Neural networks are able to approximate any continuous function on a compact set. However, it is not obvious how to quantify the error of the neural network, i.e., the remaining bias between the function and the neural network. Here, we propose the application of extreme value theory to quantify large values of the error, which are typically relevant in applications. The distribution of the error beyond some threshold is approximately generalized Pareto distributed. We provide a new estimator of the shape parameter of the Pareto distribution suitable to describe the error of neural networks. Numerical experiments are provided.

We reinterpret and propose a framework for pricing path-dependent financial derivatives by estimating the full distribution of payoffs using Distributional Reinforcement Learning (DistRL). Unlike traditional methods that focus on expected option value, our approach models the entire conditional distribution of payoffs, allowing for risk-aware pricing, tail-risk estimation, and enhanced uncertainty quantification. We demonstrate the efficacy of this method on Asian options, using quantile-based value function approximators.

Accurate option pricing is essential for effective trading and risk management in financial markets, yet it remains challenging due to market volatility and the limitations of traditional models like Black-Scholes. In this paper, we investigate the application of the Informer neural network for option pricing, leveraging its ability to capture long-term dependencies and dynamically adjust to market fluctuations. This research contributes to the field of financial forecasting by introducing Informer's efficient architecture to enhance prediction accuracy and provide a more adaptable and resilient framework compared to existing methods. Our results demonstrate that Informer outperforms traditional approaches in option pricing, advancing the capabilities of data-driven financial forecasting in this domain.

This study investigates enhancing option pricing by extending the Black-Scholes model to include stochastic volatility and interest rate variability within the Partial Differential Equation (PDE). The PDE is solved using the finite difference method. The extended Black-Scholes model and a machine learning-based LSTM model are developed and evaluated for pricing Google stock options. Both models were backtested using historical market data. While the LSTM model exhibited higher predictive accuracy, the finite difference method demonstrated superior computational efficiency. This work provides insights into model performance under varying market conditions and emphasizes the potential of hybrid approaches for robust financial modeling.

In the realm of option pricing, existing models are typically classified into principle-driven methods, such as solving partial differential equations (PDEs) that pricing function satisfies, and data-driven approaches, such as machine learning (ML) techniques that parameterize the pricing function directly. While principle-driven models offer a rigorous theoretical framework, they often rely on unrealistic assumptions, such as asset processes adhering to fixed stochastic differential equations (SDEs). Moreover, they can become computationally intensive, particularly in high-dimensional settings when analytical solutions are not available and thus numerical solutions are needed. In contrast, data-driven models excel in capturing market data trends, but they often lack alignment with core financial principles, raising concerns about interpretability and predictive accuracy, especially when dealing with limited or biased datasets. This work proposes a hybrid approach to address these limitations by integrating the strengths of both principled and data-driven methodologies. Our framework combines the theoretical rigor and interpretability of PDE-based models with the adaptability of machine learning techniques, yielding a more versatile methodology for pricing a broad spectrum of options. We validate our approach across different volatility modeling approaches-both with constant volatility (Black-Scholes) and stochastic volatility (Heston), demonstrating that our proposed framework, Finance-Informed Neural Network (FINN), not only enhances predictive accuracy but also maintains adherence to core financial principles. FINN presents a promising tool for practitioners, offering robust performance across a variety of market conditions.

We introduce a new class of neural networks designed to be convex functions of their inputs, leveraging the principle that any convex function can be represented as the supremum of the affine functions it dominates. These neural networks, inherently convex with respect to their inputs, are particularly well-suited for approximating the prices of options with convex payoffs. We detail the architecture of this, and establish theoretical convergence bounds that validate its approximation capabilities. We also introduce a \emph{scrambling} phase to improve the training of these networks. Finally, we demonstrate numerically the effectiveness of these networks in estimating prices for three types of options with convex payoffs: Basket, Bermudan, and Swing options.

Option pricing models, essential in financial mathematics and risk management, have been extensively studied and recently advanced by AI methodologies. However, American option pricing remains challenging due to the complexity of determining optimal exercise times and modeling non-linear payoffs resulting from stochastic paths. Moreover, the prevalent use of the Black-Scholes formula in hybrid models fails to accurately capture the discontinuity in the price process, limiting model performance, especially under scarce data conditions. To address these issues, this study presents a comprehensive framework for American option pricing consisting of six interrelated modules, which combine nonlinear optimization algorithms, analytical and numerical models, and neural networks to improve pricing performance. Additionally, to handle the scarce data challenge, this framework integrates the transfer learning through numerical data augmentation and a physically constrained, jump diffusion process-informed neural network to capture the leptokurtosis of the log return distribution. To increase training efficiency, a warm-up period using Bayesian optimization is designed to provide optimal data loss and physical loss coefficients. Experimental results of six case studies demonstrate the accuracy, convergence, physical effectiveness, and generalization of the framework. Moreover, the proposed model shows superior performance in pricing deep out-of-the-money options. Introduction Options are fundamental financial derivatives widely employed for risk management. The movement of option prices follows a stochastic process influenced by various factors such as the price process of the underlying assets ( S t), the strike price (K), the time-to-maturity ( T), the option type (American or European; Put ( P) or Call ( C) options), and numerous macroeconomic and market factors.

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.

Providing an arbitrage-free valuation formula and specifying risk-neutral dynamics are essentially two sides of the same coin in option pricing. Yet, the modeling methodology has been leaning towards the latter for decades. That is, the invention of an option pricing model typically starts with proposing a stochastic process that is a martingale for the underlying asset, so that the corresponding risk-neural measure is constructed, and henceforth the arbitrage-free option valuation can be determined either analytically or numerically. Such a methodology was established through the pioneering work of Bachelier [4] and Black and Scholes [9], and since then, almost all of the prevailing models have been invented along this paradigm. The list includes but is not limited to local volatility models by Dupire [17], Cox [14], stochastic volatility models by Heston [20], Hagan et al. [18], Bates [8], jump-diffusion models by Merton [28], Kou [24], and other models built upon Lévy processes by Madan et al. [26], Barndorff-Nielsen [7]. Nonetheless, the reverse approach, which first provides an arbitrage-free valuation formula as in Carr and Madan [11], Davis and Hobson [15] and then finds the underlying martingale supporting the formula, is still possible, as noted in [21, 27]. In recent work, Carr and Torricelli [12] starts with one particular pricing formula that yields logistically distributed marginals. Although there is no underlying Lévy process that produces such marginals, by allowing the increment to be nonstationary, an additive logistic process can be constructed to support that pricing formula.

One of the most discussed problems in the financial world is stock option pricing. The Black-Scholes Equation is a Parabolic Partial Differential Equation which provides an option pricing model. The present work proposes an approach based on Neural Networks to solve the Black-Scholes Equations. Real-world data from the stock options market were used as the initial boundary to solve the Black-Scholes Equation. In particular, times series of call options prices of Brazilian companies Petrobras and Vale were employed. The results indicate that the network can learn to solve the Black-Sholes Equation for a specific real-world stock options time series. The experimental results showed that the Neural network option pricing based on the Black-Sholes Equation solution can reach an option pricing forecasting more accurate than the traditional Black-Sholes analytical solutions. The experimental results making it possible to use this methodology to make short-term call option price forecasts in options markets.