IntroductionOfPtOracle.md

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Introduction of PT Oracle

In Pendle system, $PT$ can be freely traded from and to $SY$ utilizing our AMM. With the built-in TWAP oracle library, the geometric mean price of $PT$ in terms of SY or asset can be derived from our PendleMarket contracts fully on-chain. Please refer to the StandardizedYield doc for more details of SY & asset

Oracle design

Pendle's oracle implementation is inspired from the idea of UniswapV3 Oracle (see here) with a slight difference in how we define the cumulative rate. In short, our oracle stores the cumulative logarithm of implied APY (the interest rate implied by $PT/asset$ pricing). From the cumulative logarithm of Implied APY, we can calculate the geometric mean of Implied APY, which will be used to derive the mean $PT$ price.

In a way, the Pendle AMM contract has a built-in oracle of interest rate, which can be used to derive $PT$ prices.

Formulas

Our oracle storage is in the following form:

struct Observation {
    // the block timestamp of the observation
    uint32 blockTimestamp;
    // the tick logarithm accumulator, i.e., ln(impliedRate) * time elapsed since the pool was first initialized
    uint216 lnImpliedRateCumulative;
    // whether or not the observation is initialized
    bool initialized;
}

The geometric mean price of $PT$ for the time interval of $[t_0, t_1]$ is:

$$ lnImpliedRate = \frac{lnImpliedRateCumulative_1 - lnImpliedRateCumulative_0}{t_1 - t_0} $$

$$ impliedRate = e^{lnImpliedRate} $$

$$ assetToPtPrice = impliedRate^{\frac{timeToMaturity}{oneYear}} $$

$$ ptToAssetPrice = 1 / assetToPtPrice $$