International Scholarly Research Network ISRN Economics Volume 2012, Article ID 481856, 5 pages doi:10.5402/2012/481856 Research Article Futures Hedges under Basis Heteroscedasticity Subhankar Nayak and Jacques A. Schnabel School of Business & Economics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 Correspondence should be addressed to Jacques A. Schnabel, jschnabel@wlu.ca Received 30 October 2012; Accepted 26 November 2012 Academic Editors: J. H. Haslag, T. Kuosmanen, and J. Zarnikau Copyright © 2012 S. Nayak and J. A. Schnabel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 assump- tion 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 autoregres- sive 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 remov- ing 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 Q 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 exhibits this type of heteroscedasticity. Define S 0 and S 1 as the spot rates or the values of the for- eign currency that prevail as the start and end, respectively, of the period. Employing direct quotation, all exchange rates considered in this paper are given as the number of domestic currency units per unit of the foreign currency examined. Similarly, define F 0 and F 1 as the futures rates that prevail at the start and end, respectively, of the period. Note that, whereas S 0 and F 0 are observed at the time the hedge is formed, S 1 and F 1 are unknown and thus are the random variables in this discussion. The existing literature on hedg- ing conventionally assumes that the basis or (S 1 −F 1 ) exhibits an unchanging variance; that is, σ 2 (S 1 − F 1 ) is a constant. Instead of the foregoing, this paper conjectures the percentage basis or the basis calculated as a percent of the spot rate; that is, e = (S 1 −F 1 )/S 1 exhibits a constant variance. Stated another way, this paper assumes that the basis itself, (S 1 − F 1 ), exhibits a variance that rises with S 1 . Thus, a