To protect LPs from front-run when oracle price is delayed and to simulate market makersβ behavior when a large order is traded, Symmetry implements an "Adjusted AMM mechanism".
Liquidity is uniformly distributed between β Ξ» -\lambda β Ξ» Ξ» \lambda Ξ»
Parameters:
oracle price before the trade
mid price before the trade
takerβs buy price before the trade
oracle price after the trade
mid price after the trade
takerβs buy price after the trade
Calculation
AMM formula is as below:
Before trade is executed, p r e m = Ξ» β
s p r β
L P prem = \frac { \lambda \cdot s} {pr \cdot LP} p re m = p r β
L P Ξ» β
s β
After trade is executed, p r e m β² = Ξ» β
s β² p r β
L P prem' =\frac {\lambda \cdot s'} {pr \cdot LP} p re m β² = p r β
L P Ξ» β
s β² β
where s β² = s + T s' = s + T s β² = s + T T T T
Therefore,
p m i d = p o r a c l e ( 1 + p r e m ) = p o r a c l e ( 1 + Ξ» β
s p r β
L P ) p_{mid} = p_{oracle}(1+prem)
= p_{oracle}
\left(1 + \frac { \lambda \cdot s}
{pr \cdot LP}\right) p mi d β = p or a c l e β ( 1 + p re m ) = p or a c l e β ( 1 + p r β
L P Ξ» β
s β ) p m i d β² = p o r a c l e β² ( 1 + p r e m β² ) = p o r a c l e β² ( 1 + Ξ» β
( s + T ) p r β
L P ) p'_{mid} = p'_{oracle}(1+prem') = p'_{oracle}
\left(1 + \frac { \lambda \cdot (s+T)}
{pr \cdot LP}\right) p mi d β² β = p or a c l e β² β ( 1 + p re m β² ) = p or a c l e β² β ( 1 + p r β
L P Ξ» β
( s + T ) β ) Besides p o r a c l e p_{oracle} p or a c l e β p m i d p_{mid} p mi d β the taker's buy price p b u y p_{buy} p b u y β p s e l l p_{sell} p se ll β
γUserβs Buy Price & Sell Priceγ
After the trade, when p m i d p_{mid} p mi d β is between p b u y p_{buy} p b u y β p s e l l p_{sell} p se ll β p m i d β² p'_{mid} p mi d β² β at a fixed speed, reaching p m i d β² p'_{mid} p mi d β² β
When p m i d β² p'_{mid} p mi d β² β exceeds either of these two prices, the crossed price moves immediately to p m i d β² p'_{mid} p mi d β² β p m i d β² p'_{mid} p mi d β² β at a fixed speed, reaching p m i d β² p'_{mid} p mi d β² β
Formally speaking, after t t t
p b u y β² = { m a x ( t β
p m i d β² + ( 60 β t ) β
p b u y 60 , p m i d β² ) t < 60 p m i d β² t β₯ 60 p s e l l β² = { m i n ( t β
p m i d β² + ( 60 β t ) β
p s e l l 60 , p m i d β² ) t < 60 p m i d β² t β₯ 60 \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*} p b u y β² β p se ll β² β β = β© β¨ β§ β ma x ( 60 t β
p mi d β² β + ( 60 β t ) β
p b u y β β , p mi d β² β ) p mi d β² β β β t < 60 t β₯ 60 β = β© β¨ β§ β min ( 60 t β
p mi d β² β + ( 60 β t ) β
p se ll β β , p mi d β² β ) p mi d β² β β β t < 60 t β₯ 60 β β γExecution Priceγ
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 T T p m i d p_{mid} p mi d β p m i d β² p'_{mid} p mi d β² β and the userβs buy price is p b u y p_{buy} p b u y β
p e x e c β
T = β« 0 T max β‘ ( v β
p m i d + ( T β v ) β
p m i d β² T , p b u y ) d v 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 p e x ec β β
