Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
dc.contributor.author
Hutzenthaler, Martin
dc.contributor.author
Jentzen, Arnulf
dc.contributor.author
von Wurstemberger, Philippe
dc.date.accessioned
2021-07-28T16:23:44Z
dc.date.available
2020-09-13T04:38:34Z
dc.date.available
2020-09-18T06:40:02Z
dc.date.available
2021-07-28T16:23:44Z
dc.date.issued
2020
dc.identifier.issn
1083-6489
dc.identifier.other
10.1214/20-EJP423
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/440178
dc.identifier.doi
10.3929/ethz-b-000440178
dc.description.abstract
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the dimension and the reciprocal of the prescribed approximation accuracy. We thereby establish, for the first time, that the numerical approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.
en_US
dc.format
application/pdf
en_US
dc.language.iso
en
en_US
dc.publisher
University of Washington
en_US
dc.rights.uri
http://creativecommons.org/licenses/by/4.0/
dc.subject
curse of dimensionality
en_US
dc.subject
high-dimensional PDEs
en_US
dc.subject
multilevel Picard method
en_US
dc.subject
semilinear KolmogorovPDEs
en_US
dc.subject
semilinear PDEs
en_US
dc.title
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
en_US
dc.type
Journal Article
dc.rights.license
Creative Commons Attribution 4.0 International
dc.date.published
2020-08-20
ethz.journal.title
Electronic Journal of Probability
ethz.journal.volume
25
en_US
ethz.journal.abbreviated
Electron. J. Probab.
ethz.pages.start
101
en_US
ethz.size
73 p.
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ethz.version.deposit
publishedVersion
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ethz.grant
Higher order numerical approximation methods for stochastic partial differential equations
en_US
ethz.identifier.wos
ethz.identifier.scopus
ethz.publication.place
Seattle, WA
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02204 - RiskLab / RiskLab
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02204 - RiskLab / RiskLab
ethz.grant.agreementno
175699
ethz.grant.fundername
SNF
ethz.grant.funderDoi
10.13039/501100001711
ethz.grant.program
Projekte MINT
ethz.relation.isNewVersionOf
20.500.11850/364419
ethz.date.deposited
2020-09-13T04:38:39Z
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WOS
ethz.eth
yes
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ethz.availability
Open access
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ethz.rosetta.installDate
2020-09-18T06:40:19Z
ethz.rosetta.lastUpdated
2022-03-29T10:47:36Z
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