Skip to main content
SpringerLink
Log in
Book cover

Sparse Grids and Applications pp 263–276Cite as

Asymptotic Expansion Around Principal Components and the Complexity of Dimension Adaptive Algorithms

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 88))

Abstract

In this short article, we describe how the correlation of typical diffusion processes arising e.g. in financial modelling can be exploited—by means of asymptotic analysis of principal components—to make Feynman-Kac PDEs of high dimension computationally tractable. We explore the links to dimension adaptive sparse grids (Gerstner and Griebel, Computing 71:65–87, 2003), anchored ANOVA decompositions and dimension-wise integration (Griebel and Holtz, J Complexity 26:455–489, 2010), and the embedding in infinite-dimensional weighted spaces (Sloan and Woźniakowski, J Complexity 14:1–33, 1998). The approach is shown to give sufficient accuracy for the valuation of index options in practice. These numerical findings are backed up by a complexity analysis that explains the independence of the computational effort of the dimension in relevant parameter regimes.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Bungartz, H.-J., Griebel, M.: Sparse grids, Acta Numer., 13, 11–23 (2004)

    Article  MathSciNet  Google Scholar 

  2. Caflisch, R.E., Morokoff, W., Owen, A.: Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Comput. Finance, 1, 27–46 (1997)

    Google Scholar 

  3. Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing, 71, 65–87 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complexity, 26, 455–489 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Griebel, M., Kuo, F. Y., Sloan, I.H.: The smoothing effect of the ANOVA decomposition. J. Complexity, 26, 523–551 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gnewuch, M.: Infinite-dimensional integration on weighted Hilbert spaces, Preprint cucs-016-10, Department of Computer Science, Columbia University (To appear in Math. Comp.) (2011)

    Google Scholar 

  7. Gnewuch, M.: Infinite-Dimensional Integration on Weighted Hilbert Spaces. Math. Comp., 81, 2175–2205 (2012)

    Article  MathSciNet  Google Scholar 

  8. Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Lin. Algebra Appl., 420, 249–275 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hilber, N., Kehtari, S., Schwab, C., Winter, C.: Wavelet finite element method for option pricing in highdimensional diffusion market models. SAM Research Report 2010-01, ETH Zürich (2010)

    Google Scholar 

  10. Niu, B., Hickernell, F.J., Müller-Gronbach, T., Ritter, K.: Deterministic multi-level algorithms for infinite-dimensional integration on \({\mathbb{R}}^{N}\). J. Complexity, 27, 331–351 (2011)

    Google Scholar 

  11. Pflaum, C., Zhou, A.: Error analysis of the combination technique. Numer. Math., 84, 327–350 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Reisinger, C.: Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. PhD Thesis, Universität Heidelberg (2004)

    Google Scholar 

  13. Reisinger, C., Wittum, G.: Efficient hierarchical approximation of high-dimensional option pricing problems. SIAM J. Sci. Comput., 29, 440–458 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Reisinger, C.: Analysis of linear difference schemes in the sparse grid combination technique. IMA J. of Numer. Anal., doi:10.1093/imanum/drs004 (2012)

    Google Scholar 

  15. Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity, 14, 1–33 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, X., Sloan, I.H.: Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput., 27, 159–183 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zenger, C.: Sparse grids. In: Hackbusch, W. (ed) Parallel Algorithms for Partial Differential Equations, Notes on Numerical Fluid Mechanics, 31, Vieweg, Braunschweig/Wiesbaden (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    Christoph Reisinger

Authors
  1. Christoph Reisinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Reisinger .

Editor information

Editors and Affiliations

  1. , Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, Bonn, 53115, Germany

    Jochen Garcke

  2. Inst. Numerische Simulation, Universität Bonn, Wegelerstr. 6, Bonn, 53115, Germany

    Michael Griebel

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Reisinger, C. (2012). Asymptotic Expansion Around Principal Components and the Complexity of Dimension Adaptive Algorithms. In: Garcke, J., Griebel, M. (eds) Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31703-3_13

Download citation

Publish with us

Policies and ethics

Search

Navigation