Cox-Ross-Rubinstein Model
The Cox-Ross-Rubinstein (CRR) model is a discrete-time lattice (binomial tree) framework for pricing options, in which the underlying asset's price evolves step-by-step through an up or down movement, enabling valuation of American options and path-dependent features that the continuous-time Black-Scholes model cannot easily handle.
Key takeaways
- The CRR model constructs a recombining binomial tree with up-factor u = e^(σ√Δt) and down-factor d = 1/u, where σ is annualized volatility and Δt is the length of each time step.
- Risk-neutral probabilities are p = (e^(rΔt) - d) / (u - d) for an up-move and (1-p) for a down-move, with r the risk-free rate; these probabilities price the option through backward induction without requiring any assumed real-world drift.
- As the number of time steps n → ∞ (and Δt → 0), the binomial tree converges to the Black-Scholes formula for European options, validating the model's consistency with continuous-time theory.
- American option pricing is the CRR model's primary advantage: at each node, the option value is the maximum of the continuation value (discounted expected future value) and the immediate exercise value.
- Model accuracy improves with more steps but increases computational cost; practitioners often use 500-1,000 steps for accurate American option pricing, with Richardson extrapolation or control variate techniques to accelerate convergence.
Explanation
Developed by John Cox, Stephen Ross, and Mark Rubinstein in 1979, the CRR binomial model provides an intuitive, discretized approach to option pricing that serves both as a pedagogical tool and a practical pricing engine for options with early exercise features. The model partitions the option's life T into n equal time steps of length Δt = T/n. At each node in the tree, the asset price S either moves up to Su = S × u or down to Sd = S × d, where u = e^(σ√Δt) and d = e^(-σ√Δt) = 1/u. This parameterization — ensuring u × d = 1 — produces a recombining tree where the up-then-down path equals the down-then-up path, so the number of terminal nodes is n+1 rather than 2^n.
The risk-neutral up-probability is p = (e^((r-q)Δt) - d) / (u - d), where q is the continuous dividend yield. This probability, derived from the no-arbitrage condition that the discounted expected price equals the forward price, makes the tree risk-neutral: the expected return on the underlying equals the risk-free rate, regardless of investors' actual risk preferences. Option prices are then computed by backward induction: starting from terminal payoffs and discounting at the risk-free rate one step at a time, with American options checked for early exercise at each node.
At each interior node for an American put: V = max(K - S, e^(-rΔt)[p × Vu + (1-p) × Vd]). The first term is the immediate exercise value; the second is the continuation value. If immediate exercise is optimal, the American option price exceeds the European price by the early exercise premium — a feature the model captures naturally.
The CRR model converges to Black-Scholes as n → ∞, but convergence can be oscillatory for options near the money. Practitioners use an odd or even number of steps strategically, or employ acceleration techniques like Richardson extrapolation (running n and 2n step trees and extrapolating to infinite steps) to improve accuracy. The model extends naturally to barrier options (nodes outside the barrier are zeroed), lookback options (tracking the maximum/minimum across paths), and compound options. However, for complex path-dependent instruments, Monte Carlo simulation is typically preferred over lattice methods.
Formula
u = e^(σ√Δt); d = 1/u; p = (e^((r-q)Δt) - d) / (u - d); American Put Node Value = max(K - S, e^(-rΔt)[p·Vu + (1-p)·Vd])
Example
Price a 1-year American put option on a stock trading at $100, with strike K = $100, annualized volatility σ = 25%, risk-free rate r = 5%, and no dividends, using a 3-step CRR tree. Δt = 1/3. u = e^(0.25 × √(1/3)) = e^(0.1443) ≈ 1.1553. d = 1/u ≈ 0.8655. p = (e^(0.05/3) - 0.8655) / (1.1553 - 0.8655) = (1.01681 - 0.8655) / 0.2898 ≈ 0.5223. Terminal nodes after 3 steps: S_uuu = 100 × 1.1553³ ≈ 154.3, S_uud = 100 × 1.1553² × 0.8655 ≈ 115.5, S_udd ≈ 86.6, S_ddd ≈ 64.9. Terminal put payoffs: 0, 0, 13.4, 35.1. Backward induction (discounting at e^(-0.05/3) ≈ 0.9835 per step) yields an American put value of approximately $6.80, compared to a Black-Scholes European put value of $6.42 — the $0.38 difference is the early exercise premium, reflecting the option to exercise early when the put is deep in the money.
Related terms
American Option Arbitrage Average Rate Option Black Scholes Model Cash Forward Sale Convergence Declaration Date Dividend Dividend Yield Monte Carlo Simulation Option Premium