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 (mark) price |
Price (Black-76)
Rails prices options with the Black-76 model, which prices on the forwardF 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) ] |
The Get Contract Pricing endpoint inverts this
model: given a candidate price
V, it solves for the implied volatility σ, then computes
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 |