Binomial Option Pricing Model

A discrete-time model that prices options by working backward through a tree of possible stock price paths.

The CRR binomial model (Cox-Ross-Rubinstein, 1979) models the stock price as moving up by factor u = e^(σ√Δt) or down by d = 1/u at each time step Δt = T/N. The risk-neutral probability of an up move is p* = (e^(rΔt) − d)/(u − d). Option values are computed via backward induction from known terminal payoffs: V = e^(−rΔt)[p*·V_u + (1−p*)·V_d]. The binomial model naturally handles American options by comparing hold vs. exercise at each node. As N → ∞, the binomial model converges to BSM.

Formula

u = e^{\sigma\sqrt{\Delta t}}, \quad p^* = \frac{e^{r\Delta t} - d}{u - d}

Related Terms

Black-Scholes-Merton Model (BSM)

The foundational option pricing formula that gives the fair value of a European call or put as a function of spot, strike, rate, volatility, and time.

Risk-Neutral Probability

The artificial probability measure under which all assets earn the risk-free rate, used to price derivatives by expectation.