Dynamic hedging strategy in incomplete market: Evidence from Shanghai fuel oil futures market

Economic Modelling 40 (2014) 81–90

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Dynamic hedging strategy in incomplete market: Evidence from Shanghai fuel oil futures market Xiaoqiang Lin a,⁎, Qiang Chen b,c, Zhenpeng Tang d a

Fujian Branch of China Construction Bank, PR China Xiamen University, PR China Economics & Management of Shanghai Jiao Tong University, PR China d School of Economics and Management, Fuzhou University, PR China b c

a r t i c l e

i n f o

Article history: Accepted 17 March 2014 Available online xxxx Keywords: Multivariate GARCH model Optimal hedge ratio Market noise conditional volatility

a b s t r a c t This paper introduces a new incomplete index and establishes a new optimal hedging model. We find that when the market micro-noise is perfectly negatively correlated with the return of futures market, market incompleteness depends on the relative level of noise volatility. Especially when noise volatility is less than the futures market yield, noise volatility will be offset by return volatility. As a result, complete optimal hedging model emerges. As an aside, it is interesting to note that as different conditional variances derived from different volatility models being applied, the hedge performance tends to be basically consistent with subtle difference: DCC–GARCH model is more likely to execute the hedging with 1:1 ratio, while other multivariate GARCH models would give a hedging ratio with greater probability less than 1:1 and is less likely to be a perfect hedge. Therefore, we believe that a simpler econometric model might produce better empirical results. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The launch of stock index futures traded on Chinese Financial Futures Exchange On April 16th, 2010, opened a new era for China's financial industry, offering a new derivative product for hedging, given current insufficient investment and financing channel in Chinese capital market. The chairman of CSRC, Guo Shu-qing has remarked that crude oil futures is likely going to be launched within 2012, and expected to be the third global crude oil trading center after the USA and the UK, in order to gain pricing power. China's futures market has developed for more than twenty years, with twenty seven futures products, however, there is still a huge gap between the development of derivative products and the needs of real economy; especially for fossil oil, as China consumes huge quantity of fossil oil, nevertheless without pricing power, which constrains the economic development; given recent volatile crude oil price movement, the market urgently needs new financial derivative for crude oil to mitigate investment risk. With only fuel oil futures currently available in the market, the lack of hedging tools and strong speculative sentiment is the major concern among many concerns by Chairman Guo. The launch of stock index futures makes certain hedging activities possible. However, there are many institutional constrains in China's market, for instance, information asymmetry between buyers and sellers, short-sale constrains, margin ⁎ Corresponding author at: Gu-Ping Road 142#, Fuzhou, Fujian Province, 350003, PR China. E-mail address: [email protected] (X. Lin).

http://dx.doi.org/10.1016/j.econmod.2014.03.022 0264-9993/© 2014 Elsevier B.V. All rights reserved.

system, price-movement limits; thus the market is incomplete, resulting in the discrepancy between futures market price and theoretical price (Buhler and Kempf, 1995; Lafuente and Novales, 2003; Miller et al., 1994; Wang, 2008; Wang and Hsu, 2006; Yadav and Pope, 1990), which constitutes market micro-noise, and compromises the effectiveness of hedging with stock index futures. In previous literatures, Chen et al. (2003) summarized optimal hedging theory. Lafuente and Novales (2003) pioneered with the noise volatility factor to analyze the optimal hedging ratio with the discrepancy between market futures price and theoretical futures price, which largely improved hedging strategy. To solve the market noise problem, Andani and Lafuente (2009), based on the model by Lafuente and Novales (2003), conducted empirical studies on number of countries with different liquidities, and found that the hedging ratio proposed by the model shows no significant improvement over a fixed 1:1 hedging strategy. We believe that the empirical results might be caused by different data, and unique traits of each market, such as institutional factors, incompleteness, etc. Market incompleteness is the starting point of this paper. It is noteworthy that market incompleteness includes the incompleteness within the market and the incompleteness between markets (Hsu and Wu, 2010). Early measurement to market incompleteness is accomplished by measuring the incompleteness of a single market through some indicator (e.g. transaction cost (Hsu and Wang, 2004)). By establishing a pricing model for stock index futures in an incomplete market, a more comprehensive indicator can be put forward, providing a more reliable method to evaluate inter-market incompleteness. On the

