Explanation

Delta-Normal Approach Limitations:

The delta-normal approach assumes linear relationships and is based on:

• Using option delta to approximate price changes
• Assuming normal distribution of returns

Why At-the-Money Options Work Best:

For the call option with strike price $40 (at-the-money):

1. Delta is closest to 0.5, providing the most accurate linear approximation
2. Gamma is highest at-the-money, but the delta-normal approach ignores gamma risk
3. Linear approximation error is minimized when the option is not too deep in-the-money or out-of-the-money

Comparison of Options:

• $35 strike (deep ITM): Delta ≈ 1.0, but large gamma makes linear approximation poor
• $40 strike (ATM): Delta ≈ 0.5, best balance for linear approximation
• $45 strike (OTM): Delta ≈ 0.2-0.3, small delta but large relative errors
• $50 strike (deep OTM): Delta ≈ 0.0-0.1, very poor linear approximation

Conclusion:

The at-the-money option (strike $40) will have the delta-normal ES closest to the true ES because the linear approximation is most accurate when the option is neither deeply in-the-money nor out-of-the-money.