Pricing Relationship with Dated Options
The price of the perpetual option with continuous funding can be expressed as a continuous sum of dated options, weighted exponentially according to the time to expiry of the dated options :
P perpetual = 1 T ⋅ ∫ 0 ∞ e − τ T ⋅ P dated ( τ ) d τ P_{\text{perpetual}} = \frac{1}{T} \cdot \int_{0}^{\infty} e^{-\frac{\tau}{T}} \cdot P_{\text{dated}}(\tau) \, d\tau P perpetual = T 1 ⋅ ∫ 0 ∞ e − T τ ⋅ P dated ( τ ) d τ where
T T T
Pricing Functions
Under Black Scholes, the continuous nature of the option results in closed-form solution for the price (this is not the case in the discrete case) under the assumption that the short-term IV σσ is flat for a given strike.
The price of a perpetual option depends on :
Type of the option (call/put)]
S S S
K K K
q q q
: The annualised instantaneous interest rate. OrbDex derives this from the perpetual future funding rate FRFR. The relationship is : r = 1 T F u t u r e ∗ F R 1 + F R r = T F u t u r e 1 ∗ 1 + F R F R r=1TFuture∗FR1+FRr=TFuture1∗1+FRFR r = 1 TF u t u re ∗ FR 1 + FR r = TF u t u re 1 ∗ 1 + FRFR T F u t u r e = 824 ∗ 365 ≈ 0.00091324 T F u t u r e = 24 ∗ 3658 ≈ 0.00091324 TFuture=824∗365≈0.00091324TFuture=24∗3658≈0.00091324 TF u t u re = 824 ∗ 365 ≈ 0.00091324 TF u t u re = 24 ∗ 3658 ≈ 0.00091324
T T TT TT T ≈ 5365 = 0.01369863014 T ≈ 3655 = 0.01369863014 T≈5365=0.01369863014T≈3655=0.01369863014 T ≈ 5365 = 0.01369863014 T ≈ 3655 = 0.01369863014
Price of a Perpetual Call Option
C = { S ⋅ A − K ⋅ B + ( S − K 1 + r ⋅ T ) if S ≥ K S ⋅ A − K ⋅ B if S < K C = \begin{cases}
S \cdot A - K \cdot B + \left(S - \frac{K}{1 + r \cdot T} \right) & \text{if } S \geq K \\
S \cdot A - K \cdot B & \text{if } S < K
\end{cases} C = { S ⋅ A − K ⋅ B + ( S − 1 + r ⋅ T K ) S ⋅ A − K ⋅ B if S ≥ K if S < K Price of a Perpetual Put Option
P = { S ⋅ A − K ⋅ B if S ≥ K S ⋅ A − K ⋅ B − ( S − K 1 + r ⋅ T ) if S < K P = \begin{cases}
S \cdot A - K \cdot B & \text{if } S \geq K \\
S \cdot A - K \cdot B - \left(S - \frac{K}{1 + r \cdot T} \right) & \text{if } S < K
\end{cases} P = { S ⋅ A − K ⋅ B S ⋅ A − K ⋅ B − ( S − 1 + r ⋅ T K ) if S ≥ K if S < K where :
A = { 1 2 ( S K ) − 1 2 ( 1 + u ) p ⋅ ( 1 u − 1 ) , if S ≥ K 1 2 ( S K ) − 1 2 ( 1 − u ) p ⋅ ( 1 u + 1 ) , if S < K A = \begin{cases}
\frac{1}{2} \left( \frac{S}{K} \right)^{-\frac{1}{2}(1+u)} p \cdot \left( \frac{1}{u} - 1 \right), & \text{if } S \geq K \\
\frac{1}{2} \left( \frac{S}{K} \right)^{-\frac{1}{2}(1-u)} p \cdot \left( \frac{1}{u} + 1 \right), & \text{if } S < K
\end{cases} A = { 2 1 ( K S ) − 2 1 ( 1 + u ) p ⋅ ( u 1 − 1 ) , 2 1 ( K S ) − 2 1 ( 1 − u ) p ⋅ ( u 1 + 1 ) , if S ≥ K if S < K B = { 1 2 ( 1 + r ⋅ T ) ( S K ) 1 2 ( 1 + ω ) q ⋅ ( 1 ω − 1 ) , if S ≥ K 1 2 ( 1 + r ⋅ T ) ( S K ) 1 2 ( 1 − ω ) q ⋅ ( 1 ω + 1 ) , if S < K B = \begin{cases}
