> For the complete documentation index, see [llms.txt](https://docs.linora.finance/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.linora.finance/trade/pricing.md).

# Pricing

#### Pricing Relationship with Dated Options <a href="#pricing-relationship-with-dated-options" id="pricing-relationship-with-dated-options"></a>

The price of the perpetual option with continuous funding can be expressed as a continuous sum of dated options, weighted exponentially according to the time to expiry of the dated options :

$$
P\_{perpetual} = \frac{1}{T} \int\_{0}^{\infty} e^{-\frac{\tau}{T}} \cdot P\_{dated}(\tau) , d\tau
$$

where $$T$$ is the funding period of the option

#### Pricing Functions <a href="#pricing-functions" id="pricing-functions"></a>

Under Black Scholes, the continuous nature of the option results in closed-form solution for the price (this is not the case in the discrete case) under the assumption that the short-term IV σσ is flat for a given strike.

The price of a perpetual option depends on :

* Type of the option (call/put)]
* $$S$$ : Spot Price of the underlying asset
* $$K$$ : Strike Price of the option
* $$q$$ : The volatility of the underlying asset
* The annualised instantaneous interest rate.

  Linora derives this from the perpetual futures funding rate $$FR$$.

  The relationship is:&#x20;

  $$
  r = \frac{1}{TF\_{future}} \cdot \frac{FR}{1 + FR}
  $$

where

$$
TF\_{future} = \frac{8}{24 \times 365} \approx 0.00091324
$$

which corresponds to the **8-hour funding period of the perpetual future**.

$$T$$ — the funding period of the option.

Currently set to **5 days**, i.e.

Price of a Perpetual Call Option

$$
C =
\begin{cases}
S \cdot A - K \cdot B + \left(S - \frac{K}{1+rT}\right) & \text{if } S \ge K \\
S \cdot A - K \cdot B & \text{if } S < K
\end{cases}
$$

<br>

$$
T \approx \frac{5}{365} = 0.01369863014
$$

Price of a Perpetual Put Option

$$
P =
\begin{cases}
S \cdot A - K \cdot B & \text{if } S \ge K \\
S \cdot A - K \cdot B - \left(S - \frac{K}{1+rT}\right) & \text{if } S < K
\end{cases}
$$

where :

$$
A =
\begin{cases}
\frac{1}{2}\left(\frac{S}{K}\right)^{-\frac{1}{2}(1+u)}
p\left(\frac{1}{u}-1\right), & \text{if } S \ge K \\
\frac{1}{2}\left(\frac{S}{K}\right)^{-\frac{1}{2}(1-u)}
p\left(\frac{1}{u}+1\right), & \text{if } S < K
\end{cases}
$$

$$
B =
\begin{cases}
\frac{1}{2(1+rT)}
\left(\frac{S}{K}\right)^{\frac{1}{2}(1+\omega)}
q\left(\frac{1}{\omega}-1\right), & \text{if } S \ge K \\
\frac{1}{2(1+rT)}
\left(\frac{S}{K}\right)^{\frac{1}{2}(1-\omega)}
q\left(\frac{1}{\omega}+1\right), & \text{if } S < K
\end{cases}
$$

$$
p = 1 + \frac{2r}{\sigma^2}
$$

$$
q = 1 - \frac{2r}{\sigma^2}
$$

$$
u = \frac{1}{p}\sqrt{p^2 + \frac{8}{\sigma^2 T}}
$$

$$
\omega = -\frac{1}{q}\sqrt{q^2 + \frac{8(1+rT)}{\sigma^2 T}}
$$

where $$Delta TV$$ is the delta of the Time Value and is equal to:

$$
\Delta\_{TV} =
\begin{cases}
A\left(1 - \frac{(1+u)p}{2}\right) - B\frac{K}{S}\frac{(1+\omega)q}{2}, & \text{if } S \ge K \\
A\left(1 - \frac{(1-u)p}{2}\right) - B\frac{K}{S}\frac{(1-\omega)q}{2}, & \text{if } S < K
\end{cases}
$$

Gamma

$$
\gamma =
\begin{cases}
\frac{A}{S}
\left\[
\frac{(1+u)}{2}p - 1
\right]
\frac{(1+u)}{2}p
----------------

\frac{BK}{S^2}
\left\[
\frac{(1+\omega)}{2}q - 1
\right]
\frac{(1+\omega)}{2}q,
& \text{if } S \ge K
\\
\frac{A}{S}
\left\[
\frac{(1-u)}{2}p - 1
\right]
\frac{(1-u)}{2}p
----------------

\frac{BK}{S^2}
\left\[
\frac{(1-\omega)}{2}q - 1
\right]
\frac{(1-\omega)}{2}q,
& \text{if } S < K
\end{cases}
$$

Vega

$$
\nu = S \cdot \frac{\partial A}{\partial \sigma} - K \cdot \frac{\partial B}{\partial \sigma}
$$

where

$$
\frac{\partial A}{\partial \sigma} =
\begin{cases}
\frac{4A}{\sigma^3}
\left(
\left(
\frac{ru}{p} - \frac{r}{pu} - \frac{2}{p^2 u r T}
\right)
\left(
\frac{1}{u^2-u} - \frac{p}{2}\log\left(\frac{S}{K}\right)
\right)
\+
\frac{r(1+u)}{2}\log\left(\frac{S}{K}\right)
\right),
& \text{if } S \ge K
\\
\frac{4A}{\sigma^3}
\left(
\left(
\frac{ru}{p} - \frac{r}{pu} - \frac{2}{p^2 u r T}
\right)
\left(
\frac{-1}{u^2+u} + \frac{p}{2}\log\left(\frac{S}{K}\right)
\right)
\+
\frac{r(1-u)}{2}\log\left(\frac{S}{K}\right)
\right),
& \text{if } S < K
\end{cases}
$$

$$
\frac{\partial B}{\partial \sigma} =
\begin{cases}
\frac{4B}{\sigma^3}
\left(
\left(
\frac{-r\omega}{q} + \frac{r}{q\omega} + \frac{2(1+rT)}{q^2 \omega T}
\right)
\left(
\frac{1}{\omega^2-\omega} + \frac{q}{2}\log\left(\frac{S}{K}\right)
\right)
\+
\frac{r(1+\omega)}{2}\log\left(\frac{S}{K}\right)
\right),
& \text{if } S \ge K
\\
\frac{4B}{\sigma^3}
\left(
\left(
\frac{r\omega}{q} - \frac{r}{q\omega} + \frac{2(1+rT)}{q^2 \omega T}
\right)
\left(
\frac{1}{\omega^2+\omega} + \frac{q}{2}\log\left(\frac{S}{K}\right)
\right)
\+
\frac{r(1-\omega)}{2}\log\left(\frac{S}{K}\right)
\right),
& \text{if } S < K
\end{cases}
$$


---

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