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Estimation of Integrated Volatility in Continuous‐Time Financial Models with Applications to Goodness‐of‐Fit Testing

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  • HOLGER DETTE
  • MARK PODOLSKIJ
  • MATHIAS VETTER

Abstract

. Properties of a specification test for the parametric form of the variance function in diffusion processes are discussed. The test is based on the estimation of certain integrals of the volatility function. If the volatility function does not depend on the variable x it is known that the corresponding statistics have an asymptotic normal distribution. However, most models of mathematical finance use a volatility function which depends on the state x. In this paper we prove that in the general case, where σ depends also on x the estimates of integrals of the volatility converge stably in law to random variables with a non‐standard limit distribution. The limit distribution depends on the diffusion process Xt itself and we use this result to develop a bootstrap test for the parametric form of the volatility function, which is consistent in the general diffusion model.

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  • Holger Dette & Mark Podolskij & Mathias Vetter, 2006. "Estimation of Integrated Volatility in Continuous‐Time Financial Models with Applications to Goodness‐of‐Fit Testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 259-278, June.
    • Handle: RePEc:bla:scjsta:v:33:y:2006:i:2:p:259-278
    DOI: 10.1111/j.1467-9469.2006.00479.x
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    1. Kleinow, Torsten, 2002. "Testing the diffusion coefficient," SFB 373 Discussion Papers 2002,38, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. Ait-Sahalia, Yacine, 1996. "Testing Continuous-Time Models of the Spot Interest Rate," The Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 385-426.
    3. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    4. Gerety, Mason S & Mulherin, J Harold, 1994. "Price Formation on Stock Exchanges: The Evolution of Trading within the Day," The Review of Financial Studies, Society for Financial Studies, vol. 7(3), pages 609-629.
    5. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    6. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    7. Barndorff-Nielsen, Ole Eiler & Graversen, Svend Erik & Jacod, Jean & Podolskij, Mark, 2004. "A central limit theorem for realised power and bipower variations of continuous semimartingales," Technical Reports 2004,51, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    10. Constantinides, George M, 1992. "A Theory of the Nominal Term Structure of Interest Rates," The Review of Financial Studies, Society for Financial Studies, vol. 5(4), pages 531-552.
    11. Valentina Corradi & Halbert White, 1999. "Specification Tests for the Variance of a Diffusion," Journal of Time Series Analysis, Wiley Blackwell, vol. 20(3), pages 253-270, May.
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