Asymptotic Limits of Black-Scholes Call Delta

Scenario

The Black-Scholes Delta of a European call option on a non-dividend paying stock is given by: $\Delta = \Phi(d_1), \quad d_1 = \frac{\ln(S/K) + \left(r + \frac{1}{2}\sigma^2\right)T}{\sigma \sqrt{T}}$

Your Task Derive the exact mathematical limit of $\Delta$ under each of the following limiting regimes and provide the financial intuition:

1. $S \to \infty$ (deep in-the-money).
2. $S \to 0^+$ (deep out-of-the-money).
3. $T \to 0^+$ (expiration limit) for three subcases: $S > K$, $S < K$, and $S = K$.
4. $\sigma \to \infty$ (extreme volatility limit).

Constraints

• $K > 0, r \ge 0, \sigma > 0, T > 0$.

Examples At expiration $T \to 0^+$, an in-the-money call ($S > K$) becomes equivalent to holding 1 share of stock ($\Delta = 1$), while an out-of-the-money call becomes worthless ($\Delta = 0$).