How do you price a European call option using a two-period binomial model?

The two-period binomial model extends the one-period approach by building a tree with three time steps (now, period 1, period 2) and working backward from the terminal payoffs. Let's walk through it completely.

Setup — Prescott Biotech European Call

• Current stock price (S): $40
• Strike price (K): $42
• Up factor (u): 1.25 (stock goes up 25%)
• Down factor (d): 0.80 (stock goes down 20%)
• Risk-free rate per period (r): 5%
• Periods: 2

Step 1: Build the Stock Price Tree

• Suu = $40 x 1.25 x 1.25 = $62.50
• Sud = $40 x 1.25 x 0.80 = $40.00
• Sdd = $40 x 0.80 x 0.80 = $25.60

Step 2: Calculate Terminal Call Payoffs

• c_uu = max(0, $62.50 - $42) = $20.50
• c_ud = max(0, $40.00 - $42) = $0.00
• c_dd = max(0, $25.60 - $42) = $0.00

Step 3: Calculate Risk-Neutral Probability

pi = (1 + r - d) / (u - d) = (1.05 - 0.80) / (1.25 - 0.80) = 0.25 / 0.45 = 0.5556

(1 - pi) = 0.4444

Step 4: Work Backward to Period 1

At each Period 1 node, the call value is the discounted expected value of the next period:

c_u = [pi x c_uu + (1 - pi) x c_ud] / (1 + r)

= [0.5556 x $20.50 + 0.4444 x $0] / 1.05

= $11.39 / 1.05 = $10.85

c_d = [pi x c_ud + (1 - pi) x c_dd] / (1 + r)

= [0.5556 x $0 + 0.4444 x $0] / 1.05 = $0.00

Step 5: Work Backward to Period 0

c_0 = [pi x c_u + (1 - pi) x c_d] / (1 + r)

= [0.5556 x $10.85 + 0.4444 x $0.00] / 1.05

= $6.03 / 1.05 = $5.74

The European call is worth $5.74.

Why Risk-Neutral Probabilities?

The risk-neutral probability (0.5556) is NOT the real-world probability of the stock going up. It's a mathematical construct that allows us to discount expected payoffs at the risk-free rate instead of figuring out the correct risk-adjusted rate. The resulting price is the same as what no-arbitrage replication would give.

Verification via Direct Formula:

c_0 = [pi^2 x c_uu + 2 x pi x (1-pi) x c_ud + (1-pi)^2 x c_dd] / (1+r)^2

= [0.3086 x $20.50 + 2 x 0.2469 x $0 + 0.1975 x $0] / (1.05)^2

= $6.33 / 1.1025 = $5.74 (confirmed)

Exam Tip: CFA Level II will typically give you u, d, and r, and expect you to calculate pi, build the payoff tree, and work backward. Practice until the steps are mechanical.

Master binomial option pricing in our CFA Level II Derivatives course.