687 lines
26 KiB
HTML
687 lines
26 KiB
HTML
<!DOCTYPE html>
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<html>
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<head>
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
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<title>PMM Algorithm Specification</title>
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<style type="text/css">
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body {
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font-family: Helvetica, arial, sans-serif;
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font-size: 14px;
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line-height: 1.6;
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padding-top: 10px;
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padding-bottom: 10px;
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background-color: white;
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padding: 30px; }
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body > *:first-child {
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margin-top: 0 !important; }
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body > *:last-child {
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margin-bottom: 0 !important; }
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a {
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a.absent {
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color: #cc0000; }
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a.anchor {
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display: block;
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padding-left: 30px;
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margin-left: -30px;
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cursor: pointer;
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position: absolute;
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top: 0;
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left: 0;
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bottom: 0; }
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h1, h2, h3, h4, h5, h6 {
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margin: 20px 0 10px;
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padding: 0;
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font-weight: bold;
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-webkit-font-smoothing: antialiased;
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cursor: text;
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position: relative; }
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h1:hover a.anchor, h2:hover a.anchor, h3:hover a.anchor, h4:hover a.anchor, h5:hover a.anchor, h6:hover a.anchor {
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body > h2:first-child {
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margin-top: 0;
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padding-top: 0; }
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margin-top: 0;
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padding-top: 0; }
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padding-top: 0; }
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li p.first {
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display: inline-block; }
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li {
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margin: 0; }
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ul, ol {
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padding-left: 30px; }
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ul :first-child, ol :first-child {
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margin-top: 0; }
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dl {
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padding: 0; }
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dl dt {
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font-size: 14px;
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font-weight: bold;
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font-style: italic;
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padding: 0;
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margin: 15px 0 5px; }
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dl dt:first-child {
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padding: 0; }
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dl dt > :first-child {
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margin-top: 0; }
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dl dt > :last-child {
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margin-bottom: 0; }
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dl dd {
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margin: 0 0 15px;
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dl dd > :last-child {
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margin-bottom: 0; }
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blockquote {
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margin-bottom: 0; }
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table tr:nth-child(2n) {
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table tr th {
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font-weight: bold;
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border: 1px solid #cccccc;
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margin: 0;
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padding: 6px 13px; }
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table tr td {
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/**
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* prism.js default theme for JavaScript, CSS and HTML
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* Based on dabblet (http://dabblet.com)
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* @author Lea Verou
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*/
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code[class*="language-"],
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pre[class*="language-"] {
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background: #f5f2f0;
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||
top: 0;
|
||
font-size: 100%;
|
||
left: -3.8em;
|
||
width: 3em; /* works for line-numbers below 1000 lines */
|
||
letter-spacing: -1px;
|
||
border-right: 1px solid #999;
|
||
|
||
-webkit-user-select: none;
|
||
-moz-user-select: none;
|
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-ms-user-select: none;
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user-select: none;
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</style>
|
||
|
||
|
||
</head>
|
||
|
||
<body>
|
||
|
||
<h1 id="toc_0">DODO PMM 算法详解</h1>
|
||
|
||
<p>DODO 是新一代由算法驱动的纯链上流动性提供方案。很多人误以为这是对 AMM(Auto Market Maker)算法的改进，事实上 DODO 提出了全新的算法，其运作机制与 AMM 截然不同，我们将其命名为 PMM(Proactive Market Maker)算法。</p>
|
||
|
||
<p>本文将全面对算法进行解析，尤其是其链上实现版本。</p>
|
||
|
||
<h2 id="toc_1">资金池回归模型</h2>
|
||
|
||
<p>PMM 的资金池由四个参数描述</p>
|
||
|
||
<ul>
|
||
<li>\(B_0\) BaseToken 回归目标</li>
|
||
<li>\(Q_0\) QuoteToken 回归目标</li>
|
||
<li>\(B\) BaseToken 余额</li>
|
||
<li>\(Q\) QuoteToken 余额</li>
|
||
</ul>
|
||
|
||
<blockquote>
|
||
<p>以ETH-USDT交易对为例，BaseToken指ETH，QuoteToken指USDT</p>
|
||
</blockquote>
|
||
|
||
<p>价格曲线由下面的公式给出</p>
|
||
|
||
<p>\[P_{margin}=iR\]
|
||
\[if \ B<B_0, \ R=1-k+(\frac{B_0}{B})^2k\]
|
||
\[if \ Q<Q_0, \ R=1/(1-k+(\frac{Q_0}{Q})^2k)\]
|
||
\[else \ R=1\]</p>
|
||
|
||
<p>可以发现PMM有三种工作场景：均衡、BaseToken匮乏、QuoteToken匮乏。</p>
|
||
|
||
<ul>
|
||
<li>没有用户交易时，资金池处于均衡状态，BaseToken和QuoteToken都处于其回归目标</li>
|
||
<li>当用户卖出BaseToken时，资金池BaseToken余额高于回归目标，而QuoteToken余额小于回归目标。此时PMM算法会尝试卖出多余的BaseToken，以使QuoteToken余额回归均衡状态。</li>
|
||
<li>当用户买入BaseToken时，资金池QuoteToken余额高于回归目标，而BaseToken余额小于回归目标。此时PMM算法会尝试卖出多余的QuoteToken，以使BaseToken余额回归均衡状态。</li>
|
||
</ul>
|
||
|
||
<p><img src="/Users/leimingda/Documents/dodo/docs/img/img.006.jpeg" alt=""></p>
|
||
|
||
<p>参数\(R\)在此过程中起到了促进回归的作用，资金池偏离均衡状态越多，\(R\)越偏离1，PMM算法给出的价格便越偏离市场价，吸引套利者帮助资金池回归均衡状态。</p>
|
||
|
||
<p>至此我们介绍完毕了PMM算法的运作机制，下一节将进入合约实现环节。</p>
|
||
|
||
<h2 id="toc_2">交易者</h2>
|
||
|
||
