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 —

  \( S(t + \Delta t) = S(t) \times u \) with probability \( p_u \)

  \( S(t + \Delta t) = S(t) \times 1 \) with probability \( 1 - p_u - p_d \)

  \( S(t + \Delta t) = S(t) \times d \) with probability \( p_d \)

For simplicity sake let’s consider that transition magnitude is same on each side, i.e
\(u = e^{\sigma \sqrt{2 \Delta t} } \, , \,\, d = e^{-\sigma \sqrt{2 \Delta t }} \, , \,\, m=1\)

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,
\(\mathbb{E}[S(t\_{i+1})|S(t\_i)] = e^{r \Delta t}S(t_i)\)

\[\mathbb{V}ar[S(t\_{i+1})|S(t\_i)] = \Delta t S(t_i)^2 \sigma ^2 + \mathcal{O}(\Delta t^\frac{3}{2})\]

\( \ddagger \) Phelim Boyle in 1986.

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

Written on January 30, 2013