Gamma

Definition

Gamma is the rate of change of delta with respect to the underlying price — the second derivative of option value. It measures how fast directional exposure itself changes as the underlying moves: high gamma means a position’s delta is unstable. Long options have positive gamma (delta moves in the holder’s favor); short options have negative gamma (exposure grows against the writer).

How it works / structure

• Formula (Black-Scholes-Merton): gamma = φ(d1) / (S·σ·√T), where φ is the standard normal density — identical for calls and puts at the same strike/expiry.
• Shape: peaks at the money and rises steeply as expiration nears; deep in- or out-of-the-money options have little gamma.
• P&L meaning: for a delta-hedged position, gamma P&L per period ≈ ½ · gamma · (ΔS)² — long gamma earns from realized movement, short gamma pays for it; the offset is time decay (greek-theta) — the two are the same tradeoff seen from opposite sides.
• Position aggregation: net gamma sums like delta (port-portfolio-greeks); a book’s gamma sign says whether it wants movement or quiet.
• Simulation parameters: per-leg gamma at each step; rebalance rules in mgmt-delta-hedging are effectively gamma-management parameters (band width vs rebalance frequency).

When it applies

Judging hedge stability (how often a delta hedge must be reset), choosing expiries (short-dated = high gamma exposure), understanding expiration-week behavior (event-opex — gamma concentrates at near-the-money strikes), and reading whether a strategy is long or short realized movement.

Risk profile & failure modes

• Short-gamma acceleration: losses on short options grow quadratically with move size — the mechanism behind “picking up pennies in front of larger losses” outcomes in short-premium strategies; every pillar-4 short-premium entry carries this.
• Expiry spike: at-the-money gamma near expiration makes P&L swing violently around the strike (pin risk, ms-expiration-exercise-assignment).
• Hedging cost drag: long gamma is paid for through theta; realized movement below the priced level makes long-gamma positions steady losers.
• Discrete-hedge residual: gamma P&L formulas assume continuous rebalancing; real hedges at intervals leave path-dependent residual risk.

Evidence & limits

Gamma’s mathematics follow the Black-Scholes-Merton framework; textbook treatment (Hull) is the standard reference. Whether long or short gamma is compensated on average is the volatility-risk-premium question — evidence lives in opt-implied-volatility and indicator-realized-vs-implied-vol, not here.

Falsifiable-thesis examples

Illustrations only, not signals:

• “A long at-the-money straddle on X held two weeks, delta-hedged daily, will profit if realized volatility exceeds entry IV by 5+ points” — falsified by replay P&L given the realized path.
• “X will close within 1% of strike K on expiration Friday” (a pin thesis) — falsified by a close outside the band.

Cross-references

• Companions: greek-delta (what gamma changes), greek-theta (what gamma costs), greek-vega
• Mechanics: event-opex, ms-expiration-exercise-assignment
• Management: mgmt-delta-hedging
• Aggregation: port-portfolio-greeks