Gamma exposure by expiration: why near-dated gamma dominates

The square-root rule

In standard pricing models, the gamma of an at-the-money option is proportional to one divided by the underlying price, the implied volatility and the square root of time to expiry. Hold price and volatility fixed and the only moving part is time. Cut the time remaining by a factor of four and at-the-money gamma doubles; cut it by a factor of thirty and gamma rises by the square root of thirty, about 5.5.

That is why gamma, per contract, is dramatically larger on expiration week than a month out. A one-year option has a gamma about nineteen times smaller than a one-day option at the money, since the square root of 365 is about 19.

The same math also concentrates near-dated gamma tightly around the current price. A short-dated option has very large gamma when it is at the money and very little gamma a few percent away, because it has almost no chance of getting there. A long-dated option's gamma is lower at the peak but spread over a much wider band of prices.

What that does to the map

Combine the two effects. Near the current price, short-dated contracts usually carry most of the gamma, even if their open interest is smaller than that of the monthlies. Away from the current price, the short-dated gamma fades quickly and longer-dated positions make up more of what remains.

So the shape of a gamma map depends heavily on which expirations it includes. A map built from all expirations will show the nearest contracts dominating the strikes around price. A map built from monthlies only will look smoother and broader. Neither is wrong; they answer different questions about different horizons.

It also explains why intraday maps can change so much. As same-day options approach their close, their gamma near the money keeps rising while out-of-the-money same-day gamma collapses. The map in the last hour can look very different from the map at the open, with nothing traded.

When monthly and quarterly positions still matter

Size can outweigh the square-root effect. A quarterly SPX expiry with very large open interest at a strike near the price can carry more gamma than the weekly contracts around it, particularly in the final days before it expires, when its own time to expiry is short.

Longer-dated positions also dominate other Greeks. Vega concentrates in long-dated options, so a book can be short gamma in the front and long vega in the back. And longer-dated hedges are more stable: they change slowly from day to day, while near-dated gamma resets every expiration.

That is why it helps to read gamma by expiration rather than only in aggregate: the near-dated layer describes the next few sessions, while the longer-dated layer describes the structure that will still be there next month.

A worked comparison

Picture two strikes at the same distance from price. The first carries 10,000 contracts of same-day open interest; the second carries 40,000 contracts of open interest expiring in a month. Using the square-root rule, each same-day contract at the money has roughly five and a half times the gamma of a month-out contract. So the 10,000 same-day contracts carry gamma roughly equivalent to 55,000 month-out contracts, more than the larger monthly position.

Move both strikes two percent away from price and the picture reverses. The same-day options, with hours left and little chance of reaching the strike, now carry very little gamma, while the monthly options, with time to get there, keep much of theirs. Near-dated gamma dominates near price and fades quickly with distance; longer-dated gamma is lower at the peak but more persistent across prices.

Practical consequences

The near-dated layer is the one most relevant to intraday behaviour and to the character of the next session. The zero gamma level, the Apex and the walls near price are often set by the nearest expiries. After they expire, those levels can shift significantly.

The longer-dated layer shows where large, persistent positioning sits. Strikes that matter in every expiry, rather than only the nearest, describe structure with more staying power. Comparing the map with and without the nearest expiries is a quick way to see which levels are likely to survive the next expiration.

It also changes how to read a level's staying power. A wall built mostly from this week's contracts has a known end date. A wall that appears in this week's, next month's and next quarter's expiries at the same strike is supported by positions that will not all disappear at once.