Binomial Tree Option Pricing › Binomial Tree Option Pricing

The Black-Scholes formula, which we derived in previous chapters, provides elegant closed-form solutions for European options. But what happens when you need to price an American put option, where early exercise might be optimal? Or an option with path-dependent features? The continuous-time framework becomes analytically intractable, and we need alternative approaches.

The binomial tree model, introduced by Cox, Ross, and Rubinstein in 1979, offers a powerful discrete-time alternative. Rather than assuming stock prices follow continuous geometric Brownian motion, the binomial model discretizes time into steps and assumes the stock price can only move up or down by fixed amounts at each step. This simplification makes the model remarkably tractable: you can price options by working backward through a tree of possible price paths, checking at each node whether early exercise is beneficial.

Binomial trees are pedagogically clear and computationally practical. As you increase the number of time steps, the binomial model converges to Black-Scholes, and gives both intuition for continuous-time results and a computational method for options that lack closed-form solutions. This chapter builds the binomial framework from first principles, shows how risk-neutral valuation applies at each node, and demonstrates pricing for both European and American options.