V3 AMM Design

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AMM is the counterparty of all trades. Major questions to think about while designing an AMM include how to improve the capital efficiency and the expected profit of LP (Liquidity Provider).
To improve the capital efficiency, AMM should decrease the slippage to attract more traders. Our resolution is the assembling of more funds near the index price. Furthermore, MCDEX involves the concept of "shared liquidity pool" - multiple AMMs with the same collateral can use one liquidity pool.
The profit of LP has to be maximized in order to attract more LP to provide liquidity. There are two ways to increase LP profit: One is to increase trader’s trading fee, and the other one is to reduce arbitrage opportunities. The MCDEX AMM V3 is more inclined towards the second one. MCDEX AMM V3 also applies other means to increase LP profit, including but not limited to the adjustments of spread and slippage, and the funding payment which is unique in perpetual swaps.
Definition
Variable
Definition
​PiP_iPi​
 (Index Price)
Index Price from Oracle.
​PbP_bPb​
 (Base Price)
AMM’s base price without including additional charges.
​NNN
 (Net position)
AMM Net Position. N>0N>0N>0
 means that AMM is on long position, N<0N<0N<0
 means that AMM is on short position.
​McM_cMc​
 (Cash)
The perpetual swap margin of AMM deposited by all LPs.
​MbM_bMb​
 (Margin Balance)
Margin balance of the AMM’s account. Mb=Mc+PiNM_b=M_c+P_i NMb​=Mc​+Pi​N
.
​MpM_pMp​
 (Position value)
Position value. Mp=｜PiN｜M_p=｜P_i N｜Mp​=｜Pi​N｜
.
​MMM
 (pool Margin)
The evaluation of the value of shared liquidity pool. Check formula 
.
​RfR_fRf​
 (Funding Rate)
Funding rate. When Rf>0R_f>0Rf​>0
, long position pays funding fee to short position. When Rf<0R_f<0Rf​<0
, short position pays funding fee to long position.
​SSS
 (Share)
Total amount of pool share.
​M\cal{M}M
 (Market set)
The market set in a shared liquidity pool.
​mmm
 (current Market)
The current trading market in the shared liquidity pool. m∈Mm∈\cal{M}m∈M
.
​TholdT_\text{hold}Thold​
 (Holding Time)
Position holding time.
​T8hT_\text{8h}T8h​
 (8 hours)
8 hours.
​ααα
 (Spread)
Risk parameter: half spread. 0≤α<10≤α<10≤α<1
 .
​β1β_1β1​
 (Open Slippage)
Risk parameter: slippage. 0<β2≤β10<β_2≤β_10<β2​≤β1​
. A bigger β1β_1β1​
 increases slippage when open position.
​β2β_2β2​
 (Close Slippage)
Risk parameter: slippage. 0<β2≤β10<β_2≤β_10<β2​≤β1​
. A bigger β2β_2β2​
 increases slippage when close position.
​γγγ
 (Funding)
Risk parameter: funding rate factor. γ≥0γ≥0γ≥0
. A bigger γγγ
 increases the funding rate.
​ΓΓΓ
 (maximum Funding)
Risk parameter: funding rate limit. Γ≥0Γ≥0Γ≥0
. A bigger ΓΓΓ
 increases the maximum funding rate.
​δδδ
 (max Discount)
Risk parameter: max discount of price when close position.
​ϕϕϕ
 (Fee)
Risk parameter: trading fee. In practice, the trading fee consists of Vault Fee, Operator Fee and LP Fee.
​λλλ
 (max Leverage)
Risk parameter: target leverage of AMM’s margin account.
MCDEX AMM V3 Pricing Strategy
MCDEX AMM V3 uses a price function to provide more liquidity near the index price. It uses a risk control function to adjust the spread and slippage. Meanwhile, AMM shares margin between perpetuals to further improve the capital efficiency. Finally, AMM provides the funding rate to adjust the premium/discount.
