Gamma in options: convexity and how delta changes with the underlying
Gamma is the second-derivative Greek. Where delta tells how an option's price moves with the underlying, gamma tells how delta itself moves. It is the source of the curvature that distinguishes options from linear instruments and the reason option positions need continuous re-hedging.
What gamma is
Mathematically, gamma is the second partial derivative of the option price with respect to the underlying price, or equivalently the first derivative of delta. Gamma has the same sign for calls and puts: long option positions have positive gamma, short option positions have negative gamma. A gamma of 0.05 means that for a one-unit rise in the underlying, the option's delta rises by 0.05.
Gamma is highest for at-the-money options near expiry. Deep in-the-money or deep out-of-the-money options have gamma close to zero because their delta is already near the boundary (one or zero) and barely changes with underlying moves. The peak gamma at the money near expiry is what produces the dramatic P&L moves seen in short-dated at-the-money options.
How it works
Positive gamma means a position becomes more sensitive to the underlying as the underlying rises, and less sensitive as it falls. This is the convexity that long option holders pay for—and short option sellers are paid to bear. The Taylor expansion of an option's price under a small underlying move includes a half-gamma-times-move-squared term, which is positive whenever the underlying moves in either direction.
Gamma scalping is the practice of capturing this convexity. A long-gamma position is delta-hedged at frequent intervals: when the underlying rises, the position's delta rises with it, and the hedger sells underlying to bring delta back to zero, locking in profit. When the underlying falls, the position's delta falls, and the hedger buys underlying. The repeated buy-low / sell-high cycle is the cash version of the half-gamma-times-move-squared term.
What the evidence shows
The gamma-scalping P&L of a delta-hedged long option position is positive whenever realised volatility exceeds implied volatility (Derman & Kani, 1994). Conversely, a delta-hedged short option position earns positive P&L when implied volatility exceeds realised. This decomposition is the cleanest empirical statement of the volatility risk premium: investors who systematically sell options earn the premium of implied over realised volatility, while bearing the convexity cost in extreme moves.
Studies of dealer P&L attribution (Bakshi, Cao & Chen, 1997) confirm that gamma is the dominant non-vol driver of short-dated option P&L for delta-hedged books, and that gamma-related P&L tends to peak around earnings and other event-driven volatility regimes.
Limitations and trade-offs
Gamma cuts both ways. Long-gamma positions earn money in volatile markets but suffer time decay (negative theta) when markets are quiet. Short-gamma positions earn time decay but lose heavily in large moves. The well-publicised blow-ups of short-volatility funds in 2018 and 2020 were classic short-gamma episodes: the strategies earned theta steadily for years, then surrendered most of those earnings in a few sessions of explosive realised volatility.
Gamma also concentrates around specific strikes. Dealer hedging behaviour around large gamma exposures can amplify underlying moves, particularly into option expiry—a phenomenon sometimes called gamma squeeze. The effect is largest in single-name equities with concentrated short-dated option open interest.
Gamma in pfolio
Options are not currently part of pfolio's investable universe, so gamma is not displayed in pfolio Insights. Investors who use options through their broker can monitor gamma via the broker's tools and supplement pfolio's portfolio-level analytics with options-specific risk metrics.
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