Liquidity Provider Fundamentals: Formulas Summary (Part 10)

This guide summarizes key formulas for managing eGHST-DAI liquidity positions in ghostDAO. All equations account for dynamic fee (DF) parameters and are categorized for easy reference.

LP Position Valuation

Total LP Position

$\displaystyle \text{Value}(LP)_{\text{total}} = 2 \times Q_D(\text{DAI})$

ghostDAO Treasury LP Position

$\displaystyle \text{Value}(LP)_{\text{treasury}} = \text{Value}(LP)_{\text{total}} \times \frac{Q_T(LP)}{\text{Total}(LP)}$

Optimal Contribution Weights

Exact Calculation (With Swap)

$\displaystyle Q(\text{DAI})_{\text{swap}} = \frac{-Q_D(\text{DAI}) \times (2-DF) + \sqrt{\big(Q_D(\text{DAI}) \times (2-DF)\big)^2 + 4 \times Q(\text{DAI})}}{2 \times (1-DF)}$

$\displaystyle Q(\text{eGHST})_{\text{swap}} = \frac{Q(\text{DAI})_{\text{swap}} \times (1-DF) \times Q_D(\text{eGHST})}{Q_D(\text{DAI}) + (1-DF) \times Q(\text{DAI})_{\text{swap}}}$

$\displaystyle Q(\text{DAI})_{\text{no swap}} = Q(\text{DAI}) - Q(\text{DAI})_{\text{swap}}$

Approximation

$\displaystyle Q(\text{DAI})_{\text{swap,approx}} \approx Q(\text{DAI}) \times \frac{1}{2-DF}$

$\displaystyle Q(\text{eGHST})_{\text{swap,approx}} \approx \frac{Q(\text{DAI})_{\text{swap,approx}} \times (1-DF) \times Q_D(\text{eGHST})}{Q_D(\text{DAI}) + (1-DF) \times Q(\text{DAI})_{\text{swap,approx}}}$

$\displaystyle Q(\text{DAI})_{\text{no swap,approx}} \approx Q(\text{DAI}) - Q(\text{DAI})_{\text{swap,approx}}$

Price Impact Analysis

Exact Price Impact

$\displaystyle \frac{P(\text{eGHST})_{\text{after contribution}}}{P(\text{eGHST})_{\text{before contribution}}} - 1 = \frac{Q(\text{DAI})}{Q_D(\text{DAI})}$

Approximate Price Impact

$\displaystyle \frac{P(\text{eGHST})_{\text{after contribution}}}{P(\text{eGHST})_{\text{before contribution}}} - 1 \approx \left(\frac{Q(\text{DAI})}{Q_D(\text{DAI})}\right)^2 \times \frac{1-DF}{(2-DF)^2} + \frac{Q(\text{DAI})}{Q_D(\text{DAI})}$

Constant Product (k)

$\displaystyle k_{\text{init}} = Q_D(\text{DAI}) \times Q_D(\text{eGHST})$

$\displaystyle k_{\text{after adding liquidity}} = \big(Q_D(\text{DAI}) + Q(\text{DAI})\big) \times \big(Q_D(\text{eGHST}) + Q(\text{eGHST})\big)$

$\displaystyle k_{\text{after removing liquidity}} = \big(Q_D(\text{DAI}) - Q(\text{DAI})\big) \times \big(Q_D(\text{eGHST}) - Q(\text{eGHST})\big)$

$\displaystyle k_{\text{after swap}} = \big(Q_D(\text{DAI}) + Q(\text{DAI})\big) \times \left(Q_D(\text{eGHST}) - \frac{(1-DF) \times Q(\text{DAI}) \times Q_D(\text{eGHST})}{Q_D(\text{DAI}) + (1-DF) \times Q(\text{DAI})}\right)$

Impermanent Loss (IL)

Relative IL

$\displaystyle \text{IL}(\% \text{ of Value(LP)}_{\text{HODL}}) = \sqrt{\frac{k_{\text{after swap}}}{k_{\text{init}}}} \times \frac{2\sqrt{d}}{d + 1} - 1$

Where:

$\displaystyle d = \frac{P(\text{eGHST})_{\text{after swap}}}{P(\text{eGHST})_{\text{init}}}$

Absolute IL Value

$\displaystyle IL = \left(\sqrt{\frac{k_{\text{after swap}}}{k_{\text{init}}}} \times \frac{2\sqrt{d}}{d+1} - 1\right) \times Value(LP)_{HODL}$

Where:

$\displaystyle Value(LP)_{HODL} = Q_D(\text{DAI}) + Q_D(\text{eGHST}) \times P(\text{eGHST})_{\text{after swap}}$

DEX Fees

Relative DEX Fees

$\displaystyle \text{DEX Fees}_{\text{Total}} (\% \text{ of Value(LP)}_{\text{after swap}}) = 1 - \sqrt{\frac{k_{\text{init}}}{k_{\text{after swap}}}}$

Absolute DEX Fees

$\displaystyle \text{DEX Fees}_{\text{Total}} = 2 \times Q(\text{DAI})_{\text{after swap}} \times \left(1 - \sqrt{\frac{k_{\text{init}}}{k_{\text{after swap}}}}\right)$

What is ghostDAO?