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This study proposes a new range-based Markov-switching dynamic conditional correlation (MSDCC) model for estimating the minimum-variance hedging ratio and comparing its hedging performance with that of alternative conventional hedging models, including the naive, OLS regression, return-based DCC, range-based DCC and return-based MS-DCC models. The empirical results show that the embedded Markov-switching adjustment in the range-based DCC model can clearly delineate uncertain exogenous shocks and make the estimated correlation process more in line with reality. Overall, in-sample and out-of sample tests indicate that the range-based MS-DCC model outperforms other static and dynamic hedging models.

The dynamic conditional correlation (DCC) model, the celebrated multivariate correlation estimation model proposed by Engle [^{1}. The DCC model also provides some advantages through its operating procedure, which enables more parsimonious parameter estimation and easy estimation. Drawing on related literatures, the current models of correlation estimation were developed to extract return data to estimate covariance processes. A few recent studies have attempted to introduce a range variable to replace the return variable for more information content and to account for efficient market theory. More information can be gathered with a range structure than with a return variable when estimating a volatility model for many separate empirical results^{2}. For example, Chou et al. [^{3}. It is natural to introduce a nonlinear mechanism into the return- and range-based DCC models to enhance feasibility. In this study, we pro- pose a new range-based regime-switching DCC model that is able to enhance the hedge effect in futures markets.

Conventional approaches to nonlinear adjustment among financial time-series models include threshold auto- regressive techniques, smooth transition and Markov-switching. We introduce the Markov-switching method proposed by Hamilton [

The literature has historically used the utility function and ordinary least squares model to discuss mini- mum-variance hedging. According to Chen et al. [^{4}. In contrast, some empirical studies argue that there are no significant improvements from employing advanced econometric techniques^{5}. According to Lien [

The rest of this article is organized as follows. In the next section, the range-based regime-switching DCC model is introduced, and minimum-variance hedging is described in the following section. In the fourth section, this study presents a hedging effectiveness measurement. The empirical results and the economic intuition of these results are reported in the fifth section. The final section provides the conclusion.

To introduce the regime-switching structure to the range-based dynamic conditional correlation process, the range-based MS-DCC model with a general S-regime can be expressed as:

where, (1c)

where Equations (1a)-(1c) represent the range-based volatility specification, ^{th} asset during the time interval t,

the standardized residual, and the scaled expected range

ation. The unconditional standard deviation of the return series k and the sampling mean of the estimated condi-

tional range of the series k are represented as

bility, which has the constraints

Equation (1d). It is intuitive to define the stationary distribution of the Markov chain as

represented as

time-varying covariance matrix. In equations (1e) and (1f), the superscript symbol represents the regime shift from i to j. In short, the range-based MS-DCC model is composed of the conditional autoregressive range (CARR) model for the conditional variance process and the Markov-switching approach for the conditional co- variance and correlation case.

The log likelihood function of the MS-DCC model is presented in this section. According to Engle [

By maximizing the QMLE of Equation (2), the parameters

1) given the filtered probabilities as inputs, determine the joint probabilities as:

2) evaluate the regime-dependent log likelihood as:

3) evaluate the log likelihood of observation t as:

4) renew the joint probabilities as:

5) calculate the filtered probabilities as:

6) renew the dynamic conditional correlation matrix through the following approximation:

7) iterate 1 to 6 until the end of the sample. The unknown parameters of MS-DCC model can be obtained with these estimation procedures.

Minimum-variance hedging is the determination of the number of futures contracts against one spot asset that will ensure the minimum variance of the hedging portfolio. Furthermore, minimum-variance hedging can be calculated as a ratio of the covariance of spot-futures returns over the variance of futures returns, namely,

where

This study selects six various minimum-variance hedge ratios for the time being:

1) Naïve: A simple hedging strategy assigning the hedge ratio equal to −1 at all times.

2) OLS: A conventional method for analyzing the minimum-variance hedge ratio, used by Ederington [

The estimated slope,

3) Return-based DCC: A classical time-varying correlation model proposed by Engle [

4) Range-based DCC: A range-based dynamic conditional correlation model developed by Chou et al. [

5) Return-based MS-DCC: A more flexible return-based dynamic conditional correlation model proposed by Pelletier [

6) Range-based MS-DCC: A range-based dynamic conditional correlation model with a regime-switching structure, which can be expressed as Equations (1a)-(1f).

