Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
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Date
2020
Publication Type
Journal Article
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yes
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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.
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published
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Journal / series
Volume
25
Pages / Article No.
101
Publisher
University of Washington
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Date collected
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Subject
curse of dimensionality; high-dimensional PDEs; multilevel Picard method; semilinear KolmogorovPDEs; semilinear PDEs
Organisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
02204 - RiskLab / RiskLab
Notes
Funding
175699 - Higher order numerical approximation methods for stochastic partial differential equations (SNF)
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