stock market volatility and the forecasting performance of stock index futures

Copyright © 2008 John Wiley & Sons, Ltd.

Stock Market Volatility and the Forecasting Performance of Stock Index Futures

JANCHUNG WANG*National Kaohsiung First University of Science and Technology, Taiwan, ROC

ABSTRACTThis study attempts to apply the general equilibrium model of stock index futures with both stochastic market volatility and stochastic interest rates to the TAIFEX and the SGX Taiwan stock index futures data, and compares the predictive power of the cost of carry and the general equilibrium models. This study also represents the fi rst attempt to investigate which of the fi ve volatility estimators can enhance the forecasting performance of the general equilibrium model. Additionally, the impact of the up-tick rule and other various explana-tory factors on mispricing is also tested using a regression framework. Overall, the general equilibrium model outperforms the cost of carry model in forecast-ing prices of the TAIFEX and the SGX futures. This fi nding indicates that in the higher volatility of the Taiwan stock market incorporating stochastic market volatility into the pricing model helps in predicting the prices of these two futures. Furthermore, the comparison results of different volatility estimators support the conclusion that the power EWMA and the GARCH(1,1) estimators can enhance the forecasting performance of the general equilibrium model compared to the other estimators. Additionally, the relaxation of the up-tick rule helps reduce the degree of mispricing. Copyright © 2008 John Wiley & Sons, Ltd.

key words forecasting performance of stock index futures; stochastic volatil-ity; GARCH; power EWMA; up-tick rule

INTRODUCTION

Until now, the cost of carry model has been the most widely used model for pricing stock index futures. This model was based on the assumption of perfect markets with non-stochastic interest rates. However, if interest rates are stochastic, Cox et al. (1981) show that futures and forward prices cannot be equal. Furthermore, the cost of carry model implies that market volatility should not have explanatory power for futures prices. However, Hill et al. (1988) argued that in a volatile market futures mispricing may be higher because stock index futures are able to digest the new information or a change of sentiment more quickly than a broad stock index can. Many previous studies have

Journal of ForecastingJ. Forecast. 28, 277–292 (2009)Published online 10 October 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/for.1101

* Correspondence to: Janchung Wang, Department of Money and Banking, National Kaohsiung First University of Science and Technology, 1 University Road, Yanchao, Kaohsiung 824, Taiwan, ROC. E-mail: [email protected]

278 J. Wang

Copyright © 2008 John Wiley & Sons, Ltd. J. Forecast. 28, 277–292 (2009) DOI: 10.1002/for

found signifi cant discrepancies between actual futures prices and theoretical prices estimated by the cost of carry model. For example, Cornell and French (1983a,b), Modest and Sunderesan (1983), and Gould (1988) found that, according to the cost of carry model, index futures contracts were underpriced relative to their theoretical values. Brenner et al. (1990) reported that the largest pricing errors in the Japanese index futures market were observed in the fi rst year of listing. Brailsford and Cusack (1997) found that the frequency of negative pricing errors was signifi cant for the cost of carry model in the individual share futures contracts listed on the Australian stock exchange. Gay and Jung (1999) observed that the market price of futures was persistently below the theoretical value of futures estimated by the cost of carry model in the Korean stock index futures market. Moreover, some studies also found a signifi cant correlation between index futures mispricing and index volatility. These related articles can be found, for example, in Yadav and Pope (1994) for the FTSE 100 index futures market, Fung and Draper (1999) for the Hang Seng index futures market, and Gay and Jung (1999) for the Korean stock index futures market.

Thus, from the above discussion, stock market volatility seems to be one of the important factors in determining stock index futures prices. Nevertheless, stock market volatility is excluded from the cost of carry model. Hemler and Longstaff (1991) incorporated both stochastic market volatility and stochastic interest rates into the pricing model, and followed the CIR (Cox et al., 1985a,b) frame-work, to develop a closed-form general equilibrium model of stock index futures (hereafter the Hemler–Longstaff model). Moreover, the empirical tests in the study of Hemler and Longstaff also indicated that their model is superior to the cost of carry model for the NYSE futures contract when the October 1987 observation is included in the sample. However, when October 1987 is excluded, the cost of carry model has lower pricing error. Hemler and Longstaff (1991) pointed out that the difference in results occurs because the October 1987 market variance estimate is much larger than the variance estimates for the other months. Thus, their model signifi cantly outperforms the cost of carry model in volatile markets. This implies that the Hemler–Longstaff model appears particularly suitable for stock markets with high price volatility. The Taiwan stock market is also characterized by high price volatility. Thus the question is whether the Hemler–Longstaff model with stochastic market volatility can effectively predict Taiwan stock index futures prices. This study attempts to apply the Hemler–Longstaff model to SGX (Singapore Exchange Limited) MSCI (Morgan Stanley Capital International) and TAIFEX (Taiwan Futures Exchange) Taiwan stock index futures data, and compares the predictive power of the cost of carry and the Hemler–Longstaff models using the in-sample and out-of-sample tests.

Next, for the Hemler–Longstaff model, the only variable that cannot be directly observed is the volatility of the underlying index. To accommodate time-varying volatility in index returns, a number of popular estimation approaches have been used, including the equally weighted moving average, the Garman and Klass (1980) estimator, the GARCH (Generalized Autoregressive Conditional Het-eroskedastic) model, the exponentially weighted moving average (EWMA), and the power EWMA of Guermat and Harris (2002). Another major task of this study is to further determine which of the fi ve volatility estimators can adequately capture the volatility of Taiwan stock prices and further enhance the forecasting performance of the Hemler–Longstaff model.