<p>对交易者来说，最重要的就是平均成交价。平均成交价是边际价格\(P_{margin}\)的积分</p>
|
||
|
||
<h3 id="toc_3">单一工作场景下的价格</h3>
|
||
|
||
<h4 id="toc_4">BaseToken匮乏场景</h4>
|
||
|
||
<p><img src="/Users/leimingda/Documents/dodo/docs/img/img.002.jpeg" alt=""></p>
|
||
|
||
<p>\[\Delta Q =\int^{B_2}_{B_1}P_{margin}dB= \int^{B_2}_{B_1}(1-k)i+i(\frac{B_0}{B})^2kdB \]
|
||
\[= i(B_2-B_1)*(1-k+k\frac{B_0^2}{B_1B_2})\]</p>
|
||
|
||
<p>平均成交价为：</p>
|
||
|
||
<p>\[P=\frac{\Delta Q}{B_2-B_1}=i*(1-k+k\frac{B_0^2}{B_1B_2})\]</p>
|
||
|
||
<p>我们发现，平均成交价只与成交前后系统的状态有关，所以买卖两种操作的价格计算方式是一样的——都是对\(P_{margin}\)进行积分。</p>
|
||
|
||
<h4 id="toc_5">QuoteToken匮乏场景</h4>
|
||
|
||
<p>为了简化用户理解，我们只提供sellBaseToken和buyBaseToken两个接口，下面来推导quoteToken匮乏场景下，给出想要买卖的baseToken数量，如何计算价格。</p>
|
||
|
||
<p>\[\Delta B = \frac{1}{i}(Q_2-Q_1)*(1-k+k\frac{Q_0^2}{Q_1Q_2})\]</p>
|
||
|
||
<p>已知\(\Delta B, Q_0, Q_1\)，求解\(Q_2\)，这是一个二次方程，转化为标准形式后：</p>
|
||
|
||
<p>\[(1-k)Q_2^2+(\frac{kQ_0^2}{Q_1}-Q_1+kQ_1-i\Delta B)Q_2-kQ_0^2=0\]</p>
|
||
|
||
<p>\[let \ a=1-k, \ b=\frac{kQ_0^2}{Q_1}-Q_1+kQ_1-i\Delta B, \ c=-kQ_0^2\]</p>
|
||
|
||
<p>因为\(Q_2>=0\)，所以舍掉一个负数根，得到\(Q_2\)的公式为</p>
|
||
|
||
<p>\[Q_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\]</p>
|
||
|
||
<p>在\(\Delta B>0\)时，\(Q_2>Q_1\); 在\(\Delta B<0\)时，\(Q_2<Q_1\); 在\(\Delta B=0\)时，\(Q_2=Q_1\)</p>
|
||
|
||
<h3 id="toc_6">Oracle价格变化的影响</h3>
|
||
|
||
<p>当系统不处于均衡态时，Oracle变化将带来盈利或亏损。举例来讲，假设现在系统处于BaseToken匮乏场景，Oracle价格提高，将导致多余的QuoteToken无法买入足够的BaseToken，提供BaseToken的lp出现亏损。同理如果Oracle价格下降，多余的QuoteToken可以买入更多的BaseToken，使BaseToken余额超过回归目标，提供BaseToken的lp有盈余。</p>
|
||
|
||
<p>为了计算新Oracle价格下的回归目标，我们进行如下推导：</p>
|
||
|
||
<p>\[\Delta Q = i(B_2-B_1)*(1-k+k\frac{B_0^2}{B_1B_2})\]</p>
|
||
|
||
<p>回归时，\(B_2=B_0\)，整理出关于\(B_0\)的二次方程</p>
|
||
|
||
<p>\[\frac{k}{B_1}B_0^2+(1-2k)B_0-[(1-k)B_1+\frac{\Delta Q}{i}] = 0\]</p>
|
||
|
||
<p>舍弃掉负数解，得到\(B_0\)的解为</p>
|
||
|
||
<p>\[B_0=B_1+B_1*\frac{\sqrt{1+\frac{4k\Delta Q}{B_1 i}}-1}{2k}\]</p>
|
||
|
||
<p>这里\(\Delta Q=Q-Q_0\). 可以严格证明，在\(\Delta Q \ge 0\)时，\(B_0\ge B_1\). 这一性质非常重要，因为它保证了BaseToken余额和QuoteToken余额不会同时大于回归目标，或同时小于回归目标。这样一来，PMM只会在文章开头提到的三种状态中转换。</p>
|
||
|
||
<p>同理我们直接给出\(Q_0\)的计算公式</p>
|
||
|
||
<p>\[Q_0=Q_1+Q_1*\frac{\sqrt{1+\frac{4k\Delta B i}{Q_1}}-1}{2k}\]</p>
|
||
|
||
<h3 id="toc_7">跨工作场景时的价格</h3>
|
||
|
||
<p>由于PMM给出的价格曲线是分段的，如果一笔交易跨越了工作场景（比如在BaseToken匮乏时用户卖出大量BaseToken，直接进入QuoteToken匮乏场景），就需要分段计算价格，并累计。</p>
|
||
|
||
<p>这一累计要求精度很高，但计算机有精度损失，所以处理起来要非常小心。合约里提供了三种工作场景下的买卖函数，总计6个。其组合方式是关键。</p>
|
||
|
||
<h2 id="toc_8">流动性提供商</h2>
|
||
|
||
<p>流动性提供商又称为lp(liquidity provider)。lp有两种操作，充值&提现。</p>
|
||
|
||
<p>在BaseToken匮乏场景下充提BaseToken，或在QuoteToken匮乏场景下充提QuoteToken，都会使得改变价格曲线。这要求我们合理地处理充提操作，以使资金池保持健康和公平。</p>
|
||
|
||
<h3 id="toc_9">充值</h3>
|
||
|
||
<p>我们以BaseToken匮乏场景为例，研究充值如何影响lp收益。</p>
|
||
|
||
<p>根据上文推导出的\(B_0\)计算公式</p>
|
||
|
||
<p>\[B_0=B_1+B_1*\frac{\sqrt{1+\frac{4k\Delta Q}{B_1 i}}-1}{2k}\]</p>
|
||
|
||
<p>充值\(b\)后，\(B_1\)增加\(b\)，而\(B_0\)的增量大于\(b\).就说明充值使得所有提供BaseToken的lp都获得了收益。这一收益的来源是，充值使价格曲线变得平缓，同样数量的\(\Delta Q\)可以购买更多BaseToken。</p>
|
||
|
||
<p>在这种情况下，lp一旦充值就会获得收益，我们称之为“充值奖励”，因为充值使得系统更接近均衡状态了。这笔奖励的本质来源是交易者付出的滑点。</p>
|
||
|
||
<p>但“充值奖励”并不是无风险套利机会，请继续阅读下一节。</p>
|
||
|
||
<h3 id="toc_10">提现</h3>
|
||
|
||
<p>与上一节同理，提现\(b\)后，\(B_1\)减少\(b\)，而\(B_0\)减少的量多于\(b\)。说明提现使所有BaseToken的lp都蒙受损失。这一损失的来源是，提现使价格曲线变得陡峭，同样数量的\(\Delta Q\)只能购买更少的BaseToken。</p>
|
||
|
||
<p>PMM算法规定，这种情况下提现需要缴纳一笔“流动性罚金”。罚金数额等于提币后造成的lp损失，这笔罚金将直接充入资金池，分给所有未撤资的lp。</p>
|
||
|
||
<p>回到上一节中的情况，如果lp充值后立刻提现，罚金将大于收益，所以不会有无风险套利的机会。这一设计倾向于未撤资lp，可以最大限度留存资金。</p>
|
||
|
||
<blockquote>
|
||
<p>值得注意的是，不论是充值奖励，还是流动性罚金，都只有在系统偏离均衡状态较远，且充提金额很大的时候才显著。
|
||
普通用户往往感知不到这一收益（损失）的存在。当然也欢迎羊毛党，在系统偏离均衡时充值赚取充值奖励，在系统回归平衡时提现来避免流动性罚金</p>
|
||
</blockquote>
|
||
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MathJax.Hub.Config({
|
||
'showProcessingMessages': false,
|
||
'messageStyle': 'none'
|
||
});
|
||
|
||
if (typeof MathJaxListener !== 'undefined') {
|
||
MathJax.Hub.Register.StartupHook('End', function () {
|
||
MathJaxListener.invokeCallbackForKey_('End');
|
||
});
|
||
}
|
||
|
||
})();
|
||
</script>
|
||
|
||
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
||
|
||
|
||
</body>
|
||
|
||
</html>
|