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premise of market incompleteness, this paper introduces a new incompleteness measurement and considers market micro-noise to build optimal hedging model. Based on continuous time diffusion model, this paper also analyzes the optimal hedging strategy with market micro-noise. To better examine the operability of the model, we utilized the trading data of fuel oil futures contracts traded on SHFE, due to the short history of Chinese futures market and the underdevelopment of the derivative market. Because China's fuel oil futures market is far from perfect, the impact of market noise on hedging with stock index futures deserves our attention. As a result, this paper employs multivariate GARCH model to examine the reasonability of the theoretical model, and attempts to discuss the characteristics of the incompleteness exhibited in China's fuel oil and stock index futures markets with estimations and simulated results. The paper is laid out as follows: the second section discusses the theoretical analysis of the optimal hedging strategy in an incomplete market considering market micro-noise; the third section discusses the optimal hedging model and evaluation; the fourth section gives the analysis of multivariate GARCH model; the fifth section shows the empirical analysis, including in-sample ex post analysis and out-ofsample ex ante analysis; and conclusion. 2. The theory model Cornell and French (1983a,b) had shown that in the perfect market, the price Ft. T⁎ at time t (contract maturing at time T) of the index futures had following relationship with spot price St :


ðr−dÞðT−t Þ

F t:T ¼ St e

ð1Þ





where, r f
;t ¼ ln ð F t:T =F t−1:T Þ and ro,t = ln(St/St − 1). When the markets are perfect, the volatility models are displaying the same process, however, the market are usually are imperfect because of the noise, the process between the markets are quite different. In order to characterize the difference of volatility models, following the method of Lafuente and Novales (2003), we introduce a second noise specific to the future market. Therefore, we can decompose the volatility model into two different parts, and depict as following. dr f ;t

ð2Þ

Here, wf,t is a Wiener process which affects the index future volatility, and wn,t is defined as new Wiener process which is related to noise. Considering the two period generation model, an investor hold one unit long position of fuel oil future; however, in order to hedge the risk, the investor short sell ht unit index future in anther market. Then we can obtain the hedge ratio ht. Therefore, the investor can make choice the properly hedge ratio ht to minimize the portfolio variance. min Var t

  dr f ;t dro;t −ht dt
 dt


ht  s:t:dr o;t ¼ μ o r o;t dt þ σ o r o;t dwo;t

 dr f ;t ¼ μ f r f ;t dt þ σ o ro;t dwo;t þ σ n ðnt Þdwn;t þ dw

ð3Þ




1 þ ρon;t δt 1 þ δ2t þ ρon;t δt

:

ð5Þ

Thus, we can see that market incompleteness increases as market micro-noise increases (Eq. (5)), which may be a result of frequent trading noise that makes hedging portfolio partially ineffective. Risk diversification won't be achieved easily if hedging performance is not ideal. 3. Evaluate the performance of the optimal hedge model There have been many assessment and theoretical discussion of various hedge strategies (Johnson, 1960; Ederington, 1979; Myers, 1991; Baillie and Myers, 1991; Kroner and Sultan, 1993; Park and Switzer, 1995; Thomas and Brooks, 2001; Meneu and Torro, 2003; Lee et al., 2006; Chang et al., 2010a,b). A common method is to use the ratio of the minimum conditional variance of hedge position and the variance of non-hedge position (VIF), as a measurement. If the minimum conditional variance of the hedge portfolio is Vmin, then we can express VIF as below (see Appendix A) 2