Fung and Draper (1999) found that relaxing the restrictions on short selling increases market effi ciency and reduces the degree and frequency of mispricing. Gay and Jung (1999) also concluded in their study of the Korean index futures market that short-sales restrictions do affect mispricing. Within the study period, two major changes in the up-tick rule occurred in Taiwan. Thus, the impact of the up-tick rule and other various explanatory factors on mispricing is also tested using a regres-sion framework.

Stock Market Volatility and Stock Index Futures 279


PRICING MODELS OF STOCK INDEX FUTURES

One common way of calculating futures prices is to use the cost of carry model. If interest rates and continuous dividend yields are non-stochastic, then in the absence of taxes and other market imper-fections the cost of carry model can be written as

F S et tr q T t= −( ) −( ) (1)

where Ft is the theoretical futures price at time t; St denotes the current stock index; r represents the risk-free interest rate; q is the dividend yield; and T − t denotes the time to expiration.

If the underlying stock index pays irregular lumpy dividends, under the concept of continuous compounding, the cost of carry model will be

F S D et t tr T t= −( ) −( ) (2)

where

D

S d w

pet

t i i

i t

r t t

i

ni=

−( )

=∑

,1

where Dt is the sum of the present values of all cash dividends distributed by the underlying com-ponent stocks at time t during the life of the futures contract; di is the cash dividend per share for stock i during the life of the futures contract; wi is the weight of stock i in the index; ti is the time that stock i pays the cash dividend; and pi,t is the price of stock i at time t.

In Taiwan, the cash dividend payouts are relatively lumpy.1 Thus, substituting the known risk-free rate r and the lumpiness of cash dividend Dt, together with the current index price St and time to maturity T − t, into the cost of carry model (2), the theoretical prices of the TAIFEX and the SGX Taiwan stock index futures can then be obtained.

The cost of carry model was developed under the assumption of non-stochastic interest rates. Moreover, the cost of carry model implies that market volatility should not have explanatory power for futures prices. Hemler and Longstaff (1991) followed the framework of CIR and developed a closed-form general equilibrium model of stock index futures with both stochastic market volatility and stochastic interest rates. The general equilibrium model of stock index futures is given by equa-tion (15) in Hemler and Longstaff (1991). The natural logarithm of the general equilibrium model yields the following regression equation:

L r Vt t t t= + + +α β λ ε (3)

where Lt = ln(Ft eqt/St); Ft is the theoretical futures price; t denotes the time to expiration (i.e., t = T − t); and Vt represents the variance of stock index returns. With a lumpiness of cash dividends, Lt will be ln(Ft/(St − Dt)). Equation (3) implies that the logarithm of the dividend-adjusted futures/spot price ratio can be regarded as a linear function of both the risk-free rate and the variance of stock

1 In Taiwan, cash dividends for the underlying component stocks are mostly paid only once per year, and are concentrated in July.

280 J. Wang


index returns. Substituting the data rt, Vt and the coeffi cient estimates a, b, and l into the regression model (3) to generate the Lt estimate, the theoretical futures price can then be inferred from Lt.

Additionally, if a = 0, b = T − t, and l = 0 (that is, market volatility should not have explanatory power) hold, equation (3) reduces to

L rt t= τ (4)

Equation (4) can be rearranged to demonstrate (1), the cost of carry model. Thus, the cost of carry model can be regarded as a special case of the Hemler–Longstaff model.

DATA AND METHODOLOGY

Data and basic statisticsThe SGX MSCI and TAIFEX Taiwan stock index futures contracts began trading on January 9, 1997 and July 21, 1998, respectively. The TAIFEX futures contract is based on the Taiwan capital-ization weighed index, which is a market-value weighted index comprising all of the common stocks (currently about 688) listed on the Taiwan Stock Exchange. The SGX futures contract uses the MSCI Taiwan index, which is composed of 99 representative stocks listed on the Taiwan Stock Exchange. The MSCI Taiwan index closely correlates with the Taiwan capitalization weighed index. Table I describes the main features of the two futures contracts and their underlying indexes.

For the two Taiwan stock index futures, the nearest maturity contracts all have signifi cant trading volume. To reduce thin trading problems, only the near-month contracts were considered in this

Table I. Main features of the TAIFEX and the SGX MSCI Taiwan stock index futures contracts

TAIFEX futures SGX futures

1. Opening date July 21, 1998 January 9, 19972. Underlying index Taiwan capitalization

weighed indexMSCI Taiwan index

3. Contract size Futures price times NT$200 Futures price times US $1004. Contract months Spot month, next calendar

month, and next three-quarter months

2 nearest serial months and 4 quarterly months on a March, June, September, and December cycle

5. Minimum price change

1 index point = NT$200 per contract

0.10 index points = US $10 per contract

6. Price limits 7% of the previous day’s settlement price

Initial price limit: 7% of the previous day’s settlement price (last for 10 minutes)

Intermediate daily price limit: 10% of the previous day’s settlement price (last for 10 minutes)

Final daily price limit: 15% of the previous day’s settlement price

7. Last trading day Third Wednesday of the delivery month

Second last business day of the contract month

8. Settlement Cash Cash

Source: Taiwan Futures Exchange (TAIFEX) and Singapore Exchange Limited (SGX).



study. The sample period covers September 4, 1998 to December 30, 2005. Moreover, to reduce the asynchroneity problem between the spot index and the futures prices, the transaction time of each daily observation for the index futures had to match with (or, at least, was nearest to) the transaction time of each daily observation for the spot index.2

30-day commercial paper rates in the secondary market are used as the proxy of risk-free interest rates. The dividend data of the underlying component stocks in the index come from the Taiwan Economic Journal and the Taiwan Security Exchange.

