Understanding Greeks

Greeks measure how option prices change. Master them to manage risk effectively.

What Are Greeks?

Greeks are sensitivity measures showing how option prices react to different variables. Named after Greek letters: Delta (Δ), Gamma (Γ), Theta (Θ), Vega (V), and Rho (ρ).

Why They Matter: Tell you exactly how much money you’ll make or lose when markets move.

The Big 5 Greeks

1. Delta (Δ) - Directional Exposure

Definition: Change in option price for $1 move in underlying.

Delta = ∂C / ∂S

Range:
- Calls: 0 to +1
- Puts: 0 to -1

Example:

• BTC Call with Delta = 0.65
• BTC moves +$1,000
• Option value changes: +$1,000 × 0.65 = +$650

Delta Interpretation

Position Delta:

• Long call: Positive delta (bullish)
• Long put: Negative delta (bearish)
• Short call: Negative delta (bearish)
• Short put: Positive delta (bullish)

2. Gamma (Γ) - Delta Acceleration

Definition: Change in Delta for $1 move in underlying.

Gamma = ∂²C / ∂S²

Range: Always positive for long options
Peak: Highest for ATM options

Example:

• Option has Delta = 0.50, Gamma = 0.02
• BTC moves +$1,000
• New Delta = 0.50 + (0.02 × 1) = 0.52

Why Gamma Matters

High Gamma (ATM options):

• Delta changes rapidly
• Profits accelerate quickly
• But can also reverse fast
• Requires active management

Low Gamma (Deep ITM/OTM):

• Delta stable
• Less volatile P&L
• Easier to manage

3. Theta (Θ) - Time Decay

Definition: Change in option value per day passing.

Theta = ∂C / ∂t

Range: Usually negative for long options
Unit: Dollars per day

Example:

• Option worth $2,000
• Theta = -$50/day
• Tomorrow (all else equal): Worth $1,950

Key Points:

• Decay accelerates near expiration
• Last 30 days = Fastest decay
• OTM options decay to zero
• ATM options have highest theta

Theta Strategies

Sell theta (collect decay):

• Covered calls
• Cash-secured puts
• Iron condors
• Want time to pass

Buy theta (pay for time):

• Long calls/puts
• Need big move to overcome decay
• Time is your enemy

4. Vega (V) - Volatility Sensitivity

Definition: Change in option value for 1% change in volatility.

Vega = ∂C / ∂σ

Range: Always positive for long options
Unit: Dollars per 1% IV change

Example:

• Option worth $2,000
• Vega = $40
• IV increases from 80% to 85% (+5%)
• New value: $2,000 + ($40 × 5) = $2,200

Vega Characteristics

High Vega:

• Longer-dated options
• ATM strikes
• High absolute dollar value at risk

Low Vega:

• Short-dated options
• Deep ITM/OTM strikes
• Less sensitive to IV changes

Implied Volatility Impact

5. Rho (ρ) - Interest Rate Sensitivity

Definition: Change in option value for 1% change in interest rates.

Rho = ∂C / ∂r

Impact: Usually small except for long-dated options

Example:

• Option worth $2,000
• Rho = $15
• Interest rates rise from 5% to 6% (+1%)
• New value: $2,000 + $15 = $2,015 (minimal change)

Note: Rho is the least important Greek for most traders. Only matters for multi-year options.

Greeks Summary Table

Real Portfolio Example

Position: Sold 10 BTC Covered Calls

Position Details:
- Spot: $50,000
- Strike: $55,000
- Days to expiry: 30
- IV: 80%
- Contracts: 10 (representing 10 BTC)

Greek Exposure (per contract):

• Delta: -0.40 (short call)
• Gamma: -0.015
• Theta: +$50/day
• Vega: -$30
• Rho: -$10

Portfolio Greeks (× 10 contracts):

• Portfolio Delta: -4.0 (equivalent to short 4 BTC)
• Portfolio Gamma: -0.15
• Portfolio Theta: +$500/day (earning $500 daily from decay)
• Portfolio Vega: -$300 (lose $300 per 1% IV increase)

Scenario Analysis

Scenario 1: BTC +$1,000

P&L from Delta: -4.0 × $1,000 = -$4,000
P&L from Gamma: -0.15 × $1,000²/2 = -$75
Total: -$4,075 (approx)

Scenario 2: 1 Day Passes (no price move)

P&L from Theta: +$500

Scenario 3: IV increases +5%

P&L from Vega: -$300 × 5 = -$1,500

Practical Applications

Delta Hedging

Want to be delta-neutral (no directional risk)?

Your Position: 10 short calls with -4.0 delta Hedge: Buy 4 BTC to offset

Short calls delta: -4.0
Long 4 BTC delta: +4.0
Net delta: 0 (neutral)

Now you profit only from theta and vega, not direction.

Gamma Scalping

Use gamma to profit from volatility:

1. Get delta-neutral
2. When market moves, delta changes (gamma)
3. Rehedge at profit
4. Repeat

Theta Harvesting

Sell options to collect daily theta:

Day 1: Collect $500 theta
Day 2: Collect $500 theta
Day 3: Collect $500 theta
...
After 30 days: $15,000 collected (if no price move)

Risk Management Rules

1. Know Your Delta

Portfolio delta = Directional exposure
- Long 100 delta = Long 1 BTC
- Short 100 delta = Short 1 BTC

2. Monitor Gamma

High gamma = Position delta can change rapidly
- Requires frequent rehedging
- Higher transaction costs

3. Respect Theta

Long options: Time is enemy, need big moves
Short options: Time is friend, want stability

4. Watch Vega

Before known events: IV rises (options expensive)
After events: IV falls (vol crush)

Next Steps

• Delta Hedging - Neutralize directional risk
• Scenario Analysis - Test multiple scenarios
• Your First Covered Call - Apply Greeks practically

Warning: Greeks are estimates based on models. Real market moves can differ from Greek predictions, especially during crises.