Derivatives (Options)

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. ...

Law-Strength Frontiers and a No-Free-Lunch Result for Law-Seeking Reinforcement Learning on Volatility Law Manifolds ArXiv ID: 2511.17304 “View on arXiv” Authors: Jian’an Zhang Abstract We study reinforcement learning (RL) on volatility surfaces through the lens of Scientific AI. We ask whether axiomatic no-arbitrage laws, imposed as soft penalties on a learned world model, can reliably align high-capacity RL agents, or mainly create Goodhart-style incentives to exploit model errors. From classical static no-arbitrage conditions we build a finite-dimensional convex volatility law manifold of admissible total-variance surfaces, together with a metric law-penalty functional and a Graceful Failure Index (GFI) that normalizes law degradation under shocks. A synthetic generator produces law-consistent trajectories, while a recurrent neural world model trained without law regularization exhibits structured off-manifold errors. On this testbed we define a Goodhart decomposition (r = r^{"\mathcal{M"}} + r^\perp), where (r^\perp) is ghost arbitrage from off-manifold prediction error. We prove a ghost-arbitrage incentive theorem for PPO-type agents, a law-strength trade-off theorem showing that stronger penalties eventually worsen P&L, and a no-free-lunch theorem: under a law-consistent world model and law-aligned strategy class, unconstrained law-seeking RL cannot Pareto-dominate structural baselines on P&L, penalties, and GFI. In experiments on an SPX/VIX-like world model, simple structural strategies form the empirical law-strength frontier, while all law-seeking RL variants underperform and move into high-penalty, high-GFI regions. Volatility thus provides a concrete case where reward shaping with verifiable penalties is insufficient for robust law alignment. ...

Error Propagation in Dynamic Programming: From Stochastic Control to Option Pricing ArXiv ID: 2509.20239 “View on arXiv” Authors: Andrea Della Vecchia, Damir Filipović Abstract 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. ...

Applying Informer for Option Pricing: A Transformer-Based Approach ArXiv ID: 2506.05565 “View on arXiv” Authors: Feliks Bańka, Jarosław A. Chudziak Abstract 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. ...

Path-dependent option pricing with two-dimensional PDE using MPDATA ArXiv ID: 2505.24435 “View on arXiv” Authors: Paweł Magnuszewski, Sylwester Arabas Abstract In this paper, we discuss a simple yet robust PDE method for evaluating path-dependent Asian-style options using the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result, all terms of the Black-Merton-Sholes equation are consistently represented using a single high-order numerical scheme for the advection operator. We detail the additional steps required to solve the two-dimensional valuation problem compared to MPDATA valuations of vanilla instruments documented in a prior study. Using test cases employing fixed-strike instruments, we validate the solutions against Monte Carlo valuations, as well as against an approximate analytical solution in which geometric instead of arithmetic averaging is used. The analysis highlights the critical importance of the MPDATA corrective steps that improve the solution over the underlying first-order “upwind” step. The introduced valuation scheme is robust: conservative, non-oscillatory, and positive-definite; yet lucid: explicit in time, engendering intuitive stability-condition interpretation and inflow/outflow boundary-condition heuristics. MPDATA is particularly well suited for two-dimensional problems as it is not a dimensionally split scheme. The documented valuation workflow also constitutes a useful two-dimensional case for testing advection schemes featuring both Monte Carlo solutions and analytic bounds. An implementation of the introduced valuation workflow, based on the PyMPDATA package and the Numba Just-In-Time compiler for Python, is provided as free and open source software. ...

Numerical analysis of a particle system for the calibrated Heston-type local stochastic volatility model ArXiv ID: 2504.14343 “View on arXiv” Authors: Unknown Abstract We analyse a Monte Carlo particle method for the simulation of the calibrated Heston-type local stochastic volatility (H-LSV) model. The common application of a kernel estimator for a conditional expectation in the calibration condition results in a McKean-Vlasov (MV) stochastic differential equation (SDE) with non-standard coefficients. The primary challenges lie in certain mean-field terms in the drift and diffusion coefficients and the $1/2$-Hölder regularity of the diffusion coefficient. We establish the well-posedness of this equation for a fixed but arbitrarily small bandwidth of the kernel estimator. Moreover, we prove a strong propagation of chaos result, ensuring convergence of the particle system under a condition on the Feller ratio and up to a critical time. For the numerical simulation, we employ an Euler-Maruyama scheme for the log-spot process and a full truncation Euler scheme for the CIR volatility process. Under certain conditions on the inputs and the Feller ratio, we prove strong convergence of the Euler-Maruyama scheme with rate $1/2$ in time, up to a logarithmic factor. Numerical experiments illustrate the convergence of the discretisation scheme and validate the propagation of chaos in practice. ...

Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model ArXiv ID: 2503.02965 “View on arXiv” Authors: Unknown Abstract Fourier-based methods are central to option pricing and hedging when the Fourier-Laplace transform of the log-price and integrated variance is available semi-explicitly. This is the case for the Volterra Stein-Stein stochastic volatility model, where the characteristic function is known analytically. However, naive evaluation of this formula can produce discontinuities due to the complex square root of a Fredholm determinant, particularly when the determinant crosses the negative real axis, leading to severe numerical instabilities. We analyze this phenomenon by characterizing the determinant’s crossing behavior for the joint Fourier-Laplace transform of integrated variance and log-price. We then derive an expression for the transform to account for such crossings and develop efficient algorithms to detect and handle them. Applied to Fourier-based pricing in the rough Stein-Stein model, our approach significantly improves accuracy while drastically reducing computational cost relative to existing methods. ...

Leveraging Machine Learning for High-Dimensional Option Pricing within the Uncertain Volatility Model ArXiv ID: 2407.13213 “View on arXiv” Authors: Unknown Abstract This paper explores the application of Machine Learning techniques for pricing high-dimensional options within the framework of the Uncertain Volatility Model (UVM). The UVM is a robust framework that accounts for the inherent unpredictability of market volatility by setting upper and lower bounds on volatility and the correlation among underlying assets. By integrating advanced Machine Learning algorithms, we aim to enhance the accuracy and efficiency of option pricing under the UVM, especially when the option price depends on a large number of variables, such as in basket or path-dependent options. In this paper, we consider two approaches based on Machine Learning. The first one, termed GTU, evolves backward in time, dynamically selecting at each time step the most expensive volatility and correlation for each market state. Specifically, it identifies the particular values of volatility and correlation that maximize the expected option value at the next time step, and therefore, an optimization problem must be solved. This is achieved through the use of Gaussian Process regression, the computation of expectations via a single step of a multidimensional tree and the Sequential Quadratic Programming optimization algorithm. The second approach, referred to as NNU, leverages neural networks and frames pricing in the UVM as a control problem. Specifically, we train a neural network to determine the most adverse volatility and correlation for each simulated market state, generated via random simulations. The option price is then obtained through Monte Carlo simulations, which are performed using the values for the uncertain parameters provided by the neural network. The numerical results demonstrate that the proposed approaches can significantly improve the precision of option pricing particularly in high-dimensional contexts. ...

Deep Penalty Methods: A Class of Deep Learning Algorithms for Solving High Dimensional Optimal Stopping Problems ArXiv ID: 2405.11392 “View on arXiv” Authors: Unknown Abstract We propose a deep learning algorithm for high dimensional optimal stopping problems. Our method is inspired by the penalty method for solving free boundary PDEs. Within our approach, the penalized PDE is approximated using the Deep BSDE framework proposed by \cite{“weinan2017deep”}, which leads us to coin the term “Deep Penalty Method (DPM)” to refer to our algorithm. We show that the error of the DPM can be bounded by the loss function and $O(\frac{“1”}λ)+O(λh) +O(\sqrt{“h”})$, where $h$ is the step size in time and $λ$ is the penalty parameter. This finding emphasizes the need for careful consideration when selecting the penalization parameter and suggests that the discretization error converges at a rate of order $\frac{“1”}{“2”}$. We validate the efficacy of the DPM through numerical tests conducted on a high-dimensional optimal stopping model in the area of American option pricing. The numerical tests confirm both the accuracy and the computational efficiency of our proposed algorithm. ...

Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging ArXiv ID: 2401.08600 “View on arXiv” Authors: Unknown Abstract We consider two data driven approaches, Reinforcement Learning (RL) and Deep Trajectory-based Stochastic Optimal Control (DTSOC) for hedging a European call option without and with transaction cost according to a quadratic hedging P&L objective at maturity (“variance-optimal hedging” or “final quadratic hedging”). We study the performance of the two approaches under various market environments (modeled via the Black-Scholes and/or the log-normal SABR model) to understand their advantages and limitations. Without transaction costs and in the Black-Scholes model, both approaches match the performance of the variance-optimal Delta hedge. In the log-normal SABR model without transaction costs, they match the performance of the variance-optimal Barlett’s Delta hedge. Agents trained on Black-Scholes trajectories with matching initial volatility but used on SABR trajectories match the performance of Bartlett’s Delta hedge in average cost, but show substantially wider variance. To apply RL approaches to these problems, P&L at maturity is written as sum of step-wise contributions and variants of RL algorithms are implemented and used that minimize expectation of second moments of such sums. ...