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

01 ABSTRACT

The author approximates the joint law of state variables via 2PI effective action and Dyson-Schwinger gap equations, summing skeletons in closed form for exponential vertices; for nonzero correlation, conditional Gaussian modes and moment matching handle integrated variance and leverage integral. The framework reduces Markovian volatility models to a few ODEs while retaining the full two-time propagator for rough models, and numerically matches Monte Carlo references for rough Bergomi, SABR and rough SABR. These findings rely on the author's benchmarks and ancillary technical supplement and require independent replication.

02 KEY FINDINGS

1. A non-perturbative Gaussian closure for stochastic volatility pricing is built from 2PI effective action and Dyson-Schwinger gap equations; exponential vertices sum skeletons in closed form, with melon level cutting Hartree error to 0.01-0.11 bp.
2. The method separates Markovian models, where the gap equation collapses to a few ODEs, from rough (Volterra) models, where the full two-time propagator is retained, based on whether the dressed inverse propagator is time-local.
3. Rough Bergomi matches Monte Carlo references near H=0.07 and rho=-0.9; SABR reaches about 0.97 bp over a 72-cell grid with maturities to ten years, and rough SABR about 0.92 bp.