Price function. The AMM V3’s price is based on the index price PiP_iPi​
 and its risk exposure, AMM completes base average fill price:
If a trader trades −ΔN-ΔN−ΔN
 contracts, AMM trades ΔNΔNΔN
 contracts. Pb‾\overline{P_b}Pb​​
 is the average fill price. PiP_iPi​
 is the index price provided by Oracle. βββ
 is the slippage parameter satisfying β>0β>0β>0
, in which a bigger βββ
 indicates a bigger slippage. N1N_1N1​
 is AMM’s position. MMM
 is the pool margin. This price function helps AMM follows the spot market price without arbitrageur. The slippage automatic decreases when LPs deposit more collateral. The derivation process of formula 
 and 
 are shown in Appendix 1 and Appendix 2 respectively.
MCDEX enables multiple AMMs to share the same liquidity pool. M\cal{M}M
 is the set of markets in a shared liquidity pool. Each AMM with index jjj
 has its own position N1j{N_1}_jN1​j​
 and shares the same margin MMM
.
Risk control function. We believe that LP takes lots of risk when holding positions. Therefore, we add several risk parameters to control the slippage, spread, trading fees and max positions.
AMM applies different slippage parameter βββ
 when the risk exposure is increasing or decreasing. When trader buys/longs against AMM, AMM is the short side (ΔN<0ΔN<0ΔN<0
). If AMM holds short (N<0N<0N<0
), the risk exposure is increasing. If AMM holds long (N>0N>0N>0
), the risk exposure is decreasing. The price will be:
​δδδ
 is max discount when AMM closes positions.
When trader sell/short against AMM, AMM is the long side (ΔN>0ΔN>0ΔN>0
). the price will be:
Note that formula 
 and 
 contains 2 segments respectively. A trading may cross 2 segments. For example, N1=10,ΔN=−15N_1=10,ΔN=-15N1​=10,ΔN=−15
. The trading can be considered as 2 parts ΔNseg1=−10,ΔNseg2=−5ΔN_{seg1}=-10,ΔN_{seg2}=-5ΔNseg1​=−10,ΔNseg2​=−5
 and denote prices as Pβseg1‾,Pβseg2‾\overline{{P_\beta}_{seg1}}, \overline{{P_\beta}_{seg2}}Pβ​seg1​​,Pβ​seg2​​
. The average price over segments is:
AMM also controls a price spread. The spread is the difference between best ask and best bid price. We define the best ask/bid price as:
​ααα
 is the half-spread parameter. PmidP_\text{mid}Pmid​
 is the mid-price of AMM which is determined by the current AMM position N1N_1N1​
. The final trading price is:
The figure below shows the curves of the prices.
price curves
Figure 1: Fill price for Trader. The slippage increases as AMM risk increases. Vice versa, the slippage decreases as AMM risk decreases.
After adjusting the spread and slippage, the trader pays trading fee to the AMM with the fee rate ϕϕϕ
. Let ΔNΔNΔN
 be the trading positions, the transaction of the fee can be described as:
The maximum position of AMM is defined by calculating the AMM’s position margin:
​λjλ_jλj​
 sets AMM’s max leverage of each market. AMM can open a position only if margin balance M_b is larger than the AMM’s position margin. Otherwise, AMM will cease providing liquidity and opening positions.
Funding rate. AMM provides the funding rate to perpetual swap. Funding payment is a significant component when perpetual swap anchors the spot price. Funding rate is the interest rate of the funding payment. If funding rate is positive, the long side in the perpetual swap pays interest to the short side. Vice versa, if funding rate is negative, the short side pays interest to the long side.
Because AMM is the counterparty of all traders, if most traders go long, then AMM goes short (N<0N<0N<0
), and funding rate is positive. Otherwise, if most traders go short, then AMM goes long (N>0N>0N>0
), and funding rate is negative. If there is a balance in the market, then AMM has no position, and funding rate is zero. We define funding rate RfR_fRf​
 as:
The funding rate RfR_fRf​
 is hard limited by ±Γ±Γ±Γ
. γγγ
 is the coefficient of funding rate. Funding rate RfR_fRf​
 is the ratio of position value that every position needs to collect per 8 hours. Hence if a trader holds N1N_1N1​
 amount of long position over ΔTΔTΔT
 amount of time, this trader needs to make a funding payment of:
Among which T8h=8 hoursT_\text{8h}=\text{8 hours}T8h​=8 hours
.
