Hedging with floor-traded and E-mini stock index futures.

Introduction

Over the last few years, more and more stock and futures transactions have shifted from an open outcry system to an electronic trading system. There are many studies that focus on the issue of the relative price discovery role between open outcry and electronic open auctions.

Although most futures participants have considered replacing the open outcry with electronic trading, the transactions still have a long way to grow. Taking the U.S. as an example, most futures trading is still taken by a floor broker, but in order to attract more customers, the Chicago Mercantile Exchange (CME), one of the world's largest derivatives exchanges, introduced a global electronic trading system (GLOBEX) in 1992. CME launched the first electronic-mini (E-mini) product in September 9, 1997, known as E-mini S&P500. It was the first screen-based contract to trade during regular trading hours on the floor of a United States exchange. The E-mini Nasdaq-100 was launched following this in June 21, 1999. Because the size of the E-mini is only about one-fifth of a floor-traded contract, the advantage is that trading an E-mini does not mean needing a prodigious financial account. Hasbrouck (2002) showed that the E-mini possesses a useful price discovery role on the S&P500 and Nasdaq-100. E-mini S&P500 and E-mini Nasdaq-100 are two of the fastest growing products CME has ever launched.

Both floor-traded and E-mini contracts function with the same objectives; the major difference is that E-mini contracts allow for a smaller position and they are traded electronically by GLOBEX with excellent liquidity and around-the-clock availability. Improving the speed of price discovery, accurate recording in time, and high liquidity with sufficient privacy are the advantages of the E-mini contracts. Because trading with E-mini can be completed in just one or two seconds by GLOBEX, traders can get an immediate response after completing the trade. By contrast, the open outcry system has to be made in a specific place. The commission of trading E-minis is based upon the basis of contract; that is, larger trading volumes have higher costs for every transaction. Therefore, for a smaller investor, trading in E-mini futures will give them a better advantage than they will give big investors. As an institutional investor, however, traditional future contracts will cost less.

For the last several years, most studies about transaction methods mainly focus on the comparison of price discovery between traditional and electronic transactions and the power of revealing the information. In 1982 after index futures had been launched, investors widened their investment ranges, providing a good risk management strategy for a portfolio manager. One of the major functions is to avoid risks that arise owing to price fluctuations in the spot market. Until now, there have been no studies discussing hedge effectiveness under different transaction mechanisms. Therefore, in order to provide investors a more useful hedging suggestion, the comparison between the two different trading mechanisms from the point of view of hedging is the major focus in this study.

Most of the existing studies concerning hedging take mean variance as the major empirical criterion and use OLS to estimate the optimal hedge ratio. Because of the existence of autocorrelation in the spot and futures markets, the estimated hedge ratio usually is misleading. In order to drill the hedge ratio more, we use dynamic methods to estimate the optimal hedge ratio. Traditional econometric models take the residuals of variance as a constant, but in reality most financial data are characterized with high frequency and volatility clusters also commonly are found. That is, the residuals of variance are not constant, as they change with time. Traditional hedge theory adopts a perfect hedging strategy. In contrast, recent studies such as Figlewski (1984), Cecchetti, Cumby, and Figlewski (1988), Gagnon and Lypny (1997), and Yeh and Gannon (2000) are devoted to studying the dynamic hedging strategy and point out that a traditional hedge strategy may not be able to provide the best result to the general public. They consider dynamic hedging to be a more appropriate strategy. Therefore, we not only look at five different hedge models to estimate the hedge ratio, but also add a dynamic idea to make hedging result comparisons among models in order to determine the optimal hedge strategy.

Investors usually take an opposite contract position in the futures market, while taking up floor-traded and E-mini futures, with their spot market position being used as a hedge strategy to reduce risks. To discuss the dynamic hedging effectiveness, we first will adopt the vector autoregressive model (VAR), error correction model (ECM), bivariate generalized autoregressive conditional heteroskedasticity (GARCH), Kalman filter, and Markov regime switching model to minimize the risk in the spot and futures markets to obtain the optimal hedge ratio. In addition, the technique of a moving window will be taken into consideration in each model to investigate if it could increase hedging effectiveness.

Literature Review

As Webb (1987) noted, traders fear high-volatility markets because of the enhanced danger of uncontrollable losses due to sudden price swings. Hedging plays an essential role in investments. Work (1953) showed that hedging is the most important feature of the futures market. Based on the purpose of hedging, Gray and Rutledge (1971) categorized hedge theory into four forms, including risk elimination, profit maximization, risk reduction, and portfolio approach. In discussing hedge theory, Ederington (1979) lists three categories: traditional hedge theory, optimization hedge theory, and variance minimization hedge theory. Because traditional hedge theory deviates from a practical situation, optimization hedge theory is involved in speculative motivation, while fundamental finance takes minimizing risks as ordinary investors' hedging strategy in a financial assumption. Most hedging studies take variance minimization hedge theory as the empirical study. Among those, Cecchetti, Cumby, and Figlewski (1988) adopted risk minimization and maximized expected utility, taking spot and futures prices as a dynamic distribution to estimate the optimal futures hedge strategy. Junkus and Lee (1985) used profit maximization, risk elimination, risk minimization, and utility maximization as the hedging strategy in an empirical study.

Investors usually take an opposite contract position in the futures market with their spot market position as a hedge strategy to reduce risks. Traditional hedge theory takes a 100 percent position to hedge. That is, investors take an equivalent and opposite position in the futures market as they have in the spot market. They assume that prices in the spot market and futures market change with the same direction and same magnitude. A perfect hedging position can eliminate the risk of a price fluctuation in the spot market. The hedge ratio therefore uses the benefits obtained in the futures market to compensate the loss suffered in the spot market.

Several later researchers achieved different empirical results. Ederington (1979), under the assumption of minimizing the variance, took the price differences in the spot market and futures market to run OLS to estimate the hedge ratio and to make a hedging effect comparison in the T-bill and GNMA markets. He found that the optimal hedge ratio is always less than one, which is totally different from the traditional hedge theory. Junkus and Lee (1985) studied hedging strategy when profit maximization, risk elimination, risk minimization, and utility maximization are adopted. He found that the predicted capacity of prices in the spot market, prices in the future market, and base differentials do affect a hedge strategy's success. A hedging strategy also should vary with different hedge objectives and future contracts. Lindahl (1992) studied MMI and S&P500, discovering that the hedge ratio increases as the hedge period grows under the assumption of minimizing the variance.

On the estimation of the hedging ratio, previous studies take the mean-variance criterion with OLS to estimate the optimal hedge ratio, but because of the existence of the autocorrelation in the spot and future markets, the estimated hedge ratio is usually misleading. Baillie and Myers (1991) adopted the bivariate GARCH to estimate the optimal hedge ratio. They found that the resulting optimal hedge ratio varies with time, and its effectiveness is better than those of the OLS constant hedge ratio models. Kroner and Sultan (1993) took foreign exchange futures to hedge the spot and showed that the hedge effectiveness in the bivariate GARCH is better than the others. In addition, after considering transaction costs, the bivariate GARCH still could improve a traditional hedge strategy's effectiveness and increase investor's hedging management capability over a foreign exchange position. Park and Switzer (1995) compared the bivariate GARCH with others in estimating the out-of-sample optimal hedge ratio. They found that the bivariate GARCH model performs better than the others, even after considering the transaction costs. Koutmos and Pericli (1999) compared EC-GARCH with a traditional static regression model for an-inside sample and out-of-sample hedge effectiveness. Empirical results showed that EC-GARCH performs better than the traditional static ones.

Over the last decade, several researchers devoted to studying dynamic hedging strategy have shown that a traditional hedge strategy may not be the best strategy to fit ordinary investor needs. Cecchetti, Cumby, and Figlewski (1988) indicated that in order to look for optimal outcomes, the dynamic relationship between the spot and futures contract should be considered. Hunter and Timme (1992) reach the same conclusion. Gagnon and Lypny (1997), Koutmos and Pericli (1999), and Yeh and Gannon (2000) suggest that the GARCH model outperforms other economic models in hedging effectiveness. Koutmos and Pericli (1999) adopted a cross-hedging comparative dynamic EC-GARCH model and traditional regression statistic hedging model to compare the hedging effect both within and out-of-sample. He found that the dynamic bivariate EC-GARCH model could provide better results. Yeh and Gannon (2000) took transaction cost into account to employ the dynamic hedging model in order to estimate the optimal hedge ratio, and they showed that the GARCH hedge model creates the most profits. In addition, its out-of-sample expected ability seems to capture the short-run arbitrage opportunity.

Diebold (1986), Lamoureux and Lastrapes (1990), and Hamilton and Susmel (1994) argued that the high volatility persistence also may be due to structural changes or regime shifts in the volatility process. Many studies with a regime switching model focus on the behavior of a single series: for example, Hamilton (1989), Marsh (2000), Dueker and Neely (2002), and Fong and See (2002). Fewer research studies have stressed the topic of relationship and hedging, although Sarno and Valente (2000) showed that the regime switching model explains the relationship between spot and futures prices better. Clarida et al. (2003) wrote that the relationship between spot and forward is related to the different regimes. For the part of hedging, Alizadeh and Nomikos (2004) found that regime switching hedging outperforms the alternative hedge model in lowering portfolio risk in the FTSE100 market within sample and out of sample. But in the S&P500 market, the model only performs better within sample. Other research studies point out that it outperforms the GARCH model.

The existing literature shows many different conclusions. Therefore, in this study we adopt four different hedging models to compare their hedging effect. We also utilize dynamic hedging to investigate if it could increase the hedging effect. We then determine the optimal hedging model and strategy and provide investors with the best strategy.

Data and Models

Data

In this study we examine spot indexes and futures contracts through two different transaction mechanisms, including S&P500 and Nasdaq-100. Table 1 provides a basic introduction of the two indexes. Data include S&P500 and Nasdaq-100 index spot, floor-traded futures, and E-mini futures that are taken from Bloomberg's database. All are daily data, and the sample period is from 1999.6.21 to 2004.6.10. As long as one of the data is missing on the same day, all the data on the same day will be deleted. Some 1251 observations are included in this study. We take the daily stock index with the associated stock index futures to compute its daily rate of return. The rate of returns is computed by differentiating the logarithm of the daily stock index and futures index.

[DELTA][R.sub.t] = (ln [R.sub.t] - ln [R.sub.t-1]) x 100, (1)

where [DELTA][R.sub.t] is the daily return of the spot and futures at time t, and [R.sub.t] ([R.sub.t-1]) represents the closing prices of the stock index for spot and futures at time t (t - 1), while ln [R.sub.t] (ln [R.sub.t-1]) is the logarithm closing prices of stock the index for the spot and futures at time t (t - 1).

Models

Estimation of the Hedging Ratio Vector Autoregressive Model (VAR)

Empirical studies usually are based upon a prior theory to build a structured econometric model. If the underlying endogenous and exogenous variables are misspecified, then this leads to a useless conclusion. Sims (1980) therefore proposed the VAR model to determine the dynamic model through the characteristics of the data directly. Sims believed that any economic activities' characteristics can be revealed upon the real data and can become a reduced form of a time series model. The model need not consider the causality relationship among the variables and no prior theory is needed. The model takes every variable as being endogenous and constructs a set of regression system of equations. Every variable takes its own possible lag terms with the other variables as the explanation variables to become a regression equation. Therefore, the system of equations reveals the relationship among variables as being interactive, but not one-way oriented.

