Binomial Option Pricing (Cox-Ross-Rubinstein)
Cox-Ross-Rubinstein Binomial Option Pricing Model · Also known as: binomial tree model, Cox-Ross-Rubinstein model, CRR model, lattice option pricing, binom opsiyon fiyatlama
The binomial option pricing model, introduced by John Cox, Stephen Ross, and Mark Rubinstein in 1979, prices options by modelling the underlying as a discrete tree in which the price moves up or down by fixed factors at each step. Working backward from the option's payoff at maturity using risk-neutral probabilities, it produces a no-arbitrage price that converges to Black-Scholes as the number of steps grows — while naturally handling American early exercise, which the closed-form formula cannot.
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When to use it
Use the binomial model when you need to price options with early-exercise features — American options, Bermudan options, and many real options — for which Black-Scholes has no closed form, or when you want a transparent, easily extended numerical scheme. It accommodates discrete dividends, time-varying volatility or rates (by adjusting the tree), and exotic payoffs more readily than the analytic formula. The trade-offs are computational: accuracy improves with more steps but so does cost, and convergence to the continuous price can be oscillatory, which refinements (averaging, control variates, or trinomial/finite-difference methods) address. For plain European options on a single underlying, Black-Scholes is faster; the binomial tree is the tool of choice once early exercise or path-dependent features enter.
Strengths & limitations
- Handles American and Bermudan early exercise naturally by checking exercise at every node.
- Transparent and intuitive, making the no-arbitrage and risk-neutral logic explicit.
- Flexible: accommodates discrete dividends, varying rates/volatility, and many exotic payoffs.
- Converges to the Black-Scholes price as the number of steps increases, providing a unifying link.
- Computationally heavier than closed-form formulas, with cost growing in the number of steps.
- Convergence can be slow or oscillatory, requiring many steps or smoothing for accuracy.
- The basic recombining tree handles only one underlying; high-dimensional options need other methods.
- Choosing u, d, and the step count involves trade-offs that affect accuracy and stability.
Frequently asked
How does the binomial model relate to Black-Scholes?
They price the same no-arbitrage value by different means. The binomial model uses a discrete up/down tree and backward induction; as the number of steps grows and Δt shrinks, the Cox-Ross-Rubinstein tree converges to geometric Brownian motion and the price approaches the Black-Scholes formula. The binomial model's advantage is handling early exercise, which Black-Scholes cannot.
What is the risk-neutral probability p?
It is the up-move weight p = (e^{rΔt} − d)/(u − d) that makes the discounted price a martingale, so the model is arbitrage-free. It is not the real-world probability of an up move; it is the artificial probability under which all assets earn the risk-free rate, which is exactly what no-arbitrage pricing requires.
Why is the binomial model preferred for American options?
Because it evaluates the option at every node, you can compare the value of continuing to hold against the payoff from exercising immediately and take the larger at each step. This makes early-exercise decisions explicit and easy to incorporate, whereas the Black-Scholes formula assumes exercise only at maturity.
Sources
- Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229–263. DOI: 10.1016/0304-405X(79)90015-1 ↗
How to cite this page
ScholarGate. (2026, June 2). Cox-Ross-Rubinstein Binomial Option Pricing Model. ScholarGate. https://scholargate.app/en/finance/binomial-option-pricing
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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