# Numerical Examples of Everlasting Option Pricing

We have proved the following pricing formulae for everlasting options under the BSM assumptions.

Let's divide the theoretical prices of everlasting call and put options,   $$C^{ever}$$and $$P^{ever}$$, into intrinsic value and time value:

$$
\begin{align\*}\&C^{ever}=\max(S-K,0)+TimeValue\_{call}\\\&P^{ever}=\max(K-S,0)+TimeValue\_{put}\\\end{align\*}
$$

The call and put options at the same strike have the same time value

$$TimeValue\_{call}$$ = $$TimeValue\_{put}$$= $$V$$, given by

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

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

The details of the math can be found in this [article](https://github.com/deri-finance/whitepaper/blob/master/Pricing_Continuously_Funded_Everlasting_Options.pdf). Here we provide some numerical examples. Let’s take the BTCUSD-50000-CALL with 7Day funding period as an example and assume volatility = 100%:

$$
\begin{align\*}
\&T = 7D\\
\&K = 50000\\
&\sigma=100%
\end{align\*}
$$

Then we have the key intermediate variable

$$
u=\sqrt{1+\frac{8}{100%^2\times7/365}}= 20.4485
$$

## **The out-of-money scenario**

Assuming BTCUSD = 40000, then we have intrinsic value  $$I=0$$, and time value

$$
V=\frac{50000}{20.4485}\left(\frac{40000}{50000}\right)^\frac{1+20.4485}{2}=
223.3667
$$

Therefore, the theoretical price of this 50000-Call is

$$
C=I+V=223.3667
$$

Every day, a long (short) position of 1BTC of this EO pays (receives)

$$
(C-I)/7=31.9095
$$

## The at-the-money scenario

Assuming BTCUSD = 50000, then we have intrinsic value $$I=0$$, and time value

$$
V=\frac{50000}{20.4485}\left(\frac{50000}{50000}\right)^\frac{1+20.4485}{2}=
2445.1621
$$

Therefore, the theoretical price of this 50000-Call is

$$
C=I+V=  2445.1621
$$

Every day, a long (short) position of 1BTC of this EO pays (receives)

$$
(C-I)/7=
349.3089
$$

## The In-the-money scenario

Assuming BTCUSD = 60000, then we have intrinsic value $$I=60000-50000=10000$$ , and time value

$$
V=\frac{50000}{20.4485}\left(\frac{60000}{50000}\right)^\frac{1-20.4485}{2}=
415.2673
$$

Therefore, the theoretical price of this 50000-Call is

$$
C=I+V=10415.2673
$$

Every day, a long (short) position of 1BTC of this EO pays (receives)

$$
(C-I)/7=(10415.2673-10000)/7=
59.3239
$$


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://docs.deri.io/library/academy/everlasting-options/numerical-examples-of-everlasting-option-pricing.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.