Table II lists descriptive statistics for the four return series. The daily standard deviation is 1.64% for the Taiwan capitalization weighed index and 1.77% for the MSCI Taiwan index over the entire sample period.3 Additionally, the volatility of futures returns is higher than that of spot returns for the two markets. The kurtosis statistic is consistently greater than the standard normal distribution value of 3, indicating that all four return series display excess kurtosis. The Jarque–Bera statistics for the four return series are statistically signifi cant at the 1% level. Thus, as is commonly found for daily equity returns, normality is signifi cantly rejected.

Parameter estimation of the Hemler–Longstaff modelThe only variable that cannot be directly observed for the Hemler–Longstaff model is the variance of stock index returns (Vt). To account for time-varying volatility in index returns, various popular estimation approaches have been used, including the equally weighted moving average, the Garman and Klass (1980) estimator, the GARCH model, the EWMA estimator, and the power EWMA of Guermat and Harris (2002). This study determines which of the fi ve volatility estimators can enhance the forecasting performance of the Hemler–Longstaff model.

Table II. Descriptive statistics

Mean SD Skewness Kurtosis Jarque–Bera Sample size

Spot returnsTAIFEX −0.00010 0.0164 0.0073 4.5576 188.339*** 1864SGX −0.00005 0.0177 0.0938 4.2809 130.086*** 1864

Futures returnsTAIFEX −0.00011 0.0187 −0.0460 5.5312 497.987*** 1864SGX −0.00004 0.0214 −0.1861 7.0829 1304.772*** 1864

Note: Asterisks denote signifi cance at the ***1% level.

2 From July 21, 1998 to December 31, 2000, the trading session for the stocks listed in the Taiwan Stock Exchange was from 9:00 a.m. to 12:00 noon, while the trading session for the TAIFEX futures was from 9:00 a.m. to 12:15 p.m. From January 1, 2001, the end of the trading session was extended to 1:30 p.m. for the stock market and to 1:45 p.m. for the TAIFEX futures market. Therefore, for each day in the sample period, the TAIFEX futures market closes 15 minutes later than the stock market. This study uses futures prices recorded closest to 12:00 noon (or 1:30 p.m.) for matching with closing price of the stock market at 12:00 noon (or 1:30 p.m.). As for the SGX futures, we repeat the same procedure.3 The number of trading days per year is about 252 for the entire sample period. Using 252 trading days per year, the esti-mated standard deviation per annum is 26.03% for the Taiwan capitalization weighed index and 28.10% for the MSCI Taiwan index. In the same period, the standard deviation per annum is approximately 20% for the S&P 500 stock index and 22% for the Nikkei 225 index. Obviously, the two Taiwanese indexes have higher volatility than the S&P 500 stock index and the Nikkei 225 index.

282 J. Wang


The simplest, and the most commonly employed in practice, is the equally weighted moving average estimator, given by

σ t ii t n

t

nR R2 2

11

1=

−−( )

= −

−

∑ (5)

where s 2t is the variance estimate on day t; Ri is the spot index return on day i; R denotes the mean

return of spot index; and n is the length of the period set to a value of 20 trading days, as suggested by Chiras and Manaster (1978). The variance per annum (Vt) should be calculated from the variance per trading day (s 2

t ) using the formula

Vt t= × ( )σ 2 number of trading days per annum (6)

Bollerslev (1986) extended the ARCH(p) model for volatility forecasting to the commonly used generalized autoregressive conditional heteroskedasticity (GARCH(p, q)) model. The GARCH model possesses the advantage of incorporating heteroskedasticity into the estimation procedure for fi nancial and economic data. According to Bollerslev (1987), GARCH(1, 1) specifi ca-tion adequately fi ts many economic time series. The GARCH(1, 1) model was tentatively specifi ed as

Rt t= +µ ε (7)

εt t tN hΩ − ( )1 0∼ , (8)

h ht t t= + +− −α α ε α0 1 12

2 1 (9)

where et denotes an error term; Ω t−1 represents the information set on day t − 1; ht is the conditional variance on day t; and a0, a1, and a2 denote the parameters from the GARCH(1, 1) estimation. Maximum likelihood estimation is used to estimate the parameters in equations (7) and (9) for the GARCH model.

A related method of estimating the conditional variance of returns is the EWMA estimator, which is as follows:

σ λσ λt t tR21

21

21= + −( )− − (10)

where l is the decay factor. In the widely used Riskmetrics value-at-risk package of JP Morgan, l is set to 0.94 for daily data (see JP Morgan, 1996). This study takes the value of l as 0.94.

Traditionally, variances have been estimated using squared close-to-close daily return. Garman and Klass (1980) developed a variance estimator which uses the daily high, low, and opening prices. They demonstrated that this estimator is considerably more effi cient than the close-to-close estima-tors, due to incorporating the dispersion of prices over the entire day, rather than merely a snapshot price at the end of the day. The Garman and Klass (1980) estimator can be expressed as

σ t t t t t t t t t t tH O L O C O H O L O2 20 511 0 019= −( ) − −( )[ ] − −( ) −( ) + −( )[ ]−. .

