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Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation
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González Cázares, Jorge and Mijatović, Aleksandar (2022) Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation. Finance and Stochastics, 26 (4). pp. 671-732. doi:10.1007/s00780-022-00486-7 ISSN 1432-1122.
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Official URL: https://doi.org/10.1007/s00780-022-00486-7
Abstract
We develop a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in Cázares and Mijatović (SBG approximation. GitHub repository. Available online at https://github.com/jorgeignaciogc/SBG.jl (2020)), exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes.
| Item Type: | Journal Article | |||||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||||||||
| SWORD Depositor: | Library Publications Router | |||||||||||||||||||||
| Library of Congress Subject Headings (LCSH): | Lévy processes, Gaussian processes, Stochastic processes, Monte Carlo method | |||||||||||||||||||||
| Journal or Publication Title: | Finance and Stochastics | |||||||||||||||||||||
| Publisher: | Springer Berlin Heidelberg | |||||||||||||||||||||
| ISSN: | 1432-1122 | |||||||||||||||||||||
| Official Date: | October 2022 | |||||||||||||||||||||
| Dates: |
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| Volume: | 26 | |||||||||||||||||||||
| Number: | 4 | |||||||||||||||||||||
| Page Range: | pp. 671-732 | |||||||||||||||||||||
| DOI: | 10.1007/s00780-022-00486-7 | |||||||||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||||||||
| Publication Status: | Published | |||||||||||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||||||||
| Date of first compliant deposit: | 25 October 2022 | |||||||||||||||||||||
| Date of first compliant Open Access: | 25 October 2022 | |||||||||||||||||||||
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