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

  • Christoph Reisinger
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 88)


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.


Sparse Grid Finite Element Scheme Basket Option Central Finite Difference Scheme Superposition Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bungartz, H.-J., Griebel, M.: Sparse grids, Acta Numer., 13, 11–23 (2004)MathSciNetCrossRefGoogle Scholar
  2. 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. 3.
    Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing, 71, 65–87 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complexity, 26, 455–489 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Griebel, M., Kuo, F. Y., Sloan, I.H.: The smoothing effect of the ANOVA decomposition. J. Complexity, 26, 523–551 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 7.
    Gnewuch, M.: Infinite-Dimensional Integration on Weighted Hilbert Spaces. Math. Comp., 81, 2175–2205 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Lin. Algebra Appl., 420, 249–275 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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. 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. 11.
    Pflaum, C., Zhou, A.: Error analysis of the combination technique. Numer. Math., 84, 327–350 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Reisinger, C.: Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. PhD Thesis, Universität Heidelberg (2004)Google Scholar
  13. 13.
    Reisinger, C., Wittum, G.: Efficient hierarchical approximation of high-dimensional option pricing problems. SIAM J. Sci. Comput., 29, 440–458 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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. 15.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity, 14, 1–33 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations