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

TL;DR

This work addresses the computational bottleneck in calibrating stochastic volatility models by learning a global pricing operator for the Heston model with a physics-informed DeepONet. A hard-constrained ansatz enforces terminal payoffs and no-arbitrage, while residual-based adaptive refinement stabilizes training across challenging regions. The DeepSVM framework delivers near-instant pricing across parameter space and matches a semi-analytic Heston pricer for option values, though the learned Greeks exhibit noise in the ATM regime, highlighting regularization challenges. The results suggest a promising path for real-time pricing in more complex settings, with future work focusing on Sobolev training and extensions to rough or high-dimensional volatility models.

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.

Figures (4)

• Figure 1: The DeepSVM architecture. The model combines a DeepONet core with a hard-constrained ansatz to enforce the terminal payoff condition exactly. Training is stabilized via residual-based adaptive refinement (RAR).
• Figure 2: Convergence of the total loss during training. The shaded regions indicate the Adam and L--BFGS phases, respectively.
• Figure 3: Top two rows: Comparison between DeepSVM and the semi-analytic Heston solution for three randomly selected parameter vectors $\boldsymbol{\mu}$. For each column we show the option price (DeepSVM vs. semi-analytic) and the corresponding absolute pricing error as a function of log-moneyness $x = \ln(S/K)$. Bottom row: For each parameter set, we show the corresponding Greeks (Delta and Gamma) computed via Autodiff from DeepSVM and from the semi-analytic model.
• Figure 4: Spatial maps of the PDE residual mean-squared error (MSE) in $(x,\nu)$ space. Top row: Residual MSE averaged over the parameter space $\boldsymbol{\mu}$ for different time-to-maturity slices $\tau$. Bottom row: Residual MSE averaged over $\tau$ for three representative parameter vectors $\boldsymbol{\mu}$. All panels share a common logarithmic color scale.