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.

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…

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…