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  • The out-of-money scenario
  • The at-the-money scenario
  • The In-the-money scenario

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Numerical Examples of Everlasting Option Pricing

PreviousIntroducing EverLasting OptionsNextPower Perpetuals

Last updated 3 years ago

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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}Ceverand PeverP^{ever}Pever, 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*}​Cever=max(Sβˆ’K,0)+TimeValuecall​Pever=max(Kβˆ’S,0)+TimeValueput​​

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

TimeValuecallTimeValue_{call}TimeValuecall​ = TimeValueputTimeValue_{put}TimeValueput​= VVV, 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}V={uK​(KS​)21βˆ’u​,uK​(KS​)21+u​,​ifΒ Sβ©ΎKifΒ S<K​

Where u=1+8Οƒ2Tu= \sqrt{1+\frac{8}{\sigma^2T}}u=1+Οƒ2T8​​.

The details of the math can be found in this . 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*}​T=7DK=50000Οƒ=100%​

Then we have the key intermediate variable

u=1+8100%2Γ—7/365=20.4485u=\sqrt{1+\frac{8}{100\%^2\times7/365}}= 20.4485u=1+100%2Γ—7/3658​​=20.4485

The out-of-money scenario

Assuming BTCUSD = 40000, then we have intrinsic value I=0I=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}= 223.3667V=20.448550000​(5000040000​)21+20.4485​=223.3667

Therefore, the theoretical price of this 50000-Call is

C=I+V=223.3667C=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(Cβˆ’I)/7=31.9095

The at-the-money scenario

Assuming BTCUSD = 50000, then we have intrinsic value I=0I=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}= 2445.1621V=20.448550000​(5000050000​)21+20.4485​=2445.1621

Therefore, the theoretical price of this 50000-Call is

C=I+V=2445.1621C=I+V= 2445.1621 C=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(Cβˆ’I)/7=349.3089

The In-the-money scenario

Assuming BTCUSD = 60000, then we have intrinsic value I=60000βˆ’50000=10000I=60000-50000=10000I=60000βˆ’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}= 415.2673V=20.448550000​(5000060000​)21βˆ’20.4485​=415.2673

Therefore, the theoretical price of this 50000-Call is

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