2026 INFORMS Conference on Financial Engineering and Fintech
November 6–7, 2026 · Stevens Institute of Technology · Hoboken
MXN
USD
THB
today: routing through the vehicle; two legs, two spreads, two fees
what they want: direct conversion
Trading demand is given; fees cover LPs’ expected losses.
Two things matter: return correlation and cross-pair volume.
We characterize when a cost-minimizing fee schedule can deter such entry.
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Not only currencies: any assets traded against cash, e.g. tokenized stocks.
A pool of reserves + a pricing rule \(R_0\cdot R_1=k\). Three things to know:
① Price = slope
S = R₀ / R₁
② Trade walks the curve
price moves → impact
③ LPs shift it out
more depth, same price
buy & hold
AMM LP: concave relative to buy & hold
exchange rate change
holding value one currency rel. to other
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.
Same total capital \(D\): split it across three pools, or pool it.
pool the capital →
each reserve backs both of its pairs
Pool rule: \(a_0^{w_0}\cdot a_1^{w_1}\cdot a_2^{w_2}=\) constant. Weights \(w_i\) = shares of the pool’s value.
each pool: 50–50 weights, capital ⅓ \(D\)
price impact \(\;\dfrac{2\Delta}{\tfrac13 D}=\dfrac{6\Delta}{D}\)
\(\Delta\) = trade size
weights ⅓ each, capital \(D\)
price impact \(\;\dfrac{3\Delta}{D}\)
\[\begin{aligned}\text{Two assets:}\quad&\mathbb E[\text{LP loss}]\simeq\tfrac12\,w_0w_1\,\sigma^{2}\,D\\[10pt]\text{Many assets:}\quad&\mathbb E[\text{LP loss}]\simeq\tfrac12\,\Phi(w)\,D,\qquad \Phi(w)=\sum_{i<j}w_iw_j\,\sigma_{ij}^{2}\end{aligned}\]
\(\sigma_{ij}\) = volatility of the \(i\)–\(j\) exchange rate. The sum includes dollar pairs.
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 \(i,j\)
Result: fee schedules that collect the same total give the same depth and the same total trading cost.
At the optimal \(F\): fees = price impact, and \(C^*(w)=\sqrt{2\Delta\,\Phi(w)\,\Gamma(w)}\).
\(\displaystyle C^*=\min_w C^*(w)=\min_w\sqrt{2\Delta\,\Phi(w)\Gamma(w)}\)
\(\varepsilon_i\): 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.
\(V_i\): 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
\[p_i a_i^*=\frac{\sqrt{\Delta V_i}}{\sigma_{iw^*}}\]
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.
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.
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.
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.
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.
Fee allocation maximizes cross-pair savings while neither dollar pair pays more than under routing.
| Cross pair | Cross-pair cost reduction | Aggregate cost reduction |
|---|---|---|
| EUR/CHF | 75% | 10.0% |
| AUD/JPY | 42% | 1.8% |
| CAD/JPY | 32% | 0.3% |
Aggregate costs include cross-pair trading and direct trading on both dollar pairs.