Analysis of the sensitivity to discrete dividends : A new approach for pricing vanillas

TL;DR

This paper introduces a new, fast, and highly accurate method for pricing European options with fixed cash dividends by deriving an exact sensitivity formula and creating a Taylor expansion-based proxy.

Contribution

The paper presents an exact analytic formula for dividend sensitivity and a novel approximation method that matches its Taylor expansion, improving computational efficiency and accuracy.

Findings

The method achieves high accuracy across practical cases.

The approximation has the same computational complexity as Black-Scholes.

Numerical tests confirm the method's precision.

Abstract

The incorporation of a dividend yield in the classical option pricing model of Black- Scholes results in a minor modification of the Black-Scholes formula, since the lognormal dynamic 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 methods have been proposed in the literature to approximate it. Here, we present a new approach. We derive an exact analytic formula for the sensitivity to dividends of an European option. We use this result to elaborate a proxy which possesses the same Taylor expansion around 0 with respect to the dividends as the exact price. The obtained approximation is very fast to compute (the same complexity than the usual Black-Scholes formula) and numerical tests show the extreme accuracy of the method for all practical cases.