Stochastic expansion for the pricing of call options with discrete dividends

In the context of an asset paying affine-type discrete dividends, we present closed analytical approximations for the pricing of European vanilla options in the Black-Scholes model with time-dependent parameters. They are obtained using a stochastic Taylor expansion around a shifted lognormal proxy model. The final formulae are respectively first, second and third order approximations w.r.t. the fixed part of the dividends. Using Cameron-Martin transformations, we provide explicit representations of the correction terms as Greeks in the Black-Scholes model. The use of Malliavin calculus enables us to provide tight error estimates for our approximations. Numerical experiments show that the current approach yields very accurate results, in particular compared to known approximations of [BGS03, VW09], and quicker than the iterated integration procedure of [HHL03] or than the binomial tree method of [VN06]. ∗This research was partially supported by University Joseph Fourier (MSTIC grant REFINE) and by Fondation du Risque Chaire Risques Financiers Société Générale Ecole Polytechnique ENPC (second author).