Option Pricing

Quantum computing for multidimensional option pricing: End-to-end pipeline ArXiv ID: 2601.04049 “View on arXiv” Authors: Julien Hok, Álvaro Leitao Abstract This work introduces an end-to-end framework for multi-asset option pricing that combines market-consistent risk-neutral density recovery with quantum-accelerated numerical integration. We first calibrate arbitrage-free marginal distributions from European option quotes using the Normal Inverse Gaussian (NIG) model, leveraging its analytical tractability and ability to capture skewness and fat tails. Marginals are coupled via a Gaussian copula to construct joint distributions. To address the computational bottleneck of the high-dimensional integration required to solve the option pricing formula, we employ Quantum Accelerated Monte Carlo (QAMC) techniques based on Quantum Amplitude Estimation (QAE), achieving quadratic convergence improvements over classical Monte Carlo (CMC) methods. Theoretical results establish accuracy bounds and query complexity for both marginal density estimation (via cosine-series expansions) and multidimensional pricing. Empirical tests on liquid equity entities (Credit Agricole, AXA, Michelin) confirm high calibration accuracy and demonstrate that QAMC requires 10-100 times fewer queries than classical methods for comparable precision. This study provides a practical route to integrate arbitrage-aware modelling with quantum computing, highlighting implications for scalability and future extensions to complex derivatives. ...

Boundary error control for numerical solution of BSDEs by the convolution-FFT method ArXiv ID: 2512.24714 “View on arXiv” Authors: Xiang Gao, Cody Hyndman Abstract We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method. ...

An Efficient Machine Learning Framework for Option Pricing via Fourier Transform ArXiv ID: 2512.16115 “View on arXiv” Authors: Liying Zhang, Ying Gao Abstract The increasing need for rapid recalibration of option pricing models in dynamic markets places stringent computational demands on data generation and valuation algorithms. In this work, we propose a hybrid algorithmic framework that integrates the smooth offset algorithm (SOA) with supervised machine learning models for the fast pricing of multiple path-independent options under exponential Lévy dynamics. Building upon the SOA-generated dataset, we train neural networks, random forests, and gradient boosted decision trees to construct surrogate pricing operators. Extensive numerical experiments demonstrate that, once trained, these surrogates achieve order-of-magnitude acceleration over direct SOA evaluation. Importantly, the proposed framework overcomes key numerical limitations inherent to fast Fourier transform-based methods, including the consistency of input data and the instability in deep out-of-the-money option pricing. ...

Constrained deep learning for pricing and hedging european options in incomplete markets ArXiv ID: 2511.20837 “View on arXiv” Authors: Nicolas Baradel Abstract In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient serving as the hedging strategy, optimized via a loss function enforcing the self-financing portfolio condition. A key challenge arises from the non-smooth nature of option payoffs (e.g., vanilla calls are non-differentiable at-the-money, while digital options are discontinuous), which conflicts with the inherent smoothness of standard neural networks. To address this, we compare unconstrained networks against constrained architectures that explicitly embed the terminal payoff condition, drawing inspiration from PDE-solving techniques. Our framework assumes two tradable assets: the underlying and a liquid call option capturing volatility dynamics. Numerical experiments evaluate the method on simple options with varying non-smoothness, the exotic Equinox option, and scenarios with market jumps for robustness. Results demonstrate superior P&L distributions, highlighting the efficacy of constrained networks in handling realistic payoffs. This work advances machine learning applications in quantitative finance by integrating boundary constraints, offering a practical tool for pricing and hedging in incomplete markets. ...

A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures ArXiv ID: 2511.06451 “View on arXiv” Authors: Jian’an Zhang Abstract We propose ARBITER, a risk-neutral neural operator for learning joint SPX-VIX term structures under no-arbitrage constraints. ARBITER maps market states to an operator that outputs implied volatility and variance curves while enforcing static arbitrage (calendar, vertical, butterfly), Lipschitz bounds, and monotonicity. The model couples operator learning with constrained decoders and is trained with extragradient-style updates plus projection. We introduce evaluation metrics for derivatives term structures (NAS, CNAS, NI, Dual-Gap, Stability Rate) and show gains over Fourier Neural Operator, DeepONet, and state-space sequence models on historical SPX and VIX data. Ablation studies indicate that tying the SPX and VIX legs reduces Dual-Gap and improves NI, Lipschitz projection stabilizes calibration, and selective state updates improve long-horizon generalization. We provide identifiability and approximation results and describe practical recipes for arbitrage-free interpolation and extrapolation across maturities and strikes. ...

