Math deep dive

Position constructing

$$EV - equity\ value;\\ DV - debt\ value;\\ PV - sum\ value\ of\ assets\ in\ liquidity\ pool\ , associated\ with\ position;\\ C - collateral\ value;\\ l - leverage; \\ S - current\ asset\ price; S_{0} - asset\ price\ at\ position\ opening;\\ r_{B} - borrowing\ rate; r_{Y} - yield\ farming\ rate;\\ T - time\ from\ opening\; \\ \\ Bottom\ indexes\ correspond\ to\ 1'st\ and\ 2'nd\ sub-positions$$

1'st subPosition (borrowing notional)

$$DV_{1}=C_{1}\times(l-1)\times exp\{\frac{r_{B1}\times T}{365}\};\\ PV_{1} = C_{1} \times l \times \sqrt{\frac{S}{S_{0}}};\\ Farming Yield Value_{1} = C_{1} \times l \times exp\{\frac{\overline{r_{Y}}\times T}{365}\};\\ Delta_{1} = \frac{\partial{EV_{1}}}{\partial{S}} = \frac{\partial{PV_{1}}}{\partial{S}} = C_{1} \times l \times \frac{1}{2\sqrt{S\times S_{0}}}$$

2'nd subPosition (borrowing asset)

$$DV_{2}=C_{2}\times(l-1)\times exp\{\frac{r_{B2}\times T}{365}\} \times \frac{S}{S_{0}};\\ PV_{2} = C_{2} \times l \times \sqrt{\frac{S}{S_{0}}};\\ Farming Yield Value_{2} = C_{2} \times l \times exp\{\frac{\overline{r_{Y}}\times T}{365}\};\\ Delta_{2} = \frac{\partial{EV_{2}}}{\partial{S}} = \frac{\partial{PV_{2}}}{\partial{S}} = \frac{C_{2} \times l} {2\sqrt{S\times S_{0}}} - \frac{C_{2}\times (l-1)}{S_{0}}\times exp\{\frac{r_{B2}\times T}{365}\};$$

Delta-neutrality condition

$$Delta = Delta_{1} + Delta_{2};\\ Delta = \frac{l \times (C_{1}+C_{2})}{2\sqrt{S S_{0}}} - \frac{C_{2}\times (l-1)}{S_{0}}\times exp\{\frac{r_{B2}\times T}{365}\};$$

Taken Delta = 0 at opening (T=0):

that's why we put N/4 in first subPosition and 3N/4 in second.

$$\frac{C_{1}}{C_{2}} = \frac{l-2}{l}$$

Generalized equation for delta of position​

$$Delta = \frac{C \times l}{2S_{0}}\times(\sqrt{\frac{S_{0}}{S}} - exp\{\frac{r_{B2}\times T}{365}\})$$

Rebalancing

Definitions in terms of typical LYF-protocol interface (Tulip, Francium, Alpaca, Tarot)

$$Position\ 1 - position\ with\ borrowed\ stablecoin;\\ Position\ 2 - position\ with\ borrowed\ asset;\\ PV_{1} - position\ 1\ value\ (in\ stablecoins);\ DV_{1} - position\ 1\ debt\ value\ (in\ stablecoins);\\ PV_{2} - position\ 2\ value\ (in\ asset\ quantity);\ DV_{2} - position\ 2\ debt\ value\ (in\ asset\ quantity);\\ S - spot\ price;$$

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Positions' and debts' values after rebalancing:

$$PV_{1}^{*} = PV_{1} +\Delta PV_{1};\ DV_{1}^{*} = DV_{1} + \Delta DV_{1};\\ PV_{2}^{*} = PV_{2} +\Delta PV_{2};\ DV_{2}^{*} = DV_{2} + \Delta DV_{2};\\$$

How to calculate those value changes:

$$\frac{DV_{1}^{*}}{PV_{1}^{*}} = \frac{2}{3};\ (subPOS1\ leverage\ ==\ 3\ after\ rabalance)\\ \frac{DV_{2}^{*}}{PV_{2}^{*}} = \frac{2}{3};\ (subPOS2\ leverage\ ==\ 3\ after\ rabalance)\\ \frac{PV_{2}^{*}}{2} + \frac{PV_{1}^{*}}{2S} - DV_{2}^{*} = 0\ (Delta-neutrality\ condition)\\ \Delta PV_{1} + \Delta PV_{2}*S - \Delta DV_{1} - \Delta DV_{2}*S =0 \ (We\ don't\ add\ cash\ from\ outside)\\$$

The result

$$\Delta PV_{1} = \frac{3}{4}(-\frac{1}{3}PV_{1}-DV_{1}+PV_{2}\times S - DV_{2}\times S)\\ \Delta DV_{1} = \frac{1}{2}(PV_{1} - 3DV_{1} + PV_{2}\times S - DV_{2}\times S)\\ \Delta PV_{2} = \frac{9}{4S}(PV_{1} - DV_{1} + \frac{5}{9}PV_{2}\times S - DV_{2}\times S)\\ \Delta DV_{2} = \frac{3}{2S}(PV_{1} - DV_{1} + PV_{2}\times S - \frac{5}{3}DV_{2}\times S)\\$$

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:

The up-diagonal line return is the reason, why we here!