A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation

Paper: arXiv 2304.01207

Authors: Stéphane Crépey, Noufel Frikha, Azar Louzi

Abstract

We propose a multilevel stochastic approximation (MLSA) scheme for the computation of the value-at-risk (VaR) and expected shortfall (ES) of a financial loss, which can only be computed via simulations conditionally on the realisation of future risk factors. Thus the problem of estimating its VaR and ES is nested in nature and can be viewed as an instance of stochastic approximation problems with biased innovations. In this framework, for a prescribed accuracy $\varepsilon$, the optimal complexity of a nested stochastic approximation algorithm is shown to be of the order $\varepsilon^{-3}$. To estimate the VaR, our MLSA algorithm attains an optimal complexity of the order $\varepsilon^{-2-δ}$, where $δ\in(0,1)$ is some parameter depending on the integrability degree of the loss, while to estimate the ES, the algorithm achieves an optimal complexity of the order $\varepsilon^{-2}|\ln{\varepsilon}|^2$. Numerical studies of the joint evolution of the error rate and the execution time demonstrate how our MLSA algorithm regains a significant amount of the performance lost due to the nested nature of the problem.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework utilizing Multilevel Monte Carlo and stochastic approximation to significantly reduce computational complexity for nested VaR and ES estimation. While theoretical innovation and mathematical derivations are strong, empirical evaluation relies primarily on numerical simulations rather than real-world market datasets.

Research Flowchart

  flowchart TD
    A[Goal: Efficiently estimate VaR and ES<br>for nested financial losses] --> B[Inputs: Future risk factors &<br>conditional loss simulations]
    B --> C[Problem Formulation: Stochastic Approximation<br>with biased innovations]
    C --> D[Methodology: Apply Multilevel<br>Stochastic Approximation MLSA]
    D --> E[Numerical Simulation: Evaluate joint error rate<br>vs. execution time]
    E --> F[Outcomes & Complexity Reduction]
    F --> G[VaR Complexity: Reduced from Oε⁻³<br>to Oε⁻²⁻δ]
    F --> H[ES Complexity: Reduced from Oε⁻³<br>to Oε⁻²|ln ε|²]