Reading: What are some useful approximations to the Black-Scholes formula? I understand that a ATM Call option can be approximated to $$ C(S,t)≈0.4Se^{−r(T−t)}σ \sqrt{T−t}$$ Also, I often hear that an ATM delta is around $\Delta = 0.5$. However, using approximation formaula of an ATM call option price, gives: $$\Delta = 0.4σ \sqrt{T−t}$$ which is significantly lower than the financial considered delta.
This question came to my mind, while I was questionning myself what would be the $\Delta$ of a financial product $F$ paying at $t_1$ an ATM Call option with maturity $T$.
$$ Flow(T) = [S(T) - S(t_1)]_+ , with 0 < t_1 < T $$
Reasonning made me to conclude that in $t_1$, the product price would be an ATM Call option with remaining maturity $ T-t_1$ , so $\Delta(t_1) = 0.4σ \sqrt{T−t_1}$.
But should it be this, or $\Delta = 0.5$ ?
What should be the $ \Delta$ at $t$ with $0 < t < t_1$ ? $\Delta(t_1) = 0.4σ \sqrt{T−t}$ ? Or should it be $0$ for $t < t_1$, and then $\Delta(t_1) = 0.5$ ?