Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

Benjamin Peherstorfer, Pablo Gómez, Hans Joachim Bungartz

Research output: Contribution to journalArticle

Abstract

This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models.

Original languageEnglish (US)
Pages (from-to)1365-1389
Number of pages25
JournalAdvances in Computational Mathematics
Volume41
Issue number5
DOIs
StatePublished - Oct 1 2015

Fingerprint

Black-Scholes Equation
Sparse Grids
Reduced Model
Heston Model
Discretization
Fidelity
Partial differential equation
Black-Scholes Model
Black-Scholes
Curse of Dimensionality
Orthogonal Decomposition
Model Reduction
Option Pricing
Pricing
Tensor
Partial differential equations
Grid
Numerical Results
Model
Tensors

Keywords

  • Black-Scholes equation
  • Option pricing
  • Proper orthogonal decomposition
  • Sparse grids

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation. / Peherstorfer, Benjamin; Gómez, Pablo; Bungartz, Hans Joachim.

In: Advances in Computational Mathematics, Vol. 41, No. 5, 01.10.2015, p. 1365-1389.

Research output: Contribution to journalArticle

Peherstorfer, Benjamin ; Gómez, Pablo ; Bungartz, Hans Joachim. / Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation. In: Advances in Computational Mathematics. 2015 ; Vol. 41, No. 5. pp. 1365-1389.
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