Position constructing
E V − e q u i t y v a l u e ; D V − d e b t v a l u e ; P V − s u m v a l u e o f a s s e t s i n l i q u i d i t y p o o l , a s s o c i a t e d w i t h p o s i t i o n ; C − c o l l a t e r a l v a l u e ; l − l e v e r a g e ; S − c u r r e n t a s s e t p r i c e ; S 0 − a s s e t p r i c e a t p o s i t i o n o p e n i n g ; r B − b o r r o w i n g r a t e ; r Y − y i e l d f a r m i n g r a t e ; T − t i m e f r o m o p e n i n g B o t t o m i n d e x e s c o r r e s p o n d t o 1 ′ s t a n d 2 ′ n d s u b − 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 − e q u i t y v a l u e ; D V − d e b t v a l u e ; P V − s u m v a l u e o f a sse t s in l i q u i d i t y p oo l , a ssoc ia t e d w i t h p os i t i o n ; C − co ll a t er a l v a l u e ; l − l e v er a g e ; S − c u rre n t a sse t p r i ce ; S 0 − a sse t p r i ce a t p os i t i o n o p e nin g ; r B − b orro w in g r a t e ; r Y − y i e l d f a r min g r a t e ; T − t im e f ro m o p e nin g B o tt o m in d e x es corres p o n d t o 1 ′ s t an d 2 ′ n d s u b − p os i t i o n s 1'st subPosition (borrowing notional)
D V 1 = C 1 × ( l − 1 ) × e x p { r B 1 × T 365 } ; P V 1 = C 1 × l × S S 0 ; F a r m i n g Y i e l d V a l u e 1 = C 1 × l × e x p { r Y ‾ × T 365 } ; D e l t a 1 = ∂ E V 1 ∂ S = ∂ P V 1 ∂ S = C 1 × l × 1 2 S × 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 = C 1 × ( l − 1 ) × e x p { 365 r B 1 × T } ; P V 1 = C 1 × l × S 0 S ; F a r min g Yi e l d Va l u e 1 = C 1 × l × e x p { 365 r Y × T } ; De lt a 1 = ∂ S ∂ E V 1 = ∂ S ∂ P V 1 = C 1 × l × 2 S × S 0 1 2'nd subPosition (borrowing asset)
D V 2 = C 2 × ( l − 1 ) × e x p { r B 2 × T 365 } × S S 0 ; P V 2 = C 2 × l × S S 0 ; F a r m i n g Y i e l d V a l u e 2 = C 2 × l × e x p { r Y ‾ × T 365 } ; D e l t a 2 = ∂ E V 2 ∂ S = ∂ P V 2 ∂ S = C 2 × l 2 S × S 0 − C 2 × ( l − 1 ) S 0 × e x p { r B 2 × 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 = C 2 × ( l − 1 ) × e x p { 365 r B 2 × T } × S 0 S ; P V 2 = C 2 × l × S 0 S ; F a r min g Yi e l d Va l u e 2 = C 2 × l × e x p { 365 r Y × T } ; De lt a 2 = ∂ S ∂ E V 2 = ∂ S ∂ P V 2 = 2 S × S 0 C 2 × l − S 0 C 2 × ( l − 1 ) × e x p { 365 r B 2 × T } ; 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 × ( C 1 + C 2 ) 2 S S 0 − C 2 × ( l − 1 ) S 0 × e x p { r B 2 × 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 + De lt a 2 ; De lt a = 2 S S 0 l × ( C 1 + C 2 ) − S 0 C 2 × ( l − 1 ) × e x p { 365 r B 2 × T } ; 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 − 2 l \frac{C_{1}}{C_{2}} = \frac{l-2}{l}
C 2 C 1 = l l − 2 Generalized equation for delta of position
D e l t a = C × l 2 S 0 × ( S 0 S − e x p { r B 2 × 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 C × l × ( S S 0 − e x p { 365 r B 2 × T }) Rebalancing
