The effectiveness of dynamic hedging: evidence from selected European stock index futures

Abstract

This paper estimates time-varying optimal hedge ratios (OHRs) using a bivariate generalized autoregressive conditional heteroscedastic (GARCH) error correction model. The GARCH specification accounts for time-varying distribution in asset returns while the error correction term preserves short-run deviations between two fundamentally linked assets. Using stock index and stock index futures from four European countries, we compare the hedging effectiveness of the GARCH error correction model with alternative hedging models that hold the OHR constant. Overall, in three out of four cases, the GARCH error correction model is shown to offer superior risk reduction compared with the competing models. Finally, we also estimate the OHRs using the GARCH-X model, which allows the error correction term to be a determinant of the time-varying volatility. The GARCH-X model performs similar to the GARCH error correction model. The results presented in this paper have important insights into the risk management of financial assets when returns distribution changes over time.

Keywords:

• stock index futures
• bivariate GARCH-X
• error correction term
• time varying minimum variance hedge ratio
• out of sample hedge effectiveness

Acknowledgements

We thank an anonymous referee for many valuable comments and suggestions. We are responsible for all remaining errors.

Notes

See Choudhry (2003) and Kroner and Sultan (1993) and a list of references therein.

This section draws extensively from Kroner and Sultan (1993).

Note that basis risk is an important source of market risk in the hedger's portfolio. Figlewski (1984) contends that basis risk is induced by the non-market component of return on the cash stock position such as dividend, hedge duration and time to expiration of the futures contract. Unanticipated change in dividend may induce variability in hedged portfolio returns. As the holding period is extended, basic risk as a fraction of total risk would decrease and hedging effectiveness would improve. Figlewski (1984) notes that basis tends to widen when market is rising and shrinks when the market falls. Furthermore, as the market develops, the importance of the equilibrium price in determining future price changes increases markedly, and the overreaction of future prices to changes in the spot index diminish, perhaps close to zero.

The OLS estimation of the hedge ratio from Equation (6) is based on the assumption of time invariant asset distributions is suggested by Ederington (1979), and Anderson and Danthine (1980). Therefore, previous studies have also computed hedge ratios using the OLS regression specification, see Brailsford, Corrigan, and Heaney (2001), Choudhry (2003, 2004), Floros and Vougas (2004), Howard and D'Antonio (1984, 1991), Myers and Thompson (1989), and Park and Switzer (1995). These studies found the estimated hedge ratios to be significantly less than unity. For example, Floros and Vougas (2004) found b* to be.9160 and.7033 in the Greek stock index futures; Park and Switzer (1995) found hedge ratios for the S&P500 index futures and Toronto 35 index futures to be 0.961 and 0.890, respectively. Choudhry (2003) reported the OLS regression based minimum variance hedge ratio to be 0.6529, 0.7632, 0.78885, 0.5998 and 0.8260 for stock indices futures markets of Australia, Germany, Hong Kong, Japan, South Africa and UK, respectively.

There are several issues involving the use of daily data that are not addressed in this study. For example, the daily data could be noisy and may be affected by institutional factors such as triple witching days when options and futures on stock indices expire simultaneously. Another known problem is the day of the week effect. We have not addressed these issues here for the sake of simplicity. However, it is assumed that the error correction term, which captures day-to-day deviations of the markets from equilibrium, may proxy for some of these stylized features.

It is well known that Augmented Dickey-Fuller (ADF) and Phillips-Perron unit root tests have low power in rejecting the null of a unit root and are prone to size distortion. Elliott, Rothenberg, and Stock (1996) proposed an alternative DF-GLS test which involves the application of a generalized least squares method to de-trend the data. In the process of performing this test, the autoregressive truncation lag length is determined by the modified Akaike Information Criterion (AIC).

The results of the DF-GLS tests indicate that each of the price indices series is non-stationary in the levels. To conserve space, these results are not reported here but are available upon request.

The lead-lag relationship between stock indices of both cash and futures markets is important in the investigation of empirical relationship between stock cash and stock futures markets. The relationship is embedded in the cost-of-carry model, the price discovery and market efficiency hypotheses. For a discussion, see Brooks, Garrett, and Hinch (1999), Hasan (2005) and a list of references therein. Brooks, Garrett, and Hinch (1999) chose four leads and lags of the returns in the spot market as dependent variable. Choudhry (2003) chose six leads and lags of the returns in the spot market as dependent variable. We have chosen five leads and lags of the returns in the testing Equation (13) to allow five business days in a given week. We have also examined the relationship in an alternative model using AIC. The results are however robust under alternative lag specifications.

These are standard tests and are left out but are available upon request.

Note the presence of time subscript (t) in the definition of correlation.

Interestingly, it appears that correlation has become much more stable starting early 2000, indicating improved pricing efficiency for these markets. Prior to 2000, the correlation was changing, though by a miniscule amount, on a daily basis. However, starting 2000, the daily correlation almost becomes constant for these countries and the fact that it coincides with the European integration is interesting. The implication of such constant correlation and its relationship with European economic integration is beyond the scope of this study.

Note that this is the most problematic aspect of computational details for estimating dynamic hedge ratios. The hedger has to determine the length of the within sample and out of sample. These decisions, in most cases, are not scientific. In the present context, the length of the within sample must be sufficiently long enough to capture the stylized facts of the two markets. In other words, the dynamic hedging program uses as many observations as possible to capture deviations from a long-run equilibrium. So, a short within sample may not be able to capture the dynamic properties of these two markets (cointegration) as they deviate from one another in the short run.

Another reason for choosing a particular sample period has to do with the way the hedger optimizes the nonlinear optimization model. The hedger needs to estimate a conditional hedging model using, say, only half of the sample observations. Once the base model converges, the hedger then adds one observation from the out of sample and re-optimizes the model. The converged model produces the hedge ratio for the next day, and this iterative process is repeated until the hedger exhausts the entire out of sample. At each step, the hedger runs the risk that the model will fail to converge. In that case, starting values would have to be changed to continue with the optimization process.

The hedger runs the risk that at any stage this computationally elegant and sophisticated hedging model may fail to converge.

Preliminary experiments using different out of sample windows resulted in disappointing results. The results are not reported here to conserve space.

We encountered significant difficulties in convergence of the iterative model for France. The GARCH-X model would only converge if we chose a within sample of 1110. As a result, caution must be applied when comparing GARCH and GARCH-X results to the ones reported earlier.

These results certainly suggest that the conditional hedging model may be unstable for out of sample forecasting of the hedge ratios. Several alternative solutions are available including a less stricter convergence criteria for the conditional hedging model and not imposing a long-run equilibrium condition (the error correction term) in a short-run model using daily data. We leave this for future research.