> For the complete documentation index, see [llms.txt](https://help.rails.xyz/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://help.rails.xyz/trading/options/black-76-and-the-greeks.md).

# Black-76 & the Greeks

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. On Rails, the underlying index price is used as that pricing reference.

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 index. At Rails, $$F = P\_{index}$$ from our [Index Price Sources](/trading/index-price-sources.md)
* $$K$$ — 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:

$$
d\_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}
$$

$$
d\_2 = d\_1 - \sigma \sqrt{T}
$$

{% hint style="info" %}
These formulas convert the forward reference price, strike, time to expiry, and implied volatility into a theoretical option value.
{% endhint %}

### 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.

{% columns %}
{% column %}
For **calls**:

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

{% column %}
For **puts**:

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

### What moves the mark price?

Option marks can update for three main reasons:

1. **The index 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 $$F$$.

{% tabs %}
{% tab title="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: $$\Delta\_{\text{put}} = -e^{-rT} N(-d\_1)$$
{% endtab %}

{% tab title="Gamma" %}
**Gamma (**$$\Gamma$$**)** measures how quickly delta changes as the underlying moves.

For both calls and puts:

$$
\Gamma = \frac{e^{-rT} n(d\_1)}{F \sigma \sqrt{T}}
$$
{% endtab %}

{% tab title="Vega" %}
**Vega (**$$v$$**)** measures sensitivity to implied volatility.

For both calls and puts:

$$
v = F e^{-rT} n(d\_1)\sqrt{T}
$$
{% endtab %}

{% tab title="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.
{% endtab %}

{% tab title="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 $$r = 0$$.
{% endtab %}
{% endtabs %}

{% hint style="info" %}
Use [Intro to Options](/trading/options/intro-to-options.md) for definitions and contract terminology. Use [Metrics Guide](/trading/options/options-metrics-guide.md) for payoff, premium, and margin formulas.
{% endhint %}