text: import{_ as a,c as l,o as n,d as t,a as s,b as e}from"./app.143f2636.js";var i="/imgs/ohm/image-20211230223018386.png";const As='{"title":"FLOAT CAPITAL (Alpha Whitepaper)","description":"","frontmatter":{},"headers":[{"level":2,"title":"\u6458\u8981","slug":"\u6458\u8981"},{"level":2,"title":"1 \u6982\u8FF0","slug":"_1-\u6982\u8FF0"},{"level":3,"title":"1.1 Alpha\u53D1\u5E03","slug":"_1-1-alpha\u53D1\u5E03"},{"level":3,"title":"1.2 \u767D\u76AE\u4E66\u7ED3\u6784","slug":"_1-2-\u767D\u76AE\u4E66\u7ED3\u6784"},{"level":2,"title":"2 \u5408\u6210\u8D44\u4EA7","slug":"_2-\u5408\u6210\u8D44\u4EA7"},{"level":3,"title":"2.1 \u6458\u8981","slug":"_2-1-\u6458\u8981"},{"level":3,"title":"2.2 \u52A8\u673A","slug":"_2-2-\u52A8\u673A"},{"level":3,"title":"2.3 \u65B9\u6CD5","slug":"_2-3-\u65B9\u6CD5"},{"level":2,"title":"3 \u6536\u76CA\u4E0E\u6FC0\u52B1","slug":"_3-\u6536\u76CA\u4E0E\u6FC0\u52B1"},{"level":3,"title":"3.1 \u6458\u8981","slug":"_3-1-\u6458\u8981"},{"level":3,"title":"3.2 \u65B9\u6CD5","slug":"_3-2-\u65B9\u6CD5"},{"level":3,"title":"3.3 \u6536\u76CA\u5206\u79BB","slug":"_3-3-\u6536\u76CA\u5206\u79BB"},{"level":2,"title":"4 \u5408\u6210\u8D44\u4EA7\u4EF7\u683C\u884C\u4E3A","slug":"_4-\u5408\u6210\u8D44\u4EA7\u4EF7\u683C\u884C\u4E3A"},{"level":3,"title":"4.1 \u6458\u8981","slug":"_4-1-\u6458\u8981"},{"level":3,"title":"4.2 \u655E\u53E3\u57FA\u7840\u4F8B\u5B50","slug":"_4-2-\u655E\u53E3\u57FA\u7840\u4F8B\u5B50"},{"level":3,"title":"4.3 Mathematical Formulation of Floating Point Exposure","slug":"_4-3-mathematical-formulation-of-floating-point-exposure"},{"level":3,"title":"4.4 Keeper","slug":"_4-4-keeper"},{"level":2,"title":"5 \u8D28\u62BC","slug":"_5-\u8D28\u62BC"},{"level":3,"title":"5.1 \u6458\u8981","slug":"_5-1-\u6458\u8981"},{"level":3,"title":"5.2 \u57FA\u7840","slug":"_5-2-\u57FA\u7840"},{"level":3,"title":"5.3 \u539F\u7406","slug":"_5-3-\u539F\u7406"},{"level":3,"title":"5.4 Theory adjusted to Solidity code","slug":"_5-4-theory-adjusted-to-solidity-code"},{"level":2,"title":"6 ALPHAFLT TOKENS","slug":"_6-alphaflt-tokens"},{"level":3,"title":"6.1 Summary","slug":"_6-1-summary"},{"level":3,"title":"6.2 Motivation","slug":"_6-2-motivation"},{"level":2,"title":"7 OTHER","slug":"_7-other"},{"level":3,"title":"7.1 Float Capital Alpha Audit","slug":"_7-1-float-capital-alpha-audit"},{"level":3,"title":"7.2 GEMS","slug":"_7-2-gems"},{"level":3,"title":"7.3 Discounts","slug":"_7-3-discounts"},{"level":3,"title":"7.4 Shifting","slug":"_7-4-shifting"},{"level":3,"title":"7.5 Governance","slug":"_7-5-governance"},{"level":3,"title":"7.6 Alpha Live performance","slug":"_7-6-alpha-live-performance"},{"level":3,"title":"7.7 Float 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& Ep_{a}=p_{I} \\\\ \\Rightarrow & E=p_{I}\\left(p_{a}\\right)^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"strut",style:{height:"1.462054em"}}),s("span",{class:"strut bottom",style:{height:"2.4241080000000004em","vertical-align":"-0.9620540000000004em"}}),s("span",{class:"base displaystyle textstyle uncramped"},[s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist"},[s("span",{style:{top:"-0.6220539999999999em"}},[s("span",{class:"fontsize-ensurer reset-size5 size5"},[s("span",{style:{"font-size":"0em"}},"\u200B")]),s("span",{class:"mord displaystyle textstyle uncramped"})]),s("span",{style:{top:"0.6020540000000003em"}},[s("span",{class:"fontsize-ensurer reset-size5 size5"},[s("span",{style:{"font-size":"0em"}},"\u200B")]),s("span",{class:"mord displaystyle textstyle 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We can expand the percentage changes in order to get the following definition of exposure:",-1),c=e(" \u{1D438} = \\frac{\\Delta I}{I} \\frac{a}{\\Delta a} "),d=s("p",null,"where \u{1D43C} is the value of the investment and a is the value of the property of the underlying asset. In this definition we assume that both values are elements of the real