American Options

Almost-Exact Simulation Scheme for Heston-type Models: Bermudan and American Option Pricing ArXiv ID: 2601.00815 “View on arXiv” Authors: Mara Kalicanin Dimitrov, Marko Dimitrov, Anatoliy Malyarenko, Ying Ni Abstract Recently, an Almost-Exact Simulation (AES) scheme was introduced for the Heston stochastic volatility model and tested for European option pricing. This paper extends this scheme for pricing Bermudan and American options under both Heston and double Heston models. The AES improves Monte Carlo simulation efficiency by using the non-central chi-square distribution for the variance process. We derive the AES scheme for the double Heston model and compare the performance of the AES schemes under both models with the Euler scheme. Our numerical experiments validate the effectiveness of the AES scheme in providing accurate option prices with reduced computational time, highlighting its robustness for both models. In particular, the AES achieves higher accuracy and computational efficiency when the number of simulation steps matches the exercise dates for Bermudan options. ...

Full grid solution for multi-asset options pricing with tensor networks ArXiv ID: 2601.00009 “View on arXiv” Authors: Lucas Arenstein, Michael Kastoryano Abstract Pricing multi-asset options via the Black-Scholes PDE is limited by the curse of dimensionality: classical full-grid solvers scale exponentially in the number of underlyings and are effectively restricted to three assets. Practitioners typically rely on Monte Carlo methods for computing complex instrument involving multiple correlated underlyings. We show that quantized tensor trains (QTT) turn the d-asset Black-Scholes PDE into a tractable high-dimensional problem on a personal computer. We construct QTT representations of the operator, payoffs, and boundary conditions with ranks that scale polynomially in d and polylogarithmically in the grid size, and build two solvers: a time-stepping algorithm for European and American options and a space-time algorithm for European options. We compute full-grid prices and Greeks for correlated basket and max-min options in three to five dimensions with high accuracy. The methods introduced can comfortably be pushed to full-grid solutions on 10-15 underlyings, with further algorithmic optimization and more compute power. ...

Deep Neural Operator Learning for Probabilistic Models ArXiv ID: 2511.07235 “View on arXiv” Authors: Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang Abstract 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. ...

Exact Terminal Condition Neural Network for American Option Pricing Based on the Black-Scholes-Merton Equations ArXiv ID: 2510.27132 “View on arXiv” Authors: Wenxuan Zhang, Yixiao Guo, Benzhuo Lu Abstract This paper proposes the Exact Terminal Condition Neural Network (ETCNN), a deep learning framework for accurately pricing American options by solving the Black-Scholes-Merton (BSM) equations. The ETCNN incorporates carefully designed functions that ensure the numerical solution not only exactly satisfies the terminal condition of the BSM equations but also matches the non-smooth and singular behavior of the option price near expiration. This method effectively addresses the challenges posed by the inequality constraints in the BSM equations and can be easily extended to high-dimensional scenarios. Additionally, input normalization is employed to maintain the homogeneity. Multiple experiments are conducted to demonstrate that the proposed method achieves high accuracy and exhibits robustness across various situations, outperforming both traditional numerical methods and other machine learning approaches. ...

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. ...

Deep Learning Option Pricing with Market Implied Volatility Surfaces ArXiv ID: 2509.05911 “View on arXiv” Authors: Lijie Ding, Egang Lu, Kin Cheung Abstract We present a deep learning framework for pricing options based on market-implied volatility surfaces. Using end-of-day S&P 500 index options quotes from 2018-2023, we construct arbitrage-free volatility surfaces and generate training data for American puts and arithmetic Asian options using QuantLib. To address the high dimensionality of volatility surfaces, we employ a variational autoencoder (VAE) that compresses volatility surfaces across maturities and strikes into a 10-dimensional latent representation. We feed these latent variables, combined with option-specific inputs such as strike and maturity, into a multilayer perceptron to predict option prices. Our model is trained in stages: first to train the VAE for volatility surface compression and reconstruction, then options pricing mapping, and finally fine-tune the entire network end-to-end. The trained pricer achieves high accuracy across American and Asian options, with prediction errors concentrated primarily near long maturities and at-the-money strikes, where absolute bid-ask price differences are known to be large. Our method offers an efficient and scalable approach requiring only a single neural network forward pass and naturally improve with additional data. By bridging volatility surface modeling and option pricing in a unified framework, it provides a fast and flexible alternative to traditional numerical approaches for exotic options. ...

