🤖 AI Summary
This work establishes a unified theoretical framework for implied, local, and learned volatility surfaces under strict no-arbitrage constraints, enabling dynamic modeling and learning. By introducing an infinite-dimensional state-space geometric structure for the volatility surface, it disentangles static no-arbitrage conditions from dynamic evolution mechanisms. The approach integrates Hilbert space dynamics, the Musiela maturity-forward transport identity, and Dupire’s local variance geometry to achieve exact modal reduction and closed-form truncation error bounds. Innovatively combining neural operators with normalizing flows, the framework ensures universal approximation while preserving no-arbitrage properties, and introduces a falsifiable empirical protocol. It further derives the covariance-optimal hedging strategy α* = (HᵀC H)⁻¹HᵀCν and exact change-of-variable formulas linking exponential local variance flows to price simplex flows, thereby supporting autonomous finite-dimensional controlled simulations.
📝 Abstract
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.