−( ) −( ) − −( )2 0 383 2H O L O C Ot t t t t t. (11)



where Ct, Ot, Ht, and Lt represent the closing price, opening price, daily high, and daily low on day t, respectively; and all prices are expressed in logarithmic form.

Guermat and Harris (2002) argued that all of the approaches to estimating the variance of returns as described above are based on sample variance of returns. When the distribution of the data is leptokurtic, these estimators place excessive weight on extreme observations. Hence, Guermat and Harris (2002) proposed a general power EWMA estimator that is robust to the leptokurtosis of the distribution of returns. Furthermore, Harris and Shen (2003) used the power EWMA to estimate the optimal hedge ratio using the FTSE 100 index futures contract. They demonstrated that the power EWMA estimator yields better reduction in the hedged portfolio variance compared to the standard rolling window and EWMA estimators, particularly during periods of high kurtosis. The power EWMA estimator of the variance is given by

σ λσ λtk

tk

tkg k R= + −( ) ( )− −1 11 (12)

g k kk

k

k

( ) =( )( )

ΓΓ

3

1

2

(13)

where Γ(·) is the gamma function, and k is a parameter that controls the kurtosis of the distribution. When k = 2, the power EWMA estimator given by (12) coincides with the EWMA estimator given by (10). When k < 2, the power EWMA estimator is more effi cient when the conditional distribution of returns is leptokurtic. This study considers k of 1.25.4 Meanwhile, the decay factor l is set to 0.94.

Comparison of two alternative pricing modelsThis study compares the relative forecasting performance of the cost of carry model and the Hemler–Longstaff model (fi ve volatility estimators) in two ways. First, following the work of Hemler and Longstaff (1991), this study examines the restrictions imposed on futures prices by both the cost of carry and Hemler–Longstaff models. The cost of carry model implies that spot volatility should not have explanatory power for futures prices, and that the coeffi cient of the risk-free rate term should equal the average contract maturity in equation (3). Thus, if the cost of carry model holds, then a = 0, b = T − t, and l = 0 in equation (3). In contrast, according to the Hemler–Longstaff model, the predicted signs of the regression coeffi cients would be a ≠ 0, b > 0, and l ≠ 0.

Second, an alternative approach to assessing model forecasting performance is via the most popular accuracy measures, mean percentage errors (MPE) and mean absolute percentage errors (MAPE). MPE and MAPE are defi ned as follows:

MPEAF

AF= −

=

∑1

1n

Ft t

tt

n

(14)

MAPEAF

AF= −

=∑1

1n

Ft t

tt

n

(15)

4 This study also considers k = 1.00, 1.50, 1.60, and 1.75. The forecasting performance for k = 1.25 is slightly better than for the other k values.

284 J. Wang


where AFt is the actual futures price on day t; n represents the number of observations; and Ft repre-sents the theoretical futures price based on the cost of carry or Hemler–Longstaff models on day t.

Additionally, to compare the performance between the cost of carry model and the Hemler–Longstaff model, as well as comparing the performance of different volatility estimators, the t-test was used to test whether the MAPE statistics generated from each model and each volatility estima-tor were signifi cantly different.

EMPIRICAL RESULTS

Table III presents the parameter estimates for the GARCH(1, 1) model. The results indicate that the parameters a0, a1, and a2 are all positive and signifi cantly different from zero, suggesting a strong heteroskedastic effect of price volatility for the two Taiwanese stock markets.

Results of testing the specifi cations of the two pricing modelsAs mentioned previously, the cost of carry model implies that a = 0, b = T − t, and l = 0 in equation (3). In contrast, if the Hemler–Longstaff model holds, the coeffi cients of equation (3) would be a ≠ 0, b > 0, and l ≠ 0. To control for autocorrelation, the regression coeffi cients of equation (3) are estimated with an iterative Cochrane–Orcutt procedure.

Table IV summarizes the regression results of testing the specifi cations of the two pricing models. First, if the Hemler–Longstaff model holds, the coeffi cient (a) should not equal zero. As shown in Table IV, all coeffi cients (a) are statistically different from zero. This fi nding supports the Hemler–Longstaff model and is contrary to the cost of carry model. Next, the cost of carry model implies that the b coeffi cient should equal the average contract maturity. The average maturity during the sample period is 0.0414 years for the TAIFEX futures and 0.0394 years for the SGX futures. Table IV reveals that all the estimates of b signifi cantly exceed the average maturity of the contract for the TAIFEX futures. For example, the difference between the b value for the Hemler–Longstaff model with power EWMA (HL-PE) and the average maturity is 0.1046, and thus is signifi cantly positive with a t-statistic of 9.43. As in the case of the SGX futures, the coeffi cients (b) differ signifi cantly from the average maturity under the Hemler–Longstaff model with GARCH(1, 1) (HL-GARCH), the Hemler–Longstaff model with Garman–Klass estimator (HL-GK), and the

Table III. Maximum likelihood estimates of the GARCH(1, 1) model

Parameter TAIFEX SGX

m 0.00033 0.00027(1.067) (0.769)

a0 2.69E−06*** 2.81E−06***

(3.333) (3.078)

a1 0.07898*** 0.06832***

(8.251) (7.519)

a2 0.91386*** 0.92467***

(88.922) (95.036)

Log-Likelihood 5265.86 5104.82

Notes: Asterisks denote signifi cance at the ***1% level. Numbers in parentheses are t values.



