dXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​Vt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0ρV=sup⁡a{f+b Vx+12σ2Vxx}\rho V=\sup_{a}\{f+b\,V_x+\tfrac12\sigma^2 V_{xx}\}ρV=asup​{f+bVx​+21​σ2Vxx​}−ut−ν Δu+H(x,∇u)=f(x,m)-u_t-\nu\,\Delta u+H(x,\nabla u)=f(x,m)−ut​−νΔu+H(x,∇u)=f(x,m)xy=kxy=kxy=kIL(r)=2r1+r−1\mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1IL(r)=1+r2r​​−1CVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i=x_i\big[(Ax)_i-x^\top Ax\big]x˙i​=xi​[(Ax)i​−x⊤Ax]C=E[e−rT(ST−K)+]C=\mathbb{E}\big[e^{-rT}(S_T-K)^+\big]C=E[e−rT(ST​−K)+]df=(ft+μfx+12σ2fxx)dt+σfx dWtdf=\big(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\big)dt+\sigma f_x\,dW_tdf=(ft​+μfx​+21​σ2fxx​)dt+σfx​dWt​∂tm−∇ ⁣⋅(m ∇pH)=ν Δm\partial_t m-\nabla\!\cdot(m\,\nabla_p H)=\nu\,\Delta m∂t​m−∇⋅(m∇p​H)=νΔmΔ=∂V∂S,Γ=∂2V∂S2\Delta=\frac{\partial V}{\partial S},\quad \Gamma=\frac{\partial^2 V}{\partial S^2}Δ=∂S∂V​,Γ=∂S2∂2V​λ(δ)=A e−κδ\lambda(\delta)=A\,e^{-\kappa\delta}λ(δ)=Ae−κδd1=ln⁡(S/K)+(r+12σ2)TσTd_{1}=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}d1​=σT​ln(S/K)+(r+21​σ2)T​u(x)=sup⁡τ E[e−rτg(Xτ)]u(x)=\sup_{\tau}\,\mathbb{E}\big[e^{-r\tau}g(X_\tau)\big]u(x)=τsup​E[e−rτg(Xτ​)]κ=γ σ2/η\kappa=\sqrt{\gamma\,\sigma^2/\eta}κ=γσ2/η​E[dWt2]=dt\mathbb{E}[dW_t^2]=dtE[dWt2​]=dtJ(π)=E[∑tγtrt]J(\pi)=\mathbb{E}\big[\textstyle\sum_t \gamma^t r_t\big]J(π)=E[∑t​γtrt​]dXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​Vt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0ρV=sup⁡a{f+b Vx+12σ2Vxx}\rho V=\sup_{a}\{f+b\,V_x+\tfrac12\sigma^2 V_{xx}\}ρV=asup​{f+bVx​+21​σ2Vxx​}−ut−ν Δu+H(x,∇u)=f(x,m)-u_t-\nu\,\Delta u+H(x,\nabla u)=f(x,m)−ut​−νΔu+H(x,∇u)=f(x,m)xy=kxy=kxy=kIL(r)=2r1+r−1\mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1IL(r)=1+r2r​​−1CVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i=x_i\big[(Ax)_i-x^\top Ax\big]x˙i​=xi​[(Ax)i​−x⊤Ax]C=E[e−rT(ST−K)+]C=\mathbb{E}\big[e^{-rT}(S_T-K)^+\big]C=E[e−rT(ST​−K)+]df=(ft+μfx+12σ2fxx)dt+σfx dWtdf=\big(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\big)dt+\sigma f_x\,dW_tdf=(ft​+μfx​+21​σ2fxx​)dt+σfx​dWt​∂tm−∇ ⁣⋅(m ∇pH)=ν Δm\partial_t m-\nabla\!