An efficient method for option pricing with discrete dividend payment

This paper deals with the construction of a numerical solution of the Black–Scholes equation modeling option pricing with a discrete dividend payment. This model is a partial differential equation with two variables: the underlying asset and the time to maturity, and involves the shifted Dirac delta function centered at the dividend payment date. This generalized function is suitable for approximation by means of sequences of ordinary functions. By applying a semidiscretization technique on the asset, a numerical solution is obtained and the independence of the considered sequence in a wide class of delta defining sequences is proved. From the study of the influence of the spatial step hh, it follows that the difference between the numerical solution for hh and h/2h/2 is O(h2)O(h2) as h⟶0h⟶0. The proposed method is useful for different discrete dividend types like a dividend of present value D0D0, a constant yield dividend or an arbitrary underlying asset-dependent yield dividend payment. Several illustrative examples are included.