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Home / Financial Tools / Binomial Option Pricing Lattice

Graduate-Level Modeling · Derivatives Pricing

Binomial Option Pricing Lattice (Cox-Ross-Rubinstein)

Black-Scholes gives you a closed-form number and no intuition for where it comes from. The binomial lattice builds the same price node by node, and shows exactly where an American option's early exercise boundary actually sits.

How To Use This Model

Reading This Tool

Set the option parameters and lattice steps, then choose European or American exercise.

The lattice shows the option's value at every node, computed by backward induction. Switch to American style and watch which nodes flip to early exercise. Full theory in the Methodology tab.

Option & Lattice Inputs

Lattice steps are capped at 8 for legibility, this introduces modest discretization error versus a production-grade 500+ step lattice. The Insights tab shows how the price converges as steps increase.

Option Value At Root Node

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Lattice-Derived Option Price

$0.00

Up Factor (u)

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Risk-Neutral Probability (p)

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Option Value Lattice (Rows = Time Step, Cols = Node)

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Convergence To Black-Scholes

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Early Exercise Premium

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Reading The Lattice Grid

Each column in a row represents one possible path to that node, more up-moves means a higher stock price. The root (top-left) is today's option value. Follow any diagonal path rightward and you're watching one specific simulated price path unfold through time, with the option's value recalculated at each junction via backward induction from the known terminal payoffs.

Discrete-Time Derivatives Pricing

The Core Formulas

u = eσ√Δt, d = 1/u
p = (erΔt − d) / (u − d)
Vi,j = e−rΔt [p·Vi+1,j + (1−p)·Vi+1,j+1]
American: Vi,j = max(continuation, intrinsic value at that node)

The tree is built forward for stock prices, then option values are computed backward from the known terminal payoffs at maturity, discounting at the risk-free rate under the risk-neutral probability p, not the real-world probability of an up-move.

When To Actually Use This Model

  • Pricing American-style options, where Black-Scholes has no closed form and early exercise must be evaluated explicitly at each node.
  • Teaching risk-neutral valuation and backward induction in a derivatives or financial engineering course.
  • Options on assets with discrete dividends, which the lattice handles more naturally than closed-form adjustments.
  • Building intuition for how volatility and time-to-maturity propagate through a price tree before moving to continuous-time models.

Key Assumptions & Limitations

  • Assumes constant volatility and a constant risk-free rate across the full life of the option.
  • Assumes no dividends, arbitrage-free markets, and frictionless trading (no transaction costs or bid-ask spread).
  • Low step counts (used here for visual legibility) introduce discretization error; production pricing systems typically use hundreds of steps.
  • The Cox-Ross-Rubinstein parameterization is one of several valid ways to set u, d, and p; Jarrow-Rudd and other parameterizations exist.

Foundational Reference

Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option Pricing: A Simplified Approach. Journal of Financial Economics, 7(3), 229-263.

Want the closed-form Black-Scholes comparison?

Open The Black-Scholes Tool →