GitBook: [#175] No subject

SUMMARY.md

@@ -4,23 +4,23 @@
 
 ## 💱 Trading, Lending and Borrowing
 
-* [Our Design](trading-lending-and-borrowing/design.md)
+* [Our Design](trading-lending-and-borrowing/platform-architecture-that-supports-ecosystem-development.md)
 * [Trading Pool](trading-lending-and-borrowing/trading-pool.md)
 * [Collateral Rebalancing Pool](trading-lending-and-borrowing/collateral-rebalancing-pool.md)
-* [Yield Token Pool](trading-lending-and-borrowing/yield-token-pool.md)
+* [Yield Token Pool](trading-lending-and-borrowing/automated-market-making-designed-for-lending-protocols.md)
 * [Vault](trading-lending-and-borrowing/vault.md)
 
 ## 🌉 Bridge
 
-* [What is Bridge?](bridge/what-is-bridge.md)
-* [How does it work?](bridge/how-does-it-work.md)
+* [What is Bridge?](bridge/platform-architecture-that-supports-ecosystem-development.md)
+* [Understanding the Bridge](bridge/understanding-the-bridge.md)
 * [Using the Bridge](bridge/using-the-bridge.md)
 
 ## Whitepaper
 
-* [Automated Market Making of Trading Pool](whitepaper/trading-pool.md)
-* [Automated Market Making of Yield Token Pool](whitepaper/yield-token-pool.md)
-* [Automated Market Making of Collateral Rebalancing Pool](whitepaper/collateral-rebalancing-pool.md)
+* [Automated Market Making of Trading Pool](whitepaper/automated-market-making-of-alex/README.md)
+* [Automated Market Making of Yield Token Pool](whitepaper/automated-market-making-of-alex/automated-market-making-of-alex.md)
+* [Automated Market Making of Collateral Rebalancing Pool](whitepaper/automated-market-making-of-collateral-rebalancing-pool.md)
 * [Dive Into Collateral Rebalancing Pool!](whitepaper/dive-into-collateral-rebalancing-pool.md)
 
 ## Developers

bridge/understanding-the-bridge.md

@@ -1,6 +1,6 @@
-# How does it work?
+# Understanding the Bridge
 
-As mentioned [before](what-is-bridge.md), ALEX Bridge partners with Wrapped to bring a bridge experience that is both secure and responsible.
+As mentioned [before](platform-architecture-that-supports-ecosystem-development.md), ALEX Bridge partners with Wrapped to bring a bridge experience that is both secure and responsible.
 
 ## Custory of assets and monitoring of reserves
 

trading-lending-and-borrowing/automated-market-making-designed-for-lending-protocols.md

@@ -12,12 +12,12 @@ Here is the general story. Borrowers and lenders enter a loan contract. Specific
 
 In mathematical terms, the interest rate r is calculated as $$p_{t}=\frac{1}{e^{rt}}$$, where $$p_{t}$$ is the spot price of ayToken and the interest rate is assumed to compound. The formula utilises one of the most fundamental principles in asset pricing: an asset's present value is the asset's discounted future value. Thus, in our simplified example, $$t$$= 1 and $$r=\log\frac{1}{0.9}\approx10\%$$.
 
-## Automated Market Making \(AMM\) Protocol
+## Automated Market Making (AMM) Protocol
 
 When designing the AMM protocol, ALEX believes the following:
 
-1. AMMs are mathematically neat and reflect economic demand and supply. For example, price should increase when demand is high or supply is low; 
-2. AMMs are a type of mean, which remains constant during trading activities. This approach is also adopted by popular platforms such as _Uniswap,_ which employ algorithmic means; and 
+1. AMMs are mathematically neat and reflect economic demand and supply. For example, price should increase when demand is high or supply is low;
+2. AMMs are a type of mean, which remains constant during trading activities. This approach is also adopted by popular platforms such as _Uniswap,_ which employ algorithmic means; and
 3. AMMs can be interpreted through the lens of modern finance theory. Doing so enables ALEX to grow and draw comparisons with conventional finance.
 
 After extensive research, our beliefs led us to an AMM first proposed by _YieldSpace_. While we appreciate the mathematical beauty of their derivation, we adapt it in several ways with _ALEX_. For example, we replace a simple interest rate with a compounding interest rate. This change is in line with standard uses in financial pricing and modelling since Black and Scholes. We also develop a new capital efficiency scheme, as explained below.
@@ -32,7 +32,7 @@ where $$x$$, $$y$$, $$t$$ and $$L$$ are, respectively, the balance of "Token", b
 
 Our design depicts an AMM in the of a form of a generalised mean. It makes economic sense because the shape of the curve is decreasing and convex. It incorporates time to maturity $$t$$, which is explicitly built-in to derive ayToken's spot price. We refer readers to our [white paper](https://docs.alexgo.io/whitepaper/automated-market-making-of-alex) for detail.
 
-## Liquidity Providers \(LP\) and Capital Efficiency
+## Liquidity Providers (LP) and Capital Efficiency
 
 LPs deposit both ayToken and Token in a pool to facilitate trading activities. LPs are typically ready to market-make on all possible scenarios of interest rate movements ranging from $$-\infty$$ to $$+\infty$$. However, part of the interest rates curve or movements will never be considered by market participants. On example is the part where the interest rate is negative. Although negative rates can be introduced in the fiat world by central bankers as monetary policy tool, yield farmers in the crypto world are still longing everything to be positive. In ALEX, positive rate refers to spot price of ayToken not exceeding 1 and ayToken reserve is larger than Token.
 
