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

Paper: arXiv 2510.23461

Authors: Riccardo Gozzo

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced stochastic calculus, SDE discretizations (QE scheme), and adaptive multilevel splitting theory, indicating high mathematical density. It provides rigorous empirical validation with a full open-source Rcpp implementation, comparison benchmarks, and quantitative results (200-fold gains), demonstrating strong backtest-ready rigor.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Address computational inefficiency<br>in pricing rare-event binary options"] --> B["Proposed Methodology<br>Adaptive Multilevel Splitting AMS"]
  
  B --> C{"Model & Input Setup"}
  C --> D["Black-Scholes & Heston Dynamics"]
  C --> E["Discontinuous Payoff Structures"]
  
  D & E --> F["Core AMS Process<br>Splits simulation into conditional events<br>Biased variance reduction"]
  
  F --> G["Computational Experimentation<br>European, Asian, Barrier, Multidim"]
  
  G --> H{"Outcomes & Findings<br>vs Standard Monte Carlo"}
  
  H -- Efficiency Gain --> I["200-fold improvement<br>in deep out-of-the-money regimes"]
  H -- Methodology Validity --> J["Unbiased estimator preserved<br>Robust to importance function"]
  
  I & J --> K["First Application of AMS<br>to Derivative Pricing<br>Open-source Rcpp Implementation"]

Cited References (57)

1. Stochastic Calculus for Finance II: Continuous-Time Models (2004) — link
2. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (2004) — link
3. Monte Carlo methods in financial engineering (2004) — link
4. Rare Event Simulation (2015) — link
5. Variance Reduction Techniques in Monte Carlo Methods (2010)
6. Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options (1999) — link
7. Derivatives, Market Liquidity, and Infrastructural Finance (2025)
8. Parametric heat wave insurance (2023) — link

0 of 57 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.