We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining.

This paper investigates theoretical and methodological foundations for stochastic optimal control (SOC) in discrete time. We start formulating the control problem in a general dynamic programming framework, introducing the mathematical structure needed for a detailed convergence analysis. The associate value function is estimated through a sequence of approximations combining nonparametric regression methods and Monte Carlo subsampling. The regression step is performed within reproducing kernel Hilbert spaces (RKHSs), exploiting the classical KRR algorithm, while Monte Carlo sampling methods are introduced to estimate the continuation value. To assess the accuracy of our value function estimator, we propose a natural error decomposition and rigorously control the resulting error terms at each time step. We then analyze how this error propagates backward in time-from maturity to the initial stage-a relatively underexplored aspect of the SOC literature. Finally, we illustrate how our analysis naturally applies to a key financial application: the pricing of American options.

In this research, we explore neural network-based methods for pricing multidimensional American put options under the BlackScholes and Heston model, extending up to five dimensions. We focus on two approaches: the Time Deep Gradient Flow (TDGF) method and the Deep Galerkin Method (DGM). We extend the TDGF method to handle the free-boundary partial differential equation inherent in American options. We carefully design the sampling strategy during training to enhance performance. Both TDGF and DGM achieve high accuracy while outperforming conventional Monte Carlo methods in terms of computational speed. In particular, TDGF tends to be faster during training than DGM.

This study investigates the application of machine learning algorithms, particularly in the context of pricing American options using Monte Carlo simulations. Traditional models, such as the Black-Scholes-Merton framework, often fail to adequately address the complexities of American options, which include the ability for early exercise and non-linear payoff structures. By leveraging Monte Carlo methods in conjunction Least Square Method machine learning was used. This research aims to improve the accuracy and efficiency of option pricing. The study evaluates several machine learning models, including neural networks and decision trees, highlighting their potential to outperform traditional approaches. The results from applying machine learning algorithm in LSM indicate that integrating machine learning with Monte Carlo simulations can enhance pricing accuracy and provide more robust predictions, offering significant insights into quantitative finance by merging classical financial theories with modern computational techniques. The dataset was split into features and the target variable representing bid prices, with an 80-20 train-validation split. LSTM and GRU models were constructed using TensorFlow's Keras API, each with four hidden layers of 200 neurons and an output layer for bid price prediction, optimized with the Adam optimizer and MSE loss function. The GRU model outperformed the LSTM model across all evaluated metrics, demonstrating lower mean absolute error, mean squared error, and root mean squared error, along with greater stability and efficiency in training.

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.

The primary goal of option pricing is to work out the to the Black-Scholes equation for pricing American and European probability of whether the option is "in-the-money" or "outof-money" options. We test our approach on both simulated as when it is exercised. Option pricing is crucial well as real market data, compare it to analytical/numerical for traders, investors, and financial institutions in making benchmarks. Our model is able to accurately capture informed decisions about buying, selling, or hedging risks the price behavior on simulation data, while also exhibiting against certain underlying assets. Precise estimation of the reasonable performance for market data (with an improvement option price helps stabilize the financial market, as financial of 30% over benchmark). We also experiment with portfolios and strategies are adjusted according to the the architecture and learning process of our PINN model changes in the option price [2]. The problem of robust to provide more understanding of convergence and stability option pricing becomes even more pressing in the current issues that impact performance.

In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Hur\'e-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided including: Amerasian option under the Black-Scholes model, American option with a path-dependent geometric mean payoff function, and the Shiryaev's optimal stopping problem.

In this paper, we introduce two methods to solve the American-style option pricing problem and its dual form at the same time using neural networks. Without applying nested Monte Carlo, the first method uses a series of neural networks to simultaneously compute both the lower and upper bounds of the option price, and the second one accomplishes the same goal with one global network. The avoidance of extra simulations and the use of neural networks significantly reduce the computational complexity and allow us to price Bermudan options with frequent exercise opportunities in high dimensions, as illustrated by the provided numerical experiments. As a by-product, these methods also derive a hedging strategy for the option, which can also be used as a control variate for variance reduction.

This paper presents new machine learning approaches to approximate the solution of optimal stopping problems. The key idea of these methods is to use neural networks, where the hidden layers are generated randomly and only the last layer is trained, in order to approximate the continuation value. Our approaches are applicable for high dimensional problems where the existing approaches become increasingly impractical. In addition, since our approaches can be optimized using a simple linear regression, they are very easy to implement and theoretical guarantees can be provided. In Markovian examples our randomized reinforcement learning approach and in non-Markovian examples our randomized recurrent neural network approach outperform the state-of-the-art and other relevant machine learning approaches.

We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.