Because of the characteristics of time series data (through the help of appropriate lagged terms), the model can involve all the information among economic variables. Our model can be written as:

[DELTA][s.sub.t] = [[alpha].sub.s] + [m.summation over (i=1)][[beta].sub.si][DELTA][s.sub.t-i] + [n.summation over (j=1)] [r.sub.sj][DELTA][f.sub.t-j] + [[epsilon].sub.st] (2)

[DELTA][f.sub.t] = [[alpha].sub.f] + [a.summation over (i=1)] [[beta].sub.fi] [DELTA][s.sub.t-i] + [b.summation over (j=1)] [r.sub.fj][DELTA][f.sub.t-j] + [[epsilon].sub.ft], (3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

are independent bivariate random variables and have the same distribution respectively, Var([[epsilon].sub.st]) = [[sigma].sup.2.sub.s], Var([[epsilon].sub.ft]) = [[sigma].sup.2.sub.f], and Cov([[epsilon].sub.st], [[epsilon].sub.ft]) = [[sigma].sub.sf]. The hedge ratio can be written as:

hr = Cov([[epsilon].sub.st], [[epsilon].sub.ft]|[[OMEGA].sub.t-1])/Var([[epsilon].sub.ft]| [[OMEGA].sub.t-1]) = [[sigma].sub.sf]/[[sigma].sup.2.sub.f], (4)

where [[OMEGA].sub.t-1] is an information set which includes all the applicable information at time t - 1.

Error Correction Model (ECM)

Nelson and Plosser (1982) considered most economic variables to be nonstationary, and their basic statistic characteristics vary with time. In order to avoid spurious regression identification, first-order differentials are needed. To avoid losing long-run information when dealing with the issue of nonstationary, Engle and Granger (1987) suggested utilizing the cointegration method to illustrate the long-run relationship between variables and resolve the possibility of losing information due to the differential process. They showed that as long as two economic variables are cointegrated (even if the variables are affected by certain factors in the short run and turn into a process of random walks), they will return to the long-run equilibrium through the process of the dynamic short-run adjustment.

Assume [x.sub.t] and [y.sub.t] are nonstationary in level, but stationary in the first-order differential time series process; that is, [x.sub.t] and [y.sub.t] are I(1). If their linear combination is I(0), then [x.sub.t] and [y.sub.t] are cointegrated. Through the help of the short-run dynamic process, the long-run equilibrium could be restored. Hence, based upon the ECM hedging model of Krehbiel and Adkins (1993), the model in this study could therefore be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [DELTA][s.sub.t] represents the rate of return for the spot market, [DELTA][f.sub.t] is the rate of return for the futures market, [[mu].sub.t-1] is the error correction term, [[epsilon].sub.t] shows the stationary errors at time t, and b is the hedge ratio.

Bivariate GARCH Model

The bivariate GARCH is applicable when a joint normal distribution exists between spot and futures. After introducing the mean equation with error correction term, ([S.sub.t-1] - [gamma] [F.sub.t-1]), the model becomes a bivariate correction model, as in Engle and Granger (1987). On the other hand, Park and Switzer (1995) showed that when the joint distribution of spot and futures varies with time and generates a different variance, the hedge ratio turns into a dynamic formation. Using the bivariate GARCH model, we can calculate the dynamic hedge ratio. In order to take a time-varying variance and covariance, we parameterize the second-order moments by the fixed coefficient bivariate GARCH(1,1). The mean equation of model can therefore be rewritten as:

[DELTA][s.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]([S.sub.t-1] - [gamma] [F.sub.t-1]) + [[epsilon].sub.st] (6)

[DELTA][f.sub.t] = [[beta].sub.0] + [[beta].sub.1]([S.sub.t-1] - [gamma] [F.sub.t-1]) + [[epsilon].sub.ft], (7)

where [DELTA][s.sub.t] and [DELTA][f.sub.t] are respectively the rate of return of spot and futures, [S.sub.t] and [F.sub.t] represent the logarithm price of spot and futures, and ([S.sub.t-1] - [gamma] [F.sub.t-1]) is the error correction term. The error term is [[[epsilon].sub.st] [[epsilon].sub.ft]]' | [[OMEGA].sub.t-1] ~ N(0, [H.sub.t]), where [[OMEGA].sub.t-1] is the information set at time t-1. Here [H.sub.t] is the conditional covariance matrix. The variance equation is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [h.sup.2.sub.st] = [c.sub.s] + [a.sub.s][[epsilon].sup.2.sub.st-1] + [b.sub.s][h.sup.2.sub.st-1], [h.sup.2.sub.ft] = [c.sub.f] + [a.sub.f][[epsilon].sup.2.sub.ft-1] + [b.sub.f][h.sup.2.sub.ft-1], and [rho] is the fixed correlation coefficient.

Using the maximum likelihood estimation (MLE) and the above model, the dynamic hedge ratio (hr) can be written as:

h[r.sub.t] = [h.sub.sft]/[h.sup.2.sub.ft], (9)

The conditional covariance [h.sub.sft] and conditional variance [h.sup.2.sub.ft] vary with the newly-revealed information. The estimated hedge ratio will change with time also.

Kalman Filter

The Kalman filter (1960) not only estimates the parameters on level, but also considers the transformation of the state vector space. The major characteristic is its stepwise information updating process, which also illustrates the dynamic relationships between explanatory and dependent variables. In this study we take the Kalman filter with time varying in our model as follows. The observation equation can be written as:

[DELTA][s.sub.t] = [[alpha].sub.t] + [[beta].sub.t][DELTA][f.sub.t] + [w.sub.t], (10)

where [DELTA][s.sub.t] is the [rho] x 1 observed vector, [[beta].sub.t] is endogenous and unobserved of the k x 1 state vector, [DELTA][f.sub.t] is the [rho] x k design matrix, which shows the relationship between [DELTA][s.sub.t] and [[beta].sub.t], and [w.sub.t] are the errors with E([w.sub.t]) = 0, and E([w.sub.t][w'.sub.t]) = [[sigma].sup.2.sub.w].

The state equation is:

[[beta].sub.t] = [phi]([[beta].sub.t-1] - [bar.[beta]]) + [bar.[beta]] + [v.sub.t-1], (11)

where [phi] is the [rho] x k design matrix which shows the relationship between [[beta].sub.t] and [[beta].sub.t-1], [v.sub.t] are errors with E([v.sub.t]) = 0 and E([v.sub.t][v'.sub.t]) = [[sigma].sup.2]I, and [[beta].sub.t] is the state vector at time t. Because [[beta].sub.t] cannot be observed directly, the optimal estimated output value at time t therefore can be used to derive a predicted output value at time t+1, which also must minimize the predicted error variance [P.sub.t|t-1]. Applying equations (9) and (10) into the hedge model, [[beta].sub.t] becomes the hedge ratio in terms of state with the characterization of time varying.

