Deep Dive

Understanding price action from first principles

Constant Product

We start by defining a simple constant product market, commonly used to trade tokens. The market is composed of two reserves $(R_0, R_1)$ of $token0$ and $token1$ respectively, and a constraint:

$$(R_0 + x) \cdot (R_1 - y) = R_0 ⋅ R_1$$

When $x$ amount of $token0$ is inputted to the market, $y$ $token1$ comes out, constrained by the product of the reserves.

solving for y gives us the amount out:

$$y = R_1 - \frac{R_0 \cdot R_1}{R_0 + x}$$

From this, we can derive the market trading function $\kappa : (R_0, R_1, x) -> (R_0′, R_1′)$:

$$R_0′ = R_0 + x$$

$$R_1′ = \frac{R_0 \cdot R_1}{R_0 + x}$$

that describes market dynamics when traded.

KIRU Bonding Curve

We instantiate $token0 =$ ETH and $token1 =$ KIRU.

We introduce a scale parameter $α: [0, 1]$

$$α(t) = 1 - (α(0) * \frac{R_1(t)}{R_1(0)})$$

The parameter decrease in intensity as KIRU reserves deplete to ensure gradual smoothing of the scaling effect.

We can then introduce the trading function $Γ^+ : (R_0, R_1, α, x) -> (R_0′, R_1′)$ for buy orders, defined as:

$$(αR_0′, αR_1′) = \kappa(α \cdot R_0, α \cdot R_1, x)$$

$$R_0′ = R_0 + x$$

$$R_1′ = \frac{αR_1′}{αR_0′} ⋅ R_0′$$

The new KIRU reserve is computed using the virtual reserves ratio, and used to compute the amount out $y$:

$$y = R_1 - \alpha R_1\prime$$

The excess KIRU coming from the difference between the virtual effective price and the effective price is then burned.

The transition function for sell orders stay the same, enjoying full liquidity when exiting.

Spot price

We introduce the spot price formula:

$$spot = R_0 / R_1$$

Applying it to the trading functions.

For $\kappa$:

$$spot^\kappa = \dfrac{ (R_0 + x)^2 }{ R_0 \cdot R_1 }$$

and $\Gamma$:

$$spot^\Gamma = \dfrac{ ( \alpha R_0 + x )^2 }{ \alpha^2 R_0 \cdot R_1}$$

To study price action between trading functions, we simplify $\frac{spot\Gamma}{spot\kappa}$:

$$\left( \frac{ \alpha R_0 + x }{ \alpha R_0 + \alpha x } \right)^2$$

Expressing the difference of the two terms in terms of $x$ and $\alpha$:

$$\begin{align*} \text{Numerator} - \text{Denominator} &= ( \alpha R_0 + x ) - ( \alpha R_0 + \alpha x ) \\ &= x - \alpha x \\ &= x ( 1 - \alpha ) \end{align*}$$

We notice that the difference increases as $\alpha$ decreases from 1 to 0.

$$\begin{align*} \text{Denominator} - \text{Numerator} &= ( \alpha R_0 + \alpha x ) - ( \alpha R_0 + x ) \\ &= \alpha x - x \\ &= x ( \alpha - 1 ) \end{align*}$$

Since $x > 0$ and $\alpha \le 1$, the ratio is always positive, proving that over equal volume, $\Gamma^+$ implies a greater price increase than the classical constant-product, scaled by $\alpha$.