Adaptive Multilevel Splitting for Option Simulation and Pricing

Simulation and pricing routines for rare-event options using Adaptive Multilevel Splitting and standard Monte Carlo under Black-Scholes and Heston models. Core routines are implemented in C++ via Rcpp and RcppArmadillo with lightweight R wrappers.

amsSim

Adaptive Multilevel Splitting (AMS) simulation tools for rare-event option pricing and path generation in continuous-time models.
Core routines are implemented in C++ (Rcpp/RcppArmadillo) for speed.

Status: under active development; preparing for CRAN submission.

Features

• Fast Monte-Carlo path simulation:
  • Black–Scholes (exact)
  • Heston variants (Euler / Milstein / QE Andersen 2008)
• Adaptive Multilevel Splitting estimator for rare events
• Clean R interface with vectorised inputs
• Minimal dependencies (Rcpp, RcppArmadillo)

Installation

Install the development version from GitHub:

Option A: via remotes

install.packages("remotes") remotes::install_github("RiccardoGozzo/amsSim")

Option B: via devtools

install.packages("devtools")

devtools::install_github("RiccardoGozzo/amsSim")

Option C: via pak (fast)

install.packages("pak", repos = "https://r-lib.github.io/p/pak/dev/")

pak::pak("RiccardoGozzo/amsSim")

Quick start

library(amsSim)

Black–Scholes toy run (fast)

set.seed(1) res <- simulate_AMS(1, n = 500, t = 1, p = 252, r = 0.03, sigma = 0.2, S0 = 1, rho = NULL) str(res)

AMS example (small, <5s)

set.seed(1) out <- AMS(model = 2, type = 3, funz = 1, n = 500, t = 1, p = 252, r = 0.03, sigma = 0.2, rho = -0.5, S0 = 1, rim = 0, Lmax = 0.5, strike = 1.3, K = 200) str(out)

Main functions

simulate_AMS(model, n, t, p, r, sigma, S0, rho = NULL, rim = 0L, v0 = 0.04)

Returns simulated paths: model 1 (Black–Scholes): matrix/list with S of size n x (p - rim + 1) Heston models (2–4): list with S, V

AMS(model, type, funz, n, t, p, r, sigma, S0, rho = NULL, rim = 0L, v0 = 0.04, Lmax = 0, strike = 1, K = 1L)

Runs the adaptive splitting loop and returns list(price, std).

System Requirements

• R version >= 4.0
• Rcpp (>= 1.0.0), RcppArmadillo (>= 0.11)
• Compilers:
  • macOS: Xcode command line tools (clang++)
  • Linux: g++ (>= 9), make, and BLAS/LAPACK libraries
  • Windows: Rtools (>= 4.0)

Contributing

Contributions are welcome!
If you wish to contribute, please:

1. Fork the repository on GitHub.
2. Create a new branch for your feature/bugfix.
3. Add clear documentation and, if possible, unit tests.
4. Submit a pull request.

License

This package is released under the MIT License.
See the file LICENSE for details.

Citation

If you use amsSim in your research, please cite it as follows:

Gozzo, R. (2025). Adaptive Multilevel Splitting: First Application to Rare-Event Derivative Pricing Available at: https://arxiv.org/html/2510.23461v1