ð6Þ

This equation does a better job illustrating how the effectiveness of the optimal hedge strategy with micro-noise changes along with the change of the market incompleteness. By analyzing proposition Eq. (6), we can find that when market incompleteness reaches 1, the optimal hedge strategy with noise becomes ineffective, because the underlying asset is entirely unrelated with stock index futures, which enables the building of an effective hedge portfolio. Also, we found that when the market incompleteness is either too low or too high, the optimal hedge strategy with noise displaces excellent effectiveness. As matter of fact, when MI ≥ 1, which means a high degree of market incompleteness, a simple fixed 1:1 hedge strategy is not only unable to effectively hedge, but also amplify the risk of the hedge portfolio. 4. Volatility model framework

Following the method introduced by Lafuente and Novales (2003), we obtain the optimal hedge ratio (Chen et al., 2013): ht ¼

1 þ ρon;t δt MI ¼ 1− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ δ2t þ ρon;t δt

VIF ¼ 2MI−MI :





 ¼ μ f r f ;t dt þ σ f r f ;t dw f ;t ¼ μ f r f ;t dt þ σ o r o;t dwo;t þ σ n ðnt Þdwn;t :

It suggests a nonlinear relationship between the optimal hedging ratio and market micro-noise. The consideration of market noise will naturally lead to market incompleteness. Regarding market incompleteness, Hsu and Wang (2004) provides an explicit explanation, which built a mathematical theory model with noise and market incompleteness. Based on the model proposed by Hsu and Wang (2004), the invest portfolio ratio ht = σo(ro,t)/σf(rf,t) can be applied repeatedly to make the return of the hedging portfolio free of risk (σp,t = 0). However, with market incompleteness, perfect hedge is unlikely, because of which, Hsu and Wang (2004) proposed σp,t/σo,t as a measurement for market incompleteness. Also, the more the incompleteness in the market, the bigger the σp,t becomes; by selecting the optimal hedging ratio, we can minimize the volatility variance of the hedging portfolio, and furthermore, obtain market incompleteness measurement MI.

ð4Þ

Here, Corr(εot, εnt) = ρon,t and δt = σn(nt)/σo(ro,t) denote the relative ratio of the specific market noise. Consequently, we can easily make conclusion that there exist some relationship between optimal hedge ratio and market noise from Eq. (4).

In this section, we only briefly introduce a simple GARCH model, and univariate GARCH model, such as CCC, DCC, Diag-BEKK, Full-BEKK, Scalar-BEKK, etc. Given the constrain of the length, we will not list all of them here. The main purpose of these models is to obtain the conditional volatility, of course, with consideration of historical volatility that is used in the process of estimating the optimal hedge ratio. Different variances estimated by different models differ greatly, thus we reconsidered these models to make comparison of optimal hedge ratios from in and out-of-sample data. These variances play important role in optimal estimation, especially the computation of optimal ratio itself.

X. Lin et al. / Economic Modelling 40 (2014) 81–90

4.1. Univariate GARCH models

83

Table 1 The descriptive statistics.

Extended the work of Engle (1982), Bollerslev (1986) showed that the generalized autoregressive conditional heteroskedasticity (GARCH) was a useful tool to capture volatility in financial markets, then also showed that GARCH(1, 1) model work well in the empirical analysis. Given the advantage of GARCH (1,1), Hansen and Lunde (2005) showed that the model was outperformed when investigating the exchange rate. Interestingly, Sadorsky (2006) revealed that the GARCH(1,1) model could capture the crude oil volatility well. In our paper, we show the standard GARCH(1, 1) model as follows, r t ¼ μ t þ εt ¼ μ t þ σ t zt ; zt ∼ NIDð0; 1Þ; 2 2 2 σ t ¼ ω þ αε t−1 þ βσ t−1 :

ð7Þ

Here, μt denotes the conditional mean and σt2 the conditional variance with parameter restrictions ω N 0, α N 0, β N 0 and α + β b 1. And then, considering the asymmetric leverage effect, GJR and EGARCH models were developed by Glosten et al. (1993) and Nelson (1991). And the GJR(1, 1) model is

Mean Maximum Minimum Standard deviation Skewness Kurtosis Jarque–Bera ADF PP KPSS

China Securities Index 300 (CSI 300)

