Futures Hedges under Basis Heteroscedasticity

Minimum variance and mean-variance optimizing hedges are developed when basis risk exhibits heteroscedasticity; that is, the variance of the difference between spot and futures prices is not constant but rises with the level of spot prices. Two different hedging objectives are modeled and optimized. The resulting optimality conditions are then interpreted both analytically and intuitively. Simulations are run to determine whether the model proposed here is superior to the traditional model in terms of minimizing the hedger’s terminal wealth. The resulting hedge ratios are shown to differ from those that are obtained for the traditional homoscedastic basis case, but consistent with the extant theoretical paradigm, the demand for futures contacts is dichotomized into pure hedging and pure speculative components. The simulations demonstrate that, under the statistical assumptions invoked, the proposed model implies uniformly less hedging and a lower variance of terminal wealth compared with the traditional model. 1. Introduction In the usual textbook treatment of the hedging decision, the basis or the difference between spot and futures prices is assumed to exhibit a constant variance. Examples are provided by Duffie [1], Hull [2], and Stulz [3]. The empirical research that has questioned the validity of this assumption of basis homoscedasticity has sought remediation by invoking sophisticated econometric techniques that do not entail the unchanging variance requirement, for example, the autoregressive conditional heteroscedastic (ARCH) model invoked by Park and Bera [4] and the generalized autoregressive conditional heteroscedastic (GARCH) model proposed by Kalimipalli and Susmel [5]. This paper takes a different tack to the issue by examining how the hedging decision itself changes as a result of removing the restriction that the basis exhibits a fixed variance. Formulas for the minimum variance and the mean-variance optimizing hedges are developed when the variance of the basis is assumed to increase with the level of spot prices. 2. Basis Heteroscedasticity This paper examines hedging in the traditional context of a single-period decision model where the hedge is formed at time 0, and the hedge is terminated or lifted at time 1. To provide a concrete decision context, the specific case of a firm wishing to hedge a foreign currency receivable amounting to that is projected to be received at time 1 is considered. The foregoing is consistent with the empirical evidence provided by Grabbe [6], who argues that the basis for foreign currency contracts