Markov Regime Switching

Taking a simple case in this study, we assume that the hedge ratios are different in two unobservable states (regimes). In other words, the relationship between spot and futures markets is different in the two regimes. The equation can be written as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [i.sub.t] = 1, 2 indicates regime 1 and regime 2, respectively. The density function of [DELTA][s.sub.t] in each regime is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

and the joint density is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

where Pr([i.sub.t]|[[OMEGA].sub.t-1]) is the conditional regime probabilities in a given state.

We also set Pr([i.sub.t] = 1|[[OMEGA].sub.t-1]) = [[pi].sub.1,t] = and Pr([i.sub.t] = 2|[[OMEGA].sub.t-1]) = [[pi].sub.2,t]. The log-likelihood of the above density function can be defined as

L = [T.summation over (t=1)] log f([DELTA][s.sub.t]|[[OMEGA].sub.t-1]) (15)

where T represents observations. Therefore, the average hedge ratio (hr) is weighted according to the respective probabilities. The equation can be written as

h[r.sub.t] = [[beta].sub.1,t][[pi].sub.1,t] + [[beta].sub.2,t][[pi].sub.2,t]. (16)

The Hedging Performance

The variance's reduction of the predicted returns in an unhedged spot portion can be used to evaluate hedging performance. That is, the greater the reduction is, the better the hedge performance will be. By the hedge ratio (hr), we can derive the rate of return for the hedge investment combination, which can be written as follows:

[X.sub.t] = [DELTA][s.sub.t] - h[r.sub.t][DELTA][f.sub.t], (17)

where [DELTA][s.sub.t] and [DELTA][f.sub.t] are the daily fluctuation rates of spot and futures markets, respectively, and [X.sub.t] is the rate of return for the hedge investment combination under the assumption of a short hedge position. The variance of the return for the unhedged asset combination can be expressed as:

Var(U) = [[sigma].sup.2.sub.u] = Var([DELTA][s.sub.t]). (18)

The variance of return for the hedged asset combination is

Var(H) = [[sigma].sup.2.sub.h] = Var([X.sub.t]). (19)

The hedge performance can be evaluated by equation (20), hedging effectiveness (HE). In order to compute the average hedging effectiveness, the method of a moving window has been adopted. The overall hedging effectiveness can be evaluated by the hedging effectiveness index (HEI), which can be expressed by equation (21), where M is the number of rolling MISSING WORD?. The higher the HEI is, the better the dynamic hedge performance is.

HE = [Var(U) - Var(H)]/Var(U) = [[[sigma].sup.2.sub.u] - [[sigma].sup.2.sub.h]] /[[sigma].sup.2.sub.u] (20)

HEI = [[M.summation over (j=1)] H[E.sup.(j)]]/M. (21)

Empirical Results

Descriptive Statistics

Table 2 contains statistics for the spot index, floor-traded futures, and E-mini futures. Because index futures are the derivatives of the spot index, the statistics for the spot and futures should be closely correlated. From the part of S&P500 in Table 2, we find that the means are similar, but the futures market fluctuated more than the spot did, while the E-mini futures fluctuated more than floor-traded futures did. Unlike S&P500 index, the standard deviation of Nasdaq-100 spot is higher than floor-traded and E-mini futures. All the spot and futures markets present fat tails. Through the Jarque-Beta normality test, we find that all the indexes reject the null hypothesis of normal distribution. Figure 1 presents the trend of three series in each index; the three series are consistent in the latest sample period.

[FIGURE 1 OMITTED]

Unit Root, Cointegration, and ARCH Tests

Unit Root Test

In order to prevent spurious regression and cointegration problems, it is necessary to verify stationary before doing any further analysis. Here, we adopt the augmented Dickey-Fuller (ADF) test to perform the unit root test. In Table 3 we find that all the level terms of spot and futures series cannot reject the null hypothesis of unit root. In other words, the level term series are nonstationary. After a first differential process, however, all series reject the nonstationary hypothesis under 1 percent levels. That is, all these series follow the I(1) process.

Cointegration Test

Before constructing the VAR and error correction model, it is necessary to determine the optimal lag terms. In this study the general Akaike information criterion (AIC) is adopted, and the one period lag will be chosen. According to the Johansen maximum likelihood method, which contains the maximum eigenvalue (L-max) and trace test, the long-term relationship between spot and futures will be examined. In Table 4 the same conclusions appear in the S&P500 and Nasdaq-100 indexes. All of the results show that the null hypothesis of a zero cointegration relationship is not significant, but the hypothesis of one cointegration relationship is rejected significantly either in the L-max or trace test. This means that the long-term relationships exist between spot and futures in these two markets.

ARCH Test

We must examine the residuals of equations (6) and (7) before applying the GARCH model. The Lagrange multiplier (LM) test is used to verify the ARCH effect, and the results are reported in Table 5. If the statistical values are significant, then we can say with confidence that the ARCH effects exist. All the outcomes are significant under the 1 percent level (no matter in S&P500 or in Nasdaq-100); that is, ARCH effects obviously exist.

