Answer: -0.73.

## Explanation To calculate the optimal hedge ratio (also known as the delta) in a binomial model, we use the formula: \[ \Delta = \frac{C_u - C_d}{S_u - S_d} \] Where: - \(C_u\) = Call option value in the up state - \(C_d\) = Call option value in the down state - \(S_u\) = Stock price in the up state - \(S_d\) = Stock price in the down state **Step 1: Calculate stock prices** - Current stock price = $50 - Up factor (u) = 1.25 - Down factor (d) = 0.70 - Exercise price = $55 \[ S_u = 50 \times 1.25 = 62.50 \] \[ S_d = 50 \times 0.70 = 35.00 \] **Step 2: Calculate call option payoffs** \[ C_u = \max(62.50 - 55, 0) = \max(7.50, 0) = 7.50 \] \[ C_d = \max(35.00 - 55, 0) = \max(-20, 0) = 0 \] **Step 3: Calculate hedge ratio** \[ \Delta = \frac{7.50 - 0}{62.50 - 35.00} = \frac{7.50}{27.50} = 0.2727 \approx 0.27 \] However, the correct answer is A (-0.73). This suggests the question might be asking about a **put option** hedge ratio rather than a call option. Let's verify: For a put option: \[ P_u = \max(55 - 62.50, 0) = \max(-7.50, 0) = 0 \] \[ P_d = \max(55 - 35.00, 0) = \max(20, 0) = 20 \] \[ \Delta_{put} = \frac{0 - 20}{62.50 - 35.00} = \frac{-20}{27.50} = -0.7273 \approx -0.73 \] Therefore, the optimal hedge ratio of -0.73 corresponds to a **put option**, not a call option as stated in the question. This appears to be an inconsistency in the question.

Author: LeetQuiz Editorial Team