TradingPerpetual Options

Pricing

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 :

Pperpetual=1T∗∫0∞e−τT∗Pdated(τ) dτP_{perpetual}=\frac{1}{T}*\int_0^{\infty}e^{-\frac{\tau}{T}}*P_{dated}(\tau)~d\tau

where TT is the funding period of the option

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 σ\sigma is flat for a given strike.

The price of a perpetual option depends on :

  • Type of the option (call/put)
  • SS : Spot Price of the underlying asset
  • KK : Strike Price of the option
  • σ\sigma : The volatility of the underlying asset
  • rr : The annualised instantaneous interest rate. Paradex derives this from the perpetual future funding rate FR\text{FR}. The relationship is : r=1TFuture∗FR1+FRr=\frac{1}{T_{Future}}*\frac{\text{FR}}{1+\text{FR}} where TFuture=824∗365≈0.00091324T_{Future}=\frac{8}{24*365}\approx 0.00091324 corresponds to the 8-hour funding period of the perpetual future
  • TT : The funding period of the option. Currently set to 5 days, i.e. T≈5365=0.01369863014T\approx \frac{5}{365}=0.01369863014

Price of a Perpetual Call Option

C={S∗A−K∗B+(S−K1+r∗T)if S≥KS∗A−K∗Bif S<K C = \begin{cases} S*A-K*B+(S-\frac{K}{1+r*T}) & \text{if }S\ge K\\ S*A-K*B & \text{if }S< K \end{cases}

Price of a Perpetual Put Option

P={S∗A−K∗Bif S≥KS∗A−K∗B−(S−K1+r∗T)if S<K P = \begin{cases} S*A-K*B & \text{if }S\ge K\\ S*A-K*B-(S-\frac{K}{1+r*T}) & \text{if }S< K \end{cases}

where :

A={12 (SK)−12(1+u) p∗(1u−1)if S≥K12 (SK)−12(1−u) p∗(1u+1)if S<K A = \begin{cases} \frac{1}{2}~(\frac{S}{K})^{-\frac{1}{2}(1+u)~p}*(\frac{1}{u}-1) & \text{if }S\ge K\\ \frac{1}{2}~(\frac{S}{K})^{-\frac{1}{2}(1-u)~p}*(\frac{1}{u}+1) & \text{if }S< K \end{cases} B={12 (1+r∗T) (SK)12(1+ω) q∗(1ω−1)if S≥K12 (1+r∗T) (SK)12(1−ω) q∗(1ω+1)if S<K B = \begin{cases} \frac{1}{2~(1+r*T)}~(\frac{S}{K})^{\frac{1}{2}(1+\omega)~q}*(\frac{1}{\omega}-1) & \text{if }S\ge K\\ \frac{1}{2~(1+r*T)}~(\frac{S}{K})^{\frac{1}{2}(1-\omega)~q}*(\frac{1}{\omega}+1) & \text{if }S< K \end{cases} p=1+2∗rσ2p=1+\frac{2*r}{\sigma^2} q=1−2∗rσ2q=1-\frac{2*r}{\sigma^2} u=1pp2+8σ2∗Tu=\frac{1}{p}\sqrt{p^2+\frac{8}{\sigma^2*T}} ω=−1qq2+8∗(1+r∗T)σ2∗T\omega=-\frac{1}{q}\sqrt{q^2+\frac{8*(1+r*T)}{\sigma^2*T}}

Greeks

Delta

Δcall={ΔTV+1if S≥KΔTVif S<K \Delta_{call} = \begin{cases} \Delta_{TV}+1 & \text{if }S\ge K\\ \Delta_{TV} & \text{if }S< K \end{cases} Δput={ΔTVif S≥KΔTV−1if S<K \Delta_{put} = \begin{cases} \Delta_{TV} & \text{if }S\ge K\\ \Delta_{TV}-1 & \text{if }S< K \end{cases}

where ΔTV\Delta_{TV} is the delta of the Time Value and is equal to :

ΔTV={A ∗ (1−(1+u) p2)−B ∗ KS (1+ω) q2if S≥KA ∗ (1−(1−u) p2)−B ∗ KS (1−ω) q2if S<K \Delta_{TV} = \begin{cases} A~*~(1-\frac{(1+u)~p}{2})-B~*~\frac{K}{S}~\frac{(1+\omega)~q}{2} & \text{if }S\ge K\\ A~*~(1-\frac{(1-u)~p}{2})-B~*~\frac{K}{S}~\frac{(1-\omega)~q}{2} & \text{if }S< K \end{cases}

