The Greeks

The Greeks measure how an option's value responds to changes in the underlying, time, volatility and rates — the vocabulary of managing an options book.

1. The idea

Delta ( $Δ = \partial C / \partial S$ ) measures how much the option price moves per unit move in the underlying, and doubles as the number of shares needed to hedge the option (delta hedging). Calls have delta between 0 and 1; puts between -1 and 0.

Gamma ( $Γ = \partial^{2} C / \partial S^{2}$ ) measures how fast delta itself changes as the stock moves — it is largest for at-the-money options near expiry, and is the reason delta hedges must be rebalanced. Vega measures sensitivity to volatility, always positive for long options since more volatility increases the value of optionality.

Theta measures the rate of value decay as time passes (usually negative for long options — time decay), and Rho measures sensitivity to interest rates, typically the least important Greek for short-dated equity options.

2. Why it matters

Trading desks manage options portfolios by keeping Greeks within limits; being asked to reason about the sign and magnitude of a Greek under a given market move is one of the most common interview formats.

3. Core mathematics

Delta

Δ = \frac{\partial C}{\partial S}

Gamma

Γ = \frac{\partial ^{2} C}{\partial S ^{2}} = \frac{\partial Δ}{\partial S}

Vega

Vega = \frac{\partial C}{\partial σ}

Theta

Θ = \frac{\partial C}{\partial t}

usually negative for long options

4. How to recognize it

Think of this when…

'How does the option price change if the stock moves by $1' (delta) or 'if volatility rises' (vega)
Questions about rebalancing a delta hedge (gamma)
'Time decay' or 'theta bleed' language

5. Worked example

Worked example

A call has delta 0.6 and gamma 0.05. If the stock rises by $2, approximate the new delta and the option's price change.

1
Approximate the change in delta using gamma.
$Δ new \approx 0.6 + 0.05 (2) = 0.70$
2
Approximate the option's price change using a second-order (delta-gamma) expansion.
$Δ C \approx Δ \cdot Δ S + \frac{1}{2} Γ (Δ S)^{2} = 0.6 (2) + 0.5 (0.05) (4)$
3
Evaluate.
$Δ C \approx 1.2 + 0.1 = 1.3$
Answer: New delta ≈ 0.70; option price increases by about $1.30

6. Common mistakes

Assuming delta stays constant over large stock moves, ignoring gamma
Forgetting that long options always have positive vega and (usually) negative theta
Confusing the sign of delta for puts (negative) with calls (positive)

7. Quick check

Gamma measures:

A long call option typically has:

8. Practice

4 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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