Fast trees for options with discrete dividends∗

The valuation of options using a binomial non-recombining tree with discrete dividends can be intricate. This paper proposes three different enhancements that can be used alone or combined to value American options with discrete dividends using a nonrecombining binomial tree. These methods are compared in terms of both speed and accuracy with a large sample of options with one to four discrete dividends. This comparison shows that the best results can be achieved by the simultaneous use of the three enhancements. These enhancements when used together result in significant speed/accuracy gains in the order of up to 200 times for call options and 50 times for put options. These techniques allow the use of a non-recombining binomial tree with very good accuracy for valuing options with up to four discrete dividends in a timely manner. ∗The support of the Portuguese Foundation for Science and Technology Project PTDC/GES/78033/2006 is acknowledged. We thank Ana Carvalho for her helpful comments. †School of Economics and Management. University of Minho. 4710-057 Braga (Portugal). E-mail: [email protected]; Phone: +351 253 601 923; Fax:+351 253 601 380. ‡Corresponding author. School of Economics and Management. University of Minho. 4710-057 Braga (Portugal). E-mail: [email protected]; Phone: +351 253 601 923; Fax:+351 253 601 380. Fast trees for options with discrete dividends American options valuation using binomial lattices can be cumbersome. Several authors have suggested different approaches to speed the computation or to increase the speed of convergence. Among others, some authors suggested ways to reduce the number of the nodes in the tree, thus speeding up the computation time (e.g.: Baule and Wilkens [2004]). One example of the latter approach is a paper by Curran [1995] that has been largely ignored in the literature. One advantage of Curran’s [1995] method is that it is not an approximation to the value of a binomial tree, since it produces exactly the same result as Cox, Ross, and Rubinstein’s [1979] (CRR) tree with the same number of steps at a fraction of the computation time. The gains of speeding the computation of a binomial tree are more relevant for valuing options with underling assets that pay discrete dividends. In such cases the binomial trees do not recombine and the number of nodes rapidly explode even for a small number of time steps. There are several approximations that allow the use of recombining binomial trees, but all of them can in some occasions produce large valuation errors (usually they occur when a dividend is paid at the very beginning of the option life), or require a very large number of steps to avoid such valuation errors.1 The advantages of using a non-recombining binomial tree are twofold: first it considers the true stochastic process for the underlying asset which excludes arbitrage opportunities; and secondly it eliminates large valuation errors irrespective of when the dividends occur. This, in turn, results in a accurate valuation of such options. The problem of these trees is that the number of nodes in the tree grows exponentially with the number of dividends. This explains why usually a recombining approximation is used instead of the non-recombining tree. Unfortunately the method proposed by Curran [1995] is not directly applied to options on assets which pay discrete dividends. In light of the techniques proposed by Curran [1995], which in turn are based on the work of Kim and Byun [1994], this paper adjusts their acceleration techniques to the valuation of American put options and also proposes a different accelerated binomial method to the valuation of American call options. We also suggest two other improvements with very good results. One, called here Adapted Binomial, consists in making a time step to coincide with the ex-dividend dates. The other is to apply the Black and Scholes [1973] formula to obtain the continuation value in the last steps of the binomial tree. These adaptations along with the improvements here suggested result in significant speed/accuracy gains in the order of up to 200 times for call options and 50 times for put options. These techniques allow the use of a non-recombining binomial tree with very good Examples of such methods are: Schroder [1988], Hull and White [1988], Harvey and Whaley [1992], Wilmott, Dewynne, and Howison [1998], Vellekoop and Nieuwenhuis [2006].