Why now? The SEC opens the door to AMMs

17 Sept 2026

“…trade tokenized NMS stock using innovative permissioned automated market makers and liquidity pools”

Our question: how should a pool with several assets be designed?

Source: SEC press release 2026-90, 17 September 2026

Why does it matter?
A firm in Mexico pays a supplier in Thailand

MXN

USD

THB

spread + fee

spread + fee

today: dollar routing — two legs, two spreads, two fees

what they want: direct conversion

This paper

One pool, many assets: how to design it, and when does it beat routing?

Design: the asset mix and trading fees that minimize total trading costs

Trading demand is given; fees cover LPs’ expected losses.

A necessary and sufficient condition for beating routing

Two things matter: how assets move together, and cross-pair flow.

Fee schedules that deter entry by a dollar-pair competitor

We characterize when a cost-minimizing fee schedule can deter such entry.

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FX Trading: Three Structures

MXN

USD

THB

Routing

All volume in two deep dollar pools. MXN–THB: two legs.

Three separate pools

One pool per pair. Capital and volume are split.

One shared pool

All three in one pool. Every pair trades directly.

In all three: LPs break even; the designer minimizes total trading cost.

What an AMM is

A pool of reserves + a pricing rule: \(a_0\cdot a_1=\) constant. Three things to know:

Only the imbalance matters

A trade ("swap") moves the price along the curve.

• Round trip: the pool ends where it started. No positional gain or loss before fees.
• Only the net imbalance matters, and it maps 1:1 to the period’s return.

Capital multiplexing — the key intuition

Same total capital \(D\): split it across three pools, or pool it.

Relative-price risk determines LP losses

\[\begin{aligned}\text{Two assets:}\quad&\mathbb E[\text{LP loss}]\simeq\tfrac12\,w_0w_1\,\sigma^{2}\,D\\[10pt]\text{Many assets:}\quad&\mathbb E[\text{LP loss}]\simeq\tfrac12\,\Phi(w)\,D,\qquad \Phi(w)=\sum_{i<j}w_iw_j\,\sigma_{ij}^{2}\end{aligned}\]

\(\sigma_{ij}\) = volatility of the \(i\)–\(j\) exchange rate. The sum includes dollar pairs.

• Higher correlation \(\rho_{ij}\) of dollar returns → lower cross-rate volatility \(\sigma_{ij}\), at given individual volatilities.
• → lower expected LP losses per dollar of capital, at given weights.

For given weights: minimizing the total trading cost

LPs break even: fees collected \(F\) = LP losses \(\tfrac12\Phi D\)  →  depth \(D=2F/\Phi\).

Traders pay fees plus price impact:

\[C(F)=\underbrace{F}_{\text{fees}}+\underbrace{\frac{\Gamma\,\Delta}{D}}_{\text{price impact}}=F+\frac{\Delta\,\Phi\,\Gamma}{2F}\]

\(\displaystyle F=\sum_{i<j}f_{ij}V_{ij}\),     \(\displaystyle\Gamma(w)=\sum_{i<j}V_{ij}\Big(\frac{1}{2w_i}+\frac{1}{2w_j}\Big)\)

\(f_{ij}\) = fee rate, \(V_{ij}\) = volume traded on pair \(i,j\)

Result: fee schedules that collect the same total give the same depth and the same total trading cost. Two assets: one fee. Many assets: many schedules.

Optimal fees and trading costs

Higher fees → more depth → less price impact. But traders pay the fees.

At the optimum: total fees collected = total price-impact cost.

Minimized total trading cost, per period:

\[\begin{aligned}\text{Single pair:}\quad&C^*=\sigma\sqrt{\Delta\,Q}\qquad\big(\text{per dollar traded: }\sigma\sqrt{\Delta/Q}\,\big)\\[10pt]\text{Many assets:}\quad&C^*(w)=\sqrt{2\,\Delta\,\Phi(w)\,\Gamma(w)}\end{aligned}\]

\(\sigma\) = volatility · \(Q\) = volume · \(\Delta\) = representative trade size
\(\Phi\): weighted sum of all pairs’ exchange-rate variances
\(\Gamma\): volume-weighted sum of all pairs’ price-impact coefficients

Characterizing the optimal pool

For each asset mix \(w\) (each asset’s share of the pool’s value), first find the cost-minimizing fees.

Two assets

Optimized cost is the same for any weights.

One pair: one optimal fee.

Optimizing the fee completes the cost minimization.

Many assets

Optimized cost \(C^*(w)=\sqrt{2\,\Delta\,\Phi(w)\,\Gamma(w)}\) depends on the weights.

For given weights, many fee schedules reach the same minimum.

Choose the weights: minimize \(\Phi(w)\,\Gamma(w)\) → lowest total trading cost.

Choose the fee schedule: who pays, and whether entry can be deterred.

The general condition

Theorem: a multi-asset pool can be designed to achieve lower total trading costs than dollar routing if and only if \(T > 0\).

\[T=\underbrace{\sum_{i<j}\sqrt{Q_iQ_j}\,\rho_{ij}}_{\text{how currencies move together}}\;+\;\underbrace{\sum_{i<j}q_{ij}}_{\text{cross-pair volume}}\]

\(Q_i\) = total volume that currency \(i\)'s dollar market serves under routing
\(\rho_{ij}\) = correlation of dollar returns · \(q_{ij}\) = direct cross-pair demand
Sums over non-dollar currencies.

• This existence condition depends only on correlations and volumes.

