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The Greeks - Risk Metrics Mastery

Level 2: Intermediate | Module 2.4 | Time: 3 hours

🎯 Learning Objectives

By the end of this module, you will:

  • Master all five Greeks (Delta, Gamma, Theta, Vega, Rho)
  • Understand why they matter for risk management
  • Learn Greek interactions (Gamma-Theta trade-off)
  • Calculate position Greeks at portfolio level
  • Implement practical hedging strategies

Prerequisites: Volatility Analysis

What Are “The Greeks”?

The Greeks are measures of how an option’s price changes in response to changes in underlying market factors.

Why They’re Called “Greeks”

Most use Greek letters (Δ, Γ, Θ, ν, ρ) from mathematics.

Exception: “Vega” (ν) isn’t actually a Greek letter - it’s made up! But everyone uses it anyway.

The Five Essential Greeks

Greek  | Letter | Measures Sensitivity To
-------|--------|---------------------------
Delta  | Δ      | Underlying price change
Gamma  | Γ      | Delta change (acceleration)
Theta  | Θ      | Time decay
Vega   | ν      | Volatility change
Rho    | ρ      | Interest rate change

Think of Greeks as your option’s dashboard:

  • Delta: Speedometer (how fast you’re moving with the underlying)
  • Gamma: Accelerator (how fast your delta changes)
  • Theta: Fuel gauge (how much you’re losing to time)
  • Vega: Volatility gauge (how sensitive to market fear)
  • Rho: Interest rate gauge (usually least important)

Delta (Δ) - The Directional Exposure

Definition

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

Formula:
Δ = ∂C/∂S (partial derivative of option price with respect to spot)

English:
If Bitcoin rises $1, how much does my option gain/lose?

Delta Ranges

Call Options:
- Deep OTM: Δ ≈ 0.05 (barely moves with underlying)
- ATM: Δ ≈ 0.50 (moves half as much as underlying)
- Deep ITM: Δ ≈ 1.00 (moves 1:1 with underlying)
- Range: 0 to +1

Put Options:
- Deep OTM: Δ ≈ -0.05
- ATM: Δ ≈ -0.50
- Deep ITM: Δ ≈ -1.00
- Range: -1 to 0

Underlying Asset:
- Owning 1 BTC: Δ = +1.00
- Short 1 BTC: Δ = -1.00

Practical Example

Bitcoin at $50,000
You own: $55,000 call with Delta = 0.35

Scenario: Bitcoin rises to $51,000 (+$1,000)

Option value change:
ΔPrice ≈ Delta × Underlying Move
ΔPrice ≈ 0.35 × $1,000 = $350

Your call option gains ~$350

Verification:
If you owned Bitcoin directly (+1.00 delta):
Gain = 1.00 × $1,000 = $1,000

With 0.35 delta call:
Gain = 0.35 × $1,000 = $350 (35% of direct ownership)

Delta as Probability

Rough approximation: Delta ≈ Probability of finishing ITM

$55,000 call on $50,000 Bitcoin with Delta = 0.35

Interpretation:
~35% chance this call finishes in-the-money at expiration

This is why:
- OTM options have low delta (low probability)
- ITM options have high delta (high probability)
- ATM options have ~0.50 delta (~50% chance)

Gamma (Γ) - The Acceleration

Definition

Gamma measures how much delta changes for a $1 change in the underlying asset price.

Formula:
Γ = ∂Δ/∂S (partial derivative of delta with respect to spot)

English:
How fast does my delta change as the underlying moves?

Why Gamma Matters

Delta isn’t constant - it changes!

Example:

Initial state:
Bitcoin: $50,000
Call: $55,000 strike, Delta = 0.35, Gamma = 0.02

Bitcoin rises to $51,000:

New Delta = Old Delta + (Gamma × Move)
New Delta = 0.35 + (0.02 × $1,000)
New Delta = 0.35 + 20
New Delta = 0.55

Your delta increased from 0.35 to 0.55!
Now the option is MORE sensitive to moves.

Gamma Profile by Strike

        Gamma
          ↑
      0.03|     .
          |    / \
      0.02|   /   \
          |  /     \
      0.01| /       \
          |/         \____
          |________________→ Strike Price
         OTM   ATM    ITM

Gamma is HIGHEST for ATM options
Gamma is LOWEST for deep ITM/OTM options

Why?

  • ATM options: Tiny moves change from OTM → ITM dramatically
  • Deep ITM/OTM: Already decided, small moves don’t matter

Gamma and Time

Gamma increases as expiration approaches (for ATM options)

       Gamma
         ↑
     0.05|           •  (1 day to expiry)
         |       •      (7 days)
     0.03|    •         (30 days)
         | •            (90 days)
     0.01|•             (180 days)
         |______________→ Days to Expiration

"Gamma risk" explodes near expiration!

