Delta of an At-the-Money Binary Option at Expiry
What is the Delta of an at-the-money binary option (payoff $0$ for $S_T < 100$ and $1$ for $S_T \ge 100$) as it approaches expiry?
Answer
The Delta becomes infinite as the option approaches expiry.
As we move toward expiry, the price of the binary option converges to its payoff structure, which has a sharp discontinuity at $S_T = 100$. Since the price jumps instantaneously from $0$ to $1$ at this point, the slope (Delta) is infinite.
Alternatively, suppose the underlying price changes by a small $\varepsilon$ around $100$. Then the change in the option price is approximately $100 - 0 = 1$, so
\[ \Delta \approx \frac{100 - 0}{\varepsilon} = \frac{1}{\varepsilon}, \]which can be made arbitrarily large as $\varepsilon \to 0$.
Notes and comments
Comment 1: Note that delta need not lie between $-1$ and $0$ (for puts) or between $0$ and $1$ (for calls). These ranges apply only to standard calls and puts.