Virtual Balances

Introduction

To guarantee the safety of the pool, there are some conditions which need to be fulfilled when adding liquidity to KyberSwap Classic pools:

  1. After LP contributions, the token price remains unchanged.

  2. PminP_{min} and PmaxP_{max} are also unchanged after LP contributions.

In KyberSwap, the pool for pair X-Y needs to maintain 4 parameters:

  1. The initial amount of token XX that is used for amplification, denoted by x0x_0​

  2. The initial amount of token YY that is used for amplification, denoted by y0y_0​

  3. The change in token XX amount after trading activities, denoted by Δx0Δx_0​

  4. The change in token YY amount after trading activities, denoted by Δy0Δy_0​

Therefore, the real balances and virtual balances of the reserves are:

Real Balances

x=x0+Δx0y=y0+Δy0x = x_0 + \Delta x_0 \\ y = y_0 + \Delta y_0

Virtual Balances

x=ax0+Δx0y=ay0+Δy0x' = a \cdot x_0 + \Delta x_0 \\ y' = a \cdot y_0 + \Delta y_0

where aa is the amplification factor.

The constant product xy=(ax0+Δx0)(ay0+Δy0)=kx' \cdot y' = (a \cdot x_0 + \Delta x_0) \cdot (a \cdot y_0 + \Delta y_0) = k'. Note that PminP_{min} and PmaxP_{max} at this time are:

{Pmin=(y0ay0)2kPmax=k(x0ax0)2\begin{cases} P_{min} = \cfrac{(y_0 \cdot a - y_0)^2}{k'} \\ \\ P_{max} = \cfrac{k'}{(x_0 \cdot a - x_0)^2} \end{cases}

The current price: P=yx=ay0+Δy0ax0+Δx0P = \cfrac{y'}{x'} = \cfrac{a \cdot y_0 + \Delta y_0}{a \cdot x_0 + \Delta x_0}

Liquidity Providers have to contribute in the same proportion for all 4 amount types. We denote the contribution ratio to be bb. LPs have to contribute x_1 + \Delta x_1$, $y_1 + \Delta y_1in which:

{x1=bx0Δx1=bΔx0y1=by0Δy1=bΔy0\begin{cases} x_1 = b \cdot x_0 \\ \Delta x_1 = b \cdot \Delta x_0 \\ y_1 = b \cdot y_0 \\ \Delta y_1 = b \cdot \Delta y_0 \end{cases}

The real balances and virtual balances of the reserve after contribution are:

Real Balances

x=(x0+x1)+(Δx0+Δx1)=(b+1)(x0+Δx0)y=(y0+y1)+(Δy0+Δy1)=(b+1)(y0+Δy0) x = (x_0 + x_1) + (\Delta x_0 + \Delta x_1) = (b + 1) \cdot (x_0 + \Delta x_0) \\ y = (y_0 + y_1) + (\Delta y_0 + \Delta y_1) = (b + 1) \cdot (y_0 + \Delta y_0)

Virtual Balances

x=a(x0+x1)+(Δx0+Δx1)=(b+1)(ax0+Δx0)y=a(y0+y1)+(Δy0+Δy1)=(b+1)(ay0+Δy0) x' = a \cdot (x_0 + x_1) + (\Delta x_0 + \Delta x_1) = (b + 1) \cdot (a \cdot x_0 + \Delta x_0) \\ y' = a \cdot (y_0 + y_1) + (\Delta y_0 + \Delta y_1) = (b + 1) \cdot (a \cdot y_0 + \Delta y_0)

The constant product, after the LP contribution, becomes:

xy=(b+1)2(ax0+Δx0)(ay0+Δy0)=(b+1)2kx' \cdot y' = (b + 1)^2 \cdot (a \cdot x_0 + \Delta x_0) \cdot (a \cdot y_0 + \Delta y_0) = (b + 1)^2 \cdot k'

PminP_{min} and PmaxP_{max} at this time are:

{Pmin=((y0+y1)a(y0+y1))2(b+1)2k=(y0ay0)2kPmax=(b+1)2k((x0+x1)a(x0+x1))2=(x0ax0)2k\begin{cases} P_{min} = \cfrac{((y_0 + y_1) \cdot a - (y_0 + y_1))^2}{(b + 1)^2 \cdot k'} = \cfrac{(y_0 \cdot a - y_0)^2}{k'} \\ P_{max} = \cfrac{(b + 1)^2 \cdot k'}{((x_0 + x_1) \cdot a - (x_0 + x_1))^2} = \cfrac{(x_0 \cdot a - x_0)^2}{k'} \end{cases}

The current price is updated to be P=yx=(ay0+Δy0)(b+1)(ax0+Δx0)(b+1)=ay0+Δy0ax0+Δx0P = \cfrac{y'}{x'} = \cfrac{(a \cdot y_0 + \Delta y_0) \cdot (b + 1)}{(a \cdot x_0 + \Delta x_0) \cdot (b + 1)} = \cfrac{a \cdot y_0 + \Delta y_0}{a \cdot x_0 + \Delta x_0}

We see that after LP contributes, the current price, PminP_{min} and PmaxP_{max} are unchanged. It is similar in the case of LPs withdrawals, where the ratio bb is negative.

Example

  • Initially, the first LP put 100 XX and 100 YY to the reserve, we have: x=100,y=100,Δx=0,Δy=0x = 100, y = 100, \Delta x = 0, \Delta y = 0.

  • A user trades 20 XX for 15 YY, so we have the updated parameters: x=100,y=100,Δx=20,Δy=15x = 100, y = 100, \Delta x = 20, \Delta y = −15.

  • Suppose an LP wants to contribute 20% of the current token amounts in the pool, so he should deposit:

    0.2100+0.220=24(X)0.2100+0.2(15)=17(Y) 0.2 · 100 + 0.2 · 20 = 24 (X) \\ 0.2 · 100 + 0.2 · (−15) = 17 (Y)

ie. deposit 24X and 17Y tokens.

The parameters are then updated to be: x = 120$, $y = 120$, $\Delta x = 24$, $\Delta y = −18.

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