Long vs Short Gamma

Long Gamma (Bought Options):
✅ Delta helps you (increases with favorable moves)
✅ Self-hedging (convexity benefit)
❌ Pay theta (time decay)

Example:
Buy call, Delta = 0.40
BTC up $1,000 → Delta now 0.60
BTC up another $1,000 → Delta now 0.75
Accelerating gains! ✅

Short Gamma (Sold Options):
❌ Delta hurts you (increases with adverse moves)
❌ Needs constant rehedging
✅ Collect theta (time decay)

Example:
Sell call, Delta = -0.40
BTC up $1,000 → Delta now -0.60 (worse exposure!)
BTC up another $1,000 → Delta now -0.75 (even worse!)
Accelerating losses! ❌

Theta (Θ) - The Time Decay

Definition

Theta measures how much an option loses in value each day due to time passing (all else equal).

Formula:
Θ = ∂C/∂T (partial derivative of option price with respect to time)

English:
How much value do I lose tomorrow just from time passing?

Always Negative for Long Options

You bought a $55,000 call for $5,000
Theta = -$80 per day

Tomorrow (all else equal):
Option value = $5,000 - $80 = $4,920

Day after:
Option value = $4,920 - $80 = $4,840

Time is your enemy when you buy options!

Theta Decay Profile

     Option Value
          ↑
    $5,000|•
          | \_
    $3,000|   \__
          |      \____
    $1,000|           \______
          |                  •______ → $0
          |_____________________________→ Time
         90d   60d   30d   7d   1d   0d

Decay accelerates as expiration approaches!

Theta by Moneyness

       Theta
($ lost/day)
         ↑
         |      .
      80|     / \
         |    /   \
      60|   /     \
         |  /       \
      40| /         \
         |/           \___
         |________________→ Strike
        OTM   ATM    ITM

ATM options have HIGHEST theta (most time value to lose)
Deep ITM/OTM have LOWER theta (mostly intrinsic or worthless)

The Gamma-Theta Trade-Off

The most important Greek relationship:

Long Options (Positive Gamma, Negative Theta):
✅ Gamma: Profit from big moves (convexity)
❌ Theta: Lose money daily (decay)

Trade-off:
If market moves big → Gamma gains > Theta losses ✅
If market stays flat → Theta losses dominate ❌

Short Options (Negative Gamma, Positive Theta):
❌ Gamma: Lose from big moves
✅ Theta: Earn money daily (premium decay)

Trade-off:
If market stays flat → Theta gains > Gamma risks ✅
If market moves big → Gamma losses dominate ❌

Golden Rule: Can’t have both positive gamma AND positive theta!

Vega (ν) - Volatility Sensitivity

Definition

Vega measures how much an option’s price changes for a 1% change in implied volatility.

Formula:
ν = ∂C/∂σ (partial derivative with respect to volatility)

English:
If implied volatility increases 1%, how much does my option gain?

Always Positive for Long Options

You own: $55,000 call, value = $5,000
Vega = $60 per 1% IV change

Implied volatility increases from 70% to 80% (+10%):

Option value change = Vega × IV Change
                    = $60 × 10
                    = +$600

New option value = $5,000 + $600 = $5,600

You profited from volatility increase!

Vega Profile

        Vega
     ($/1% IV)
          ↑
       80|     •
          |    / \
       60|   /   \
          |  /     \
       40| /       \
          |/         \___
          |________________→ Strike
         OTM   ATM    ITM

ATM options have HIGHEST vega (most sensitive to vol)

Vega and Time

Vega is HIGHER for longer-dated options

        Vega
          ↑
      100|•             (365 days)
          |  •          (180 days)
       60|    •         (90 days)
          |      •      (30 days)
       20|        •     (7 days)
          |__________→ Days to Expiration

More time = more uncertainty = more sensitive to vol changes

Vega in Action: Event Risk

Before Bitcoin ETF announcement:
Call option value: $5,000
Implied volatility: 60%
Vega: $75

ETF APPROVED (positive news):
Bitcoin spikes +10% ✅
BUT implied volatility crashes to 40% (-20%) ❌

Option value change:
From price move (delta): +$3,500 (assuming delta 0.35)
From vol crush (vega): -$75 × 20 = -$1,500

Net change: +$3,500 - $1,500 = +$2,000

You were right on direction but vol crush ate half your gains!

This is why buying options before events can be tricky.

Rho (ρ) - Interest Rate Sensitivity

Definition

Rho measures how much an option’s price changes for a 1% change in interest rates.

Formula:
ρ = ∂C/∂r (partial derivative with respect to risk-free rate)

English:
If interest rates increase 1%, how much does my option value change?

