Accurate approximation formulas for stock options with discrete dividends

Abstract

Pricing options on a stock that pays discrete dividends has not been satisfactorily settled in the literature. Frishling (2002) shows that there are three different models to model stock price with discrete dividends, but only one of these models is close to reality and generates consistent option prices. We follow Frishling (2002) by calling this model Model 3. Unfortunately, there is no analytical option pricing formula for Model 3, and many popular numerical methods such as trees are inefficient when used to implement Model 3. A new stock price model is proposed in this article. To guarantee that the option prices generated by this new model are close to those generated by Model 3, the distributions of the new model at exdividend dates and maturity approximate the distributions of Model 3 at those dates. To achieve this, a discrete dividend in Model 3 is replaced by a continuous dividend yield that can be represented as a function of discrete dividends and stock returns in the new model. Thus, the new model follows a lognormal diffusion process and the analytical option pricing formulas can be easily derived. Numerical experiments show that our analytical pricing formulas provide accurate pricing results.