Hemler–Longstaff model with moving average (HL-MA). Thus, in general, the results seem to be contrary to the cost of carry model. Finally, if the Hemler–Longstaff model determines futures prices, spot volatility should have explanatory power. Table IV indicates that all of the l coeffi cients, except for the SGX futures under the HL-MA estimator, differ signifi cantly from zero. This fi nding is consistent with the Hemler–Longstaff model. Overall, the regression results support the specifi cation of the Hemler–Longstaff model for both the TAIFEX and SGX futures contracts.

Forecasting performance of the two pricing modelsTable V lists in-sample comparisons regarding the model pricing errors for the two futures markets. From the ‘percentage error’ column, both models overprice the TAIFEX and SGX futures contracts. Meanwhile, Table V also shows that for every volatility estimator the magnitude of MPE of the Hemler–Longstaff model is clearly smaller than that of the cost of carry model. For example, the cost of carry model overprices the SGX futures contract by an average of −0.1216%, compared to a minimum average overestimate of −0.0024% for HL-PE and HL-GARCH.

Table V also compares the MAPEs of the cost of carry and Hemler–Longstaff models. This study further uses the t-test to examine whether the MAPE statistics generated from each model and each volatility estimator differ signifi cantly. Table VI lists the results. For every volatility estimator, the Hemler–Longstaff model outperforms the cost of carry model in forecasting prices of the TAIFEX

Table IV. Cost of carry model versus Hemler–Longstaff model

Equation (3): Lt = a + brt + lVt + et

a b l R2 DW

TAIFEXHL-PE −0.0015*** 0.1460*** −0.0319*** 0.483 2.04

(−3.61) (12.91) (−6.01)HL-GARCH 0.0019*** 0.1400*** −0.0253*** 0.482 2.03

(−4.69) (12.82) (−6.02)HL-EWMA −0.0014*** 0.1647*** −0.0403*** 0.482 2.02

(−3.44) (14.36) (−7.05)HL-GK −0.0027*** 0.1263*** −0.0229* 0.482 2.04

(−7.29) (11.96) (−1.86)HL-MA −0.0027*** 0.1503*** −0.0153* 0.481 2.04

(−2.78) (5.17) (−1.82)SGXHL-PE −0.0021*** 0.0401*** 0.0105** 0.250 2.04

(−4.80) (3.65) (2.24)HL-GARCH −0.0011*** 0.0579*** −0.0072* 0.249 2.03

(−2.79) (5.40) (−1.81)HL-EWMA −0.0019*** 0.0429*** 0.0082* 0.248 2.02

(−4.39) (3.89) (1.68)HL-GK −0.0010*** 0.0638*** −0.0188*** 0.250 2.03

(−2.89) (6.24) (−5.41)HL-MA −0.0016** 0.0724*** −0.0047 0.241 2.04

(−2.29) (3.54) (−0.81)

Note: HL-PE, HL-GARCH, HL-EWMA, HL-GK, and HL-MA represent the Hemler–Longstaff model with power EWMA, Hemler–Longstaff model with GARCH(1,1), Hemler–Longstaff model with EWMA, Hemler–Longstaff model with Garman–Klass estimator, and Hemler–Longstaff model with moving average, respectively. Numbers in parentheses are t values. For two-tailed test, asterisks denote signifi cance at the *10%, **5%, and ***1% levels. DW denotes the Durbin-Watson statistic.

286 J. Wang


and the SGX futures. As expected, the incorporation of stochastic market volatility helps in predict-ing the TAIFEX and SGX futures prices because the two underlying stock indexes have the higher volatility.

Another main task of this study was to decide which of the fi ve volatility estimators can capture the volatility of Taiwan stock prices and further improve the forecasting performance of the Hemler–Longstaff model. Table V lists the MAPE of each of fi ve volatility estimators examined here. Using the TAIFEX futures, the Hemler–Longstaff model with moving average (HL-MA) performs worst. The MAPE of HL-MA is 0.5054%, and is signifi cantly larger than those of the other estimators. Next, the MAPE when the Hemler–Longstaff model with EWMA (HL-EWMA) is used is 0.4568%. The reduction in MAPE (relative to HL-MA), shown in Table VI, is statistically signifi cant at the 1% level, based on a t-test of the mean difference. Finally, from Tables V and VI, the Power EWMA

Table V. In-sample comparisons for the pricing errors of the cost of carry and Hemler–Longstaff models

n TAIFEX SGX

Percentage error Absolute percentage error


Mean (%) SD (%) Mean (%) SD (%) Mean (%) SD (%) Mean (%) SD (%)

CCM 1864 −0.0800 0.8515 0.6259 0.5827 −0.1216 0.8061 0.5183 0.6292HL-PE 1862 −0.0019 0.6119 0.4328 0.4324 −0.0024 0.7002 0.4320 0.5438HL-GARCH 1862 −0.0015 0.6123 0.4373 0.4389 −0.0024 0.7005 0.4289 0.5418HL-EWMA 1862 −0.0017 0.6293 0.4568 0.4561 −0.0025 0.7097 0.4630 0.5588HL-GK 1864 −0.0014 0.6294 0.4786 0.4781 −0.0030 0.7098 0.4819 0.5890HL-MA 1844 −0.0033 0.8120 0.5054 0.5120 −0.0031 0.7935 0.4946 0.6039

Note: CCM represents the cost of carry model; n represents the number of observations.