Definitions in terms of typical LYF-protocol interface (Tulip, Francium, Alpaca, Tarot)
P o s i t i o n 1 − p o s i t i o n w i t h b o r r o w e d s t a b l e c o i n ; P o s i t i o n 2 − p o s i t i o n w i t h b o r r o w e d a s s e t ; P V 1 − p o s i t i o n 1 v a l u e ( i n s t a b l e c o i n s ) ; D V 1 − p o s i t i o n 1 d e b t v a l u e ( i n s t a b l e c o i n s ) ; P V 2 − p o s i t i o n 2 v a l u e ( i n a s s e t q u a n t i t y ) ; D V 2 − p o s i t i o n 2 d e b t v a l u e ( i n a s s e t q u a n t i t y ) ; S − s p o t 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 1 − p os i t i o n w i t h b orro w e d s t ab l eco in ; P os i t i o n 2 − p os i t i o n w i t h b orro w e d a sse t ; P V 1 − p os i t i o n 1 v a l u e ( in s t ab l eco in s ) ; D V 1 − p os i t i o n 1 d e b t v a l u e ( in s t ab l eco in s ) ; P V 2 − p os i t i o n 2 v a l u e ( in a sse t q u an t i t y ) ; D V 2 − p os i t i o n 2 d e b t v a l u e ( in a sse t q u an t i t y ) ; S − s p o t p r i ce ;
Positions' and debts' values after rebalancing:
P V 1 ∗ = P V 1 + Δ P V 1 ; D V 1 ∗ = D V 1 + Δ D V 1 ; P V 2 ∗ = P V 2 + Δ P V 2 ; D V 2 ∗ = D V 2 + Δ 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 ∗ = P V 1 + Δ P V 1 ; D V 1 ∗ = D V 1 + Δ D V 1 ; P V 2 ∗ = P V 2 + Δ P V 2 ; D V 2 ∗ = D V 2 + Δ D V 2 ; How to calculate those value changes:
D V 1 ∗ P V 1 ∗ = 2 3 ; ( s u b P O S 1 l e v e r a g e = = 3 a f t e r r a b a l a n c e ) D V 2 ∗ P V 2 ∗ = 2 3 ; ( s u b P O S 2 l e v e r a g e = = 3 a f t e r r a b a l a n c e ) P V 2 ∗ 2 + P V 1 ∗ 2 S − D V 2 ∗ = 0 ( D e l t a − n e u t r a l i t y c o n d i t i o n ) Δ P V 1 + Δ P V 2 ∗ S − Δ D V 1 − Δ D V 2 ∗ S = 0 ( W e d o n ′ t a d d c a s h f r o m 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 ∗ D V 1 ∗ = 3 2 ; ( s u b POS 1 l e v er a g e == 3 a f t er r aba l an ce ) P V 2 ∗ D V 2 ∗ = 3 2 ; ( s u b POS 2 l e v er a g e == 3 a f t er r aba l an ce ) 2 P V 2 ∗ + 2 S P V 1 ∗ − D V 2 ∗ = 0 ( De lt a − n e u t r a l i t y co n d i t i o n ) Δ P V 1 + Δ P V 2 ∗ S − Δ D V 1 − Δ D V 2 ∗ S = 0 ( W e d o n ′ t a dd c a s h f ro m o u t s i d e ) The result
Δ P V 1 = 3 4 ( − 1 3 P V 1 − D V 1 + P V 2 × S − D V 2 × S ) Δ D V 1 = 1 2 ( P V 1 − 3 D V 1 + P V 2 × S − D V 2 × S ) Δ P V 2 = 9 4 S ( P V 1 − D V 1 + 5 9 P V 2 × S − D V 2 × S ) Δ D V 2 = 3 2 S ( P V 1 − D V 1 + P V 2 × S − 5 3 D V 2 × 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)\\ Δ P V 1 = 4 3 ( − 3 1 P V 1 − D V 1 + P V 2 × S − D V 2 × S ) Δ D V 1 = 2 1 ( P V 1 − 3 D V 1 + P V 2 × S − D V 2 × S ) Δ P V 2 = 4 S 9 ( P V 1 − D V 1 + 9 5 P V 2 × S − D V 2 × S ) Δ D V 2 = 2 S 3 ( P V 1 − D V 1 + P V 2 × S − 3 5 D V 2 × 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:
