Paper: arXiv 2611.12899

Authors: Anastasis Kratsios, Yannick Limmer, Hans Buehler, Blanka Horvath, Raeid Saqur

Abstract

Neural operators (NOs) have been introduced as deep learning-powered solutions to the computational bottleneck of solving families of problems. Once trained, an NO can rapidly output approximate solutions, avoiding the need to rerun classical solvers. However, standard NO architectures cannot guarantee static no-arbitrage when predicting European call-price surfaces from market or model features and, as we verify experimentally, can yield negative PnL in downstream trading tasks. We introduce a certifiably arbitrage-free (AF) and universal encoder-decoder NO framework, continuously mapping inputs from a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ over any fixed subset of $[0,\infty)^2$ to a convex combination of arbitrage-free European call-price surfaces by continuously solving an approximate nearest-neighbour problem. We show that a nearest arbitrage-free correction of an unconstrained predicted surface need not exist, implying that no-arbitrage cannot, in general, be restored through post-processing. Further, we show that standard NOs, such as the FNO and certain encoder-decoder NOs, admit parameter configurations whose outputs violate static no-arbitrage. Thus, out-of-the-box NOs generally cannot be certifiably restricted to arbitrage-free configurations, but may only be guided towards them through PINN-type regularization during training.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly novel and mathematically sophisticated framework for certifiably arbitrage-free option pricing using neural operators. While the theoretical contributions are strong, the empirical rigor, though present, could be further enhanced with more extensive real-world backtesting details. The overall contribution is significant for quantitative finance.

Research Flowchart

  flowchart TD
    A[Research Goal: Certifiably Arbitrage-Free NOs for European Call-Price Surfaces] --> B{Problem: Standard NOs violate no-arbitrage};
    B --> C[Methodology: AFNO Encoder-Decoder Framework];
    C --> D{Inputs: Market/Model Features & RKHS Mapping};
    D --> E[Computational Process: Convex combination of AF surfaces via approximate nearest-neighbor];
    E --> F[Key Finding 1: AFNO guarantees static no-arbitrage];
    F --> G[Key Finding 2: Nearest AF correction may not exist; Post-processing insufficient];
    G --> H[Key Finding 3: Standard NOs (e.g., FNO) cannot be certifiably restricted to AF configurations].