FX Transactions today and in days past

• 90% of FX transactions pass through a "vehicle" currency even when neither counterparty needs USD
• Krugman (1984); Rey (2001) — a dominant currency emerges endogenously whenever it minimizes transaction costs through thick-market externalities.
• OUR QUESTIONS: can DeFi tools do one better? How to design such tools?

gold &silver

sterling

USD

???

pre-1800

British imperial trade

post-Bretton Woods
"Pax" Americana
1944-

multi-lateral stablecoins

The World of MONEY is in flux

Stablecoin FX infrastructure
is being built

• GENIUS Act (2025) — U.S. framework for payment stablecoins
• Project Mariana (BIS, 2023) — three-currency AMM proof-of-concept for wholesale CBDC FX
• Project Rialto (BIS, 2024–25) — applies the Mariana FX layer to retail cross-border payments
• Major banks launching multi-currency stablecoin products

The World's Dependence on the USD is a Risk and has a Cost

• USD-vehicle status has geopolitical risks
• USD-routing status quo has cost because every non-USD pair pays twice — two spreads, two fees
• May be a burden on smaller firms and smaller economies

Two things are happening at once

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

Can a multi-currency AMM pool beat USD Routing?

Formalize Optimal AMM Design for Multi-Asset Pool

Micro-founded mechanism: AMM attracts volume \(\to\) attracts liquidity \(\to\) lowers per-trade costs

Identify the cross-subsidy

Identify parameter region where the AMM designer prefers vehicle routing but cross-pair traders don't

Scale to many currencies

Identify parameter region where multi-asset pool improves vehicle routing; extend to \(n\) small currencies

1

2

3

What an AMM is

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

Constant Product AMM

Liquidity pool

\(R_0\) (USD)

\(R_1\) (MXN)

liquidity invariant \[R_0\cdot R_1=k\]

marginal price \[S=\frac{R_0}{R_1}\]

pool depth (at market price)

\[D=R_0+S\cdot R_1=2R_0\]

larger pool \(\to\) smaller price impact 

define a trade's price impact:

\[\frac{S^{\text{paid}}-S^{\text{initial}}}{S^{\text{initial}}}\]

\(S^{\text{paid}}\): average rate over the whole trade — the VWAP of "walking" the pool

Only the imbalance matters

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

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

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.

More broadly: DEX liquidity provision

Constant product is not the only curve

Constant product (Uniswap) is the prevalent AMM — but other pricing curves exist.

Decentralized Exchanges for Stablecoins  ·  Huang, Rostova & Song

• LOB venues (CEX) and AMM venues (DEX) coexist
• DEX uses a flatter, low-convexity curve (Curve protocol)
• Theory + transaction-level empirics: venue structure shapes peg deviations

Flatter curves suit same-currency stable pairs (USDT–USDC).

This paper builds on our "Learning from DeFi"

We take this to FX: view passive, competitive V2-style LP as a conservative benchmark for the status quo.

A model of an AMM: the AMM fee that minimizes trading cost (price impact + fee) → a measure of per-period adverse selection, calibrated to equities.

FX Trading: Three Structures

MXN

USD

THB

Status Quo (SQ)

Two bilateral USD pools. MXN/THB route through USD

Three Bilateral Pools (BP)

One pool per pair. No forced routing but fragmented capital and volume

Single multilateral pool (MP)

all three currencies in one pool; direct trading of all pairs

Same approach as single pool: LP break even, cost-minimizing planner

Capital multiplexing — the key intuition

Split the capital and each pair gets a book only ⅓ as deep — price impact rises. Pool it, and every dollar backs all three pairs.

⅓ D

⅓ D

⅓ D

USD/MXN
USD/THB
MXN/THB

capital split 3 ways

pool the capital →

D

every dollar backs all 3 pairs

price impact, split capital:

\[\frac{2\Delta}{D/3}=\frac{6\Delta}{D}\]

price impact, pooled:

\[\frac{2\Delta}{\frac{2}{3}D}=\frac{3\Delta}{D}\]

Multi-asset AMM: the setup behind the intuition

Bilateral pools vs one multilateral pool

① LPs commit liquidity

until fee revenue covers expected IL — this pins down depth D.

② Designer picks fee F

to minimize price impact + fee — depth substitutes out, leaving the cost.

Bilateral pools vs one multilateral pool

① LPs commit liquidity

until fee revenue covers expected IL — this pins down depth D.

② Designer picks fee F

to minimize price impact + fee — depth substitutes out, leaving the cost.

