Dyson-Schwinger Effective-Action Methods for Rough Volatility: A Correlation-Response Architecture for Calibration, Exotics and Risk

Abstract: We develop a non-perturbative framework for stochastic-volatility option pricing organised by the two-particle-irreducible (2PI) effective action and the Dyson-Schwinger gap equations of quantum field theory. In log-price, log-volatility or Lamperti coordinates, the joint law of the state variables is approximated by a self-consistent Gaussian whose mean and effective diffusion follow from the 2PI stationarity conditions, with the drift Jacobian given by statistical linearisation. The smile-generating exponential and CEV interactions are evaluated through the exact Gaussian moment-generating function rather than a Taylor cut. Whether the dressed inverse propagator is local in time separates Markovian models, where the gap equation collapses to a few ODEs, from rough (Volterra) models, where the full two-time propagator is retained and the characteristic function becomes a Gaussian integral over the log-variance field. Across exp-OU, SABR, rough Bergomi and rough SABR, the resulting deterministic engines match PDE or quasi-Monte-Carlo references from sub-basis-point (exp-OU) through single-digit basis points (SABR, rough Bergomi) to tens of basis points (rough SABR), with rough Heston as an exactly-transformable control. Conditional on the volatility field, forward-start smiles and continuously-monitored barriers reduce to field-only quadratures, in rough Bergomi directly on the native Volterra field, and the causal response block yields the full impulse-vega curve in one contraction at one-to-two orders of magnitude below bump-and-revalue. A technical supplement with complete derivations, extended benchmarks and secondary applications (stochastic-rate FX local volatility, quadratic Gaussian Volterra variance and the arbitrage-free FX triangle) is provided as an ancillary file.