arXiv subjects

Chiheb Ben Hammouda

Publications and source records attributed to Chiheb Ben Hammouda.

16 recordsLinked to original sources

arXiv2026

Single- and Multilevel Quadrature with Error Control for Fourier Pricing under the Rough Heston Model

Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is $O(Δt^p)$, that evaluating the characteristic function once costs $O(Δt^{-β})$, and that the algebraic Gauss-Laguerre quadrature error is $O(N^{-s_{SL}/2})$, where $s_{SL}$ is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires $O(ε^{-(β/p+2/s_{SL})})$ computational work to achieve accuracy $ε$, whereas the proposed multilevel method requires $O(ε^{-β/p})$ computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations.

q-fin.CP↗

arXiv2026

Data-Driven Stochastic Optimal Control for Intraday Electricity Trading by Renewable Producers

The rapid growth of weather-dependent renewable generation increases price volatility and imbalance penalty risk in power markets, creating the need for advanced quantitative trading strategies. We develop a data-driven continuous-time stochastic optimal control framework for intraday electricity trading using stochastic differential equations with drift terms ensuring mean reversion to deterministic forecast trajectories. Production follows a Jacobi diffusion, while prices follow an asymmetric jump-diffusion to reflect the heavy-tailed behavior observed in intraday markets. The framework accounts for realistic market features by incorporating gate closure and energy-based imbalance settlement over the delivery window, where the path-dependent imbalance cost is handled by state augmentation to preserve the Markovian structure. The value function is characterized via the dynamic programming principle by a three-stage sequence of two linear Kolmogorov backward equations and a nonlinear Hamilton-Jacobi-Bellman partial integro-differential equation. To solve this problem efficiently, we propose a monotone IMEX finite-difference scheme with operator splitting, semi-implicit linearization, and a differential formulation for the jump operator. Numerical experiments based on German market data indicate that, under the provided forecasts, the computed strategy outperforms the TWAP benchmark and approaches the perfect-foresight benchmark. Sensitivity experiments further show how jump intensity, delivery-window length, and trading horizon affect the trading policy and the resulting profit-and-loss distribution.

q-fin.MF↗

arXiv2026

A Damped SWIFT Method for European Option Pricing: Coefficients Decay, Truncation, and Error Analysis

We introduce a damped variant of the Shannon Wavelet Inverse Fourier Technique (SWIFT) for pricing European options when the characteristic function of the underlying model is available. The key idea is to apply an exponential damping transformation to the payoff, which enables the direct computation of Fourier coefficients in the frequency domain without introducing an additional physical-domain truncation parameter. We provide a rigorous analysis of the decay of these coefficients by exploiting the singularity structure of the associated Fourier transforms. For light-tailed models, we obtain Gaussian-type decay estimates, while for semi-heavy and heavy-tailed models whose singularities are poles, algebraic branch points, or logarithmic branch points, we derive exponential decay bounds with explicit polynomial prefactors. The resulting sharp bounds make it possible to truncate the Fourier series without relying on the cumulants of the underlying density, which are often unavailable or difficult to compute in practice. We further derive an error decomposition separating projection, truncation, and quadrature errors, and translate the analysis into practical rules for selecting the damping parameter, resolution level, and truncation range. Numerical experiments demonstrate that the proposed approach consistently improves the accuracy of the original SWIFT method while requiring a significantly smaller number of Fourier coefficients and remaining stable in cases where the undamped method deteriorates.

math.NA↗

arXiv2026

Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Options

Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. Fourier methods leverage the regularity properties of the integrand in the Fourier domain to accurately and rapidly value options that typically lack regularity in the physical domain. However, most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. To overcome this difficulty, this work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{R}^d$, requires a domain transformation to $[0,1]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and hence deteriorate the performance of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on boundary growth conditions on the transformed integrand. The proposed transformation preserves sufficient regularity of the original integrand for fast convergence of the RQMC method. To validate our analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over MC or TP in the Fourier domain, and over MC in the physical domain for options with up to 15 assets.

q-fin.CP↗

arXiv2026

SigMA: Path Signatures and Multi-head Attention for Learning Parameters in fBm-driven SDEs

Stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) are increasingly used to model systems with rough dynamics and long-range dependence, such as those arising in quantitative finance and reliability engineering. However, these processes are non-Markovian and lack a semimartingale structure, rendering many classical parameter estimation techniques inapplicable or computationally intractable beyond very specific cases. This work investigates two central questions: (i) whether integrating path signatures into deep learning architectures can improve the trade-off between estimation accuracy and model complexity, and (ii) what constitutes an effective architecture for leveraging signatures as feature maps. We introduce SigMA (Signature Multi-head Attention), a neural architecture that integrates path signatures with multi-head self-attention, supported by a convolutional preprocessing layer and a multilayer perceptron for effective feature encoding. SigMA learns model parameters from synthetically generated paths of fBm-driven SDEs, including fractional Brownian motion, fractional Ornstein-Uhlenbeck, and rough Heston models, with a particular focus on estimating the Hurst parameter and on joint multi-parameter inference, and it generalizes robustly to unseen trajectories. Extensive experiments on synthetic data and two real-world datasets (i.e., equity-index realized volatility and Li-ion battery degradation) show that SigMA consistently outperforms CNN, LSTM, vanilla Transformer, and Deep Signature baselines in accuracy, robustness, and model compactness. These results demonstrate that combining signature transforms with attention-based architectures provides an effective and scalable framework for parameter inference in stochastic systems with rough or persistent temporal structure.

cs.LG↗

arXiv2026

Single- and Multi-Level Fourier-RQMC Methods for Multivariate Shortfall Risk

Multivariate shortfall risk measures provide a principled framework for quantifying systemic risk and determining capital allocations prior to aggregation in interconnected financial systems. Despite their well established theoretical properties, the numerical estimation of multivariate shortfall risk and the corresponding optimal allocations remains computationally challenging, as existing Monte Carlo based approaches can be numerically expensive due to slow convergence. In this work, we develop a new class of single and multilevel numerical algorithms for estimating multivariate shortfall risk and the associated optimal allocations, based on a combination of Fourier inversion techniques and randomized quasi Monte Carlo (RQMC) sampling. Rather than operating in physical space, our approach evaluates the relevant expectations appearing in the risk constraint and its optimization in the frequency domain, where the integrands exhibit enhanced smoothness properties that are well suited for RQMC integration. We establish a rigorous mathematical framework for the resulting Fourier RQMC estimators, including convergence analysis and computational complexity bounds. Beyond the single level method, we introduce a multilevel RQMC scheme that exploits the geometric convergence of the underlying deterministic optimization algorithm to reduce computational cost while preserving accuracy. Numerical experiments demonstrate that the proposed Fourier RQMC methods outperform sample average approximation and stochastic optimization benchmarks in terms of accuracy and computational cost across a range of models for the risk factors and loss structures. Consistent with the theoretical analysis, these results demonstrate improved asymptotic convergence and complexity rates relative to the benchmark methods, with additional savings achieved through the proposed multilevel RQMC construction.

q-fin.CP↗

arXiv2025

Filtered Markovian Projection: Dimensionality Reduction in Filtering for Stochastic Reaction Networks

Stochastic reaction networks (SRNs) model stochastic effects for various applications, including intracellular chemical or biological processes and epidemiology. A typical challenge in practical problems modeled by SRNs is that only a few state variables can be dynamically observed. Given the measurement trajectories, one can estimate the conditional probability distribution of unobserved (hidden) state variables by solving a stochastic filtering problem. In this setting, the conditional distribution evolves over time according to an extensive or potentially infinite-dimensional system of coupled ordinary differential equations with jumps, known as the filtering equation. The current numerical filtering techniques, such as the filtered finite state projection (D'Ambrosio et al., 2022), are hindered by the curse of dimensionality, significantly affecting their computational performance. To address these limitations, we propose to use a dimensionality reduction technique based on the Markovian projection (MP), initially introduced for forward problems (Ben Hammouda et al., 2024). In this work, we explore how to adapt the existing MP approach to the filtering problem and introduce a novel version of the MP, the Filtered MP, that guarantees the consistency of the resulting estimator. The novel consistent MP filter employs a reduced-variance particle filter for estimating the jump intensities of the projected model and solves the filtering equations in a low-dimensional space. The analysis and empirical results highlight the superior computational efficiency of projection methods compared to the existing filtered finite state projection in the large dimensional setting.

