Binomial Tree (1-Step)

Binomial Tree: 1-Step Model

The binomial tree model is a discrete-time approach to options pricing that doesn't require advanced mathematics — just basic probability and discounting.

The 1-Step Setup

Over one time period T, the stock price can either:

• Go up by factor u: $S_u = S \cdot u$
• Go down by factor d: $S_d = S \cdot d$

CRR Parameterization

The Cox-Ross-Rubinstein (CRR) parameterization ties the tree to Black-Scholes volatility:

$u = e^{\sigma\sqrt{T}}, \quad d = \frac{1}{u} = e^{-\sigma\sqrt{T}}$

Risk-Neutral Probability

Under the risk-neutral measure, the probability of an up-move is:

$p = \frac{e^{rT} - d}{u - d}$

This ensures the expected return equals the risk-free rate.

Pricing the Option

1. Compute terminal payoffs: $C_u = \max(S_u - K, 0)$ and $C_d = \max(S_d - K, 0)$
2. Discount the expected payoff: $C = e^{-rT}(p \cdot C_u + (1-p) \cdot C_d)$

Why Use Binomial?

• Intuitive and easy to implement
• Easily extended to American options (allow early exercise at each node)
• Converges to Black-Scholes as the number of steps increases