Paper: arXiv 2609.37741

Authors: Frédéric Pauquay

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents an exceptionally complex and novel mathematical framework derived from quantum field theory for option pricing. While the empirical validation is strong with benchmarks, the primary contribution lies in the theoretical development and its application to rough volatility models. The clarity is good given the complexity, but the specialized jargon makes it challenging for a broader quant finance audience.

Research Flowchart

  flowchart TD
    A[Research Goal: Non-perturbative Stochastic Volatility Option Pricing] --> B{Methodology: 2PI Effective Action & Dyson-Schwinger Equations};
    B --> C{Key Concepts: Self-consistent Gaussian Approx., Statistical Linearization, Exact Gaussian Moment-Generating Function};
    C --> D[Model Types & Inputs: Exp-OU, SABR, Rough Bergomi, Rough SABR, Rough Heston];
    D --> E{Computational Process: Solving Gap Equations (ODEs/Full Propagator) & Characteristic Function via Gaussian Integral};
    E --> F[Outcomes: High Accuracy Engine (sub-bps to tens of bps), Efficient Exotics Pricing (Forward-Start Smiles, Barriers), Fast Impulse-Vega];