Thesis

Asymptotic Normality of the Adaptive Multilevel Splitting Estimator for Conditional Value-at-Risk

Provable, single-run confidence intervals for tail risk deep in the loss distribution, from an interacting-particle estimator and a vectorised C++ engine.

last updated 09 June 2026

Abstract

Conditional Value-at-Risk averages the loss beyond the Value-at-Risk quantile. It is coherent where Value-at-Risk is not, so it never penalises diversification, and Basel III ties market-risk capital to it. The losses that set it are rare, and crude Monte Carlo needs about ten billion paths to reach one percent accuracy on a one-in-a-million event, far past any practical budget.

This thesis studies adaptive multilevel splitting, an interacting-particle method that sets each level from an order statistic of the cohort rather than a fixed threshold. The main result is a central limit theorem for the resulting CVaR estimator under explicit regularity conditions for geometrically-ergodic driving processes, built by composing the splitting empirical-process functional CLT with the functional delta method through Hadamard differentiability of the CVaR map over a VC-subgraph Donsker class of level-set indicators.

A disjoint-ancestral-lines U-statistic reads the limiting variance off the particle genealogy in a single run, so every estimate carries a root-N confidence interval at no extra cost. The argument widens from CVaR to the coherent spectral risk measures, and the estimator is realised in a cache-aligned, vectorised C++ engine spanning Black-Scholes and the Heston model under Euler, quadratic-exponential, and Broadie-Kaya discretisations.

Research Framework

Rare-Event Problem

The tail that defines CVaR is rare, so crude Monte Carlo cost grows like one over the probability and exhausts any budget before reaching the deep quantiles that set capital.

Splitting Estimator

A fixed population of trajectories advances between adaptive levels set by cohort order statistics, killing laggards and branching survivors, which turns the one-over-probability growth into growth polynomial in the log-probability.

Limit Theory

Asymptotic normality carries the splitting empirical-process CLT through the CVaR functional by the delta method, and the inverse-quantile term cancels, so the limiting variance needs no estimate of the loss density at the quantile.

Variance and Coherence

One pass over the particle genealogy gives a consistent variance estimate through disjoint ancestral lines, and the same limit theory covers the wider family of coherent spectral risk measures.