T = β« 0 T β max ( T v β
p mi d β + ( T β v ) β
p mi d β² β β , p b u y β ) d v Therefore, the execution price for userβs buy order is
p e x e c = { p b u y p m i d β² β€ p b u y ( p b u y β p m i d ) β
p b u y + ( p m i d β² β p b u y ) β
p m i d β² + p b u y 2 p m i d β² β p m i d p m i d β² > p b u y 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} p e x ec β = β© β¨ β§ β p b u y β p mi d β² β β p mi d β ( p b u y β β p mi d β ) β
p b u y β + ( p mi d β² β β p b u y β ) β
2 p mi d β² β + p b u y β β β β p mi d β² β β€ p b u y β p mi d β² β > p b u y β β Same as above, the execution notional for userβs sell order is
p e x e c β
T = β« 0 T min β‘ ( v β
p m i d + ( T β v ) β
p m i d β² T , p s e l l ) d v 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 p e x ec β β
T = β« 0 T β min ( T v β
p mi d β + ( T β v ) β
p mi d β² β β , p se ll β ) d v Example
Initial State
L P LP L P
Ξ± \alpha Ξ±
Ξ» \lambda Ξ»
p r pr p r
p o r a c l e p_{oracle} p or a c l e β
s = 0
p m i d p_{mid} p mi d β p b u y p_{buy} p b u y β p s e l l p_{sell} p se ll β
Trade #1 at time t = 0s
State before the trade
s = 0 s = 0 s = 0
p m i d = p b u y = p s e l l = 20000 p_{mid} = p_{buy} = p_{sell} = 20000 p mi d β = p b u y β = p se ll β = 20000
User sells $40,000,000
s β² = β 40 , 000 , 000 s' = -40,000,000 s β² = β 40 , 000 , 000
p m i d β² = 20000 Γ ( 1 + 1 Γ 0.05 Γ β 40 , 000 , 000 0.5 Γ 100 , 000 , 000 ) = 19200 p'_{mid} = 20000 \times \left( 1 + \frac{1 \times 0.05 \times -40,000,000}{0.5 \times 100,000,000} \right) = 19200 p mi d β² β = 20000 Γ ( 1 + 0.5 Γ 100 , 000 , 000 1 Γ 0.05 Γβ 40 , 000 , 000 β ) = 19200
p β² s e l l = min β‘ ( p s e l l , p m i d β² ) = 19200 p'{sell} = \min(p_{sell}, p'_{mid}) = 19200 p β² se ll = min ( p se ll β , p mi d β² β ) = 19200
The trade volume is evenly distributed between p s e l l p_{sell} p se ll β p s e l l β² p'_{sell} p se ll β² β
p e x e c = p s e l l + p s e l l β² 2 = 19600 p_{exec} = \frac{p_{sell} + p'_{sell}}{2} = 19600 p e x ec β = 2 p se ll β + p se ll β² β β = 19600
p b u y β² = p b u y = 20000 p'_{buy} = p_{buy} = 20000 p b u y β² β = p b u y β = 20000
Summary
Before this trade
After this trade
Trade #2 at time t = 15s
State before the trade
s = β 40 , 000 , 000 s = -40,000,000 s = β 40 , 000 , 000
p m i d = p s e l l = 19200 p_{mid} = p_{sell} = 19200 p mi d β = p se ll β = 19200
p b u y = p m i d Γ 15 + 20000 Γ ( 60 β 15 ) 60 = 19800 p_{buy} = \frac{p_{mid} \times 15 + 20000 \times (60 - 15)}{60} = 19800 p b u y β = 60 p mi d β Γ 15 + 20000 Γ ( 60 β 15 ) β = 19800
User sells $20,000,000
s β² = β 60 , 000 , 000 s' = -60,000,000 s β² = β 60 , 000 , 000
p m i d β² = 18800 p'_{mid} = 18800 p mi d β² β = 18800
p s e l l β² = min β‘ ( p s e l l , p m i d β² ) = 18800 p'_{sell} = \min(p_{sell}, p'_{mid}) = 18800 p se ll β² β = min ( p se ll β , p mi d β² β ) = 18800