\cdot(m\,\nabla_p H)=\nu\,\Delta m∂t​m−∇⋅(m∇p​H)=νΔmΔ=∂V∂S,Γ=∂2V∂S2\Delta=\frac{\partial V}{\partial S},\quad \Gamma=\frac{\partial^2 V}{\partial S^2}Δ=∂S∂V​,Γ=∂S2∂2V​λ(δ)=A e−κδ\lambda(\delta)=A\,e^{-\kappa\delta}λ(δ)=Ae−κδd1=ln⁡(S/K)+(r+12σ2)TσTd_{1}=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}d1​=σT​ln(S/K)+(r+21​σ2)T​u(x)=sup⁡τ E[e−rτg(Xτ)]u(x)=\sup_{\tau}\,\mathbb{E}\big[e^{-r\tau}g(X_\tau)\big]u(x)=τsup​E[e−rτg(Xτ​)]κ=γ σ2/η\kappa=\sqrt{\gamma\,\sigma^2/\eta}κ=γσ2/η​E[dWt2]=dt\mathbb{E}[dW_t^2]=dtE[dWt2​]=dtJ(π)=E[∑tγtrt]J(\pi)=\mathbb{E}\big[\textstyle\sum_t \gamma^t r_t\big]J(π)=E[∑t​γtrt​]dXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​Vt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0ρV=sup⁡a{f+b Vx+12σ2Vxx}\rho V=\sup_{a}\{f+b\,V_x+\tfrac12\sigma^2 V_{xx}\}ρV=asup​{f+bVx​+21​σ2Vxx​}−ut−ν Δu+H(x,∇u)=f(x,m)-u_t-\nu\,\Delta u+H(x,\nabla u)=f(x,m)−ut​−νΔu+H(x,∇u)=f(x,m)xy=kxy=kxy=kIL(r)=2r1+r−1\mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1IL(r)=1+r2r​​−1CVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i=x_i\big[(Ax)_i-x^\top Ax\big]x˙i​=xi​[(Ax)i​−x⊤Ax]C=E[e−rT(ST−K)+]C=\mathbb{E}\big[e^{-rT}(S_T-K)^+\big]C=E[e−rT(ST​−K)+]df=(ft+μfx+12σ2fxx)dt+σfx dWtdf=\big(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\big)dt+\sigma f_x\,dW_tdf=(ft​+μfx​+21​σ2fxx​)dt+σfx​dWt​∂tm−∇ ⁣⋅(m ∇pH)=ν Δm\partial_t m-\nabla\!\cdot(m\,\nabla_p H)=\nu\,\Delta m∂t​m−∇⋅(m∇p​H)=νΔmΔ=∂V∂S,Γ=∂2V∂S2\Delta=\frac{\partial V}{\partial S},\quad \Gamma=\frac{\partial^2 V}{\partial S^2}Δ=∂S∂V​,Γ=∂S2∂2V​λ(δ)=A e−κδ\lambda(\delta)=A\,e^{-\kappa\delta}λ(δ)=Ae−κδd1=ln⁡(S/K)+(r+12σ2)TσTd_{1}=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}d1​=σT​ln(S/K)+(r+21​σ2)T​u(x)=sup⁡τ E[e−rτg(Xτ)]u(x)=\sup_{\tau}\,\mathbb{E}\big[e^{-r\tau}g(X_\tau)\big]u(x)=τsup​E[e−rτg(Xτ​)]κ=γ σ2/η\kappa=\sqrt{\gamma\,\sigma^2/\eta}κ=γσ2/η​E[dWt2]=dt\mathbb{E}[dW_t^2]=dtE[dWt2​]=dtJ(π)=E[∑tγtrt]J(\pi)=\mathbb{E}\big[\textstyle\sum_t \gamma^t r_t\big]J(π)=E[∑t​γtrt​]
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POL Labs

