Binomial Tree Model

Derivatives & Options · intermediate · CC-BY-4.0

The binomial tree model is a discrete-time option pricing methodology that models the underlying asset's price as moving up or down by specified factors at each time step, building a recombining lattice of possible prices and working backward from expiration to value the option through risk-neutral probability weighting at each node.

Key takeaways

The binomial model approximates continuous-time option pricing (Black-Scholes) and converges to it as the number of time steps approaches infinity.
At each node, the underlying price either increases by factor u or decreases by factor d; risk-neutral probabilities p and 1-p are calculated to prevent arbitrage.
The model naturally accommodates American-style options (early exercise), dividend payments, and changing parameters over time — areas where Black-Scholes closed-form solutions are inadequate.
Cox-Ross-Rubinstein (CRR) parameterization — the most common formulation — sets u = e^(σ√Δt), d = 1/u, ensuring the tree recombines and that the model is calibrated to the volatility parameter.
Computational complexity grows exponentially with the number of steps, but recombining tree structures reduce this to O(n²) for n time steps.

Explanation

The binomial option pricing model was developed by Cox, Ross, and Rubinstein in their seminal 1979 paper and immediately became the preferred tool for pricing American options, for which no simple closed-form solution exists. The model's key insight is that option pricing can be reduced to a backward induction problem on a discrete lattice of possible asset prices, using risk-neutral probabilities that eliminate the need to estimate expected returns.

In the CRR formulation, the time to expiration T is divided into n equal intervals of length Δt = T/n. At each step, the underlying price S can move up to S×u or down to S×d, where u = e^(σ√Δt) and d = e^(−σ√Δt) = 1/u. The risk-neutral up-probability is p = (e^(r×Δt) − d) / (u − d), where r is the risk-free rate. This probability is derived by constructing a one-period hedge portfolio and invoking the no-arbitrage condition — the hedge portfolio must earn the risk-free rate.

At the terminal nodes (time T), the option payoff is computed directly: max(S_T − K, 0) for a call, max(K − S_T, 0) for a put. The model then works backward through the tree. At each non-terminal node, the option value is: V_t = e^(−r×Δt) × [p × V_u + (1−p) × V_d], where V_u and V_d are option values at the up and down child nodes. For American options, this continuation value is compared to the immediate exercise value at each node, and the maximum is selected — this comparison cannot be made with Black-Scholes.

The binomial tree is particularly well-suited for pricing options with path-dependent early exercise features, discrete dividends, and time-varying parameters. For example, a company stock paying a known discrete dividend D at time t_D can be incorporated by subtracting the present value of the dividend from the current stock price before building the tree (the 'prepaid forward' approach). Trinomial trees extend the model to three branches per node (up, flat, down), improving convergence speed for a given number of steps and providing greater flexibility in calibrating the tree to market conditions.

For practitioners, the binomial model is the standard tool for valuing stock options in compensation plans (where the American feature and discrete dividends are material), warrants, real options in capital budgeting, and convertible bond analysis. It is also the foundation for more complex lattice-based models used in interest rate derivatives pricing.

Formula

u = e^(σ × √Δt), d = 1/u (CRR)
p = (e^(r × Δt) - d) / (u - d)
Node value: V = e^(-r × Δt) × [p × V_up + (1-p) × V_down]
American: V_node = max(Exercise Value, Continuation Value)

Example

A European call option on a non-dividend-paying stock: S = $100, K = $100, T = 1 year, r = 5%, σ = 20%. Using a two-step CRR binomial tree (Δt = 0.5): u = e^(0.20×√0.5) = 1.1503; d = 1/u = 0.8694; p = (e^(0.05×0.5) − 0.8694)/(1.1503 − 0.8694) = (1.0253 − 0.8694)/0.2809 = 0.5549. Terminal nodes: S_uu = 100×1.1503² = 132.32, S_ud = 100, S_dd = 75.36. Call payoffs: 32.32, 0, 0. Back one step: V_u = e^(−0.025) × (0.5549×32.32 + 0.4451×0) = 0.9753×17.93 = 17.49; V_d = e^(−0.025) × (0.5549×0 + 0.4451×0) = 0. Today: V = e^(−0.025) × (0.5549×17.49 + 0.4451×0) = 0.9753×9.70 = 9.46. The Black-Scholes value for the same option is approximately $10.45; the two-step binomial underestimates this due to coarse discretization, but converges rapidly with more steps.

Binomial Tree Model

Key takeaways

Explanation

Formula

Example

Related terms