The binomial model prices an option from a simpler assumption than Black-Scholes: the underlying can only move to one of two prices — up or down — over the period.
How it works
The risk-neutral probability of the up move is derived from the risk-free rate and the up/down factors; discounting the probability-weighted expected payoff back to today gives the option value.
What this does not include
This does not include the multi-step binomial tree, which chains many of these one-step calculations together and converges toward the Black-Scholes result as the number of steps grows — this calculator shows just the foundational single-step building block.
How to use this calculator
- Enter spot price, strike price, up and down factors, risk-free rate, and time to expiry.
A worked example
Spot price $100, strike $100, up factor 1.1, down factor 0.9, 5% risk-free rate, 1 year to expiry: option value = $7.19, risk-neutral probability of the up move = 75.64%.
What the variables mean
| Variable | Meaning |
|---|---|
| Spot price, strike price | Current price and option strike |
| Up factor, down factor | Multipliers describing the two possible price moves in one step |
| Risk-free rate | The risk-free interest rate |
Edge cases worth knowing
The risk-neutral probability isn’t the same as the real-world probability of the price moving up. It’s a mathematical construct that makes the pricing formula arbitrage-free, distinct from an actual forecast of market direction.
The up factor must exceed the down factor for the model to be valid — otherwise there’s no genuine “up” move to price against, so the calculator declines to show a result when that ordering is reversed.
Frequently asked questions
Why is it called “risk-neutral” probability?
It’s not the real-world probability of the stock actually going up — it’s a mathematical construct that makes the discounted expected payoff match a no-arbitrage price, regardless of investors’ actual risk preferences.
How does this relate to Black-Scholes?
As the number of steps in a multi-step binomial tree increases toward infinity, the model’s result converges to the Black-Scholes price — the two are connected, not competing, models.
Can the binomial model price American-style options?
Yes — unlike the closed-form Black-Scholes formula, a multi-step binomial tree can check for early-exercise value at every node, making it well-suited for American options.