CSI 300 index futures

Shanghai fuel oil futures

−6.36E−06 0.0174 −0.0245 0.00206 −0.297 11.429 37,411.22⁎⁎⁎ −113.7408⁎⁎⁎ −115.514⁎⁎⁎

−8.17E−06 0.03051 −0.0238 0.00216 0.9801 24.921 253,787.4⁎⁎⁎ −54.198⁎⁎⁎ −113.732⁎⁎⁎

5.02E−06 0.06325 −0.0429 0.00166 5.413 347.573 62,271,422⁎⁎⁎ −116.272⁎⁎⁎ −116.276⁎⁎⁎

0.237

0.231

0.069

Notes: The Jarque–Bera statistic is used to the null hypothesis of normality in sample distribution. ADF statistic represents the Augmented Dickey–Fuller unit root test based on the AIC criterion. PP and KPSS denote Phillips–Perron and Kwiatkowski–Phillips– Schmidt–Shin unit root tests, respectively. ⁎⁎⁎ Indicates the 1% confidence level.

5. Empirical results 5.1. Preliminary analysis

2

2

2

σ t ¼ ω þ ½α þ γIðεt−1 b 0Þε t−1 þ βσ t−1 ;

ð8Þ

where I(.) is an indicator function. γ is the asymmetric leverage coefficient to describe volatility leverage effect. The EGARCH(1,1) model is described as follows,

 2 2 log σ t ¼ ω þ αzt−1 þ γðjzt−1 j−Ejzt−1 jÞ þ β log σ t−1 :

ð9Þ

4.2. Multivariate GARCH models The multivariate GARCH models are usually applied to the study of the relations between the volatilities and co-volatilities (Kearney and Patton, 2000). Another advantage of multivariate GARCH models is employed to compute the time-varying hedge ratios (Lien and Tse, 2002). The constant conditional correlation GARCH (CCC–GARCH) model introduced by Bollerslev (1990) is one of the most popular models because of its ease of model estimation. However, a constant correlation seems too restrictive, the BEKK–GARCH model with time varying correlations first presented by Engle and Kroner (1995), have obtained more and more attention in the literatures. Consequently, we provide the BEKK–GARCH model for estimation. And the bivariate BEKK–GARCH model1 is written as follows, 0

0

0

0

H t ¼ Ω Ω þ A εt−1 εt−1 A þ B H t−1 B where, the individual elements for the matrices Ω, A and B are given as,  A¼

α 11 α 21

  β11 α 12 ; B¼ α 22 β21

  ω11 β12 ; Ω¼ ω21 β22

 0 : ω22

Here, the coefficients α12, α21, β12 and β21 can reflect the volatility transmission and spillover between two markets.

1 In this paper, we employ the bivariate BEKK–GARCH model for future markets in Chinese markets, rather than consider the multivariate GARCH model. Because the bivariate BEKK–GARCH (11 parameters to be estimated) is easier to estimate than multivariate GARCH model (65 parameters to be estimated).

After years' development of crude oil spot market, mainland China still have not build a crude oil futures market. Domestic institution investors have not found a reliable hedge tool that can be put in practice, also have not been able to found derivative products that can be used to manage excessive volatility and systematic risk in the market. Therefore, with the approval of the State Council and the authorization of the China Securities Regulatory Commission (CSRC), the China Financial Futures Exchange (CFFEX) was estimated on September 2006 in Shanghai. And then, on 16 April 2010, CSI 300 stock index futures were listed on the CFFEX2. The famous index in China include one seventh of all stocks listed on Shanghai and Shenzhen stock markets and about 70% of the markets' value. The data that this paper uses is 5-minute high frequency data of SHFE fuel oil futures and CSI 300 stock index futures, from April 16th, 2010 to April 15th, 2011. The data enables us to calculate logarithmic return series of SHFE fuel oil futures and CSI 300 stock index futures. Table 1 presents the descriptive statistics for the return of China Securities Index 300 (CSI 300), CSI 300 index futures and Shanghai fuel oil futures. The ADF statistics show the rejections of null hypothesis of unit root in the return series at 1% significance level, indicating that those return series are stationary, and also evidenced by PP and KPSS statistics. Return skewness and kurtosis tell us that all three exhibit positive skewness and leptokurtic, and SHFE fuel oil return is of greater positive skewness and leptokurtic than stock index futures. Also, both markets have low level of returns, as zero-return null hypothesis cannot be denied. Both markets have similar level of return and average volatility, however, SHFE fuel oil futures market obviously contains more unstable factors or extreme value shocks than stock index futures market, which to some extent means that SHFE fuel oil futures market is subject to greater impact from market noise than stock index futures market. Fig. 1 can give the same conclusion. Because stock index futures are a natural safe-haven, their trading volume and frequency is quite large compared with SHFE fuel oil futures, which make their volatility different. CSI 300 as the main contract of CSI 300 index futures, its return is apparently more volatile. These derivative products, given the current lack of hedge tools in Chinese financial market, deserve our attention, as their instrumentality stands out especially in the age of volatile stock market.