Bear in mind that the funding rate provided by AMM in formula 
 is always into the favorable direction for AMM. Specifically, AMM collects funding payments when it longs or shorts. AMM does not collect funding payment only when there is a tie (AMM has no position).
Features of the Pricing Strategy
In formula 
 we defined the base price of AMM V3. The formula has 3 major features: 1) The introduction of index price PiP_iPi​
, so that AMM’s price automatically follows the market price; 2) The assembling of liquidity near the index price, decreasing the slippage; 3) The solely dependent on margin balance to avoid the situation in which LP provides two types of assets.
The introduction of index price. This function introduces index price PiP_iPi​
 to direct AMM pricing. For instance, when the spot price increases drastically, AMM would not have to rely on arbitrageurs to give price but make use of the updated market price instead. In such way arbitrage opportunities can be reduced to avoid unnecessary loss in LP investment.
Slippage increases along with the increment of risk exposure. When the net position NNN
 held by AMM increases, the AMM risk exposure will increase as well. Thus, the AMM pricing will gradually moves away from index price PiP_iPi​
, and the slippage depends on the current utilization rate PiNM\frac{P_i N} {M}MPi​N​
. When the utilization rate is low, the price will be close to PiP_iPi​
. Otherwise, the slippage will increase.
Slippage is decreased near the index price. AMM V3 introduces the slippage parameter βββ
 to drastically decrease the slippage. In the constant product AMM function, the slippage is relatively high in order to make sure that AMM covers 0 to ∞ price range. The V3 AMM function flattens the price curve near index price. While it improves user experience, capital efficiency is also maximized so that the fund provided by LP could be concentrated to fulfill the trading demand near the index price.
price compare
Figure 2: Comparison with the constant product formula. The blue line is the AMM V3 price which decreases the slippage.
LP only provides collateral into margin account. In perpetual swap, trader and LP only need to provide collateral into margin account but not the underlying asset. This differentiates perpetual swap from spot trade.
In an ETH-USDC perpetual swap, the collateral is USDC, and the underlying asset is ETH. When trader longs or shorts, the fluctuation of PNL is reflected by the collateral USDC in the margin account instead of ETH.
An essential characteristic of AMM V3 is that LP only needs to provide the collateral token to add liquidity. This characteristic simplifies the operation of LP and calculation of PNL. With constant product AMM, LP needs to provide two types of tokens, hence the PNL of LP depends on the value of these two assets, making the PNL calculation of LP unclear.
Limited Market Making Depth. The market making depth of AMM V3 is limited by the max leverage. When AMM’s position reaches the upper limit, AMM will only provide one side of liquidity. For example, if the long position of AMM reaches the limit, AMM will refuse the trader to sell anymore.
LP Profit and Risk Exposure
LP profit mainly comes from spread, funding payment, and transaction fee.
Spread. There is a spread between the AMM’s best bid and best ask. When traders and arbitrageurs trade against AMM, this spread will cause AMM to profit. As N increases, AMM’s risk increases. A higher risk is designed to result in a larger spread, which then lead to a higher profit for LP.
Funding payment. Funding payment is the interest that trader pays to LP when trader goes long or short. When the long side and the short side happen to be identical, it can be interpreted as the two sides lending money from each other and the interests cancel out. When the amount of long side and short side are different, AMM is the only asset lender. Therefore, AMM deserves the interest from net position.
Trading fee. Most AMMs charge fee at a fixed ratio when trader trades with AMM. MCDEX AMM V3 is inclined towards decreasing the trading fee and arranging the distribution of profit through the methods discussed earlier.
Other income. Technically LP receives liquidation penalty as well. But since it is a function of the perpetual swap, we will not discussion it in this passage.
The major risk of LP comes from the position it holds. If the index price PiP_iPi​
 moves toward an unfavorable direction for AMM post trade, AMM will suffer a lost. Certain traders could take advantage of the publicity of AMM. MCDEX AMM V3 introduces the concept of an "operator". An operator is the creator of the perpetual swap as well as the one who manages risk parameters so that the risk on LP can be limited. In return, Operator receives part of LP’s profit.
Trading Procedure of AMM
This section is a complete step-by-step trading procedure of AMM. If a trader trades -ΔN amount of perpetual swaps, AMM, the counterparty thus trades ΔN amount of perpetual swaps. Then:
1.Read price PiP_iPi​
 from Oracle. Realize the funding payment using formula 
.
