One step

Over a short interval \(\Delta t\) the stock either rises by a factor \(u\) or falls by a factor \(d\). A portfolio of one option and \(-\Delta\) shares can be made riskless, and a riskless portfolio must earn \(r\). Solving gives the risk-neutral probability of an up move,

$$ p = \frac{e^{(r-q)\Delta t} - d}{u - d} $$

and the option value one step earlier as the discounted expectation:

$$ V = e^{-r\Delta t}\left[ p\, V_u + (1 - p)\, V_d \right] $$

Many steps

Chain the steps and the tree recombines: an up then a down lands on the same node as a down then an up, so \(N\) steps produce only \(N + 1\) terminal nodes instead of \(2^N\). Fill in the payoff at expiry and roll back one level at a time. For an American option, at every node compare holding with exercising and keep the larger.

To see it happen, watch the tree on the tree page: it grows from today's price to the prices at expiry, then the value rolls back to today node by node, with each node's formula and numbers.

Choosing \(u\), \(d\) and \(p\)

Any choice that matches the mean and variance of the lognormal return over \(\Delta t\) converges to Black-Scholes. The classic choices:

• Cox-Ross-Rubinstein: \(u = e^{\sigma\sqrt{\Delta t}}\), \(d = 1/u\). The tree is centred on the spot.
• Jarrow-Rudd: \(p = \tfrac{1}{2}\) with the drift built into \(u\) and \(d\).
• Tian: matches the third moment as well.
• Leisen-Reimer: chooses \(p\) through an inversion of the binomial distribution so that a node lands exactly on the strike.

The first three converge at order \(1/N\) and oscillate, because the strike moves between nodes as \(N\) changes. Leisen-Reimer converges smoothly at order \(1/N^2\): a few dozen steps give the accuracy the others need thousands for.

Trinomial trees

Add a middle branch where the stock stays put. With \(u = e^{\sigma\sqrt{2\Delta t}}\) the three probabilities match the first two moments and stay positive. A trinomial tree is the same thing as an explicit finite-difference grid, which is why it converges a little faster per step than a binomial tree.

Greeks from the tree

Delta is the slope between the two nodes after the first step, and Gamma the change in that slope across the three nodes after the second. Theta comes from the middle node two steps in, which sits at the original spot in a Cox-Ross-Rubinstein tree. No bumping needed.