With regard to the hedging effectiveness measure, this study employs the variance reduction and further calcu- lates the percentage variance reduction. In addition, this study calculates the economic benefits including the expected daily utility and the value-at-risk (VaR) estimate. The investor faces the mean-variance expected daily utility function proposed by Kroner and Sultan [

, where. (16)

where

where

This study selects two stock indices with different weighting schemes for model testing at this stage, namely, the value-weighted S&P 500 index, the equal-weighted NIKKEI 225 index, and their corresponding futures con- tracts. The sample period is from January 3, 2000 to June 29, 2011 for the empirical study. One goal of this study is to detect using these two different weighted indices whether the range-based MS-DCC model outper- forms competing models. The daily high, low, and close price for the S&P 500 and NIKKEI 225 are obtained from Datastream.

Descriptive statistics for daily returns and ranges data are reported in

Billio and Caporin [

in

high-correlation regime, and the smaller coefficient of

The estimated transition probability of remaining in the low-correlation state

dices; however, the estimated transition probability of remaining in the high-correlation state

from low to high correlation

225, but the expected transition period from high to low correlation

for the S&P 500 and 0.03 years for the NIKKEI 225^{6}. This finding indicates that the expected transition period from high to low correlation is shorter than that of the reverse and that the correlation is relatively stable in the long term, although shocks may cause the correlation to oscillate violently and move from a low to high state. This study also calculates the steady-state probabilities of the Markov process as a benchmark. The estimated result of the range-based MS-DCC model indicates that the steady-state probability that the correlation of the S&P 500 will move to a low (high) state in the next period is 0.810 (0.190) and that the probability that the cor- relation of the NIKKEI 225 will move to a low (high) state in the next period is 0.998 (0.002). In brief, the probability of the expected correlation for either stock index moving to a low state is over 80%. With regard to the dynamic correlation process, the high-correlation state has a larger value of

. Descriptive statistics for the daily returns and ranges of spot and futures of S&P 500 and of NIKKEI 225 (2000.1.3-2011.6.29)

S&P 500 | NIKKEI 225 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Spot | Futures | Spot | Futures | |||||||

Return | Range | Return | Range | Return | Range | Return | Range | |||

Mean | −0.004 | 1.528 | −0.004 | 1.533 | −0.023 | 1.541 | −0.023 | 1.738 | ||

Median | 0.054 | 1.235 | 0.064 | 1.248 | 0.004 | 1.324 | 0.062 | 1.432 | ||

Maximum | 10.957 | 10.904 | 13.197 | 11.639 | 13.235 | 13.763 | 18.812 | 24.144 | ||

Minimum | −9.470 | 0.239 | −10.400 | 0.000 | −12.111 | 0.236 | −14.003 | 0.256 | ||

Std. Dev. | 1.361 | 1.117 | 1.371 | 1.124 | 1.616 | 0.992 | 1.677 | 1.375 | ||

Skewness | −0.113 | 3.049 | 0.039 | 3.089 | −0.401 | 3.643 | −0.257 | 5.616 | ||

Kurtosis | 10.670 | 18.278 | 12.837 | 18.905 | 9.606 | 29.130 | 14.897 | 58.917 | ||

Bera-Jarque | 7072.398 | 32505.10 | 11624.49 | 34974.54 | 5198.131 | 86370.71 | 16645.12 | 381803.60 | ||

observation | 2882 | 2883 | 2882 | 2883 | 2816 | 2817 | 2816 | 2817 | ||

Notes: The Bera-Jarque is the statistic for normality testing.

. The estimation of range- and return-based dynamic conditional correlation model of daily spot and futures of S&P 500 and of NIKKEI 225 (2000.1.3-2011.6.29)

Univariate: | range-based | |
---|---|---|

return-based | ||

DCC: |

Univariate:

range-based

return-based

DCC:

Panel A: | S&P 500 | NIKKEI 225 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Spot | Futures | Spot | Futures | |||||||||

univariate | CARR | GARCH | CARR | GARCH | CARR | GARCH | CARR | GARCH | ||||

0.022 | 0.013 | 0.021 | 0.016 | 0.031 | 0.041 | 0.038 | 0.046 | |||||

(0.006) | (0.006) | (0.005) | (0.007) | (0.008) | (0.011) | (0.009) | (0.012) | |||||

0.169 | 0.079 | 0.178 | 0.083 | 0.168 | 0.109 | 0.175 | 0.105 | |||||

(0.013) | (0.012) | (0.013) | (0.012) | (0.020) | (0.020) | (0.020) | (0.018) | |||||

