# Terminal node probabilities for a trinomial tree

I need to model the probability distribution of the values generated from a recombining trinomial tree after n steps

i.e. a recombining tree with three forward steps (up, down or no change) from each node, and an inverse relationship between the up and down multipliers

Are there formulae available for the probabilities associated with each terminal node? Any help would be much appreciated

## 1 Answer

The probability after $N$ steps to have $u$ up, $d$ down and $N-u-d$ steps with no change is a multinomal distribution: $$P(u,d,N) = {p_{up}}^{u} * {p_{down}}^{d} * {p_{stay}}^{N-u-d} * N!/(u!*d!*(N-u-d)! )$$ Take the sum over all combinations with the same $m=u-d$. Than the probability to land at $m$ steps up from your starting point after $N$ steps, with $-N \leq m \leq N$ is: $$P(m, N) = \sum_{d=max(0,-m)}^{floor((N-m)/2)} { {p_{up}}^{m+d} * {p_{down}}^{d} * {p_{stay}}^{N-2d-m} * N!/(d!*(m+d)!*(N-2d-m)! ) }$$