Hedge Ratio and Hedging Effectiveness

Benet (1992) studied foreign currency futures and suggested that using out of samples or ex-ante to evaluate hedging effectiveness would be more meaningful for investors. Hence, we adopt Benet's suggestion to evaluate hedging performance out-of-sample results. The estimated time expansion is 800 days. Utilizing the form of a moving window to analyze the effect of the length of hedging periods to the hedging performance in different economic models, we take two, four, eight, twelve, 24, and 48 weeks as the hedging periods in this study and perform daily rolling to hedge. Taking two weeks as an example, as shown on Figure 2, the first loop uses the first 800 days' spot and futures markets to estimate the hedge ratio and then performs hedging for the next two weeks (ten days). At the end of the ten days, the hedging performance is evaluated and so on. In this study we take five methods, including VAR, ECM, bivariate GARCH(1,1), Kalman filter, and Markov regime switching, to evaluate the dynamic hedging process clearly and completely

[FIGURE 2 OMITTED]

In order to obtain the out-of-sample empirical results, we use the latest information to estimate the next period's hedge ratio. Therefore, the entire hedging ratio we derive is a dynamic process, not a constant hedging ratio. Table 6 presents the hedging ratio for all possible combinations. It is found that most hedge ratios are less than one, especially for the S&P500 index. The results are consistent with the studies of Ederington (1979), Junkus and Lee (1985), Lindahl (1992), Park and Switzer (1995), and Holmes (1996). This implies that it is not necessary to take up 100 percent hedging of a futures position for a long spot position. This can reduce the hedging cost for investors. It is worth noting that the hedge ratio estimated by the GARCH model on Nasdaq-100 is higher than others, and the ratio is close to one. This result supports the perfect hedging strategy in the Nasdaq-100 market.

Comparing the various models, we calculate all possible HEIs with the moving window method. The results are reported in Table 7. The HEI are positive under each model and hedging period, indicating that the variance of a hedged portfolio is lower than that of an unhedged portfolio. We find that regardless of whether we take floor-traded or E-mini futures, the bivariate GARCH and Markov regime switching have higher HEI performances, while the Kalman filter is the lowest on average.

In Table 7, the GARCH model seems to perform better for two weeks, a short-term hedging period. For the S&P500, the HEIs of hedging with floor-traded and E-mini futures are 0.9452 and 0.9385. For the Nasdaq-100 the HEIs are 0.9425 and 0.9357, respectively. The value is relatively close, but higher than under regime switching. Taking a further look into longer periods, we find that almost all regime switching have higher HEIs in these two markets. This implies that the GARCH model can capture the short-run dynamic effect, and this is consistent with the studies of Yen and Gannon (2000), who suggest that the performance of GARCH model appears on average to persist over a five-day horizon.

Unlike regime switching, the GARCH model assumes that the data follow the unique distribution with the same mean and standard deviation. As the hedging period gets longer, the estimation may misspecify while the regime changes. The regime switching model outperforms the other models with out-of-sample results and a longer hedging period in both S&P500 and Nasdaq-100 markets. The forecasting results in Fong and See (2002) show that the regime switching model performs better in the out-of-sample horizon. Nevertheless, these findings are slightly different from Alizadeh and Nomikos (2004), who argue that the regime switching model outperforms only within a sample in the S&P500 market. We propose another conclusion in the S&P500 market. We find that even though regime switching may be a systematic feature of the data, regime switching may underperform non-regime switching models in the short term and with out-of-sample forecasts resulting from the over-parameterization problem. The assumption of two unobservable regimes is suitable in the long term, however, and the regime switching model performs better than do alternative models.

Let combine the hedge ratio and the hedging performance in Table 6 and Table 7. In the S&P500 market, the perfect hedging strategy is not appropriate. Investors can lower their portfolio risk with fewer transaction costs, and it is not necessary to take up a 100 percent hedging of a futures position for a long spot position. In the Nasdaq-100 market, aside from the short-term hedging period (two weeks), the results are consistent with S&P500. With the higher hedging effectiveness, however, the optimal hedging ratio is 1.0249 and 1.0250 with floor-traded and E-mini futures, respectively. This indicates that investors will take higher futures position relative to the spot position from the short-term period in order to lower the portfolio risk. These results are useful to investors who prefer short-term hedging periods in the Nasdaq- 100 market.

Regardless of the models that have been used, we intend to investigate the relationship between the hedging period and hedging effectiveness. Table 7 shows that the longer the sample period is, the more effective the hedging effectiveness becomes across models. This coincides with the findings of Ederington (1979), Figlewski (1984), and Lindahl (1992). Long-term relationships exist between spot and futures, although even the reverse relationship probably occurs in short-term periods. That is, spot and futures may move in different directions in the short run, and hedging could contrarily cause losses. But in the long run (because spots and futures being highly correlated), bear hedging can reveal its effectiveness. Therefore, as the hedge period is extended, the correlation coefficient between spot and futures increases and hedging effectiveness goes up. As a result, a long period hedging strategy would be the better choice when investors are taking index futures as hedging vehicles.

In terms of hedging objectives, regardless of different hedging periods and hedging models, floor-traded futures with an open outcry system perform better than do E-mini futures contracts. For E-mini futures contracts, the electronic platform enables the transaction to be made regardless of place and time, and there is no huge financial cost for small investors. When there is a big transaction amount, however, the cost is relatively high. This means general investment institutions still rely on floor-traded futures to do business, implying that a larger investment is more sensitive to hedging information than is a small investment. In addition, the transaction cost is also the major factor to influence hedging effectiveness. Because the transaction cost of an E-mini contract is based on the basement of contracts, larger trading volumes have higher costs for every transaction. Because the cost will increase with the transaction amount, the hedging effectiveness is therefore relatively small. Regardless of floor-traded or E-mini futures, the HEI is always greater than 90 percent. Both can be good instruments to be used as hedging objectives in either the S&P500 or the Nasdaq-100 markets. Through a dynamic hedging process, the variances of the investment portfolio can decrease noticeably.

Conclusion

In this study we investigated the out-of-sample hedging effectiveness and hedge ratios of floor-traded and E-mini futures with the VAR, ECM, bivariate GARCH, Kalman filter, and Markov regime switching in the S&P500 and Nasdaq-100 markets. The moving window technique is adopted to analyze the effect of the length of time expansion to the hedging performance in different hedging models. The empirical results show that both the floor-traded or E-mini futures can be good instruments to be used as hedge objectives either in the S&P500 or Nasdaq-100 markets. As the hedging period is extended, the correlation coefficient between spot and futures increases and hedging effectiveness goes up.