Table VI. In-sample results of statistical tests for differences in MAPE between the pricing models

TAIFEX SGX

CCM vs. HL-PE 11.594*** 4.580***CCM vs. HL-GARCH 11.266*** 4.676***CCM vs. HL-EWMA 9.958*** 2.864***CCM vs. HL-GK 8.514*** 1.843*CCM vs. HL-MA 6.757*** 1.183HL-MA vs. HL-PE 4.701*** 3.405***HL-MA vs. HL-GARCH 4.385*** 3.501***HL-MA vs. HL-EWMA 3.077*** 1.668HL-MA vs. HL-GK 1.657* 0.655HL-GK vs. HL-PE 3.097*** 2.770***HL-GK vs. HL-GARCH 2.777*** 2.867***HL-GK vs. HL-EWMA 0.774 1.013HL-EWMA vs. HL-PE 1.661 1.793*HL-EWMA vs. HL-GARCH 1.342 1.891*HL-GARCH vs. HL-PE 0.315 −0.096

Note: For two-tailed test, asterisks denote signifi cance at the *10%, **5%, and ***1% levels, respectively.



and GARCH estimators tested in this study were the most successful in reducing the pricing error. The mean absolute percentage errors of HL-GARCH and HL-PE are 0.4373% and 0.4328%, respec-tively. As illustrated in Table VI, the MAPEs of HL-GARCH and HL-PE do not differ signifi cantly. Meanwhile, HL-GARCH and HL-PE have smaller MAPEs compared to the other volatility estima-tors. Regarding the SGX futures, similarly, the GARCH and the power EWMA estimators offered lower absolute percentage errors than the other estimators. Thus, the results of the SGX futures are consistent with those of the TAIFEX futures.

In-sample comparisons provide us with a broad picture of relative performance of the two pricing models. However, these resultant forecasting performances are not practical since investors do not know future return distributions of spot index and index futures. Thus, this study also conducts out-of-sample comparisons of forecasting performance. For an out-of-sample test, the sample period is broken into two approximately equal length subsamples. The fi rst is the estimation period, which uses the sample period from September 4, 1998 through April 10, 2002, and the second is the fore-casting period, which uses the sample period from April 11, 2002 through December 30, 2005. Moreover, the empirical implementation of the out-of-sample performance involves a three-stage procedure. First, for the Hemler–Longstaff model, the three parameters a, b, and l in regression model (3) are estimated using the estimation period. Next, the predicted futures prices are based on the forecasting period. Finally, the forecasting accuracy of the cost of carry model and the Hemler–Longstaff model (fi ve volatility estimators) is based on MPE and MAPE values. Table VII lists the out-of-sample MPE and MAPE values of the two pricing models. In line with the in-sample results, the Hemler–Longstaff model also outperforms the cost of carry model for both the TAIFEX and SGX futures contracts.

Table VII also lists the out-of-sample MAPE values of each of fi ve volatility estimators examined here. For the TAIFEX futures market, the GARCH model had the lowest MAPE value (0.3594%), followed closely by the power EWMA (0.3632%), the EWMA (0.3860%), and the Garman–Klass estimator (0.4077%). The moving average estimator had a much higher MAPE value (0.4206%). This work further uses the t-test to examine whether the out-of-sample MAPE statistics generated from each estimator are signifi cantly different. The results are reported in Table VIII. The GARCH model yields a signifi cantly lower MAPE value compared to the other estimators (except for the power EWMA). Additionally, there is no signifi cant difference in out-of-sample performance

Table VII. Out-of-sample comparisons for the pricing errors of the cost of carry and Hemler–Longstaff models

N TAIFEX SGX



Mean (%) SD (%) Mean (%) SD (%) Mean (%) SD (%) Mean (%) SD (%)

CCM 932 −0.2166 0.6273 0.4916 0.4456 −0.1913 0.6113 0.4193 0.4840HL-PE 932 −0.0075 0.4572 0.3632 0.3091 −0.0217 0.5040 0.3490 0.3209HL-GARCH 932 −0.0055 0.4528 0.3594 0.3064 −0.0237 0.4998 0.3360 0.3127HL-EWMA 932 −0.0073 0.4645 0.3860 0.3220 −0.0275 0.5112 0.3746 0.3373HL-GK 932 −0.0083 0.4673 0.4077 0.3438 −0.0316 0.5062 0.3852 0.3471HL-MA 932 −0.0096 0.6016 0.4206 0.3578 −0.0479 0.5975 0.4105 0.3821

Notes: CCM represents the cost of carry model; N is the number of out-of-sample observations.

288 J. Wang


between the GARCH and the power EWMA. In the case of the SGX futures market, like the fore-casting results of the TAIFEX futures market, the GARCH model ranks fi rst, with the smallest MAPE value (0.3360%), while the power EWMA estimator is a close second, with a MAPE value of 0.3490%.

