Dethroning the Dollar?
Multi-Currency AMMs for FX

Katya Malinova, Degroote School of Business, McMaster University
Andreas Park, University of Toronto

OMI-SBS Conference on FinTech and Blockchain Economics 2026

Overview of the Talk

1. Some motivation why you should listen to me
    
2. Principles of "optimal" AMM design
    
3. Optimally design an multi-currency AMM
    
4. Analysis of the multi-currency AMM: is it better than a bilateral one and if so, when?
    
5. Empirical calibration: does it make sense in practice?

Some motivation: why you should listen to me

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

Principles for "optimal" AMM design

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

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.

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"

Closed-form
from returns + volume, no intraday data

Square-root law
optimal fee \(\approx \sigma\sqrt{\frac{\Delta}{V}}\)  — the price-impact law, as adverse-selection compensation

A lower bound
same scale as spreads, ~30% below — prices only adverse selection

AMM as a limit order book
the pricing schedule can be interpreted as a continuous order book

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.

The Fee Pins Down the Depth of the Pool

Break-even condition: fee revenue return = expected IL on LP capital 

Step 1

Pool Designer picks fee \(F\)

A protocol parameter, set at the pool level.

Step 2

LPs commit liquidity until dpeth equates \(F\) to expected IL

Competitive break-even provision \(\to\) equilibrium pool size \(V^*(F)\)

Step 3

Traders face price impact (liquidity driven) and fee

Deeper pool \(\to\) lower PI for given trade

All-in trading cost = price impact + fee. 

The pool designer chooses F to minimize the sum (loose idea: pre-empt entry of competitor).

liquidity is increasing in fee \(\to\) price impact is decreasing in fee \(\to\) interior optimal \(F\)*

*we solve model for exogenous noise volume for simplicity; condition for endogenous volume is that elasticity relative to fee is \(<1\)

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

Large Volume Approximation

When epoch volume is large relative to any single trade we can approximate:

Optimal fee

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

Liquidity/depth
 

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

Per-trade cost

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

\(\sigma=\) epoch-specific return volatility (captures adverse selection risk)

\(Q=\) expected noise volume

\(\Delta=\) representative order size

Same adverse-selection logic as Kyle (1985) and Glosten (1985), in AMM form.

Trading Costs in an optimally calibrated bilateral AMM

now starts the FX paper; this was the level-setting warm-up

Optimally designing an multi-currency AMM

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.

Multi-Token AMM Maths

\(R_n\)

\(R_0\)

\(R_1\)

Capital Multiplexing

The Key Intuition

USD/MXN
1/3

USD/THB
1/3

TBH/MXN
1/3

USD \(\cdot\) THB \(\cdot\) MXN

every dollar backs all 3 pairs

price impact on any pair: \[\frac{2\Delta}{\frac{1}{3}V_{\text{tot}}}=6\cdot \frac{\Delta}{V_{\text{tot}}}\]

price impact on any pair: \[3\cdot \frac{\Delta}{V_{\text{tot}}}\]

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 \(\Sigma:=\sigma_{01}^2+\sigma_{02}^2+\sigma_{12}^2\)
\[\mathbb{E}[-\text{IL}]=\frac{\Sigma}{18}\]

Liquidity Provider Decision

Define \(Q\) as the total expected noise volume in all currencies in terms of the numeraire.

The LP breakeven condition is \[\text{fee}\cdot\text{volume}=\text{initial pool value}\cdot \text{proportional loss}~\Leftrightarrow~f\cdot Q=V_0\cdot\mathbb{E}[-IPL]\]

Pool Designer Decision

Loosely, pool designer wants lowest cost to attract volume
The total cost is price impact plus fee

\[c^\text{multi}(f)=\frac{\Delta\Sigma}{6Q}\frac{1}{f}+f~~~~~~c^\text{pair}(f_{ij})=\frac{\Delta \sigma^2_{ij}}{4Q_{ij}}\frac{1}{f_{if}}+f_{ij}\]

equilibrium price impact \(\approx\) \(2\times f\)

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.

Analysis of the multi-currency AMM: is it better than a bilateral one and if so, when?

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

A useful parametrization for empirical work

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

USD/THB

USD/MXN

MXN/THB

volume \(Q\)
volatility \(\sigma\)

volume \(Q\)
volatility \(\sigma\)

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

Two knobs

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

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

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

An important practical extension: Optimal Weights

weighted pools may yield better outcomes: \(R_0^\alpha R_1^\beta R_2^\beta=k\) with \(\beta=(1-\alpha)/2\)

Proposition: For given \(v,s\) and \(s<2\), the optimal weight for the numeraire is \(\alpha(v,s)=\frac{s}{\sqrt{(1+2v)(4-s^2)}}\).

generally speaking, when optimally weighting the pool, we expand the region of parameters where multi-lateral pools are optimal

​Empirical calibration: does it make sense in practice?

Bilateral vs. MultiLateral Pools for Major Currencies

Disclaimer: these currencies trade against one another, vehicle routing is not the issue here

Vehicle Routing vs. MultiLateral Pools for Small Currencies

based on bilateral trade data from the IMF

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)

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