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

TL;DR

The paper tackles the computational bottleneck of pricing binary options in rare-event regimes by adapting adaptive multilevel splitting (AMS) to finance. It demonstrates how AMS, with an adaptive sequence of levels and trajectory cloning, yields unbiased probability estimates for rare digital payoffs under Black–Scholes and Heston dynamics, and it analyzes sensitivity to the number of trajectories and resampling rate. Across European, Asian, barrier, and multi-asset digital options, AMS achieves substantial speedups—up to about 200x over standard Monte Carlo—while maintaining accuracy and robustness against the choice of importance function. A new open-source C++ (Rcpp) implementation, amsSim, supports multiple discretizations and payoff types, making AMS a practical tool for rare-event pricing and risk assessment with potential extensions to VaR, stress testing, and more complex stochastic models. The work establishes AMS as a scalable, unbiased, and highly efficient alternative to traditional variance-reduction methods in financial derivatives pricing.

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.

Figures (4)

• Figure 1: Illustration of the first two iterations of the AMS algorithm with population size $N=3$ and killing parameter $K=1$. The horizontal dashed lines represent successive adaptive levels $Z^{(1)}$ and $Z^{(2)}$, while the top dashed line denotes the rare level $L_{\max}$. All trajectories start from the common initial point $X_0$. At the first branching level, the trajectory with the lowest score is killed; one of the two better trajectories is cloned and its clone is restarted from the time at which the original path first crosses $Z^{(1)}$, after which it is resimulated using new random increments (coloured path). The same procedure is repeated at the second branching level. The figure shows how the algorithm progressively reallocates computational effort to trajectories that move closer to the rare-event level $L_{\max}$.
• Figure 2: Normalized variance (vertical axis) as a function of the discard fraction k (horizontal axis).
• Figure 3: Computational time (log scale) as a function of relative accuracy for different simulation methods for the Heston digital call and digital up-and-in barrier call; numerical values are reported in Tables \ref{['tab:heston_digital_call']}, \ref{['tab:heston_digital_put']}
• Figure 4: Computational time (log scale) as a function of relative accuracy for different simulation methods for the Black-Scholes and Heston digital Asian call; numerical values are reported in Tables \ref{['tab:bs_Asian_digital_call']}, \ref{['tab:heston_Asian_digital_call']}

Theorems & Definitions (1)

• Remark 4.1: Unbiasedness of AMS for digital options