@@ -42,15 +42,14 @@ Inspired by _Uniswap v3_, ALEX employs virtual tokens - part of the assets that
 
 Figure 1 illustrates an example of adopting virtual tokens in the event of positive interest rate. The blue line is the standard AMM. The blue dot marks an equal balance of Token and ayToken of $$y_{v}$$, meaning there is no, or a 0%, interest rate. $$y_{v}$$ is the boundary amount, as any amount lower than it will never be touched by LP to avoid negative rate, which is represented by blue dashed line. Thus, $$y_{v}$$ is virtual token reserve. Effectively, LP is market-making on the red line, which shifts the blue line lower by $$y_{v}$$. When ayToken is depleted as shown by red dashed line, trading activities are suspended.
 
-A numerical example provided in Table 1 shows capital efficiency with respect to various interest rate, assuming $$t$$= 0.5 and $$L$$= 20 for illustration's sake.  When the current interest rate$$r$$= 10%, LPs are required to deposit 95 token and 105 ayToken according to standard AMM. However, if the interest rate is floored at 0%, LP only needs to contribute 5 ayToken, as the rest 100 ayToken would be virtual. This is a decent saving more than 90%.
+A numerical example provided in Table 1 shows capital efficiency with respect to various interest rate, assuming $$t$$= 0.5 and $$L$$= 20 for illustration's sake. When the current interest rate$$r$$= 10%, LPs are required to deposit 95 token and 105 ayToken according to standard AMM. However, if the interest rate is floored at 0%, LP only needs to contribute 5 ayToken, as the rest 100 ayToken would be virtual. This is a decent saving more than 90%.
 
 ![Table 1: Capital Efficiency when Interest Rate is Floored at 0](../.gitbook/assets/cectable3.png)
 
 ## Yield Curve and Yield Farming
 
-By expressing the interest rate as $$p_{t}=\frac{1}{e^{rt}}$$, i.e. $$r=-\frac{1}{t}\log p_{t}$$, we can obtain a series of interest rates from trading pool prices with respect to various maturities, based on which we are able to build a yield curve. The Yield curve is the benchmark tool for modelling risk-free rates in conventional finance. The shape of the curve dictates expectations about future interest rate path, which helps market participants understand market behaviours and trends. Currently we might be able to build a Bitcoin yield curve from Bitcoin futures listed on the Chicago Mercantile Exchange \(CME\). However, not only is the exchange heavily regulated, its trading volume is skewed to the very short dated front end contracts lasting several months only. ALEX aims to offer future contracts up to 1y when the platform goes live. Should markets mature, ALEX may extend to longer tenors.
+By expressing the interest rate as $$p_{t}=\frac{1}{e^{rt}}$$, i.e. $$r=-\frac{1}{t}\log p_{t}$$, we can obtain a series of interest rates from trading pool prices with respect to various maturities, based on which we are able to build a yield curve. The Yield curve is the benchmark tool for modelling risk-free rates in conventional finance. The shape of the curve dictates expectations about future interest rate path, which helps market participants understand market behaviours and trends. Currently we might be able to build a Bitcoin yield curve from Bitcoin futures listed on the Chicago Mercantile Exchange (CME). However, not only is the exchange heavily regulated, its trading volume is skewed to the very short dated front end contracts lasting several months only. ALEX aims to offer future contracts up to 1y when the platform goes live. Should markets mature, ALEX may extend to longer tenors.
 
 Yield farmers can benefit from understanding the yield curve by purchasing ayToken whose tenor corresponds to high interest rates and selling ayToken whose tenor associates with low interest rates. This is a typical “carry" strategy.
 
 Last but not least, based on the development of the yield curve and solid design work of our AMM, ALEX will be able to provide more products. Specifically, ALEX will be able to offer derivatives, including options and structured products, building on and extending a large amount of literatures and applications in conventional finance.
-

trading-lending-and-borrowing/collateral-rebalancing-pool.md

@@ -1,6 +1,6 @@
 # Collateral Rebalancing Pool
 
-Please refer to our [white paper](../whitepaper/collateral-rebalancing-pool.md) for a more rigorous treatment on the subject.
+Please refer to our [white paper](../whitepaper/automated-market-making-of-collateral-rebalancing-pool.md) for a more rigorous treatment on the subject.
 
 Collateral Rebalancing Pool ("CRP") uses [Weighted Equation](https://docs.alexgo.io/protocol/platform-architecture-that-supports-ecosystem-development) and dynamically rebalances between Token and Collateral.
 

trading-lending-and-borrowing/platform-architecture-that-supports-ecosystem-development.md

@@ -26,7 +26,7 @@ As the price of each token changes, arbitrageurs rebalance the pool by making tr
 
 ### Yield Token Equation
 
-Yield Token Equation drives [Yield Token Pool](yield-token-pool.md). It follows [Yield Space](https://yield.is/YieldSpace.pdf) and is designed specifically to facilitate efficient trading between ayToken and Token. Our main contribution is to extend the model to allow for capital efficiency from liquidity provision perspective (inspired by [Uniswap V3](https://uniswap.org/whitepaper-v3.pdf)).
+Yield Token Equation drives [Yield Token Pool](automated-market-making-designed-for-lending-protocols.md). It follows [Yield Space](https://yield.is/YieldSpace.pdf) and is designed specifically to facilitate efficient trading between ayToken and Token. Our main contribution is to extend the model to allow for capital efficiency from liquidity provision perspective (inspired by [Uniswap V3](https://uniswap.org/whitepaper-v3.pdf)).
 
