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Constant Value Trading Protocol Introduction

$$\text{Value before commodity exchange} = \text{Value after commodity exchange} \\$$

The reason for exchange is the mismatch between quantity and value, automatically matched through market mechanisms

🔹 Step 1: Definition Introduction

Let:

• $V_a$: Current total value of Token A (unit: e.g., USD)
• $Q_a$: Current total quantity of Token A
• $V_b$, $Q_b$: Total value and quantity of Token B
• $Δa$: Quantity of A that the user wants to invest
• $ΔV$: Value transferred in this exchange (from A to B)
• $Δb$: Quantity of B exchanged from the pool based on value transfer

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🔹 Step 2: Formula Under Equal Value Conditions

$$Δb = \frac{Q_b}{Q_a + Δa} \cdot Δa$$

Split commodity b according to the quantity of a, then exchange Δa portions

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🔹 Step 3: Derivation of Exchange Formula Under Unequal Value Conditions

$$ΔV = \frac{V_a}{Q_a + {{Δa}\over 2}} \cdot Δa \\$$ $$Δb = \frac{Q_b}{V_b + {{ΔV}\over 2}} \cdot ΔV \\$$

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🔹 Step 4: Update Token A and B Values and Quantities After Trading

	Token A	Token B
Value	$V_a$	$V_b$
Quantity	$Q_a + Δa$	$Q_b - Δb$

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🔹 Step 5: New Exchange Ratio After Trading

• After trading completion, new marginal values:

$$P_a^{\text{new}} = \frac{V_a}{Q_a + Δa} \quad , \quad P_b^{\text{new}} = \frac{V_b}{Q_b - Δb}$$

• New exchange ratio:

$$R_{a \to b} = \frac{P_a^{\text{new}}}{P_b^{\text{new}}}$$

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