Gamma
Gamma (Γ) is the second-order sensitivity of an option's price to changes in the price of the underlying asset, measuring the rate of change of the option's delta for a one-unit move in the underlying. As the first derivative of delta with respect to the spot price, gamma quantifies the convexity of an option's value relative to its underlying, and is a critical risk measure for options traders managing delta-hedged portfolios.
Key takeaways
- Gamma is always positive for both calls and puts (for a long position), reflecting the beneficial convexity of option payoffs: as the underlying moves in a favorable direction, delta increases (for calls) or becomes more negative (for puts), accelerating the rate of option value appreciation.
- Gamma is highest for at-the-money options close to expiration, where the probability of expiring in-the-money is most sensitive to small moves in the underlying; deep in-the-money or out-of-the-money options have low gamma as their delta is already near 1 or 0 respectively.
- Theta-gamma trade-off is the fundamental tension in long options positions: long gamma (beneficial convexity) comes at the cost of long theta (time decay eroding option value daily); a delta-neutral long options portfolio profits from large moves but loses money from time decay if the underlying is quiet.
- Gamma hedging—adjusting delta hedges to account for second-order price sensitivity—requires dynamic rebalancing as the underlying price moves, generating transaction costs; the rate of hedge rebalancing is proportional to gamma, making high-gamma positions more expensive to hedge dynamically.
- Dealer gamma positioning has become a key concept in equity market microstructure: when options dealers are 'long gamma' (net positive gamma from short put/call positions from customers), they sell into rallies and buy dips to maintain delta neutrality, dampening volatility; when 'short gamma,' they buy rallies and sell dips, amplifying volatility.
Explanation
Gamma occupies a central position in options risk management, quantifying the curvature or convexity of an option's value relative to the underlying price in a way that delta—a first-order measure—cannot. While delta tells a trader how much an option position will gain or lose for a given move in the underlying, gamma tells the trader how quickly that delta estimate becomes stale as the underlying moves. Understanding gamma is essential for managing dynamic hedging programs, assessing the risk of large market moves, and understanding the market microstructure dynamics created by the aggregate gamma positions of options market makers.
Mathematically, gamma is the partial second derivative of the option's price with respect to the underlying price: Γ = ∂²C/∂S². For a European call option in the Black-Scholes framework, gamma equals N'(d₁) / (S × σ × √T), where N'(d₁) is the standard normal probability density function evaluated at d₁, S is the current underlying price, σ is implied volatility, and T is time to expiration. This formula reveals several important properties: gamma is highest when the option is at-the-money (where N'(d₁) is maximized), decreases as the option moves further in- or out-of-the-money, and increases as expiration approaches for at-the-money options (the √T term in the denominator shrinks). For deep in- or out-of-the-money options near expiration, gamma approaches zero rapidly.
The gamma-theta relationship defines one of the most fundamental trade-offs in options trading. For a long options position (long calls or long puts), gamma is positive—the position benefits from large moves in either direction—but theta is negative—the position loses value as time passes. This relationship arises directly from the Black-Scholes partial differential equation, which shows that delta-neutral option positions must satisfy: Theta + (1/2) × Gamma × S² × σ² = r × Option Value. In rough terms, a long gamma position 'pays for' its positive convexity through daily time decay: if the underlying is quiet, the time decay loss outweighs the convexity benefit, but if the underlying makes large moves, the gamma profit exceeds the accumulated theta loss.
Gamma scalping is the dynamic hedging strategy that attempts to extract profit from gamma by systematically rebalancing the delta hedge as the underlying moves. A market maker who is long gamma will re-hedge their delta after each significant underlying move, selling when prices rise (because their delta has increased) and buying when prices fall (because their delta has decreased)—effectively buying low and selling high on each oscillation. The profitability of gamma scalping depends on realized volatility exceeding the implied volatility at which the options were purchased; if realized volatility is lower than implied volatility, the theta cost exceeds the gamma profits.
The market-wide gamma positioning of options dealers has emerged as a significant driver of short-term equity market dynamics. When dealers hold large inventories of options sold to end users (making dealers short options and therefore short gamma), market moves trigger procyclical delta hedging that amplifies volatility: a market decline forces dealers to sell more equity futures to maintain delta neutrality, accelerating the decline. Conversely, when dealers are net long gamma (having sold options to buyers who are the end users), their countercyclical hedging dampens moves. Research by firms including SpotGamma and Nomura's cross-asset derivatives team has shown that aggregate dealer gamma positioning correlates with market volatility patterns, particularly around major options expiration dates (such as monthly and quarterly SPX expiries).
Formula
Γ = ∂²C/∂S² = N'(d₁) / (S × σ × √T), where d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
Example
A trader holds 100 long at-the-money call options on a $100 stock, each with a delta of 0.50 and a gamma of 0.04. The position's aggregate delta is 100 × 0.50 = 50 shares (equivalent to being long 50 shares). The aggregate gamma is 100 × 0.04 = 4.0. If the stock rises from $100 to $101 (a $1 move), the position's delta increases by the gamma: new delta ≈ 50 + (4.0 × $1) = 54 shares. The trader's delta-hedged position, which started short 50 shares of stock to be delta-neutral, is now long 4 net deltas—a profit-generating position from the favorable gamma. If the stock then falls back to $100, the delta returns to 50, and the trader re-hedges by selling the 4 shares acquired, capturing a profit of $4 (4 shares × $1 gain from $100 to $101 average, sold at $101 during the re-hedge). Over time, this scalping of gamma profits generates income proportional to realized volatility, partially or fully offsetting the theta decay on the long options position.
Related terms
At The Money Backwardation Binary Option Call Option Class Of Options Convexity Delta Delta Hedge Diagonal Spread Equity Gamma Scalping Hedging