Part VI · Dealers · Chapter 6D

Gamma

Why the hedge keeps coming undone

1 · Gamma, briefly

Gamma is the partial derivative of delta with respect to spot, which makes it the second derivative of the option's price with respect to spot. Saying those words and understanding what gamma does are different things, so this chapter shows it rather than tells it, starting from the Black-Scholes formula:

Here $S$ is the price of the underlying, $\sigma$ the implied volatility, $T$ the time to expiry in years, $q$ the dividend yield and $d_1$ the z-score of Chapter 6B. The function $\varphi$ is the standard normal density: the height of the bell curve, not the area under it, which is the cumulative distribution function (CDF) $N$. Chapter 5C draws three facts from this formula:

1. Gamma is the same for a call and a put with the same strike. By put-call parity (Chapter 5F), their deltas differ by a constant, so they change with spot at the same rate.
2. Gamma is largest at the money (ATM). The density $\varphi(d_1)$ peaks when $d_1 = 0$, which happens approximately when spot equals the strike.
3. Gamma grows as $1/\sqrt{T}$. As time to expiry shrinks toward zero, at-the-money gamma diverges to infinity. This is not a mathematical curiosity but the dominant feature of modern intraday microstructure, because 0DTE options live exactly in this regime (Chapter 9B).

2 · The short-gamma dealer chases the move

Take a dealer that has sold 1,000 at-the-money calls, strike $100, with 30 days to expiry. At first it hedges by buying shares equal to its net short delta, as in Chapter 6C. As spot moves, delta changes, and the existing share position becomes wrong. The dealer must rebalance, and gamma decides which way.

The dealer is short gamma, so its forced trades go in the same direction as the price move. A rise in spot lifts the calls' delta and leaves the dealer short more share-equivalents, so it buys, while a fall makes it sell. It buys high and sells low, and over a stretch of time the losses from doing so are set against the theta it collects.

Put together, the short dealer's P&L over each short interval $dt$, in which spot moves by $dS$, is the negative of the hedged long option's P&L in Chapter 5C:

The first term is the theta income, $\tfrac12\Gamma S^2\sigma^2 dt$. The second is the cost of rebalancing, $\tfrac12\Gamma(dS)^2$: the buy-high, sell-low loss of short gamma. The dealer wins when the market moves less than the implied volatility it sold, and loses when it moves more.

The simulator follows this dealer along one simulated path of 50 daily bars. The dealer holds delta × 100,000 shares as its hedge, and the calls were sold at 20% IV. Press play, or drag the bar slider from 0 to 50, and use pause and reset as you need. The bars along the top are the shares the dealer buys (up) or sells (down) on each bar, drawn at full height from 3,000 shares. The readouts give spot, the current delta, the shares traded on the bar, the theta income so far, the gamma rebalancing cost so far and the net P&L. The calls run out at bar 30, deep in the money, so from there the hedge stays at 100,000 shares and trades nothing. One simplification: theta accrues a trading day per bar, while the calls' time to expiry falls a calendar day per bar.

bar

0/50

play pause reset

spot = $100.00

current delta = 0.534

shares this bar = 0

theta income = $0

gamma rebal cost = $0

net dealer P&L = $0

Realized stayed below implied: the rebalancing cost so far is no more than the theta income.

Watch the flow bars line up with the price. On bar 1 spot slips to $99.21, delta falls from 0.534 to 0.478, and the dealer sells 5,645 shares. On bar 13 a 1.4% drop to $98.84 makes it sell 12,754. On bar 24 a 2% jump to $102.70 makes it buy 24,311 shares, and that single bar costs about $16,900 in rebalancing, more than any one bar's theta income. By bar 30 delta is 1.000 and trading stops. At bar 50 the theta income is $189,826, the rebalancing cost $60,766 and the net P&L +$129,060, and the readout says realized stayed below implied. The path is drawn with a daily noise of 1.1%, about 17.5% a year. While the calls were alive it moved less, about 0.7% a day, or 11% a year. Both are below the 20% at which the calls were sold, so on this path the dealer won.

