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 e v e r C^{ever} C e v er P e v e r P^{ever} P e v er
C e v e r = max β‘ ( S β K , 0 ) + T i m e V a l u e c a l l P e v e r = max β‘ ( K β S , 0 ) + T i m e V a l u e p u t \begin{align*}&C^{ever}=\max(S-K,0)+TimeValue_{call}\\&P^{ever}=\max(K-S,0)+TimeValue_{put}\\\end{align*} β C e v er = max ( S β K , 0 ) + T im e Va l u e c a ll β P e v er = max ( K β S , 0 ) + T im e Va l u e p u t β β The call and put options at the same strike have the same time value
T i m e V a l u e c a l l TimeValue_{call} T im e Va l u e c a ll β T i m e V a l u e p u t TimeValue_{put} T im e Va l u e p u t β V V V
V = { K u ( S K ) 1 β u 2 , ifΒ S β©Ύ K K u ( S K ) 1 + u 2 , ifΒ S < K 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} V = { u K β ( K S β ) 2 1 β u β , u K β ( K S β ) 2 1 + u β , β if Β S β©Ύ K if Β S < K β Where u = 1 + 8 Ο 2 T u= \sqrt{1+\frac{8}{\sigma^2T}} u = 1 + Ο 2 T 8 β β
The details of the math can be found in this article . Here we provide some numerical examples. Letβs take the BTCUSD-50000-CALL with 7Day funding period as an example and assume volatility = 100%:
T = 7 D K = 50000 Ο = 100 % \begin{align*}
&T = 7D\\
&K = 50000\\
&\sigma=100\%
\end{align*} β T = 7 D K = 50000 Ο = 100% β Then we have the key intermediate variable
u = 1 + 8 100 % 2 Γ 7 / 365 = 20.4485 u=\sqrt{1+\frac{8}{100\%^2\times7/365}}= 20.4485 u = 1 + 100 % 2 Γ 7/365 8 β β = 20.4485 The out-of-money scenario
Assuming BTCUSD = 40000, then we have intrinsic value I = 0 I=0 I = 0
V = 50000 20.4485 ( 40000 50000 ) 1 + 20.4485 2 = 223.3667 V=\frac{50000}{20.4485}\left(\frac{40000}{50000}\right)^\frac{1+20.4485}{2}=
223.3667 V = 20.4485 50000 β ( 50000 40000 β ) 2 1 + 20.4485 β = 223.3667 Therefore, the theoretical price of this 50000-Call is
C = I + V = 223.3667 C=I+V=223.3667 C = I + V = 223.3667 Every day, a long (short) position of 1BTC of this EO pays (receives)
( C β I ) / 7 = 31.9095 (C-I)/7=31.9095 ( C β I ) /7 = 31.9095 The at-the-money scenario
Assuming BTCUSD = 50000, then we have intrinsic value I = 0 I=0 I = 0
V = 50000 20.4485 ( 50000 50000 ) 1 + 20.4485 2 = 2445.1621 V=\frac{50000}{20.4485}\left(\frac{50000}{50000}\right)^\frac{1+20.4485}{2}=
2445.1621 V = 20.4485 50000 β ( 50000 50000 β ) 2 1 + 20.4485 β = 2445.1621 Therefore, the theoretical price of this 50000-Call is
C = I + V = 2445.1621 C=I+V= 2445.1621
C = I + V = 2445.1621 Every day, a long (short) position of 1BTC of this EO pays (receives)
( C β I ) / 7 = 349.3089 (C-I)/7=
349.3089 ( C β I ) /7 = 349.3089 The In-the-money scenario
Assuming BTCUSD = 60000, then we have intrinsic value I = 60000 β 50000 = 10000 I=60000-50000=10000 I = 60000 β 50000 = 10000
V = 50000 20.4485 ( 60000 50000 ) 1 β 20.4485 2 = 415.2673 V=\frac{50000}{20.4485}\left(\frac{60000}{50000}\right)^\frac{1-20.4485}{2}=
415.2673 V = 20.4485 50000 β ( 50000 60000 β ) 2 1 β 20.4485 β = 415.2673 Therefore, the theoretical price of this 50000-Call is
C = I + V = 10415.2673 C=I+V=10415.2673 C = I + V = 10415.2673 Every day, a long (short) position of 1BTC of this EO pays (receives)
( C β I ) / 7 = ( 10415.2673 β 10000 ) / 7 = 59.3239 (C-I)/7=(10415.2673-10000)/7=
59.3239 ( C β I ) /7 = ( 10415.2673 β 10000 ) /7 = 59.3239 