Options with discrete deterministic dividends

in which CBS is the usual European call price of an option without dividends. The function φ(S0, S, td) is the log-normal no dividend density function. The integral representation given in (1) does not have an explicit solution. It can be approximated with a numerical integration method, which may be complicated due to the right side boundary at infinity. Another possibility is to solve the Black-Scholes equation with numerical techniques, because the integral representation (1) is based on the Black-Scholes equation. Suppose there is one dividend payment at time td within the lifetime of the option, then the time-interval is split into two parts: [0, T ] ⊃ [0, t− d ]∪[t d , T ], whereby t d − t− d → 0. The condition V (S(t d ), t d ) = V (S(t− d ), t− d ) has to be fulfilled [7]. The asset price will change by the dividend payment, S(t d ) = S(t− d ) − D. So at td, the condition reads: V (S, t d ) = V (S −D, t− d ). (3)