Very High Order Lattice Methods for One Factor Models

Lattice methods are often used to value derivative instruments. Multinomial lattice methods can in principle converge to the true value of the derivative to very high order. In this paper we describe how very high order multinomial lattices can be constructed and implemented when the SDE followed by the underlying state variable can be solved. We illustrate with comparisons between methods with branching order 3, 7, 11, 15 and 19 applied to a geometric Brownian motion. Incorporating both a terminal correction and appropriate truncation methods we find for the heptanomial lattice convergence rates at its theoretical maximum for European style options. With 50 time steps per year our errors are $O\\left( 10^{-11}\\right) $. With 100 time steps per year our errors are $O\\left( 10^{-13}\\right) $, approaching the practical limit of the accuracy obtainable in our implementation. We discuss alternative methods of enabling the heptanomial lattice to achieve high convergence rates for payoff functions with a finite number of critical points. As an example we value a binary option to a high degree of accuracy. We also investigate applications to American and Bermudan options. Based on our comparisons, we conclude that the heptanomial lattice is the fastest and most accurate of the lattices of higher order, and recommend its use as standard in many one factor lattice implementations.