Options Gamma Calculator
Compute Gamma, the rate of change of Delta, and see why at-the-money options are hardest to hedge.
Γ What is the Options Gamma Calculator?
Gamma is the second of the Black-Scholes "Greeks", measuring the rate of change of an option's Delta as the underlying's price moves. If Delta tells you the option's current speed relative to the stock, Gamma tells you its acceleration: how quickly that speed itself changes. Gamma is a single number that applies identically to a call and a put sharing the same strike, expiry, and underlying, a direct consequence of put-call parity.
Options market makers and professional traders track Gamma constantly because it governs how often a delta-neutral hedge needs to be rebalanced. A position with high Gamma sees its Delta swing quickly as the stock moves, so a hedge that was neutral a moment ago can drift out of balance fast, forcing frequent rebalancing trades. Retail traders encounter Gamma indirectly through "gamma squeezes", the well-publicized phenomenon where heavy call buying near a strike forces market makers who sold those calls to buy the underlying stock as it rises (to stay delta-hedged), amplifying the very move that triggered it.
A common misconception is that Gamma matters only to option sellers. In fact, long option holders have positive Gamma, which works in their favor: as the stock moves favorably, Delta increases (for calls) or decreases (for puts) in the helpful direction, accelerating gains. It is short option positions that carry negative Gamma and the associated acceleration of losses, which is why option sellers, not buyers, are the ones who actively manage Gamma risk.
This calculator is useful because it computes Gamma directly from the same five Black-Scholes inputs used across this site's option tools, expresses it as a position-level figure per 100 shares, estimates the dollar convexity P&L for a realistic 1% move, and charts how Gamma peaks near the strike price, the single most important visual intuition for understanding hedging difficulty.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — At-the-Money Stock Option
A stock trades at $100, with a $100 strike, 1 year to expiry, a 5% risk-free rate, and 20% volatility.
Example 2 — In-the-Money Option
A stock trades at $150, with a $140 strike, 6 months to expiry, a 4% risk-free rate, and 25% volatility.
Example 3 — Out-of-the-Money Option with Dividends
A dividend-paying stock trades at $50, with a $55 strike, 3 months to expiry, a 3% risk-free rate, 30% volatility, and a 1% dividend yield.
❓ Frequently Asked Questions
🔗 Related Calculators
What is Gamma in options trading?
Gamma measures how much an option's Delta changes for a $1 move in the underlying's price. It is the second derivative of the option price with respect to the underlying, or equivalently, the first derivative of Delta. Gamma is the same for calls and puts sharing the same strike, expiry, and underlying.
What is the formula for Gamma?
Gamma = e^(-qT) x phi(d1) / (S x sigma x sqrt(T)), where phi is the standard normal probability density function (not the cumulative distribution N used for price and Delta), and d1 = [ln(S/K) + (r - q + sigma^2/2)T] / (sigma x sqrt(T)).
Why is Gamma the same for calls and puts?
Delta_call minus Delta_put equals e^(-qT) exactly (a constant, from put-call parity), which does not depend on the stock price. Since Gamma is the derivative of Delta with respect to the stock price, and the derivative of a constant is zero, the two Deltas must change at exactly the same rate, giving identical Gamma.
Why does Gamma peak for at-the-money options?
At the money, small moves in the stock have the biggest effect on whether the option ends up in or out of the money, so Delta swings fastest there. Far from the strike, the option is already effectively certain to expire in or out of the money, so Delta barely moves and Gamma is small.
What does negative Gamma mean for an options seller?
A short option position has negative Gamma, meaning its Delta becomes more unfavorable as the underlying moves against the position, accelerating losses. This is the core risk options sellers manage through frequent re-hedging or position limits.
How is Gamma used in delta hedging?
A delta-neutral hedge only stays neutral for small moves, because Gamma causes Delta to drift as the underlying moves. Traders monitor Gamma to know how often they need to rebalance their hedge; high Gamma positions require more frequent rebalancing.
Why does Gamma spike near expiration for at-the-money options?
As expiry approaches, an at-the-money option's fate (finish in or out of the money) becomes increasingly binary, so Delta swings from near 0 to near 1 (or -1) over a shrinking price range, pushing Gamma sharply higher right before expiration.
What is 'dollar Gamma' or convexity P&L?
Dollar Gamma converts raw Gamma into an actual profit-or-loss estimate for a given percentage move in the underlying, using the Taylor expansion term 0.5 x Gamma x (price change)^2. This calculator's 'Convexity P&L (1% Move)' output shows exactly this for a 1% move.
Does time to expiry always increase Gamma?
No. For at-the-money options, less time to expiry increases Gamma (it spikes near expiration). For away-from-the-money options, more time to expiry can increase Gamma since it gives the option a realistic chance to move toward the strike. The relationship depends on moneyness.
Can Gamma ever be negative for a long option position?
No. Both call and put Gamma are always positive (or zero at the theoretical limits) for a long option position, since phi(d1), S, sigma, and T are all positive. Only short option positions carry negative Gamma.