Pricing & Greeks

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
S	Index 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.

Expected Price Move

The expected move of the index price by an expiry, published per expiry as expectedPriceMove on the Contracts Stream and from Get Expirations.

Term	Formula
Expected move	S · σ · √T

σ is the expiry’s at-the-money implied volatility, published as atmIV — its IV curve interpolated at the forward F. The result is one standard deviation, so roughly 68% of outcomes fall within that distance of S.