2.The margin MMM
 could change in circumstances like AMM has a PNL caused by the price change or AMM receives funding payment or trading fee. Use formula 
 to calculate this PNL and update the MMM
.
3.Based on AMM position and trader’s operating direction, there are 4 possibilities. Use formula 
 or 
 to choose slippage parameter βββ
.
4.Use 
 to calculate the average fill price.
5.Make sure that the new position does not exceed the upper limit defined by formula 
.
6.Pay the fee. Use formula 
.
Peripheral
There are some peripheral issues with the pricing strategy.
Deposit
LP deposits collateral token to the liquidity pool to earn trading fees and market making profits. AMM’s margin MMM
 increases and slippage decreases when LP deposits.
In order to track the proportion of LP in the pool, AMM mints share tokens when LP deposits into AMM. When LP withdraw from AMM, AMM burns share tokens and transfer collateral tokens back to the LP.
AMM keeps the increasing ratio of MMM
 as same as share tokens. Given the depositing collateral www
 and the total shares SSS
, AMM mints sss
 shares. The AMM’s margin MMM
 satisfies:
Withdrawal Penalty
Withdrawing from liquidity pool increases the slippage and causes LP to suffer losses. AMM reduces the losses by charging withdrawal penalty fees.
When LP withdraw from AMM, AMM first calculates the current margin M1=M(Mc,N)M_1=M(M_c,N)M1​=M(Mc​,N)
 and calculate the new margin M2=M1S−sSM_2=M_1 \frac {S-s} {S}M2​=M1​SS−s​
. Then we can calculate the withdrawal collateral w:
If AMM’s position is 0, all collateral of AMM can be withdrawn. But if the position is not 0, the www
 will be smaller. We consider this reduced collateral as withdrawal penalty. The derivation can be found in Appendix 4.
Appendix 1 The Average Fill Price
The AMM V3’s price is based on the index price PiP_iPi​
 and its risk exposure, AMM completes base price:
During the transaction, the more important thing is the average fill price when trading ΔNΔNΔN
 positions. Let the AMM position before and after trade be N1N_1N1​
 and N2N_2N2​
. The average fill price can be considered as the used position value divided by the trading positions:
When a trader trades −ΔN-ΔN−ΔN
 contracts, AMM trades ΔNΔNΔN
 contracts. So N2=N1+ΔNN_2=N_1+ΔNN2​=N1​+ΔN
. We can now express the average fill price as:
Appendix 2 Update M to Realize PNL Caused by Index Change
Assume no trading fees are taken, when PiP_iPi​
 from Oracle remains unchanged, the trade does not affect the margin MMM
, so the curve of the pricing function stays the same. But MMM
 will be affected in the following situations: AMM has the PNL when index price PiP_iPi​
 changes; AMM receives funding payment; LP add or remove liquidity. Therefore, AMM needs to keep MMM
 updated.
For example, let’s say that AMM holds long (N>0N>0N>0
), if the index price PiP_iPi​
 decreases, there will be a loss on the long side. This loss will drag down MMM
 and the slippage will increase. If trader goes long to make AMM suffer a margin closeout, AMM will have to take another loss due to the change of slippage. These two losses will result in the loss of LP.
The mechanism of updating MMM
 is to simulate trades that make AMM close its position. The resulted McM_cMc​
 will be the new MMM
. Assume the current position is N1N_1N1​
, and becomes N2=0N_2=0N2​=0
 after a simulated closing. Use 
 to calculate the change of McM_cMc​
 for each market. We get:
Solve equation 
 we get:
Among them Δ=Mb2−2(∑j∈MβjPij2Nj2)\Delta=M_b^2-2(\sum_{j \in \cal{M}} \beta_j {P_i}_j^2 N_j^2)Δ=Mb2​−2(∑j∈M​βj​Pi​j2​Nj2​)
​
Due to the fact that margin balance is positive, Δ≥0\Delta \geq 0Δ≥0
 can also be detected by checking McM_cMc​
 before every trading:
When Δ≥0\Delta \geq 0Δ≥0
, there are 2 solutions, among which formula 
 has an unfavorable characteristic: if N>0N>0N>0
 MMM
 decreases when PiP_iPi​
 increases, which is inappropriate for long position. So, formula 
 is disregarded and formula 
 is favored. AMM will avoid this situation by limiting the maximum position, we will discuss this method in Appendix 3.
