Moments and transition probability of Trinomial Tree

I just switched from using WP-Latex to MathJax since Chrome is soon going to have built-in support for MathJax and it is easier to recover the tex codes for the reader.

To test out the functionality, I wanted to write down some equations befitting to the occasion. I scraped through my notes to see if there was anything interesting and worth posting. I stumbled upon Trinomial Tree ( \ddagger ). Now it is not that common and haven’t seen much usage of Trinomial Tree outside finance (One of the ways for Options Pricing), nonetheless I thought it would be interesting to refresh these formulas, you never know when you might need them again.

Trinomial Tree, a special case of binomial tree where instead of two we have three branches, lets call them Up ( u ) , Middle ( m ) and Down (d ) . Transition of each step could then be defined as —

  S(t+Δt)=S(t)×u with probability pu

  S(t+Δt)=S(t)×1 with probability 1pupd

  S(t+Δt)=S(t)×d with probability pd

For simplicity sake let’s consider that transition magnitude is same on each side, i.e
u=eσ2Δt,d=eσ2Δt,m=1

then the transition probability is given by
p_u = \left ( \frac{e^\frac{r \Delta t}{2} – e^{-\sigma\sqrt{\frac{r \Delta t}{2}}}}{e^{\sigma\sqrt{\frac{r \Delta t}{2}}} – e^{-\sigma\sqrt{\frac{r \Delta t}{2}}}} \right ) ^2

p_u = \left ( \frac{ e^{\sigma\sqrt{\frac{r \Delta t}{2}}} + e^\frac{r \Delta t}{2} }{e^{\sigma\sqrt{\frac{r \Delta t}{2}}} – e^{-\sigma\sqrt{\frac{r \Delta t}{2}}}} \right ) ^2 p\_m = 1 – p\_u – p_d

from above equations we can derive the **moments **as given below,
E[S(t_i+1)|S(t_i)]=erΔtS(ti)

Var[S(t_i+1)|S(t_i)]=ΔtS(ti)2σ2+O(Δt32)

Phelim Boyle in 1986.

An excellent 23 slide background on Trinomial including other forms besides Boyle’s.

Written on January 30, 2013