numbers.",-1),u=s("h4",{id:"_4-3-2-market-rebalance-equation",tabindex:"-1"},[e("4.3.2 Market rebalance equation. "),s("a",{class:"header-anchor",href:"#_4-3-2-market-rebalance-equation","aria-hidden":"true",target:"_blank",rel:"noopener"},"#")],-1),y=s("p",null,"Suppose we have a market whose long & short synthetic tokens track the value of some property of some underlying asset, a. Suppose also that the total value that has been invested in the long side of the market is l, and the total value in the short side is s. Consider the case where there is a change in the price of the underlying asset, \u{1D6FF}a. If the change is positive then value needs to shift from the long side to the short side, and vice versa when the change is negative. The value that is shifted between the long and short sides needs to be such that the maximum exposure for both sides is 100%. The following update equation is the one that Float uses in order to achieve this:",-1),z=s("p",null,[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",null,[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"\u0394"),s("mi",null,"l"),s("mo",null,"="),s("mi",null,"m"),s("mi",null,"i"),s("mi",null,"n"),s("mo",null,"("),s("mi",null,"l"),s("mo",{separator:"true"},","),s("mi",null,"s"),s("mo",null,")"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"\u0394"),s("mi",null,"a")]),s("mrow",null,[s("mi",null,"a")])])]),s("annotation",{encoding:"application/x-tex"},"\\Delta l = min (l, s) \\frac{\\Delta a}{a} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"strut",style:{height:"1.36033em"}}),s("span",{class:"strut bottom",style:{height:"2.04633em","vertical-align":"-0.686em"}}),s("span",{class:"base displaystyle textstyle uncramped"},[s("span",{class:"mord mathrm"},"\u0394"),s("span",{class:"mord mathit",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mrel"},"="),s("span",{class:"mord mathit"},"m"),s("span",{class:"mord mathit"},"i"),s("span",{class:"mord mathit"},"n"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathit",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mpunct"},","),s("span",{class:"mord mathit"},"s"),s("span",{class:"mclose"},")"),s("span",{class:"mord reset-textstyle displaystyle textstyle uncramped"},[s("span",{class:"sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist"},[s("span",{style:{top:"0.686em"}},[s("span",{class:"fontsize-ensurer reset-size5 size5"},[s("span",{style:{"font-size":"0em"}},"\u200B")]),s("span",{class:"reset-textstyle textstyle cramped"},[s("span",{class:"mord textstyle cramped"},[s("span",{class:"mord mathit"},"a")])])]),s("span",{style:{top:"-0.22999999999999998em"}},[s("span",{class:"fontsize-ensurer reset-size5 size5"},[s("span",{style:{"font-size":"0em"}},"\u200B")]),s("span",{class:"reset-textstyle textstyle uncramped frac-line"})]),s("span",{style:{top:"-0.677em"}},[s("span",{class:"fontsize-ensurer reset-size5 size5"},[s("span",{style:{"font-size":"0em"}},"\u200B")]),s("span",{class:"reset-textstyle textstyle uncramped"},[s("span",{class:"mord textstyle uncramped"},[s("span",{class:"mord mathrm"},"\u0394"),s("span",{class:"mord mathit"},"a")])])]),s("span",{class:"baseline-fix"},[s("span",{class:"fontsize-ensurer reset-size5 size5"},[s("span",{style:{"font-size":"0em"}},"\u200B")]),e("\u200B")])])]),s("span",{class:"sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"})])])])])])],-1),h=e(" \\Delta s = \u2212min (l, s) \\frac{\\Delta a}{a} "),f=s("p",null,"You can see that the exposure is always less than 100% by plugging the above values into the exposure equation (1):",-1),x=e(" \u{1D438}_l = \\frac{\\Delta l}{l} \\frac{a}{\\Delta a} \u{1D438}_l =\\frac{min (l, s)}{l} \\leq 1 \u{1D438}_s =\\frac{\\Lambda s}{s} \\frac{a}{\\Lambda a} \u{1D438}_s =\\frac{\u2212min(l, s)}{l} \\leq 1 "),g=s("h4",{id:"_4-3-3-differential-equations",tabindex:"-1"},[e("4.3.3 Differential equations. 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