0.816 | 0.912 | 0.808 | 0.907 | 0.812 | 0.877 | 0.803 | 0.880 | |||||

(0.014) | (0.011) | (0.014) | (0.011) | (0.022) | (0.019) | (0.023) | (0.017) | |||||

Panel B: | S&P 500 | NIKKEI 225 | ||||||||||

DCC | Range-based | Return-based | Range-based | Return-based | ||||||||

0.136 (0.010) | 0.015 (0.001) | 0.046 (0.002) | 0.039 (0.001) | |||||||||

0.229 (0.031) | 0.980 (0.002) | 0.945 (0.002) | 0.955 (0.002) | |||||||||

Notes: The number in parentheses is robust standard error proposed by Bollerslev and Wooldridge [

ing that the high-correlation regime follows major financial events, especially in the case of the S&P 500, makes intuitive sense7.

. The estimation of range- and return-based markov switching dynamic conditional correlation model for daily spot and futures of S&P 500 and NIKKEI 225 (2000.1.3-2011.6.29)

Univariate: | Range-based | |
---|---|---|

Return-based | ||

MS-DCC: |

Univariate:

Range-based

Return-based

MS-DCC:

S&P 500 | NIKKEI 225 | |||||
---|---|---|---|---|---|---|

Regime | Low Correlation | High Correlation | Low Correlation | High Correlation | ||

Panel A: Range-based | ||||||

0.052 (0.004) | 0.075 (0.001) | 0.074 (0.002) | 0.335 (0.095) | |||

0.931 (0.007) | 0.268 (0.006) | 0.906 (0.003) | 0.377 (0.131) | |||

0.999 (0.194) | 0.997 (0.077) | 0.999 (0.424) | 0.860 (0.117) | |||

0.810 | 0.190 | 0.998 | 0.002 | |||

Panel B: Return-based | ||||||

0.030 (0.001) | 0.147 (0.042) | 0.043 (0.002) | 0.158 (0.012) | |||

0.957 (0.003) | −0.023 (0.083) | 0.955 (0.002) | 0.373 (0.042) | |||

0.994 (0.160) | 0.879 (0.170) | 0.999 (0.085) | 0.968 (0.105) | |||

0.954 | 0.046 | 0.990 | 0.010 | |||

Notes: The number in parentheses is robust standard error proposed by Bollerslev and Wooldridge [

The smoothed probability of low correlation regime for S&P 500 and NIKKEI 225 (2000.1.3 - 2011.6.29). This figure plots the smoothed probability estimated by range-based Markov-switching dynamic conditional correlation model

^{8}. The correlations estimated by the range-based MS-DCC are less volatile than those estimated by the range-based DCC for the S&P 500, but the estimated results are the opposite for the NIKKEI 225. This finding indicates that the use of range data could lead to correlation estimates with a wider range compared to those produced using return data. Furthermore, considering a regime-switching method for the range-based DCC model could make the correlation estimates more flexible, as the regime- switching method can capture reactions to variation with greater precision.

^{9}. In the case of given parameters for the VaR estimate, the initial value and the quantile of normal distribution are assumed to be ^{10}. However, it is not appropriate to make a conclusion with regard to out-of-sample performance with these empirical results. So far, the use of variance as the proxy for risk has two flaws: trading position and the proxy chosen. It is necessary to discuss the hedging performance of different trading positions to compare hedging methods, as the short and long positions present different hedging performance (see for instance, Cotter and Hanly [^{11}. Hedgers are concerned with negative losses; therefore, this study uses semi-variance as an alternative risk proxy to replace the variance variable.

. In Panel A, all of the effectiveness criteria for the short hedger indicate that the range- based MS-DCC model is the most effective hedging strategy for the S&P 500. However, the naïve hedging strategy shows the best performance for the NIKKEI 225. This result from Panel A of Table 5 is almost identic- al to that from Panel B of Table 4, as both of these calculations are based on short hedgers. Comparing Panel A of Table 5 with Panel B of Table 4, the VaR figures calculated from the semi-standard deviation for each hedg- ing strategy are larger than the corresponding half-VaR figures estimated using the standard deviation

The constant OLS, return-based MS-DCC, range-based DCC and MS-DCC hedge ratios for S&P 500 and NIKKEI 225 (2000.1.3-2011.6.29). This figure plots four hedge ratios estimated from constant OLS (HR_OLS), return-based MS-DCC (HR_MSDCCG), range-based DCC (HR_DCCC) and MS-DCC (HR_ MSDCCC) models

