0.00635 and 0.28067 for copula strategies where the Clayton copula is excluded) for Hong Kong, 0.00106 and 0.19505(0.00125 and 0.19424) for Japan, 0.01309 and 0.85786 (0.01256 and 0.85274) for Korea, 0.00243 and 0.22083 (0.00247and 0.21808) for Singapore and 0.00048 and 0.38288 (0.00090 and 0.38029) for Taiwan. Compared with the un-hedgedreturns, the portfolio returns of the copula strategies are about 0.010 lower for Hong Kong, 0.033 lower for Korea,0.003 lower for Singapore and 0.015 higher for Japan and 0.014 higher for Taiwan; however, the portfolio variancesfor the copula strategy are consistently lower. The performance of the copula strategies uniformly outperforms thatfor the one-to-one and OLS strategies in terms of generating higher return and lower return variances (except OLS ofJapan and Singapore). Compared with the DCC model, the copula strategies consistently have higher portfolio returnsfor Singapore and lower variance for Hong Kong. The Clayton copula strategy is clearly the inferior one among copulastrategies with highest variance among such models in all markets. This suggests that the Clayton copula does notcapture the observed data dependence as well as the other copula models. If we exclude the Clayton model, the copulastrategies generally have superior hedge performance; e.g. the average portfolio variances of the copula strategies areuniformly lower than those of the DCC model for all markets. Among the five different copula strategies, the modelswith symmetric dependence (the Gaussian or Student’s-t copulas) outperform the others in terms of minimum varianceand variance reduction for all markets; except that the asymmetric Mixture copula strategy performs well for Korea.

Regarding the percentage of variance reduction in detail, the copula strategies are the most effective except for afew cases. After excluding the Clayton copula, compared with the un-hedged, one-to-one, OLS and DCC, the averageportfolio variance of each copula strategy is about 89.7%, 26.6%, 2.2% and 1.9% lower for Hong Kong, 90.7%, 5.8%,−0.1% and 0.2% lower for Japan, 82.8%, 31.9%, 2.3% and 0.6% lower for Korea, 82.2%, 12.4%, −2.7% and 0.1%lower for Singapore and 203.3%, 25.9%, 2.0% and 0.2% lower for Taiwan. The magnitudes of variance reductionfor copula strategies also correspond with the stabilities of each market. That is, the more volatile the stock market,the more variance reduction for the copula strategies. For the volatile markets such as Korea, Taiwan, Hong Kong,the copula strategies are the best choices of dependence structure to construct time-varying hedge ratios, in terms ofminimum variance; for the relatively stable markets of Japan and Singapore, the OLS strategy performs well. The DCCmodel was not optimal in any market in terms of variance reduction or mean return.

In terms of average portfolio returns, the best copula strategy is Gumbel for Hong Kong and Taiwan, Gaussian forJapan, Clayton for Korea and Mixture copula for Singapore, respectively; while in terms of variance reduction, thebest is Gaussian for Hong Kong, Japan and Taiwan, Student’s-t for Singapore and the Mixture for Korea.

According to these two measures, we conclude that copula hedge strategies seem superior to traditional ones suchas one-to-one and OLS, since the copula methods have higher mean returns and lower portfolio variances (except OLSof Japan and Singapore). In comparison with DCC, copula strategies still outperform for all markets, after excludingthe Clayton copula. Among the five different copula strategies, the models with Gaussian, Student’s-t and Mixturecopulas outperform the others on variance reduction, while from the viewpoint of mean portfolio returns each copulastrategy has its own merits. Consequently, the elliptical Gaussian copula hedge strategies seem like the better choices

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2621Table 5Hedge performance.

