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# Price Stability

How arbitrage keeps BD-Stable's price-stable
BD-Stables can always be minted and redeemed from the system for the value of its peg. This allows arbitragers to balance the demand and supply of BD-Stables in the open market. If the market price of a particular BD-Stable is above the price of its peg, then there is an arbitrage opportunity to mint this BD-Stable tokens by placing the value of the peg into the system per BD-Stable and selling the minted BD-Stable for over the peg price in the open market. At all times, in order to mint BD-Stables, a user must place the peg worth of value into the system. The difference is simply what proportion of collateral and BDX makes up that value.
When BD-Stable is in the 100% collateral phase, 100% of the value that is put into the system to mint it is collateral. As the protocol moves into the fractional phase, part of the value that enters into the system during minting becomes BDX (which is then burned from circulation). For example, in a 98% collateral ratio, every BD-Stable minted requires 98% of the value of its peg in collateral and burning 2% of its peg value in BDX. In a 97% collateral ratio, every BD-Stable minted requires 97% of the value of its peg in collateral and burning 3% of its peg value in BDX, and so on. If the market price of BD-Stable is below the price range of its peg, then there is an arbitrage opportunity to redeem BD-Stable tokens by purchasing them cheaply on the open market and redeeming BD-Stable for the worth of its peg's value from the system. At all times, a user is able to redeem BD-Stable for its peg's worth of value from the system. The difference is simply what proportion of the collateral and BDX is returned to the redeemer. When BD-Stable is in the 100% collateral phase, 100% of the value returned from redeeming BD-Stable is collateral. As the protocol moves into the fractional phase, part of the value that leaves the system during redemption becomes BDX (which is minted to give to the redeeming user). For example, in a 98% collateral ratio, every BD-Stable can be redeemed for 98% of collateral and 2% of minted BDX. In a 97% collateral ratio, every BD-Stable can be redeemed for 97% of collateral and 3% of minted BDX. The BD-Stables redemption process is seamless, easy to understand, and economically sound. During the 100% phase, it is trivially simple.
During the fractional-algorithmic phase, as BD-Stables are minted, BDX is burned. As BD-Stables are redeemed, BDX is minted. As long as there is demand for BD-Stables, redeeming it for collateral plus BDX simply initiates minting of a similar amount of BD-Stables into circulation on the other end (which burns a similar amount of BDX). Thus, the BDX token’s value is determined by the demand for BD-Stables. The value that accrues to the BDX market cap is the summation of the non-collateralized value of the BD-Stables market cap. This is the summation of all past and future shaded areas under the curve displayed as follows.
The demand-supply curve illustrates how minting and redeeming BD-Stables keeps the price stabilized (q is quantity, p is price). At CD0 the price of BD-Stable is p0=peg at q0​. If there is more demand for BD-Stables, the curve shifts right to CD1​ and a new price, p1​, for the same quantity q0​. In order to recover the price to the peg, new BD-Stable must be minted until q1 ​is reached and the p0 ​price is recovered. Since market capitalization is calculated as price times quantity, the market cap of BD-Stable at q0 ​is the blue square. The market cap of BD-Stable at q1 ​is the sum of the areas of the blue square and green square. Notice that in this example the new market cap of BD-Stable would have been the same if the quantity did not increase because the increase in demand is simply reflected in the price, p1​. Given an increase in demand, the market cap increases either through an increase in price or increase in quantity (at a stable price). This is clear because the red square and green square have the same area and thus would have added the same amount of value in market cap. Note: the semi-shaded portion in the green square denotes the total value of BDX that would be burned if the new quantity of BD-Stables was generated at a hypothetical collateral ratio of 66%. This is important to visualize because BDX market cap is intrinsically linked to demand for BD-Stables. Lastly, it’s important to note that Blindex is an agnostic protocol. It makes no assumptions about what collateral ratio the market will settle on in the long-term. It could be the case that users simply do not have confidence in a stablecoin with 0% collateral that’s entirely algorithmic. The protocol does not make any assumptions about what that ratio is and instead keeps the ratio at what the market demands for pricing BD-Stable at the value of its peg. It could be the case that the protocol only ever reaches, for example, a 60% collateral ratio and only 40% of the BD-Stable supply is algorithmically stabilized while over half of it is backed by collateral. The protocol only adjusts the collateral ratio as a result of demand for more BD-Stables and changes in BD-Stables price. When the price of BD-Stable falls below its peg, the protocol recollateralizes and increases the ratio until confidence is restored and the price recovers. It will not decollateralize the ratio unless demand for BD-Stables increases again. It could even be possible that BD-Stables become entirely algorithmic but then recollateralizes to a substantial collateral ratio should market conditions demand. We believe this deterministic and reflexive protocol is the most elegant way to measure the market’s confidence in a non-backed stablecoin. Previous algorithmic stablecoin attempts had no collateral within the system on day 1 (and never used collateral in any way). Such previous attempts did not address the lack of market confidence in an algorithmic stablecoin on day 1.
The logic above derives from the original FRAX protocol. Blindex introduces 2 new elements to the protocol: Effective Collateral Ratio and Effective BDX Coverage Ratio which modify the protocol when a shortage of collateral or BDX occurs. The goal of these changes is to provide more just collateral and BDX distribution in a situation when many users simultaneously decide to redeem or buyback.