This is the destabilizing pattern that dominates regimes of negative gamma exposure (GEX) at the market level. Every dollar of move generates more flow in the direction of the move, which generates more move. That feedback loop is what makes Volmageddon (February 2018), March 2020 and single-stock gamma squeezes like GME (GameStop) possible (Chapters 8H and 8I). It also explains why such episodes feel so violent: not because of news, but because of mechanical positive feedback.

3 · The long-gamma mirror

The opposite case is a dealer that is long gamma. It buys dips and sells rallies, the pattern that stabilizes most days in modern markets and gives positive-GEX regimes their mean-reverting feel.

Studies of dealers' SPX positions find them long gamma on average in the 0DTE era. The Dealer Mechanics Lab explains this by retail selling those options and dealers buying them. It reads the result as partly why intraday SPX volatility has compressed even as headline option volume has exploded. The flow data point elsewhere: customers' 0DTE buying and selling are about balanced. The long gamma comes mainly from longer-dated positions that become 0DTEs as they near expiry (Chapter 9B).

Mirror the short-gamma example, with the signs reversed, at SPX scale, so that it lines up with the dollar-gamma figures later in this chapter.

Spot rising makes the long-gamma dealer sell, which presses the rally back toward where it started. Spot falling makes it buy, which absorbs the fall. The pattern reverses the short-gamma case at every step:

The short-gamma dealer's forced trades reinforce the move, selling into drops and buying into rallies, while the long-gamma dealer's forced trades fade it. It is the same 8,000-share rebalance per 1% move with the opposite sign. This is the structural source of the stabilizing dynamic in positive-GEX regimes. Nobody is providing liquidity out of judgment: the arithmetic of delta hedging requires the opposite trade when the dealer's gamma has the other sign.

The long-gamma dealer's trades also make money. Selling after rises and buying after falls earns $\tfrac12\Gamma(dS)^2$ per move, the mirror of the short dealer's cost, and harvesting it is called gamma scalping. The long dealer pays for the privilege in theta, so it profits only if the market moves more than the implied volatility it paid for.

The bridge to Chapter 8A is that long gamma damps and short gamma amplifies. The aggregate dealer book in any product is a sum of both signs across all dealers, strikes and expiries. When that sum is positive, with dealers net long gamma, the market behaves like the long-gamma case writ large, and when it is negative, like the short-gamma case. Chapter 8A performs exactly this aggregation, collapsing thousands of long- and short-gamma positions into a single signed number that classifies the prevailing regime.

4 · Dollar gamma: the unit that matters

Raw gamma is measured in delta per dollar of spot move, which is hard to compare across underlyings at different prices. The standardized unit dealers actually care about is dollar gamma per 1% move:

To see why this form, start from a 1% move in price, $\Delta S = 0.01\,S$. The dealer's delta changes by $\Gamma\,\Delta S = 0.01\,\Gamma S$ for each share the options are written on, and to hedge it transacts that many shares. The dollar value of that trade is $(0.01\,\Gamma S) \times S = 0.01\,\Gamma S^2$ per share. (6D.3) is that per-share figure. For one contract, multiply by the multiplier of 100, and for a whole strike, by the open interest (OI) as well: $0.01\,\Gamma S^2 \times 100 \times \text{OI}$. Written per share, without the constant 100, it is the canonical dollar gamma per 1% used in every GEX model from SqueezeMetrics onward.

Dollar gamma also prices the rebalancing itself. The trade a 1% move forces is worth $\$\Gamma_{1\%}$, and the hedger's gain or loss on it is half the trade times the move, the area of a triangle. So for a move of $p$ percent the gain or loss is

This is $\tfrac12\Gamma(dS)^2$ written in dollar gamma. For one SPX contract and a 1% move, that is $\tfrac12 \times 46{,}200 \times 0.01 = \$231$, earned by a long-gamma hedger and paid by a short one. Chapter 6E builds its hedging arithmetic on this.

