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Education / Track II · The Greeks / L.06

Gamma: The Convexity of Delta

Gamma is the Greek that breaks naive intuition. It is small when nothing is happening and brutal when everything is. Almost every options blow-up in history involves someone underestimating gamma.

15 min read · Lesson 6 of 24

What gamma measures

Gamma is the rate of change of delta with respect to a move in the underlying. If your call has a delta of 0.40 and a gamma of 0.05, then a $1 rise in the stock will push the delta to roughly 0.45. A $1 fall will push it to 0.35.

Gamma is the curvature of the option's price. Delta is the slope. Gamma tells you how much that slope is itself bending.

Why gamma is a real risk

If delta were constant, you could hedge an option position by holding a fixed number of shares of the underlying, and the hedge would work. But because delta moves (because of gamma), the hedge stops being neutral the moment the stock moves.

This means delta-hedged option positions are not really riskless. They are exposed to realized volatility. The bigger the actual moves, the more rebalancing is required, and the more gamma matters. This is why options are sometimes called "a bet on volatility" even when you take no directional view.

$60 $80 $100 $120 $140 0 0.02 0.04 Underlying Price Gamma Peak at ATM Gamma vs Underlying Price · Strike $100 · 90 days · IV 25%
Fig. 06.1 Gamma peaks at the strike and decays in both directions. ATM options have the most gamma.

Gamma is highest at the money and near expiry

Two facts about gamma matter most in practice:

  1. Gamma is highest at the strike. Far OTM and far ITM options have low gamma. Their deltas barely move because they are essentially already "decided" (worthless or a stock equivalent).
  2. Gamma rises sharply as expiry approaches. A 7-day ATM option has many times more gamma than a 90-day ATM option.
$80 $90 $100 $110 $120 0 0.05 0.10 0.15 Underlying Price Gamma 7d 30d 60d 90d Gamma at Different Times to Expiry · Strike $100
Fig. 06.2 Gamma profile across days to expiry. Short-dated ATM options have explosive gamma. The 7-day curve is several times taller than the 90-day curve.

The combination is what makes the last week before expiry so dangerous for short-gamma positions. A short ATM straddle that looks safe with three weeks left can become catastrophic in the final two days. The same dollar move in the stock causes a much larger P&L swing.

Long gamma vs short gamma

If you are long an option, you are long gamma. The position benefits from large moves in either direction. Your delta keeps adjusting in your favor. If you are short an option, you are short gamma. You are paid premium up front, but you bleed money on every meaningful move.

This asymmetry is the entire mechanic behind volatility trading. Long gamma is a long volatility position. Short gamma is a short volatility position. The connection is direct.

The intuition you want
Long gamma is like owning insurance on movement. You pay theta as the cost of carrying it. When realized movement exceeds what you paid for, you make money. Short gamma is the opposite: you are the insurance company. You collect premium but eat losses every time the underlying actually moves.

Gamma scalping

Market makers and option dealers run delta-hedged books. Because their positions have gamma, their delta drifts as the underlying moves. Re-hedging that delta forces them to buy when the stock falls (because falling stock makes their delta more negative) and sell when it rises. This is called gamma scalping.

If you are long gamma, gamma scalping generates real cash. The bigger the moves, the more cash. If you are short gamma, the same activity bleeds cash. Every re-hedge locks in a small loss. Over many small re-hedges in a volatile market, the losses compound.

Why short-gamma blow-ups happen

The classic options disaster is a trader who has been short gamma for months, collecting steady theta as the market drifts sideways, and then gets caught in a sudden move. The losses come fast and they come at an accelerating rate, because gamma itself is increasing as the move develops.

This is not a statistical artifact. It is the math of the position. Short-gamma positions are designed to lose more on big days than they gain on quiet days. They survive by relying on the days being quiet.

Short gamma is the trade that pays you slowly and bills you all at once.

How traders think about gamma

Quantitative options traders rarely talk about "the option price went up." They talk about gamma exposure. "Net long gamma" means the book benefits from movement. "Short gamma into the close" means the book is exposed to a late-day reversal. The framing is volatility-first, direction-second.

Track 5 of this curriculum returns to gamma at the dealer level (L.20). For now, just know that every option you trade carries gamma, and that gamma is what makes options behave non-linearly when things get interesting.

What you carry forward