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Lesson 6: Greeks 101 (Risk in Four Knobs)

Promise: Translate "options are complicated" into 4 intuitive sensitivities.

What Are Greeks?​

Greeks measure how an option's price changes when different factors move. They're called "Greeks" because they use Greek letters.

💡

Greeks are not magic. They are local slopes.

Each Greek answers a simple question:

Greek
Question It Answers
Delta (Δ)
How much does the option move when spot moves?
Gamma (Γ)
How much does delta change when spot moves?
Theta (Θ)
How much does the option lose each day?
Vega (ν)
How much does the option move when IV moves?

Delta (Δ): Direction Exposure​

Delta measures the option's sensitivity to spot price movement.

Δ≈∂V∂S\Delta \approx \frac{\partial V}{\partial S}
Position
Delta Range
Meaning
Long Call
0 to +1
Profits when spot rises
Long Put
-1 to 0
Profits when spot falls
Short Call
-1 to 0
Profits when spot falls
Short Put
0 to +1
Profits when spot rises

Interpretation: A delta of 0.5 means the option price moves ~$0.50 for every $1 move in the underlying.

See Delta In Action​

Click or drag to move spot price. Toggle between position types and adjust time/volatility.

Position:
Days to Expiry30d
1d90d
Implied Volatility50%
10%150%
Positive delta: profits when price rises. Click or drag to move spot price.
ITMOTMATM1.00.50.00.00.0$70kStrike $100k$130kSpot PriceDelta
Spot Price
$100.0k
Delta
+0.540
Moneyness
ATM
Hedge Ratio
54%
Gamma Risk
Low
Delta = +0.540 → For every $1,000 move in spot, the option moves $540 in the same direction

Delta by Moneyness​

MoneynessCall DeltaPut Delta
Deep ITM~1.0~-1.0
ATM~0.5~-0.5
Deep OTM~0~0

Gamma (Γ): How Delta Changes​

Gamma measures how much delta changes when spot moves.

Γ=∂Δ∂S\Gamma = \frac{\partial \Delta}{\partial S}
Property
Implication
Always positive for long options
Delta moves in your favor when spot moves
Highest near ATM
ATM options are most "twitchy"
Increases near expiry
Short-dated ATM = high gamma risk
💡

Gamma is why short-dated ATM options feel "twitchy."

Why gamma matters: If you're short options, gamma works against you. Large spot moves cause your delta exposure to grow in the wrong direction.

See Gamma In Action​

Notice how gamma peaks at ATM and explodes near expiry:

Position:
Days to Expiry30d
1d90d
Implied Volatility50%
10%150%
Long gamma: Delta moves in your favor. Price rises → delta increases. Price falls → delta decreases.
ATM (max gamma)OTMITM+00$70kStrike $100k$130kSpot PriceGamma
Spot Price
$100.0k
Γ per $1k
+0.03
Moneyness
ATM
Risk Level
Low
Γ = +0.03 per $1k → If spot moves $1,000, delta changes by ~0.03

Theta (Θ): Time Decay​

Theta measures how much the option loses per day from time passing.

Θ=∂V∂t\Theta = \frac{\partial V}{\partial t}
Position
Theta Sign
Meaning
Long options
Negative
Lose value over time
Short options
Positive
Gain value over time

Theta accelerates near expiry: An ATM option loses more per day in its final week than in its first month.

See Theta In Action​

Drag the slider to simulate time passing. Notice how decay accelerates near expiry:

Option Type
What this means
KSOTMITMSpot = Strike (at the money)
Day 0 Value
$2,000
Day 0 Value
$2,000
Cumulative Loss
$0
$0$1k$2kToday30dExpiryDaily decay−$17/day
Drag to simulate time passingDay 0
060 days
ATM options decay fastest - they have the most extrinsic value and highest uncertainty about finishing ITM or OTM.

Vega (ν): Volatility Sensitivity​

Vega measures how much the option price changes when IV moves.

ν=∂V∂σ\nu = \frac{\partial V}{\partial \sigma}
Property
Implication
Always positive for long options
Higher IV = higher option price
Highest for ATM options
ATM is most sensitive to IV
Higher for longer-dated
More time = more vol exposure

Interpretation: Vega of 50 means the option price moves $50 for a 1% (1 vol point) change in IV.

See Vega In Action​

Notice how vega peaks at ATM and increases with time to expiry:

Position:
Days to Expiry30d
1d90d
Implied Volatility50%
10%150%
Long vega: Profit when IV rises. ATM options have highest vega. Longer-dated = more vega.
ATM (max vega)OTMITM+00$70kStrike $100k$130kSpot PriceVega
Spot Price
$100.0k
Vega
+$11,380
Moneyness
ATM
Time Effect
Medium
Vega = +$11,380 → If IV moves 1%, the option price changes by $11,380. ATM has maximum vega exposure.

Greeks Summary Table​

Greek
Measures
Long Option
Short Option
Delta
Spot exposure
+ calls, - puts
Opposite
Gamma
Delta convexity
+ (good)
- (bad)
Theta
Time decay
- (costs)
+ (earns)
Vega
IV sensitivity
+ (want high)
- (want low)

The Greeks Trade-Off​

There's no free lunch in options. Each Greek represents a trade-off:

If You Want...You Accept...
Long gamma (delta moves in your favor)Negative theta (pay time decay)
Positive theta (collect premium)Short gamma (delta moves against you)
Long vega (profit from IV rise)Pay more premium upfront
Key Insight

Long options have negative theta but positive gamma (you pay for convexity). Short options have positive theta but negative gamma (you collect premium but face blow-up risk).

Common Mistakes​

MistakeCorrection
Believing delta is constantDelta changes as spot moves. That's what gamma measures.
Ignoring gamma near expiryShort-dated ATM options have extreme gamma. Small moves cause big delta changes.
Treating theta as linearTheta accelerates near expiry. The last week decays fastest.
Thinking Greeks are independentThey interact. Gamma affects how delta changes, which affects P&L.

Test your understanding before moving on.

Q: If delta is 0.3 and BTC moves +$1000, what is the approximate option price change?
Q: Where is gamma highest: deep ITM, deep OTM, or near ATM?
Q: Which Greek makes option price sensitive to IV?

💡 Tip: Try answering each question yourself before revealing the answer.

See Also​

Navigation: ← Lesson 5: Implied Volatility | Lesson 7: Basic Strategies →