Pin Risk At Expiry

Most of the time an option's fate is obvious well before expiry: deep in the money, it will be exercised; far out, it won't. The hard case is when the stock finishes the day within a few cents of the strike. Nobody, including the option holder, knows for certain whether it will be exercised, and the market maker who hedged it has to decide, before that certainty exists, how many shares to hold.

A coin balanced on its edge

Picture a coin spinning down to rest exactly on its edge. You had to place your bet — heads or tails — before it settled, because trading stops at the close. If it topples your way, your hedge was right. If it topples the other way, you wake up holding shares you don't want, or missing shares you needed, with the market already open and moving against you. That is pin risk: your hedge was built for a probability, but the outcome is binary, and by the time you know which side won, it's too late to trade.

Why gamma is the culprit

$$\Gamma = \frac{\phi(d_1)}{S\,\sigma\sqrt{T}}$$

In plain English: gamma measures how fast delta changes as the stock moves, and it grows as time to expiry $T$ shrinks, because $\sqrt{T}$ sits in the denominator. Far from expiry, delta creeps from 0 toward 1 gradually as the stock rises. On the last day, that entire journey compresses into a razor-thin band around the strike — delta effectively jumps from near 0 to near 1 within pennies of $K$.

Worked example 1: how much gamma explodes. Take an at-the-money option, $S=K=100$ (a $100 stock), $\sigma=20\%$, and set $r=0$. With one day left, $T=1/365=0.00274$, so $\sigma\sqrt{T}=0.20\times0.0523=0.01047$, and $\Gamma=0.3989/(100\times0.01047)=0.381$. Compare that to the same option with 30 days left, where the identical arithmetic gives $\Gamma\approx0.0695$ (see Deriving The Black-Scholes PDE). Gamma is roughly 5.5 times larger on the final day — delta near the strike swings almost violently for the smallest tick in the stock.

Drag the stock price near $100 in the payoff above and notice how the outcome (exercise or not) flips over a range of only a few cents — that flip is exactly what the market maker can't see coming before the close.

Worked example 2: what you're stuck with. You've sold 10 call contracts (1,000 shares of exposure) struck at $50, and you delta-hedged by holding 500 shares long, assuming roughly 50% odds of assignment. The stock closes at $50.02 — technically in the money — so all 10 contracts are assigned: you must deliver 1,000 shares, but you only own 500. You're short 500 shares over the weekend, unhedged. Had the stock closed at $49.98 instead, no assignment happens and you're left long 500 shares you never wanted, also unhedged. Either way, if the stock gaps 3% before you can trade again, that's $500 \times 50 \times 0.03 = 750$, i.e. $750 of pure, unpriced luck — money made or lost for no reason connected to your view of the stock.

What this means in practice

Market makers with a large open interest sitting on a pin strike often trade extra size right into the close specifically to flatten the exposure before it becomes a binary bet — buying or selling stock to reduce reliance on guessing assignment. Index options mostly sidestep the physical-delivery version of this problem by cash-settling, but they still face settlement-price uncertainty, since the official settlement value (often an opening-print average, not the prior close) can differ from wherever the book was last marked. Zero-DTE And Weekly Options turn this from a monthly nuisance into a daily one.