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

Paper: arXiv 2602.06424

Authors: Chiheb Ben Hammouda, Truong Ngoc Nguyen

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework combining Fourier inversion and randomized quasi-Monte Carlo (RQMC) methods for multivariate shortfall risk, supported by comprehensive convergence proofs and computational complexity analysis. While the numerical experiments effectively showcase performance against standard SAA and stochastic approximation benchmarks, the paper remains primarily theoretical without real-world financial data backtests.

Research Flowchart

  flowchart TD
    A[Research Goal: Fast, Accurate Estimation of<br>Multivariate Shortfall Risk & Allocations] --> B[Inputs: Risk Factor Models & Interconnected Loss Structures]
    B --> C[Method: Transform Risk Expectations<br>from Physical to Frequency Domain via Fourier Inversion]
    C --> D[Sampling: Apply Randomized Quasi-Monte Carlo<br>RQMC on Smooth Frequency Integrands]
    D --> E[Optimization: Single-Level & Multilevel RQMC<br>Leveraging Geometric Convergence]
    E --> F[Outcomes: Outperforms SAA & Stochastic Benchmarks<br>Higher Accuracy & Reduced Complexity]

Cited References (59)

1. Recent Advances in Randomized Quasi - Monte Carlo Methods (2002) — link
2. Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation (2022) — link
3. Quasi- Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi - Asset Options — → 2403.02832
4. Nonasymptotic Convergence Rate of Quasi - Monte Carlo : Applications to Linear Elliptic PDEs with Lognormal Coefficients and Importance Samplings (2026)
5. Hierarchical adaptive sparse grids and quasi- Monte Carlo for option pricing under the rough Bergomi model (2020)
6. Numerical smoothing with hierarchical adaptive sparse grids and quasi- Monte Carlo methods for efficient option pricing (2023) — link
7. Numerical Smoothing with Multilevel Monte Carlo for Efficient Option Pricing and Density Estimation (2020) — link
8. Achieving High Convergence Rates by Quasi - Monte Carlo and Importance Sampling for Unbounded Integrands (2024) — link

1 of 59 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.