DeepSVM: Learning Stochastic Volatility Models with Physics-Informed Deep Operator Networks

Paper: arXiv 2512.07162

Authors: Kieran A. Malandain, Selim Kalici, Hakob Chakhoyan

Abstract

Real-time calibration of stochastic volatility models (SVMs) is computationally bottlenecked by the need to repeatedly solve coupled partial differential equations (PDEs). In this work, we propose DeepSVM, a physics-informed Deep Operator Network (PI-DeepONet) designed to learn the solution operator of the Heston model across its entire parameter space. Unlike standard data-driven deep learning (DL) approaches, DeepSVM requires no labelled training data. Rather, we employ a hard-constrained ansatz that enforces terminal payoffs and static no-arbitrage conditions by design. Furthermore, we use Residual-based Adaptive Refinement (RAR) to stabilize training in difficult regions subject to high gradients. Overall, DeepSVM achieves a final training loss of $10^{"-5"}$ and predicts highly accurate option prices across a range of typical market dynamics. While pricing accuracy is high, we find that the model’s derivatives (Greeks) exhibit noise in the at-the-money (ATM) regime, highlighting the specific need for higher-order regularization in physics-informed operator learning.

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 demonstrates high mathematical complexity through advanced operator learning theory, PDE transformations, and physics-informed neural network architecture. Empirical rigor is solid with detailed implementation specs (Sobol sequences, adaptive refinement) and validation results, though it lacks real-market backtesting or code availability, placing it between lab research and street-ready application.

Research Flowchart

  flowchart TD
  A["Research Goal<br/>Real-time calibration of<br/>Stochastic Volatility Models"] --> B["Methodology<br/>Physics-Informed Deep Operator Network<br/>PI-DeepONet"]
  B --> C["Data/Input Sources<br/>No labeled data<br/>Heston PDE residuals & boundary conditions"]
  C --> D["Computational Processes<br/>1. Hard-constrained ansatz<br/>2. Residual-based Adaptive Refinement<br/>3. Backpropagation on PDE loss"]
  D --> E["Key Findings/Outcomes<br/>Accurate pricing (Loss ~10^-5)<br/>Noisy Greeks in ATM regime<br/>Need for higher-order regularization"]

Cited References (14)

1. Physics-Informed Neural Networks (PINNs) in Finance — link
2. Option Pricing in the Heston Model with Physics Inspired Neural Networks — link
3. Financial Machine Learning (2023) — link
4. Deep Learning for Solving and Estimating Dynamic Macro-finance Models — → 2305.09783
5. Towards Fast Option Pricing PDE Solvers Powered by PIELM — link
6. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators (2021)
7. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993)
8. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019)

1 of 14 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.