Can someone walk through the binomial option pricing model with a two-period example?

The binomial model is one of the most important quantitative tools in CFA Level II Derivatives. Let's build a two-period tree from scratch.

Setup: Meridian Tech stock trades at S₀ = $80. Each period, the stock can move:

• Up by factor u = 1.20 (20% increase)
• Down by factor d = 0.833 (1/u, ~16.7% decrease)
• Risk-free rate per period r = 5%
• We're pricing a European call with strike K = $85

Step 1: Build the Stock Price Tree

Period 0	Period 1	Period 2
		Suu = 80 × 1.20 × 1.20 = $115.20
	Su = 80 × 1.20 = $96
S₀ = $80		Sud = 80 × 1.20 × 0.833 = $80.00
	Sd = 80 × 0.833 = $66.64
		Sdd = 80 × 0.833 × 0.833 = $55.47

Step 2: Calculate Risk-Neutral Probability p = (1 + r − d) / (u − d) = (1.05 − 0.833) / (1.20 − 0.833) = 0.217 / 0.367 = 0.5913

Step 3: Calculate Option Payoffs at Expiration (Period 2)

• cuu = max(115.20 − 85, 0) = $30.20
• cud = max(80.00 − 85, 0) = $0.00
• cdd = max(55.47 − 85, 0) = $0.00

Step 4: Backward Induction to Period 1

• cu = [p × cuu + (1−p) × cud] / (1+r) = [0.5913 × 30.20 + 0.4087 × 0.00] / 1.05 = 17.86 / 1.05 = $17.01
• cd = [p × cud + (1−p) × cdd] / (1+r) = [0.5913 × 0.00 + 0.4087 × 0.00] / 1.05 = $0.00

Step 5: Backward Induction to Period 0

• c₀ = [p × cu + (1−p) × cd] / (1+r) = [0.5913 × 17.01 + 0.4087 × 0.00] / 1.05 = 10.06 / 1.05 = $9.58

The European call option is worth $9.58 today.

Key Insights:

1. The risk-neutral probability p is NOT the actual probability of the stock going up — it's the probability that makes the expected return equal to the risk-free rate.
2. Backward induction means you work from the terminal payoffs back to the present, discounting at the risk-free rate.
3. For American options, at each node you'd compare the backward-induced value against early exercise value and take the maximum.

Practice more binomial pricing with our CFA Level II question bank.