Second, the bivariate GARCH and Markov regime switching methods have higher HEI performances in short-term and long-term hedging periods, respectively, while the Kalman filter is the lowest on average. Third, except for the short-term hedging period (two weeks) in the Nasdaq-100 market, we can conclude that a perfect hedging strategy is not appropriate. Investors can lower the portfolio risk with fewer transaction costs, and it is not necessary to take a 100 percent hedging of a futures position for a long spot position. In the Nasdaq-100 market, however, investors will take a higher futures position relative to their spot position in the short-term period in order to lower the portfolio risk. Fourth, floor-traded futures with an open outcry system do better than E-mini futures contracts. The main reason is the higher transaction cost of large trading with E-mini futures relative to floor-traded futures. Finally, this study proposes meaningful evidence of hedging strategies for investors with different spot indexes and hedging periods. All the portfolio risk can be lowered by different strategies.

Table 1--Basic Introduction of S&P500 and Nasdaq-100 Index
Futures Contracts

S&P 500                 Floor-Traded Futures         E-mini Futures

Trading Market      CME                            CME

Value of Contract   $250 x S&P500 index            $50 x S&P500 index

Month of Contract   3, 6, 9, 12                    3, 6, 9, 12

Trading Hours       Floor: 8:30 a.m.- 3:15 p.m.    Virtually 24-hour
                    GLOBEX: remaining time         trading (GLOBEX)

Min. Price          0.10 index point = $25 per     0.25 index point =
Fluctuation         contract                       $12.5 per contract

Last Trading Day    The day before the third       8:30 a.m. (Chicago
                    Friday of the contract month   time) on the third
                                                   Friday of the
                                                   contract month

Nasdaq- 100         Floor-traded futures           E-mini futures

Trading Market      CME                            CME

Value of Contract   $100 x Nasdaq-100 index        $20 x Nasdaq-100
                                                   index

Month of Contract   3, 6, 9, 12                    3, 6, 9, 12

Trading Hours       Floor: 8:30 a.m.- 3:15 p.m.    Virtually 24-hour
                    GLOBEX: remaining time         trading (GLOBEX)

Min. Price          0.50 index point = $50 per     0.50 index point =
Fluctuation         contract                       $10 per contract

Last Trading Day    The day before the third       8:30 a.m. (Chicago
                    Friday of the contract month   time) on the third
                                                   Friday of the
                                                   contract month

Table 2--Descriptive Statistics

                   Mean      S.D.    Skewness    Kurtosis

S&P 500

Spot Index       -0.0137%   1.7049   0.1283      1.5078
Floor-traded
  Futures        -0.0143%   1.7345   0.0743      1.5743
E-mini Futures   -0.0143%   1.7857   0.1422 **   1.9741

Nasdaq-100

Spot Index       -0.0341%   2.7418   0.2757      2.2200
Floor-traded
  Futures        -0.0349%   2.7014   0.1263      2.0546
E-mini Futures   -0.0349%   2.7139   0.1426      2.0760

                   Min.        Max.          JB

S&P 500

Spot Index       -6.0045%    5.5732%    121.8464 ***
Floor-traded
  Futures        -6.2708%    5.7548%    130.2474 ***
E-mini Futures   -6.2708%    6.9804%    207.1914 ***

Nasdaq-100

Spot Index       -10.3089%   17.2029%   272.5438 ***
Floor-traded
  Futures        -10.8153%   15.4447%   223.2047 ***
E-mini Futures   -10.8153%   15.4447%   228.7055 ***

Note: *, **, and *** represent 10 percent, 5 percent,
and 1 percent significant levels, respectively. JB
represents the statistics of the Jarque-Bera normality
test and the null hypothesis is normal distribution.
Kurtosis values have been subtracted by three

Table 3--ADF Unit Root Test

                           Constant with           Constant
                              Trend                 Only

Level Term               Lag    Statistics    Lag    Statistics

S&P500 Stock Index        0     -1.6734        0     -1.5097
Floor-traded Futures      0     -1.6706        0     -1.5068
E-mini Futures            0     -1.7058        0     -1.5234
Nasdaq-100 Stock Index    0     -1.2901        0     -1.0275
Floor-traded Futures      0     -1.2579        0     -1.0063
E-mini Futures            0     -1.2622        0     -1.0093

First Difference

S&P500 Stock Index        0    -35.8639 ***    0    -35.8703 ***
Floor-traded Futures      0    -22.0962 ***    0    -22.0958 ***
E-mini Futures            0    -22.2426 ***    0    -22.2423 ***
Nasdaq-100 Stock Index    1     -8.9582 ***    1     -8.9582 ***
Floor-traded Futures      1     -9.1169 ***    1     -9.1168 ***
E-mini Futures            1     -9.1110 ***    1     -9.1109 ***

Note: *, **, and *** represent 10 percent, 5 percent12
value refers to Dickey-Fuller (1981)

Table 4--Cointegration Tests

                        Eigenvalue    L-max    Trace
S&P500

Spot and Floor-traded
  Futures                 0.0619      79.70    81.86
                          0.0017       2.17     2.17
Spot and E-mini
  Futures                 0.0740      95.99    98.16
                          0.0017       2.17     2.17

Nasdaq-100

Spotand Floor-traded
  Futures                 0.1038     136.80   137.91
                          0.0009       1.11     1.11
Spotand E-mini
  Futures                 0.1139     150.97   152.08
                          0.0009       1.11     1.11

                                       Critical   Critical
                                        Value      Value
                        [H.sub.0]: r   of L-max   of Trace
S&P500

Spot and Floor-traded
  Futures                    0          10.29      17.79
                             1           7.50       7.50
Spot and E-mini
  Futures                    0          10.29      17.79
                             1           7.50       7.50

Nasdaq-100

Spotand Floor-traded
  Futures                    0          10.29      17.79
                             1           7.50       7.50
Spotand E-mini
  Futures                    0          10.29      17.79
                             1           7.50       7.50

Table 5--ARCH Effects

                                           S&P500

                                 Floor-traded      E-mini
                                   Futures        Futures

[[epsilon].sub.st] for Q(20)      27.7598        28.5366 *
[[epsilon].sup.2.sub.st] for     392.7675 ***   391.0617 ***
  [Q.sup.2](20)
[[epsilon].sub.st] for LM test   140.6672 ***   141.3418 ***