Overall, from Tables V–VIII, both in-sample and out-of-sample results support the conclusion that, among the volatility estimators examined, the Hemler–Longstaff model with GARCH and the Hemler–Longstaff model with power EWMA provide the best forecasting performance for both two futures markets. Table II shows that the spot returns of both markets display excess kurtosis. Perhaps because as demonstrated by Guermat and Harris (2002), the power EWMA approach can better capture the leptokurtic distribution of returns, this approach helps enhance the forecasting perfor-mance of the Hemler–Longstaff model. Additionally, as indicated in Table III, the parameter estima-tion of the GARCH(1, 1) model shows high signifi cance of several parameters, suggesting that the GARCH(1, 1) structure captures the heteroskedastic effect of price volatility.5 Hence the forecasting performance of the Hemler–Longstaff model is also improved using the GARCH(1, 1) model.

Explanatory factors affecting the pricing errorTo further understand the relationship between the mispricing and various explanatory factors, the absolute percentage errors (APE) are regressed on the APEs of 1-day and 2-day lags, time to maturity (T − t), futures trading volume (Vol), and up-tick rule. The following regression thus is estimated:

APE APE APE Volt t t t tT t D= + + + −( ) + + +− −β β β β β β ε0 11 1 12 2 2 3 4 (16)

where D represents a dummy variable associated with the change in the up-tick rule.Gay and Jung (1999) observed that the market price of futures was consistently below the theo-

retical value estimated by the cost of carry model for the Korean stock index futures market. Thus, the APEs of the 1-day and 2-day lags are included to examine the persistence of mispricing. Table

5 Lin et al. (1999) indicated that the volatility of the Taiwan stock market has heteroskedastic and serially correlated effects.

Table VIII. Out-of-sample results of statistical tests for differences in MAPE between the pricing models

TAIFEX SGX

CCM vs. HL-PE 7.228*** 3.694***CCM vs. HL-GARCH 7.464*** 4.412***CCM vs. HL-EWMA 5.863*** 2.312**CCM vs. HL-GK 4.553*** 1.749*CCM vs. HL-MA 3.794*** 0.438HL-MA vs. HL-PE 3.704*** 3.758***HL-MA vs. HL-GARCH 3.965*** 4.602***HL-MA vs. HL-EWMA 2.192** 2.146**HL-MA vs. HL-GK 0.795 1.495HL-GK vs. HL-PE 2.935*** 2.334***HL-GK vs. HL-GARCH 3.199*** 3.212***HL-GK vs. HL-EWMA 1.403 0.666HL-EWMA vs. HL-PE 1.560 1.677*HL-EWMA vs. HL-GARCH 1.828* 2.561***HL-GARCH vs. HL-PE −0.268 −0.886

Note: For two-tailed test, asterisks denote signifi cance at the *10%, **5%, and ***1% levels, respectively.



IX reports the results of the regression in (16) for the TAIFEX and SGX futures. For the cost of carry model and the Hemler–Longstaff model under all fi ve volatility estimators, the positive and signifi cant coeffi cients on the APEs of the 1-day and 2-day lags suggest persistent mispricing during the sample period for both futures markets.

A longer term to maturity of the futures may create additional uncertainty regarding dividends and future market volatility. Hence, the absolute pricing error increases with the time to maturity. Previous studies (e.g., Cakici and Chatterjee, 1991; Yadav and Pope, 1994; Brailsford and Cusack, 1997; Fung and Draper, 1999; Gay and Jung, 1999) have found a positive correlation between APE and (T − t). As shown in Table IX, all of the estimated coeffi cients (b2) on the time to maturity are positive and signifi cant, indicating that the mispricing increases with time to maturity. These results are also consistent with the fi ndings of previous studies.

The third possible explanatory factor relates to futures trading volume. Bessembinder and Seguin (1992) demonstrated that active futures markets improve spot market liquidity. Thus, heavy futures trading volume may represent an effi cient market in which arbitrage opportunities do not occur. In this case, a negative relationship exists between pricing errors and futures trading volume. Con-versely, it is based on the argument that arbitrage signals may attract trading volume. Thus, a posi-tive relationship is expected between pricing errors and futures trading volume. Table IX shows that

Table IX. Regression results for the effect of various factors on the absolute percentage errors

Equation (16): APEt = b0 + b11APEt−1 + b12APEt−2 + b2(T − t) + b3Volt + b4D + et

b0 b11 b12 b2 b3 b4 R2

TAIFEXCCM 0.002*** 0.421*** 0.188*** 0.019*** −9.86E−09 −0.001*** 0.337

(7.13) (8.32) (4.53) (4.53) (−1.08) (−2.73)HL-PE 0.002*** 0.235*** 0.192*** 0.015*** −5.67E−09 −0.001*** 0.144

(8.57) (10.38) (8.47) (4.30) (−0.73) (−3.53)HL-GARCH 0.002*** 0.237*** 0.192*** 0.014*** −4.79E−09 −0.001*** 0.145

(8.48) (10.49) (8.47) (4.33) (−0.62) (−3.53)HL-EWMA 0.003*** 0.257*** 0.181*** 0.016*** −9.17E−09 −0.001*** 0.158

(8.67) (11.30) (7.96) (4.3) (−1.16) (−3.58)HL-GK 0.002*** 0.259*** 0.185*** 0.015*** −6.47E−09 −0.001*** 0.162

(8.54) (11.49) (8.20) (4.37) (−0.83) (−3.65)HL-MA 0.002*** 0.398*** 0.197*** 0.014*** −1.44E−08 −0.001*** 0.307

(8.05) (17.53) (8.68) (3.36) (−1.58) (−2.75)

