Explanation

To understand why option C is correct, let's analyze the Greeks:

Gamma (Γ)

• Gamma measures the rate of change of delta with respect to changes in the underlying asset price
• Gamma is highest for at-the-money options with short time to expiration
• The 1-month option has higher gamma than the 4-month option
• To reduce gamma: We need to sell options with high gamma (1-month option)

Vega (ν)

• Vega measures sensitivity to changes in implied volatility
• Vega is higher for options with longer time to expiration
• The 4-month option has higher vega than the 1-month option
• To increase vega: We need to buy options with high vega (4-month option)

Analysis of Option C: "Sell the 1-month option and buy the 4-month option"

• Selling 1-month option: Reduces gamma (since short-term options have high gamma)
• Buying 4-month option: Increases vega (since long-term options have high vega)

This combination achieves both objectives:

• Gamma decreases (from selling high-gamma 1-month option)
• Vega increases (from buying high-vega 4-month option)

The other options don't achieve both objectives simultaneously:

• A: Buying both would increase both gamma and vega
• B: Would increase gamma (from buying 1-month) and decrease vega (from selling 4-month)
• D: Selling both would decrease both gamma and vega