Local-Volatility Calibration via Differentiable Chernoff–Remizov Resolvents is the closing entry of the SpectralNet v1–v6 program (2026), which explored constructive Chernoff–Remizov approximations (Remizov 2025, Vladikavkaz Math. J. ) across three problem families: neural architectures for classification (v1–v3), PDE inverse problems in Darcy flow (v4–v5), and calibration in quantitative finance (v6). Abstract. Local-volatility calibration from vanilla option data is attractive in principle but fragile in practice: classical Dupire-style reconstructions depend on differentiating noisy implied-volatility surfaces, while direct inverse formulations become unstable when the observed payoffs are non-smooth. We study an operator-theoretic alternative based on differentiable Chernoff–Remizov approximations for the Black–Scholes generator. Our initial semigroup-inverse formulation with option payoffs fails for a structural reason: the call payoff violates the bounded-smooth (UCb) hypotheses of Theorem 6 in Remizov (2025), and the gradient signal collapses through long Jacobian chains. We therefore replace the semigroup map with a resolvent formulation, where the probe function is smooth and the inverse becomes forcing-like. This resolves the gradient-collapse mechanism and yields stable synthetic calibration with sigma-error 1. 80% and resolvent fit 0. 78%. For market data we use the pipeline: Market Data -> IV Surface -> Carr–Madan -> Laplace -> Remizov Inverse. The Carr–Madan representation bridges smooth probes with traded option quotes; Laplace weighting acts as an integral regulariser that is absent in local second-derivative baselines. On a four-asset market test (SPY, QQQ, AAPL, NVDA) the Remizov fit stays below 8% on all tested assets, whereas Dupire diverges with relative L2 disagreement 0. 59–0. 80 and wing errors up to 5. 04. A short-horizon SPY delta-hedging stress test shows 50. 7% variance reduction relative to Dupire. Contributions. 1. We identify the semigroup-inverse formulation with non-smooth option payoffs as a genuine boundary case of Theorem 6 (Remizov 2025), not a tuning failure. 2. The resolvent reformulation, using smooth Gaussian probes linked to market quotes via Carr–Madan integration, restores a mathematically admissible and numerically stable inverse problem. 3. Real-market validation across four assets and a delta-hedging stress test document a principled robustness-first alternative to Dupire in precisely the regimes where local differentiation is least trustworthy. Scope and honest limitations. The calibrated object is a one-dimensional sigma (y) slice on a log-spot grid at a single maturity, not a full sigma (y, T) surface. The method is approximately 50–125x slower than Dupire, so it targets daily calibration, risk management and wing-sensitive downstream tasks rather than intraday recalibration. Extensions to non-stationary generators and to L2-valued probes are discussed as open directions but not claimed here. Position in the SpectralNet v1–v6 program. This is the sixth and concluding preprint of the series. Together with v4 (Differentiable Chernoff–Remizov Trajectories for Inverse Problems, 1D Darcy) and v5 (Spectral Evaluation of 2D Chernoff–Remizov Trajectories, 2D Darcy), v6 completes the empirical map of where Chernoff–Remizov constructions provide a usable differentiable forward/inverse object and where their hypotheses bite back.
Sergey Shpital (Tue,) studied this question.