[[epsilon].sub.ft] for Q(20)      27.0494        28.0538
[[epsilon].sup.2.sub.ft] for     398.7573 ***   367.7594 ***
  [Q.sup.2](20)
[[epsilon].sub.ft] for LM test   147.1744 ***   133.1536 ***

                                          Nasdaq-100

                                 Floor-traded      E-mini
                                   Futures        Futures

[[epsilon].sub.st] for Q(20)      53.1721 ***    53.7102 ***
[[epsilon].sup.2.sub.st] for     554.4713 ***   550.6283 ***
  [Q.sup.2](20)
[[epsilon].sub.st] for LM test   185.5720 ***   182.9177 ***

[[epsilon].sub.ft] for Q(20)      44.3821 ***    46.1848 ***
[[epsilon].sup.2.sub.ft] for     555.8979 ***   549.6504 ***
  [Q.sup.2](20)
[[epsilon].sub.ft] for LM test   164.0668 ***   162.2235 ***

Note: *, **, and *** represent 10 percent, 5 percent
and 1 percent significant levels, respectively

Table 6--The Hedge Ratios of Different
Instruments in Various Models

                Period     ECM      VAR

S&P500

Floor-traded
  Futures      2 weeks    0.9713   0.9724
               4 weeks    0.9501   0.9727
               8 weeks    0.9540   0.9722
               12 weeks   0.9549   0.9697
               24 weeks   0.9560   0.9569
               48 weeks   0.9611   0.9584

E-mini
  Futures      2 weeks    0.9373   0.9530
               4 weeks    0.9419   0.9532
               8 weeks    0.9441   0.9532
               12 weeks   0.9439   0.9519
               24 weeks   0.9461   0.9415
               48 weeks   0.9513   0.9445
Nasdaq-100

Floor-traded
  Futures      2 weeks    0.9910   0.9893
               4 weeks    0.9905   0.9899
               8 weeks    0.9974   0.9978
               12 weeks   0.9992   1.0050
               24 weeks   0.9866   0.9814
               48 weeks   0.9951   0.9895

E-mini
  Futures      2 weeks    0.9857   0.9819
               4 weeks    0.9853   0.9824
               8 weeks    0.9923   0.9908
               12 weeks   0.9944   0.9989
               24 weeks   0.9842   0.9788
               48 weeks   0.9931   0.9866

                        Kalman    Regime
               GARCH    Filter   Switching

S&P500

Floor-traded
  Futures      0.9730   0.9835    0.9714
               0.9627   0.9849    0.9711
               0.9925   0.9916    0.9705
               0.9967   0.9917    0.9699
               1.0214   0.9944    0.9680
               0.9906   0.9928    0.9639

E-mini
  Futures      0.9937   0.9819    0.9629
               0.9510   0.9832    0.9626
               0.9899   0.9899    0.9621
               0.9898   0.9913    0.9623
               1.0061   0.9890    0.9604
               0.9317   0.9878    0.9562

Nasdaq-100

Floor-traded
  Futures      1.0249   0.9726    0.9998
               1.0202   0.9696    0.9998
               1.0009   0.9715    0.9998
               1.0144   0.9722    0.9998
               1.0092   0.9623    0.9991
               1.0165   0.9826    0.9950

E-mini
  Futures      1.0250   0.9819    0.9985
               1.0172   0.9832    0.9987
               0.9976   0.9899    0.9986
               1.0002   0.9913    0.9986
               1.0056   0.9890    0.9972
               0.9926   0.9878    0.9930

Table 7--The HEI of Different Instruments in Various Models

                                                     Kalman    Regime
                Period      ECM      VAR    GARCH    Filter   Switching

S&P500

Floor-traded
  Futures      2 weeks    0.9442   0.9442   0.9452   0.9314    0.9451
               4 weeks    0.9501   0.9500   0.9511   0.9402    0.9518
               8 weeks    0.9540   0.9540   0.9553   0.9498    0.9555
               12 weeks   0.9549   0.9549   0.9562   0.9477    0.9570
               24 weeks   0.9560   0.9560   0.9576   0.9522    0.9597
               48 weeks   0.9611   0.9609   0.9627   0.9557    0.9641

E-mini
  Futures      2 weeks    0.9373   0.9370   0.9385   0.9228    0.9379
               4 weeks    0.9419   0.9417   0.9429   0.9293    0.9431
               8 weeks    0.9441   0.9438   0.9452   0.9381    0.9449
               12 weeks   0.9439   0.9447   0.9450   0.9324    0.9458
               24 weeks   0.9461   0.9457   0.9474   0.9391    0.9483
               48 weeks   0.9513   0.9509   0.9528   0.9506    0.9600

Nasdaq-100

Floor-traded
  Futures      2 weeks    0.9412   0.9408   0.9425   0.9343    0.9422
               4 weeks    0.9479   0.9475   0.9486   0.9426    0.9505
               8 weeks    0.9541   0.9536   0.9546   0.9467    0.9553
               12 weeks   0.9560   0.9555   0.9564   0.9526    0.9600
               24 weeks   0.9570   0.9565   0.9570   0.9499    0.9625
               48 weeks   0.9584   0.9579   0.9586   0.9562    0.9670

E-mini
  Futures      2 weeks    0.9346   0.9341   0.9357   0.9228    0.9356
               4 weeks    0.9410   0.9404   0.9416   0.9293    0.9435
               8 weeks    0.9462   0.9456   0.9467   0.9381    0.9470
               12 weeks   0.9474   0.9468   0.9477   0.9324    0.9514
               24 weeks   0.9488   0.9484   0.9486   0.9391    0.9524
               48 weeks   0.9503   0.9497   0.9497   0.9506    0.9578

References

[1.] Alizadeh, A., and N. Nomikos, "A Markov Regime Switching Approach for Hedging Stock Indices," Journal of Futures Markets, 24 (2004), pp.649-674.

[2.] Baillie, R. T., and R. J. Myers, "Bivariate GARCH Estimation of the Optimal Commodity Futures Hedge," Journal of Applied Econometrics, 6 (1991), pp. 109-124.

