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

Tom P. Davis
The Journal of Derivatives Fall 2021, 29 (1) 49-69; DOI: https://doi.org/10.3905/jod.2021.1.130
Tom P. Davis
is director of fixed income and derivatives research at FactSet Research Systems in London, UK
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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.

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The Journal of Derivatives: 29 (1)
The Journal of Derivatives
Vol. 29, Issue 1
Fall 2021
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Evergreen Trees: The Likelihood Ratio Method for Binomial and Trinomial Trees
Tom P. Davis
The Journal of Derivatives Aug 2021, 29 (1) 49-69; DOI: 10.3905/jod.2021.1.130

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Evergreen Trees: The Likelihood Ratio Method for Binomial and Trinomial Trees
Tom P. Davis
The Journal of Derivatives Aug 2021, 29 (1) 49-69; DOI: 10.3905/jod.2021.1.130
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  • Article
    • Abstract
    • BACKGROUND AND GENERAL FRAMEWORK
    • THE BINOMIAL TREE
    • THE TRINOMIAL TREE
    • CONCLUSIONS
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    • APPENDIX
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