Binomial Trees for Options Pricing

The binomial model is a discrete-time approach to options pricing that's more flexible than Black-Scholes and can handle American options, dividends, and changing volatility.

How Binomial Trees Work

The binomial model assumes that stock prices follow a multiplicative random walk where each period the stock can move up or down by specific factors.

Key Parameters

• u = up factor = e^(σ√Δt)
• d = down factor = 1/u = e^(-σ√Δt)
• p = risk-neutral probability = (e^(rΔt) - d) / (u - d)
• Δt = time step = T/n (where n = number of periods)

Building the Tree

1. Start with current stock price S₀
2. At each node, stock can move to Su or Sd
3. Continue until expiration
4. Calculate option values at expiration
5. Work backward using discounted expected values

Step-by-Step Example

Setup:

• Stock price: $100
• Strike price: $105 (call option)
• Time to expiration: 6 months
• Risk-free rate: 5%
• Volatility: 20%
• Periods: 2

Step 1: Calculate Parameters

Δt = 0.5/2 = 0.25
u = e^(0.20√0.25) = e^0.10 = 1.105
d = 1/1.105 = 0.905
p = (e^(0.05×0.25) - 0.905) / (1.105 - 0.905) = 0.563

Step 2: Build Stock Price Tree

Period 0:     $100
Period 1:    $110.5    $90.5
Period 2:   $122.1  $100.0  $81.9

Step 3: Calculate Option Values at Expiration

Period 2: max(S-K, 0)
$122.1 → $17.1
$100.0 → $0.0
$81.9  → $0.0

Step 4: Work Backward

Period 1:
Up node:   e^(-0.05×0.25) × [0.563×17.1 + 0.437×0.0] = $9.52
Down node: e^(-0.05×0.25) × [0.563×0.0 + 0.437×0.0] = $0.00

Period 0:
e^(-0.05×0.25) × [0.563×9.52 + 0.437×0.0] = $5.30

Result: Call option value = $5.30

American vs European Options

European Options

• Can only be exercised at expiration
• Use standard backward induction

American Options

• Can be exercised at any time
• At each node, compare continuation value vs immediate exercise value
• Take the maximum: max(continuation_value, intrinsic_value)

Advantages of Binomial Model

✅ Flexible: Handles American options, dividends, changing parameters
✅ Intuitive: Easy to understand and explain
✅ Convergent: Approaches Black-Scholes as periods increase
✅ Numerical: No closed-form solution required

Common Interview Questions

Question 1: Basic Tree Construction

"Build a 1-period binomial tree to price a 3-month European put option with strike $95 when the stock is $100, volatility is 25%, and risk-free rate is 4%."

Question 2: American Exercise

"Explain how you would modify the binomial tree to handle early exercise for an American put option."

Question 3: Convergence

"How does the binomial model relate to the Black-Scholes model?"

Answer: As the number of periods approaches infinity, the binomial model converges to Black-Scholes.

Implementation Tips

Choosing Number of Periods

• More periods = more accuracy but more computation
• Rule of thumb: At least 30-50 periods for reasonable accuracy
• For American options, may need 100+ periods

Memory Optimization

• Don't store entire tree if not needed
• Can compute values on-the-fly
• For large trees, use iterative rather than recursive approach

Handling Dividends

• Reduce stock price by present value of dividends at each ex-dividend date
• Or model as continuous dividend yield

Practice Problem

Challenge: Price an American put option with the following parameters:

• Current stock price: $50
• Strike price: $52
• Time to expiration: 4 months
• Volatility: 30%
• Risk-free rate: 6%
• Use a 4-period binomial tree

Approach:

1. Calculate u, d, p for Δt = 1/3 month
2. Build stock price tree
3. Calculate put values at expiration
4. Work backward, checking for early exercise at each node
5. Compare with European put value

Related Topics

Binomial trees are a favorite topic for quant interviews at firms like Jane Street, Citadel, and IMC.