vAMM (Virtual AMM)

Overview

The Virtual Automated Market Maker (vAMM) system is the core of Black Market's exchange. vAMM facilitates exchange between sSPL and VALOR tokens to ensure price stability and adequate liquidity.

The below models can be updated due to the balance update

Key Definitions and Variables

• Volatility Parameter: Parameter to control volatility. Calculated with C and k value

• k: Constant to give minimum stability to the system

• $S$: Total supply of specific sSPL tokens.

• $S_{old}$: Total supply of specific sSPL tokens before the calculation.

• $S_{new}$: Total supply of specific sSPL tokens after the calculation.

• $\Delta S$: represents the trade-induced change in supply.

• $P_{int}$: Price of the sSPL when initialized.

• $P_{old}$: Price of the sSPL before calculation.

• $P_{new}$: Price of the sSPL after calculation.

• $P_{avg}$: The price of the sSPL applies to a specific trade.

Models

The vAMM model dynamically adjusts the token price to provide appropriate stability and volatility while ensuring symmetry in trading.

Symmetric Model

vAMM calculates the price after the trade using the symmetric model.

• Calculation based on sSPL amount

$P_{new} = P_{old} \times (1 + \frac{\Delta S}{k + S_{old}} )$

• Calculate based on VALOR budget

$\Delta S = \frac{B}{\frac{P_{old} + P_{new}}{2}} = \frac{2B}{P_{old} + P_{new}}$

$\therefore P_{new} = \sqrt{P_{old}^2 + \frac{2B \cdot P_{old}}{k + S_{old}}}$

Average Price Model

vAMM swap transaction works based on the average price before the trade and the adjusted price after the trade.

$P_{avg} = \frac{P_{old} + P_{new}}{2}$

Price Adjustment Due to External Changes

If the supply of sSPL tokens changes outside of vAMM trades, the system adjusts the price using the following equation:

$P_{\text{new}} = \begin{cases} P_{\text{int}} + \frac{(P_{\text{old}} - P_{\text{int}})(2k + S_{\text{old}})}{2k + S_{\text{new}}} & \text{if} \,\, P_{\text{old}} > P_{\text{int}} \\ P_{\text{int}} - \frac{(P_{\text{int}} - P_{\text{old}})(2k + S_{\text{old}})}{2k + S_{\text{new}}} & \text{if} \,\, P_{\text{old}} \le P_{\text{int}} \end{cases}$

$k = 100000 / P_{int}$

Initial price

Token	Initial Price ($VALOR)
Voodoo Doll ($VD)	0.46118
Gold Teeth ($GT)	0.72776
JB Whiskey ($JBW)	0.94118
Canteen ($CT)	1.07882
G Badge ($GB)	1.55294
Holy Water ($HW)	2.32353
Used Engine ($UE)	3.41176
Enhanced Bullet ($EB)	3.43529
Oil Lighter Case ($OLC)	6.82353
Oil ($OIL)	0.00750
MRE ($MRE)	0.00750

Examples

1. Swap:

   1. Buy: Paying VALOR to buy 1 sSPL

      • Stability constant $k = 10$ sSPL (after buying)

      • Current supply $S_{old} = 0$ sSPL

      • New supply $S_{new} = 1$ sSPL

      • Current price $P_{old} = 10$ VALOR

      • New price $P_{new} = 10 \times (1 + \frac{1}{10 + 0} ) = 11$ VALOR

      • Average price $P_{avg} = \frac{10 + 11}{2} = 10.5$ VALOR

      • Fee: $10.5 * 0.0005 = 0.0525$ VALOR

      • Result: Paying 10.5525 VALOR to buy 1 sSPL

   2. Sell Transaction: Selling 1 sSPL to get VALOR

      • Stability constant $k = 10$ sSPL (after buying)

      • Current supply $S_{old} = 1$ sSPL (after buying)

      • New supply $S_{new} = 0$ sSPL

      • Current price $P_{old} = 11$ VALOR (after buying)

      • New price $P_{new} = 11 \times ( 1 - \frac{1}{10 + 1} ) = 10$ VALOR

      • Average price $P_{avg} = \frac{11 + 10}{2} = 10.5$ VALOR

      • Fee: $10.5 * 0.0005 = 0.0525$ VALOR

      • Result: Receiving 10.5525 VALOR for selling 1 sSPL

2. Price Adjustment:

   

   1. On Mint:

      • Stability constant $k = 10$ sSPL (after buying)

        • Current supply $S_{old} = 1$ sSPL

        • Supply after mint $S_{new} = 5$ sSPL

        • Initial price $P_{int} = 10$ VALOR

        • Current price $P_{old} = 11$ VALOR

        • Adjusted price $P_{new} = \frac{|11 - 10|(1 + 2 \times 10)}{5 + 2 \times 10} + 10 = 10.84$ VALOR

   2. On Burn:

      • Stability constant $k = 10$ sSPL (after buying)

        • Existing supply $S_{old} = 5$ sSPL

        • Supply after burn $S_{new} = 1$ sSPL

        • Initial price $P_{int} = 10$ VALOR

        • Current price $P_{old} = 10.84$ VALOR

        • Adjusted price $P_{new} = \frac{|10.84 - 10|(5 + 2 \times 10)}{1 + 2 \times 10} + 10 = 11$ VALOR