## Collateral Ratio

The protocol adjusts the collateral ratio during times of BD-Stables expansion and retraction. During times of expansion, the protocol decollateralizes (lowers the ratio) the system so that less collateral and more BDX must be deposited to mint BD-Stables. This lowers the amount of collateral backing all BD-Stables. During times of retraction, the protocol recollateralizes (increases the ratio). This increases the ratio of collateral in the system as a proportion of BD-Stables supply, increasing market confidence in BD-Stables as its backing increases. At genesis, the protocol adjusts the collateral ratio once every hour by a step of .25%. When a particular BD-Stable is above its peg, the function lowers the collateral ratio by one step per hour and when the price of BD-Stable is below its peg, the function increases the collateral ratio by one step per hour. This means that if BD-Stable price is over its peg for the majority of the time through some time frame, then the net movement of the collateral ratio is decreasing. If BD-Stable price is under its peg for the majority of the time, then the collateral ratio is increasing toward 100% on average. In a future protocol update, the price feeds for collateral can be deprecated and the minting process can be moved to an auction-based system to limit reliance on price data and further decentralize the protocol. In such an update, the protocol would run with no price data required for any asset including BD-Stables and BDX. Minting and redemptions would happen through open auction blocks where bidders post the highest/lowest ratio of collateral plus BDX they are willing to mint/redeem BD-Stables for. This auction arrangement would lead to collateral price discovery from within the system itself and not require any price information via oracles. Another possible design instead of auctions could be using PID-controllers to provide arbitrage opportunities for minting and redeeming BD-Stables similar to how a Uniswap trading pair incentivizes pool assets to keep a constant ratio that converges to their open market target price.

## Effective Collateral Ratio

Effective Collateral Ratio was introduced to equalize users chances to withdraw collateral ratio. The formula for Effective Collateral Ratio is:
$efC_r = C_v / BD_s$
where
$C_v$
is collateral value in all collateral pools for a given BD-Stable expressed in underlying fiat currency
$BD_s$
is total supply of this BD-Stable​
Effective Collateral Ratio (efCR) replaces Collateral Ratio (CR) in Buyback and Redemption process when efCR < CR.
In result user will get less collateral (and more BDX) then CR suggests. This approach prevents users leaving the protocol early from getting unfair advantage over those to leave later and could otherwise be left with no collateral.

## Effective BDX Coverage Ratio

BDX is a deflationary token - there will only be 21M BDX tokens minted. In the original FRAX protocol, the amount of FXS rewarded to the user was (1-CR). This won't work with BDX due to its deflationary nature.
In Blindex, every BD-Stable is supplied with a number of BDX tokens (decided by governance). These tokens are released to users in redemption and recollateralization processes.
At any moment we can calculate the total value of BDX need to support BD-Stable collateralization:
$BDX_n = (BD_s *(1- min(CR, efCR)))/BDX_p$
where
$BDX_n$
is BDX value needed to support BD-Stable collateralization
$BD_s$
is total supply of this BD-Stable​
$CR$
is collateral ratio
$efCR$
​ is effective collateral ratio
$BDX_p$
​ is BDX/BD-Stable price
If we need more BDX then is assigned to a particular BD-Stable, Effective BDX Coverage Ratio is equal to:
$efBDXC_r = min(1, BDX_s/BDX_n)$
where
$BDX_s$
is BDX supply which belongs the this BD-Stable
$BDX_n$
is BDX value needed to support BD-Stable collateralization
When there is excessive amount of BDX assigned to the BD-Stable, the Effective BDX Coverage Ratio = 100%.
In result user will get less BDX. This approach prevents users leaving the protocol early from getting unfair advantage over those to leave later and could otherwise be left with no BDX.