The trade volume is evenly distributed between p s e l l p_{sell} p se ll β p s e l l β² p'_{sell} p se ll β² β
p e x e c = p s e l l + p s e l l β² 2 = 19000 p_{exec} = \frac{p_{sell} + p'_{sell}}{2} = 19000 p e x ec β = 2 p se ll β + p se ll β² β β = 19000
p b u y β² = p b u y = 19800 p'_{buy} = p_{buy} = 19800 p b u y β² β = p b u y β = 19800
Summary
After the last trade
Before this trade
After this trade
Trade #3 at time t = 39s (24 seconds later)
State before the trade
s = β 60 , 000 , 000 s = -60,000,000 s = β 60 , 000 , 000
p m i d = p s e l l = 18800 p_{mid} = p_{sell} = 18800 p mi d β = p se ll β = 18800
p b u y = p m i d Γ 24 + 19800 Γ ( 60 β 24 ) 60 = 19400 p_{buy} = \frac{p_{mid} \times 24 + 19800 \times (60 - 24)}{60} = 19400 p b u y β = 60 p mi d β Γ 24 + 19800 Γ ( 60 β 24 ) β = 19400
User buys $10,000,000
s β² = β 50 , 000 , 000 s'=-50,000,000 s β² = β 50 , 000 , 000
p m i d β² = 19000 p'_{mid} = 19000 p mi d β² β = 19000
p b u y β² = max β‘ ( p b u y , p m i d β² ) = 19400 p'_{buy} = \max(p_{buy}, p'_{mid}) = 19400 p b u y β² β = max ( p b u y β , p mi d β² β ) = 19400
The trade is completely executed at p b u y p_{buy} p b u y β
p e x e c = p b u y = 19400 p_{exec} = p_{buy} = 19400 p e x ec β = p b u y β = 19400
p s e l l β² = p s e l l = 18800 p'_{sell} = p_{sell} = 18800 p se ll β² β = p se ll β = 18800
Summary
After the last trade
Before this trade
After this trade
Trade #4 at time t = 54s (15 seconds later)
State before the trade
s = β 50 , 000 , 000 s = -50,000,000 s = β 50 , 000 , 000
p m i d = 19000 p_{mid} = 19000 p mi d β = 19000
p b u y = p m i d Γ 15 + 19400 Γ ( 60 β 15 ) 60 = 19300 p_{buy} = \frac{p_{mid} \times 15 + 19400 \times (60 - 15)}{60} = 19300 p b u y β = 60 p mi d β Γ 15 + 19400 Γ ( 60 β 15 ) β = 19300
p s e l l = p m i d Γ 15 + 18800 Γ ( 60 β 15 ) 60 = 18850 p_{sell} = \frac{p_{mid} \times 15 + 18800 \times (60 - 15)}{60} = 18850 p se ll β = 60 p mi d β Γ 15 + 18800 Γ ( 60 β 15 ) β = 18850
User buys $50,000,000
s β² = 0 s' = 0 s β² = 0
p m i d β² = 20000 p'_{mid} = 20000 p mi d β² β = 20000
p b u y β² = max β‘ ( p b u y , p m i d β² ) = 20000 p'_{buy} = \max(p_{buy}, p'_{mid}) = 20000 p b u y β² β = max ( p b u y β , p mi d β² β ) = 20000
A fraction of the trade volume is executed at p b u y p_{buy} p b u y β p b u y p_{buy} p b u y β p b u y β² p'_{buy} p b u y β² β
p e x e c = ( ( p b u y β p m i d ) Γ p b u y + ( p b u y β² β p b u y ) Γ p b u y + p b u y β² 2 ) Γ· ( p b u y β² β p m i d ) = 19545 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 e x ec β = ( ( p b u y β β p mi d β ) Γ p b u y β + ( p b u y β² β β p b u y β ) Γ 2 p b u y β + p b u y β² β β ) Γ· ( p b u y β² β β p mi d β ) = 19545
p s e l l β² = p s e l l = 18850 p'_{sell} = p_{sell} = 18850 p se ll β² β = p se ll β = 18850
Summary
After the last trade
Before this trade
After this trade
