AMM for continuous liquidation／deliquidation (LLAM

figure 2:dependence of the loss on the price shift relative to the liquidation threshold . time window for the observation is 3 days

In this design、If some one borrows against collateral、even at liquidation threshold、And the price of collateral dips And bounces

在这个设计中，假设有人用抵押借款，即使在清算阈值下，抵押物的价格下降，也会反弹。不会发生明确的损失。例如，根据2017年9月以来利用ETH/USD的历史数据中断的模拟，如果CDP在搁置三天后降至清算价格的10%以下

AMM for continuous liquidation/de liquidation(ll amma)无需继续/清算清算的AMM(ll amma)the core idea of the stable coin design is lisif the price of collateral is higha user has deposits all in eth It converts to USD . this is very different from traditional AMM designs where one HHE

浮动货币设计的中心思想是Lending-Liquidating AMM算法。这个想法是在抵押品(如ETH)和稳定物(此处称为USD)之间停止转换。假设抵押品的价格很高。用户的贷款全部是ETH，但价格下降后，将转换为USD稳定。这与激进的AMM设计大不相同。保守的AMM设计是把USD稳定硬币放在下面(AMM曲线的上半部分)，ETH放在下面(AMM曲线的下半部分)。

the below description doesn ' t serve as fully self-consistent rigorous proofs . a lot of that(特殊场合)Are obtained from

下面的描述并不能作为完整的自洽的松散证明。很多东西(特别是不变的)是从各种层面思考得到的。为了完整的数学描述需要进一步的研究，但上述描述被认为足以支持在中实施。

this is only possible with an external price Oracle . in a nutshell，If one makes a typical AMM(for example with a bonding curve being a piece ofit is somewhat similar to avoided crossing(also called landau-zener transition)In quantum physics(though only as an idea:mathethe range where the liquidity is concentrated is called band here，At the constant po band has liquidity from PCD to PCU . we seek for PCD(po)in addition，Let ' s defi fine prices pand pbeing prices where p(po)=po，and p (po)=

要经过外部伙食价才能完成。简而言之，如果一组生成典型的AMM(例如粘合曲线为双曲线)，使“中心价格”从(例如)下降到下降，令牌将从(例如)USD“绝热”转换为ETH，同时提供了两种方法。这有点像量子物理学中的“交叉回避”(也称为Landau-Zener转移)。(当然，对这个过程的数学描述可能大不相同。）

活动性集的范围在这里称为“带”，在一定的po带中具有从PCD到PCU的活动性。我们只寻找pcd(po)和pcu(po)作为po的函数，因为函数比线性更陡，所以增减速度比po快(图4)。另外，让我们将价格p和p定义为p(po)=po和p(po)=po的价格，定义为绝热极限的频带两端(例如p=po)。

figure 3:behavior of an " AMM with an external price source "。external price p center determines a price around which Liquidity is form Liquidity concentrated from prices PCD to PCU、PCD p center pcu。when current price p is out of range between PCD and PCU、Amm is either fully in stable coin(when at PCU)或fully in collateral (when atwhen PCD p PCU、AMM price

We start from a number of bandswhere、similarly to uniswap 3、hyperbolic shape of The bonding curve is preserved by adding virtual balances to

在一些波段末端，像Uniswap3一样，增加了“虚拟余额”，保留了粘合曲线的双曲线形状。例如，USD的数量是X，ETH的数量是Y，因此“增强的”常数-产品不变性如下：

we also can denote x 0x F and y 0y g So that the invariant can be written as a familiar I=x 0y 0 . however，F and g do not stay constant:theat a given po，f and g are constant across the band . as mentioned beford We denote pas the top price of the band and pas the bottom protomwe defi fine a(a measure of concentration of liquide)

我们还可以表示X0X F和Y0Y G

the property we are looking for is such that higher price po should lead to even higher price at the same balances，So that the current market priceit is possible to find many ways to say

我们要找的属性如下。更高的价格po应该从相同的余额延续到更高的价格。因此，今后的市长/市场价格(平均会跟随po)低于这个价格，乐队在当地以ETH的方向买卖(其他方向也是如此)。可以找到很多满足的方法，但我们需要这样的方法。

where y 0 is a P 0-dependent measure of deposits in the current band，denominated in eth，defi fined in such a way that when current price Pthen if we substities

其中，y0是与P0相关的未来频带贷款的折价手段，以ETH为单位定义，以后价格P、p和po彼此相等时，定义为y=y0、x=0(参见图4中po=p的点)。那么，假设我们交换那一刻的Y。

price is equal to dx0/dy0 which then for a constant-product invariant is:

