Adaptive Multilevel Stochastic Approximation of the Value-at-Risk

ArXiv ID: 2408.06531 “View on arXiv”

Authors: Unknown

Abstract

Crépey, Frikha, and Louzi (2023) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The optimal complexity of the scheme is in $O({"\varepsilon"}^{"-5/2"})$, ${"\varepsilon"} > 0$ being a prescribed accuracy, which is suboptimal when compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity of the Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its optimal complexity is in $O({"\varepsilon"}^{"-2"}|\ln {"\varepsilon"}|^{“5/2”})$. Our theoretical analysis is exemplified through numerical experiments.

Keywords: Value-at-Risk (VaR), Multilevel Stochastic Approximation, Monte Carlo, Heaviside Function, Risk Measurement

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, featuring advanced stochastic calculus, multilevel methods, and complexity analysis, but relies on generic toy models for validation without backtest-ready datasets or code. It targets theoretical improvements over existing Monte Carlo methods rather than direct trading application.

  flowchart TD
    A["Research Goal<br>Design optimal multilevel stochastic approximation<br>for Value-at-Risk computation"] --> B["Key Methodology<br>Adaptive multilevel stochastic approximation<br>with dynamic inner sample selection"]
    B --> C["Data/Inputs<br>Monte Carlo simulatable<br>financial loss process"]
    C --> D["Computational Process<br>Recursive biased gradient evaluation<br>mitigating Heaviside discontinuity"]
    D --> E["Key Findings<br>Optimal complexity: O(ε⁻²|ln ε|⁵ᐟ²)<br>(Improved from O(ε⁻⁵ᐟ²))<br>Validated through numerical experiments"]