2

More detail can be found in http://www.cffex.com.cn/en_new/sspz/hs300zs/.

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X. Lin et al. / Economic Modelling 40 (2014) 81–90

China Securities Index 300 (CSI 300) 0.02

CSI 300 index futures 0.04

0.01

0.03

0.005

0.02

0

return

return

0.015

-0.005

0.01

0

-0.01 -0.01 -0.015 -0.02

-0.02 -0.025

-0.03 Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

China Securities Index 300 (CSI 300)

CSI 300 index futures

Shanghai fuel oil 0.08

0.06

0.04

return

0.02

0

-0.02

-0.04

-0.06

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Shanghai fuel oil Fig. 1. The return series of China Securities Index 300, CSI 300 index, and Shanghai fuel oil, respectively.

5.2. The segmentation of bear/bull Chinese future markets Fig. 2 indicates bear and bull future markets for China Securities Index 300, CSI 300 index, and Shanghai fuel oil futures with almost the same time point for peaks and valleys. The segmentation bull and bear markets criterion method can be found in Alexander (1964), Fama and Blume (1966), and Sweeney (1988). Just as Chang et al. (2010a,b) said: “When asset prices rise above a certain level from their previous local low, persistent forces subsequently make the price go in the same direction. The period of the previous local low to the next local high is called a bullish market. Similarly, the local high exhibits when the upper pattern reverses. The period from the peak to the next local low is called a bearish market.” Therefore, we divide the high frequency data into three different periods in Fig. 2. The sample in our paper identifies period (2010/6– 2010/11) as bullish markets, identifies period (2010/4–2010/6) as bearish market, and identifies period (2010/11–2011/4) as adjustment market. The correlation between CSI 300 index futures and Shanghai fuel oil futures (showed in Table 3) is obviously different during the bull and bear markets. The relationship between CSI 300 index futures and Shanghai fuel oil futures is the key factor for building hedging

strategies. Therefore, it is reasonable to assume that the same hedging strategies can provide different performances across bull and bear trend periods. (See Table 2.) 5.3. Dynamic correlation coefficients The constant conditional correlation GARCH (CCC–GARCH) assumes a constant correlation throughout the estimation process, which is a very strong restrictive condition; while BEKK–GARCH solves this constant correlation problem by deriving a time varying correlations. In order to obtain the optimal hedge ratio, we must ensure that a high-quality correlation relation can be obtained during the estimation process of the model, which is a necessary precondition for any hedge models. In this paper, we first estimated the dynamic correlation between CSI 300 index futures and Shanghai fuel oil futures. Fig. 3 demonstrates a close correlation between the two, with maximum value exceeding 0.85, and minimum value less than 0.5.3 Therefore, we have a sound reason to believe that CSI 300 index futures posts itself as a plausible derivative product for hedging purpose under the circumstance of scarce hedge tools available in China. 3 In this paper, we employ the bivariate BEKK–GARCH model for correlation estimation. Because the bivariate BEKK–GARCH (11 parameters to be estimated) is easier to estimate than multivariate GARCH model N65 parameters to be estimated).