A useful parametrization 

Real FX is not symmetric: dollar pairs dominate volume, cross pairs are (or would be) thinly traded

USD/THB

USD/MXN

MXN/THB

volume \(V\)
volatility \(\sigma\)

volume \(V\)
volatility \(\sigma\)

volume \(v\times V\)
volatility \(s\times\sigma\)

Two "knobs"

\(v\)   cross-pair volume, relative to a vehicle pair

\(s\)   cross-pair volatility, relative to a vehicle pair  

— typically \(v<1\), cross pairs are thin

FX Trading: Three Structures

MXN

USD

THB

Status Quo (SQ)

Two bilateral USD pools. MXN/THB route through USD

Three Bilateral Pools (BP)

One pool per pair. No forced routing but fragmented capital and volume

Single multilateral pool (MP)

all three currencies in one pool; direct trading of all pairs

narket designer considers the volume-weighted costs for all groups

MXN/THB

Future directions and extensions

Many small currencies: which to pool?

• Pool currencies that move together against the dollar
• Higher correlation against the dollar lowers cross-pair volatility
• The "high enough correlation" threshold for when "small collective pool" beats the vehicles routing.
  • This threshold collapses as the number of small currencies grows.

Optimal weights (AMM curve design)

• Equal thirds isn't optimal
• Tilting toward the dollar does better

More broadly: multi-asset AMM

•  letting LPs choose which pools to create -- which assets to place into a single pool

More Multi-Token AMM Maths

Impermanent loss:

\[\text{IL}=\frac{\text{pool value at }T -\text{pool value buy-and-hold at} T}{\text{pool value at start}}\]

\[=\left(\prod_{i=1}^nx_i\right)^{\frac{1}{n+1}}-\frac{1}{n+1}\left(1+\sum_{i=1}^nx_i\right)\]

gross return of currency i relative to numeraire 0: \[x_i:=\frac{S_i(T)}{S_i(0)}\] 

Proposition: For three currencies with \(E[x_i]=1\), \(x_i:=1+\epsilon_i,\),  \(\sigma_{ij}^2:=Var(\epsilon_i-\epsilon_j)\), and \(\Sigma:=\sigma_{01}^2+\sigma_{02}^2+\sigma_{12}^2\)
\[\mathbb{E}[-\text{IL}]=\frac{\Sigma}{18}\]

Multi-token AMM: setup and price impact

liquidity invariant (equal weights \(w_i=\tfrac1n\)) \[\prod_{i} R_i^{\,w_i}=k\]

marginal rate vs numeraire \((S_0\equiv1)\) \[S_i=\frac{R_0}{R_i}\]

pool value (at market prices) \[V=\sum_{i} S_i R_i = n\,R_0\]

3-currency closed form below; for general \(n\), Li, Park, Singh & Veneris (2026) give a numerical algorithm for optimal weights and pool construction.

Lemma (equal thirds). At market-consistent prices each currency holds one third of pool value: \[S_i R_i=\frac{V}{3}\]

Proposition (price impact). A trade \(i\to j\) of size \(\Delta\) hits a sub-pool of value \(\tfrac23 V\), so \[\Pi=\frac{2\Delta}{\tfrac23 V}=\frac{3\Delta}{V}\]

Looks worse than the bilateral \(2\Delta/V\) — but here \(V\) is the whole pool. (next slide)

Large-volume approximation

When period volume is large relative to any single trade:

Optimal fee

\[F^{*}=\frac{\sigma}{2}\sqrt{\frac{\Delta}{V}}\]

Pool depth

\[D^{*}=\frac{4}{\sigma}\sqrt{\Delta V}\]

Per-trade cost

\[c^{*}=\sigma\sqrt{\frac{\Delta}{V}}\]

\(\sigma\) = period return volatility — captures adverse selection, reminiscent of the square-root price-impact law

\(V\) = expected "noise" (balanced) volume  ·  \(\Delta\) = representative order size  ·  \(D\) = pool depth

FX Trading: Three Structures

Status Quo (SQ)

Two USD pools; MXN/THB routes through USD.

Three Bilateral Pools (BP)

One pool per pair; capital fragmented.

One Multilateral Pool (MP)

All three in one pool; direct trading.

Similar analysis as one pool — LPs break even, planner sets the cost-minimizing fee.

Benchmark: every pair is identical: volume \(V\), volatility \(\sigma\). We relax this with  later.

Trading Costs in an optimally calibrated bilateral AMM