Matching Sensitivities to Discrete Dividends: A New Approach for Pricing Vanillas

Abstract

The incorporation of a dividend yield in the classical Black-Scholes framework results in a minor modification of the well-known formula, since the lognormality of the underlying asset is preserved. However, market-makers prefer to work with cash dividends with fixed value instead of a dividend yield. Since there is no closed-form solution for the price of a European call in this case, many approximations have been proposed in the literature. Here, we present a new approach based on an exact analytical formula for the sensitivity to dividends of a European option. We use this result to elaborate a price proxy having the same Taylor expansion as the exact price with respect to the dividends. The resulting proxy has no computational complexity (the same as for the usual Black-Scholes formula), and numerical tests show the excellent accuracy of the method for all practical cases. Copyright © 2011 Wilmott Magazine Ltd.