Terminal Probabilities in a Recombining Trinomial Tree

Summary

The document derives the probability of reaching a terminal position in a recombining trinomial tree after a fixed number of steps. Each step can move up, move down, or stay unchanged, with corresponding probabilities. First, it gives the multinomial probability for a particular count of up, down, and unchanged steps, accounting for the number of ways those steps can be ordered.

To find the probability of a terminal position, the method sums the probabilities of all step-count combinations that produce the same net displacement: up moves minus down moves. The stated bounds identify the feasible combinations for each displacement. This provides a direct formula for the distribution across terminal nodes when the step probabilities are specified. The explanation assumes the step probabilities apply throughout the tree and does not discuss how to estimate them, dependencies between steps, or extensions to state-dependent probabilities.

Key ideas

• The probability of a specific combination of up, down, and unchanged moves follows a multinomial distribution.
• Different step sequences can reach the same terminal node and must be combined.
• A terminal node's net displacement is the number of up moves minus the number of down moves.
• Summing over feasible move counts gives the terminal probability for each displacement.

Tags

# Terminal node probabilities for a trinomial tree


# 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

## Answer by Ami44 (score 2)

https://quant.stackexchange.com/a/29900

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: $$ P(m, N) = \sum_{d=max(0,-m)}^{floor((N-m)/2)} { {p_{up}}^{m+d} * {p_{down}}^{d} * {p_{stay}}^{N-2d-m} * N!/(d!*(m+d)!*(N-2d-m)! ) } $$

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