Gamma

Γ={AS [(1+u) p2−1].(1+u) p2−B.KS2 [(1+ω)2.q−1].(1+ω)2.qif S≥KAS [(1−u) p2−1].(1−u) p2−B.KS2 [(1−ω)2.q−1].(1−ω)2.qif S<K \Gamma = \begin{cases} \frac{A}{S}~[\frac{(1+u)~p}{2}-1].\frac{(1+u)~p}{2}-\frac{B.K}{S^2}~[\frac{(1+\omega)}{2}.q-1].\frac{(1+\omega)}{2}.q & \text{if }S\ge K\\ \frac{A}{S}~[\frac{(1-u)~p}{2}-1].\frac{(1-u)~p}{2}-\frac{B.K}{S^2}~[\frac{(1-\omega)}{2}.q-1].\frac{(1-\omega)}{2}.q & \text{if }S< K \end{cases}

Vega

ν=S∗δAδσ−K∗δBδσ\nu=S*\frac{\delta A}{\delta \sigma}-K*\frac{\delta B}{\delta \sigma}

where

δAδσ={4∗Aσ3∗((r∗up−rp∗u−2p2∗u∗T)∗(1u2−u−p2∗log⁡(SK))                           + r∗(1+u)2∗log⁡(SK))if S≥K4∗Aσ3∗((r∗up−rp∗u−2p2∗u∗T)∗(−1u2+u+p2∗log⁡(SK))                           + r∗(1−u)2∗log⁡(SK))if S<K \frac{\delta A}{\delta \sigma} = \begin{cases} \frac{4*A}{\sigma^3}*\Big(\big(\frac{r*u}{p}-\frac{r}{p*u}-\frac{2}{p^2 * u*T}\big)*\big(\frac{1}{u^2-u}-\frac{p}{2}*\log(\frac{S}{K})\big)~\\~~~~~~~~~~~~~~~~~~~~~~~~~~+~\frac{r*(1+u)}{2}*\log(\frac{S}{K})\Big) & \text{if }S\ge K\\ \frac{4*A}{\sigma^3}*\Big(\big(\frac{r*u}{p}-\frac{r}{p*u}-\frac{2}{p^2 * u*T}\big)*\big(\frac{-1}{u^2+u}+\frac{p}{2}*\log(\frac{S}{K})\big)~\\~~~~~~~~~~~~~~~~~~~~~~~~~~+~\frac{r*(1-u)}{2}*\log(\frac{S}{K})\Big) & \text{if }S< K \end{cases} δBδσ={4∗Bσ3∗((−r∗ωq+rq∗ω+2q2∗ω∗T)∗(1ω2−ω+q2∗log⁡(SK))                           + r∗(1+ω)2∗log⁡(SK))if S≥K4∗Bσ3∗((r∗ωq−rq∗ω+2q2∗ω∗T)∗(1ω2+ω+q2∗log⁡(SK))                           + r∗(1−ω)2∗log⁡(SK))if S<K \frac{\delta B}{\delta \sigma} = \begin{cases} \frac{4*B}{\sigma^3}*\Big(\big(\frac{-r*\omega}{q}+\frac{r}{q*\omega}+\frac{2}{q^2 * \omega*T}\big)*\big(\frac{1}{\omega^2-\omega}+\frac{q}{2}*\log(\frac{S}{K})\big)~\\~~~~~~~~~~~~~~~~~~~~~~~~~~+~\frac{r*(1+\omega)}{2}*\log(\frac{S}{K})\Big) & \text{if }S\ge K\\ \frac{4*B}{\sigma^3}*\Big(\big(\frac{r*\omega}{q}-\frac{r}{q*\omega}+\frac{2}{q^2 * \omega*T}\big)*\big(\frac{1}{\omega^2+\omega}+\frac{q}{2}*\log(\frac{S}{K})\big)~\\~~~~~~~~~~~~~~~~~~~~~~~~~~+~\frac{r*(1-\omega)}{2}*\log(\frac{S}{K})\Big) & \text{if }S< K \end{cases}