Usually the Least Important Greek

Why Rho doesn't matter much:

1. Interest rates change SLOWLY (months/years, not days)
2. Impact is SMALL for short-term options
3. Other Greeks dominate

Example:
30-day Bitcoin call
Rho = $15 per 1% rate change

Fed raises rates 0.25% (25 basis points):
Option value change = $15 × 0.25 = $3.75

Compared to:
- Daily theta: -$80 (20x bigger!)
- Delta on $1k move: +$350 (100x bigger!)
- 10% vol change: +$600 (160x bigger!)

Conclusion: Rho is noise for short-term traders

When Rho Matters

Long-dated options (LEAPS):
- 1-2 year options
- Interest rates have time to compound
- Rho becomes meaningful

Example:
2-year Bitcoin call
Rho = $800 per 1% rate change

Rate increase 2% over 2 years:
Impact = $800 × 2 = $1,600 (now material!)

Portfolio Greeks (Position Level)

Aggregating Greeks

Your total risk is the sum of all individual position Greeks.

Portfolio:
Position 1: Long 5 calls, Delta = 0.40 each
Position 2: Long 10 puts, Delta = -0.30 each
Position 3: Long 2 BTC (underlying)

Portfolio Delta:
= (5 × 0.40) + (10 × -0.30) + (2 × 1.00)
= 2.00 - 3.00 + 2.00
= +1.00

Interpretation:
Portfolio acts like owning 1 Bitcoin
If BTC rises $1,000, portfolio gains ~$1,000

Position Greek Dashboard Example

Position Summary:

Position                | Quantity | Delta  | Gamma | Theta | Vega
------------------------|----------|--------|-------|-------|------
Long $55k Call          | +10      | 0.35   | 0.02  | -80   | 60
Short $60k Call         | -10      | -0.20  | -0.01 | +50   | -40
Long $45k Put           | +5       | -0.25  | 0.018 | -60   | 50
Long Bitcoin            | +2       | 1.00   | 0     | 0     | 0

Portfolio Total (per underlying unit):
                                  +0.55  +0.028  -900  +320

Interpretation:
✅ Net Delta: +0.55 (bullish directional bias)
✅ Net Gamma: +0.028 (benefit from big moves)
❌ Net Theta: -$900/day (paying for time)
✅ Net Vega: +320 (benefit from vol increase)

This is a volatility-bullish position.

Hedging with Greeks

Goal: Delta-Neutral Portfolio

Eliminate directional risk, isolate other Greeks.

Current Position:
Long 100 calls at $55k, Delta = 0.40 each
Portfolio Delta = 100 × 0.40 = +40

To neutralize:
Need -40 delta

Option A: Short 40 BTC (each has delta = 1.00)
40 × -1.00 = -40 delta ✅

Option B: Sell 200 calls at $60k (delta = 0.20 each)
200 × -0.20 = -40 delta ✅

Result: Delta-neutral (profit from gamma/vega, not direction)

Gamma Scalping

Profit from gamma while delta-neutral.

Setup:
Long 100 calls (positive gamma, delta-neutral via hedge)

Day 1:
BTC = $50k, Portfolio Delta = 0 (hedged)

BTC rises to $51k:
Gamma kicks in → Delta now +5 (positive delta)

Action: Sell 5 BTC to rehedge to delta-neutral
Sell at $51k → Book $5,000 profit ✅

BTC falls to $50k:
Delta now -5 (negative delta)

Action: Buy 5 BTC to rehedge
Buy at $50k → Cost $5,000

Net: Sold high ($51k), bought low ($50k)
Profit: $5,000 from gamma scalping!

This is how market makers profit from volatility.

Greek Interactions: The Big Picture

The Greek Relationship Matrix

Long Call:
Delta: + (benefit from price increase)
Gamma: + (delta increases with favorable moves)
Theta: - (lose value daily)
Vega: + (benefit from vol increase)

Long Put:
Delta: - (benefit from price decrease)
Gamma: + (delta decreases with favorable moves)
Theta: - (lose value daily)
Vega: + (benefit from vol increase)

Short Call:
Delta: - (lose from price increase)
Gamma: - (delta worsens with adverse moves)
Theta: + (earn value daily)
Vega: - (hurt by vol increase)

Short Put:
Delta: + (lose from price decrease)
Gamma: - (delta worsens with adverse moves)
Theta: + (earn value daily)
Vega: - (hurt by vol increase)

Key Trade-Offs

1. Gamma vs Theta (The Big One)

Can't have both!
Buy options → Get gamma, pay theta
Sell options → Get theta, pay (gamma risk)

2. Vega vs Theta (Time Decay Relationship)

Longer time → Higher vega, lower theta/day
Shorter time → Lower vega, higher theta/day

3. Gamma vs Vega (Leverage Relationship)

Short-dated ATM → High gamma, low vega
Long-dated ATM → Lower gamma, higher vega

Practical Greek Management

Scenario 1: Earnings Trade (High Vega Exposure)