Three currencies: correlation and cross-pair flow

With two non-dollar currencies, the theorem’s \(T\) becomes:

\[T=\sqrt{Q_1Q_2}\,(\rho+\beta),\qquad \beta=\frac{q_{12}}{\sqrt{Q_1Q_2}}\]

\(\rho\): correlation of dollar returns
\(\beta\): cross-pair flow intensity; with equal volumes, the cross-pair share of demand

A pool can be designed to beat routing if and only if \(\rho>-\beta\).

More cross-pair demand increases the savings from direct trading. These savings can outweigh the extra LP compensation that negative correlation requires.

A concrete design: \(ρ = 0.5, β = 0.25\)

Capital weights

USD: 44.7%
MXN: 27.6%
THB: 27.6%

Total trading cost relative to routing

Routing: 100
Equal thirds (BIS Mariana): 93.5
Optimal weights: 92.6

The optimized pool lowers total trading costs by 7.4%.

Equal thirds already saves 6.5%. Stronger correlation or a larger cross-pair share lowers the optimal dollar weight.

Symmetric illustration: both non-dollar currencies have the same total demand and volatility.

The value of choosing the weights

Above: equal thirds (BIS Project Mariana’s design) beats routing.

Above: the optimally weighted pool beats routing: \(\rho > -\beta\).

Blue band, the design margin: equal thirds loses to routing; well-chosen weights win.

With no cross-pair flow (\(\beta = 0\)): equal thirds needs \(\rho > 0.5\); optimal weights need only \(\rho > 0\).

Symmetric case: same demand and volatility.

Dollar-invoicing proxy: small cross-pair flow intensity

3,164 pairs, 92 currencies

About 70% pass
30% fail ρ > −β.

Failures: near-zero intensity, slightly negative correlation.

Pass = some cheaper design exists. Not measured savings.

\(\beta=q_{12}/\sqrt{Q_1Q_2}\). The test also holds for unequal volumes and volatilities. Data: Refinitiv 2019–24; IMF DOTS 2019–23 (goods trade).

Deterring dollar-pair entry: general conditions

A dollar–peso entrant attracts only direct dollar–peso demand. To deter it:

1. Room on every dollar pair: the pool’s price impact alone does not exceed the entrant’s best total cost. The gap caps that pair’s fee.
2. Enough fee revenue: within these limits, the pool can raise the fees needed to compensate LPs at the cost-minimizing depth.

Dollar-pair fees are capped by their respective gaps. Each cross-pair fee is capped by the sum of its two dollar-pair fees, because traders can route inside the pool.

Result: a cost-minimizing fee schedule that deters dollar-pair entry exists if and only if conditions 1 and 2 hold. In general, \(T>0\) guarantees neither.

A fee schedule that deters entry

Result: in the symmetric three-currency case, for \(-\beta<\rho\), the cost-minimizing pool can be supported by fees that deter both dollar-pair entrants.

One such fee schedule:

USD–MXN: \(f\)   ·   USD–THB: \(f\)   ·   MXN–THB: \(2f\)

• \(f\) is set so that total fees collected equal half the minimized total trading cost.
• Total trading cost stays at its minimum.
• Cross-pair traders pay the same fee directly as over two dollar legs.

Which currencies should trade together?

Optimized pool cost from primitives

\[K(S,\widehat Q)=\sqrt{\Delta}\min_{\substack{\omega_i\geq0\\\sum_i\omega_i=1}}\sum_{i\in S}\sqrt{\widehat Q_i}\,\sqrt{\operatorname{Var}\!\left(\varepsilon_i-\sum_{j\in S}\omega_j\varepsilon_j\right)}.\]

Demand and trade routes determine the volume \(\widehat Q_i\) served by each currency reserve.

Return covariances determine risk relative to the reference return.

The formula optimizes over pool weights and fee revenue. Add optimized pool costs to compare arrangements.

Demand and return covariances (i.e., model primitives) determine which pools are cheapest.

Reserve: Which currencies should trade together?   1/4

One example: four currencies

USD, EUR, CHF and JPY

Each pair has demand \(q\), except CHF–JPY, which has zero demand.

\[\varepsilon_{\mathrm{USD}}=0,\quad\varepsilon_{\mathrm{EUR}}=X,\quad\varepsilon_{\mathrm{CHF}}=X+Y,\quad\varepsilon_{\mathrm{JPY}}=\eta X+Z.\]

The shocks are independent, with

\[\operatorname{Var}(X)=\sigma^2,\qquad\operatorname{Var}(Y)=\operatorname{Var}(Z)=\sigma^2/25.\]

EUR and CHF move closely together.

At \(\eta=0\), JPY moves independently. At \(\eta=1\), it shares their common shock.

Reserve: Which currencies should trade together?   2/4

Should JPY join the pool?

All reserves and fees are optimized.

Costs in units of \(\sigma\sqrt{\Delta q}\).

Independent JPY: keep it separate. Save 34.15% relative to USD routing.

Correlated JPY: include it. Save 50.93%.

Winners among the five partitions with USD in every pool. Three are shown.

Reserve: Which currencies should trade together?   3/4

Can another vehicle currency help?

Keep \(\eta=0\). Compare bilateral arrangements.

\[\{\mathrm{USD,EUR}\},\qquad\{\mathrm{EUR,CHF}\},\qquad\{\mathrm{USD,JPY}\}.\]

EUR serves USD–CHF trades. USD serves EUR–JPY trades.

Different currency pairs can benefit from different vehicles.
The mixed arrangement saves 33.54% relative to USD routing (the {USD, EUR, CHF}; {USD, JPY} configuration is still better!).

Costs in units of \(\sigma\sqrt{\Delta q}\).