[Paper Review] Single- and Multi-Level Fourier-RQMC Methods for Multivariate Shortfall Risk

Chiheb Ben Hammouda, Truong Nguyen|arXiv (Cornell University)|Feb 6, 2026

Risk and Portfolio Optimization0 citations

TL;DR

The paper develops single-level and multilevel Fourier-RQMC methods to efficiently estimate multivariate shortfall risk and optimal capital allocations, using Fourier inversion and RQMC to achieve improved convergence and computational complexity.

ABSTRACT

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.

Motivation & Objective

• Advance numerical estimation of Multivariate Shortfall Risk Measures (MSRM) and corresponding allocations.
• Incorporate Fourier inversion and randomized quasi–Monte Carlo (RQMC) to exploit smoother frequency-domain integrands.
• Provide rigorous error and computational complexity analysis for single-level and multilevel schemes.
• Preserve regularity of integrands along optimization trajectories through adaptive damping and domain transforms.
• Develop a scalable optimization framework (SQP-based) integrated with Fourier–RQMC surrogates.

Proposed method

• Represent MSRM expectations and gradients in the frequency domain via Fourier transforms of the loss function and its derivatives.
• Use admissible contour shifts (damping) to ensure integrability and smoothness of the Fourier integrands.
• Decompose high-dimensional Fourier integrals into finite sums of lower-dimensional componentwise integrals based on interaction order.
• Apply single-level Fourier–RQMC to estimate g(m), ∇g(m), ∇²g(m) within an SQP framework.
• Extend to a multilevel Fourier–RQMC scheme that leverages geometric convergence of the optimization to reduce cost.
• Solve the resulting SQP subproblems with a damping-aware line search and SLSQP implementation.

Experimental results

Research questions

• RQ1 Can MSRM and the associated optimal allocations be efficiently estimated using frequency-domain representations and RQMC sampling?
• RQ2 How do damping rules and domain transformations affect the regularity and accuracy of Fourier-based MSRM estimators?
• RQ3 Does a multilevel RQMC construction reduce computational cost while preserving accuracy along optimization trajectories?
• RQ4 What are the convergence and complexity properties of single-level versus multilevel Fourier–RQMC in estimating MSRM?
• RQ5 How do these methods compare to SAA and SA benchmarks across different loss structures and dimensions?

Key findings

• Fourier–RQMC estimators provide improved asymptotic convergence and complexity rates compared with benchmark Monte Carlo approaches.
• A multilevel RQMC construction achieves additional reductions in computational cost by exploiting the optimization’s geometric convergence.
• Adaptive damping and regularized update rules maintain robustness of the integration along the optimization trajectory.
• Domain transformations tailored for RQMC preserve regularity and improve boundary handling during integration.
• Numerical experiments show Fourier–RQMC outperforming standard benchmarks across various loss models and dimensions.