Binomial Option Pricing Calculator
Option value from a Cox-Ross-Rubinstein tree, including American early exercise, with Black-Scholes alongside for comparison.
Results
What this tool does
The binomial model prices an option by splitting its life into steps and letting the share go up or down at each one, then working backwards from every possible ending to today. It is slower than Black-Scholes and far more flexible: because it revisits every node, it can ask whether exercising early would be worth more than waiting — something no closed formula can do. Set the steps high and the European price converges on Black-Scholes, which is shown next to it. The difference between the American and European figures is what the right to act early is worth.
Formula
Cox-Ross-Rubinstein: u = e^(σ√Δt), d = 1÷u, p = (e^(rΔt) − d) ÷ (u − d) · backward induction through the tree
Variables
| Symbol | Meaning | Unit |
|---|---|---|
spot | Current price | — |
strike | Strike price | — |
rate | Risk-free rate | % |
vol | Volatility | % |
years | Years | — |
steps | Steps in the tree | — |
kind | Call or put | — |
style | When it can be exercised | — |
OP | What the option is worth | — |
BS | Black-Scholes, for comparison | — |
GP | Distance from Black-Scholes | — |
EU | Same option, exercisable only at the end | — |
EP | What early exercise is worth | — |
IV | Intrinsic value | — |
TV | Value that is only time | — |
UF | Up step | — |
DF | Down step | — |
RP | Risk-neutral chance of going up | % |
SY | Length of one step | — |
NS | Steps | — |
Worked example
- Current price100
- Strike price100
- Risk-free rate5 %
- Volatility20 %
- Years1
- Steps in the tree50
- Call or putcall
- When it can be exercisedeuropean
- What the option is worth10.4107
- Black-Scholes, for comparison10.4506
- Distance from Black-Scholes-0.0399
- Same option, exercisable only at the end10.4107
- What early exercise is worth0.0000
- Intrinsic value0.0000
- Value that is only time10.4107
- Up step1.028688
- Down step0.972112
- Risk-neutral chance of going up51.0614 %
- Length of one step0.020000
- Steps50
Limitations
- This is an informational calculator, not personalised financial advice. Rates, fees, taxes and contract conditions vary between institutions and countries.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- The tool works with whatever currency you use for the inputs; it does not convert between currencies.
Frequently asked questions
Why use a tree when Black-Scholes exists?
Because Black-Scholes only prices an option that can be exercised at the end. Most options on individual shares can be exercised at any time, and that extra freedom has a value which no closed formula captures. The tree handles it naturally: at every node it asks whether exercising now beats holding on, and takes the better of the two. Increase the steps and the European price converges on Black-Scholes, which is a useful check that the tree is working — the two figures are shown side by side for exactly that reason.
How many steps should I use?
Fifty is enough to see the shape; a few hundred is enough for the answer to stop moving in the fourth decimal. The convergence is not smooth — it oscillates above and below the true value as the steps increase, because whether a node lands exactly on the strike matters. That is why doubling the steps sometimes makes the answer look worse before it gets better. If you are comparing against Black-Scholes, use several hundred and look at the trend rather than any single figure.
What does the model assume?
That volatility is a single constant number for the whole life of the option, that the share pays no dividend, that you can borrow and lend freely at the risk-free rate, and that trading costs nothing. None of those is true. Volatility in particular is not observable — the figure you type in is a forecast, and the price that comes out is only as good as it. Real option markets price different strikes at different implied volatilities precisely because the model's assumption does not hold.