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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Nov 4, 2026Research

The Binance drain, followed to the last wallet: $6.3B across ten chains, and what it cost to outrun the freeze

by Dipa · 15 min

6.32B USD left Binance on 29 October 2026: 3.36B is still with the attacker, 2.76B is frozen, and outrunning the freeze cost up to 39.6% over market.

Nov 4, 2026Data

DeFi in October 2026: Solana, Ethereum and Base

by Agustin Munoz Gonzalez · 8 min

Every month we pull the same set of numbers for Solana, Ethereum and Base and put them side by side. October was a month where DEX volume across all chains fell 20.9%, Solana lost share for the fourth month in a row, and Ethereum's monthly gain came from the three days after the…

Jun 19, 2026Research

BIT-0011: One Safety Dial, Two Opposing Jobs, No Valid Setting

by Agustin Munoz Gonzalez · 6 min

At POL we study market and mechanism design on-chain, and this post is part of that research. Everything here comes from the spec (PR 27) and an on-chain read of the network. BIT-0011 is an open draft, not a final design. --- I'll get straight to the point, so if you bail here…

Jun 8, 2026Research

What Are PropAMMs, and Why Do They Matter?

by Agustin Munoz Gonzalez · 5 min

At POL we study DeFi market structure, and this post is part of that ongoing research. The mechanisms and numbers are sourced from public data and the teams’ own docs. Given the buzz PropAMMs have picked up, I’m going to borrow a few minutes of your time (I’ll pay you back with…

Jun 8, 2026Research

PropAMMs on Ethereum: Beating the 12-Second Staleness Problem

by Agustin Munoz Gonzalez · 7 min

At POL we study DeFi market structure, and this post is part of our ongoing research on how PropAMMs are arriving on Ethereum via Titan. The mechanisms and numbers are sourced from public data and Titan’s own docs. Last updated: 8 Jun 2026. This is a live, fast-moving topic…

Jun 8, 2026Research

Why Are There So Many Ways to Trade Crypto? A Map of DeFi Venues

by Agustin Munoz Gonzalez · 6 min

The DeFi universe is confusing. When I first started reading about this stuff, I had a hard time sorting out what kinds of platforms even exist. Why would I ever leave the comfort of Binance? This post is the map I wish I’d had back then. It’s current as of today, but like any…

Oct 15, 2025Research

Curve vs Fluid

by Agustin Munoz Gonzalez · 20 min

Decentralized exchanges (DEXes) are the lifeblood of DeFi, but not all DEXes play the same role. Two notable models have emerged in the stablecoin trading arena: Curve - the long-standing stablecoin liquidity pioneer , and Fluid – a newer capital-efficient upstart . Both aim to…

Sep 23, 2025Research

Research

Stochastic Risk Modeling for CDP Defaults

by Dipa · 3 min

Collateralized Debt Positions (CDPs) are the backbone of many DeFi lending and stablecoin systems, such as Sky’s USDS. They allow users to borrow against crypto collateral while keeping their exposure to the underlying asset. To protect solvency, protocols impose…

Sep 16, 2025Research

Research

Fee Mechanisms Beyond EIP-1559

by Dipa · 4 min

When Ethereum adopted EIP-1559, it changed not only the economics of transaction fees but also the way researchers describe them. Instead of talking about “gas auctions,” the mechanism can be understood through three simple building blocks: allocation, payment, and distribution…

Sep 9, 2025Research

Research

Token Supply Regimes & Market Microstructure

by Dipa · 5 min

In Proof-of-Stake (PoS), the mix of cash-flows—burned base fees, validator tips & MEV, and net issuance—determines who gets paid (stakers vs. non-stakers), how volatile those payments are, and how much risk compensation the token demands. We: 1. write an accounting identity that…

Aug 18, 2025Research

When the Pool Fights Back (Part II): Approximating Equilibria in Complex MFGs

by Agustin Munoz Gonzalez · 6 min

Introduction: From Theory Breakdown to Smart Workarounds In \[Part I], When the Pool Fights Back (Part I): Modeling Liquidity with Transaction Costs we introduced a more realistic mean field game (MFG) model for decentralized trading, where agents interact with a liquidity pool…

Aug 18, 2025Research

When the Pool Fights Back (Part I): Modeling Liquidity with Transaction Costs

by Agustin Munoz Gonzalez · 5 min

Exploring how a simple tweak such as adding transaction costs turns a tractable mean field game into a complex dance of strategic interaction.

Aug 18, 2025Research

What if Uniswap Was a Game?

by Agustin Munoz Gonzalez · 7 min

When I first encountered mean field games , I was struck by their elegance: a tool to study strategic behavior among many agents without modeling each one individually. At the time, I was also working on quantitative strategies inside liquidity pools . It didn’t take long to…

Aug 18, 2025Research

Impermanent Loss from a Quantitative Perspective: Static Replication and Hedging via Options

by Agustin Munoz Gonzalez · 7 min

Introduction Automated Market Makers (AMMs) have revolutionized decentralized finance (DeFi), enabling permissionless, on-chain token swaps without relying on traditional order books. Among the most widely adopted designs is the Constant Product Market Maker (CPMM) , the engine…

Jul 15, 2025Research

Orbital: A New Frontier in Stablecoin Liquidity

by Agustin Munoz Gonzalez · 8 min

Introduction In DeFi’s short but explosive history, a few architectural innovations have fundamentally reshaped how we allocate capital. Uniswap made market making permissionless. Curve optimized it for stablecoins, assuming 1:1 pegs. Morpho gave lenders the power to choose which…

Apr 11, 2025Research

Building Agent-Based Models for DeFi: A Practical Bottom-Up Guide

by Agustin Munoz Gonzalez · 10 min

1. Introduction Agent-Based Modeling (ABM) is an increasingly valuable tool for studying complex decentralized systems, particularly in DeFi, where markets are driven by heterogeneous agents and emergent behavior. At POL Finance, ABMs are a core part of the research methodology…

Mar 31, 2025Research

Static Hedging of Impermanent Loss in Constant Product AMMs

by Agustin Munoz Gonzalez · 4 min

What Is Impermanent Loss and Why It Matters In decentralized finance (DeFi), liquidity providers (LPs) are the backbone of automated market makers (AMMs) like Uniswap, Balancer, or Sushi. But being an LP comes with a hidden risk: impermanent loss . Impermanent loss happens when…

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