Options Greeks Explained: Delta, Gamma, Theta, Vega, Rho

What are Options Greeks?

Options Greeks are sensitivity measures that describe how the price of an options contract changes in response to different factors — the underlying price, time to expiry, volatility, and interest rates. Traders use Greeks to understand, hedge, and manage options positions.

Delta (Δ) — Sensitivity to Underlying Price

Delta measures how much an option's price changes for a $1 move in the underlying asset.

• Call options: Delta ranges from 0 to 1
• Put options: Delta ranges from −1 to 0
• At-the-money options: Delta ≈ ±0.50
• Deep in-the-money options: Delta approaches ±1

Delta is also used as an approximation for the probability the option expires in the money. A 0.25 delta call has roughly a 25% chance of expiring in the money.

Gamma (Γ) — Rate of Change of Delta

Gamma measures how much Delta changes for a $1 move in the underlying. It's the second derivative of the option price with respect to the underlying.

• Gamma is highest for at-the-money options near expiry
• Long options (calls or puts) have positive Gamma
• Short options have negative Gamma
• High Gamma positions need frequent delta-hedging rebalancing

Gamma risk is highest in the final days before expiration — a position that looks delta-neutral can become significantly directional overnight.

Theta (Θ) — Time Decay

Theta measures the daily loss in option value due to the passage of time, all else equal. It's almost always negative for long options (options lose value as expiration approaches).

• Theta accelerates as expiration approaches (non-linear decay)
• ATM options experience the most absolute theta decay
• Selling options (short theta) generates income — but also creates Gamma risk

A position with Theta of −$50 loses approximately $50 per day from time decay, assuming no other factors change.

Vega (ν) — Sensitivity to Implied Volatility

Vega measures how much the option price changes for a 1% change in implied volatility (IV). Vega is positive for long options — higher IV means higher option prices.

• Vega is highest for long-dated, at-the-money options
• Buying options before a catalyst (earnings, FOMC) captures Vega — if IV rises
• Selling options when IV is high (Vega-short) is the basis of volatility selling strategies

Rho (ρ) — Sensitivity to Interest Rates

Rho measures how much the option price changes for a 1% change in the risk-free interest rate. Rho is typically small and is the least-traded Greek for short-dated options, but becomes material for long-dated LEAPS or when rates move significantly.

• Calls have positive Rho (benefit from higher rates)
• Puts have negative Rho (hurt by higher rates)

Computing Greeks with ClaudeFinLab

ClaudeFinLab's Black-Scholes Local Volatility Options Pricer computes all five Greeks for any options position:

{
  "mcpServers": {
    "claudefinlab-market": {
      "url": "https://claudefinancelab.com/market/sse",
      "headers": { "Authorization": "Bearer YOUR_API_KEY" }
    }
  }
}

Then ask Claude: "Price a European call on AAPL: spot $195, strike $200, 30 days to expiry, 25% implied vol, 4.5% risk-free rate. Give me all five Greeks."

Greek Summary Table

Greek	Measures	Long call	Long put
Delta	Price sensitivity to underlying	0 to +1	−1 to 0
Gamma	Delta sensitivity to underlying	Positive	Positive
Theta	Daily time decay	Negative	Negative
Vega	IV sensitivity	Positive	Positive
Rho	Rate sensitivity	Positive	Negative