波动率曲面的数学理论
The Mathematics of Volatility Surfaces
AI总结：
本文构建了隐含、局部与学习型波动率曲面的统一数学理论，推导了相关恒等式、对冲公式等，形成涵盖多方面的框架及实证方案。
AI中文摘要：
本文发展了隐含波动率曲面、局部波动率曲面与学习型波动率曲面的统一数学理论。总方差 $w_t(k,\tau)=\tau\sigma_t^2(k,\tau)$ 是受正性、日历单调性和蝶式微分不等式约束的无限维状态。我们建立了该无套利集合的拓扑结构与切几何，并证明在主动约束下非退化高斯冲击以趋近于1/2的概率出现。因此，精确不变性要求与无套利流形相切、反射或受限。我们将此静态不变性问题与动态无套利分离，推导了Musiela到期-传输恒等式，并确定了额外的固定合约鞅约束。我们构建了希尔伯特空间动力学，证明了带闭式截断误差的精确模态约化，推导了Karhunen--Loève因子，将投资组合导数识别为vega场，并得到协方差最优对冲 $\alpha^\ast=(H^\ast C H)^{-1}H^\ast C\nu$。局部波动率图表完善了几何：Dupire局部方差是日历与蝶式约束泛函的比值 $a=\partial_\tau w/g[w]$。神经算子通过单纯形与锥头提供无套利通用近似，归一化流映射则添加了易处理的条件密度：我们推导了指数局部波动率流与可逆断价价格单纯形流的精确变量替换公式，同时给出了真实函数空间所需的拟不变性条件。最后， fading-signature场编码了曲面历史并产生自主有限维受控动力学。该成果形成了一个涵盖表示、动力学、无套利、降维、似然学习、模拟与对冲的框架，以及一个可证伪的实证方案。
英文摘要：
This paper develops a unified mathematical theory of implied, local, and learned volatility surfaces. Total variance $w_t(k,τ)=τσ_t^2(k,τ)$ is an infinite-dimensional state constrained by positivity, calendar monotonicity, and the butterfly differential inequality. We establish the topology and tangent geometry of this arbitrage set and prove that a nondegenerate Gaussian shock at an active constraint exits with probability tending to one half. Exact invariance therefore requires tangency, reflection, or confinement to an arbitrage-free manifold. We separate this static invariance problem from dynamic no-arbitrage, derive the Musiela maturity-transport identity, and identify the additional fixed-contract martingale restriction. We formulate Hilbert-space dynamics, prove an exact modal reduction with closed-form truncation error, derive Karhunen--Loève factors, identify the portfolio derivative as a vega field, and obtain the covariance-optimal hedge $α^\ast=(H^\ast C H)^{-1}H^\ast Cν$. The local-volatility chart completes the geometry: Dupire local variance is the ratio $a=\partial_τw/g[w]$ of the calendar and butterfly constraint functionals. Neural operators provide arbitrage-free universal approximation through simplex and cone heads. Normalizing-flow maps then add tractable conditional densities: we derive exact change-of-variables formulae for exponential local-variance flows and invertible stick-breaking price-simplex flows, while stating the quasi-invariance conditions required in genuine function space. Finally, fading-signature fields encode surface history and yield autonomous finite-dimensional controlled dynamics. The result is one framework for representation, dynamics, arbitrage, dimension reduction, likelihood-based learning, simulation, and hedging, together with a falsifiable empirical protocol.