Pre-earnings:
- IV elevated (80%)
- You expect IV crush post-earnings

Strategy: Short vega (collect premium)

Execution:
Sell iron condor:
- Net vega: -300 (short vol)
- Net theta: +$200/day (collect decay)
- Net delta: ~0 (neutral)

Post-earnings:
IV drops 80% → 50% (-30%)
Profit from vega: $300 × 30 = $9,000 ✅

Risk: If stock moves big, gamma losses could exceed vega gains
View: Bitcoin going to $60k (bullish)

Strategy: Long delta, but minimize theta

Execution:
Buy deep ITM call:
- Strike: $45k (Bitcoin at $50k)
- Delta: 0.85 (high directional exposure)
- Theta: -$20/day (low decay)
- Cost: $6,000

Alternative (worse):
Buy ATM call:
- Strike: $50k
- Delta: 0.50 (less directional exposure)
- Theta: -$80/day (high decay!)
- Cost: $5,000

Cheaper but theta kills you if move takes time.

Scenario 3: Expecting Big Move (Long Gamma)

View: Huge move coming, direction uncertain

Strategy: Long gamma, accept negative theta

Execution:
Buy ATM straddle:
- Buy $50k call + $50k put
- Net delta: ~0 (direction-neutral)
- Net gamma: +0.04 (high!)
- Net theta: -$160/day (expensive)
- Net vega: +120 (also benefit from vol spike)

Result:
Big move either way → Gamma profits exceed theta losses

Practice Exercise: Calculate Portfolio Greeks

Given Positions

Position A: Long 20 calls
Strike: $55,000
Delta: 0.38, Gamma: 0.022, Theta: -75, Vega: 58

Position B: Short 30 calls
Strike: $60,000
Delta: -0.18, Gamma: -0.015, Theta: +48, Vega: -42

Position C: Long 3 Bitcoin
Delta: 1.00 each

Calculate: Portfolio Greeks
Click for solution
Position A Total:
Delta: 20 × 0.38 = +7.6
Gamma: 20 × 0.022 = +0.44
Theta: 20 × -75 = -1,500
Vega: 20 × 58 = +1,160

Position B Total:
Delta: 30 × -0.18 = -5.4
Gamma: 30 × -0.015 = -0.45
Theta: 30 × +48 = +1,440
Vega: 30 × -42 = -1,260

Position C Total:
Delta: 3 × 1.00 = +3.0
Gamma: 3 × 0 = 0
Theta: 3 × 0 = 0
Vega: 3 × 0 = 0

Portfolio Totals:
Net Delta: 7.6 - 5.4 + 3.0 = +5.2
Net Gamma: 0.44 - 0.45 + 0 = -0.01
Net Theta: -1,500 + 1,440 + 0 = -60
Net Vega: 1,160 - 1,260 + 0 = -100

Analysis:
✅ Bullish (delta +5.2 → gains if BTC rises)
⚠️ Slightly negative gamma (risky if big moves)
⚠️ Slight theta decay (-$60/day)
❌ Short vega (loses if IV increases)

This is a directional bullish bet with short vol exposure.
Risk: Volatility spike hurts you.

Key Takeaways

1. Greeks quantify option risk precisely

  • Delta: Directional (price sensitivity)
  • Gamma: Acceleration (delta change)
  • Theta: Time decay (daily loss)
  • Vega: Volatility sensitivity
  • Rho: Interest rate (usually minor)

2. Greeks interact with each other

  • Gamma ↔ Theta (fundamental trade-off)
  • Vega ↔ Time (longer = more vega)
  • Gamma ↔ Moneyness (ATM highest)

3. Portfolio Greeks = Sum of position Greeks

  • Aggregate to understand total risk
  • Hedge at portfolio level
  • Monitor continuously

4. Long options: Pay theta, get gamma + vega

  • Need big moves or vol increase to profit
  • Time is your enemy

5. Short options: Collect theta, pay (gamma risk + vega risk)

  • Profit from time decay and stability
  • Big moves or vol spikes hurt you

6. Greeks enable sophisticated risk management

  • Delta hedging (remove directional risk)
  • Gamma scalping (profit from rehedging)
  • Vega trading (vol strategies)

What’s Next?

You’ve mastered the Greeks! You now understand:

  • ✅ All five Greeks and what they measure
  • ✅ Greek interactions and trade-offs
  • ✅ Portfolio-level Greek aggregation
  • ✅ Practical hedging strategies

Ready to implement delta-neutral strategies?

Continue to: Delta Hedging →

Learn how to become delta-neutral and profit from gamma, theta, and vega.

Tools & Resources

Greek Calculators:

Visualization:

  • Greek charts by strike
  • Greek evolution over time
  • Sensitivity heat maps

Next Module: Delta Hedging →

Related Topics:

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