Position constructing
E V β e q u i t y Β v a l u e ; D V β d e b t Β v a l u e ; P V β s u m Β v a l u e Β o f Β a s s e t s Β i n Β l i q u i d i t y Β p o o l Β , a s s o c i a t e d Β w i t h Β p o s i t i o n ; C β c o l l a t e r a l Β v a l u e ; l β l e v e r a g e ; S β c u r r e n t Β a s s e t Β p r i c e ; S 0 β a s s e t Β p r i c e Β a t Β p o s i t i o n Β o p e n i n g ; r B β b o r r o w i n g Β r a t e ; r Y β y i e l d Β f a r m i n g Β r a t e ; T β t i m e Β f r o m Β o p e n i n g β
β B o t t o m Β i n d e x e s Β c o r r e s p o n d Β t o Β 1 β² s t Β a n d Β 2 β² n d Β s u b β p o s i t i o n s EV - equity\ value;\\ DV - debt\ value;\\ PV - sum\ value\ of\ assets\ in\ liquidity\ pool\ , associated\ with\ position;\\
C - collateral\ value;\\
l - leverage; \\
S - current\ asset\ price; S_{0} - asset\ price\ at\ position\ opening;\\
r_{B} - borrowing\ rate; r_{Y} - yield\ farming\ rate;\\
T - time\ from\ opening\;
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Bottom\ indexes\ correspond\ to\ 1'st\ and\ 2'nd\ sub-positions E V β e q u i t y Β v a l u e ; D V β d e b t Β v a l u e ; P V β s u m Β v a l u e Β o f Β a sse t s Β in Β l i q u i d i t y Β p oo l Β , a ssoc ia t e d Β w i t h Β p os i t i o n ; C β co ll a t er a l Β v a l u e ; l β l e v er a g e ; S β c u rre n t Β a sse t Β p r i ce ; S 0 β β a sse t Β p r i ce Β a t Β p os i t i o n Β o p e nin g ; r B β β b orro w in g Β r a t e ; r Y β β y i e l d Β f a r min g Β r a t e ; T β t im e Β f ro m Β o p e nin g B o tt o m Β in d e x es Β corres p o n d Β t o Β 1 β² s t Β an d Β 2 β² n d Β s u b β p os i t i o n s 1'st subPosition (borrowing notional)
D V 1 = C 1 Γ ( l β 1 ) Γ e x p { r B 1 Γ T 365 } ; P V 1 = C 1 Γ l Γ S S 0 ; F a r m i n g Y i e l d V a l u e 1 = C 1 Γ l Γ e x p { r Y βΎ Γ T 365 } ; D e l t a 1 = β E V 1 β S = β P V 1 β S = C 1 Γ l Γ 1 2 S Γ S 0 DV_{1}=C_{1}\times(l-1)\times exp\{\frac{r_{B1}\times T}{365}\};\\
PV_{1} = C_{1} \times l \times \sqrt{\frac{S}{S_{0}}};\\
Farming Yield Value_{1} = C_{1} \times l \times exp\{\frac{\overline{r_{Y}}\times T}{365}\};\\
Delta_{1} = \frac{\partial{EV_{1}}}{\partial{S}} = \frac{\partial{PV_{1}}}{\partial{S}} = C_{1} \times l \times
\frac{1}{2\sqrt{S\times S_{0}}} D V 1 β = C 1 β Γ ( l β 1 ) Γ e x p { 365 r B 1 β Γ T β } ; P V 1 β = C 1 β Γ l Γ S 0 β S β β ; F a r min g Yi e l d Va l u e 1 β = C 1 β Γ l Γ e x p { 365 r Y β β Γ T β } ; De lt a 1 β = β S β E V 1 β β = β S β P V 1 β β = C 1 β Γ l Γ 2 S Γ S 0 β β 1 β 2'nd subPosition (borrowing asset)
D V 2 = C 2 Γ ( l β 1 ) Γ e x p { r B 2 Γ T 365 } Γ S S 0 ; P V 2 = C 2 Γ l Γ S S 0 ; F a r m i n g Y i e l d V a l u e 2 = C 2 Γ l Γ e x p { r Y βΎ Γ T 365 } ; D e l t a 2 = β E V 2 β S = β P V 2 β S = C 2 Γ l 2 S Γ S 0 β C 2 Γ ( l β 1 ) S 0 Γ e x p { r B 2 Γ T 365 } ; DV_{2}=C_{2}\times(l-1)\times exp\{\frac{r_{B2}\times T}{365}\} \times \frac{S}{S_{0}};\\
PV_{2} = C_{2} \times l \times \sqrt{\frac{S}{S_{0}}};\\
