The Latent Generator

We introduce the latent generator: a dynamical operator \(M\) that is not directly observed, but is inferred from observable data. In general, the latent generator is the law behind a family of snapshots: the operator that generates the observables rather than merely fitting them pointwise. In this paper, finance is the first application. From observed option prices, we infer the latent generator behind the volatility surface, without assuming any parametric model (no Heston, no SABR, no Black--Scholes). The recovery exploits the fact that the spectral pricing map --- from generator to option prices --- is LINEAR: \(\text{Price}(K, T) = e^{-rT}\sum_k A_k(T)\,G_k(K)\) where \(A(T) = e^{MT}A(0)\) and \(G_k(K)\) are precomputed payoff coefficients. Inversion proceeds in two steps: (1) from prices at multiple strikes, recover the spectral density \(A(T)\) via linear least squares; (2) from densities at multiple maturities, recover \(M\) via constrained optimization. In the current package benchmark, a single \(12 \times 12\) generator recovered from 52 synthetic option prices (13 strikes \(\times\) 4 maturities) reprices 8 holdout options at unseen strikes and maturities with holdout RMSE \(8.9 \times 10^{-4}\), while preserving spectral quality from \(\rho_{\mathrm{spec}} = 1.72\) to \(\hat{\rho}_{\mathrm{spec}} = 1.91\). Unlike Heston (5 parameters) or SABR (4 parameters), which require per-maturity recalibration, the recovered generator provides one object for all maturities: \(e^{MT}\) at any \(T\) from a single \(M\). The generator is arbitrage-free by construction (dissipative: all eigenvalues \(\leq 0\)), and its eigenvalues reveal the market's implied time scales: mean reversion speed, skew dynamics rate, and tail decay rate.