\frac{1}{2(1 + r \cdot T)} \left( \frac{S}{K} \right)^{\frac{1}{2}(1+\omega)} q \cdot \left( \frac{1}{\omega} - 1 \right), & \text{if } S \geq K \\
\frac{1}{2(1 + r \cdot T)} \left( \frac{S}{K} \right)^{\frac{1}{2}(1-\omega)} q \cdot \left( \frac{1}{\omega} + 1 \right), & \text{if } S < K
\end{cases} B = ⎩ ⎨ ⎧ 2 ( 1 + r ⋅ T ) 1 ( K S ) 2 1 ( 1 + ω ) q ⋅ ( ω 1 − 1 ) , 2 ( 1 + r ⋅ T ) 1 ( K S ) 2 1 ( 1 − ω ) q ⋅ ( ω 1 + 1 ) , if S ≥ K if S < K p = 1 + 2 ⋅ r σ 2 q = 1 − 2 ⋅ r σ 2 p = 1 + \frac{2 \cdot r}{\sigma^2}
\quad\quad
q = 1 - \frac{2 \cdot r}{\sigma^2} p = 1 + σ 2 2 ⋅ r q = 1 − σ 2 2 ⋅ r u = 1 p p 2 + 8 σ 2 ⋅ T ω = − 1 q q 2 + 8 ⋅ ( 1 + r ⋅ T ) σ 2 ⋅ T u = \frac{1}{p} \sqrt{p^2 + \frac{8}{\sigma^2 \cdot T}}
\quad\quad
\omega = -\frac{1}{q} \sqrt{q^2 + \frac{8 \cdot (1 + r \cdot T)}{\sigma^2 \cdot T}} u = p 1 p 2 + σ 2 ⋅ T 8 ω = − q 1 q 2 + σ 2 ⋅ T 8 ⋅ ( 1 + r ⋅ T ) where
Δ T V Δ T V Δ
T
V
Δ
TV Δ T V Δ T V
Δ T V = { A ⋅ ( 1 − ( 1 + u ) ⋅ p 2 ) − B ⋅ K S ⋅ ( 1 + ω ) ⋅ q 2 , if S ≥ K A ⋅ ( 1 − ( 1 − u ) ⋅ p 2 ) − B ⋅ K S ⋅ ( 1 − ω ) ⋅ q 2 , if S < K \Delta_{TV} =
\begin{cases}
A \cdot \left( 1 - \frac{(1 + u) \cdot p}{2} \right) - B \cdot \frac{K}{S} \cdot \frac{(1 + \omega) \cdot q}{2}, & \text{if } S \geq K \\
A \cdot \left( 1 - \frac{(1 - u) \cdot p}{2} \right) - B \cdot \frac{K}{S} \cdot \frac{(1 - \omega) \cdot q}{2}, & \text{if } S < K
\end{cases} Δ T V = ⎩ ⎨ ⎧ A ⋅ ( 1 − 2 ( 1 + u ) ⋅ p ) − B ⋅ S K ⋅ 2 ( 1 + ω ) ⋅ q , A ⋅ ( 1 − 2 ( 1 − u ) ⋅ p ) − B ⋅ S K ⋅ 2 ( 1 − ω ) ⋅ q , if S ≥ K if S < K Gamma
Γ = { A S [ ( 1 + u ) 2 ⋅ p − 1 ] ⋅ ( 1 + u ) 2 ⋅ p − B ⋅ K S 2 [ ( 1 + ω ) 2 ⋅ q − 1 ] ⋅ ( 1 + ω ) 2 ⋅ q , if S ≥ K A S [ ( 1 − u ) 2 ⋅ p − 1 ] ⋅ ( 1 − u ) 2 ⋅ p − B ⋅ K S 2 [ ( 1 − ω ) 2 ⋅ q − 1 ] ⋅ ( 1 − ω ) 2 ⋅ q , if S < K \Gamma =
\begin{cases}
\frac{A}{S} \left[ \frac{(1 + u)}{2} \cdot p - 1 \right] \cdot \frac{(1 + u)}{2} \cdot p - \frac{B \cdot K}{S^2} \left[ \frac{(1 + \omega)}{2} \cdot q - 1 \right] \cdot \frac{(1 + \omega)}{2} \cdot q, & \text{if } S \geq K \\
\frac{A}{S} \left[ \frac{(1 - u)}{2} \cdot p - 1 \right] \cdot \frac{(1 - u)}{2} \cdot p - \frac{B \cdot K}{S^2} \left[ \frac{(1 - \omega)}{2} \cdot q - 1 \right] \cdot \frac{(1 - \omega)}{2} \cdot q, & \text{if } S < K
\end{cases} Γ = ⎩ ⎨ ⎧ S A [ 2 ( 1 + u ) ⋅ p − 1 ] ⋅ 2 ( 1 + u ) ⋅ p − S 2 B ⋅ K [ 2 ( 1 + ω ) ⋅ q − 1 ] ⋅ 2 ( 1 + ω ) ⋅ q , S A [ 2 ( 1 − u ) ⋅ p − 1 ] ⋅ 2 ( 1 − u ) ⋅ p − S 2 B ⋅ K [ 2 ( 1 − ω ) ⋅ q − 1 ] ⋅ 2 ( 1 − ω ) ⋅ q , if S ≥ K if S < K Vega
ν = S ∗ δ A δ σ − K ∗ δ B δ σ \nu = S \ast \frac{\delta A}{\delta \sigma} - K \ast \frac{\delta B}{\delta \sigma} ν = S ∗ δ σ δ A − K ∗ δ σ δ B where