X. Lin et al. / Economic Modelling 40 (2014) 81–90

China Securities Index 300 (CSI300)

85

CSI 300 index futures 3800

3600

3600

3400

3400 3200

index

index

3200 3000

3000 2800 2800 2600

2400

2600

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

2400

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

China Securities Index 300 (CSI 300)

CSI 300 index futures

Shanghai fuel oil 5200

5000

index

4800

4600

4400

4200

4000

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Shanghai fuel oil Fig. 2. Graphical representation of price index.

To obtain different time varying hedge ratios, we need different conditional variances that can effectively construct different hedge ratios. One of the major purposes of this paper is to analyze the performance, specially out-sample performance, of various hedge strategies by constructing different time varying hedge ratios. We obtain different conditional variances (see Figs. 4 and 5) via Multivariate GARCH models (CCC, DCC, Diag-BEKK, Full-BEKK, Scalar-BEKK). We found that in Fig. 4, the conditional variances of Full-BEKK and Scalar-BEKK in CSI 300 index

futures (obtained from Multivariate GARCH models) are obviously larger than DCC and Diag-BEKK, as a sign of greater difference in volatilities. In Fig. 5, we found that the conditional variances of Full-BEKK and Scalar-BEKK in Shanghai fuel oil futures (Obtained from Multivariate GARCH models) are apparently smaller than DCC and Diag-BEKK, as a sign of insignificant difference in volatilities. This difference determines the inconsistency in out-sample and in-sample hedge performance. Figs. 4 and 5 remarkably demonstrate the hedge performance.

Table 3 Comparison of in-sample hedging effectiveness. Table 2 Standard deviation and correlation in Bull/Bear market. Bear period

Bull period

Compared to non-hedge strategy (%) Adjustment period

Hedging model (multivariate GARCH family)

Bear period

CCC DCC Diag-BEKK Full-BEKK Scalar-BEKK

4.124 2.255 3.672 1.723 4.385

Shanghai fuel oil futures Standard deviation

0.002653

Standard deviation Correlationa

0.001620 0.8002

0.002252

0.001807

CSI 300 index futures

a

0.001972 0.6245

0.001381 0.7731

Represents the correlations between CSI 300 index futures and Shanghai fuel oil futures.

Bull period

Adjustment period

Shanghai fuel oil futures 4.617 2.991 5.105 3.068 6.512

4.511 2.455 4.226 2.665 4.867

Note: This table presents an in-sample comparison of hedging effectiveness of shanghai fuel oil, under different alternative models. Greater numerical values imply higher hedging effectiveness.

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X. Lin et al. / Economic Modelling 40 (2014) 81–90 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55

CSI 300 index futures-Shanghai fuel oil futures

0.5 0.45

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Dynamic correlation coefficients Fig. 3. Dynamic correlation coefficients.

5.4. In-sample performance Table 3 shows a summary of in-sample hedging effectiveness. Hedge strategies perform differently during bullish and bearish markets across the future markets. In bullish market, hedging is more effective than during bearish market; and during market adjustment period, hedging 2.5

x 10

is also more effective than during bearish market, but inferior to bullish market. Thereby, we conclude that hedging works best when the market is bullish. Hedging effectiveness for shanghai fuel oil is typically higher in the increasing pattern than in the decreasing pattern of price movements. Obviously, The hedging effectiveness for shanghai fuel oil is almost better that of the bullish than the bearish period across models. The optimal model leads to the highest hedging effectiveness (see Table 3). Scalar-BEKK has the most outstanding performance in either bearish or bullish markets, while Full-BEKK performs worst. However, the most interesting point is that the performance of CCC–GARCH model is eye-catching, from which we can see that a hedge model with constant correlation is not the worst performing model. In real practice, this observation, to some extent, assures us a satisfactory hedge performance from model with constant relative hedge ratio when market prospect is unclear. This shows that a complex model does not always outperform a simple one. 5.5. Out-sample performance Out-of-sample hedging strategies are obtained from traditional hedge model using a rolling window of 22 business days (almost about 1 month). We choose the estimation period, and the estimation is rolled forward as time going by adding one from the next subsample and dropping first sample, to characterize the volatility variances. And by the rolling method, we have the same sample when estimating the data. That is to say, the estimation sample is rolled forward by one

