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

TL;DR

This work develops an optimization-aware Fourier–RQMC framework to efficiently estimate the Multivariate Shortfall Risk Measure and the corresponding optimal capital allocations under a pre-aggregation risk model. By performing risk evaluation in the frequency domain, the authors leverage smoother integrands and, together with randomized quasi–Monte Carlo sampling and adaptive domain transformations, obtain provable convergence and favorable complexity. A key innovation is the iteration-indexed multilevel Fourier–RQMC construction, which uses differences between successive optimization iterates to achieve substantial variance reduction and cost savings. Theoretical results include error and complexity bounds for both single-level and multilevel schemes, and numerical experiments across exponential and QPC loss families with Gaussian and NIG distributions demonstrate superior accuracy and computational efficiency over SAA and SA benchmarks. The framework is robust along the optimization trajectory and extends naturally to other multivariate risk measures with Fourier representations.

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.

Figures (12)

• Figure 3.1: Effect of the damping parameter ${\mathbf{K}_{1,1}^{(0)}}$ on the QPC loss integrand component $h^{(0)}_{1,1}$ for a $10$-dimensional Gaussian loss vector (Example in Section \ref{['subsec:10D_gaussian_qcl']}).
• Figure 3.2: Unregularized optimal damping selection for the QPC loss with a $3$-dimensional NIG loss vector (Example in Section \ref{['subsec:3D_NIG_qcl']}).
• Figure 3.3: Regularized damping selection for the QPC loss with a $3$-dimensional NIG loss vector (parameter setting in Section \ref{['subsec:3D_NIG_qcl']}), using the anisotropic weighting matrix $\boldsymbol W_{1,2}$. Compared to Figure \ref{['fig:damping_subplots']}, regularization shifts the optimal damping away from the boundary of the analyticity strip, yielding a smoother and better-conditioned integrand $h^{(0)}_{1,2}$.
• Figure 4.1: Transformed integrand component $\widetilde{h}^{(0)}_{1,1}$ for the QPC loss and a $3$-dimensional NIG loss vector, with $\mathbf K^{(0)}_{1,1}=4.5$ and $\mathbf m_{1,1}=-0.8$ (parameter setting in Section \ref{['subsec:3D_NIG_qcl']}).
• Figure 4.2: 1D slice of the transformed integrand component ${\widetilde{h}_{2,(3,4)}^{(0)}}$ with $\mathbf{K}_{2,(3,4)}^{(0)} = \left[2.763,0.523\right], \mathbf{m}_{2,(3,4)} =\left[0.255,0.105\right]$ for the QPC loss and a $10$-dimensional Gaussian loss vector (parameter setting in Section \ref{['subsec:10D_gaussian_qcl']}).

Theorems & Definitions (66)

• Definition 2.2: Multivariate Loss Function
• Example 2.3
• Definition 2.5: Multivariate shortfall risk
• Definition 2.6
• Theorem 2.8: Theorem 3.4 in armenti_multivariate_2018
• Remark 2.9
• Remark 2.10
• Remark 2.11: Interchanging Differentiation and Expectation
• Corollary 2.12: Uniqueness of the optimal allocation
• proof