Numerical valuation of European options under two-asset infinite-activity exponential Lévy models ArXiv ID: 2511.02700 “View on arXiv” Authors: Massimiliano Moda, Karel J. in ’t Hout, Michèle Vanmaele, Fred Espen Benth Abstract We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian theta-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal favourable second-order convergence of our method whenever the exponential Lévy process has finite-variation. ...

Black-Scholes Model, comparison between Analytical Solution and Numerical Analysis ArXiv ID: 2510.27277 “View on arXiv” Authors: Francesco Romaggi Abstract The main purpose of this article is to give a general overview and understanding of the first widely used option-pricing model, the Black-Scholes model. The history and context are presented, with the usefulness and implications in the economics world. A brief review of fundamental calculus concepts is introduced to derive and solve the model. The equation is then resolved using both an analytical (variable separation) and a numerical method (finite differences). Conclusions are drawn in order to understand how Black-Scholes is employed nowadays. At the end a handy appendix (A) is written with some economics notions to ease the reader’s comprehension of the paper; furthermore a second appendix (B) is given with some code scripts, to allow the reader to put in practice some concepts. ...

Quantum Machine Learning methods for Fourier-based distribution estimation with application in option pricing ArXiv ID: 2510.19494 “View on arXiv” Authors: Fernando Alonso, Álvaro Leitao, Carlos Vázquez Abstract The ongoing progress in quantum technologies has fueled a sustained exploration of their potential applications across various domains. One particularly promising field is quantitative finance, where a central challenge is the pricing of financial derivatives-traditionally addressed through Monte Carlo integration techniques. In this work, we introduce two hybrid classical-quantum methods to address the option pricing problem. These approaches rely on reconstructing Fourier series representations of statistical distributions from the outputs of Quantum Machine Learning (QML) models based on Parametrized Quantum Circuits (PQCs). We analyze the impact of data size and PQC dimensionality on performance. Quantum Accelerated Monte Carlo (QAMC) is employed as a benchmark to quantitatively assess the proposed models in terms of computational cost and accuracy in the extraction of Fourier coefficients. Through the numerical experiments, we show that the proposed methods achieve remarkable accuracy, becoming a competitive quantum alternative for derivatives valuation. ...

Semi-analytical pricing of American options with hybrid dividends via integral equations and the GIT method ArXiv ID: 2510.18159 “View on arXiv” Authors: Andrey Itkin Abstract This paper introduces a semi-analytical method for pricing American options on assets (stocks, ETFs) that pay discrete and/or continuous dividends. The problem is notoriously complex because discrete dividends create abrupt price drops and affect the optimal exercise timing, making traditional continuous-dividend models unsuitable. Our approach utilizes the Generalized Integral Transform (GIT) method introduced by the author and his co-authors in a number of papers, which transforms the pricing problem from a complex partial differential equation with a free boundary into an integral Volterra equation of the second or first kind. In this paper we illustrate this approach by considering a popular GBM model that accounts for discrete cash and proportional dividends using Dirac delta functions. By reframing the problem as an integral equation, we can sequentially solve for the option price and the early exercise boundary, effectively handling the discontinuities caused by the dividends. Our methodology provides a powerful alternative to standard numerical techniques like binomial trees or finite difference methods, which can struggle with the jump conditions of discrete dividends by losing accuracy or performance. Several examples demonstrate that the GIT method is highly accurate and computationally efficient, bypassing the need for extensive computational grids or complex backward induction steps. ...

Can Machine Learning Algorithms Outperform Traditional Models for Option Pricing? ArXiv ID: 2510.01446 “View on arXiv” Authors: Georgy Milyushkov Abstract This study investigates the application of machine learning techniques, specifically Neural Networks, Random Forests, and CatBoost for option pricing, in comparison to traditional models such as Black-Scholes and Heston Model. Using both synthetically generated data and real market option data, each model is evaluated in predicting the option price. The results show that machine learning models can capture complex, non-linear relationships in option prices and, in several cases, outperform both Black-Scholes and Heston models. These findings highlight the potential of data-driven methods to improve pricing accuracy and better reflect market dynamics. ...