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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2 posts.

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

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…