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Answer: CAD 1.23
## 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 ``` ### 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.**
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A quantitative analyst on a bank's corporate equity derivatives desk is pricing a 1-year barrier option on a stock. The option has a down-and-in feature specifying that the price in 6 months should be at or below CAD 52.50 for the call option to come into existence. The current price of the stock is CAD 55.00, the strike price of the option is CAD 50.00, the implied volatility of the stock is 12% and the risk-free rate is 1.40%. Using a two-step binomial tree approach, what is the price of the option?
A
CAD 1.23
B
CAD 2.58
C
CAD 5.28
D
CAD 6.51
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