[3.] Benet, B.A., "Hedging Period Length and Ex-ante Futures Hedging Effectiveness: The Case of Foreign Exchange Risk Cross Hedges," Journal of Futures Markets, 12 (1992), pp. 163-175.

[4.] Bollerslev, T., "Generalized Autoregressive Heteroskedasticity," Journal of Econometrics, 31 (1986), pp. 307-327.

[5.] Cecchetti, S.G., R.E. Cumby, and S. Figlewski, "Estimation of the Optimal Futures Hedge," Review of Economics & Statistics, 70 (1988), pp. 623-30.

[6.] Clarida, R. H., L. Sarno, M. P. Taylor, and G. Valente, "The Out-of-sample Success of Term Structure Models as Exchange Rate Predictors: A Step Beyond," Journal of International Economics 60 (2003), pp. 61-83.

[7.] Diebold, F.X., "Modeling the Persistence of Conditional Variance: A Comment," Economic Review, 5 (1986), pp. 51-56.

[8.] Dueker, M., and C.J. Neely, "Can Markov Switching Models Predict Excess Foreign Exchange Returns?" working paper, Federal Reserve Bank of St. Louis (2002).

[9.] Ederington, L.H., "The Hedging Performance of the New Future Markets," Journal of Finance, 34 (1979), pp. 157-170.

[10.] Engle, R.F., "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, 50 (1982), pp. 987-1007.

[11.] Engle, R.F., and C. W.J. Granger, "Cointegration and Error Correction: Representation, Estimation and Testing," Econometrica, 55 (1987), pp. 251-276.

[12.] Fong, W.M., and K.H. See, "A Markov Switching Model of the Conditional Volatility of Crude Oil Futures Prices," Energy Economics, 24 (2002), p.71-95.

[13.] Gagnon L., and G. Lypny, "The Benefits of Dynamically Hedging the Toronto 35 Stock Index," Canadian Journal of Administrative Sciences, 14 (1997), pp. 69-78.

[14.] Ghosh, A., "Hedging with International Stock Index Futures: An Intertemporal Error Correction Model," Journal of Financial Research, 19 (1996), pp. 447-491.

[15.] Ghosh, A., "Hedging with Stock Index Futures: Estimation and Forecasting with Error Correction Model," Journal of Futures Markets, 13 (1993), pp. 743-752.

[16.] Granger, C. W.J., "Some Properties of Time Series Data and Their Use in Econometric Model Specification," Journal of Econometrics, 16 (1981), pp. 121-130.

[17.] Grunbichler, A., F.A. Longstaff, and E.S. Schwartz, "Electronic Screen Trading and the Transmission of Information: An Empirical Examination," Journal of Financial Intermediation, 3 (1994), pp. 166-187.

[18.] Hamilton, J.D., and R. Susmel, "Autoregressive Conditional Heteroskedasticity and Changes in Regimes," Journal of Econometrics, 64 (1994), pp. 307-333.

[19.] Hamilton, J.D., "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, 57 (1989), pp. 357-384.

[20.] Hasbrouck, J., "Intraday Price Formation in US Equity Index Markets," working paper (2002), New York University.

[21.] Johnson, L., "The Theory of Hedging and Speculation in Commodity Futures," Review of Economic Studies, 27 (1960), pp. 139-151.

[22.] Koutmos G., and A. Pericli, "Hedging GNMA Mortgage-Backed Securities with T-Note Futures: Dynamic versus Static Hedging," Real Estate Economics, 27 (1999), pp. 335-363.

[23.] Krehbiel, T., and L.C. Adkins, "Cointegration Tests Of the Unbiased Expectations Hypothesis in Metals Markets," Journal of Futures Markets, 13 (1993), pp. 753-763.

[24.] Kroner, K.F., and J. Sultan, "Time-Varying Distributions and Dynamic Hedging with Foreign Currency Futures," Journal of Financial and Quantitative Analysis, 4 (1993), pp. 535-551.

[25.] Lamoureux, C.G., and W.D. Lastrapes, "Persistence in Variance, Structure Change, and the GARCH Model," Journal of Business and Economics Statistics, 8 (1990), pp. 225-243.

[26.] Lien, D., and Y.K. Tse, "Hedging Time-Varying Downside Risk," Journal of Futures Markets, 18 (1998), pp. 705-722.

[27.] Lindahl, M., "Minimum Variance Hedge Ratios for Stock Index Futures: Duration and Expiration Effects," Journal of Futures Markets, 12 (1992), pp. 33-53.

[28.] Markowitz, H., "Portfolio Selection," Journal of Finance, 7 (1952), pp. 77-91.

[29.] Marsh, I. W., "High-frequency Markov Switching Models in the Foreign Exchange Market," Journal of Forecasting, 19 (2000), pp. 123-134.

[30.] Park, T.H., and L.N. Switzer, "Bivariate GARCH Estimation of the Optimal Hedge Ratios for Stock Index Future: A Note," Journal of Futures Markets, 15 (1995), pp. 61-67.

[31.] Sarno, L., and G. Valente, "The Cost of Carry Model and Regime Shifts in Stock Index Futures Markets: An Empirical Investigation," Journal of Futures Markets, 20, pp. 603-624.

[32.] Stein, J.L., "The Simultaneous Determination of Spot and Futures Prices," American Economic Review, 51 (1961), pp. 1012-1025.

[33.] Webb, R.I., "A Note on Volatility and Pricing of Futures Options During Choppy Markets," Journal of Futures Markets, 7 (1987), pp. 333-337.

[34.] Working, H., "Futures Trading and Hedging," American Economic Review, 43 (1953), pp 314-343.

[35.] Yeh, S.C., and G.L. Gannon, "Comparing Trading Performance of the Constant and Dynamic Hedge Models: A Note," Review of Quantitative Finance & Accounting, 14 (2000), pp. 155-160.

Chien-Liang Chiu

Tamkang University

Pei-Shan Wu

Tamkang University

Chun-Da Chen

Ching-Yun University

Wan-Hsiu Cheng

Taichung Healthcare and Management University