 For example, if a pool is configured to trade between 0% and 10% APY, the capital efficiency can improve to 40x compared to when the yield can trade between $$-\infty$$ and $$+\infty$$.
 

trading-lending-and-borrowing/trading-pool.md

@@ -1,6 +1,6 @@
 # Trading Pool
 
-Please refer to our [white paper](../whitepaper/trading-pool.md) for a more rigorous treatment on the subject.
+Please refer to our [white paper](../whitepaper/automated-market-making-of-alex/) for a more rigorous treatment on the subject.
 
 At ALEX, we build DeFi primitives targeting developers looking to build ecosystem on Bitcoin, enabled by [Stacks](https://www.stacks.co/). As such, we focus on trading, lending and borrowing of crypto assets with Bitcoin as the settlement layer and Stacks as the smart contract layer. At the core of this focus is the automated market making ("AMM") protocol, which allows users to exchange one crypto asset with another trustlessly.
 

whitepaper/automated-market-making-of-alex/README.md

@@ -21,7 +21,7 @@ AMM protocol, which provides liquidity algorithmically, is the core engine of De
 * monotonically decreasing, i.e. $$\frac{dg(x_1)}{dx_1}<0$$. This is because price is often defined as $$-\frac{dg(x_1)}{dx_1}$$. A decreasing function ensures price to be positive.
 * convex, i.e. $$\frac{d^2g(x_1)}{dx_1^2} \geq 0$$. This is equivalent to say that $$-\frac{dg(x_1)}{dx_1}$$ is a non-increasing function of $$x_1$$. It is within the expectation of economic theory of demand and supply, as more reserve of $$x_1$$ means declining price.
 
-Meanwhile, $$f$$ can usually be interpreted as a form of mean, for example, [mStable](https://docs.mstable.org) relates to arithmetic mean, where $$x_1+x_2=L$$ (constant sum formula); one of the most popular platforms [Uniswap](https://uniswap.org/whitepaper-v3.pdf) relates to geometric mean, where $$x_1 x_2=L$$ (constant product formula); [Balancer](https://balancer.fi/whitepaper.pdf), which our [collateral rebalancing pool](collateral-rebalancing-pool.md) employs, applies weighted geometric mean. Its AMM is $$x_1^{w_1} x_2^{w_2}=L$$ where $$w_1$$ and $$w_2$$ are fixed weights. ALEX AMM extends these to create a generalised mean.
+Meanwhile, $$f$$ can usually be interpreted as a form of mean, for example, [mStable](https://docs.mstable.org) relates to arithmetic mean, where $$x_1+x_2=L$$ (constant sum formula); one of the most popular platforms [Uniswap](https://uniswap.org/whitepaper-v3.pdf) relates to geometric mean, where $$x_1 x_2=L$$ (constant product formula); [Balancer](https://balancer.fi/whitepaper.pdf), which our [collateral rebalancing pool](../automated-market-making-of-collateral-rebalancing-pool.md) employs, applies weighted geometric mean. Its AMM is $$x_1^{w_1} x_2^{w_2}=L$$ where $$w_1$$ and $$w_2$$ are fixed weights. ALEX AMM extends these to create a generalised mean.
 
 ### ALEX AMM
 
@@ -40,9 +40,9 @@ $$
 -\frac{dx_2}{dx_1}&=\left(\frac{x_2}{x_1} \right)^{t}\end{split}
 $$
 
-This equation is regarded reasonable as AMM, because (i) function $$g$$ where $$x_2=g(x_1)$$ is monotonically decreasing and convex; and (ii) The boundary value of $$t=0$$ and $$t=1$$ corresponds to constant sum and constant product formula respectively. When $$t$$ decreases from 1 to 0, price $$-\frac{dg(x_1)}{x_1}$$ gradually converges to 1, i.e. the curve converges from constant product to constant sum (see [Appendix 1](trading-pool.md#appendix-1-generalised-mean-when-d-2) for the relevant proofs).
+This equation is regarded reasonable as AMM, because (i) function $$g$$ where $$x_2=g(x_1)$$ is monotonically decreasing and convex; and (ii) The boundary value of $$t=0$$ and $$t=1$$ corresponds to constant sum and constant product formula respectively. When $$t$$ decreases from 1 to 0, price $$-\frac{dg(x_1)}{x_1}$$ gradually converges to 1, i.e. the curve converges from constant product to constant sum (see [Appendix 1](./#appendix-1-generalised-mean-when-d-2) for the relevant proofs).
 
-Though purely theoretical at this stage, [Appendix 2](trading-pool.md#appendix-2-liquidity-mapping-to-uniswap-v3) maps $$L$$ to the liquidity distribution of [Uniswap V3](https://uniswap.org/whitepaper-v3.pdf). This is motivated by an independent research from [Paradigm](https://www.paradigm.xyz/2021/06/uniswap-v3-the-universal-amm/).
+Though purely theoretical at this stage, [Appendix 2](./#appendix-2-liquidity-mapping-to-uniswap-v3) maps $$L$$ to the liquidity distribution of [Uniswap V3](https://uniswap.org/whitepaper-v3.pdf). This is motivated by an independent research from [Paradigm](https://www.paradigm.xyz/2021/06/uniswap-v3-the-universal-amm/).
 
