Greek Conventions

How Lavender defines and scales its Greeks.

Signs

Greek	Calls	Puts
Delta	0 to +1	-1 to 0
Gamma	Always positive	Always positive
Theta	Typically negative	Typically negative
Vega	Always positive	Always positive
Rho	Positive	Negative

Units

Greek	Expression	Unit
Delta	\(\partial V / \partial S\)	Per $1 spot move
Gamma	\(\partial^2 V / \partial S^2\)	Per $1 spot move
Theta	\(\partial V / \partial t\)	Per calendar day
Vega	\(\partial V / \partial \sigma\)	Per 1% vol (0.01)
Rho	\(\partial V / \partial r\)	Per 1% rate (0.01)

Time decay: theta and decay

Lavender provides two views of time decay:

• Theta — the conventional partial derivative \(\partial V / \partial t\), per calendar day. This is what most systems report and what standard models produce.
• Decay — the expected price change from now to the same time on the next trading day, accounting for weekends and holidays. This is the number that tells you what your position will actually lose overnight.

The two values are close on a typical weekday, but diverge significantly on Friday afternoons (when decay spans the weekend) and around holidays. Theta is available on all endpoints; decay is available on the Lavender API.

Put-call parity and borrow rates

Lavender solves for the implied borrow rate per expiry to enforce put-call parity. This produces a calibrated forward price and consistent Greeks across calls and puts, even in hard-to-borrow names where standard models that assume zero or fixed borrow rates produce inconsistent results.

Implied Volatility

IV is expressed as an annualized decimal:

• 0.25 = 25% annualized IV
• 1.50 = 150% annualized IV

Vendor-specific scaling

Eight of nine vendor compatibility layers pass through vega and rho without conversion — they all use the same per-1% convention as the Lavender API. The one exception is ThetaData, which returns vega, rho, and epsilon per unit (raw BSM) — 100× larger than per-1%.

Extended Greeks

Lavender computes higher-order Greeks beyond the standard five. These are available through the Lavender API and select vendor compatibility layers.

Second-Order Greeks

Greek	Expression	What it measures	When it matters
Vanna	\(\partial^2 V / \partial S\,\partial \sigma\)	How delta changes as vol moves	Vol-of-vol exposure; skew trading; risk reversals
Volga	\(\partial^2 V / \partial \sigma^2\)	How vega changes as vol moves	OTM wing positions; convexity in vol; straddle P&L in vol spikes
Charm	\(\partial^2 V / \partial S\,\partial t\)	How delta decays over time	Delta-hedging frequency; overnight delta drift
Veta	\(\partial^2 V / \partial \sigma\,\partial t\)	How vega decays over time	Term structure trades; calendar spreads
Vera	\(\partial^2 V / \partial \sigma\,\partial r\)	Cross-sensitivity of vol and rates	Long-dated options where both rate and vol assumptions matter

Third-Order Greeks

Greek	Expression	What it measures	When it matters
Speed	\(\partial^3 V / \partial S^3\)	How gamma changes as spot moves	Large delta-hedged positions; gamma P&L asymmetry
Zomma	\(\partial^3 V / \partial S^2\,\partial \sigma\)	How gamma changes as vol moves	Gamma exposure in vol regimes; crash risk
Color	\(\partial^3 V / \partial S^2\,\partial t\)	How gamma decays over time	Pin risk near expiry; gamma scalping horizon
Ultima	\(\partial^3 V / \partial \sigma^3\)	How volga changes as vol moves	Deep OTM tail risk; vol-of-vol-of-vol

Additional First-Order Greeks

Greek	Expression	What it measures
Decay	—	Expected price change to same time on next trading day
Epsilon	\(\partial V / \partial q\)	Sensitivity to dividend yield
Lambda	\(\Delta \cdot S / V\)	Leverage ratio (percent option move per percent spot move)