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

Research report

WORK IN PROGRESS

Paper proposes a multi-level stochastic approximation model (MLSA) which nests numerous stochastic approximations at varying granularity levels within one another to generate fine-tuned forecasting values, especially in value at risk and expected shortfall scenarios.

Nested stochastic approximation functions are much more accurate for complex, high-dimensional analytical scenarios than single-function approximations/models.
MLSAs are likely the next step after Monte-Carlo simulations in Concrete’s use case when Monte Carlo simulations are computationally inefficient.

Implementing an MLSA, especially one as complex as this, is non-trivial to say the least. MLSAs are beneficial for extremely large and complex portfolios and requires a skilled team to implement. Monte Carlo based approaches are better suited for Concrete’s use cases prior to significant scale and model complexity.
This paper provides no empirical evidence to support the model’s efficacy. As such, it takes significant effort to motivate a trial run of such a model for Concrete.

Methods and outputs.

This paper proposes the MLSA and accompanying parameterization algorithms applied to VaR and ES calculations. The models are not tested on any historical or live data, but pure and applied mathematical proofs and cases are provided.
Value at risk (VaR) and expected shortfall (ES) are battle-tested, widely-used measures of risk for leveraged firms. They measure the expected losses or exposure to losses for a portfolio if faced with extremal market scenarios (as defined by a confidence interval past 95% generally). Crucially, VaR and ES calculations are linked via a convex optimization problem.
Multi-level stochastic approximation algorithms (MLSAs) are extensions of multi-level Monte Carlo path simulations, but differ from that in that they model stochastic processes at a range of resolution depths. They then combine these outputs to form complex forecasting outputs that are flexible and computationally efficient.
The paper outlines the parameterization of successive approximation granularity levels and which are nested and proven to converge. The tuning of these models and the approximation algorithms they optimize towards allow for the fine-tuning of MLSAs to complex portfolios with highly volatile assets.