 ## Trading Formulae
 
@@ -164,7 +164,7 @@ $$
 \begin{split} &\frac{dy}{dr}=L^{\frac{1}{1-t}}\frac{e^{-(1-t)r}}{(1+e^{-(1-t)r})^{\frac{2-t}{1-t}}}\\ &L_{\text{Uniswap}}=\frac{2}{t}L^{\frac{1}{1-t}}\left(e^{\frac{r(1-t)}{2}}+e^{\frac{-r(1-t)}{2}}\right)^{\frac{-2+t}{1-t}}\\ &=\frac{2}{t}L^{\frac{1}{1-t}}\big\{2\cosh\left[\frac{r(1-t)}{2}\right]\big\}^{\frac{-2+t}{1-t}} \end{split}
 $$
 
-![Figure 1](<../.gitbook/assets/liquidity (2) (2) (2) (2) (2) (2) (2) (1).png>)
+![Figure 1](<../../.gitbook/assets/liquidity (2) (2) (2) (2) (2) (2) (2) (1).png>)
 
 Figure 1 plots $$L_{\text{Uniswap}}$$ against $$r$$ (which is proportional to $$p$$) regarding various levels of $$t$$. When $$0<t<1$$, $$L_{\text{Uniswap}}$$ is symmetric around 0% at which the maximum reaches . This is because
 

whitepaper/automated-market-making-of-alex/automated-market-making-of-alex.md

@@ -45,7 +45,7 @@ $$
 x_1^p+x_2^p=L
 $$
 
-This equation is regarded reasonable as AMM, because (i) function $$g$$ where $$x_2=g(x_1)$$ is monotonically decreasing and convex; and (ii) The boundary value of $$p=1$$ and $$p=0$$ corresponds to constant sum and constant product formula respectively. When $$p$$ increases from 0 to 1, price $$-\frac{dg(x_1)}{x_1}$$ gradually converges to 1. This is what ALEX hopes to achieve when forward becomes spot. This also means that $$p$$ is somehow related to time to maturity. Please refer to [Appendix 1](yield-token-pool.md#appendix-1-generalised-mean-when-d-2) for a detailed discussion.
+This equation is regarded reasonable as AMM, because (i) function $$g$$ where $$x_2=g(x_1)$$ is monotonically decreasing and convex; and (ii) The boundary value of $$p=1$$ and $$p=0$$ corresponds to constant sum and constant product formula respectively. When $$p$$ increases from 0 to 1, price $$-\frac{dg(x_1)}{x_1}$$ gradually converges to 1. This is what ALEX hopes to achieve when forward becomes spot. This also means that $$p$$ is somehow related to time to maturity. Please refer to [Appendix 1](automated-market-making-of-alex.md#appendix-1-generalised-mean-when-d-2) for a detailed discussion.
 
 In the benchmark research piece by [Yield Space](https://yield.is/YieldSpace.pdf), the invariant function above is formalised from the perspective of zero coupon bond. $$p$$ is replaced by $$1-t$$ where $$t$$ is time to maturity and $$L$$ is a function of $$t$$, so that
 
@@ -79,7 +79,7 @@ ALEX's implied interest rate is compound. Not only does the compound rate allow
 
 Using notations above, the invariant function is rewritten as $$x^{1-t}+y^{1-t}=L$$ with the differential equation $$-\frac{dy}{dx}=\left(\frac{y}{x} \right)^t$$. Unless specified, we assume $$L$$ constant and call it invariant constant. This means that $$t$$ is fixed and there is no minting or burning coins. In practise, liquidity providers can add or reduce liquidity, and $$L$$ needs to be recalibrated daily when $$t$$ changes.
 
-Though purely theoretical at this stage, [Appendix 2](yield-token-pool.md#appendix-2-liquidity-mapping-to-uniswap-v3) maps $$L$$ to the liquidity distribution of [Uniswap V3](https://uniswap.org/whitepaper-v3.pdf). This is motivated by an independent research from [Paradigm](https://www.paradigm.xyz/2021/06/uniswap-v3-the-universal-amm/).
+Though purely theoretical at this stage, [Appendix 2](automated-market-making-of-alex.md#appendix-2-liquidity-mapping-to-uniswap-v3) maps $$L$$ to the liquidity distribution of [Uniswap V3](https://uniswap.org/whitepaper-v3.pdf). This is motivated by an independent research from [Paradigm](https://www.paradigm.xyz/2021/06/uniswap-v3-the-universal-amm/).
 
 ## Trading Formulae
 
@@ -185,7 +185,7 @@ $$
 
 Figure 1 illustrates the example above of $$t$$= 0.5 and $$L$$= 20 by displaying two sets of curves: Invariant Function Curve (“IFC") satisfying $$x^{1-t}+y^{1-t}=L$$ and Capital Efficiency Curve (“CEC") satisfying $$x^{1-t}+(y+y_v)^{1-t}=L$$. Intuitively CEC is attained by lowering IFC by $$y_v$$= 100.
 
-![Figure 1](../.gitbook/assets/cecjing.png)
+![Figure 1](../../.gitbook/assets/cecjing.png)
 
 #### Initialisation
 
@@ -239,7 +239,7 @@ $$
 \begin{split} &x_{v}=\left[\frac{L}{1+e^{(1-t)r_{u}}}\right]^{\frac{1}{1-t}}\\ &y_{v}=\left[\frac{L}{1+e^{-(1-t)r_{l}}}\right]^{\frac{1}{1-t}} \end{split}
 $$
 
-See [Appendix 3](yield-token-pool.md#appendix-3-derivation-of-actual-and-virtual-token-reserve) for a detailed derivation of virtual, as well as actual token reserve.
+See [Appendix 3](automated-market-making-of-alex.md#appendix-3-derivation-of-actual-and-virtual-token-reserve) for a detailed derivation of virtual, as well as actual token reserve.
 
 Similar to the case of 0% floor, minting or burning coins would result in invariant constant changing from $$L$$ to $$k^{1-t}L$$. Meanwhile, both actual and virtual Token and ayToken would grow proportionally by $$k$$, as they are linear function of $$L^{\frac{1}{1-t}}$$.
 
@@ -247,7 +247,7 @@ Similar to the case of 0% floor, minting or burning coins would result in invari
 
 We aim to show here how virtual token is able to assist liquidity providers to efficiently manage capital.
 
