CRR Binomial Tree Lab
📚 How Does the CRR Binomial Tree Work? (Click to expand)
The Big Picture
The Cox-Ross-Rubinstein (CRR) binomial tree is a way to price American options, which can be exercised at any time before expiration. The key insight: work backwards from expiration to today, and at each node, ask "Should I exercise now or wait?"
Step 1: Building the Stock Price Tree (Forward in Time)
We model how the stock price could evolve by breaking time into N steps. At each step, the stock can move:
- UP by a factor u (multiply by u)
- DOWN by a factor d (multiply by d)
Starting at $100 with u=1.10 and d=1/1.10≈0.909:
- UP: $100 × 1.10 = $110 (10% gain)
- DOWN: $100 × 0.909 = $90.90 (9.1% loss)
Why asymmetric?
Notice the up move (+10%) seems bigger than the down move (-9.1%). But in multiplicative terms, they're perfectly symmetric!
- Up-then-down: $100 × 1.10 × 0.909 = $100 ✓
- Down-then-up: $100 × 0.909 × 1.10 = $100 ✓
This is because stock prices are lognormal, not normal. A 10% gain followed by a 10% loss does NOT get you back to breakeven ($100 → $110 → $99). But u and d=1/u perfectly cancel out!
The CRR Parameters - Where Do They Come From?
Time Step (dt)
dt = T / N
If we have 1 year to expiration and use 100 steps, each step represents dt = 1/100 = 0.01 years (about 3.65 days).
Up Factor (u)
u = e^(σ√dt)
Why this formula? We want the tree to match the stock's volatility (σ). In continuous time, stock returns follow a lognormal distribution with volatility σ. The factor e^(σ√dt) gives us the correct up-move size so that over many steps, the tree's volatility matches the market's implied volatility.
Risk-Neutral Probability (p)
p = (e^(r×dt) - d) / (u - d)
This is the most subtle parameter! It's NOT the real-world probability of the stock going up. Instead, it's the probability in a "risk-neutral world" where investors don't demand extra return for taking risk.
Step 2: Backward Induction (The Magic!)
Now we step backwards through time, one step at a time. At each node, we calculate the option value by considering two choices:
Option 1: HOLD (Continue)
Hold Value = e^(-r×dt) × [p × ValueUP + (1-p) × ValueDOWN]
Option 2: EXERCISE Now
Exercise Value = max(0, intrinsic value)
The Decision Rule
American Option Value = max(Hold Value, Exercise Value)
🎯 Try it yourself!
Build a tree with 3-5 steps using the controls below. Click on nodes to see the Hold vs Exercise decision at each point. Watch how the algorithm finds the optimal exercise strategy!