"Adjusted" AMM

To protect LPs from front-run when oracle price is delayed and to simulate market makers’ behavior when a large order is traded, Symmetry implements an "Adjusted AMM mechanism".

Liquidity is uniformly distributed between $-\lambda$ and. Outside this range, LP provides extra linear liquidity, which has the same liquidity distribution between and $\lambda$.

Parameters:

Symbol	Meaning
$p_{oracle}$	oracle price before the trade
$p_{mid}$	mid price before the trade
$p_{buy}$	taker’s buy price before the trade
$p_{sell}$	taker’s sell price before the trade. $p_{sell}\leq p_{buy}$
$p'_{oracle}$	oracle price after the trade
$p'_{mid}$	mid price after the trade
$p'_{buy}$	taker’s buy price after the trade
$p'_{sell}$	taker’s sell price after the trade. $p'_{sell}\leq p'_{buy}$

Calculation

AMM formula is as below:

Before trade is executed, $prem = \frac { \lambda \cdot s} {pr \cdot LP}$,

After trade is executed, $prem' =\frac {\lambda \cdot s'} {pr \cdot LP}$,

where $s' = s + T$, $T$ is trade quantity.

Therefore,

$$p_{mid} = p_{oracle}(1+prem) = p_{oracle} \left(1 + \frac { \lambda \cdot s} {pr \cdot LP}\right)$$

$$p'_{mid} = p'_{oracle}(1+prem') = p'_{oracle} \left(1 + \frac { \lambda \cdot (s+T)} {pr \cdot LP}\right)$$

Besides $p_{oracle}$ and $p_{mid}$, smart contract also keeps track of the taker's buy price $p_{buy}$ and sell price $p_{sell}$.

【User’s Buy Price & Sell Price】

After the trade, when $p_{mid}$ is between $p_{buy}$ and $p_{sell}$., both prices gradually move towards $p'_{mid}$at a fixed speed, reaching $p'_{mid}$ after 60 seconds.

When $p'_{mid}$ exceeds either of these two prices, the crossed price moves immediately to $p'_{mid}$. And the other one will gradually move towards $p'_{mid}$ at a fixed speed, reaching $p'_{mid}$ after 60 seconds.

Formally speaking, after $t$ seconds following a trade, these two prices change as follows.

$$\begin{align*} p'_{buy} &= \begin{dcases} max\left(\frac{t \cdot p'_{mid} + (60 - t) \cdot p_{buy}}{60}, p'_{mid} \right) && t < 60 \\ p'_{mid} && t \ge 60\\ \end{dcases} \\ p'_{sell} &= \begin{dcases} min \left(\frac{t \cdot p'_{mid} + (60 - t) \cdot p_{sell}}{60},p'_{mid} \right) && t < 60 \\ p'_{mid} && t \ge 60\\ \end{dcases} \end{align*}$$

【Execution Price】

A trade is executed as if it is divided into infinite tiny pieces, which are executed one by one. For example, assume that a buy trade of volume $T$ changes the mid price from $p_{mid}$ to $p'_{mid}$ and the user’s buy price is $p_{buy}$ before the trade. Its turnover is

$$p_{exec} \cdot T = \int_0^T \max\left( \frac{v \cdot p_{mid} + (T - v) \cdot p'_{mid}}{T}, p_{buy} \right) \mathrm{d}v$$

Therefore, the execution price for user’s buy order is

$$p_{exec} = \begin{dcases} p_{buy} & p'_{mid} \le p_{buy} \\ \frac{(p_{buy} - p_{mid}) \cdot p_{buy} + (p'_{mid} - p_{buy}) \cdot \frac{p'_{mid} + p_{buy}}{2}}{p'_{mid} - p_{mid}} & p'_{mid} > p_{buy} \end{dcases}$$

Same as above, the execution notional for user’s sell order is

$$p_{exec} \cdot T = \int_0^T \min\left( \frac{v \cdot p_{mid} + (T - v) \cdot p'_{mid}}{T}, p_{sell} \right) \mathrm{d}v$$

