Binomial Option Pricing

The binomial option pricing model values options by constructing a discrete tree where at each time step, the underlying stock can move up or down. Starting from expiration and working backward, the model calculates option value at each node as the probability-weighted average of future values, discounted to present value. The binomial model can handle american-options (early exercise), dividends, and other features Black-Scholes model cannot, making it more flexible though less elegant.

The binomial tree structure

In the simplest binomial model:

• Start at today’s stock price S
• At each step, stock can move up by factor u or down by factor d
• Continue for n steps to expiration date
• At expiration, calculate option payoff for each possible final price
• Work backward: at each node, value = probability-weighted discounted payoffs

Example:

• Stock at $100
• 1 year to expiration
• u = 1.1 (up 10%), d = 0.9 (down 10%)
• 2 steps (semi-annual)

Tree:

• Year 0: $100
• Year 0.5: $110 or $90
• Year 1: $121, $99, or $81

For each final state, calculate option payoff (e.g., $110 − $100 = $10 for a $100 call if price is $110 at expiration).

Backward recursion

At each node working backward:

Node value = [p × up_value + (1−p) × down_value] / (1 + r)

Where p is the risk-neutral probability of an up move (derived from volatility).

For american-options, also check: should I exercise now or hold? Take the max of exercise value or hold value.

Convergence to Black-Scholes

As the number of steps increases, the binomial model’s prices converge to Black-Scholes model prices. With 500 steps, binomial prices are nearly identical to Black-Scholes for european-options.

This convergence validates both models and allows practitioners to choose based on implementation needs.

Early exercise and American options

The binomial model shines for american-options, where early exercise can be optimal. At each node, the model checks: is it better to exercise now or hold?

For a call option with dividend, you might exercise just before the dividend goes ex-date. The binomial model captures this.

Black-Scholes model cannot, requiring binomial-option-pricing or other numerical methods.

Tuning volatility and time steps

The up/down factors are set based on volatility:

u = e^(σ√Δt) d = 1/u

Where σ is volatility and Δt is the time step.

Higher volatility → larger up/down moves. More steps → finer granularity.

Advantages and limitations

Advantages:

• Flexible; handles early exercise, dividends, exotic features
• Intuitive; visualize price paths
• Converges to Black-Scholes for confirmation

Limitations:

• Computationally intensive (500+ steps for decent accuracy)
• Assumes discrete jumps; real prices continuous
• Harder to compute Greeks numerically

See also