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.

The Heston stochastic volatility model is a widely used tool in financial mathematics for pricing European options. However, its calibration remains computationally intensive and sensitive to local minima due to the model's nonlinear structure and high-dimensional parameter space. This paper introduces a hybrid deep learning-based framework that enhances both the computational efficiency and the accuracy of the calibration procedure. The proposed approach integrates two supervised feedforward neural networks: the Price Approximator Network (PAN), which approximates the option price surface based on strike and moneyness inputs, and the Calibration Correction Network (CCN), which refines the Heston model's output by correcting systematic pricing errors. Experimental results on real S\&P 500 option data demonstrate that the deep learning approach outperforms traditional calibration techniques across multiple error metrics, achieving faster convergence and superior generalization in both in-sample and out-of-sample settings. This framework offers a practical and robust solution for real-time financial model calibration.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper introduces a GP-Vol model to flexibly capture the time-dependent changes in variance, and develops a new online algorithm for fully Bayesian inference under the model. The paper is clearly written, the developed inference method seems technically sound, and the presented results look promising. My opinion on the model itself, using a non-parametric approach such as using the GP prior on the transition function (as in the paper), seems, though, a bit an obvious way of extending the prior work developed in the finance area. So, I wouldn't put too high grade on the paper in terms of its originality.

This paper contributes to the existing literature on hedging American options with Deep Reinforcement Learning (DRL). The study first investigates hyperparameter impact on hedging performance, considering learning rates, training episodes, neural network architectures, training steps, and transaction cost penalty functions. Results highlight the importance of avoiding certain combinations, such as high learning rates with a high number of training episodes or low learning rates with few training episodes and emphasize the significance of utilizing moderate values for optimal outcomes. Additionally, the paper warns against excessive training steps to prevent instability and demonstrates the superiority of a quadratic transaction cost penalty function over a linear version. This study then expands upon the work of Pickard et al. (2024), who utilize a Chebyshev interpolation option pricing method to train DRL agents with market calibrated stochastic volatility models. While the results of Pickard et al. (2024) showed that these DRL agents achieve satisfactory performance on empirical asset paths, this study introduces a novel approach where new agents at weekly intervals to newly calibrated stochastic volatility models. Results show DRL agents re-trained using weekly market data surpass the performance of those trained solely on the sale date. Furthermore, the paper demonstrates that both single-train and weekly-train DRL agents outperform the Black-Scholes Delta method at transaction costs of 1% and 3%. This practical relevance suggests that practitioners can leverage readily available market data to train DRL agents for effective hedging of options in their portfolios.

This article leverages deep reinforcement learning (DRL) to hedge American put options, utilizing the deep deterministic policy gradient (DDPG) method. The agents are first trained and tested with Geometric Brownian Motion (GBM) asset paths and demonstrate superior performance over traditional strategies like the Black-Scholes (BS) Delta, particularly in the presence of transaction costs. To assess the real-world applicability of DRL hedging, a second round of experiments uses a market calibrated stochastic volatility model to train DRL agents. Specifically, 80 put options across 8 symbols are collected, stochastic volatility model coefficients are calibrated for each symbol, and a DRL agent is trained for each of the 80 options by simulating paths of the respective calibrated model. Not only do DRL agents outperform the BS Delta method when testing is conducted using the same calibrated stochastic volatility model data from training, but DRL agents achieves better results when hedging the true asset path that occurred between the option sale date and the maturity. As such, not only does this study present the first DRL agents tailored for American put option hedging, but results on both simulated and empirical market testing data also suggest the optimality of DRL agents over the BS Delta method in real-world scenarios. Finally, note that this study employs a model-agnostic Chebyshev interpolation method to provide DRL agents with option prices at each time step when a stochastic volatility model is used, thereby providing a general framework for an easy extension to more complex underlying asset processes.

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].

We study Merton's expected utility maximization problem in an incomplete market, characterized by a factor process in addition to the stock price process, where all the model primitives are unknown. We take the reinforcement learning (RL) approach to learn optimal portfolio policies directly by exploring the unknown market, without attempting to estimate the model parameters. Based on the entropy-regularization framework for general continuous-time RL formulated in Wang et al. (2020), we propose a recursive weighting scheme on exploration that endogenously discounts the current exploration reward by the past accumulative amount of exploration. Such a recursive regularization restores the optimality of Gaussian exploration. However, contrary to the existing results, the optimal Gaussian policy turns out to be biased in general, due to the interwinding needs for hedging and for exploration. We present an asymptotic analysis of the resulting errors to show how the level of exploration affects the learned policies. Furthermore, we establish a policy improvement theorem and design several RL algorithms to learn Merton's optimal strategies. At last, we carry out both simulation and empirical studies with a stochastic volatility environment to demonstrate the efficiency and robustness of the RL algorithms in comparison to the conventional plug-in method.

We present a novel approach to approximate Gaussian and mixture-of-Gaussians filtering. Our method relies on a variational approximation via a gradient-flow representation. The gradient flow is derived from a Kullback--Leibler discrepancy minimization on the space of probability distributions equipped with the Wasserstein metric. We outline the general method and show its competitiveness in posterior representation and parameter estimation on two state-space models for which Gaussian approximations typically fail: systems with multiplicative noise and multi-modal state distributions.

Deep learning algorithms have recently shown to be a successful tool in estimating parameters of statistical models for which simulation is easy, but likelihood computation is challenging. But the success of these approaches depends on simulating parameters that sufficiently reproduce the observed data, and, at present, there is a lack of efficient methods to produce these simulations. We develop new black-box procedures to estimate parameters of statistical models based only on weak parameter structure assumptions. For well-structured likelihoods with frequent occurrences, such as in time series, this is achieved by pre-training a deep neural network on an extensive simulated database that covers a wide range of data sizes. For other types of complex dependencies, an iterative algorithm guides simulations to the correct parameter region in multiple rounds. These approaches can successfully estimate and quantify the uncertainty of parameters from non-Gaussian models with complex spatial and temporal dependencies. The success of our methods is a first step towards a fully flexible automatic black-box estimation framework.

Volatility clustering is a common phenomenon in financial time series. Typically, linear models can be used to describe the temporal autocorrelation of the (logarithmic) variance of returns. Considering the difficulty in estimating this model, we construct a Dynamic Bayesian Network, which utilizes the conjugate prior relation of normal-gamma and gamma-gamma, so that its posterior form locally remains unchanged at each node. This makes it possible to find approximate solutions using variational methods quickly. Furthermore, we ensure that the volatility expressed by the model is an independent incremental process after inserting dummy gamma nodes between adjacent time steps. We have found that this model has two advantages: 1) It can be proved that it can express heavier tails than Gaussians, i.e., have positive excess kurtosis, compared to popular linear models. 2) If the variational inference(VI) is used for state estimation, it runs much faster than Monte Carlo(MC) methods since the calculation of the posterior uses only basic arithmetic operations. And its convergence process is deterministic. We tested the model, named Gam-Chain, using recent Crypto, Nasdaq, and Forex records of varying resolutions. The results show that: 1) In the same case of using MC, this model can achieve comparable state estimation results with the regular lognormal chain. 2) In the case of only using VI, this model can obtain accuracy that are slightly worse than MC, but still acceptable in practice; 3) Only using VI, the running time of Gam-Chain, in general case, can be reduced to below 5% of that based on the lognormal chain via MC.