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
then the transition probability is given by
from above equations we can derive the **moments **as given below,
\( \ddagger \) Phelim Boyle in 1986.
An excellent 23 slide background on Trinomial including other forms besides Boyle’s.