How do the BSM Greeks (delta, gamma, vega, theta, rho) measure option sensitivity, and which matters most?

The Greeks are partial derivatives of the Black-Scholes-Merton (BSM) option pricing formula. Each one measures sensitivity to a different input variable. Together they give a complete picture of an option position's risk profile.

The Five Greeks

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Worked Example — Albright Technologies (S = $50, K =$50, T = 0.5, sigma = 35%, r = 4%)

BSM call price = $4.82

Greek	Value	Interpretation
Delta	+0.56	If stock moves +$1, call gains ~$0.56
Gamma	+0.038	If stock moves +$1, delta increases from 0.56 to 0.598
Vega	+0.139	If volatility rises 1%, call gains ~$0.139
Theta	-0.018	Call loses ~$0.018 per day from time decay
Rho	+0.115	If rates rise 1%, call gains ~$0.115

Deep Dive on Each Greek:

Delta (Most Important for Hedging):

• Call delta: 0 to +1 (ATM calls have delta near 0.5)
• Put delta: -1 to 0 (ATM puts have delta near -0.5)
• Delta hedging: To hedge 100 long calls with delta 0.56, short 56 shares of stock
• Delta changes as the stock moves (that's what gamma captures)

Gamma (Rate of Change of Delta):

• Highest for ATM options near expiration
• Gamma risk: If you're delta-hedged but gamma is high, a large stock move causes your hedge to become incorrect quickly
• Long options have positive gamma (beneficial — delta moves in your favor)
• Short options have negative gamma (dangerous — delta moves against you)

Vega (Volatility Sensitivity):

• Both calls and puts have positive vega (higher volatility = higher option value)
• Highest for ATM options with long time to expiry
• Not technically a Greek letter but universally used
• A position with $500,000 vega gains$500K per 1% volatility increase

Theta (Time Decay):

• Both calls and puts typically have negative theta (time passing hurts option holders)
• Accelerates near expiration (as discussed earlier)
• Theta is the 'cost' of owning gamma — there's a gamma-theta tradeoff

Rho (Interest Rate Sensitivity):

• Calls have positive rho (higher rates increase call value)
• Puts have negative rho (higher rates decrease put value)
• Usually the least important Greek for short-dated options

The Gamma-Theta Tradeoff:

For an ATM option: Gamma x Theta is approximately constant. High gamma (good for the holder — profits from large moves) comes with high theta (bad for the holder — loses money every day from time decay). This is one of the fundamental tradeoffs in options trading.

Exam Tip: CFA Level II may describe a portfolio's Greek exposures and ask you to identify the dominant risk. ATM options near expiry have high gamma and theta; long-dated ATM options have high vega.

Master the Greeks with our CFA Level II Derivatives course.