Option Gamma Calculator
Gamma is the second-order derivative of an option's price with respect to the underlying asset price, and equivalently the first derivative of delta. It tells you how quickly your delta exposure changes as the stock moves. A position with gamma of 0.05 will see its delta increase by 0.05 for each $1 gain in the stock. Gamma is always positive for both long calls and long puts, meaning long option holders benefit from convexity: their delta increases in favorable directions. Gamma is highest for at-the-money options near expiration, making those options the most sensitive to rapid moves. Market makers who are "short gamma" from selling options face rapidly growing delta exposure and must hedge aggressively during large market moves.
Gamma formula
d1 = [ln(S/K) + (r + sigma^2/2) * T] / (sigma * sqrt(T))
phi(d1) = (1/sqrt(2*pi)) * e^(-d1^2/2) [standard normal PDF]
Gamma = phi(d1) / (S * sigma * sqrt(T))
Where phi() is the standard normal probability density function, S is the stock price, K is the strike price, r is the risk-free rate (decimal), T is years to expiration, and sigma is annualized volatility (decimal). Gamma is identical for calls and puts with the same inputs.
Gamma in practice
- Gamma peaks at-the-money and increases as expiration approaches, making short-dated ATM options the most gamma-intensive.
- Long options (calls and puts) have positive gamma; short options have negative gamma.
- A gamma of 0.05 means the delta of your option changes by 0.05 for every $1 move in the stock.
- To convert gamma to a dollar P&L estimate for a $1 move: 0.5 * gamma * (price change)^2 * contracts * 100.
- Gamma scalping involves repeatedly delta-hedging a long gamma position to capture realized volatility.
Frequently asked questions
What is option gamma?
Gamma measures the rate of change of an option's delta for a $1 change in the underlying asset's price. If a call has delta 0.50 and gamma 0.05, a $1 rise in the stock changes the delta to approximately 0.55.
Which options have the highest gamma?
At-the-money options have the highest gamma, especially as expiration approaches. Deep in-the-money or deep out-of-the-money options have near-zero gamma because their delta barely changes with small price movements.
Is gamma the same for calls and puts?
Yes. By put-call parity, calls and puts with the same strike, expiration, underlying, and other parameters have identical gamma values. Gamma is always positive for long option positions (calls and puts).
What does high gamma mean for a trader?
High gamma means the delta of your position changes rapidly with stock price movements. Long gamma positions profit from large moves in either direction but lose time value. Short gamma positions collect premium but risk large losses from big moves.
How is gamma used in hedging?
Delta-hedged portfolios must be rebalanced as the stock price moves because delta changes (gamma effect). Gamma quantifies how frequently and how much the hedge needs to be adjusted. Gamma-neutral portfolios minimize rebalancing needs.
Official sources
- Black, F. and Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3): jstor.org/stable/1831029.
- CBOE Options Institute: cboe.com/education.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.