Closed formula for options with discrete dividends and its derivatives

We present a closed pricing formula for European options under the Black-Scholes model and formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and by expressing the spatial derivatives as expectations under special measures, as in Carr, together with an unusual change of measure technique that relies on the replacement of the initial condition. The closed formulas are attained for the case where no dividend payment policy is considered. Despite its small practical relevance, a digital dividend policy case is also considered which yields approximation formulas. The results are readily extensible to time dependent volatility models but no so for local-vol type models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis using the formulas here obtained. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.