Gamma (Options)

Definition

Gamma (Γ) measures how much an option’s delta changes for a one-point move in the underlying asset. It is the second derivative of an option’s price with respect to the underlying price and quantifies convexity—how the rate of change of the option’s price itself changes as the underlying moves.

Key points:
* Gamma is highest for at‑the‑money options and for options close to expiration.
* Long option positions (calls or puts) have positive gamma; short option positions have negative gamma.
* Intuition: delta is the option’s “speed”; gamma is its “acceleration.”

How gamma works

Delta tells how much an option’s price moves for a small change in the underlying. Gamma shows how delta itself will change as the underlying moves. For a small underlying move ΔS:

Change in delta ≈ Gamma × ΔS

Example: if an option has delta = 0.40 and gamma = 0.10, a $1 increase in the underlying raises the delta to 0.50.

Gamma is largest when an option is at the money because small price moves change moneyness most dramatically. As an option moves deep in‑ or out‑of‑the‑money, gamma falls toward zero. Near-term expirations amplify gamma compared with longer‑dated options.

Third-order sensitivities (e.g., “color”) measure how gamma itself changes and are used by traders managing very large or precision hedged portfolios.

Uses of gamma

• Forecasting how an option’s delta will evolve with underlying price moves.
• Designing hedges that remain effective as market prices change.
• Managing trading strategies that profit from or are exposed to changes in volatility and price movement behavior.

Gamma is typically computed with pricing models (Black‑Scholes, numerical methods) or financial software; traders may use approximate finite‑difference calculations for quick estimates.

Example

Stock price = $10
Option delta = 0.50
Option gamma = 0.10

For each $1 move in the stock:
* If stock rises $1 → delta becomes 0.60
* If stock falls $1 → delta becomes 0.40

Gamma hedging and delta‑gamma hedging

• Gamma hedging aims to keep portfolio gamma near zero (gamma‑neutral), often in combination with delta hedging (delta‑neutral).
• Delta‑gamma hedging attempts to make both net delta and net gamma close to zero, reducing sensitivity to small and moderate moves in the underlying.
• Maintaining such hedges requires ongoing rebalancing as prices and Greeks change; hedging can be costly and imperfect, especially during large or rapid moves.

Long gamma vs short gamma

• Long gamma: delta moves favorably with price swings (you can sell delta as prices rise and buy delta as prices fall). Long gamma benefits from larger realized volatility and enables a buy‑low, sell‑high pattern if executed well.
• Short gamma: exposes the holder to potentially compounding losses as the underlying moves. Short gamma positions require continuous rebalancing; adverse moves and reversals can produce large losses.

Key takeaways

• Gamma is the rate of change of an option’s delta per one‑point move in the underlying.
• It is highest for at‑the‑money and near‑expiry options.
• Long options have positive gamma; short options have negative gamma.
• Traders use gamma to plan hedges and to understand how an option’s risk profile will evolve as prices move.