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, CeverC^{ever}and PeverP^{ever}, into intrinsic value and time value:

Cever=max⁑(Sβˆ’K,0)+TimeValuecallPever=max⁑(Kβˆ’S,0)+TimeValueput\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

TimeValuecallTimeValue_{call} = TimeValueputTimeValue_{put}= VV, given by

V={Ku(SK)1βˆ’u2,ifΒ Sβ©ΎKKu(SK)1+u2,ifΒ S<KV= \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=1+8Οƒ2Tu= \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%:

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

Then we have the key intermediate variable

u=1+8100%2Γ—7/365=20.4485u=\sqrt{1+\frac{8}{100\%^2\times7/365}}= 20.4485

The out-of-money scenario

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

V=5000020.4485(4000050000)1+20.44852=223.3667V=\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=223.3667C=I+V=223.3667

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

(Cβˆ’I)/7=31.9095(C-I)/7=31.9095

The at-the-money scenario

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

V=5000020.4485(5000050000)1+20.44852=2445.1621V=\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=2445.1621C=I+V= 2445.1621

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

(Cβˆ’I)/7=349.3089(C-I)/7= 349.3089

The In-the-money scenario

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

V=5000020.4485(6000050000)1βˆ’20.44852=415.2673V=\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=10415.2673C=I+V=10415.2673

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

(Cβˆ’I)/7=(10415.2673βˆ’10000)/7=59.3239(C-I)/7=(10415.2673-10000)/7= 59.3239

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