-![Figure 2](<../.gitbook/assets/cectable2 (1).png>)
+![Figure 2](<../../.gitbook/assets/cectable2 (1).png>)
 
 In Figure 2, assume lower bound is 0%, whereas upper bound is 50%. We also set $$t$$= 0.5 and $$L$$= 20. If interest rate is 0%, $$L$$= 20 means holding equal amount of Token and ayToken of 100 each $$\left(100^{0.5}+100^{0.5}=20\right)$$. The figure compares actual holding of Token and ayToken with and without cap and floor.
 
@@ -325,7 +325,7 @@ $$
 \begin{split} &\frac{dy}{dr}=L^{\frac{1}{1-t}}\frac{e^{-(1-t)r}}{(1+e^{-(1-t)r})^{\frac{2-t}{1-t}}}\\ &L_{\text{Uniswap}}=\frac{2}{t}L^{\frac{1}{1-t}}\left(e^{\frac{r(1-t)}{2}}+e^{\frac{-r(1-t)}{2}}\right)^{\frac{-2+t}{1-t}}\\ &=\frac{2}{t}L^{\frac{1}{1-t}}\big\{2\cosh\left[\frac{r(1-t)}{2}\right]\big\}^{\frac{-2+t}{1-t}} \end{split}
 $$
 
-![Figure 3](<../.gitbook/assets/liquidity (2) (2) (2) (2) (2) (2) (2) (1).png>)
+![Figure 3](<../../.gitbook/assets/liquidity (2) (2) (2) (2) (2) (2) (2) (1).png>)
 
 Figure 3 plots $$L_{\text{Uniswap}}$$ against interest rate $$r$$ regarding various levels of $$t$$. When $$0<t<1$$, $$L_{\text{Uniswap}}$$ is symmetric around 0% at which the maximum reaches . This is because
 

whitepaper/automated-market-making-of-collateral-rebalancing-pool.md

@@ -2,25 +2,25 @@
 
 ## Abstract
 
-Collateral pools of many DeFi platforms typically comprise a single asset reflecting a borrower’s crypto ownership and risk appetite. While single asset pools benefit from asset appreciation, a pool's value can diminish swiftly in volatile market conditions. The increasing chance of default and potential shortening of the loan term affect both borrowers and lenders who enter fixed-term and fixed-rate contract hoping to remove uncertainties. 
+Collateral pools of many DeFi platforms typically comprise a single asset reflecting a borrower’s crypto ownership and risk appetite. While single asset pools benefit from asset appreciation, a pool's value can diminish swiftly in volatile market conditions. The increasing chance of default and potential shortening of the loan term affect both borrowers and lenders who enter fixed-term and fixed-rate contract hoping to remove uncertainties.
 
-Unlike many DeFi platforms, ALEX uses diversified rather than single collateral pools. Diversified pools consist of a risky asset and risk-free asset. As a result, diversified pools reduce default risk while maintaining potential upside gains. Diversified pools are similar to  portfolios with equity and government bonds, where the former represents the risky asset while the latter represents the risk-free asset. Importantly, diversified collateral pools are systematically managed. An algorithmic engine dynamically adjusts the split of the risky and risk-free asset in the diversified pool, based on Black-Scholes' model.
+Unlike many DeFi platforms, ALEX uses diversified rather than single collateral pools. Diversified pools consist of a risky asset and risk-free asset. As a result, diversified pools reduce default risk while maintaining potential upside gains. Diversified pools are similar to portfolios with equity and government bonds, where the former represents the risky asset while the latter represents the risk-free asset. Importantly, diversified collateral pools are systematically managed. An algorithmic engine dynamically adjusts the split of the risky and risk-free asset in the diversified pool, based on Black-Scholes' model.
 
 ALEX's algorithmic engine and diversified pools minimize the risk of a borrower defaulting. The result is a smoother lending and borrowing experience. Parties have more peace of mind, are interrupted less frequently, and achieve robust returns even in volatile market conditions. This represents a radical improvement over existing alternatives.
 
 ## Introduction
 
-Protocols for loanable funds \(PLF\) enable borrowing and lending activities. Examples PLFs are Compound and Aave on Ethereum. The lender provides a token in need and earns interest in return. The borrower deposits collateral and gets access to a preferred asset. The borrower must pay back the borrowed asset in due time. Protocols enabling this borrowing and lending functionality are incredibly useful. Simply, they enable the present consumption on future earnings. This idea is powerful, and has been at the core of DeFi's rise, including the rise of protocols such as Uniswap.
+Protocols for loanable funds (PLF) enable borrowing and lending activities. Examples PLFs are Compound and Aave on Ethereum. The lender provides a token in need and earns interest in return. The borrower deposits collateral and gets access to a preferred asset. The borrower must pay back the borrowed asset in due time. Protocols enabling this borrowing and lending functionality are incredibly useful. Simply, they enable the present consumption on future earnings. This idea is powerful, and has been at the core of DeFi's rise, including the rise of protocols such as Uniswap.
 
-However, one of the risks posed to market participants in existing PLFs is default risk. While each loan must be secured with collateral, the price of crypto collateral can fluctuate wildly and quickly. As a result, many PLFs ask borrows to significantly overcollateralise their positions. Overcollateralisation refers to value of collateralised assets being higher than the value of the loaned assets. The proportion of the collateral value to loaned value is often called "collateralisation ratio" \(CR\), the inverse of "loan to value" \(LTV\). For simplicity sake, we use the term LTV throughout the paper. The higher the LTV is, the more likely the default occurs. A LTV larger than 1 means the value of collateral cannot cover the value of the loaned asset.
+However, one of the risks posed to market participants in existing PLFs is default risk. While each loan must be secured with collateral, the price of crypto collateral can fluctuate wildly and quickly. As a result, many PLFs ask borrows to significantly overcollateralise their positions. Overcollateralisation refers to value of collateralised assets being higher than the value of the loaned assets. The proportion of the collateral value to loaned value is often called "collateralisation ratio" (CR), the inverse of "loan to value" (LTV). For simplicity sake, we use the term LTV throughout the paper. The higher the LTV is, the more likely the default occurs. A LTV larger than 1 means the value of collateral cannot cover the value of the loaned asset.
 
