An accurate approximation formula for pricing European options with discrete dividend payments

In this article, two relevant problems related to pricing European options with discrete dividend under the classic Black-Scholes framework are considered. For the case when a discrete dividend payment is proportional to the underlying asset value, we discuss an interesting phenomenon observed; the option price is independent of the dividend payment date. This appears to be at odds with one's intuition that dividend amount, as well as the dividend date, should both affect the price of a European call or put option. We reveal the fundamental reasons, from both mathematical and financial viewpoints, why this occurs. When the amount of the discrete dividend is fixed, we provide an approximation formula for European option prices, with only one-dimensional integrals involved. It should be noted that our formula is a general one since it can not only be applied when there is only a single dividend, but also be suitable for the case of multiple dividends.