Gamma

Motivation: why this matters in quant finance

Gamma (

\Gamma

) is the second-order sensitivity of an option's price to the underlying:

\Gamma := \frac{\partial^2 V}{\partial S^2} = \frac{\partial\Delta}{\partial S}.

Gamma measures how fast delta changes. It's the curvature of the option price as a function of the underlying. Gamma appears everywhere in option trading:

Residual risk after delta-hedging. A delta-hedged book is exposed to gamma: when the underlying moves, the hedge needs rebalancing, and the mismatch shows up as gamma P&L.
Hedging frequency economics. The optimal rebalancing frequency trades gamma risk against transaction costs — high-gamma positions need frequent hedging.
Convexity = gamma. Convex (long-gamma) positions gain from volatility; concave (short-gamma) positions lose. This is the mathematical content of "long vol ≈ long gamma."
Pin risk. At expiry, gamma of ATM options blows up — near-the-money positions see extreme gamma swings on the last trading day, known as pin risk.
Gamma scalping. Market makers long gamma systematically capture the "volatility premium" by repeatedly delta-rebalancing as the stock oscillates.

The informal idea

Gamma is how much your delta changes per $1 move in the underlying. If $\Delta = 0.5$ and $\Gamma = 0.02$ , then after a $1 rise in the stock, $\Delta \approx 0.52$ .

A long option position has positive gamma — the payoff is convex in

S

, so delta rises as

S

rises. Short options have negative gamma — you've sold convexity.

Why it matters practically: after delta-hedging, your residual P&L over a small time $dt$ is

d\Pi \approx \tfrac{1}{2}\Gamma(dS)^2 - \Theta\,dt \text{ (long option, delta-hedged)}.

So long gamma pays you when the stock moves (quadratic payoff) but costs you time decay. Short gamma is the reverse: you collect theta but bleed on large moves.

Black-Scholes gamma

For a European call (or put — they share gamma via put-call parity) under Black-Scholes:

\Gamma = \frac{\phi(d_1)e^{-q(T-t)}}{S\sigma\sqrt{T-t}},

where

\phi

is the standard normal PDF and

d_1

is as in the delta lesson.

Properties

Call gamma = put gamma. Because they differ by a linear function of $S$ (parity), second derivatives are equal.
Always positive for long vanillas. $\phi > 0, S > 0, \sigma > 0$ ; so $\Gamma > 0$ for long calls/puts.
Maximised ATM. $\phi(d_1)$ peaks at $d_1 = 0$ , so gamma is largest for ATM options.
Diverges as $T - t \to 0$ for ATM options. The $1/\sqrt{T-t}$ factor makes ATM gamma blow up near expiry — the "pin risk" phenomenon.
Decays for deep ITM/OTM. $\phi(d_1)$ decays rapidly away from the strike.

Gamma and vol

A crucial identity in Black-Scholes is the gamma-theta-vega relationship. The BS PDE is

\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \tfrac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV.

Setting $\Delta = \partial V/\partial S$ , $\Gamma = \partial^2 V/\partial S^2$ , $\Theta = \partial V/\partial t$ :

\Theta + rS\Delta + \tfrac{1}{2}\sigma^2 S^2\Gamma = rV.

For a delta-hedged portfolio (

\Delta

cancelled by stock holding) and ignoring the

rV

funding term,

\Theta \approx -\tfrac{1}{2}\sigma^2 S^2\Gamma

. Theta and gamma are linked by volatility. Long gamma ⇔ short theta; you pay time decay for the right to collect variance.

Proof sketch

$\Delta_{\text{call}} = e^{-q(T-t)}\Phi(d_1)$ . Differentiating:

\frac{\partial\Delta}{\partial S} = e^{-q(T-t)}\phi(d_1)\cdot \frac{\partial d_1}{\partial S} = \frac{e^{-q(T-t)}\phi(d_1)}{S\sigma\sqrt{T-t}}.

Numerical example

$S = 100, K = 100, T = 0.5, r = 0.05, q = 0, \sigma = 0.2$ :

$d_1 \approx 0.247$ , $\phi(0.247) \approx 0.3866$ .

\Gamma = \frac{0.3866}{100\cdot 0.2\cdot \sqrt{0.5}} = \frac{0.3866}{14.14} \approx 0.0273.

So a $1 move in the stock changes delta by $\approx 0.0273$ . If $S$ rises from 100 to 105: $\Delta$ rises by roughly $0.0273\times 5 = 0.137$ , giving a new delta of $\approx 0.735$ (compared to direct computation $\approx 0.731$ ).

Gamma P&L and the Black-Scholes PDE

For a delta-hedged portfolio with value $\Pi = V - \Delta S$ (short one option, long $\Delta$ shares):

d\Pi = dV - \Delta dS.

Applying Itô's lemma to $V(S, t)$ :

dV = \Theta\,dt + \Delta\,dS + \tfrac{1}{2}\Gamma\,(dS)^2.

So $d\Pi = \Theta\,dt + \tfrac{1}{2}\Gamma(dS)^2$ .

For GBM with realised volatility $\sigma_R$ : $(dS)^2 = \sigma_R^2 S^2\,dt$ . The expected P&L per $dt$ is

\mathbb{E}[d\Pi] = \Theta\,dt + \tfrac{1}{2}\Gamma\sigma_R^2 S^2\,dt.

Using BS theta (valid under $\sigma = \sigma_{\text{imp}}$ ): $\Theta = -\tfrac{1}{2}\sigma_{\text{imp}}^2 S^2\Gamma$ (ignoring small rate terms). So

\mathbb{E}[d\Pi] = \tfrac{1}{2}\Gamma S^2(\sigma_R^2 - \sigma_{\text{imp}}^2)\,dt.

This is the gamma scalping formula: a long-gamma trader profits when realised volatility exceeds implied volatility. Short gamma = negative carry on the same mismatch. This formula is the quantitative basis for the "long vol / short vol" trading lingo.

Common pitfalls

"Gamma is the rate of change of delta with respect to time." No — that's charm (

\partial\Delta/\partial t

), a less commonly tracked Greek. Gamma is spatial curvature.

"Short gamma is always bad." It depends on the premium. Short-gamma trades (e.g. selling straddles, covered calls) are often profitable on average because implied vol typically exceeds realised vol (variance risk premium). You collect theta in exchange for tail risk.

"Gamma is a moneyness metric." Not exactly — gamma is maximised ATM, but zero ATM and ITM values of gamma both mean "far from the strike." So the magnitude of gamma is a measure of ATM-ness, not ITM-ness.

"ATM gamma is infinite at expiry." Only the limit as

T - t \to 0

with

S = K

; for any finite

T - t

, gamma is finite. But practically, gamma can get so large near expiry that rebalancing becomes infeasible.

"Delta + gamma fully describes option risk." Only to second order. For large moves, you also need higher-order Greeks (speed, colour) and cross-terms with volatility (vanna, volga). First-second order is only a local linearisation.

Where this goes next

Delta: First-order hedge ratio.
Vega: Volatility sensitivity (first-order in $\sigma$ ).
Itô's Lemma: Gamma is the second-order term of Itô's lemma applied to $V(S, t)$ .
The Derivation of the Black-Scholes Formula: Delta-gamma-theta relation built in.
Implied Volatility: Gamma-scalping exploits the realised-vs-implied vol spread.