TradingCalcAmerican Option Calculator
OPTIONS · COX-ROSS-RUBINSTEIN BINOMIAL TREE WITH EARLY EXERCISE
American put or call price on a binomial tree, with delta, gamma, theta, an early-exercise flag and the European price beside it.
Volatility, rate and dividend yield are annual, in percent. Steps is the size of the binomial tree (50 to 2000): more steps are slower and closer to the exact value.
An American option can be exercised before expiry, and that has a price
Black-Scholes prices a European option, one that can only be exercised on its expiry date. Exchange-traded stock and ETF options in the US are American: the holder may exercise on any day. That right is worth something whenever exercising early can beat waiting, which happens for puts on a falling price with a positive interest rate, and for calls on an asset that pays a dividend. There is no closed formula for it, so the price comes from a binomial tree that checks the early-exercise choice at every node.
How the tree works
The life of the option is cut into N equal steps of dt = T / N. At each step the price moves up by a factor u or down by d (the Cox-Ross-Rubinstein tree), then the value is rolled back from expiry to today:
- u = exp(vol × √dt), d = 1 / u
- p = (exp((r − q) × dt) − d) / (u − d)
- hold = exp(−r × dt) × (p × Vup + (1 − p) × Vdown)
- V = max(hold, intrinsic value at that node)
Here r is the interest rate and q the dividend yield. Delta and gamma are read from the nodes of the first and second step, theta from the middle node of the second step against today's value.
Worked example
A put on a $100 stock, strike $110, 90 days, 30% volatility, 5% rate, no dividend:
- American put, 500 steps: $11.7509
- European put (Black-Scholes): $11.4550
- Early-exercise premium: $0.2959
- Delta −0.7131
How the step count changes accuracy
A tree with few steps is a coarse picture of the price. The same put costs $11.7650 at 50 steps, $11.7509 at 500 and $11.7508 at 2,000. The page shows the tree error next to the result: the same tree run without early exercise, minus the exact Black-Scholes European price. The American value carries an error of a similar size, and it shrinks as the steps grow, though not smoothly: an even and an odd step count can land on different sides of the exact value. 500 steps is a good default; use 2,000 when a cent matters.
What it assumes
Constant volatility and a constant rate. The dividend is a continuous yield, not a list of dated payments, so a single large dividend just before an ex-date is smoothed out. A negative interest rate is not accepted. The early-exercise flag is a reading of the tree: next to the exercise boundary a different number of steps can flip it. The tree was checked against an independent Python implementation and against QuantLib's binomial engine, the Barone-Adesi-Whaley approximation and a 5,001-step Leisen-Reimer tree, on 1,000 random options.
Where to go next
A European option, or the Greeks in closed form? Use the Black-Scholes Calculator. Sizing the position that goes with the option? Run it through the Pre-Trade Check.
Use via API or MCP
This calculation is available as a deterministic API call: useful for bots or AI agents that need an American option price with its early-exercise flag, not a European approximation.
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What is the difference between an American and a European option?
A European option can be exercised only on its expiry date, an American option on any day up to it. Because the holder has more choices, an American option is worth at least as much as the European one with the same terms. The difference is the early-exercise premium, shown next to the result.
When does early exercise pay?
For a put, when the price has fallen far below the strike and the interest on the cash you would receive is worth more than the remaining chance of a further drop. For a call, almost only when the asset pays a dividend: with no dividend an American call is never exercised early and costs the same as the European call.
Why does it use a tree and not a formula?
No exact closed formula exists for an American put. A binomial tree checks the exercise choice at every possible price on every step and converges to the exact value as the steps grow. The approximations in closed form, such as Barone-Adesi-Whaley, are faster but can be off by more.
How many steps should I use?
The default of 500 is close to the exact value for most options. More steps cost more time and shrink the error, though not smoothly. The tree error line shows what the chosen step count costs on the European price of the same option, which is a good guide to the American one.
Does it handle dividends?
It takes a continuous dividend yield in percent per year, the usual approximation for an index, an ETF or a stock with regular dividends. Dated cash dividends are not modelled, and a very large single dividend before the ex-date is smoothed over the whole life of the option.
Can I use it for crypto options?
Deribit and Binance crypto options are European, so the Black-Scholes Calculator is the right model there. This calculator is for American-style contracts, such as US stock and ETF options.