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.