Binomial Tree Option Pricing › Summary

Binomial trees give an intuitive and flexible framework for option pricing that links the pricing idea to computation.

The key concepts from this chapter include:

• Discrete-time modeling: Stock prices evolve through discrete up and down moves, with the Cox-Ross-Rubinstein parameterization $u = e^{\sigma\sqrt{\Delta t}}$ and $d = 1/u$ ensuring convergence to geometric Brownian motion.

• Risk-neutral valuation: The risk-neutral probability $p = \frac{e^{r\Delta t} - d}{u - d}$ allows pricing as a discounted expected payoff without knowing the true probability of up moves.

• Backward induction: Option prices are computed by working backward from known terminal payoffs, applying the one-step pricing formula at each node.

• American options: Early exercise is handled by comparing exercise value to continuation value at each node, making binomial trees the method of choice for American-style derivatives.

• Convergence to Black-Scholes: As the number of steps increases, binomial prices converge to Black-Scholes at rate $O(1/N)$, with Richardson extrapolation improving accuracy.

The binomial framework extends naturally to dividend-paying stocks, exotic features, and is a foundation for understanding more advanced numerical methods. In the next chapter, we'll explore Monte Carlo simulation, which handles high-dimensional problems and path-dependent options that are challenging for tree-based methods.