Estimation of partial differential equations with applications in finance

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...

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Bibliographic Details
Published in:Journal of econometrics 2008-06, Vol.144 (2), p.392-408
Main Author: Kristensen, Dennis
Format: Article
Language:English
Subjects:
Applications
Asset pricing
Asymptotic methods
Derivative pricing
Derivatives
Diffusion
Diffusion theory
Distribution
Economic models
Error
Estimation
Exact sciences and technology
Insurance, economics, finance
Mathematical analysis
Mathematics
Nonparametric
Numerical analysis
Numerical analysis. Scientific computation
Option pricing
Parameter estimation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Probability and statistics
Sciences and techniques of general use
Securities prices
Statistics
Studies
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Summary: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.
ISSN:0304-4076
1872-6895
DOI:10.1016/j.jeconom.2008.04.004