Pricing & Greeks - Rails Trading Developer API Documentation

The formulas Rails uses to price options and compute Greeks. For a conceptual overview, see the Pricing & Greeks guide.

Notation

Symbol	Meaning
`F`	Forward price of the underlying
`K`	Strike price
`σ`	Volatility (implied volatility, annualized as a decimal — `0.55` = 55%)
`r`	Annualized risk-free rate
`T`	Time to expiry, in years
`N(x)`	Standard normal cumulative distribution function (CDF)
`φ(x)`	Standard normal probability density function (PDF)
`V`	Option price

Price (Black-76)

Rails prices options with the Black-76 model, which prices on the forward F rather than spot.

Term	Formula
`d1`	`[ ln(F / K) + (σ² / 2) · T ] / (σ · √T)`
`d2`	`d1 − σ · √T`
Call	`V = e^(−rT) · [ F · N(d1) − K · N(d2) ]`
Put	`V = e^(−rT) · [ K · N(−d2) − F · N(−d1) ]`

To go the other way — from a market price V to its implied volatility σ — invert this model numerically (solve for the σ that reproduces V), then feed σ into the Greeks below.

Greeks

Greek	Formula	Notes
Delta	Call: `e^(−rT) · N(d1)` · Put: `−e^(−rT) · N(−d1)`	Per `1` change in `F`
Gamma	`e^(−rT) · φ(d1) / (F · σ · √T)`	Same for calls and puts
Vega	`e^(−rT) · F · φ(d1) · √T`	Per `1.0` (100%) change in `σ`
Theta	`−[ e^(−rT) · F · φ(d1) · σ ] / (2 · √T) + r · V`	Per year
Rho	`−T · V`	Per `1.0` (100%) change in `r`

Normalization

The raw formulas above are normalized before they are published on the Contracts Stream and REST responses:

Greek	Published as	Conversion
Vega	Per 1% IV change	raw `÷ 100`
Theta	Daily decay	raw `÷ 365`
Rho	Per 1% rate change	raw `÷ 100`

Delta and gamma are published as computed.

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