如果价格与dx0 /dy0相同，则有关恒定产品不变的内容如下：

one can substitute situations where po=por po=pwith x=0 or y=0 correspndingly to verify that the above formulation

我们可以用x=0或y=0代替po=p或PO=P 的情况，证明上述公式是自愿的。

Typically for a band，we know p and，hence，p ，po，constant a，and also x and y(current deposits in the be)it can be found by solving the quadratic equation for the invariant:

一般来说，对于一个乐队，我们知道p，所以我们也知道p，po，常数A，X和Y(乐队目前的贷款)。为了计算剩下的一切，我们必须找到yo。可以通过处置不变的二次方程来找到。

which turns into the quadratic equation against yo:

这将成为yo的二次方程。

In the smart contract，we solve this quadratic equation in get _ y0 function。

在智能合同中，我们在get_y0函数中处理这个二次方程。

while Oracle price po stays constant，the AMM works in a normal way，e . g . sells ETH when going up/Buys ETH when going down . by simply subply

在预测器价格po不变的情况下，AMM在下跌时以ETH销售/下跌时购买ETH等一般方式负有义务。X=0 AC为“当前下降”的价格PCD或y=0 AC为“当前下降”的价格PCU值通过变量方程进行区分，po的当前值和p的当前值以下的AMM价格如下：

another practically important question is:if price changes up or down so slowly that the Oracle price po is fully capable to follow it adiabatic aswhile it ' s not an immediately trivial mathematical problem to solve，nume

另一个主要的实际效果是，价格变化太慢，假设预测器价格po可以绝热地(在一个波段内)跟上，现在给定值X和Y，在p=po的末尾得到几y的ETH(假设价格下降)，这不是可以立即处置的数学效果，但数值计算显示出相当繁杂的答案。

we will use these results when evaluating safety of the loan as well as the potential losses of the AMM。

在评估贷款的成果AMM的潜在损失时，我们将运用这些结果。

now we have a description of one band . we split all the price space into bands which touch other with prices pand pso that if we se

总是我们有一个替身的描述。我们将价格p和P 相互接触的所有价格空间分配给多个乐队。因此，设置基本价格pbase，频带编号N:

it is possible to prove that the solution of eq . 7 and eq . 5 for any band gives:

任何乐队都可以证明官方7和官方5的解法都有可能丢失。

which shows that there are no gaps between the bands .

这表示乐队之间没有空隙。

trades occur while preserving the invariant from eq . 1，However the current price inside the AMM shifts when the price po:it goes up when po googo

买卖发生时，将保留公式1的不变性。但是，如果价格为po，AMM外部的当前价格将发生变化。po下降时会下降，反之亦然(立方系数)。可以在公式8中看到。

lla MMA vs stablecoinstablecoin is a CDP where one borrows stable coin Against a volatile collateral(crypto currency，for example，aggeThe collateral is loaded into llamma in such a price range(such bands)that if price of collateral goes down relatively The eth gets convertely

稳定货币是人们用不稳定的抵押品(如ETH)借入稳定货币的CDP。抵押物加载在LLAMMA的价格范围内(这种波段)，如果抵押物的价格下跌相对缓慢，ETH将转换为足以掩盖CDP封锁的稳定货币。(这可以通过自我清算发生，大致经过外部清算，如果抵押率太接近风险限制，几乎不会被封锁，等待价格反弹。) (阿尔伯特爱因斯坦，抵押物、抵押物、抵押物、抵押物、抵押物、抵押物)

when a user deposits collateral and borrows a stable coin，The lla MMA smart contract calculates The bands where to locate The collateral . when The user already has enough USD to cover the loan . the amount of stable coins which can be obtained can be calculated using a public get _ x _ de It get a health method returns a ratio of get _ x _ down to debt plus the value increase in collateral when the price is well above " liquidation

当用户拿出抵押物借用稳定货币时，LLAMMA智能合同将计算有担保的频带。如果抵押物的价格发生变化，其末端将转换为稳定货币。当系统处于“水下”时，用户有足够的美元支付贷款。可以用通用的get_x_down方法计算得到的稳定肺数。提供过于接近清算阈值的数字，外部清算人可以参与(一般来说，不应在抵押品价格下跌和横盘后几天甚至几周内发生)。甚至，如果抵押品价格从未下降，或者相对快速的公开从未下降，就永远不会发生)。(约翰f肯尼迪，Northern Exposure，《财富》)当价格远高于“清算”时，安康方法是get_x_down和债务的比率加上抵押物的价值。