-5

1.8

x 10-5

1.6 2 1.4 1.2 1.5 1 0.8 1 0.6 0.4

0.5

0.2 0

0

6

x 10

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Conditional variance(DCC-GARCH)

Conditional variance(Diag BEKK-GARCH)

-4

6

5

5

4

4

3

3

2

2

1

1

0

x 10-4

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Conditional variance( Full BEKK-GARCH)

0

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Conditional variance( Scalar BEKK-GARCH)

Fig. 4. CSI 300 index futures conditional variance (obtained from multivariate GARCH models).

X. Lin et al. / Economic Modelling 40 (2014) 81–90 4

x 10-3

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

x 10

-3

0

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Conditional variance(DCC-GARCH)

Conditional variance( Diag BEKK-GARCH)

-3

2.5

x 10

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

87

0

x 10-3

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

Conditional variance( Full BEKK-GARCH)

Conditional variance( scalar BEKK-GARCH)

Fig. 5. Shanghai fuel oil futures conditional variance (obtained from multivariate GARCH models).

step (sometimes, a trading day), while the total length of the estimation sample is kept constant (Koopman et al., 2005). Obviously, hedging effectiveness for out-of-sample analysis in traditional method is much reliable and applicable for practical use, and then Table 4 shows the empirical results that are consistent with the empirical results presented in Table 3. Also in Tables 3 and 4, we continue to use a traditional hedge method to make comparison; which already becomes a classic method when it comes to evaluating models (Chang et al., 2010a,b). In the estimation process, we utilized the findings to re-estimate the optimal hedge ratio, for the purpose of evaluating the feasibility and superiority of the model constructed in this paper.

from the result estimated by Full-BEKK. The discrepancy between market volatility estimations deviates our understanding of the market condition, which in turn leads to different results in estimating optimal hedge ratio; however, this method provides us a framework that gives us an opportunity to better understand the degree of market incompleteness. When market incompleteness is understood, it is natural to probe the problem of market micro-noise. Because the two estimated conditional variances for measuring market incompleteness are so different, it is absolutely crucial to have an in-depth knowledge of

5.6. Market noise analysis and hedge performance

Table 4 Comparison of out-sample hedging effectiveness.

The proposal of σp,t/σo,t being the measurement of market incompleteness based on Hsu and Wang (2004) reveals the interdependent relationship between market incompleteness and market micro-noise. Before examining the optimal ratio, it is necessary to investigate market incompleteness, for further examining the performance of the optimal ratio. Therefore we can illustrate the degree of market incompleteness (see Fig. 64). The degree of market incompleteness calculated with conditional variance estimated by DCC–GARCH apparently differs 4

For saving space, we do not show the results by the models of DCC, Diag-BEKK, ScalarBEKK. They can be obtained upon request from the corresponding author.

Compared to non-hedge strategy (%) Hedging model (multivariate GARCH family)

Bear period

CCC DCC Diag-BEKK Full-BEKK Scalar-BEKK

4.004 2.475 3.225 2.112 4.117

Bull period

Adjustment period

Shanghai fuel oil futures 4.854 2.781 4.799 2.336 5.996

4.578 2.444 4.646 2.275 4.659

Note: This table presents an out-sample comparison of hedging effectiveness of Shanghai fuel oil, under different alternative models. Greater numerical values imply higher hedging effectiveness.

