Binomial Tree Model

Why the binomial tree?

The binomial tree model, introduced by Cox, Ross, and Rubinstein (CRR) in 1979, is one of the most intuitive and widely used methods for pricing options. It builds a discrete-time, discrete-space model of the stock price using a multiplicative Random Walk and then prices derivatives by backward induction under the risk-neutral measure.

Its power lies in its flexibility: unlike the Black-Scholes formula, which only handles European options in closed form, the binomial tree can price American options, exotic options, and options on assets with dividends — all by modifying the backward recursion.

Moreover, the binomial tree provides a concrete, visual illustration of how risk-neutral pricing works. Every concept in continuous-time finance — delta hedging, risk-neutral valuation, replication — has a transparent discrete-time counterpart in the tree.

One-Step Binomial Model

Setup

Consider a stock with current price $S_0$ . Over one time step $\Delta t$ , the stock can either:

Move up to $S_0 u$ with probability $p$
Move down to $S_0 d$ with probability $1 - p$

where $u > 1$ and $0 < d < 1$ are the up and down factors. A risk-free bond grows at rate $r$ , so $1 invested becomes $e^{r\Delta t}$ after one step.

For the model to be arbitrage-free, we need:

d < e^{r\Delta t} < u

If the bond always grew faster than the stock (even in the up state), or if the stock always beat the bond, there would be a risk-free profit — an arbitrage.

Risk-neutral pricing

The key insight is that the option can be replicated by a portfolio of stock and bond. Rather than working out the replicating portfolio explicitly, we can equivalently find the risk-neutral probability

\tilde{p}

such that the expected discounted stock price equals the current price:

S_0 = e^{-r\Delta t}\left[\tilde{p} \cdot S_0 u + (1 - \tilde{p}) \cdot S_0 d\right]

Solving:

\tilde{p} = \frac{e^{r\Delta t} - d}{u - d}

This

\tilde{p}

is not the real-world probability of an up-move. It is the probability under the risk-neutral measure

\mathbb{Q}

that makes the discounted stock price a martingale. The connection between real-world and risk-neutral probabilities is formalised by the change of measure.

The value of any derivative with payoff $V_u$ (up state) and $V_d$ (down state) is then:

V_0 = e^{-r\Delta t}\left[\tilde{p} \cdot V_u + (1 - \tilde{p}) \cdot V_d\right]

Example: One-step European call

Let $S_0 = 100$ , $u = 1.1$ , $d = 0.9$ , $r = 5\%$ per year, $\Delta t = 1$ year, and strike $K = 100$ .

Stock prices at expiry:

Up: $S_u = 100 \times 1.1 = 110$
Down: $S_d = 100 \times 0.9 = 90$

Call payoffs:

Up: $V_u = \max(110 - 100, 0) = 10$
Down: $V_d = \max(90 - 100, 0) = 0$

Risk-neutral probability:

\tilde{p} = \frac{e^{0.05} - 0.9}{1.1 - 0.9} = \frac{1.05127 - 0.9}{0.2} \approx 0.7564

Call price:

V_0 = e^{-0.05}\left[0.7564 \times 10 + 0.2436 \times 0\right] = 0.9512 \times 7.564 \approx 7.19

Multi-Step Binomial Tree

Construction

For $n$ time steps, the tree is built by applying the up/down factors repeatedly. After $n$ steps, the stock price at node $(n, j)$ (step $n$ , $j$ up-moves) is:

S_{n,j} = S_0 \cdot u^j \cdot d^{n-j}, \quad j = 0, 1, \dots, n

This is a recombining tree: an up-then-down move arrives at the same node as a down-then-up move (since $ud = du$ ), giving $n + 1$ terminal nodes instead of $2^n$ .

Backward induction

European options are priced by working backward from the terminal payoffs:

At maturity (step $n$ ): $V_{n,j} = \text{payoff}(S_{n,j})$
At each earlier node: $V_{i,j} = e^{-r\Delta t}\left[\tilde{p} \cdot V_{i+1,j+1} + (1 - \tilde{p}) \cdot V_{i+1,j}\right]$

American options add the early exercise condition:

V_{i,j} = \max\left(\text{payoff}(S_{i,j}), \; e^{-r\Delta t}\left[\tilde{p} \cdot V_{i+1,j+1} + (1 - \tilde{p}) \cdot V_{i+1,j}\right]\right)

At every node, the holder compares the immediate exercise value against the continuation value. This is why the binomial tree is the standard tool for American option pricing — the early exercise decision is naturally embedded in the recursion.

CRR Parameterisation

The Cox-Ross-Rubinstein choice of parameters ensures the tree converges to Geometric Brownian Motion as

n \to \infty

u = e^{\sigma\sqrt{\Delta t}}, \quad d = e^{-\sigma\sqrt{\Delta t}} = \frac{1}{u}

\tilde{p} = \frac{e^{r\Delta t} - d}{u - d}

where $\Delta t = T/n$ , $\sigma$ is the annualised volatility, and $r$ is the continuously compounded risk-free rate.

Why these values?

The parameterisation is chosen so that the first two moments of the log-return match those of GBM:

\mathbb{E}[\ln Z_i] = \tilde{p}\ln u + (1-\tilde{p})\ln d \approx \left(r - \frac{\sigma^2}{2}\right)\Delta t

\text{Var}(\ln Z_i) = \tilde{p}(1-\tilde{p})(\ln u - \ln d)^2 \approx \sigma^2 \Delta t

n \to \infty

, the sum

\sum_{i=1}^n \ln Z_i

converges to a Normal Distribution by the Central Limit Theorem, so

S_T

converges to a Log-Normal Distribution — exactly the distribution assumed by Black-Scholes.

Convergence to Black-Scholes

As $n \to \infty$ , the binomial tree price converges to the Black-Scholes price. This can be verified by expressing the binomial price as:

C_n = e^{-rT}\sum_{j=a}^{n} \binom{n}{j}\tilde{p}^j(1-\tilde{p})^{n-j}\left(S_0 u^j d^{n-j} - K\right)

where

a

is the minimum number of up-moves for the option to be in-the-money. By the Central Limit Theorem (specifically, the De Moivre-Laplace theorem), the binomial sums converge to the normal CDF

\Phi

, recovering:

C = S_0\Phi(d_1) - Ke^{-rT}\Phi(d_2)

This convergence is the discrete-to-continuous bridge: the multiplicative Random Walk becomes Brownian Motion, the binomial probabilities become normal probabilities, and the backward recursion becomes the Black-Scholes PDE.

The Greeks from the Tree

The binomial tree also provides numerical estimates of the option Greeks:

Delta (sensitivity to stock price):

\Delta = \frac{V_{1,1} - V_{1,0}}{S_{1,1} - S_{1,0}} = \frac{V_u - V_d}{S_0 u - S_0 d}

Gamma (sensitivity of delta to stock price):

\Gamma = \frac{\Delta_{\text{up}} - \Delta_{\text{down}}}{(S_0 u^2 - S_0 d^2)/2}

Theta (sensitivity to time):

\Theta = \frac{V_{2,1} - V_0}{2\Delta t}

These finite-difference approximations improve as $n$ increases, converging to the continuous-time Greeks.