-In variable rate platforms, such as Aave, collateral in the form of more liquid assets tends to have a higher LTV. In fixed-rate fixed-term protocols, such as YieldSpace, using ETH as collateral to borrow Dai requires a LTV of 67%. In the event that the portfolio is underfunded, three scenarios could emerge in the existing protocols: \(i\) a borrower could top up the collateral asset to stay afloat; \(ii\) a borrower could return some of the borrowing asset to decrease the LTV; and \(iii\) the loan could be unwound by a third party such as liquidator if the borrower defaults. A third party unwinds a borrower's position by paying back the loan, and in return earns certain fees. In cases when a collateral asset is illiquid, fees can be as high as 15% on Aave, representing a significant penalty to defaulting borrowers. This also poses disruption to borrowing/lending activity, as a pre-agreed loan is terminated early.
+In variable rate platforms, such as Aave, collateral in the form of more liquid assets tends to have a higher LTV. In fixed-rate fixed-term protocols, such as YieldSpace, using ETH as collateral to borrow Dai requires a LTV of 67%. In the event that the portfolio is underfunded, three scenarios could emerge in the existing protocols: (i) a borrower could top up the collateral asset to stay afloat; (ii) a borrower could return some of the borrowing asset to decrease the LTV; and (iii) the loan could be unwound by a third party such as liquidator if the borrower defaults. A third party unwinds a borrower's position by paying back the loan, and in return earns certain fees. In cases when a collateral asset is illiquid, fees can be as high as 15% on Aave, representing a significant penalty to defaulting borrowers. This also poses disruption to borrowing/lending activity, as a pre-agreed loan is terminated early.
 
 ALEX abolishes liquidation. ALEX keeps the loan active until maturity, regardless of market condition, solving problems plaguing many existing PLFs. The basis for ALEX's superior solution is based on an innovative combination of asset management and collateral pools. This is how it works.
 
 First, unlike many others, ALEX does not use a static collateral pool with a single asset. Instead, ALEX maintains robust performance in the collateral pool by splitting the deposited asset between a risky and a riskless asset. The collateral pool systematically rebalances the allocation of these two assets based on market conditions. Typically, the better the performance of one asset relative to the other asset, the higher its relative allocation. In mathematical terms, weight is calculated and modified from option delta derived from Black-Scholes model.
 
-So the collateral pool consists of two assets. This opens up new opportunities. For example, ALEX can enable borrowers to gain additional income by engaging in automated market marking \(AMM\). Automated market making helps guarantee a constant proportion of assets in the collateral pool. The AMM takes the form a geometric mean, made popular by [Balancer](https://balancer.fi/whitepaper.pdf). The notion that users _get paid_ is different from much of conventional finance, where portfolio holders are typically required to pay fees to rebalance the portfolio.
+So the collateral pool consists of two assets. This opens up new opportunities. For example, ALEX can enable borrowers to gain additional income by engaging in automated market marking (AMM). Automated market making helps guarantee a constant proportion of assets in the collateral pool. The AMM takes the form a geometric mean, made popular by [Balancer](https://balancer.fi/whitepaper.pdf). The notion that users _get paid_ is different from much of conventional finance, where portfolio holders are typically required to pay fees to rebalance the portfolio.
 
 Importantly, in market downturns, a risky asset may constantly depreciate. In these cases, ALEX's relative allocation of a riskless asset will gradually increase by design. In the event of a loan close to being under-collateralized or defaulting, ALEX converts the remaining portion of the risky asset so that only the riskless asset remains. This ensures no interruption to borrowing/lending activities. Similarly, the agreed rate and maturity remains unaffected. This is different from liquidation. In other platforms, liquidations usually unwind the loan partially, or even fully, resulting in the early termination of a loan. ALEX's design allows for borrowing and lending activity to unfold without the interruptions that plague many of ALEX's alternatives.
 
@@ -30,13 +30,13 @@ Most loanable funds assume a single asset in the collateral pool. While this typ
 
 In conventional finance, whether or not to hold a risky asset depends on investors' risk appetite and their perception of the market environment. A "risk on" market environment entices investors to purchase risky assets and to seek larger returns. In our view, collateral pools made up of single assets are more suitable during "risk on" environments. In these environments, "risk-seeking" borrowers worry less about defaulting and believe risky assets will rally further. This contrasts with "risk-off" environments, when market uncertainty increases. Investors become more risk-averse and tend to hold riskless assets which exhibit small volatility and smaller return.
 
-Many investors would like to profit in both "risk on" and "risk off" periods by holding a diversified portfolio comprised of both risky and riskless assets. A typical example is a portfolio consisting of an S&P 500 index and of US Treasury bonds. Diversification is essential to portfolio management. Diversification ensures portfolios are not overly exposed to one specific asset. Diversifications reduces unsystematic risk. Thus, diversification is a core reason why ALEX creates collateral pools with more than one asset: diversified collateral pools reduce pool volatility while enhancing returns. 
+Many investors would like to profit in both "risk on" and "risk off" periods by holding a diversified portfolio comprised of both risky and riskless assets. A typical example is a portfolio consisting of an S\&P 500 index and of US Treasury bonds. Diversification is essential to portfolio management. Diversification ensures portfolios are not overly exposed to one specific asset. Diversifications reduces unsystematic risk. Thus, diversification is a core reason why ALEX creates collateral pools with more than one asset: diversified collateral pools reduce pool volatility while enhancing returns.
 