The units lab carries one option's gamma through every step. Its sliders set spot and strike (3,000 to 8,000, both starting at 6,000), days to expiry (1 to 365, starting at 30) and IV (5% to 100%, starting at 18%). Two more set the open interest (100 to 500,000, starting at 10,000) and the multiplier (1 to 1,000, starting at 100), and the rate is 4%. A convention button puts the dealer short the calls ("0DTE-era", the default) or long them ("JPM-era"), the two vendor conventions of Chapter 8B. The table walks from gamma per share to dollar gamma across the open interest, signed from the dealer's side.

spot S

6,000

strike K

6,000

DTE

30d

IV σ

18%

open interest

10,000

multiplier

100

convention 0DTE-era (call short) JPM-era (call long)

Unit	Formula	Value
Γ per share	Black-Scholes gamma	1.28 × 10⁻³
Γ per contract	Γ × multiplier	1.28 × 10⁻¹
$Γ per $1 (1 contract)	Γ × S × multiplier	$769.99
$Γ per 1% (1 contract)	Γ × S² × 0.01 × multiplier	$46.20k
Unsigned $Γ per 1%	× open interest	$461.99M
Signed (dealer perspective)	× dealer sign (−1)	−⁠$461.99M

Short γ, destabilizing. Signed from the dealer's side, the open interest carries −⁠$461.99M of dollar gamma per 1% move.

At the defaults the table reads a gamma of $1.28 \times 10^{-3}$ per share and $1.28 \times 10^{-1}$ per contract, $769.99 of dollar gamma per $1 of spot and $46.20k per 1%, the $46,000 of the worked example. Across 10,000 contracts of open interest that is $461.99M per 1%, signed −$461.99M under the default convention: "Short γ, destabilizing". Now set the multiplier to 10. One SPY contract is 100 shares at about a tenth of the index, the same exposure as 10 units of SPX, and the per-1% figure drops to $4.62k, as in Step 2. Cut days to expiry to 1 and one contract's dollar gamma jumps to $254.03k, 5.5 times the 30-day figure. Raise it to 365 and it falls to $12.67k. The convention button flips only the sign, because gamma itself is never negative: the sign is an assumption about which side of the calls the dealer holds.

Aggregate dealer dollar gamma across the listed SPX book typically ranges from $20B to $100B per 1% move on a normal day. That is the scale of mechanical hedging flow potentially sitting in dealer inventory. A 1% move in SPX could generate tens of billions of dollars of forced rebalancing, depending on the sign of dealer net gamma. Chapter 8A formalizes this into the GEX metric, and Chapter 8D shows how it can characterize the market's regime. The magnitude of dollar gamma tells you the potential size of mechanical flow, and its sign tells you the direction. Chapter 6A gives the more skeptical reading of the same force from Part X.

5 · What the mechanism predicts

If the framework so far is correct, several predictions follow, and each has support:

1. Intraday momentum should appear in markets dominated by short-gamma dealer positioning. Baltussen, Da, Lammers and Martens (2021) document this across more than 60 futures markets (Chapter 6C).
2. Intraday mean reversion should appear in markets dominated by long-gamma dealer positioning. Modern SPX intraday character has compressed measurably since the growth of 0DTE, consistent with the Cboe and Dim-Eraker-Vilkov finding that 0DTE dealer positioning is net long gamma on average (Chapter 8B).
3. Volatility should be self-reinforcing in short-gamma regimes. The Volmageddon and March 2020 case studies of Chapter 8H document exactly this.
4. The transition between regimes, the "gamma flip", should be marked by a visible change of character. The gamma flip is the index level at which dealers' net gamma changes sign. Practitioner experience and the academic literature both confirm the change, though the precise identification of the flip level is contested across vendor methodologies.

The framework is empirically supported, but not in the sense that gamma squeezes account for everything. It is supported in the sense that gamma hedging accounts for measurable effects of tens of basis points, with occasional larger episodes during stress. Part VIII quantifies this, Part IX applies it and Part XIII sizes positions for it.

6 · Check yourself