American Option Pricing Calculator
Price American calls and puts via a Cox-Ross-Rubinstein binomial tree. Handles early exercise and dividend-paying underlyings — the two cases the Black-Scholes formula cannot.
Results
For this call (183 days, 2.0% dividend yield), the early-exercise premium is essentially zero — the American and European prices match. This is the textbook result for calls on non-dividend-paying stocks: the right to exercise early has no economic value, so the American option is worth the same as the European one.
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Frequently asked questions
When to use a binomial tree instead of Black-Scholes
The Black-Scholes-Merton formula is exact, fast, and has a closed form — but only for European options on non-dividend-paying stocks. Real markets violate both assumptions constantly: most US-listed equity options are American-style (exerciseable any time before expiration), and most stocks pay dividends. The Cox-Ross-Rubinstein binomial tree handles both cases by working backward through a discrete tree of possible price paths and, at each node, taking the maximum of (exercise now) and (continue holding).
How the model works
The binomial tree partitions the time to expiration into n equal steps. At each step the stock can move up by a factor u or down by a factor d; with log-normal returns and matching the underlying volatility, u = exp(σ·√Δt) and d = 1/u. The risk-neutral probability of an up move is p = (exp((r-q)·Δt) − d) / (u − d).
At each terminal node, the option payoff is its intrinsic value. Working backward, the value at each interior node is the maximum of (a) exercise immediately and take the intrinsic value, or (b) hold and take the discounted risk-neutral expected value of the two child nodes. The price at the root of the tree is today's option value.
The early-exercise premium, in dollars
This calculator shows the binomial price and the equivalent Black-Scholes price side-by-side. The difference is the early-exercise premium — the dollar amount you would pay extra for the American feature. Three observations:
- American call on a non-dividend stock: premium is exactly zero (proven theorem — never optimal to exercise early). Use Black-Scholes instead, it's faster and exact.
- American call on a dividend-paying stock: premium can be material (especially right before ex-dividend), worth using the binomial tree.
- American put: premium can be material, especially when in-the-money. Always check if it matters for your use case.
Convergence and step count
The binomial tree converges to the continuous-time price as the number of steps → ∞. Convergence is roughly O(1/n) but oscillates, so doubling steps doesn't halve the error. For most practical purposes, 100-200 steps gives accuracy within a cent. Near-the- money short-dated options converge slowest; very deep ITM/OTM options converge fastest.
What this calculator does NOT do
- Discrete dividends — modeled here as a continuous yield. For known discrete dividends on specific dates, you need a tree that models cash dividends explicitly.
- Stochastic volatility — the tree assumes constant σ. Real markets have volatility smiles and skews; for that, use the API's volatility-surface endpoint.
- Path-dependent payoffs — for barrier, Asian, or lookback options, use the dedicated exotic endpoints in the API.