This finding indicates that using semi-variance to substitute for the variance is meaningful, as the hedge portfo- lio has a significantly asymmetric return distribution. In terms of long hedgers, the empirical results mostly in- dicate the outperformance of the range-based MS-DCC model over competing models for both stock indices, as shown in Panel B of

. Hedging effectiveness of range-based Markov switching dynamic conditional correlation model against the alter- native hedge ratio models for daily spot and futures of S&P 500 and NIKKEI 225 (2000.1.3-2011.6.29)

S&P 500 | NIKKEI 225 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean return | Variance | VI(%) | Daily utility | VaR(5%) | Mean return | Variance | VI(%) | Daily utility | VaR(5%) | ||

Panel A: In-sample hedging effectiveness | |||||||||||

Unhedged | −0.0037 | 1.8534 | N/A | −7.4173 | −2,239,521 | −0.0235 | 2.6139 | N/A | −10.4791 | −2,659,567 | |

Naïve | 0.0004 | 0.0781 | 95.7861 | −0.3120 | −459,764 | <0.0001 | 0.2826 | 89.1886 | −1.1304 | −874,446 | |

OLS | 0.0002 | 0.0766 | 95.8670 | −0.3062 | −455,371 | −0.0020 | 0.2619 | 89.9805 | −1.0496 | −841,916 | |

DCC-G | −0.0010 | 0.0795 | 95.7106 | −0.3190 | −463,773 | −0.0048 | 0.2629 | 89.9422 | −1.0564 | −843,400 | |

DCC-C | 0.0003 | 0.0752 | 95.9426 | −0.3005 | −451,182 | 0.0006 | 0.2549 | 90.2483 | −1.0190 | −830,552 | |

MS-DCC-G | −0.0010 | 0.0794 | 95.7160 | −0.3186 | −463,670 | −0.0028 | 0.2490 | 90.4740 | −0.9988 | −820,873 | |

MS-DCC-C | 0.0007 | 0.0749 | 95.9588 | −0.2989 | −450,144 | 0.0011 | 0.2489 | 90.4778 | −0.9945 | −820,776 | |

Panel B: Out-of-sample hedging effectiveness | |||||||||||

Unhedge | 0.0731 | 1.1003 | N/A | −4.3281 | −1,725,564 | −0.0068 | 2.0418 | N/A | −8.1740 | −2,350,555 | |

Naïve | 0.0001 | 0.0353 | 96.7918 | −0.1411 | −308,877 | 0.0002 | 0.0749 | 96.3317 | −0.2994 | −450,337 | |

OLS | 0.0020 | 0.0350 | 96.8190 | −0.1380 | −307,844 | −0.0004 | 0.0971 | 95.2444 | −0.3888 | −512,638 | |

DCC-G | 0.0017 | 0.0344 | 96.8736 | −0.1359 | −305,100 | −0.0002 | 0.0821 | 95.9790 | −0.3286 | −471,281 | |

DCC-C | 0.0009 | 0.0346 | 96.8554 | −0.1375 | −305,866 | 0.0023 | 0.0835 | 95.9105 | −0.3317 | −475,393 | |

MS-DCC-G | −0.0034 | 0.0392 | 96.4373 | −0.1602 | −325,763 | −0.0003 | 0.0790 | 96.1309 | −0.3163 | −462,437 | |

MS-DCC-C | 0.0002 | 0.0334 | 96.9645 | −0.1334 | −300,678 | 0.0003 | 0.0787 | 96.1456 | −0.3145 | −461,597 |

Notes: VI is the variance improvement of unhedged model against the other competing models, and it is calculated as: [Var(unhedged)- Var(model_{i})]/Var(unhedged). Daily utility is calculated by the mean-variance utility function and the coefficient of risk aversion is 4. VaR(5%) is the value at risk.