SGXCCM 0.001*** 0.310*** 0.144*** 0.022*** 2.46E−08 −0.001*** 0.169

(5.18) (13.63) (6.36) (4.10) (1.62) (−2.91)HL-PE 0.002*** 0.196*** 0.145*** 0.011** 1.79E−08 −0.001*** 0.083

(7.45) (8.62) (6.39) (2.27) (1.57) (−3.43)HL-GARCH 0.002*** 0.194*** 0.145*** 0.011** 1.81E−08 −0.001*** 0.082

(7.42) (8.54) (6.42) (2.29) (1.58) (−3.48)HL-EWMA 0.003*** 0.250*** 0.083*** 0.010** 1.53E−08 −0.001*** 0.091

(7.93) (10.91) (3.64) (2.13) (1.34) (−3.43)HL-GK 0.003*** 0.252*** 0.084*** 0.010** 1.47E−08 −0.001*** 0.093

(7.95) (11.01) (3.66) (2.10) (1.29) (−3.47)HL-MA 0.002*** 0.294*** 0.115*** 0.017*** 1.71E−08 −0.001*** 0.138

(6.79) (12.79) (4.99) (3.22) (1.39) (−3.04)

Note: Numbers in parentheses are t values. Asterisks denote signifi cance at the **5% and ***1% levels, respectively.

290 J. Wang


the relationship between the absolute percentage error and futures trading volume is insignifi cantly negative for the TAIFEX futures but insignifi cantly positive for the SGX futures. These fi ndings are consistent across both pricing models. As discussed earlier, because of the existence of confl icting hypotheses regarding the sign of this variable, the empirical results are diffi cult to distinguish among the hypotheses.

Finally, exactly how the up-tick rule impacts the mispricing series is also tested. The up-tick rule increases transaction costs and reduces the convenience of index arbitrage. Additionally, Fung and Draper (1999) found that relaxing the restrictions on short selling increases market effi ciency and reduces the degree and frequency of mispricing. Within the study period, two major changes in short-sales restrictions occurred in Taiwan. The up-tick rule, which stipulated that all TAIFEX index stocks (currently about 688) could only be sold short at prices above those of their previous trade, was established on September 4, 1998. From May 16, 2005, 50 blue-chip stocks within the TSEC (Taiwan Stock Exchange Corporation) Taiwan 50 index were allowed to be sold short at prices below their previous trades.6 This study uses a dummy variable to distinguish the change in the up-tick rule, with D = 1 representing the period (from May 16, 2005 to December 30, 2005) with no up-tick rule on 50 blue-chip stocks, and 0 indicating otherwise. All coeffi cients (b4) are signifi cantly negative in Table IX. The result shows that the relaxation of the up-tick rule reduces the level of mispricing. This result is also consistent with the fi nding of Fung and Draper (1999).

CONCLUSIONS

The empirical implication of the Hemler–Longstaff model is that this model appears especially suit-able when stock markets have high price volatility. Notably, the Taiwan stock market has high price volatility. Thus, this study attempts to apply the Hemler–Longstaff model to the SGX and the TAIFEX Taiwan stock index futures data, and examines the power of the cost of carry and Hemler–Longstaff models in predicting the prices of the two futures. Furthermore, for the Hemler–Longstaff model, the only variable that cannot be directly observed is the volatility of the underlying index. This study also represents the fi rst attempt to investigate which of the fi ve volatility estimators can enhance the forecasting performance of the Hemler–Longstaff model. Additionally, the impact of the up-tick rule and other various explanatory factors on mispricing is also examined using a regres-sion framework.

The results of testing the two pricing models specifi cations support the Hemler–Longstaff model. Moreover, both in-sample and out-of-sample results indicate that the Hemler–Longstaff model out-performs the cost of carry model in forecasting prices of the TAIFEX and the SGX futures. Thus, as expected, in the higher volatility of the Taiwan stock market, the Hemler–Longstaff model with stochastic market volatility can accurately predict the prices of these two futures. Furthermore, the comparison results of different volatility estimators support the conclusion that the power EWMA and the GARCH(1, 1) estimators can provide better forecasts of future market volatility and improve the forecasting performance of the Hemler–Longstaff model compared to the other estimators. This fi nding demonstrates that, compared to the other estimators, the power EWMA appears to more

6 TSEC Taiwan 50 index comprises 50 of the most highly capitalized blue-chip stocks listed on the Taiwan Stock Exchange, and represents almost 70% of the capitalization of the Taiwanese stock market. The correlation between the Taiwan capi-talization weighed index (that is, the TAIFEX index) and the Taiwan 50 index is as high as 95%.



accurately capture the leptokurtic distribution of returns, and the GARCH(1, 1) can better capture the heteroskedastic effect of price volatility. Finally, the pricing errors are positively related to the lagged pricing errors, suggesting persistent mispricing. Meanwhile, the relaxation of the up-tick rule helps reduce the extent of mispricing.

ACKNOWLEDGEMENT

The author would like to thank the National Science Council of the Republic of China, Taiwan for fi nancially supporting this research under Contract No. NSC 96-2416-H-327-014.

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Author’s biography:Janchung Wang is an associate professor of Finance in the Department of Money and Banking at the National Kaohsiung First University of Science and Technology, Taiwan, ROC. His major research fi elds are futures hedging, futures pricing, and index arbitrage.

Author’s address:Janchung Wang, Department of Money and Banking, National Kaohsiung First University of Science and Tech-nology, 1 University Road, Yanchao, Kaohsiung 824, Taiwan, ROC.

stock market volatility and the forecasting performance of stock index futures

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