American options valuation in time-dependent jump-diffusion models via integral equations and characteristic functions ArXiv ID: 2506.18210 “View on arXiv” Authors: Andrey Itkin Abstract Despite significant advancements in machine learning for derivative pricing, the efficient and accurate valuation of American options remains a persistent challenge due to complex exercise boundaries, near-expiry behavior, and intricate contractual features. This paper extends a semi-analytical approach for pricing American options in time-inhomogeneous models, including pure diffusions, jump-diffusions, and Levy processes. Building on prior work, we derive and solve Volterra integral equations of the second kind to determine the exercise boundary explicitly, offering a computationally superior alternative to traditional finite-difference and Monte Carlo methods. We address key open problems: (1) extending the decomposition method, i.e. splitting the American option price into its European counterpart and an early exercise premium, to general jump-diffusion and Levy models; (2) handling cases where closed-form transition densities are unavailable by leveraging characteristic functions via, e.g., the COS method; and (3) generalizing the framework to multidimensional diffusions. Numerical examples demonstrate the method’s efficiency and robustness. Our results underscore the advantages of the integral equation approach for large-scale industrial applications, while resolving some limitations of existing techniques. ...

Floating exercise boundaries for American options in time-inhomogeneous models ArXiv ID: 2502.00740 “View on arXiv” Authors: Unknown Abstract This paper examines a semi-analytical approach for pricing American options in time-inhomogeneous models characterized by negative interest rates (for equity/FX) or negative convenience yields (for commodities/cryptocurrencies). Under such conditions, exercise boundaries may exhibit a “floating” structure - dynamically appearing and disappearing. For example, a second exercise boundary could emerge within the computational domain and subsequently both could collapse, demanding specialized pricing methodologies. ...

Numerical analysis of American option pricing in a two-asset jump-diffusion model ArXiv ID: 2410.04745 “View on arXiv” Authors: Unknown Abstract This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives, a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals. We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is expressed as a 2-D convolution integral involving the Green’s function of the PIDE. We derive an infinite series representation of this Green’s function, where each term is non-negative and computable. This facilitates the numerical approximation of the PIDE solution through a monotone integration method. To enhance efficiency, we develop an implementation of this monotone scheme via FFTs, exploiting the Toeplitz matrix structure. The proposed method is proved to be both $\ell_{"\infty"} $-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach. ...

KANOP: A Data-Efficient Option Pricing Model using Kolmogorov-Arnold Networks ArXiv ID: 2410.00419 “View on arXiv” Authors: Unknown Abstract Inspired by the recently proposed Kolmogorov-Arnold Networks (KANs), we introduce the KAN-based Option Pricing (KANOP) model to value American-style options, building on the conventional Least Square Monte Carlo (LSMC) algorithm. KANs, which are based on Kolmogorov-Arnold representation theorem, offer a data-efficient alternative to traditional Multi-Layer Perceptrons, requiring fewer hidden layers to achieve a higher level of performance. By leveraging the flexibility of KANs, KANOP provides a learnable alternative to the conventional set of basis functions used in the LSMC model, allowing the model to adapt to the pricing task and effectively estimate the expected continuation value. Using examples of standard American and Asian-American options, we demonstrate that KANOP produces more reliable option value estimates, both for single-dimensional cases and in more complex scenarios involving multiple input variables. The delta estimated by the KANOP model is also more accurate than that obtained using conventional basis functions, which is crucial for effective option hedging. Graphical illustrations further validate KANOP’s ability to accurately model the expected continuation value for American-style options. ...