Gamma - The Rate of Change of Delta

By EC Assets Research Team, Derivatives Strategy · Published 19 May 2026 · Updated 15 June 2026

Gamma — Gamma measures how an option's delta changes as the underlying price moves. It is the second derivative of option price with respect to underlying price, and the source of both convexity benefit and dynamic-hedging cost.

Definition

Gamma is the second derivative of option price with respect to the underlying asset price. While delta measures how much the option price changes per unit change in the underlying, gamma measures how delta itself changes per unit change in the underlying. It is, in effect, the rate of change of the rate of change.

Mathematically: Γ = ∂Δ/∂S = ∂²V/∂S². For a Black-Scholes call option, gamma equals φ(d₁) / (S × σ × √T), where φ is the standard normal probability density function and d₁ is the standard Black-Scholes term.

Operationally, gamma captures the convexity of option payoffs. Long options (calls or puts) have positive gamma - their payoffs are convex in the underlying. Short options have negative gamma - their payoffs are concave. This convexity is the source of much of the strategic value (and risk) of options.

When Gamma is Highest

Three structural features drive gamma magnitude:

The combination produces the highest gamma for at-the-money options near expiry - these are also the options that change in value most dramatically with small moves in the underlying, which is the operational manifestation of high gamma.

Long Gamma vs Short Gamma

[!key] The strategic decision between long-gamma and short-gamma positions is the most important single trade in volatility markets. Long-gamma positions (long calls or puts, long straddles) profit when realised volatility exceeds implied volatility. Short-gamma positions (short options, short straddles) profit when realised volatility is below implied volatility. Most professional volatility trading is fundamentally a bet on whether implied volatility will be realised.

The 1-day asymmetry: long gamma is comfortable when nothing happens (small loss equal to theta) but produces large gains when underlying moves. Short gamma is comfortable when nothing happens (premium kept) but produces large losses when underlying moves.

Dynamic Hedging and Gamma

Gamma exposure interacts with delta-hedging strategies in important ways. A market-maker who sells an option must delta-hedge to remain market-neutral. The dynamic of this hedging illustrates gamma's economic role:

Short option (negative gamma). Market-maker sells a call, becomes short the underlying through delta. As underlying rises, delta becomes more negative - market-maker must buy underlying (at higher prices). As underlying falls, delta becomes more positive - market-maker must sell underlying (at lower prices). The market-maker's hedging activity systematically loses money proportional to realised volatility; the option premium received compensates for this loss.

Long option (positive gamma). Reverse dynamic. Market-maker who buys a call must short the underlying as delta hedge. As underlying rises, delta becomes more positive - market-maker sells more underlying (at higher prices). As underlying falls, delta becomes more negative - market-maker buys underlying (at lower prices). The hedging activity systematically makes money proportional to realised volatility; the option premium paid is the cost.

The Volatility Risk Premium and Gamma

The structural relationship between long and short gamma underlies the volatility risk premium discussed in other articles. Option sellers (short gamma) demand a premium (implied vol > expected realised vol) as compensation for taking the convex risk. Option buyers (long gamma) pay the premium as the cost of convex protection.

Over long horizons, the volatility risk premium has paid option sellers a positive expected return - implied volatility has averaged 3-4 vol points above realised. But the premium is paid in years and lost in crashes. The 2018 Volmageddon and February 2020 produced large losses for systematic short-gamma strategies that had appeared profitable for years.

Common Misconceptions

"Higher gamma is better." Higher gamma means more convexity but also higher cost. Out-of-the-money options have lower gamma but lower premium; at-the-money options have higher gamma but higher premium. The relevant question is gamma per dollar of premium, not absolute gamma level.

"Gamma is the same as volatility exposure." Related but distinct. Vega captures sensitivity to changes in implied volatility (the price of options). Gamma captures sensitivity to realised price moves of the underlying. A position can have high vega and zero gamma (e.g., specific calendar spreads) or vice versa.

"Long-gamma positions are always safer." Cheaper to recover from but more expensive to maintain. Long-gamma positions bleed via theta when nothing happens; the strategy only works if eventually the market moves enough to make the gamma profit exceed the theta cost.

Gamma Across Maturities and Moneyness

A practical illustration of how gamma varies. Consider S&P 500 options with the underlying at 5,000:

At-the-money options near expiry have the highest gamma. This is why short-dated, at-the-money options react most violently to small underlying moves.

Pin Risk

[!warning] Pin risk is a specific gamma-related phenomenon at expiration. When the underlying closes near a major option strike at expiration (the "pin"), large gamma exposures held by market makers must be hedged through purchases or sales. The hedging activity can push the underlying further toward the strike, creating self-reinforcing dynamics. Famous examples: Tesla pinning at $200, multiple AAPL pinning episodes around earnings expirations. Institutional traders with large positions near major strikes near expiration must be alert to pin dynamics that can materially affect short-term price behaviour.

References

1. Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson.
2. Natenberg, S. (2015). Option Volatility and Pricing (2nd ed.). McGraw-Hill.
3. Chicago Board Options Exchange (CBOE). Options education materials. (https://www.cboe.com/education)