Farming Yield Value_{2} = C_{2} \times l \times exp\{\frac{\overline{r_{Y}}\times T}{365}\};\\
Delta_{2} = \frac{\partial{EV_{2}}}{\partial{S}} = \frac{\partial{PV_{2}}}{\partial{S}} = \frac{C_{2} \times l}
{2\sqrt{S\times S_{0}}} - \frac{C_{2}\times (l-1)}{S_{0}}\times exp\{\frac{r_{B2}\times T}{365}\}; D V 2 β = C 2 β Γ ( l β 1 ) Γ e x p { 365 r B 2 β Γ T β } Γ S 0 β S β ; P V 2 β = C 2 β Γ l Γ S 0 β S β β ; F a r min g Yi e l d Va l u e 2 β = C 2 β Γ l Γ e x p { 365 r Y β β Γ T β } ; De lt a 2 β = β S β E V 2 β β = β S β P V 2 β β = 2 S Γ S 0 β β C 2 β Γ l β β S 0 β C 2 β Γ ( l β 1 ) β Γ e x p { 365 r B 2 β Γ T β } ; Delta-neutrality condition
D e l t a = D e l t a 1 + D e l t a 2 ; D e l t a = l Γ ( C 1 + C 2 ) 2 S S 0 β C 2 Γ ( l β 1 ) S 0 Γ e x p { r B 2 Γ T 365 } ; Delta = Delta_{1} + Delta_{2};\\
Delta = \frac{l \times (C_{1}+C_{2})}{2\sqrt{S S_{0}}} - \frac{C_{2}\times (l-1)}{S_{0}}\times exp\{\frac{r_{B2}\times T}{365}\};
De lt a = De lt a 1 β + De lt a 2 β ; De lt a = 2 S S 0 β β l Γ ( C 1 β + C 2 β ) β β S 0 β C 2 β Γ ( l β 1 ) β Γ e x p { 365 r B 2 β Γ T β } ; Taken Delta = 0 at opening (T=0):
that's why we put N/4 in first subPosition and 3N/4 in second.
C 1 C 2 = l β 2 l \frac{C_{1}}{C_{2}} = \frac{l-2}{l}
C 2 β C 1 β β = l l β 2 β Generalized equation for delta of positionβ
D e l t a = C Γ l 2 S 0 Γ ( S 0 S β e x p { r B 2 Γ T 365 } ) Delta = \frac{C \times l}{2S_{0}}\times(\sqrt{\frac{S_{0}}{S}} - exp\{\frac{r_{B2}\times T}{365}\}) De lt a = 2 S 0 β C Γ l β Γ ( S S 0 β β β β e x p { 365 r B 2 β Γ T β }) Rebalancing
Definitions in terms of typical LYF-protocol interface (Tulip, Francium, Alpaca, Tarot)
P o s i t i o n Β 1 β p o s i t i o n Β w i t h Β b o r r o w e d Β s t a b l e c o i n ; P o s i t i o n Β 2 β p o s i t i o n Β w i t h Β b o r r o w e d Β a s s e t ; P V 1 β p o s i t i o n Β 1 Β v a l u e Β ( i n Β s t a b l e c o i n s ) ; Β D V 1 β p o s i t i o n Β 1 Β d e b t Β v a l u e Β ( i n Β s t a b l e c o i n s ) ; P V 2 β p o s i t i o n Β 2 Β v a l u e Β ( i n Β a s s e t Β q u a n t i t y ) ; Β D V 2 β p o s i t i o n Β 2 Β d e b t Β v a l u e Β ( i n Β a s s e t Β q u a n t i t y ) ; S β s p o t Β p r i c e ; Position\ 1 - position\ with\ borrowed\ stablecoin;\\
Position\ 2 - position\ with\ borrowed\ asset;\\
PV_{1} - position\ 1\ value\ (in\ stablecoins);\ DV_{1} - position\ 1\ debt\ value\ (in\ stablecoins);\\
PV_{2} - position\ 2\ value\ (in\ asset\ quantity);\ DV_{2} - position\ 2\ debt\ value\ (in\ asset\ quantity);\\
S - spot\ price; P os i t i o n Β 1 β p os i t i o n Β w i t h Β b orro w e d Β s t ab l eco in ; P os i t i o n Β 2 β p os i t i o n Β w i t h Β b orro w e d Β a sse t ; P V 1 β β p os i t i o n Β 1 Β v a l u e Β ( in Β s t ab l eco in s ) ; Β D V 1 β β p os i t i o n Β 1 Β d e b t Β v a l u e Β ( in Β s t ab l eco in s ) ; P V 2 β β p os i t i o n Β 2 Β v a l u e Β ( in Β a sse t Β q u an t i t y ) ; Β D V 2 β β p os i t i o n Β 2 Β d e b t Β v a l u e Β ( in Β a sse t Β q u an t i t y ) ; S β s p o t Β p r i ce ; β
