> For the complete documentation index, see [llms.txt](https://tidal-flats.gitbook.io/tidal-flats-whitepaper-en/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tidal-flats.gitbook.io/tidal-flats-whitepaper-en/whitepaper/black-market-2-road-to-fidelia/black-market-dex/vamm-virtual-amm.md).
# 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.
{% hint style="info" %}
The below models can be updated due to the balance update
{% endhint %}
### 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
---
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