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

MXN

USD

THB

spread + fee

spread + fee

today: routing through the vehicle; 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: return correlation and cross-pair volume.

Fee schedules that deter entry by a dollar-pair competitor

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

1

2

3

What an AMM is

A pool of reserves + a pricing rule \(R_0\cdot R_1=k\). Three things to know:

Arbitrageurs pick off the pool whenever fundamentals move

buy & hold

AMM LP: concave relative to buy & hold

exchange rate change

holding value one currency rel. to other

"Impermanent loss" = adverse selection

Proposition: Adverse selection, priced from returns + volume alone.

Our closed form (Learning from DeFi):

\(\mathbb{E}[\text{IL}] = \dfrac{\sigma^2}{8}\)

The AMM fee must cover it.

FX trading: three structures

Capital multiplexing — the key intuition

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

Relative-price risk determines the loss

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

\(\sigma_{ij}\) = 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:

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

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

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

Next task: find the cost-minimizing capital weights

\(\displaystyle C^*=\min_w C^*(w)=\min_w\sqrt{2\Delta\,\Phi(w)\Gamma(w)}\)

\(\varepsilon_i\): currency \(i\)’s dollar return; \(\varepsilon_0=0\) for the dollar.

\(w_j\): capital weight of currency \(j\). The pool portfolio’s return is \(\displaystyle\sum_j w_j\varepsilon_j\).

Currency relative to the pool

\(\displaystyle \sigma_{iw}=\sqrt{\operatorname{Var}\!\left(\varepsilon_i-\sum_j w_j\varepsilon_j\right)}\)

We can express the minimum trading cost using trade size \(\Delta\), currency volumes, and these pool-relative volatilities.

Optimal capital weights and trading costs

\(V_i\): total volume involving currency \(i\), including cross-pair volume.

\(\sigma_{iw}\): currency \(i\)’s volatility relative to the pool portfolio.   \(\Delta\): representative trade size.

Theorem. Optimal capital weights minimize the volume-weighted sum of relative-return volatilities:

\[C^*=\min_w\sum_{i=0}^{N}\sqrt{\Delta V_i}\,\sigma_{iw}\]

For a minimizing vector \(w^*\) of capital weights, optimal deposits ($) are

\[p_i a_i^*=\frac{\sqrt{\Delta V_i}}{\sigma_{iw^*}}\]

• For three currencies, we can express the weights in closed form

When does pooling beat vehicle routing?

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

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

\(V_i\) = Total volume involving currency \(i\), including cross-pair volume.
\(\rho_{ij}\) = correlation of dollar returns · \(v_{ij}\) = direct cross-pair volume
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:

\[T=\sqrt{V_1V_2}\,(\rho+\beta),\qquad \beta=\frac{v_{12}}{\sqrt{V_1V_2}}\]

\(\rho\): correlation of dollar returns
\(\beta\): cross-pair volume relative to \(\sqrt{V_1V_2}\)

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

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

The value of choosing the weights

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

Symmetric case: same volume and volatility for both non-dollar currencies.

Where the BIS cross pairs sit

All 13 pairs clear \(\rho > -\beta\): an optimally weighted multi-asset pool beats dollar routing.

Dashed and solid lines: symmetric benchmark. Data: BIS Triennial Survey 2025; Refinitiv 2019–24. 

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.

Cross-pair savings (select) without raising dollar-pair costs

Fee allocation maximizes cross-pair savings while neither dollar pair pays more than under routing.

Aggregate costs include cross-pair trading and direct trading on both dollar pairs.

Take home: when multi-asset pools are superior

• Shared deposits can support direct trading without fragmenting liquidity across bilateral markets.
• An expression based on primitives (volatility and volume) determines
  • lowest possible trading costs
  • feasibility of a pool to beat vehicle routing
• The pooling condition uses correlations and cross-pair volumes: positive correlation is good, sufficient cross-pair volume can offset negative correlation risk
• Fee allocation can be used to defend the pool against dedicated dollar-market entry \(\to\) ex ante condition available