Positions' and debts' values after rebalancing:
P V 1 β = P V 1 + Ξ P V 1 ; Β D V 1 β = D V 1 + Ξ D V 1 ; P V 2 β = P V 2 + Ξ P V 2 ; Β D V 2 β = D V 2 + Ξ D V 2 ; PV_{1}^{*} = PV_{1} +\Delta PV_{1};\ DV_{1}^{*} = DV_{1} + \Delta DV_{1};\\
PV_{2}^{*} = PV_{2} +\Delta PV_{2};\ DV_{2}^{*} = DV_{2} + \Delta DV_{2};\\ P V 1 β β = P V 1 β + Ξ P V 1 β ; Β D V 1 β β = D V 1 β + Ξ D V 1 β ; P V 2 β β = P V 2 β + Ξ P V 2 β ; Β D V 2 β β = D V 2 β + Ξ D V 2 β ; How to calculate those value changes:
D V 1 β P V 1 β = 2 3 ; Β ( s u b P O S 1 Β l e v e r a g e Β = = Β 3 Β a f t e r Β r a b a l a n c e ) D V 2 β P V 2 β = 2 3 ; Β ( s u b P O S 2 Β l e v e r a g e Β = = Β 3 Β a f t e r Β r a b a l a n c e ) P V 2 β 2 + P V 1 β 2 S β D V 2 β = 0 Β ( D e l t a β n e u t r a l i t y Β c o n d i t i o n ) Ξ P V 1 + Ξ P V 2 β S β Ξ D V 1 β Ξ D V 2 β S = 0 Β ( W e Β d o n β² t Β a d d Β c a s h Β f r o m Β o u t s i d e ) \frac{DV_{1}^{*}}{PV_{1}^{*}} = \frac{2}{3};\ (subPOS1\ leverage\ ==\ 3\ after\ rabalance)\\
\frac{DV_{2}^{*}}{PV_{2}^{*}} = \frac{2}{3};\ (subPOS2\ leverage\ ==\ 3\ after\ rabalance)\\
\frac{PV_{2}^{*}}{2} + \frac{PV_{1}^{*}}{2S} - DV_{2}^{*} = 0\ (Delta-neutrality\ condition)\\
\Delta PV_{1} + \Delta PV_{2}*S - \Delta DV_{1} - \Delta DV_{2}*S =0 \ (We\ don't\ add\ cash\ from\ outside)\\ P V 1 β β D V 1 β β β = 3 2 β ; Β ( s u b POS 1 Β l e v er a g e Β == Β 3 Β a f t er Β r aba l an ce ) P V 2 β β D V 2 β β β = 3 2 β ; Β ( s u b POS 2 Β l e v er a g e Β == Β 3 Β a f t er Β r aba l an ce ) 2 P V 2 β β β + 2 S P V 1 β β β β D V 2 β β = 0 Β ( De lt a β n e u t r a l i t y Β co n d i t i o n ) Ξ P V 1 β + Ξ P V 2 β β S β Ξ D V 1 β β Ξ D V 2 β β S = 0 Β ( W e Β d o n β² t Β a dd Β c a s h Β f ro m Β o u t s i d e ) The result
Ξ P V 1 = 3 4 ( β 1 3 P V 1 β D V 1 + P V 2 Γ S β D V 2 Γ S ) Ξ D V 1 = 1 2 ( P V 1 β 3 D V 1 + P V 2 Γ S β D V 2 Γ S ) Ξ P V 2 = 9 4 S ( P V 1 β D V 1 + 5 9 P V 2 Γ S β D V 2 Γ S ) Ξ D V 2 = 3 2 S ( P V 1 β D V 1 + P V 2 Γ S β 5 3 D V 2 Γ S ) \Delta PV_{1} = \frac{3}{4}(-\frac{1}{3}PV_{1}-DV_{1}+PV_{2}\times S - DV_{2}\times S)\\
\Delta DV_{1} = \frac{1}{2}(PV_{1} - 3DV_{1} + PV_{2}\times S - DV_{2}\times S)\\
\Delta PV_{2} = \frac{9}{4S}(PV_{1} - DV_{1} + \frac{5}{9}PV_{2}\times S - DV_{2}\times S)\\
\Delta DV_{2} = \frac{3}{2S}(PV_{1} - DV_{1} + PV_{2}\times S - \frac{5}{3}DV_{2}\times S)\\ Ξ P V 1 β = 4 3 β ( β 3 1 β P V 1 β β D V 1 β + P V 2 β Γ S β D V 2 β Γ S ) Ξ D V 1 β = 2 1 β ( P V 1 β β 3 D V 1 β + P V 2 β Γ S β D V 2 β Γ S ) Ξ P V 2 β = 4 S 9 β ( P V 1 β β D V 1 β + 9 5 β P V 2 β Γ S β D V 2 β Γ S ) Ξ D V 2 β = 2 S 3 β ( P V 1 β β D V 1 β + P V 2 β Γ S β 3 5 β D V 2 β Γ S ) This cash flows totally define the algorythm of rebalancing in general. It can be implemented in any LYF protocol after fitting to protocol's interface.
Here is the backtesting result of sample strategy, using Francium's SOL-USD pool characteristics with 3'rd leverage:
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