δ A δ σ = { 4 ∗ A σ 3 ∗ ( ( r ∗ u p − r p ∗ u − 2 p 2 ∗ u ∗ r ∗ T ) ∗ ( 1 u 2 − u − p 2 ∗ log ( S K ) ) + r ∗ ( 1 + u ) 2 ∗ log ( S K ) ) , if S ≥ K 4 ∗ A σ 3 ∗ ( ( r ∗ u p − r p ∗ u − 2 p 2 ∗ u ∗ r ∗ T ) ∗ ( − 1 u 2 + u + p 2 ∗ log ( S K ) ) + r ∗ ( 1 − u ) 2 ∗ log ( S K ) ) , if S < K \frac{\delta A}{\delta \sigma} =
\begin{cases}
\frac{4 \ast A}{\sigma^3} \ast \left( \left( \frac{r \ast u}{p} - \frac{r}{p \ast u} - \frac{2}{p^2 \ast u \ast r \ast T} \right) \ast \left( \frac{1}{u^2 - u} - \frac{p}{2} \ast \log \left( \frac{S}{K} \right) \right) + \frac{r \ast (1+u)}{2} \ast \log \left( \frac{S}{K} \right) \right), & \text{if } S \geq K \\
\frac{4 \ast A}{\sigma^3} \ast \left( \left( \frac{r \ast u}{p} - \frac{r}{p \ast u} - \frac{2}{p^2 \ast u \ast r \ast T} \right) \ast \left( \frac{-1}{u^2 + u} + \frac{p}{2} \ast \log \left( \frac{S}{K} \right) \right) + \frac{r \ast (1 - u)}{2} \ast \log \left( \frac{S}{K} \right) \right), & \text{if } S < K
\end{cases} δ σ δ A = ⎩ ⎨ ⎧ σ 3 4 ∗ A ∗ ( ( p r ∗ u − p ∗ u r − p 2 ∗ u ∗ r ∗ T 2 ) ∗ ( u 2 − u 1 − 2 p ∗ log ( K S ) ) + 2 r ∗ ( 1 + u ) ∗ log ( K S ) ) , σ 3 4 ∗ A ∗ ( ( p r ∗ u − p ∗ u r − p 2 ∗ u ∗ r ∗ T 2 ) ∗ ( u 2 + u − 1 + 2 p ∗ log ( K S ) ) + 2 r ∗ ( 1 − u ) ∗ log ( K S ) ) , if S ≥ K if S < K δ B δ σ = { 4 ∗ B σ 3 ∗ ( ( − r ∗ ω q + r q ∗ ω + 2 ( 1 + r ∗ T ) q 2 ∗ ω ∗ T ) ∗ ( 1 ω 2 − ω + q 2 ∗ log ( S K ) ) + r ∗ ( 1 + ω ) 2 ∗ log ( S K ) ) , if S ≥ K 4 ∗ B σ 3 ∗ ( ( r ∗ ω q − r q ∗ ω + 2 ( 1 + r ∗ T ) q 2 ∗ ω ∗ T ) ∗ ( 1 ω 2 + ω + q 2 ∗ log ( S K ) ) + r ∗ ( 1 − ω ) 2 ∗ log ( S K ) ) , if S < K \frac{\delta B}{\delta \sigma} =
\begin{cases}
\frac{4 \ast B}{\sigma^3} \ast \left( \left( \frac{-r \ast \omega}{q} + \frac{r}{q \ast \omega} + \frac{2(1 + r \ast T)}{q^2 \ast \omega \ast T} \right) \ast \left( \frac{1}{\omega^2 - \omega} + \frac{q}{2} \ast \log \left( \frac{S}{K} \right) \right) + \frac{r \ast (1 + \omega)}{2} \ast \log \left( \frac{S}{K} \right) \right), & \text{if } S \geq K \\
\frac{4 \ast B}{\sigma^3} \ast \left( \left( \frac{r \ast \omega}{q} - \frac{r}{q \ast \omega} + \frac{2(1 + r \ast T)}{q^2 \ast \omega \ast T} \right) \ast \left( \frac{1}{\omega^2 + \omega} + \frac{q}{2} \ast \log \left( \frac{S}{K} \right) \right) + \frac{r \ast (1 - \omega)}{2} \ast \log \left( \frac{S}{K} \right) \right), & \text{if } S < K
\end{cases} δ σ δ B = ⎩ ⎨ ⎧ σ 3 4 ∗ B ∗ ( ( q − r ∗ ω + q ∗ ω r + q 2 ∗ ω ∗ T 2 ( 1 + r ∗ T ) ) ∗ ( ω 2 − ω 1 + 2 q ∗ log ( K S ) ) + 2 r ∗ ( 1 + ω ) ∗ log ( K S ) ) , σ 3 4 ∗ B ∗ ( ( q r ∗ ω − q ∗ ω r + q 2 ∗ ω ∗ T 2 ( 1 + r ∗ T ) ) ∗ ( ω 2 + ω 1 + 2 q ∗ log ( K S ) ) + 2 r ∗ ( 1 − ω ) ∗ log ( K S ) ) , if S ≥ K if S < K 