Binomial Tree (N-Step)

N-Step Binomial Tree

The 1-step model is too coarse. With n steps, we subdivide T into n intervals of length Δt = T/n, giving a much more accurate approximation of the continuous Black-Scholes model.

CRR Parameters for N Steps

$\Delta t = T/n$ $u = e^{\sigma\sqrt{\Delta t}}, \quad d = 1/u$ $p = \frac{e^{r\Delta t} - d}{u - d}$

Tree Structure

At step n, the stock can be at any of n+1 values:

$S_{n,j} = S \cdot u^{n-2j} \quad \text{for } j = 0, 1, \ldots, n$

Backward Induction Algorithm

1. Compute terminal payoffs at time n for all n+1 nodes
2. Work backwards: at each step, the value at node j is: $V_j = e^{-r\Delta t}(p \cdot V_j^{\text{up}} + (1-p) \cdot V_j^{\text{down}})$
3. Repeat until you reach the root node (t=0)

Convergence

As n → ∞, the binomial price converges to the Black-Scholes price. With n = 100 steps, the error is typically less than 0.05%.

The key insight: more steps = finer time grid = better approximation of continuous-time Brownian motion.