Options Gamma¶
OVERVIEW¶
Gamma measures the rate of change of an option's delta for every $1 move in the underlying stock. While delta tells you how much an option price moves per $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.
Key Characteristic
Long option positions (calls and puts purchased) always carry positive gamma. Short option positions always carry 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:
| Variable | Definition |
|---|---|
φ(d1) |
Standard normal probability density function evaluated at d1 |
S |
Underlying asset price |
σ |
Implied volatility |
T |
Time to expiration in years |
Call vs. Put Gamma
Call gamma equals put gamma for the same strike and expiration. Gamma is sign-independent across call/put.
Position gamma is calculated as:
WORKED EXAMPLE¶
Assume AAPL trades at $180, with the $180-strike call, 30 days to expiration, 25% implied volatility, 4.5% risk-free rate, and zero dividend.
| Input | Value |
|---|---|
| d1 | ≈ 0.09 |
| φ(d1) | ≈ 0.397 |
| σ√T | 0.25 × √(30/365) ≈ 0.0716 |
| Γ | 0.397 / (180 × 0.0716) ≈ 0.0308 |
With delta at approximately 0.53:
| AAPL Price | Estimated Delta |
|---|---|
| $180 (base) | 0.53 |
| $181 (+$1) | ≈ 0.56 |
| $183 (+$3) | ≈ 0.62 |
Position Gamma in Practice
A trader long ten contracts holds position gamma of 0.0308 × 100 × 10 = 30.8. Every $1 move in AAPL adds 30.8 shares of delta exposure.
HOW TO USE¶
Gamma is applied differently depending on your role and strategy.
| Use Case | Description |
|---|---|
| Market maker delta hedging | Dealers short near-dated calls must buy stock as it rises and sell as it falls, amplifying price moves |
| Gamma squeeze identification | Retail traders monitor dealer hedging flows in heavily-optioned names (e.g., GME, TSLA) |
| Income strategy sizing | Iron condor and short strangle sellers track short gamma to size positions against tail moves |
Gamma Explosion
Gamma rises sharply in the final week before expiration. Weekly options can whipsaw violently around expiration as a result. Size positions accordingly.
LIMITATIONS AND COMMON MISCONCEPTIONS¶
Gamma Is a Snapshot
Gamma changes continuously with price, time, and implied volatility. Never treat a single gamma reading as static.
| Misconception | Reality |
|---|---|
| Gamma indicates directional risk | A gamma-positive straddle profits from any large move; direction is irrelevant |
| Positive gamma guarantees profit on a move | Long gamma but short vega positions can lose if IV collapses even as the stock moves |
| Gamma alone is sufficient for risk assessment | Gamma must be combined with delta, theta, and vega for a complete picture |
| High gamma always means high risk | Context matters — positive gamma caps losses on purchased options, negative gamma on short positions does not |