5 posts tagged with "mathematical-finance"

A liquidity pool is a portfolio. Its reserve and price updates define an investment strategy: which assets it accumulates, which it releases, and the terms on which it changes its composition.

Multiswap's post-trade elasticity model expresses that strategy with one elasticity parameter, while retaining a rich accounting and safety structure. A natural extension is to give the portfolio an explicit preferred composition. The question is whether that preference can enter through a coherent valuation rule, preserve the post-trade framework, and strengthen the response to scarce inventory.

This article develops one such candidate. Its central result is an exact identity: the composition-sensitive part of portfolio valuation equals reference-priced holdings discounted by relative entropy from the preferred composition. The resulting prices favor acquiring underrepresented assets, retain exact invariance under uniform reserve scaling, and develop stronger elasticity near depletion.

USD.cav needs to stand on its own as a market for dollar-denominated assets. Its role as a connector to other pools matters, but the first question is whether traders receive competitive quotes while the pool retains enough inventory to keep making markets.

The first historical experiment identifies a useful boundary: scale elasticity $e_s=0.995$ is too low for consistently competitive large hub trades in this three-asset comparison. At $e_s=0.997$, Multiswap wins most large quotes, but a recorded DAI price spike draws down almost all of its DAI inventory. Increasing elasticity to $0.999$ improves quotes further and makes that inventory response more extreme.

This article shows the experiment, the charts, and the code behind that conclusion. Each venue starts with the same quantities of every asset and the same prices. Curve uses a star of two-token pools; Multiswap uses one shared reserve portfolio.

Multiswap prices are gradients of a concave projective potential. Post-trade execution therefore produces a nonnegative Bregman divergence. For a closed swap, that divergence accumulates in LP Token backing. For a partial liquidity action, it finances the permitted change in the price surface. For externally owned Surplus, it combines with an explicit value flow across the Reserve boundary.

These are not three unrelated safety arguments. They are special cases of one exact balance law:

Projective-potential change equals Bregman production plus Reserve boundary value flow plus price-surface reset.

This article derives that law from the native Multiswap state and applies it to swaps, liquidity operations, and externally owned Surplus.

Multiswap's Post-Trade Elasticity Model gives every token a reserve, a scale, and a marginal price. This article asks whether the same structure can support two directional claims associated with a Reserve Asset:

• BULL, whose return exceeds the Reserve Asset's return when that return is positive and falls more when it is negative;
• BEAR, whose return has the opposite sign and may have greater than one-for-one short exposure.

The accounting must come first. BULL and BEAR are both positive claims against the pool, so both are credit accounts. They are economically opposite because their prices respond differently, not because their ledger polarities differ. Once that distinction is fixed, double-entry accounting and post-trade value-flow conservation reduce the claim problem to a small system of exact finite equations.

The resulting research model has a general form for any homogeneous elasticity. When price elasticity and scale elasticity both equal one half, the claim solve becomes quadratic. A constant relative-claim elasticity $\kappa$ remains optional: $\kappa=2$ gives a particularly symmetric log-return geometry, while $\kappa>2$ is necessary if both BULL and BEAR must have greater than one-for-one exposure.

Multiswap is a multi-asset exchange built from three native quantities: reserve, scale, and marginal price. The Post-Trade Elasticity Model specifies how those quantities change under an atomic swap, how LP Token liquidity actions extend the same state space, and which internal safety properties follow from the resulting coefficient dynamics.

The model has one central distinction:

• a token can move along a fixed elastic curve because its reserve changes;
• an operation can move the token to a different elastic curve because its coefficient changes.

This distinction unifies the reserve-to-price homomorphism with the complete price law. Swaps keep every Reserve Asset coefficient fixed. Liquidity actions can change the coefficients of nonparticipating Reserve Assets through one common multiplier. The complete transformation remains multiplicative. The reserve ratio describes the fixed-coefficient case, while the coefficient ratio extends the same structure to liquidity actions.

This article develops the complete framework from first principles. It covers atomic $m$-to-$n$ swaps, finite-step divergence, market depth, user-controlled splitting, LP Token liquidity, the exact liquidity-dispersion boundary, gauge-invariant safety, a qualified value-flow entropy, parameter changes, deposit phases, and the current implementation boundary.