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X. Lin et al. / Economic Modelling 40 (2014) 81–90 2

0.9971

1.8 0.9971 1.6 0.9971

1.4 1.2

0.9971

1 0.9971

0.8 0.6

0.997

0.4 0.997

0.2 0

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

0.997

market imperfection(full-bekk)

Jun.2010 Aug.2010 Oct.2010 Dec.2010 Feb.2011 Apr.2011

market imperfection(DCC)

Note: For saving space, we do not show the results by the models of DCC, Diag-BEKK, Scalar-BEKK. They can be obtained upon request from the corresponding author. Fig. 6. Market imperfection (incompleteness). Note: For saving space, we do not show the results by the models of DCC, Diag-BEKK, Scalar-BEKK. They can be obtained upon request from the corresponding author.

market micro-noise in the estimation process (see Fig. 7). From Fig. 7, the relative ratio of the noise in terms of the common noise, we found that the market noise estimated by DCC–GARCH is narrower, while the market noise estimated by Full-BEKK is wider, which demonstrates the influence that different conditional variance estimations could have done on optimal hedge ratio. However, Full-BEKK has been known being superior in term of measuring market noise. These estimations partially rely on the correlation coefficient between fuel oil future and the noise (see Fig. 8). Now we know that two models can produce very different correlation coefficient, which might be the result of inconsistent model specification during the process of the two estimation models. The micro-noise estimated by DCC–GARCH is negatively correlated with market return, which means that market incompleteness is determined by the relative level of the volatility of market noise and return. Noise volatility could be offset by market volatility if noise volatility is less than market return volatility; thereby, the optimal hedge model would perform as intended. The estimation by Full-BEKK model shows that a complete negative correlation

between market micro-noise and market return does not exist, and noise volatility is less than market return volatility. It is interesting to note that the hedge performance with conditional variance estimations by the two models is basically the same (see Fig. 9), with subtle difference that DCC–GARCH is more likely to reach perfect hedge with 1:1 ratio, while Full-BEKK is more like to reach perfect hedge with ratio less than 1:1, which may be caused by the difference in market micro-noise. Interestingly, the CCC–GARCH is also likely to reach perfect hedge with 1:1 ratio, and the evidences are also provided in Fig. 10. Comparing with the empirical results presented in Tables 3 and 4, our estimated results cover the content of Tables 3 and 4, with extension that shows that we can conduct hedging at any moment. The empirical results not only consider noise (variance), but also take into account of factors including conditional variance, inter-market correlation and market incompleteness. In additional, we depict the time-varying process in measuring hedge performance, which provides theoretical framework for market practice.

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Fig. 7. The relative ratio of the noise in terms of the common noise (δt = σn(nt)/σo(ro,t)).

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Fig. 8. The correlation coefficient between oil future and the noise.

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Hedging performance(full-BEKK) Fig. 9. Hedging performance.

6. Conclusion This paper, on the premise of market incompleteness, introduced a new market incompleteness measurement, at the same time, considered -0.2

market micro-noise to build optimal hedging model. Based on continuous time diffusion model, this paper also analyzed the optimal hedging strategy with market micro-noise, and examined the impact of market incompleteness. This paper emphasized the difference of hedging 2

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Fig. 10. The correlation coefficient and hedging performance.

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effectiveness under bull and bear markets. This paper found that market micro-noise is negatively related with the return of shanghai fuel oil future; market incompleteness is determined by the relative level of noise volatility and the volatility of fuel oil futures return; and more importantly, there is a greater chance that noise volatility is less than the market return volatility (δt b 1), as noise volatility will be offset by market return volatility, and optimal hedge model could be easily obtained. An interesting finding is that different conditional variance estimations from various volatility models give a basically consistent hedge performance, only with subtle difference. DCC–GARCH model is more likely to reach perfect hedge with 1:1 ratio, while other multivariate GARCH models are less likely to reach perfect hedge, and even they do, the hedge ratio to achieve perfect hedge is more likely to be less than 1:1. These results if applied can be beneficial to market practice, and we have sound reason to believe that a simple model could rather produce better practical results. Appendix A The minimum conditional variance of the hedge portfolio Vmin can be obtained as follows, and for more detail see our working paper (Chen et al., 2013,



 2
2
V min ¼ σ o r o;t þ ht σ f r f ;t −2ht ρof ;t σ o r o;t σ f r f ;t



 2
2
¼ σ o r o;t þ ht σ f r f ;t þ 2ht σ o r o;t σ f r f ;t


− 2ht ρof ;t 1−ρof ;t σ f r f ;t


 2
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