Greek Conventions

How Lavender defines and scales its Greeks.

Applies to Terminal too

These conventions are identical in Lavender Terminal — it runs on the same engine, so every number it shows follows the signs, units, and scaling on this page.

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 (spot-based)
Gamma	\(\partial^2 V / \partial S^2\)	Per $1 spot move (spot-based)
Theta	\(\partial V / \partial t\)	Per calendar day (annual ÷365), forward-fixed
Decay	—	Next-trading-day reprice (spot-fixed) — see below
Vega	\(\partial V / \partial \sigma\)	Per 1 vol point — $ change per 1% absolute IV move (stored ×0.01)
Rho	\(\partial V / \partial r\)	Per 1% rate move (×0.01)
Epsilon	\(\partial V / \partial q\)	Per 1% borrow / dividend-yield move (×0.01)

Delta and gamma are spot-based — reported per $1 move in spot, so delta sums correctly for hedging even when the forward diverges from spot (e.g., high-borrow or dividend-paying stocks).

Lavender reports spot delta: the forward Greek scaled by \(F/S\) (forward over spot). Under high borrow or negative carry the forward sits below spot, so this scaling can push |delta| above 1 — a deep-in-the-money call on a hard-to-borrow name can read a delta greater than 1.0. That is expected, not an error: it reflects the true per-$1-spot sensitivity of the position.

Vega is reported per 1 vol point — the dollar change for a 1% absolute move in implied volatility (e.g. 25% → 26%). It is stored scaled by ×0.01 relative to the raw per-unit-vol derivative. Rho and epsilon follow the same convention: per a 1% move in the rate / borrow, scaled ×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.

Borrow rates and the forward

Lavender accounts for stock-borrow cost when it builds each forward, producing 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.

Near-expiry behavior

For 0-DTE and sub-one-day options, the time-derivative Greeks (veta, charm, color) are taken to their at-expiry limit over the remaining time rather than a full next-trading-day step: vega and gamma go to 0, and delta steps to the intrinsic spot step. This keeps the time Greeks well-behaved as a contract approaches its final hours instead of blowing up as the remaining tenor shrinks toward zero.

Portfolio and Dollar Greeks

The Greeks above are per-contract. Terminal's Risk view sums a book across strikes, expiries, and underlyings, so it shows dollar Greeks (the $-prefixed columns) — the position Greek scaled to dollars, so it can be added across names when the raw per-contract number can't:

$delta	$gamma	$vega	$theta
\(\Delta \times 100 \times \text{qty} \times S\)	\(\Gamma \times 100 \times \text{qty} \times S\)	\(\nu \times 100 \times \text{qty}\)	\(\theta \times 100 \times \text{qty}\)

Delta and gamma carry the extra \(\times S\); vega and theta are already dollar-denominated. A raw gamma of 2.8 on a $300 name isn't the same risk as 2.8 on a $30 name — dollar Greeks make them comparable, which is why the Risk view uses them.

Reconciling a vendor's raw gamma

A raw position gamma (e.g. IBKR's "Portfolio Gamma" = \(\gamma \times \text{position}\)) omits the spot, so $gamma = raw \(\times S\). A 1-lot AAPL call at \(S \approx 313\) reads $gamma ≈ 874 vs a raw ≈ 2.79 (\(874 / 313\)). Same Greek, different unit.

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.

Two base conventions run through the higher-order Greeks:

• Time-derivative Greeks — charm, veta, and color measure how delta, vega, and gamma decay as time passes. They are reported per calendar day, on the same basis as theta. (veta is per unit vol — raw, like the other higher-order cross-greeks; only first-order vega/rho/epsilon are per 1%.)
• Vol-curvature Greeks — vanna and volga (vomma) measure curvature in volatility and are reported per unit (raw model output), not per-1% like first-order vega. (Per-1% scaling of these higher-order greeks would round to zero at typical display precision.)

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)