 ## AMM: Geometric Mean Market Maker
 
 AMMs are the key drivers behind many DeFi trading platforms. We discuss its general features in our [first white paper](https://docs.alexgo.io/whitepaper/automated-market-making-of-alex). Our AMM design adopts a "generalised mean market maker". This design is powerful because it incorporates time to maturity features while aligning various AMM derivations with traditional financial pricing theory.
 
-In the collateral pool, as weights of the underlying assets change regularly, we employ an AMM which has embedded weights in its expression: the geometric mean market maker \(GMMM\). Each weight of the GMMM corresponds to the proportion of a relevant asset's value to the whole portfolio's value. This is a desirable property for any portfolio manager who sets target weights for a portfolio's assets.
+In the collateral pool, as weights of the underlying assets change regularly, we employ an AMM which has embedded weights in its expression: the geometric mean market maker (GMMM). Each weight of the GMMM corresponds to the proportion of a relevant asset's value to the whole portfolio's value. This is a desirable property for any portfolio manager who sets target weights for a portfolio's assets.
 
 GMMMs were first introduced by Balancer. A GMMM represents an extension to the AMM of the popular AMM platform Uniswap. Uniswap's AMM is a special case of Balancer's GMMM by imposing weights of 50% each on two assets in a given pool.
 
@@ -57,10 +57,7 @@ $$
 Denote the pool value as $$v(t)=x(t)p_{x}(t)+y(t)p_{y}(t)$$. Combining with a no-arbitrage condition, we can show that:
 
 $$
-\begin{split}
-w_{x}(t)&=\frac{x(t)p_{x}(t)}{v(t)}\\
-w_{y}(t)&=\frac{y(t)p_{y}(t)}{v(t)}\\
-\end{split}
+\begin{split} w_{x}(t)&=\frac{x(t)p_{x}(t)}{v(t)}\\ w_{y}(t)&=\frac{y(t)p_{y}(t)}{v(t)}\\ \end{split}
 $$
 
 This means that a pool's weight represents the underlying asset value in proportion to the pool's value.
@@ -90,15 +87,13 @@ $$
 L(t_{i})=x(t_{\tilde{i}})^{w_{x}(t_{i})}\times y(t_{\tilde{i}})^{w_{y}(t_{i})}
 $$
 
-3. Compute token balance to align the price with the market. This involves arbitrageurs:
+1. Compute token balance to align the price with the market. This involves arbitrageurs:
 
 $$
-\begin{split}
-x(t_{i})&=L(t_{i})\left(\frac{w_{x}(t_{i})}{w_{y}(t_{i})}\frac{p_{y}(t_{i})}{p_{x}(t_{i})}\right)^{w_{y}(t_{i})}\\y(t_{i})&=L(t_{i})\left(\frac{w_{y}(t_{i})}{w_{x}(t_{i})}\frac{p_{x}(t_{i})}{p_{y}(t_{i})}\right)^{w_{x}(t_{i})}
-\end{split}
+\begin{split} x(t_{i})&=L(t_{i})\left(\frac{w_{x}(t_{i})}{w_{y}(t_{i})}\frac{p_{y}(t_{i})}{p_{x}(t_{i})}\right)^{w_{y}(t_{i})}\\y(t_{i})&=L(t_{i})\left(\frac{w_{y}(t_{i})}{w_{x}(t_{i})}\frac{p_{x}(t_{i})}{p_{y}(t_{i})}\right)^{w_{x}(t_{i})} \end{split}
 $$
 
-4. Calculate portfolio value:
+1. Calculate portfolio value:
 
 $$
 v(t_{i})=x(t_{i})p_{x}(t_{i})+y(t_{i})p_{y}(t_{i})
@@ -112,7 +107,7 @@ When a loan expires at $$t_{k}=T$$, the remaining balance of $$x(t_{k})$$ and $$
 
 ALEX's rebalancing collateral pool is dynamics. It integrates the concept of asset management with collateral management. By holding and dynamically managing both a risky asset and a riskless asset, several benefits result. When the risky asset depreciates, the collateral pool increasingly holds more of the riskless rather than the risky asset, dynamically reducing the threat of undercollatisation. On the contrary, a higher weight is dynamically assigned to a risky asset when the its price surges, ensuring that the collateral pool captures most of the upside gains. This dynamic is similar to a call option, whose upside is protected whereas its downside is limited. With ALEX, no actual option is involved however, and thus the borrower is not required to pay any expensive option premiums. Options premiums are significant with many cryptoassets because many cryptoassets are very volatile.
 
-In the current version, a collateral pool's allocation mechanism has close ties with option delta. Option delta measures the sensitivity of the option's valuation to the underlying asset price movement. The delta of a call option ranges between 0 and 1 depending on an asset's spot price and the option's strike price, as shown in Figure 1. The higher the spot price, the larger the delta. Therefore, the more weight would be assigned to risky asset. When the strike price is set to be the same as asset spot price \(at-the-money option\), the delta is around 0.5. In ALEX's design, this means holding an equal amount of the risky and of the riskless assets.
+In the current version, a collateral pool's allocation mechanism has close ties with option delta. Option delta measures the sensitivity of the option's valuation to the underlying asset price movement. The delta of a call option ranges between 0 and 1 depending on an asset's spot price and the option's strike price, as shown in Figure 1. The higher the spot price, the larger the delta. Therefore, the more weight would be assigned to risky asset. When the strike price is set to be the same as asset spot price (at-the-money option), the delta is around 0.5. In ALEX's design, this means holding an equal amount of the risky and of the riskless assets.
 
