Exercise: ATM Gamma Explosion Near Expiry

Prerequisites: Gamma

Problem

A market maker is short a large position in a 1-week ATM call ($S_0 = K = 100, r = 0.05, \sigma = 0.2, q = 0$). As expiry approaches, gamma explodes and hedging becomes increasingly costly.

1. Compute and tabulate ATM gamma ($S = K = 100$) at times to expiry $T - t \in \{0.02, 0.01, 0.005, 0.001, 0.0001\}$ (in years). Note the growth rate.

2. Show analytically that $\Gamma_{\text{ATM}} \sim 1/\sqrt{T - t}$ as $T - t \to 0$ (with $q = 0, r \approx 0$). What's the exact constant?

3. Pin-risk scenario. Suppose at $t = T - \epsilon$ (e.g. $\epsilon = 0.001$ years $\approx 6$ hours before expiry), the stock is at $100$ and a $0.50 move in either direction would change delta by how much?
4. Why does this create operational risk? Explain the feedback mechanism: as expiry approaches for an ATM option, delta jumps from near-0 to near-1 over an arbitrarily small move in $S$. How does this cause a market maker who has sold the option to get squeezed?

Hint

Use $\Gamma = e^{-qT}\phi(d_1)/(S\sigma\sqrt{T-t})$. For ATM with $q = r = 0$: $d_1 = \tfrac{1}{2}\sigma\sqrt{T-t} \to 0$, so $\phi(d_1) \to \phi(0) = 1/\sqrt{2\pi}$.