A boutique quantitative research firm for the hardest problems in finance.

Bring us a problem worth solving

Who we are

POL Labs is a boutique quantitative research firm for the hardest problems in finance. We are a group of mathematicians, economists and engineers, and we do what most researchers never do: we take our results out of the papers and run them in live markets.

What holds POL Labs together is the team. We are PhDs from the Universidad de Buenos Aires (UBA), from researchers fresh out of their doctorate to professors with decades in the field: scientists in love with the work, and rigorous about it. We choose our people as carefully as our problems.

Part of POL Finance Group, a holding building financial technology.

What we do

Some of the mathematics we work in

Derivative pricingVt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0

Pricing and hedging options: a listed call, a structured note, an on-chain options vault.

Automated market makingx y=k,IL(r)=2r1+r−1x\,y=k,\qquad \mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1xy=k,IL(r)=1+r2r​​−1

The constant-product pool and the loss an LP takes to price moves, the core of on-chain liquidity.

Tail riskCVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβ

Worst-case loss under stress: capital buffers, collateral haircuts, a lending protocol's solvency.

Optimal controlρ V=sup⁡a{ f+b Vx+12σ2Vxx }\rho\,V=\sup_{a}\big\{\,f+b\,V_x+\tfrac12\sigma^2 V_{xx}\,\big\}ρV=asup​{f+bVx​+21​σ2Vxx​}

Optimal execution and dynamic hedging: working a large order, or rebalancing an LP position over time.

Game theoryx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i = x_i\big[(Ax)_i - x^\top A x\big]x˙i​=xi​[(Ax)i​−x⊤Ax]

Strategic agents in equilibrium: auction and fee-mechanism design, MEV and block-building competition.

Stochastic dynamicsdXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​

Simulating a price, rate or volatility path: an equity, a funding rate, an on-chain oracle feed.

Method

We take ownership of the problem. We work inside it, in constant contact, and we stay until it works.

  1. Formalize. Cast the market as a well-posed model: state variables, dynamics, constraints, objective. No hand-waving.
  2. Solve. Closed form where the model admits it, high-fidelity convergent numerics where it does not. No approximating the problem away.
  3. Stress. Calibrate to data, validate out of sample, then break it against adversarial and tail scenarios over thousands of Monte Carlo paths.
  4. Ship. Production code and a written derivation, reproducible end to end, to the standard of a peer-reviewed paper. Not a deck.

We take on a few problems at a time, and we choose them. We would rather solve one hard thing completely than touch ten halfway.

We choose our problems carefully. Tell us about yours →

Publications

A selection of the group's peer-reviewed and preprint work.

Finance and markets

  • A New Framework for Modelling Liquidity Pools as Mean Field Games · 2024
  • Static Hedging of Impermanent Loss in Constant-Product AMMs · 2024
  • Liquidity Pools as Mean Field Games with Transaction Costs · 2025
  • Complex Markets and Mean Field Games: Beyond Basic Models · 2026
  • Learning, Mean Field Approximations, and Phase Transitions in Auction Models · 2024
  • Kinetic Theory of Active Particles Meets Auction Theory · 2024
  • Competition Level as a Key Parameter in Renewable Energy Auctions · 2021
  • A Game Theoretic Model of Wealth Distribution · 2018

Game theory

  • Evolutionary Game Theory in Mixed Strategies: from Microscopic Interactions to Kinetic Equations · 2020
  • Replicator Dynamics for Continuous Strategies · 2024
  • Random Multi-Player Games · 2022
  • Co-evolution of Viruses and Games · 2026
  • Monty Hall Game: a Host with Limited Budget · 2014

Opinion and social dynamics

  • The Undecided Have the Key: Interaction-Driven Opinion Dynamics · 2015
  • Opinion Formation Models with Heterogeneous Persuasion and Zealotry · 2018
  • Measure-Valued Opinion Dynamics · 2020
  • Modeling Opinion Dynamics: Theoretical Analysis and Continuous Approximation · 2017
  • Role of Voting Intention in Public Opinion Polarization · 2020
  • Schelling–Voter Model: an Application to Language Competition · 2013

Epidemics, control and analysis

  • Optimal Control for a SIR Epidemic Model with Limited Quarantine · 2022
  • SIR Dynamics with Vaccination in a Large Configuration Model · 2021
  • Coupling Epidemiological Models with Social Dynamics · 2021
  • Lower Bounds for Eigenvalues of the One-Dimensional p-Laplacian · 2004
  • Estimates for Eigenvalues of Quasilinear Elliptic Systems · 2006
  • Blow-up for Parabolic and Hyperbolic Problems with Variable Exponents · 2009
  • Lyapunov-Type Inequalities: With Applications to Eigenvalue Problems · 2013

Need the paper or more detail on any of these? Ask us and we will send it.

Careers

We are always hiring exceptional talent.

A problem

Optimal spread under staleness. A proprietary AMM streams preconfirmed quotes, refreshing its price every Δ. Between refreshes, the quote is stale against a reference price that follows

dSt=σ St dWtdS_t=\sigma\,S_t\,dW_tdSt​=σSt​dWt​

Flow arriving in a window is a fraction φ informed, trading in the direction the reference has moved, and the rest is noise. The AMM posts a symmetric half-spread s.

Find the tightest half-spread at which the AMM breaks even against the informed flow, and show how it scales with volatility σ, staleness Δ, and the informed fraction φ.

If you can solve it, send your solution alongside your CV to admin@pol.finance.

POL LabsA research company of POL Finance Group.