# Trinomial tree

Canonical: https://web4.xpay.bi/en/wiki/wp-en-20262149/

Language: en

Published: 2026-10-25T00:09:15.381Z

Updated: 2026-10-25T00:09:15.381Z

The trinomial tree is a lattice-based computational model used in financial mathematics to price options on equity. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing.

## Key facts

## Overview

The trinomial tree is a lattice-based computational model used in financial mathematics to price options on equity. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing.  Trinomial trees are also deployed

for fixed income and interest rate derivatives; see under Lattice model (finance).

## Formula

Under the trinomial method, the underlying stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path. These values are found by multiplying the value at the current node by the appropriate factor

u

\{\\displaystyle u\\,\}

,

d

\{\\displaystyle d\\,\}

or

m

\{\\displaystyle m\\,\}

where

u         =

e

σ

2                 Δ                 t

\{\\displaystyle u=e^\{\\sigma \{\\sqrt \{2\\Delta t\}\}\}\}

d         =

e

−             σ

2                 Δ                 t

=

## Application

The trinomial model is considered to produce more accurate results than the binomial model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For vanilla options, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For exotic options the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.

## See also

Binomial options pricing model Valuation of options Option: Model implementation Korn–Kreer–Lenssen model Implied trinomial tree Lattice model (finance) § Interest rate derivatives

## External links

Phelim Boyle, 1986. "Option Valuation Using a Three-Jump Process", International Options Journal 3, 7–12. Rubinstein, M. (2000). "On the Relation Between Binomial and Trinomial Option Pricing Models". Journal of Derivatives. 8 (2): 47–50. CiteSeerX 10.1.1.43.5394. doi:10.3905/jod.2000.319149. Archived from the original on June 22, 2007. \{\{cite journal\}\}: Cite uses deprecated parameter |citeseerx= (help) Paul Clifford et al. 2010. Pricing Options Using Trinomial Trees, University of Warwick Tero Haahtela, 2010. "Recombining Trinomial Tree for Real Option Valuation with Changing Volatility", Aalto University, Working Paper Series. Ralf Korn, Markus Kreer and Mark Lenssen, 1998. "Pricing of european options when the underlying stock price follows a linear birth-death process", Stochastic Models Vol. 14(3), pp 647 – 662 Peter Hoadley. Trinomial Tree Option Calculator (Tree Visualized)

## Sources

- Trinomial tree · Wikipedia: https://en.wikipedia.org/?curid=20262149

Source: Wikipedia contributors — https://en.wikipedia.org/?curid=20262149&oldid=1317997257

Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/

Adaptation notice: Reformatted for XPay; the source may be abridged, restructured or translated. Consult the cited revision for the original.