From 22c06906b4e11611540fba2c2b841573cc883d4a Mon Sep 17 00:00:00 2001
From: Colin Roberts <colin@primitive.xyz>
Date: Thu, 22 Feb 2024 22:27:07 -0700
Subject: [PATCH] docs: constant sum (#18)

* docs: `ConstantSum` README

* edit: `LogNormal` README
---
 src/ConstantSum/README.md | 83 +++++++++++++++++++++++++++++++++++++++
 src/LogNormal/README.md   |  2 +-
 2 files changed, 84 insertions(+), 1 deletion(-)

diff --git a/src/ConstantSum/README.md b/src/ConstantSum/README.md
index e69de29b..6bf2e6a9 100644
--- a/src/ConstantSum/README.md
+++ b/src/ConstantSum/README.md
@@ -0,0 +1,83 @@
+# Constant Sum Market Maker
+This will be all the background needed to understand the `GeometricMean` DFMM.
+
+## Conceptual Overview
+The `ConstantSum` DFMM gives the LP a portfolio that will allow exchange of a pair of tokens at a single price.
+We can allow this price to be dynamically chosen.
+
+## Core
+We mark reserves as:
+- $x \equiv \mathtt{rX}$
+- $y \equiv \mathtt{rY}$
+
+`ConstantSum` has one variable parameter:
+- $P \equiv \mathtt{price}$ 
+
+The **trading function** is:
+$$
+\boxed{\varphi(x,y,L;P) = \frac{x}{L} + \frac{y}{LP} -1}
+$$
+where $L$ is the **liquidity** of the pool. 
+
+## Price
+The reported price of the pool given the reseres is $P$.
+
+## Pool initialization
+The `ConstantSum` pool can be initialized with any given price and any given value of reserves. 
+A user may supply $(x_0,y_0,P)$, then we find that:
+$$
+L_0 = x_0 + \frac{y_0}{P}
+$$
+
+## Swap 
+We require that the trading function remain invariant when a swap is applied, that is:
+$$
+\frac{x+\Delta_X}{L + \Delta_L} + \frac{y+\Delta_Y}{P(L + \Delta_L)}-1 = 0
+$$
+where either $\Delta_X$ or $\Delta_Y$ is given by user input and the $\Delta_L$ comes from fees.
+
+### Trade in $\Delta_X$ for $\Delta_Y$
+If we want to trade in $\Delta_X$ for $\Delta_Y$, 
+we first accumulate fees by taking 
+$$
+\Delta_L = (1-\gamma) \Delta_X.
+$$
+Then we can use our invariant equation and solve for $\Delta_Y$ in terms of $\Delta_X$ to get:
+$$
+\boxed{\Delta_Y = \gamma P \Delta_X}
+$$
+
+### Trade in $\Delta_Y$ for $\Delta_X$
+If we want to trade in $\Delta_X$ for $\Delta_Y$, 
+we first accumulate fees by taking 
+$$
+\Delta_L = \frac{1-\gamma}{P}\Delta_Y.
+$$
+Then we can use our invariant equation and solve for $\Delta_X$ in terms of $\Delta_Y$ to get:
+$$
+\boxed{\Delta_X = \frac{\gamma}{P} \Delta_Y}
+$$
+
+## Allocations and Deallocations
+Allocations and deallocations should not change the price of a pool and since this pool only quotes a single price, any amount of reserves can be allocated at any time.
+We need only compute the new $L$.
+Specifically:
+$$
+L + \Delta_L = x+\Delta_X + \frac{y+\Delta_Y}{P}
+$$
+
+
+## Value Function via $L$ and $S$
+Given that we treat $Y$ as the numeraire, we know that the portfolio value of a pool when $X$ is at price $S$ is:
+$$
+V = Sx(S) + y(S)
+$$
+
+In this case, the value function is that of a limit order and follows:
+$$
+V(L,S) = \begin{cases}
+LS & \text{if } S \leq P \\
+LP & \text{if } S \geq P
+\end{cases}
+$$
+
diff --git a/src/LogNormal/README.md b/src/LogNormal/README.md
index 5ded076d..6ecfadc6 100644
--- a/src/LogNormal/README.md
+++ b/src/LogNormal/README.md
@@ -126,7 +126,7 @@ $$
 If we want to trade in $\Delta_X$ for $\Delta_Y$, 
 we first accumulate fees by taking 
 $$
-\Delta_L = (1-\gamma) \Delta_X.
+\Delta_L = \frac{1-\gamma}{\mu} \Delta_Y.
 $$
 Then we can use our invariant equation and solve for $\Delta_X$ in terms of $\Delta_Y$ to get:
 $$