Geometrically Convergent Simulation of the Extrema of Levy Processes

• 成果类型: Article
• 署名作者: Gonzalez Cazares, Jorge Ignacio; Mijatovic, Aleksandar; Uribe Bravo, Geronimo
• 署名单位: Alan Turing Institute; University of Warwick; Universidad Nacional Autonoma de Mexico
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1163
• 发表日期: 2022
• 页码: 1141-1168
• 关键词: multilevel monte-carlo; continuity correction; barrier options; supremum; discrete

摘要

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum, and the time of the supremum of a general Levy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in L-p (for any p >= 1) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct nonasymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e., of order epsilon(-2) if the mean squared error is at most epsilon(2)) for locally Lipschitz and barrier-type functions of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.