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Yield Math — the logit-curve AMM

The Yield-Market is the AMM that prices PT against SY (Principal Token vs. yield-bearing token). Pesalo uses a logit curve — the same family Pendle introduced for yield trading. The shape concentrates liquidity around 1:1 (PT and SY are almost interchangeable for small trades) and naturally maps price ↔ implied APY.

The pool state​

A Yield-Market pool tracks two reserves:

  • pt_reserve — PT held by the pool.
  • sy_reserve — SY held by the pool.
  • total_pt — current total supply of PT (PT minted by the Splitter).
  • maturity_ts — Unix timestamp at which PT becomes 1:1 redeemable for SY.

The price function​

The pool's mid-price is computed from the proportion of PT in the pool:

p = pt_reserve / (pt_reserve + sy_reserve)   // ∈ (0, 1)
implied_rate = ln(p / (1 - p)) / years_to_maturity

This is the logit transform: logit(p) = ln(p / (1-p)). As p → 0.5, implied rate → 0. As p → 1, implied rate → +∞ (PT very cheap, AMM happy to sell it). As p → 0, implied rate → -∞.

In practice the pool stays in a narrow band around p ≈ 0.5 + small.

The trade function​

Given the pool wants to maintain a target rate curve, swapping dSY for dPT works out to a fixed-point computation in yield-math:

pub fn swap_sy_for_pt(
    pool: &Pool,
    sy_in: i128,        // amount of SY the user is supplying
    time_to_maturity: i128,
) -> SwapResult {
    // 1. Solve the logit-curve invariant for the new pool state.
    // 2. Return (pt_out, fee_in_sy).
}

The library is pure (no Soroban state) so it's testable in isolation. There's a snapshot suite in contracts/yield-market/test_snapshots/ that pins the exact pt_out for several sy_in × years_left × pool_state combinations.

Why logit specifically​

Three reasons:

  1. Price → APY is just a logit/exp identity. No iterative solve, no Brent's method. The boost UI can show "you'll lock 12.5% APY" in real time.
  2. Liquidity concentrates near 1:1. PT ≈ SY for small swaps, which matches real user behavior: most boosts are sized to small fractions of pool depth.
  3. The curve has natural asymptotes at p = 0 and p = 1. The AMM can never run dry without the implied rate going to infinity, which trader sniping would arbitrage away first.

Fees​

A small swap fee (default 0.3% of input SY, configurable per pool at deploy time) is charged on every swap. Half goes to LPs, half is held as protocol revenue (no governance / payout flow yet — it just sits in the pool).

Open work​

  • Two-sided swap: today PT is bought by selling YT (boost) or sold to redeem SY (unboost). A direct PT/SY swap is implemented but not surfaced in the mobile UI.
  • LP UI: liquidity is currently seeded by a script. The protocol supports add_liquidity / remove_liquidity; an LP screen would let users deposit SY + PT pairs and earn the fee share.