Explanation

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

Key Information:

• Number of options: 100,000
• Stock price (S) = $50
• Strike price (K) = $49
• Time to maturity (T) = 0.25 years (3 months)
• Volatility (σ) = 20%
• Risk-free rate (r) = 5%

Delta Calculation: The delta of a call option is given by N(d₁), where:

d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)

Let's calculate: d₁ = [ln(50/49) + (0.05 + 0.20²/2)×0.25] / (0.20×√0.25) d₁ = [ln(1.02041) + (0.05 + 0.02)×0.25] / (0.20×0.5) d₁ = [0.02020 + 0.0175] / 0.10 d₁ = 0.0377 / 0.10 = 0.377

Delta = N(0.377) ≈ 0.647

Hedge Position:

• Bank sold 100,000 call options
• Each option has delta of approximately 0.647
• Total delta exposure = -100,000 × 0.647 = -64,700
• To hedge, need to buy 64,700 shares
• Rounded to nearest thousand: 65,000 shares

Therefore, the bank should buy 65,000 shares to delta hedge the short call position.