Mean Variance Difference in mean returns Percentage of variance reduction

Un-hedged One-to-one OLS DCC Un-hedged One-to-one OLS DCC

Hong KongCopula strategy Normal 0.00611 0.28057 −0.01032 0.00661 0.00382 −0.00181 −89.70% −26.45% −2.24% −1.90%

Student’s-t 0.00606 0.28061 −0.01037 0.00656 0.00377 −0.00186 −89.69% −26.44% −2.23% −1.88%Clayton 0.00631 0.28532 −0.01012 0.00681 0.00402 −0.00161 −89.52% −25.20% −0.59% −0.24%Gumbel 0.00709 0.28076 −0.00934 0.00759 0.00480 −0.00083 −89.69% −26.40% −2.18% −1.83%Mixture 0.00615 0.28074 −0.01028 0.00665 0.00386 −0.00177 −89.69% −26.40% −2.18% −1.84%

Average with Clayton 0.00634 0.28160 −0.01009 0.00684 0.00405 −0.00158 −89.66% −26.18% −1.88% −1.54%Average without Clayton 0.00635 0.28067 −0.01008 0.00685 0.00406 −0.00157 −89.69% −26.42% −2.21% −1.86%Traditional strategy Un-hedged 0.01643 2.72300

One-to-one −0.00050 0.38145OLS 0.00229 0.28701DCC 0.00792 0.28600

JapanCopula strategy Normal 0.00133 0.19417 0.01557 0.00130 0.00235 0.00021 −90.65% −5.83% 0.01% −0.20%

Student’s-t 0.00127 0.19427 0.01551 0.00124 0.00229 0.00015 −90.65% −5.78% 0.06% −0.14%Clayton 0.00033 0.19829 0.01457 0.00030 0.00135 −0.00079 −90.45% −3.83% 2.13% 1.92%Gumbel 0.00115 0.19425 0.01539 0.00112 0.00217 0.00003 −90.65% −5.79% 0.05% −0.15%Mixture 0.00124 0.19428 0.01548 0.00121 0.00226 0.00012 −90.64% −5.77% 0.07% −0.14%

Average with Clayton 0.00106 0.19505 0.01530 0.00103 0.00208 −0.00006 −90.61% −5.40% 0.46% 0.26%Average without Clayton 0.00125 0.19424 0.01549 0.00122 0.00227 0.00013 −90.65% −5.79% 0.05% −0.16%Traditional strategy Un-hedged −0.01424 2.07671

One-to-one 0.00003 0.20618OLS −0.00102 0.19415DCC 0.00112 0.19455

KoreaCopula strategy Normal 0.01273 0.85304 −0.03326 0.02085 0.00822 −0.00038 −82.82% −31.88% −2.22% −0.56%

Student’s-t 0.01229 0.85204 −0.03370 0.02041 0.00778 −0.00082 −82.84% −31.96% −2.34% −0.68%Clayton 0.01521 0.87834 −0.03078 0.02333 0.01070 0.00210 −82.31% −29.86% 0.68% 2.39%Gumbel 0.01297 0.85383 −0.03302 0.02109 0.00846 −0.00014 −82.80% −31.82% −2.13% −0.47%Mixture 0.01226 0.85203 −0.03373 0.02038 0.00775 −0.00085 −82.84% −31.96% −2.34% −0.68%

Average with Clayton 0.01309 0.85786 −0.03290 0.02121 0.00858 −0.00002 −82.72% −31.50% −1.67% 0.00%Average without Clayton 0.01256 0.85274 −0.03343 0.02068 0.00805 −0.00055 −82.83% −31.90% −2.26% −0.60%Traditional strategy Un-hedged 0.04599 4.96523

One-to-one −0.00812 1.25227OLS 0.00451 0.87245DCC 0.01311 0.85784

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Mean Variance Difference in mean returns Percentage of variance reduction

Un-hedged One-to-one OLS DCC Un-hedged One-to-one OLS DCC

SingaporeCopula strategy Normal 0.00235 0.21820 −0.00301 0.00211 0.00129 0.00016 −82.16% −12.36% 2.74% −0.06%

