Explanation

To delta hedge a short call position, the bank needs to buy shares to offset the negative delta from the short calls.

Given information:

• Short position in call options on 100,000 equities
• Current stock price (S) = $50
• Strike price (K) = $49
• Time to maturity (T) = 3 months = 0.25 years
• Volatility (σ) = 20%
• Risk-free rate (r) = 5%

Delta calculation: For a call option, delta = N(d₁)

First calculate d₁: $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ $d_1 = \frac{\ln(50/49) + (0.05 + 0.20^2/2) \times 0.25}{0.20 \times \sqrt{0.25}}$ $d_1 = \frac{\ln(1.0204) + (0.05 + 0.02) \times 0.25}{0.20 \times 0.5}$ $d_1 = \frac{0.0202 + 0.0175}{0.10} = \frac{0.0377}{0.10} = 0.377$

Using standard normal distribution: N(0.377) ≈ 0.647

Total delta hedge:

• Number of options = 100,000
• Delta per option = 0.647
• Total delta = 100,000 × 0.647 = 64,700

Since the bank is short calls (negative delta), they need to buy approximately 65,000 shares to become delta neutral.

Therefore, the correct answer is A: Buy 65,000 shares.