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Rank Reduction of Correlation Matrices by Majorization

Author

Listed:
  • Raoul Pietersz

    (Erasmus University Rotterdam)

  • Patrick J. F. Groenen

    (Erasmus University Rotterdam)

Abstract

A novel algorithm is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix. The algorithm is based on majorization and, therefore, it is globally convergent. The algorithm is computationally efficient, is straightforward to implement, and can handle arbitrary weights on the entries of the correlation matrix. A simulation study suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. The problem of rank reduction of correlation matrices occurs when pricing a derivative dependent on a large number of assets, where the asset prices are modelled as correlated log-normal processes. Mainly, such an application concerns interest rates.

Suggested Citation

  • Raoul Pietersz & Patrick J. F. Groenen, 2005. "Rank Reduction of Correlation Matrices by Majorization," Finance 0502006, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpfi:0502006
    Note: Type of Document - pdf; pages: 29
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    References listed on IDEAS

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    1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. Igor Grubisic & Raoul Pietersz, 2005. "Efficient Rank Reduction of Correlation Matrices," Finance 0502007, University Library of Munich, Germany.
    3. Henk Kiers & Patrick Groenen, 1996. "A monotonically convergent algorithm for orthogonal congruence rotation," Psychometrika, Springer;The Psychometric Society, vol. 61(2), pages 375-389, June.
    4. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-430, March.
    5. Alan Brace & Dariusz G¸atarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 127-155, April.
    6. Frank de Jong & Joost Driessen & Antoon Pelsser, 2004. "On the Information in the Interest Rate Term Structure and Option Prices," Review of Derivatives Research, Springer, vol. 7(2), pages 99-127, August.
    7. Willard I. Zangwill, 1969. "Convergence Conditions for Nonlinear Programming Algorithms," Management Science, INFORMS, vol. 16(1), pages 1-13, September.
    8. Kiers, Henk A. L., 2002. "Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 157-170, November.
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    Citations

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    Cited by:

    1. Kohei Adachi, 2011. "Constrained principal component analysis of standardized data for biplots with unit-length variable vectors," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 5(1), pages 23-36, April.
    2. Raoul Pietersz & Antoon Pelsser, 2010. "A comparison of single factor Markov-functional and multi factor market models," Review of Derivatives Research, Springer, vol. 13(3), pages 245-272, October.
    3. Hebert, Pierre-Alexandre & Masson, Marie-Helene & Denoeux, Thierry, 2006. "Fuzzy multidimensional scaling," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 335-359, November.
    4. Shujun Bi & Le Han & Shaohua Pan, 2013. "Approximation of rank function and its application to the nearest low-rank correlation matrix," Journal of Global Optimization, Springer, vol. 57(4), pages 1113-1137, December.
    5. Sudhanshu K Mishra, 2013. "Global Optimization of Some Difficult Benchmark Functions by Host-Parasite Coevolutionary Algorithm," Economics Bulletin, AccessEcon, vol. 33(1), pages 1-18.
    6. Harry Oviedo, 2023. "Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold," Mathematics, MDPI, vol. 11(11), pages 1-17, May.
    7. Qingna Li & Houduo Qi & Naihua Xiu, 2011. "Block relaxation and majorization methods for the nearest correlation matrix with factor structure," Computational Optimization and Applications, Springer, vol. 50(2), pages 327-349, October.
    8. Anders Løland & Ragnar Bang Huseby & Nils Lid Hjort & Arnoldo Frigessi, 2013. "Statistical Corrections of Invalid Correlation Matrices," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 807-824, December.
    9. Igor Grubisic & Raoul Pietersz, 2005. "Efficient Rank Reduction of Correlation Matrices," Finance 0502007, University Library of Munich, Germany.
    10. Mishra, SK, 2004. "Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm," MPRA Paper 1783, University Library of Munich, Germany.
    11. Zhu, Xiaojing, 2015. "Computing the nearest low-rank correlation matrix by a simplified SQP algorithm," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 404-414.
    12. Raoul Pietersz & Marcel Regenmortel, 2006. "Generic market models," Finance and Stochastics, Springer, vol. 10(4), pages 507-528, December.
      • Raoul Pietersz & Marcel van Regenmortel, 2005. "Generic Market Models," Finance 0502009, University Library of Munich, Germany.
      • Pietersz, R. & van Regenmortel, M., 2005. "Generic Market Models," ERIM Report Series Research in Management ERS-2005-010-F&A, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    13. Yulan Liu & Shujun Bi & Shaohua Pan, 2018. "Equivalent Lipschitz surrogates for zero-norm and rank optimization problems," Journal of Global Optimization, Springer, vol. 72(4), pages 679-704, December.
    14. Yitian Qian & Shaohua Pan & Yulan Liu, 2023. "Calmness of partial perturbation to composite rank constraint systems and its applications," Journal of Global Optimization, Springer, vol. 85(4), pages 867-889, April.
    15. Alexander Tchernitser & Dmitri Rubisov, 2009. "Robust estimation of historical volatility and correlations in risk management," Quantitative Finance, Taylor & Francis Journals, vol. 9(1), pages 43-54.
    16. Mishra, SK, 2007. "Completing correlation matrices of arbitrary order by differential evolution method of global optimization: A Fortran program," MPRA Paper 2000, University Library of Munich, Germany.

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    More about this item

    Keywords

    rank; correlation matrix; majorization; lognormal price processes;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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