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Example

Initial State

$LP$ = 100,000,000 (assume it does not change in this example)

$\alpha$= 1

$\lambda$ = 0.05

$pr$ = 0.5

$p_{oracle}$ = 20000

s = 0

$p_{mid}$ = $p_{buy}$ = $p_{sell}$ = 20000

Trade #1 at time t = 0s

State before the trade

$s = 0$

$p_{mid} = p_{buy} = p_{sell} = 20000$

User sells $40,000,000

$s' = -40,000,000$

$p'_{mid} = 20000 \times \left( 1 + \frac{1 \times 0.05 \times -40,000,000}{0.5 \times 100,000,000} \right) = 19200$

$p'{sell} = \min(p_{sell}, p'_{mid}) = 19200$

The trade volume is evenly distributed between $p_{sell}$ and $p'_{sell}$.

$p_{exec} = \frac{p_{sell} + p'_{sell}}{2} = 19600$

$p'_{buy} = p_{buy} = 20000$

Summary

	Before this trade	After this trade
Buy Price	0%	0%
Mid Price	0%	-4%
Sell Price	0%	-4%

Trade #2 at time t = 15s

State before the trade

$s = -40,000,000$

$p_{mid} = p_{sell} = 19200$

$p_{buy} = \frac{p_{mid} \times 15 + 20000 \times (60 - 15)}{60} = 19800$

User sells $20,000,000

$s' = -60,000,000$

$p'_{mid} = 18800$

$p'_{sell} = \min(p_{sell}, p'_{mid}) = 18800$

The trade volume is evenly distributed between $p_{sell}$ and $p'_{sell}$.

$p_{exec} = \frac{p_{sell} + p'_{sell}}{2} = 19000$

$p'_{buy} = p_{buy} = 19800$

Summary

	After the last trade	Before this trade	After this trade
Buy Price	0%	-1%	-1%
Mid	-4%	-4%	-6%
Sell Price	-4%	-4%	-6%

Trade #3 at time t = 39s (24 seconds later)

State before the trade

$s = -60,000,000$

$p_{mid} = p_{sell} = 18800$

$p_{buy} = \frac{p_{mid} \times 24 + 19800 \times (60 - 24)}{60} = 19400$

User buys $10,000,000

$s'=-50,000,000$

$p'_{mid} = 19000$

$p'_{buy} = \max(p_{buy}, p'_{mid}) = 19400$

The trade is completely executed at $p_{buy}$, which does not move in this trade

$p_{exec} = p_{buy} = 19400$

$p'_{sell} = p_{sell} = 18800$

Summary

	After the last trade	Before this trade	After this trade
Buy Price	-1%	-3%	-3%
Mid	-6%	-6%	-5%
Sell Price	-6%	-6%	-6%

Trade #4 at time t = 54s (15 seconds later)

State before the trade

$s = -50,000,000$

$p_{mid} = 19000$

$p_{buy} = \frac{p_{mid} \times 15 + 19400 \times (60 - 15)}{60} = 19300$

$p_{sell} = \frac{p_{mid} \times 15 + 18800 \times (60 - 15)}{60} = 18850$

User buys $50,000,000

$s' = 0$

$p'_{mid} = 20000$

$p'_{buy} = \max(p_{buy}, p'_{mid}) = 20000$

A fraction of the trade volume is executed at $p_{buy}$ and the rest is evenly distributed between $p_{buy}$ and $p'_{buy}$.

$p_{exec} = \left((p_{buy} - p_{mid}) \times p_{buy} + (p'_{buy} - p_{buy}) \times \frac{p_{buy} + p'_{buy}}{2} \right) \div (p'_{buy} - p_{mid}) = 19545$

$p'_{sell} = p_{sell} = 18850$

Summary

	After the last trade	Before this trade	After this trade
Buy Price	-3%	-3.5%	0%
Mid	-5%	-5%	0%
Sell Price	-6%	-5.75%	-5.75%