Efficient pricing of vanilla and exotic options with multiple discrete dividends using finite-difference method

A modern system for trading on electronic markets should calculate theoretical prices of thousands of vanilla and exotic options in real time. It is much trickier to satisfy this requirement if underlying securities pay discrete dividends during option lifetime, because exact closed formulas are usually not available in this case and more time-consuming numerical procedures are employed. In this paper we address pricing of vanilla European and American options, as well as Asian options, all with multiple discrete dividends. We describe a common approach for all setups based on finite-difference solution of Black-Scholes (BS) partial differential equation (PDE) using Crank-Nicolson (CN) scheme with Rannacher time stepping. If an exact or approximate closed formula is available for the option price without dividends, we apply it as a final condition on the last ex-dividend date instead of a standard payoff function on the expiration date. This original contribution substantially improves numerical results for European and exotic options, because such final conditions are smooth, while payoffs are not, which is crucial for CN scheme. Besides, the computation time is shortened proportionally to the reduction of time domain. The approach is efficient also for Asian options with dividends paid before the averaging period, because it eliminates the need in extra dimension for the average underlying price in BS PDE. 

Highlights: Generic framework to price vanilla and exotic options with dividends is developed; Crank-Nicholson scheme with Rannacher time stepping provides stability; Final condition is moved to last dividend date if analytical solution is available.