When a stable coin charges interest，this should be reflected in the AMM，too。this is done by adjusting All the grid of prices . so、When a stable coin charges interest rate r和All the grid of prices in the AMM shifts upwards

稳定货币收取利息时，这应该反映在AMM上。应该反映出来。这是通过所有调整价格的网格来实现的。因此，当稳定货币收到利率R时，AMM的所有价格网格都将与相同的利率R一起向上移动。这是通过基本价格乘数完成的。因此，直接接受的利率是正数，乘数会随着时间的推移而下降。

when we calculate get _ x _ down or get _ y _ up，we are first looking for the amounts of stable coin and collateral xAnd yif current price moves to the current price po。then we look at how much stable coin or collateral we get if po adiabatic ally changes to either the look St price of the lowest band，or the highest price

当我们计算get_x_down或get_y_up时，我们首先要找到的是，如果当前价格移动到当前价格po的稳定货币和抵押品x还有y的数量。然后，如果po单列地更改为最低区间的最低价格或最高区间的最低价格，我们将区分失去几个稳定货币或抵押品。这样，我们可能会失去目前不依赖价格的几个稳定货币的准则，这对抵抗夹层攻击至关重要。* * * *

it is important to point out that the llamma uses po defined as eth/USD price as a price source，And our stable coin could be traded under the pegif PS1、then price in the llamma is p po。

根据需求，LLAMMA使用ETH/USD价格定义的po作为价格根源，我们的稳定货币可以在联动下(PS1)或超额联动下(PS1)停止买卖。如果是Ps1，则LLAMMA的价格为PPO。

In adiabatic approximation，p=po/PS，and all the collateralstablecoin conversion would happen at a higher Oracle price/as

绝热近似法中p=po/ps，所有担保品稳定货币的转换都发生在高预测器价格上。好像预测器价格低，如下所示。

At this price，the amount of stable coins obtained at conversion is higher by factor of 1/PS(if PS 1)。

在这个价格下，转换时获得的稳定货币数在PS1的情况下高于1/ps的系数。

it is less desirable to have PS 1 for prolonged times，and for that we will use the stabilizer(see next)

长时间以来，PS1不太做梦，所以会使用稳定器(见下一章)。

automatic stabilizer and monetary policy自动稳定器和货币政策when example，because of the increased demand for stable coin(PS 1)，There is peg)deposit it to The stable swap pool single-sided in such a way that The final price after this is still no less than 1 . when PS1，The peg keeper is all

Ps1点(例如，随着对稳定货币的需求增加)，稳定货币和可赎回的参考货币或LP令牌之间的不对称贷款由stableswap Curve池形成的固定存储。一旦成为PS1，PegKeeper合同将允许铸造无担保稳定货币，单方面只能取出stableswap胶水，最终价格仍应在1以上。在Ps1中，PegKeeper允许提取(不对称)，可以燃烧稳定肺。

these actions cause price PS to quickly depreciate when it ' s higher than 1 and appreciate if lower than 1 because asymmetric deposits and withdrate

这些行为导致价格PS高于1时的快速升值、低于1时的升值、不对称存款和提款，从而改变了价格。虽然这部分“铸币”没有担保，但稳定货币似乎是由稳定货币池的活动性隐性担保支持的。整个铸币/关机周期在最后似乎是有益的，提供了稳定性。

let ' s denote the amount of stable coin minted to the stabilizer(debt)As dst and the function which calculates necessary amount of redeementts In order to keep reserves not very large，we use the“slow”mechanism of stabilization via varying the borrow r:

让我们用dst表示铸造在稳定器(负债)上的稳定货币数，并在stableswap AMM get_dx中用fdx()表示计算准备稳定货币所需的可偿还USD数的函数。而且，为了保持“储存”不是很大

where h is the change in PS at which the rate r changes by factor of 2(higher PS leads to lower)。the amount of stabilizer debt dst will equilibrate at different value depending on the rate at PS=1 r0。therefore，设置手册(Wecan)be reducing r0 while dst/supply is larger than some target number(for example

其中H是PS的变化，速度R的变化是2倍(PS越高，R越低)。稳定器负债dst的数量根据ps=1 r0的速度，根据不同的值进行平衡。因此，当dst/supply大于特定目的数字(例如5%)时，可以添加r0(例如5%)。

conclusion/summary the presented mechanisms can，hopefully，solve the risk iness of liquidations for stable coin-making and borrowing

希望提出的机制能够处置为了制造稳定货币和贷款手腕而停止清算的危险。此外，稳定器和自动货币政策机制有助于维持价格锚，而无需维持过大的PSM(Peg Stability Module锚稳定性模块)。