Binomial Option Pricing Model
Valuing an option by mapping out every up-and-down move on a price tree, one step at a time.
Definition
The binomial option pricing model values an option by breaking the time to expiry into many small steps. At each step the underlying can move only up or down by a fixed factor, forming a branching tree. The option's payoff is calculated at every final node, then discounted back through the tree using risk-neutral probabilities to arrive at today's fair value. The most common version is the Cox-Ross-Rubinstein (CRR) model.
Formula
For one step, the value is the risk-neutral expected payoff discounted at the risk-free rate: p = (e^(r·dt) − d) / (u − d), where u and d are the up and down factors and p is the risk-neutral probability of an up move. The option price equals e^(−r·dt) × [p·C_up + (1−p)·C_down], applied recursively from the final nodes back to the root.
Why it matters
Unlike a single closed-form equation, the tree lets you check the option's value at every intermediate node. That makes it the natural tool for pricing American-style options, which can be exercised before expiry, and for instruments with dividends or changing parameters. As the number of steps grows, the binomial price converges to the Black-Scholes value.
Example
Suppose a stock trades at 100 and over one step can rise to 110 (u) or fall to 91 (d). With a strike of 100, the call pays 10 in the up state and 0 in the down state. Computing the risk-neutral probability and discounting that expected payoff gives the call's fair value today. Add more steps and the tree becomes a finer, more accurate picture.
See it live
Watch how model-driven values line up with real market premiums on TradePulse's live option chain.