Adaptive Multilevel Splitting: First Application to Rare-Event Derivative Pricing

TL;DR

This paper introduces an adaptive multilevel splitting method for efficiently pricing rare-event financial derivatives, significantly reducing computational costs while maintaining unbiased estimates, with broad applicability demonstrated through various option types.

Contribution

It presents the first application of AMS to derivative pricing, adapting it for financial models and demonstrating substantial efficiency gains over standard Monte Carlo methods.

Findings

Up to 200-fold reduction in computational effort

Robust performance across different models and importance functions

Unbiased estimates maintained in rare-event regimes

Abstract

This work investigates the computational burden of pricing binary options in rare event regimes and introduces an adaptation of the adaptive multilevel splitting (AMS) method for financial derivatives. Standard Monte Carlo becomes inefficient for deep out-of-the-money binaries due to discontinuous payoffs and extremely small exercise probabilities, requiring prohibitively large sample sizes for accurate estimation. The proposed AMS framework reformulates the rare-event problem as a sequence of conditional events and is applied under both Black-Scholes and Heston dynamics. Numerical experiments cover European, Asian, and up-and-in barrier digital options, together with a multidimensional digital payoff designed as a stress test. Across all contracts, AMS achieves substantial gains, reaching up to 200-fold improvements over standard Monte Carlo, while preserving unbiasedness and showing robust performance with respect to the choice of importance function. To the best of our knowledge, this is the first application of AMS to derivative pricing. An open-source Rcpp implementation is provided, supporting multiple discretisation schemes and alternative importance functions.