# Price Stability

How arbitrage keeps FRAX price-stable

FRAX can always be minted and redeemed from the system for $1 of value. This allows arbitragers to balance the demand and supply of FRAX in the open market. If the market price of FRAX is above the price target of$1, then there is an arbitrage opportunity to mint FRAX tokens by placing $1 of value into the system per FRAX and sell the minted FRAX for over$1 in the open market. At all times in order to mint new FRAX a user must place $1 worth of value into the system. The difference is simply what proportion of collateral and FXS makes up that$1 of value. When FRAX is in the 100% collateral phase, 100% of the value that is put into the system to mint FRAX is collateral. As the protocol moves into the fractional phase, part of the value that enters into the system during minting becomes FXS (which is then burned from circulation). For example, in a 98% collateral ratio, every FRAX minted requires $.98 of collateral and burning$.02 of FXS. In a 97% collateral ratio, every FRAX minted requires $.97 of collateral and burning$.03 of FXS, and so on.

If the market price of FRAX is below the price range of $1, then there is an arbitrage opportunity to redeem FRAX tokens by purchasing cheaply on the open market and redeeming FRAX for$1 of value from the system. At all times, a user is able to redeem FRAX for $1 worth of value from the system. The difference is simply what proportion of the collateral and FXS is returned to the redeemer. When FRAX is in the 100% collateral phase, 100% of the value returned from redeeming FRAX is collateral. As the protocol moves into the fractional phase, part of the value that leaves the system during redemption becomes FXS (which is minted to give to the redeeming user). For example, in a 98% collateral ratio, every FRAX can be redeemed for$.98 of collateral and $.02 of minted FXS. In a 97% collateral ratio, every FRAX can be redeemed for$.97 of collateral and $.03 of minted FXS. The FRAX redemption process is seamless, easy to understand, and economically sound. During the 100% phase, it is trivially simple. During the fractional-algorithmic phase, as FRAX is minted, FXS is burned. As FRAX is redeemed, FXS is minted. As long as there is demand for FRAX, redeeming it for collateral plus FXS simply initiates minting of a similar amount of FRAX into circulation on the other end (which burns a similar amount of FXS). Thus, the FXS token’s value is determined by the demand for FRAX. The value that accrues to the FXS market cap is the summation of the non-collateralized value of FRAX’s market cap. This is the summation of all past and future shaded areas under the curve displayed as follows. The demand-supply curve illustrates how minting and redeeming FRAX keeps the price stabilized ($q$ is quantity, $p$ is price). At $CD_0$ the price of FRAX is at $q_0$. If there is more demand for FRAX, the curve shifts right to $CD_1$ and a new price, $p_1$, for the same quantity $q_0$. In order to recover the price to$1, new FRAX must be minted until $q_1$ is reached and the $p_0$ price is recovered. Since market capitalization is calculated as price times quantity, the market cap of FRAX at $q_0$ is the blue square. The market cap of FRAX at $q_1$ is the sum of the areas of the blue square and green square. Notice that in this example the new market cap of FRAX would have been the same if the quantity did not increase because the increase in demand is simply reflected in the price, $p_1$. Given an increase in demand, 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 FXS shares that would be burned if the new quantity of FRAX was generated at a hypothetical collateral ratio of 66%. This is important to visualize because FXS market cap is intrinsically linked to demand for FRAX.

Lastly, it’s important to note that Frax 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 FRAX at $1. It could be the case that the protocol only ever reaches, for example, a 60% collateral ratio and only 40% of the FRAX 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 FRAX and changes in FRAX price. When the price of FRAX falls below$1, the protocol recollateralizes and increases the ratio until confidence is restored and the price recovers. It will not decollateralize the ratio unless demand for FRAX increases again. It could even be possible that FRAX becomes 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. It should be noted that even USD, which Frax is pegged to, was not a fiat currency until it had global prominence.

# Collateral Ratio

The protocol adjusts the collateral ratio during times of FRAX expansion and retraction. During times of expansion, the protocol decollateralizes (lowers the ratio) the system so that less collateral and more FXS must be deposited to mint FRAX. This lowers the amount of collateral backing all FRAX. During times of retraction, the protocol recollateralizes (increases the ratio). This increases the ratio of collateral in the system as a proportion of FRAX supply, increasing market confidence in FRAX as its backing increases.

At genesis, the protocol adjusts the collateral ratio once every hour by a step of .25%. When FRAX is at or above $1, the function lowers the collateral ratio by one step per hour and when the price of FRAX is below$1, the function increases the collateral ratio by one step per hour. This means that if FRAX price is at or over $1 a majority of the time through some time frame, then the net movement of the collateral ratio is decreasing. If FRAX price is under$1 a 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 FRAX and FXS. Minting and redemptions would happen through open auction blocks where bidders post the highest/lowest ratio of collateral plus FXS they are willing to mint/redeem FRAX 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 FRAX similar to how a Uniswap trading pair incentivizes pool assets to keep a constant ratio that converges to their open market target price.

# PIDController (update)

As of Feburary 2021, the system uses a PIDController to control the collateral ratio according to the change in the growth ratio, defined as such:

$G_r$ is the growth ratio

$Z_i$is the supply of FXS provided as liquidity to a pair on a decentralized AMM (Uniswap, Sushiswap, etc.)

$a_i$to $a_n$are the FXS pairs on the AMMs

$P_z$is the price of FXS

$F$is the total supply of FRAX

At its core, the growth ratio measures how much FXS liquidity there is against the overall supply of FRAX. The reasoning is that the higher the growth ratio, the more FRAX that could be redeemed with less overall percentage change in the FXS supply. If redeemers were to sell their FXS minted from redeemed FRAX, a higher growth ratio would imply less price slippage on FXS and thus less likelihood of any undesireable negative feedback loops.

As the collateral ratio is changed by the change in the growth ratio, a low overall CR implies more preceeding periods of net positive growth ratio change than net negative. This can be caused by periods of sustained positive FXS price increases, redemptions of FRAX that do not affect the FXS price from the newly minted FXS, or more FXS liquidity entering AMMs.

The motivation for the growth ratio is to take in the signal of the market cap of FRAX and FXS, such that a change in the collateral ratio can be supported by current conditions. For example, a situation of \$5 million of FXS liquidity with 50 million outstanding FRAX is much less fragile than one with the same FXS liquidity but 500 million outstanding FRAX.

In the previous model, looking only at the price of FRAX to change the collateral ratio was sufficient for the protocol's bootstrap phase, but the recent growth and current size of the system merits a change in the model to consider the growth ratio to allow for more accurrate feedback. The new system still uses a price band, but only adjusts the collateral ratio up or down when the price of FRAX is outside of the targeted band.