Student’s-t 0.00255 0.21756 −0.00281 0.00231 0.00149 0.00036 −82.21% −12.62% 2.44% −0.35%Clayton 0.00229 0.23182 −0.00307 0.00205 0.00123 0.00010 −81.04% −6.89% 9.15% 6.18%Gumbel 0.00239 0.21876 −0.00297 0.00215 0.00133 0.00020 −82.11% −12.13% 3.00% 0.20%Mixture 0.00259 0.21781 −0.00277 0.00235 0.00153 0.00040 −82.19% −12.52% 2.56% −0.24%

Average with Clayton 0.00243 0.22083 −0.00293 0.00219 0.00137 0.00024 −81.94% −11.30% 3.98% 1.15%Average without Clayton 0.00247 0.21808 −0.00289 0.00223 0.00141 0.00028 −82.17% −12.41% 2.69% −0.11%Traditional strategy Un-hedged 0.00536 1.22278

One-to-one 0.00024 0.24897OLS 0.00106 0.21238DCC 0.00219 0.21833

TaiwanCopula strategy Normal 0.00084 0.38021 0.01473 0.00117 0.00366 −0.00014 −203.27% −25.96% −1.99% −0.25%

Student’s-t 0.00082 0.38023 0.01471 0.00115 0.00364 −0.00016 −203.27% −25.95% −1.99% −0.24%Clayton −0.00120 0.39327 0.01269 −0.00087 0.00162 −0.00218 −202.20% −23.42% 1.37% 3.18%Gumbel 0.00102 0.38040 0.01491 0.00135 0.00384 0.00004 −203.25% −25.92% −1.94% −0.20%Mixture 0.00092 0.38031 0.01481 0.00125 0.00374 −0.00006 −203.26% −25.94% −1.97% −0.22%

Average with Clayton 0.00048 0.38288 0.01437 0.00081 0.00330 −0.00050 −203.05% −25.44% −1.30% 0.45%Average without Clayton 0.00090 0.38029 0.01479 0.00123 0.00372 −0.00008 −203.26% −25.94% −1.97% −0.23%Traditional strategy Un-hedged −0.01389 2.86574

One-to-one −0.00033 0.51351OLS −0.00282 0.38794DCC 0.00098 0.38115


for investors to help avoid risks associated with portfolio returns. In particular, the asymmetric Mixture copula modelseemed to add (very) small value in terms of variance reduction for the Korean market only.

5. Conclusion

This paper employs a time-varying Copula-TGARCH model to evaluate the optimal hedge ratio for stock returnson spot and futures for five East Asian markets, from 1998 to 2005. Compared with the benchmarks of the un-hedged,one-to-one, OLS and DCC hedge ratios, we find that Copula-TGARCH time-varying hedge ratios are quite effectivein reducing risks in portfolio returns. Therein, the Gaussian and Mixture copulas provide the best-performed hedgeratios for risk reduction for Hong Kong, Taiwan and Korea, implying copula hedge strategies are suitable for thosevolatile markets. For stable markets such as Japan and Singapore, complicated models such DCC and copula do notprove effective risk reducers compared with OLS, although these models did generate higher returns than OLS in thesemarkets.

Copula methodology is quite flexible to model the dependence between two markets or risk factors. One can furtheruse different copula functions from the presenting paper to mimic the different dependence structures. As to theselection of a specific copula function in improving the hedging performance, more research is needed.

Acknowledgements

We thank the Editor (Professor Dave Allen) and a referee for comments that improved the paper. Cathy Chen issupported by grant 95-2118-M-035-001 of National Science Council (NSC) of Taiwan and grant 94GB67 of Feng ChiaUniversity. Richard Gerlach, while on research visits to Feng Chia University, was supported by the grant 94GB67of Feng Chia University and the University of Sydney via a new staff grant. Part of the work of Cathy Chen wasundertaken during a research visit to the University of Sydney.

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Optimal dynamic hedging via copula-threshold-GARCH models