Binomial Option Pricing Calculator: American and European

Cox-Ross-Rubinstein option model

Binomial Option Pricing Calculator

Estimate American or European call and put values with a recombining CRR binomial tree, continuous dividend yield and an early-exercise comparison.

American value is checked at every tree node.
Choose the payoff at expiration.
Formatting only; no currency conversion is performed.
Current price in the selected display currency.
Exercise price in the same currency.
Enter calendar days or years according to the next field.
Days are divided by 365 to obtain model years.
Annualized standard deviation, not a daily percentage.
Continuously compounded; negative rates are accepted.
Continuous yield. Enter 0 when no dividend is modeled.
Use 5–1,000. More steps usually improve convergence but take longer.

This is a model estimate under constant volatility, rate and continuous dividend yield. It is not a live option quote or exercise recommendation.

CRR binomial option pricing formula

Δt = T ÷ N; u = eσ√Δt; d = 1 ÷ u

p = [e(r−q)Δt − d] ÷ (u − d)

Continuation value = e−rΔt × [pVup + (1−p)Vdown]

An American node equals the greater of continuation value and immediate intrinsic value. A European node uses continuation value only.

S is the current underlying price, K is strike, T is time in years, N is the number of steps, r is the continuously compounded risk-free rate, q is continuous dividend yield and σ is annual volatility. Terminal call payoff is max(ST − K, 0); terminal put payoff is max(K − ST, 0).

Verified examples

One-year at-the-money call

With S = 100, K = 100, one year, volatility 20%, risk-free rate 5%, dividend yield 0% and 200 steps, the CRR European call is approximately 10.4406. The American call has the same value to displayed precision because early exercise is not beneficial in this no-dividend example.

One-year at-the-money American put

With the same assumptions and 200 steps, the American put is approximately 6.0864, compared with a European binomial value of approximately 5.5635. The difference is the model’s early-exercise premium.

American versus European options

American style allows exercise at every modeled step. European style allows exercise only at expiration. For a non-dividend-paying call, the model normally produces no early-exercise premium. American puts and dividend-paying calls can have a positive premium because immediate exercise may sometimes exceed continuation value.

How to use the convergence table

A binomial tree is discrete, so the estimate can move as the step count changes. Compare the half-step, selected-step and double-step rows. Close values suggest numerical convergence under the same assumptions; they do not prove that market inputs or the model are correct.

Limits and assumptions

  • The model uses a recombining Cox-Ross-Rubinstein tree with constant volatility, rate and continuous dividend yield.
  • It does not model discrete dividend dates, volatility smiles, changing rates, transaction costs, taxes, liquidity, spreads or assignment procedures.
  • The risk-neutral probability must remain strictly between 0 and 1. Extreme input combinations can violate this no-arbitrage condition for the chosen step size.
  • Spot and strike support six decimals up to 100,000,000. Time is limited to 10 years, volatility to 200% and the tree to 1,000 steps.
  • More steps can reduce discretization error but do not make uncertain volatility, rates or dividends known.
  • The early-exercise premium is the American tree value minus the European tree value at the same step count; it is not a broker fee or market premium.

Risk note: Small changes in volatility, time, dividends and rates can materially change option values. Exercise decisions can also depend on taxes, borrow costs, financing and market microstructure not modeled here. This calculator is educational and is not investment advice.

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