math.NA↗

arXiv2024

Lagrangian Relaxation for Continuous-Time Optimal Control of Coupled Hydrothermal Power Systems Including Storage Capacity and a Cascade of Hydropower Systems with Time Delays

This work considers a short-term, continuous time setting characterized by a coupled power supply system controlled exclusively by a single provider and comprising a cascade of hydropower systems (dams), fossil fuel power stations, and a storage capacity modeled by a single large battery. Cascaded hydropower generators introduce time-delay effects in the state dynamics, which are modeled with differential equations, making it impossible to use classical dynamic programming. We address this issue by introducing a novel Lagrangian relaxation technique over continuous-time constraints, constructing a nearly optimal policy efficiently. This approach yields a convex, nonsmooth optimization dual problem to recover the optimal Lagrangian multipliers, which is numerically solved using a limited memory bundle method. At each step of the dual optimization, we need to solve an optimization subproblem. Given the current values of the Lagrangian multipliers, the time delays are no longer active, and we can solve a corresponding nonlinear Hamilton--Jacobi--Bellman (HJB) Partial Differential Equation (PDE) for the optimization subproblem. The HJB PDE solver provides both the current value of the dual function and its subgradient, and is trivially parallelizable over the state space for each time step. To handle the infinite-dimensional nature of the Lagrange multipliers, we design an adaptive refinement strategy to control the duality gap. Furthermore, we use a penalization technique for the constructed admissible primal solution to smooth the controls while achieving a sufficiently small duality gap. Numerical results based on the Uruguayan power system demonstrate the efficiency of the proposed mathematical models and numerical approach.

math.OC↗

arXiv2024

Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach

We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of (Ben Hammouda et al., 2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete $L^2$ regression. By solving the resulting projected Hamilton-Jacobi-Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

math.NA↗

arXiv2023

Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models

Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are competitive compared to alternative techniques because the integrand in the frequency space often has a higher regularity than that in the physical space. However, when designing a numerical quadrature method for most Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of high dimensionality. We propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas to address these challenges. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed optimization rule. Second, we employ sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. The extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some Lévy models demonstrates the advantages of adaptivity and the damping rule on the numerical complexity of quadrature methods. Moreover, for the tested two-asset examples, the proposed approach outperforms the COS method in terms of computational time. Finally, we show significant speed-up compared to the Monte Carlo method for up to six dimensions.

q-fin.CP↗

arXiv2023

Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities

The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in (Quantitative Finance, 23(2), 209-227, 2023), in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence, and consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.

q-fin.CP↗

arXiv2023

Learning-Based Importance Sampling via Stochastic Optimal Control for Stochastic Reaction Networks

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.

math.NA↗

arXiv2022

Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing

When approximating the expectations of a functional of a solution to a stochastic differential equation, the numerical performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and improve the regularity structure of the problem, we consider cases in which analytic smoothing (bias-free mollification) cannot be performed and introduce a novel numerical smoothing approach by combining a root-finding method with a one-dimensional numerical integration with respect to a single well-chosen variable. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (ie., the Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, focusing on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we demonstrate the advantages of combining numerical smoothing with the ASGQ and QMC methods over these methods without smoothing and the Monte Carlo approach. Finally, our approach is generic and can be applied to solve a broad class of problems, particularly approximating distribution functions, computing financial Greeks, and estimating risk quantities.

q-fin.CP↗

arXiv2020

Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to \textit{catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from $β=1$ for the standard case (without IS), to $β=1+δ$, where $0<δ<1$ is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $\text{TOL}$, this results in an improvement of the complexity from $\mathcal{O}\left(\text{TOL}^{-2} \log(\text{TOL})^2\right)$ in the standard case to $\mathcal{O}\left(\text{TOL}^{-2}\right)$, which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

math.NA↗

arXiv2020

Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model

The rough Bergomi (rBergomi) model, introduced recently in [5], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits with empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

q-fin.CP↗

arXiv2016

Multilevel Hybrid Split-Step Implicit Tau-Leap

In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (Explicit-TL) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012). This method uses split-step implicit tau-leap (SSI-TL) at levels where the SSI-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.

math.NA↗