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. 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)

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