When Δ<0\Delta < 0Δ<0
, the huge loss of AMM makes it incapable of going back to the zero position status using β\betaβ
 . The solution is to specify a β\betaβ
 by the algorithm just like adding a slippage to the current trading market mmm
. Meanwhile, the AMM would stop adding more position to prevent further loss. The purpose changing β\betaβ
 is to allow AMM to provide single sided liquidity in dangerous situation. To get a safe β\betaβ
, let Δ=0\Delta = 0Δ=0
 and we can get:
The βsafe\beta_\text{safe}βsafe​
 has a minimum value min{βsafe}=0min\{\beta_\text{safe}\}=0min{βsafe​}=0
. For the sake of simplicity, we set β=0\beta=0β=0
. Substitute the β\betaβ
 into the price formula 
, we get P‾=Pi\overline{P}=P_iP=Pi​
.
In summary:
Appendix 3 Maximum AMM Positions
AMM has a limited market making depth to keep the position under the max leverage when the position increases. The maximum position is subject to:
Condition 1: keep a positive price in 
.
Condition 2: The margin balance is limited by leverages according to 
.
Condition 3: Prevent Δ<0 in formula 
.
The first condition is already solved the first edge point:
To find the other edge points, we can simulate an opening position process assuming the AMM has no position until one of the above conditions is break. The position N2N_2N2​
 at this moment will be the maximum positions it can holds.
Let N1=0,Mc1=MN_1=0, M_{c1}=MN1​=0,Mc1​=M
 to simulate the situation when AMM’s position is 0. We assume only the current trading market mmm
 is increasing the position until N2N_2N2​
 while the other markets j,j∈M,j≠mj,j∈M,j≠mj,j∈M,j=m
 keep their positions unchanged. The final cash Mc2M_{c2}Mc2​
 will be changed to:
Solving condition 2: The effect of max leverage λλλ
 can be expressed as:
Solve this equation we get the maximum long position is:
The maximum short position is:
Solving condition 3: Let Δ=0Δ=0Δ=0
 in formula 
 while substitute McM_cMc​
 as Mc2M_{c2}Mc2​
, we get the maximum long position is:
The maximum short position is:
If 2M2−∑j∈M,j≠mβjPij2Nj2<02M^2-\sum_{j \in \cal{M}, j≠m} \beta_j {P_i}_j^2 N_j^2<02M2−∑j∈M,j=m​βj​Pi​j2​Nj2​<0
, the other markets j,j∈M,j≠mj, j∈M,j≠mj,j∈M,j=m
 already reach the maximum positions. In this case, the current trading market cannot open position anymore.
In summary: The maximum long position NmaxN_\text{max}Nmax​
 and maximum short position NminN_\text{min}Nmin​
 are:
Appendix 4 Withdrawal Penalty
When AMM’s position does not equal to 0, there is a risk-free means for LP to arbitrage: The LP can go long, withdraw, go short, he will profit without bearing any risk. To prevent this situation, we can fix MMM
 before and after withdrawal (except for the withdrawal part).
Given the amount of shares that LP wishes to burn is s and the collateral he will withdraw is www
, the AMM’s position is NNN
, the total shares is SSS
 and AMM’s margin is MMM
. AMM first calculates the current margin M1=M(Mc,N)M_1=M(M_c,N)M1​=M(Mc​,N)
 and calculate the new margin M2=M1S−sSM_2=M_1 \frac{S-s} {S}M2​=M1​SS−s​
 . We can get the withdrawal collateral www
 by solving M(Mc−w,N)=M2M(M_c-w,N)=M_2M(Mc​−w,N)=M2​
:
The withdrawal request may fail if LP removes too much liquidity and violates formula 
. We can get the minimum MMM
 in formula 
 by substitute McM_cMc​
 in it to formula 
. So that M2M_2M2​
 must larger than the minimum MMM
 to satisfies formula 
.