Explanation

This is a down-and-in barrier call option that only becomes active if the stock price falls to or below CAD 52.50 within 6 months.

Step 1: Construct the Binomial Tree

• Time step: 0.5 years (6 months)
• Up factor: u = exp(σ√Δt) = exp(12% × √0.5) = 1.0886
• Down factor: d = 1/u = 0.9186
• Risk-neutral probability of up move: p = (exp(rΔt) - d)/(u - d) = (exp(1.40% × 0.5) - 0.9186)/(1.0886 - 0.9186) = 0.5201
• Risk-neutral probability of down move: 1 - p = 0.4799

Step 2: Stock Price Tree

          65.18
         /
    59.87
   /     \
55.00   55.00
   \     /
    50.52
      \
      46.41

          65.18
         /
    59.87
   /     \
55.00   55.00
   \     /
    50.52
      \
      46.41

Step 3: Option Valuation

• The barrier is triggered at CAD 52.50
• Only the branch from 50.52 (which is ≤ 52.50) triggers the knock-in
• From 50.52 node:
  • Up move to 55.00: payoff = max(55 - 50, 0) = 5.00
  • Down move to 46.41: payoff = max(46.41 - 50, 0) = 0.00
  • Value at 50.52 = exp(-1.40% × 0.5) × (0.5201 × 5.00 + 0.4799 × 0.00) = 2.58

Step 4: Current Option Price

• Only reaches 50.52 with probability of down move from initial node: 0.4799
• Current price = exp(-1.40% × 0.5) × 0.4799 × 2.58 = 1.23

Therefore, the correct price of the down-and-in barrier call option is CAD 1.23.