 ![](../.gitbook/assets/calldelta-2.png)
 
@@ -124,7 +119,7 @@ $$
 
 where $$p(t)=\frac{p_{x}(t)}{p_{y}(t)}$$ is the price of asset $$x$$ in terms of asset $$y$$; $$K$$ is the strike price; $$r$$ is the expected return; $$\sigma$$ is implied volatility; $$T$$ is the tenor and $$N(.)$$ is the cumulative distribution of the standard normal distribution.
 
-Ideally, the relative weights of the risky and riskless assets should be updated continuously to reflect spot price movements and delta changes. In practice, we are updating the weights periodically \(e.g. daily\). However as cryptoassets typically exhibit higher volatility than other asset classes, delta changes can be significantly different between time periods. Significant movements tend to imply considerable price deviations from the market after rebalancing. This leads to significant profits for arbitrageurs, but arbitrageurs' gains are often a collateral pool losses. This is similar to impermanent loss. However, while impermanent loss is caused by token balances moving along an AMM curve, here it is caused by weight changes and the effort to preserve price.
+Ideally, the relative weights of the risky and riskless assets should be updated continuously to reflect spot price movements and delta changes. In practice, we are updating the weights periodically (e.g. daily). However as cryptoassets typically exhibit higher volatility than other asset classes, delta changes can be significantly different between time periods. Significant movements tend to imply considerable price deviations from the market after rebalancing. This leads to significant profits for arbitrageurs, but arbitrageurs' gains are often a collateral pool losses. This is similar to impermanent loss. However, while impermanent loss is caused by token balances moving along an AMM curve, here it is caused by weight changes and the effort to preserve price.
 
 ![](../.gitbook/assets/calldeltaspot-2.png)
 
@@ -160,16 +155,16 @@ Most of the contents below are discussed in the main sections. Nonetheless, we l
 
 #### Contract Initialisation
 
-* **Loan-to-Value \(LTV\)**: The ratio of the loan amount to the value of the collateral \("collateralised asset\(s\)"\). For example, if LTV is set to be 80%, a loan amount equivalent to 80 BTC requires 100 BTC as collateral. There is generally no objectively correct LTV ratio; the LTV ratio depends on the quality of the collateralised asset, as well as the market condition when the loan is taken out. **Collateralisation Ratio \(CR\)** is the inverse of LTV.
+* **Loan-to-Value (LTV)**: The ratio of the loan amount to the value of the collateral ("collateralised asset(s)"). For example, if LTV is set to be 80%, a loan amount equivalent to 80 BTC requires 100 BTC as collateral. There is generally no objectively correct LTV ratio; the LTV ratio depends on the quality of the collateralised asset, as well as the market condition when the loan is taken out. **Collateralisation Ratio (CR)** is the inverse of LTV.
 * **Tenor**: The length of time remaining before the loan expires.
-* **Strike Price**: In the Black-Scholes model, strike price refers to the price at which the contract holder can purchase the underlying security when exercising a call option, or sell the underlying security when exercising a put option. In ALEX, the strike price determines the initial split of the risky and the riskless asset. For example, for an at-the-money option, in which strike price is set to be equal to the spot price, there would be an equal split of between two assets in the pool \(i.e. ~50%\).
+* **Strike Price**: In the Black-Scholes model, strike price refers to the price at which the contract holder can purchase the underlying security when exercising a call option, or sell the underlying security when exercising a put option. In ALEX, the strike price determines the initial split of the risky and the riskless asset. For example, for an at-the-money option, in which strike price is set to be equal to the spot price, there would be an equal split of between two assets in the pool (i.e. \~50%).
 * **Implied Volatility**: In the Black-Scholes model, implied volatility is the volatility estimate of an underlying security. A crude approximation of implied volatility is historical volatility. In practise, implied volatility is usually backed out from the observed option price.
 * **Risk-free Interest Rate**: In the Black-Scholes model using risk-neutral valuation, the risk-free interest rate equals the expected return. The risk-free interest rate is usually assumed to be 0%, as future direction of the underlying security is unknown.
 
 #### Pool Rebalancing
 
 * **Rebalancing Frequency**: Theoretically, continuous rebalancing is preferred for price continuity. In practise, ALEX updates the weights periodically to avoid over-calibration.
-* **Smoothing Factor of Exponential Moving Average \(EMA\)**: EMA is an averaging method that places more weight on more recent observations. Assume y\(t\) is the observation value of y at time t, and that {y}\(t\) is the corresponding moving average, where $$\alpha$$ is the smoothing factor. Then:
+* **Smoothing Factor of Exponential Moving Average (EMA)**: EMA is an averaging method that places more weight on more recent observations. Assume y(t) is the observation value of y at time t, and that {y}(t) is the corresponding moving average, where $$\alpha$$ is the smoothing factor. Then:
 
 $$
 \hat{y}(t)=\alpha y(t)+(1-\alpha)\hat{y}(t-1)
@@ -179,4 +174,3 @@ $$
 
 * **Conversion Threshold**: This is the LTV level when the risky asset in the collateral pool is completely converted to the riskless asset to prevent the loan from under-collateralization.
 * **Reserve Premium**: The reserve premium is collected on behalf of a reserve fund. The reserve fund serves as the protocol's last resort. In the extreme event that market turmoil dries up liquidity and ALEX cannot convert all risky assets quickly enough to cover the loan amount, the protocol would cover the difference. The reserve premium is thus another mechanism to guarantee continuity of borrowing and lending activity on ALEX.
-