AMM

Basic Definitions:

Market: A platform where users can trade SY for PT, or provide liquidity. The market contains a specific number of SY and PT tokens at any given time.

• SY (Standardized Yield Token): Defined separately, SY represents a standardized yield over time. At any given time tt, the market possesses n_sy(t) SY tokens.

• PT (Principal Token): Defined in the Yield Tokenization paper, PT correlates to the SY token. At any given time tt, the market holds n_pt(t) PT tokens.

Asset: The primary asset associated with the SY token.

sy_ExchangeRate(t): This parameter indicates the rate at which one SY token can be exchanged for the underlying asset at a given time t (For example, 1 SY can swap to sy_ExchangeRate(t) asset). The exchange utilizes underlying protocols such as Haedal, SpringSUI, etc

Total Asset: The total equivalent value of the asset that the SY tokens in the market are worth at time t, abbreviated as n_asset(t):

• n_asset(t) = n_sy(t) * sy_ExchangeRate(t)

Expiry and Time Metrics:

• Expiry: The designated expiry time of the PT token.

• timeToExpiry(t): The remaining time at t until the PT token expires, calculated as timeToExpiry(t) = Expiry - t

• yearsToExpiry(t): The time to expiry expressed in years, computed as yearsToExpiry(t) = timeToExpiry(t) / (one year duration).

Market Dynamics:

• Proportion (proportion(t)): Indicates the proportion of PT in the market at time t, calculated as p(t)=n_pt(t) + n_asset(t) .

• Scalar Root: A fixed parameter for each market, adjusting the capital efficiency based on the rate scaler.

• Initial Anchor: Establishes the initial rate anchor to enhance capital efficiency around a certain interest rate.

• Fee Rate Root: A fixed market parameter determining the fees rate in terms of interest rate.

• Rate Scalar (rateScalar(t)rateScalar(t)): A parameter for adjusting the capital efficiency of the exchange rate at time tt, defined as scalarRootyearsToExpiry(t)\frac{scalarRoot}{yearsToExpiry(t)}.

• Rate Anchor (rateAnchor(t)rateAnchor(t)): Adjusts the interest rate around which trading will be most capital efficient.

Exchange Rate:

• Exchange Rate (exchangeRate(t)exchangeRate(t)): The spot exchange rate of the asset in PT, excluding any fees, where exchangeRate(t)=ln⁡(p(t)1−p(t))×rateScaler(t)+rateAnchor(t)exchangeRate(t) = \ln\left(\frac{p(t)}{1-p(t)}\right) \times rateScaler(t) + rateAnchor(t). The value of 1 PT should naturally be less than or equal to the value of 1 asset, meaning exchangeRate(t)≥1exchangeRate(t) \geq 1.

Implied Rate:

• Implied Rate for a Specific Exchange Rate (impliedRateForExchangeRate(exchangeRate0,t)impliedRateForExchangeRate(exchangeRate0, t)): The annual return rate implied by valuing PT at exchangeRate0exchangeRate0 asset, with a capital of 1 asset equaling exchangeRate0exchangeRate0 PT, leading to exchangeRate0exchangeRate0 asset after the duration of timeToExpiry(t)timeToExpiry(t).

• Exchange Rate for an Implied Rate (exchangeRateForImpliedRate(impliedRate0,t)exchangeRateForImpliedRate(impliedRate0, t)): The exchange rate equivalent to an implied rate of impliedRate0impliedRate0 at time tt.

Liquidity Tokens and Providers:

• Liquidity Token: Represents ownership in the market liquidity pool. At time tt, there are L(t)L(t) liquidity tokens available.

• Liquidity Providers: Users who contribute liquidity to the market. At any given time tt, a liquidity provider uu holds lu(t)lu(t) liquidity tokens, where lu(t)=L(t)lu(t) = L(t).