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Estimation of partial differential equations with applications in finance

By Dennis Kristensen


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. We give a number of examples of this, including the pricing of bonds and interest rate derivatives. Very often derivative prices are calculated given preliminary estimates of the diffusion model for the underlying variable. We demonstrate that the derivative prices are consistent and asymptotically normally distributed 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. Our general results have other applications in asset pricing theory and in the estimation of diffusion models; these are also discussed

Topics: HG Finance, HB Economic Theory
Publisher: Financial Markets Group, London School of Economics and Political Science
Year: 2004
OAI identifier:
Provided by: LSE Research Online

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  2. (1995). A New Approach to Maximum-Likelihood Estimation for Stochastic Differential Equations Based on Discrete Observations.
  3. (1997). A Nonparametric Approach to the Estimation of Diffusion Processes -with an Application to a Short-term Interest Rate Model. doi
  4. (1997). A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk. doi
  5. (1981). A Second Course in Stochastic Processes. doi
  6. (2004). A Semiparametric Single-Factor Model for the Term Structure of Interest Rates. Working
  7. (1987). A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. doi
  8. (1996). A Yield Curve Model of Interest Rates.
  9. (2000). Accelerated Asymptotics for Diffusion Model Estimation. Working paper, GSB,
  10. (1977). An Equilibrium Characterization of the Term Structure. doi
  11. (1998). Arbitrage Theory in Continuous Time. doi
  12. (1997). Autocorrelation-Robust Inference. doi
  13. (1992). Bandwidth Choice for Average Derivative Estimation. doi
  14. (1990). Bandwidth Choice for Density Derivatives.
  15. (1992). Bias Corrected Semiparametric Estimation. Working paper,
  16. (1992). Bond Pricing and the Term Structure of Interest Rates: A New methodology for Contingent Claims Valuation. doi
  17. (2003). Bootstrap Methods for Markov Processes. doi
  18. (1991). Brownian Motion and Stochastic Calculus, 2nd Edition. doi
  19. (2000). Continuous-Time Methods in Finance: A Review and an Assessment. doi
  20. (1986). Density Estimation for Statistics and Data Analysis. doi
  21. (1996). Dynamic Asset Pricing Theory, 3rd edition. doi
  22. (2002). Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions. Working paper, doi
  23. (2004). Estimation in Two Classes of Semiparametric Diffusion Models. Working doi
  24. (2001). Estimation of Affine Pricing Models Using the Empirical Characteristic Function. doi
  25. (2003). Fully Nonparametric Estimation of Scalar Diffusion Models. doi
  26. (1985). Functional Integration and Partial Differential Equations. doi
  27. (1987). Kernel Estimators of Regression Functions. doi
  28. (2001). Likelihood Inference for Discretely Observed Non-linear Diffusions. doi
  29. (1995). Martingale Estimating Functions for Discretely Observed Diffusion Processes. doi
  30. (2002). Maximum Likelihood Estimation of Discretely Sampled Diffusions: A ClosedForm Approximation Approach. doi
  31. (1988). Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data. doi
  32. (1996). Multivariate Local Polynomial Regression for Time Series: Uniform Strong Consistency and Rates. doi
  33. (2004). Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach. Forthcoming in doi
  34. (1995). Nonparametric Estimation of Exact Consumers Surplus and Deadweight Loss. doi
  35. (1995). Nonparametric Kernel Estimation for Semiparametric Models. doi
  36. (1998). Nonparametric Modeling of U.S. Interest Rate Term Structure Dynamics and Implications on the Prices of Derivative Securities. doi
  37. (2003). Nonparametric Option Pricing under Shape Restrictions. doi
  38. (1996). Nonparametric Pricing of Interest Rate Derivative Securities. doi
  39. (1998). Nonparametric Statistics for Stochastic Processes. doi
  40. (2003). Nonparametric Study of Solutions of Differential Equations. Working paper, ESC Toulouse.
  41. (1992). Numerical Methods for Partial Differential Equations. doi
  42. (1995). On Bandwidth Choice in Nonparametric Regression with Both Short- and Long-Range Dependent Errors. doi
  43. (1962). On Estimation of a Probability Density Function and Mode. doi
  44. (2001). On the Functional Estimation of Multivariate Diffusion Processes. Working paper,
  45. (1993). One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities. doi
  46. (1996). Optimal Bandwidth Choice for Density-Weighted Averages. doi
  47. (1976). Option Pricing when Underlying Stock Returns Are Discontinuous. doi
  48. (1997). Options, Futures and Other Derivatives. doi
  49. (1964). Partial Differential Equations of Parabolic Type. doi
  50. (1998). Partial Differential Equations. doi
  51. (1998). Quasi-Indirect Inference for Diffusion Processes. doi
  52. Scheinkman (2000a) Shape-preserving Estimation of Diffusions. Working paper,
  53. Scheinkman (2000b) Principal Components and the Long Run. Working paper, doi
  54. (1993). Simulated Moments Estimation of Markov Models of Asset Prices. doi
  55. (1993). Smoothing Bias in Density Derivative Estimation. doi
  56. (2003). Spectral GMM Estimation of Continuous-Time Processes. doi
  57. (1993). Stability of Markovian processes III: Foster-Lyapunov Criteria for Continuous-time Processes. doi
  58. (1999). Statistical Approach to Some Ill-Posed Probl e m sf o rL i n e a rP a r t i a lD i fferential Equations. Probability Theory and Related Fields doi
  59. (1986). Statistical Tests of Contingent-Claims Asset-Pricing Models. doi
  60. (1975). Stochastic Differential Equations and Applications, doi
  61. (1996). Stochastic Volatility. doi
  62. (2002). Term Premia and Interest Rate Forecasts in Affine Models. doi
  63. (1992). The Bootstrap and Edgeworth Expansion. doi
  64. (1993). The Delta Method for Nonparametric Kernel Functionals. Working paper,
  65. (2004). The Econometrics of Option Pricing. Working paper 2004s-04, CIRANO. Forthcoming in Handbook of Financial Econometrics. doi
  66. (1977). The Impact of Variance Estimation in Option Valuation Models. doi
  67. (1973). The Pricing of Options and Corporate Liabilities. doi
  68. (1997). The Statistical Properties of the Black-Scholes Option Price. doi
  69. (2004). Twicing Kernels and a Small Bias Property of Semiparametric Estimators. doi

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