How do the BSM Greeks (delta, gamma, vega, theta, rho) measure option sensitivity, and which matters most?
I'm studying CFA Level II Derivatives and the Greeks section is dense. I know delta measures the option price change per $1 move in the underlying, but I'm fuzzy on the others — especially gamma and vega. Can someone provide a practical overview of all five with a worked example?
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
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.
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