. Effectiveness short/long hedger of range-based Markov switching dynamic conditional correlation model against the alternative hedge ratio models for daily spot and futures of S&P 500 and NIKKEI 225 (2000.1.3-2011.6.29)

S&P 500 | NIKKEI 225 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean return | Semi- variance | Semi-VI (%) | Semi-daily utility | VaR(5%) | Mean return | Semi- Variance | Semi- VI(%) | Semi-daily utility | VaR(5%) | ||||||

Panel A: Short hedger | |||||||||||||||

Unhedged | 0.0731 | 0.5939 | N/A | −2.3024 | −1,194,596 | −0.0068 | 1.1538 | N/A | −4.6221 | −1,773,792 | |||||

Naïve | 0.0001 | 0.0168 | 97.1712 | −0.0670 | −213,003 | 0.0002 | 0.0416 | 96.3945 | −0.1662 | −335,294 | |||||

OLS | 0.0020 | 0.0169 | 97.1544 | −0.0657 | −212,001 | −0.0004 | 0.0547 | 95.2591 | −0.2194 | −385,278 | |||||

DCC-G | 0.0017 | 0.0166 | 97.2049 | −0.0647 | −210,223 | −0.0002 | 0.0483 | 95.8138 | −0.1933 | −361,636 | |||||

DCC-C | 0.0009 | 0.0167 | 97.1881 | −0.0660 | −211,793 | 0.0023 | 0.0474 | 95.8918 | −0.1874 | −355,963 | |||||

MS-DCC-G | −0.0034 | 0.0212 | 96.4304 | −0.0883 | −243,103 | −0.0003 | 0.0456 | 96.0478 | −0.1826 | −351,438 | |||||

MS-DCC-C | 0.0002 | 0.0162 | 97.2722 | −0.0647 | −209,414 | 0.0003 | 0.0455 | 96.0565 | −0.1816 | −350,515 | |||||

Panel B: Long hedger | |||||||||||||||

Unhedge | 0.0731 | 0.5939 | N/A | −2.3024 | −1,194,596 | 0.0068 | 1.1538 | N/A | −4.6221 | −1,773,792 | |||||

Naïve | −0.0001 | 0.0184 | 96.9018 | −0.0737 | −223,263 | −0.0002 | 0.0332 | 97.1226 | −0.1330 | −299,933 | |||||

OLS | −0.0020 | 0.0180 | 96.9692 | −0.0741 | −222,866 | 0.0004 | 0.0422 | 96.3425 | −0.1683 | −337,445 | |||||

DCC-G | −0.0017 | 0.0177 | 97.0197 | −0.0726 | −220,763 | 0.0002 | 0.0336 | 97.0879 | −0.1344 | −301,501 | |||||

DCC-C | −0.0009 | 0.0178 | 97.0029 | −0.0721 | −220,285 | −0.0023 | 0.0359 | 96.8885 | −0.1460 | −314,057 | |||||

MS-DCC-G | 0.0034 | 0.0179 | 96.9860 | −0.0682 | −216,717 | 0.0003 | 0.0333 | 97.1139 | −0.1329 | −299,909 | |||||

MS-DCC-C | −0.0002 | 0.0171 | 97.1207 | −0.0686 | −215,350 | −0.0003 | 0.0331 | 97.1312 | −0.1327 | −299,584 | |||||

Notes: Semi-variance denotes the variability of returns below the mean return. Semi-VI is the semi-variance improvement of unhedged model against the other competing models, and it is calculated as: [Semi-Var(unhedged)-Semi-Var(model_{i})]/Semi-Var(unhedged). Semi-daily utility is calculated by the mean-semi-variance utility function and the coefficient of risk aversion is 4. VaR(5%) is the value at risk calculated by the semi-standard devia- tion.

In this study, a new range-based Markov-switching dynamic conditional correlation model is proposed to ad- dress minimum variance hedging for futures. Under this specification, the range-based MS-DCC model ad- dresses flaws of the linear functional forms of the conventional conditional covariance estimation method. For the empirical study, spot and futures data of the value-weighted S&P 500 index and the equal-weighted NIKKEI 225 index are collected to estimate the range-based MS-DCC model. The estimated results show that the dy- namic correlation process is derived by both the low- and high-correlation states by means of an estimated en- dogenous transition probability for both stock indices. This finding indicates that incorporating a regime- switching mechanism into the dynamic correlation process can show more realistic variation in correlation pat- terns. In addition, the calculated transition period shows that the frequency of switching from high to low corre- lation is lower than that of the reverse. The graph of smoothed probability clearly shows that important financial events lead correlation to move to a high regime, especially in the case of the S&P 500. We believe that this ad- vantage can lead to outperformance in minimum-variance hedging. This study introduces several different cor- relation models to calculate the minimum-variance hedging ratio and then compares their hedging effectiveness in terms of three criteria. Overall, the in-sample and out-of-sample performance indicates that the use of the range-based MS-DCC model for hedging leads to superior variance (semi-variance) improvement and greater economic benefits.