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

This work analyzes the computational burden of pricing binary options in rare-event settings and introduces an adaptation of the adaptive multilevel splitting (AMS) method for financial derivatives. Standard Monte Carlo is inefficient for deep out of the money binaries due to discontinuous payoffs and low exercise probabilities, requiring very large samples for accurate estimates. An AMS scheme is developed for binary options under Black-Scholes and Heston dynamics, reformulating the rare-event problem as a sequence of conditional events. Numerical experiments compare the method to Monte Carlo and to other techniques such as antithetic variables and multilevel Monte Carlo (MLMC) across four contracts: European digital calls and puts, and Asian digital calls and puts. Results show up to a 200-fold computational gain for deep out-of-the-money cases while preserving unbiasedness. No evidence is found of prior applications of AMS to financial derivatives. The approach improves pricing efficiency for rare-event contracts such as parametric insurance and catastrophe linked securities. An open-source Rcpp implementation is provided, supporting multiple discretizations and importance functions.