Binomial Option Pricing Model, Step by Step

A lattice you can compute by hand, then in code.

The binomial option pricing model builds a tree of possible underlying prices across time, then works backward from expiration to calculate the option's fair value today. Unlike Black-Scholes, which delivers a closed-form answer instantly, the binomial model is transparent at every node — you can follow the arithmetic step by step and see exactly where the price comes from. That transparency has a practical payoff: the model handles American versus European exercise naturally, which the Black-Scholes formula cannot do. If you want to price anything with an early-exercise feature, a binomial tree is the standard starting point, and you can also test your results against the options pricing calculator on this site.

The one-step binomial model

Start with the simplest possible case: a single time step of length Δt. The underlying sits at price S today. Over the next period it either goes up to S·u or down to S·d, where u > 1 and d < 1. At expiration the option pays whatever its intrinsic value is at each of those two endpoints.

To price the option, you do not need to know the actual probability of an up move. Instead you derive the risk-neutral probability p — the probability that makes the underlying's expected return equal to the risk-free rate:

p = (e^rΔt − d) / (u − d)

where r is the continuously compounded risk-free rate. The complementary probability of a down move is simply 1 − p. The option value today is then the discounted expected payoff under these risk-neutral probabilities:

V = e^−rΔt · [p · V_u + (1 − p) · V_d]

where V_u and V_d are the option values at the up and down nodes. No arbitrage is enforced by construction: if the tree drifts faster or slower than r, you could lock in a riskless profit by trading the option against a replicating portfolio in the underlying and cash.

Choosing u and d: Cox-Ross-Rubinstein

The most widely used parameterisation is the one developed by Cox, Ross, and Rubinstein in 1979. They set:

• u = e^σ√Δt
• d = 1/u = e^−σ√Δt

where σ is the annualised volatility of the underlying. This choice is not arbitrary — it ensures that the variance of the log-return over one step matches σ²Δt, so the tree is consistent with the same volatility assumption used in Black-Scholes. The up and down factors grow as you widen each step and shrink as you use more, finer steps.

For example, with σ = 0.25, r = 0.05, and Δt = 0.5 years (a two-step tree priced over one year): u = e^0.25·√0.5 ≈ 1.1934, d ≈ 0.8380, and p = (e^0.05·0.5 − 0.8380) / (1.1934 − 0.8380) ≈ 0.5765.

Backward induction through the tree

With more than one step you build the full tree of underlying prices first, then fold it back from right to left. At each terminal node, compute the option's intrinsic value — max(S − K, 0) for a call, max(K − S, 0) for a put. Then at each interior node, apply the one-step discounting formula above to get the continuation value from the two nodes it branches into.

For a European option, the value at each interior node is simply that continuation value. For an American option, you compare the continuation value to the intrinsic value of immediate exercise and take the larger of the two:

V_node = max(intrinsic, e^−rΔt · [p · V_u + (1 − p) · V_d])

That single line is why the binomial model matters for American options. It checks at every node whether early exercise is optimal, something a closed-form formula cannot replicate.

Worked two-step example

Price a one-year European call, K = 100, S = 100, σ = 0.25, r = 0.05, using two steps of Δt = 0.5.

From the parameterisation above: u ≈ 1.1934, d ≈ 0.8380, p ≈ 0.5765.

Node	Underlying price	Call payoff / value
uu (t = 1)	100 · 1.1934² ≈ 142.42	max(142.42 − 100, 0) = 42.42
ud (t = 1)	100 · 1.1934 · 0.8380 ≈ 100.00	max(100 − 100, 0) = 0.00
dd (t = 1)	100 · 0.8380² ≈ 70.22	max(70.22 − 100, 0) = 0.00

Step back to t = 0.5. The up node value is e^−0.05·0.5 · (0.5765 · 42.42 + 0.4235 · 0.00) ≈ 23.87. The down node value is e^−0.05·0.5 · (0.5765 · 0.00 + 0.4235 · 0.00) = 0.00.

Step back to t = 0. The call price is e^−0.05·0.5 · (0.5765 · 23.87 + 0.4235 · 0.00) ≈ 13.44. The Black-Scholes closed form for the same inputs gives approximately 12.33; the two-step tree is a rough approximation, but it converges quickly as you add steps.

Convergence to Black-Scholes

As the number of steps N → ∞, the binomial tree converges to the same price that Black-Scholes produces for a European option. The logic is that the binomial distribution of terminal prices approaches the lognormal distribution assumed by Black-Scholes, and the risk-neutral drift and variance match by construction. In practice, 50–100 steps gives prices within a few cents of the closed form on standard equity options, and 200–500 steps is common for American puts where accuracy matters more.

The convergence is not perfectly smooth — prices oscillate slightly above and below the true value as N increases depending on whether the strike falls on or between nodes. A standard fix is to average the results for N and N + 1 steps.

When to use the binomial model

• American options. Early exercise on puts — and calls on dividend-paying stocks — requires a tree. There is no Black-Scholes shortcut.
• Discrete dividends. A known dividend at a specific date fits naturally into the tree by shifting node prices at the ex-dividend step.
• Employee stock options. ESOs with vesting schedules and early-exercise behaviour are routinely valued with binomial models under accounting standards.
• Building intuition. The tree makes every assumption visible and every step auditable. Before deploying any closed-form model, building a binomial tree for the same option is the clearest sanity check available.

The binomial model trades the elegance of a single formula for a step-by-step lattice that mirrors the actual decision problem an option holder faces. That makes it slower to compute but more honest about optionality — and for any instrument where the holder might rationally exercise early, it is the right tool.