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.

Despite relaxed assumptions, the proposed method possesses proven theoretical guarantees for both detection and localization.

The problem of hedging derivatives represents a central challenge in financial markets. Under Markovian models, the theory is very well developed, specifically for European-style derivatives. However, significant challenges arise when considering path-dependent options where the payoff depends on the asset's entire price path, or further still, when the underlying asset has non-Markovian dynamics, where conventional parametrized hedging approaches tend to be too restrictive or untractable. In response to these challenges, non-parametric approaches have gained a lot of popularity, and more specifically with the improvement of machine learning software and hardware, deep hedging approaches for their versatility, ease of train and ability to capture nonlinearities, see for instance B uhler et al. (2018).

This paper investigates the forecasting performance of COMEX copper futures realized volatility across various high-frequency intervals using both econometric volatility models and deep learning recurrent neural network models. The econometric models considered are GARCH and HAR, while the deep learning models include RNN (Recurrent Neural Network), LSTM (Long Short-Term Memory), and GRU (Gated Recurrent Unit). In forecasting daily realized volatility for COMEX copper futures with a rolling window approach, the econometric models, particularly HAR, outperform recurrent neural networks overall, with HAR achieving the lowest QLIKE loss function value. However, when the data is replaced with hourly high-frequency realized volatility, the deep learning models outperform the GARCH model, and HAR attains a comparable QLIKE loss function value. Despite the black-box nature of machine learning models, the deep learning models demonstrate superior forecasting performance, surpassing the fixed QLIKE value of HAR in the experiment. Moreover, as the forecast horizon extends for daily realized volatility, deep learning models gradually close the performance gap with the GARCH model in certain loss function metrics. Nonetheless, HAR remains the most effective model overall for daily realized volatility forecasting in copper futures.

A long memory and non-linear realized volatility model class is proposed for direct Value at Risk (VaR) forecasting. This model, referred to as RNN-HAR, extends the heterogeneous autoregressive (HAR) model, a framework known for efficiently capturing long memory in realized measures, by integrating a Recurrent Neural Network (RNN) to handle non-linear dynamics. Loss-based generalized Bayesian inference with Sequential Monte Carlo is employed for model estimation and sequential prediction in RNN HAR. The empirical analysis is conducted using daily closing prices and realized measures from 2000 to 2022 across 31 market indices. The proposed models one step ahead VaR forecasting performance is compared against a basic HAR model and its extensions. The results demonstrate that the proposed RNN-HAR model consistently outperforms all other models considered in the study.

The option pricing partial differential equation is reformulated as an energy minimization problem, which is approximated in a time-stepping fashion by deep artificial neural networks. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness, and adheres to a priori known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model. Stochastic volatility models have been popular in the mathematical finance literature because they allow to accurately model and reproduce the shape of implied volatility smiles for a single maturity. They require though certain modifications, such as making the parameters time-or maturity-dependent, in order to reproduce a whole volatility surface; see e.g. the comprehensive books by Gatheral [25] or Bergomi [15]. The class of rough volatility models, in which the volatility process is driven by a fractional Brownian motion, offers an attractive alternative to classical volatility models, since they allow to reproduce many stylized facts of asset and option prices with only a few (constant) parameters; see e.g. the seminal articles by Gatheral, Jaisson, and Rosenbaum [27] and Bayer, Friz, and Gatheral [9], and the recent volume by Bayer, Friz, Fukasawa, Gatheral, Jacquier, and Rosenbaum [13].

Volatility, as a measure of uncertainty, plays a crucial role in numerous financial activities such as risk management. The Econometrics and Machine Learning communities have developed two distinct approaches for financial volatility forecasting: the stochastic approach and the neural network (NN) approach. Despite their individual strengths, these methodologies have conventionally evolved in separate research trajectories with little interaction between them. This study endeavors to bridge this gap by establishing an equivalence relationship between models of the GARCH family and their corresponding NN counterparts. With the equivalence relationship established, we introduce an innovative approach, named GARCH-NN, for constructing NN-based volatility models. It obtains the NN counterparts of GARCH models and integrates them as components into an established NN architecture, thereby seamlessly infusing volatility stylized facts (SFs) inherent in the GARCH models into the neural network. We develop the GARCH-LSTM model to showcase the power of the GARCH-NN approach. Experiment results validate that amalgamating the NN counterparts of the GARCH family models into established NN models leads to enhanced outcomes compared to employing the stochastic and NN models in isolation.