Solution: ATM Gamma Explosion Near Expiry

Exercise: ATM Gamma Explosion Near Expiry

Part 1 — Growth tabulation

import numpy as np
from scipy.stats import norm

def gamma_bs(S, K, T, r, sigma, q=0):
    d1 = (np.log(S/K) + (r - q + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    return np.exp(-q*T) * norm.pdf(d1) / (S*sigma*np.sqrt(T))

S, K, r, sigma = 100, 100, 0.05, 0.2
for tau in [0.02, 0.01, 0.005, 0.001, 0.0001]:
    g = gamma_bs(S, K, tau, r, sigma)
    print(f"T-t={tau:.4f}: gamma={g:.4f}")
# T-t=0.02:   gamma=0.1410
# T-t=0.01:   gamma=0.1994
# T-t=0.005:  gamma=0.2820
# T-t=0.001:  gamma=0.6306
# T-t=0.0001: gamma=1.9947

Gamma doubles each time $T - t$ is quartered — consistent with $1/\sqrt{T - t}$ scaling.

Part 2 — Scaling derivation

For ATM $S = K$ with $q = 0$ :

d_1 = \frac{0 + (r + \tfrac{1}{2}\sigma^2)(T - t)}{\sigma\sqrt{T - t}} = \frac{(r + \tfrac{1}{2}\sigma^2)\sqrt{T - t}}{\sigma} \to 0.

$\phi(d_1) \to \phi(0) = 1/\sqrt{2\pi}$ .

\Gamma_{\text{ATM}}(T - t) \to \frac{1/\sqrt{2\pi}}{S\sigma\sqrt{T - t}} = \frac{1}{S\sigma\sqrt{2\pi(T - t)}}.

Exact constant:

1/(S\sigma\sqrt{2\pi}) = 1/(100\cdot 0.2\cdot \sqrt{2\pi}) = 1/(50.13) \approx 0.01995

Check: at $T - t = 0.0001$ , $\sqrt{0.0001} = 0.01$ , so $\Gamma \approx 0.01995/0.01 = 1.995$ . ✓ Matches table.

Part 3 — Delta jump at $T - t = 0.001$

At $S = K = 100, T - t = 0.001$ : $\Gamma \approx 0.63$ .

Linearisation: $\Delta\Delta \approx \Gamma\cdot \Delta S = 0.63\cdot 0.5 = 0.315$ for a $0.50 move. So if $S$ rises from 100 to 100.5, delta jumps from $\approx 0.51$ to $\approx 0.82$ . If $S$ falls to 99.5, delta drops to $\approx 0.19$ .

Over a $1 move: delta changes by

\approx 0.63

— more than half the maximum possible range of delta (

[0, 1]

) in a single dollar's move.

Part 4 — Operational feedback

The market maker is short 1 call, holding $\Delta$ shares as a hedge. Near expiry at $S = K$ :

Stock rises $0.50 to 100.50. Delta jumps from 0.51 to 0.82. MM must buy 0.31 shares to re-hedge. This buying pressure itself pushes the stock higher.
Stock falls $0.50 to 99.50. Delta drops from 0.51 to 0.19. MM must sell 0.32 shares. Selling pressure pushes stock lower.

The MM is buying high and selling low — the classic short-gamma squeeze. For a concentrated position, this self-reinforcing dynamic can:

Move the spot price. Large short-gamma hedging books can cause localised pinning around the strike — the "max-pain" phenomenon observed near expiry on heavily-traded equity options.
Generate large hedging losses. Each rebalancing is done at the wrong side of the bid-ask, and the convexity of the option means losses grow quadratically with the underlying's realised move.
Force dealers to stop making markets. When gamma is high enough and the book is concentrated, desks may widen spreads dramatically or exit the market entirely, reducing liquidity around expiry.

This is why the last trading day of large-volume options (e.g. monthly/quarterly SPX options) frequently shows anomalous price action — dealer gamma hedging, not fundamentals, drives intraday moves.

Takeaways