Pricing Math

The Yield Forge AMM uses a constant product formula with a twist: one side of the pool is virtual. This page explains the math.

The Invariant

$\text{PT Reserve} \times \text{Virtual Quote Reserve} = k$

• PT Reserve — actual PT tokens deposited by LPs
• Virtual Quote Reserve — a calculated value derived from the current price, not real tokens

This enables single-sided liquidity — LPs deposit only PT, and the protocol infers the quote side mathematically.

How Swaps Work

When someone buys PT with quote tokens:

1. Quote tokens enter the pool (increase virtual quote reserve)
2. PT tokens leave the pool (decrease PT reserve)
3. The constant $k$ is maintained

The price impact depends on trade size relative to reserves — larger trades move the price more.

Interest Rate Discovery

The PT price directly implies a fixed interest rate:

$P_{pt} = \frac{V(t)}{(1 + r)^{t}}$

Where $r$ is the implied APY, $t$ is time to maturity in years, and $V(t)$ is the maturity target (LP value per PT unit, stored on-chain as maturityTargetPriceBps).

This makes the Yield Forge market an on-chain interest rate oracle — the trading price reflects the market's consensus on fair yield for that pool.

Dynamic Fees

Swap fees scale dynamically with time elapsed in the cycle — starting at a base rate and increasing as the pool approaches maturity. This compensates LPs for increased impermanent loss risk as PT price converges to the maturity target.