Binomial Option Price (Cox-Ross-Rubinstein)

European option price from a recombining binomial tree: in each step the stock moves up by u = e^{σ√Δt} or down by d = 1/u with risk-neutral probability p = (e^{rΔt} − d)/(u − d); the option is valued at expiry and discounted back node by node. As the number of steps grows the price converges to Black-Scholes.

When to use: Use to see option pricing built up from first principles, and as the foundation that extends to American exercise and dividends where Black-Scholes cannot go. A few hundred steps is enough for a price within a cent or two of the closed form.

Calculator

Formula

u = e^{\sigma\sqrt{\Delta t}}, \quad d = \tfrac{1}{u}, \quad p = \frac{e^{r\Delta t} - d}{u - d}, \quad V_{t} = e^{-r\Delta t}\left[p\,V_{t+1}^{u} + (1-p)\,V_{t+1}^{d}\right]

Variables

Symbol	Name	Description	Unit
BinomialPrice	Option Price	European option value from the tree	$
S	Stock Price	Current price of the underlying	$
Strike	Strike Price	Exercise price of the option contract	$
Rf	Risk-Free Rate	Continuously compounded annual risk-free rate as a decimal	%
Sigma	Volatility	Annualized volatility of the underlying as a decimal (e.g. 0.25 for 25%)	%
T	Time to Expiration	Time to expiration in years (e.g. 0.25 for 3 months)	years
Steps	Tree Steps	Number of time steps in the tree (1 to 2,000)	integer
OptionType	Option Type	Call (1) or Put (-1) — enter 1 for a call, -1 for a put	integer

Real-Life Examples

Example 1: At-the-Money Call, 100 Steps

Stock $100, strike $100, risk-free 5%, volatility 20%, one year, 100 steps.

Given

S = $100.00Strike = $100.00Rf = 5.0000%Sigma = 20.0000%T = 1.00 yearsSteps = 100.00OptionType = 1.00

Step-by-Step

1.Δt = 1/100; u = e^(0.2 × 0.1) = 1.02020, d = 0.98020

2.p = (e^(0.0005) − 0.98020) / (1.02020 − 0.98020) = 0.50750

3.Value the 101 terminal payoffs and discount back through the tree

4.Price = $10.43

Result:$10.43

Black-Scholes gives $10.45 for the same inputs. The 100-step tree is within two cents; at 500 steps the gap is under half a cent.

Example 2: Out-of-the-Money Put, 50 Steps

Stock $50, strike $55, risk-free 3%, volatility 30%, six months, 50 steps.

Given

S = $50.00Strike = $55.00Rf = 3.0000%Sigma = 30.0000%T = 0.50 yearsSteps = 50.00OptionType = -1.00

Step-by-Step

1.Δt = 0.01; u = e^(0.3 × 0.1) = 1.03045, d = 0.97045

2.p = (e^(0.0003) − 0.97045) / (1.03045 − 0.97045) = 0.49750

3.Put payoffs max(0, 55 − S_T) at the 51 terminal nodes, discounted back

4.Price = $6.82

Result:$6.82

The put is $5 in the money with $1.82 of time value on top. Switching the option type to 1 prices the matching call at $2.63, and the two satisfy put-call parity.

Frequently Asked Questions

The error shrinks roughly in proportion to 1/steps and oscillates between odd and even counts. A hundred steps is within a few cents of Black-Scholes for ordinary inputs; five hundred is within a fraction of a cent.

Black-Scholes is the limit of this tree as the steps go to infinity. With finitely many steps the stock can only land on a discrete set of prices, so the option value is slightly off; the gap closes as steps are added.

The same tree can, by checking early exercise at every node. This formula values European exercise only, so it can be compared directly with Black-Scholes.

Related Formulas

Black-Scholes Call Price

Same family

Black-Scholes Put Price

Same family

Put-Call Parity

Option Intrinsic Value

Futures Price (Cost-of-Carry Model)