Binomial trees and early exercise

A Cox-Ross-Rubinstein tree that values American options by checking at every step whether exercising beats holding.

What it does

The tree divides the time to expiry into N steps. In each step the price moves up by a factor u or down by d = 1/u. Working backward from the payoffs at expiry, each node's value is the discounted risk-neutral average of the two nodes after it. For an American option the node is worth the larger of that holding value and the payoff from exercising right there.

The early-exercise premium is the American price minus the European price on the same tree, so the tree's discretization error largely cancels.

Why it is used

The Black-Scholes formula cannot value the right to exercise early. The tree can, and as the steps get finer its European price converges to the formula, which makes it easy to check.

Inputs

The same inputs as Black-Scholes-Merton.
Number of steps N: 500 by default, 100 to 2,000 on the page (up to 5,000 through the API).

Formulas

Δt = T / N, u = e^(σ·√Δt), d = 1 / u, p = (e^((r − q)·Δt) − d) / (u − d) Hold value = e^(−r·Δt) · [p·V_up + (1 − p)·V_down] European: V = hold value American: V = max(hold value, exercise value) Premium = V_American − V_European (same tree)

Assumptions

The same lognormal, constant-volatility model as Black-Scholes-Merton, in discrete steps.
Exercise is possible only at the tree's time steps.
When the drift is large relative to volatility, the standard CRR probability p falls outside 0 to 1 (the tree would allow arbitrage). The tree is then centered on the drift, u, d = e^((r − q)·Δt ± σ·√Δt), which converges to the same limit.
Log prices at the extreme nodes are capped to stay finite; those nodes carry negligible probability.

How to read the results

The European tree price zigzags toward the Black-Scholes value as steps are added, because the strike falls at different positions between nodes. The error shrinks roughly in proportion to 1/N.

Without dividends, an American call is never worth exercising early: exercising gives up the option's time value and the interest on the strike. Its price equals the European price (Merton, 1973). American puts, and calls on dividend-paying assets, can be worth exercising early, and the chart of value against spot shows where the American value meets the payoff line.

Limitations

The zigzag means a single tree price can be off by more than its average error; smoothing techniques (averaging adjacent N, or a Black-Scholes value at the last step) are not applied.
Discrete cash dividends, which drive most early exercise of calls in practice, are approximated by a continuous yield.
Constant volatility; the tree does not reproduce a volatility smile.

Where it can fail

Very few steps give a coarse price; with five steps the tree is a teaching example, not a price.
The exercise boundary read from the chart is limited by the tree's grid and the spacing of the spot prices shown.

Validation on current data

Hull's five-step American put, then the European tree price at increasing steps against the exact Black-Scholes value, recomputed by the running engine.

References

Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics 7(3).
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1).
Hull, J. C. Options, Futures, and Other Derivatives. Pearson (chapters on binomial trees).

Try it in the options pricer