Fast Trees for Options with Discrete Dividends

Abstract

The binomial model is a workhorse for valuing options with early exercise possibilities, including options with American or Bermudan exercise and barrier options. The procedure is straightforward when the underlying asset does not pay dividends, and it can be readily adapted to a payout at a fixed proportional rate. But discrete dividends cause trouble because they make the tree “splinter” rather than recombining at each time step. If the tree recombines, an up move followed by a down move produces the same asset price as a down move followed by an up move. But if the intermediate date corresponds to an ex-dividend date and the same fixed amount is subtracted from the up and down nodes, the tree no longer recombines at the subsequent step. The number of nodes in a non-recombining tree increases at a geometric rate, quickly leading to unreasonably long execution times. A number of alternative procedures have been developed to deal with this problem. In this article, Areal and Rodrigues adopt a new approach, accepting the splintering of the binomial lattice but then applying several techniques to accelerate the calculations. They show that their procedures are faster and more accurate than the current methods that force the tree to recombine despite discrete dividend payouts.