94. A Generalized Model for Optimum Futures Hedge Ratio

Under martingale and joint-normality assumptions, various optimal hedge ratios are identical to the minimum variance hedge ratio. As empirical studies usually reject the joint-normality assumption, we propose the generalized hyperbolic distribution as the joint log-return distribution of the spot and futures. Using the parameters in this distribution, we derive several most widely used optimal hedge ratios: minimum variance, maximum Sharpe measure, and minimum generalized semivariance. Under mild assumptions on the parameters, we find that these hedge ratios are identical. Regarding the equivalence of these optimal hedge ratios, our analysis suggests that the martingale property plays a much important role than the joint distribution assumption.

To estimate these optimal hedge ratios, we first write down the log-likelihood functions for symmetric hyperbolic distributions. Then we estimate these parameters by maximizing the log-likelihood functions. Using these MLE parameters for the generalized hyperbolic distributions, we obtain the minimum variance hedge ratio and the optimal Sharpe hedge ratio. Also based on the MLE parameters and the numerical method, we can calculate the minimum generalized semivariance hedge ratio.