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id: greeks-in-plain-english title: Greeks in Plain English summary: What delta, gamma, theta, vega, and rho actually mean in a real trade. outlook: neutral tags:

• greeks
• foundations
• beginner

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Learning goals

By the end of this lesson, you will be able to:

1. Describe each Greek in one sentence with a practical example.
2. Predict how an option responds to changes in price, time, and volatility.
3. Understand when each Greek matters most.
4. Recognize the trade-offs between Greeks in strategy selection.
5. Use Greeks for basic position sizing and risk management.

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The Greeks at a glance

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Delta (Δ): Your directional exposure

The definition

Delta measures how much the option price changes when the stock moves $1.

$$\Delta = \frac{\partial V}{\partial S}$$

Key facts

Delta by moneyness

Delta
│
1.0 ─ ─ ─ ─ ─ ─ ─ ─ ─●
│                   /
│                  /
0.5 ─ ─ ─ ─ ─ ─ ─●
│              /
│            /
0.0 ●───────
└───────────────────────→ Stock Price
   OTM      ATM       ITM

Example

You own 10 contracts of a 0.40 delta call.

Equivalent position: 10 × 100 × 0.40 = 400 shares

If stock rises $2:

$$\text{Profit} \approx 400 \times 2 = 800$$

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Gamma (Γ): How fast delta changes

The definition

Gamma measures how much delta changes when the stock moves $1.

$$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$$

Why gamma matters

• High gamma = delta changes rapidly = more convexity
• Low gamma = delta is stable = more predictable

Gamma by strike and time

Gamma over time

Gamma
│
│          ATM near expiry
│              ▲
│             /│\
│            / │ \
│           /  │  \
│    ──────    │    ──────
└─────────────────────────→ Strike
         (spot)

The gamma-theta relationship

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Theta (Θ): The cost of time

The definition

Theta measures how much option value decays per day (all else equal).

$$\Theta = \frac{\partial V}{\partial t}$$

Key facts

Theta acceleration

Theta by moneyness

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Vega (ν): Sensitivity to volatility

The definition

Vega measures how much option value changes when IV moves 1%.

$$\nu = \frac{\partial V}{\partial \sigma}$$

Key facts

Vega by expiration

Example

You own 5 contracts with vega = $0.12

IV rises from 25% to 30% (5% increase):

$$\text{Gain} = 5 \times 100 \times 0.12 \times 5 = 300$$

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Rho (ρ): Interest rate sensitivity

The definition

Rho measures option sensitivity to a 1% change in interest rates.

$$\rho = \frac{\partial V}{\partial r}$$

Key facts

When rho matters

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Greek interactions and trade-offs

The impossible trinity

You cannot have all three:

1. High gamma (convexity)
2. Low theta (cheap carry)
3. High vega (volatility exposure)

Common Greek profiles

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Using Greeks for risk management

Position sizing with delta

Target: Limit portfolio delta to ±100 shares equivalent per $10k account

If you want to buy a 0.35 delta call:

$$\text{Max contracts} = \frac{100}{0.35 \times 100} = 2.86 \approx 2 \text{ contracts}$$

Theta budget

Rule: Limit daily theta to 0.5-1% of account value

$10k account → Max$50-100 theta per day at risk

Vega exposure check

Before earnings, check your vega:

• Long vega = benefit from IV spike
• Short vega = hurt by IV spike (but benefit from crush)

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Greek formulas (Black-Scholes)Read more

Delta

$$\Delta_{\text{call}} = N(d_1)$$$$\Delta_{\text{put}} = N(d_1) - 1$$

Gamma

$$\Gamma = \frac{N'(d_1)}{S \cdot \sigma \cdot \sqrt{T}}$$

Theta

$$\Theta_{\text{call}} = -\frac{S \cdot N'(d_1) \cdot \sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2)$$

Vega

$$\nu = S \cdot \sqrt{T} \cdot N'(d_1)$$

Rho

$$\rho_{\text{call}} = KTe^{-rT}N(d_2)$$

Where:

• $N(x)$ = cumulative normal distribution
• $N'(x)$ = standard normal PDF
• $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$
• $d_2 = d_1 - \sigma\sqrt{T}$