Explanation

The delta-normal approach uses a linear approximation of option price changes based on delta (the first derivative of option price with respect to the underlying asset price). This approximation works best when the option is at-the-money because:

• At-the-money options have delta values around 0.5, and the linear approximation is most accurate when the option price changes are relatively linear
• Deep in-the-money options (strike $35) have delta close to 1.0, but the approximation may not capture gamma effects well
• Deep out-of-the-money options (strike $45 and $50) have delta close to 0, and the linear approximation becomes increasingly inaccurate due to significant gamma effects

With the current spot price of $40:

• Strike $35: Deep in-the-money (delta ≈ 0.95-1.0)
• Strike $40: At-the-money (delta ≈ 0.5)
• Strike $45: Out-of-the-money (delta ≈ 0.05-0.2)
• Strike $50: Deep out-of-the-money (delta ≈ 0.01-0.05)

The at-the-money call option (strike $40) will have the delta-normal ES closest to the true ES because the linear approximation is most accurate for options that are not too far from at-the-money.