A Closed-Form Approximation for Valuing Basket Options

Abstract

The no-arbitrage valuation .f basket options is complicated by the fact that the sum of lognormal random variables is not lognormal. This problem is shared with arithmetic Asian options as well. Various ad hoc approximation techniques have been proposed, none of them very satisfactory or accurate. In this article we suggest using the rec&rocal gamma distribution as an approximation for the state-price density (SPD) function .f the underlying basket stochastic variable. This, in turn, allows us to obtain a closed-jorm expression for the price of a basket option. The technique, when compared against a simple lognormal approximation, perjorms at its best when the correlation structure of the underlying basket exhibits a spec$c decaying pattern. As a by-product, we introduce a formal approach for assessing the goodness of _fit of candidate distributions for approximating the SPD. Finally, we present a numerical example in which we apply our formula to value (G-7) index-linked guaranteed investment certificates, which can be decomposed into a zero-coupon bond and a basket option.