@article{Kim2025, 
author = {Eungpyo Kim and Jaegi Jeon},
title = {Deterministic value iteration for perpetual American put options},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {12},
pages = {29784-29814},
keywords = {American options, perpetual option, optimal stopping, exercise boundary, value iteration, Gauss–Hermite quadrature, Black–Scholes model, Heston model},
url = {https://www.sciopen.com/article/10.3934/math.20251309},
doi = {10.3934/math.20251309},
abstract = {We introduce a deterministic, policy-targeted Bellman value-iteration framework for computing the optimal exercise boundary of perpetual American put options. Our method replaces path sampling in the Bellman operator with Gauss–Hermite quadrature and employs shape-preserving interpolation for off-grid evaluations, eliminating sampling noise and reducing computational cost. Under the Black–Scholes (BS) model, our approach recovers the analytic boundary with a mean absolute percentage error below 1.5% in approximately 19–56 seconds. The resulting policy values, evaluated via Monte Carlo simulation, deviate from the analytic benchmark by less than 0.07%. For the Heston model, where no closed-form solution exists, our method produces boundaries that differ from a high-resolution finite-difference benchmark by 1–5%. Despite these boundary deviations, the expected payoffs from the policies are remarkably close, with relative policy value gaps well below 0.2%. Notably, our method computes the boundary in about 127–180 seconds, a significant speedup compared to the 2,103–3,119 seconds required by the finite-difference method. This work presents a practical and robust alternative for optimal stopping problems, offering a compelling balance of speed and accuracy, particularly when partial differential equation (PDE) solvers are cumbersome or Monte Carlo simulation is prohibitively expensive.}
}