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Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis

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  • Nien-Lin Liu
  • Hoang-Long Ngo

Abstract

In order to study the geometry of interest rates market dynamics, Malliavin, Mancino and Recchioni [A non-parametric calibration of the HJM geometry: an application of It\^o calculus to financial statistics, {\it Japanese Journal of Mathematics}, 2, pp.55--77, 2007] introduced a scheme, which is based on the Fourier Series method, to estimate eigenvalues of a spot cross volatility matrix. In this paper, we present another estimation scheme based on the Quadratic Variation method. We first establish limit theorems for each scheme and then we use a stochastic volatility model of Heston's type to compare the effectiveness of these two schemes.

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  • Nien-Lin Liu & Hoang-Long Ngo, 2014. "Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis," Papers 1409.2214, arXiv.org.
    • Handle: RePEc:arx:papers:1409.2214
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    References listed on IDEAS

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    1. Ogawa Shigeyoshi, 2008. "Real-time scheme for the volatility estimation in the presence of microstructure noise," Monte Carlo Methods and Applications, De Gruyter, vol. 14(4), pages 331-342, January.
    2. Shigeyoshi Ogawa & Simona Sanfelici, 2011. "An Improved Two‐step Regularization Scheme for Spot Volatility Estimation," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 40(3), pages 105-132, November.
    3. Jean Jacod & Mark Podolskij, 2012. "A test for the rank of the volatility process: the random perturbation approach," CREATES Research Papers 2012-57, Department of Economics and Business Economics, Aarhus University.
    4. Alexander Alvarez & Fabien Panloup & Monique Pontier & Nicolas Savy, 2012. "Estimation of the instantaneous volatility," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 27-59, April.
    5. Maria Elvira Mancino & Paul Malliavin, 2002. "Fourier series method for measurement of multivariate volatilities," Finance and Stochastics, Springer, vol. 6(1), pages 49-61.
    6. Ole E. Barndorff-Nielsen & Neil Shephard, 2004. "Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics," Econometrica, Econometric Society, vol. 72(3), pages 885-925, May.
    7. Jean Jacod & Mark Podolskij, 2012. "A Test for the Rank of the Volatility Process: The Random Perturbation Approach," Global COE Hi-Stat Discussion Paper Series gd12-268, Institute of Economic Research, Hitotsubashi University.
    8. Ogawa, Shigeyoshi & Ngo, Hoang-Long, 2010. "Real-time estimation scheme for the spot cross volatility of jump diffusion processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(9), pages 1962-1976.
    9. Ngo Hoang-Long & Ogawa Shigeyoshi, 2009. "A central limit theorem for the functional estimation of the spot volatility," Monte Carlo Methods and Applications, De Gruyter, vol. 15(4), pages 353-380, January.
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    Cited by:

    1. Jir^o Akahori & Nien-Lin Liu & Maria Elvira Mancino & Yukie Yasuda, 2014. "The Fourier estimation method with positive semi-definite estimators," Papers 1410.0112, arXiv.org.

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