Black-76 & the Greeks

Learn how Rails calculates option mark prices and Greeks with Black-76.

Rails uses the Black-76 model to calculate option mark prices and the Greeks. Black-76 is a standard model for European options priced from a forward-style reference price.

The mark price is the reference value Rails assigns to each option for unrealized P&L, margin, and liquidations. It is derived from the model rather than the last traded price, which keeps it resistant to manipulation and thin-book noise.

Inputs

The model uses these inputs:

• $F$ — pricing reference for the underlying forward price.

• $K$ — strike price

• $T$ — time to expiry, in years

• $\sigma$ — annualized implied volatility (the mark volatility)

• $r$ — risk-free rate

• $N(x)$ — cumulative distribution function of the standard normal

• $n(x)$ — standard normal density

Black-76 pricing formulas

For a call:

$$C = e^{-rT}\left[F N(d_1) - K N(d_2)\right]$$

For a put:

$$P = e^{-rT}\left[K N(-d_2) - F N(-d_1)\right]$$

Where:

$$d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}$$

$$d_2 = d_1 - \sigma \sqrt{T}$$

These formulas convert the forward reference price, strike, time to expiry, and implied volatility into a theoretical option value.

Pricing conventions on Rails

Rate convention

The Black-76 formulas above include a discount factor $e^{-rT}$.

Rails sets $r=0$, so this simplifies to 1 and the formulas are not discounted.

Put-call parity

Black-76 implies a consistency relationship between calls and puts at the same strike and expiry:

$$C - P = e^{-rT}(F - K)$$

With the Rails rate convention $r = 0$, this becomes:

$$C - P = F - K$$

Intrinsic value floor

Rails does not let option marks fall below intrinsic value.

For calls:

$$\text{Mark}_{\text{call}} \geq \max(F - K, 0)$$

For puts:

$$\text{Mark}_{\text{put}} \geq \max(K - F, 0)$$

What moves the mark price?

Option marks can update for three main reasons:

1. The forward price changes.

2. Implied volatility changes.

3. Time passes.

In practice:

• a higher $F$ usually helps calls and hurts puts

• a higher $\sigma$ usually increases both call and put values

• a lower $T$ usually reduces time value

The Greeks

The Greeks measure how sensitive the theoretical option value is to changes in the model inputs.

Unless stated otherwise, the Greeks below are sensitivities of the Black-76 option premium. Delta and Gamma are shown with respect to $F$.

Delta

Delta ($\Delta$) measures how much the option value changes for a small move in the underlying reference price.

For a call: $\Delta_{\text{call}} = e^{-rT} N(d_1)$

For a put: $\Delta_{\text{put}} = -e^{-rT} N(-d_1)$

Gamma

Gamma ($\Gamma$) measures how quickly delta changes as the underlying moves.

For both calls and puts:

$$\Gamma = \frac{e^{-rT} n(d_1)}{F \sigma \sqrt{T}}$$

Vega

Vega ($v$) measures sensitivity to implied volatility.

For both calls and puts:

$$v = F e^{-rT} n(d_1)\sqrt{T}$$

Theta

Theta ($\Theta$) measures sensitivity to the passage of time.

For a call:

$$\Theta_{\text{call}} = -\frac{F e^{-rT} n(d_1)\sigma}{2\sqrt{T}} + r e^{-rT}\left[F N(d_1) - K N(d_2)\right]$$

For a put:

$$\Theta_{\text{put}} = -\frac{F e^{-rT} n(d_1)\sigma}{2\sqrt{T}} + r e^{-rT}\left[K N(-d_2) - F N(-d_1)\right]$$

Theta is usually negative for long options because time value tends to decay as expiry approaches.

Rho

Rho ($\rho$) measures sensitivity to interest rates.

For a call:

$$\rho_{\text{call}} = -T e^{-rT}\left[F N(d_1) - K N(d_2)\right]$$

For a put:

$$\rho_{\text{put}} = -T e^{-rT}\left[K N(-d_2) - F N(-d_1)\right]$$

Rho is usually less important in crypto options because Rails uses the convention $r = 0$.

Use Intro to Options for definitions and contract terminology. Use Metrics Guide for payoff, premium, and margin formulas.