Binomial and Trinomial Trees

A list of popular binomial and trinomial tree used in finance for pricing options.

Binomial Tree, geometric Brownian motion: Cox, Ross, Rubinstein

The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down.

    \begin{align*} u&=e^{\sigma \sqrt{t}} \\ d&=\frac{1}{u} = e^{-\sigma \sqrt{t}} \\ S_u &= S . u\\ S_d &= S. d \\ p_u&=\frac{e^{Yt}-d}{u - d}\\ p_s&=1-p_u \end{align*}

Trinomial Tree, geometric Brownian motion

The Trinomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move up, mid or down.

    \begin{align*} u&=e^{\sigma \sqrt{2t}},\;\; m=1, \;\; d= e^{-\sigma \sqrt{2t}} \\ \\ S_u &= S . u, \;\;S_m = S, \;\;S_d = S. d \\ \\ p_u&=\left( \frac{e^{Yt/2} - e^{-\sigma \sqrt{t/2}}}{e^{\sigma \sqrt{t/2}}-e^{-\sigma \sqrt{t/2}}}  \right)^2\\ p_d&=\left( \frac{e^{\sigma \sqrt{t/2}}- e^{Yt/2}  }{e^{\sigma \sqrt{t/2}}-e^{-\sigma \sqrt{t/2}}}  \right)^2\\ p_m&=1-p_u-p_d \end{align*}

Binomial Tree, geometric Brownian motion: Jarrow, Rudd

This is the Jarrow and Rudd version of the Binomial tree. The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down.

    \begin{align*} u&=e^{(Y-\frac{1}{2}\sigma^2)t +\sigma \sqrt{t}} \\ d&=e^{(Y-\frac{1}{2}\sigma^2)t - \sigma \sqrt{t}} \\ S_u &= S . u\\ S_d &= S. d \\ p_u&=p_d=\frac{1}{2} \end{align*}

Binomial Tree, geometric Brownian motion: Tian

This is the Tian version of the Binomial tree. The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down.

    \begin{align*} u&=\textstyle\frac{1}{2}\displaystyle e^{(Y+\sigma^2)t} \left(e^{\sigma^2t} +1 + \sqrt{e^{\sigma^4t^2} + 2e^{\sigma^2t}-3} \right) \\ d&=\textstyle\frac{1}{2}\displaystyle e^{(Y+\sigma^2)t} \left(e^{\sigma^2t} +1 -\sqrt{e^{\sigma^4t^2} + 2e^{\sigma^2t}-3} \right) \\ S_u &= S . u\\ S_d &= S. d \\ p_u&=\frac{e^{Yt}-d}{u - d}\\ p_d&=1-p_u \end{align*}

Binomial Tree, geometric Brownian motion: Trigeorgis

This is the Trigeorgis version of the Binomial tree. The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down.

    \begin{align*} \gamma &=Y - \textstyle\frac{1}{2}\displaystyle \sigma^2\\ x &= \sqrt{\sigma^2t+\gamma^2 t^2}\\ u&= e^x\\ d&=e^{-x} = \frac{1}{u} \\ S_u &= S . u\\ S_d &= S. d \\ p_u&=\frac{1}{2} +\frac{1}{2} \frac{\gamma t}{x}\\ p_d&=1-p_u \end{align*}