What is Options Gamma? Definition, Formula, and Example
Options gamma is the rate of change of an option's delta for a one-dollar move in the underlying — the second derivative of option price with respect to price.
Options Gamma: Plain-English Definition
Gamma is the rate of change of an option's delta for every $1 move in the underlying stock. Where delta tells you how much the option price moves for a $1 move in the stock, gamma tells you how much delta itself moves. It is the second derivative of option value with respect to the underlying price — a measure of convexity. Long option positions (calls and puts purchased) always have positive gamma. Short option positions always have negative gamma. Gamma is highest for at-the-money contracts near expiration and approaches zero as options move deep in- or out-of-the-money.
How Gamma Is Calculated
Under Black-Scholes, gamma for a European option is:
Γ = φ(d1) / (S · σ · √T)Where φ(d1) is the standard normal probability density function evaluated at d1, S is the underlying price, σ is implied volatility, and T is time to expiration in years. Call gamma equals put gamma for the same strike and expiration — gamma is sign-independent across call/put.
Position gamma = Γ × contract multiplier (100) × number of contracts × position sign.
Worked Example
AAPL trades at $180. Consider the $180-strike call, 30 days to expiration, 25% implied volatility, 4.5% risk-free rate, zero dividend for simplicity.
d1≈ 0.09φ(d1)≈ 0.397σ√T= 0.25 × √(30/365) ≈ 0.0716Γ= 0.397 / (180 × 0.0716) ≈ 0.0308
Delta for this call is roughly 0.53. If AAPL moves to $181, delta rises to approximately 0.53 + 0.0308 ≈ 0.56. If AAPL moves to $183, delta rises to roughly 0.62. A trader long ten of these contracts holds position gamma of 0.0308 × 100 × 10 = 30.8 — every $1 move in AAPL adds 30.8 shares of delta exposure.
When Traders Use Gamma
Delta-hedged market makers rebalance inventory as gamma shifts their delta exposure; a dealer short a large block of near-dated calls must buy the stock as it rises and sell as it falls, amplifying price moves. Retail traders use gamma to identify gamma squeezes — dealer hedging flows that force a feedback loop in heavily-optioned names like GME or TSLA. Income sellers of iron condors and short strangles monitor short gamma to size positions against tail moves.
Limitations and Common Misconceptions
Gamma is a snapshot — it changes with price, time, and implied volatility. It is highest at-the-money and collapses to zero deep ITM or OTM. Gamma rises sharply in the final week before expiration ("gamma explosion"), which is why weekly options whipsaw violently around expiration. Gamma does not tell you directional risk — a gamma-positive straddle profits from any large move, while a gamma-negative iron condor loses on large moves in either direction. Gamma also ignores volatility changes entirely; a trader long gamma but short vega can lose money if IV collapses even as the stock moves favorably. Reading gamma in isolation is misleading — it must be combined with delta, theta, and vega.