Evergreen Trees: The Likelihood Ratio Method for Binomial and Trinomial Trees

Abstract

Despite their age, binomial and trinomial trees are still used extensively in the financial industry to price securities with early exercise features such as American equity options and callable bonds. This technique is related to the fully explicit finite difference method used to numerically solve partial differential equations. The purpose of this article is to present an alternative mathematical derivation for binomial and trinomial trees using the path integral formalism. Recasting the tree in this light admits an extremely efficient, accurate, and novel method to calculate deltas by using the likelihood ratio method.

TOPICS: Statistical methods, derivatives, options, fixed income and structured finance

Key Findings

• ▪ Binomial and trinomial trees are derived by use of the path integral formalism and a novel quadrature technique called Shifted Gauss-Hermite Quadrature.

• ▪ The likelihood ratio method in the path integral formalism is derived, leading to an extremely fast and efficient method to calculate deltas on trees.

• ▪ Numerical results are presented which demonstrate the power of this method, which can be applied to securities with early exercise.