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Estimation of partial differential equations with applications in finance

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  • Kristensen, Dennis

Abstract

Linear parabolic partial differential equations (PDE's) and diffusion models are closely linked through the celebrated Feynman-Kac representation of solutions to PDE's. In asset pricing theory, this leads to the representation of derivative prices as solutions to PDE's. Very often implied derivative prices are calculated given preliminary estimates of the diffusion model for the underlying variable. We demonstrate that the implied derivative prices are consistent and derive their asymptotic distribution under general conditions. We apply this result to three leading cases of preliminary estimators: Nonparametric, semiparametric and fully parametric ones. In all three cases, the asymptotic distribution of the solution is derived. We demonstrate the use of these results in obtaining confidence bands and standard errors for implied prices of bonds, options and other derivatives. Our general results also are of interest for the estimation of diffusion models using either historical data of the underlying process or option prices; these issues are also discussed.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 144 (2008)
Issue (Month): 2 (June)
Pages: 392-408

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Handle: RePEc:eee:econom:v:144:y:2008:i:2:p:392-408

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Web page: http://www.elsevier.com/locate/jeconom

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Citations

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Cited by:
  1. Kristensen, Dennis, 2010. "Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models," Journal of Econometrics, Elsevier, vol. 156(2), pages 239-259, June.
  2. Kristensen, Dennis & Mele, Antonio, 2011. "Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models," Journal of Financial Economics, Elsevier, vol. 102(2), pages 390-415.
  3. Koo, Bonsoo & Linton, Oliver, 2012. "Estimation of semiparametric locally stationary diffusion models," Journal of Econometrics, Elsevier, vol. 170(1), pages 210-233.
  4. Nikolay Gospodinov & Masayuki Hirukawa, 2008. "Nonparametric Estimation of Scalar Diffusion Processes of Interest Rates Using Asymmetric Kernels," Working Papers 08011, Concordia University, Department of Economics, revised Dec 2008.
  5. Dennis Kristensen, 2010. "Semi-Nonparametric Estimation and Misspecification Testing of Diffusion Models," Discussion Papers 10-10, University of Copenhagen. Department of Economics.

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