At-the-Money Straddle Delta Neutrality and Vega Exposure

Scenario

An options trader constructs a long at-the-money (ATM) straddle consisting of 1 European call and 1 European put with identical strike $K = S_0$ and expiration $T$.

Your Task

1. Under Black-Scholes assumptions with interest rate $r = 0$, prove that the ATM straddle has net delta $\Delta_{\text{straddle}} = 2\Phi\left(\frac{1}{2}\sigma\sqrt{T}\right) - 1$, and explain why forward ATM strike $K = S_0 e^{r T}$ achieves exact delta-neutrality.
2. Derive the total Gamma $\Gamma_{\text{straddle}}$ and total Vega $\mathcal{V}_{\text{straddle}}$ of the position.
3. Given $S_0 = 100, K = 100, r = 0, \sigma = 0.25, T = 0.25$ years, compute the exact Delta, Gamma, and Vega of the straddle.

Constraints

• Black-Scholes model dynamics.

Examples For $r = 0, \sigma = 0.25, T = 0.25$, $d_1 = \frac{1}{2} \times 0.25 \times 0.5 = 0.0625$. $\Delta_C \approx 0.5249, \Delta_P = \Delta_C - 1 = -0.4751$, so spot ATM straddle has slight positive delta $+0.0498$.