Specification Tests for the Variance of a Diffusion

We propose specification tests for the variance of a diffusion that do not require complete knowledge of the functional form under the null. We first propose a test for the constancy of the variance that, under the null of constancy, has a limiting normal distribution, while under the alternative of either unconditional or conditional heteroskedasticity it diverges at an appropriate rate. We then propose a test for the null of a parametric specification against the alternative of a more general functional form. Under the null, the test has a well-defined limiting distribution, normal in the unconditional and mixed normal in the conditional heteroskedasticity case; under the alternative, it diverges.

[1]  Dean P. Foster,et al.  Asypmtotic Filtering Theory for Univariate Arch Models , 1994 .

[2]  I. Karatzas.  On the pricing of American options , 1988 .

[3]  Jean Jacod,et al.  On the estimation of the diffusion coefficient for multi-dimensional diffusion processes , 1993 .

[4]  D. Florens-zmirou.  On estimating the diffusion coefficient from discrete observations , 1993, Journal of Applied Probability.

[5]  M. Yor.  Some Aspects Of Brownian Motion , 1992 .

[6]  Walter L. Smith.  Probability and Statistics , 1959, Nature.

[7]  Yacine Ait-Sahalia.  Testing Continuous-Time Models of the Spot Interest Rate , 1995 .

[8]  Yacine Aït-Sahalia.  Nonparametric Pricing of Interest Rate Derivative Securities , 1996 .

[9]  D. Florens-Zmirou.  Estimation de la variance d'une diffusion à partir d'une observation discrétisée , 1989 .

[10]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[11]  L. Hansen,et al.  Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes , 1993 .

[12]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[13]  A. Kemna,et al.  A pricing method for options based on average asset values , 1990 .

[14]  Ruth J. Williams,et al.  Introduction to Stochastic Integration , 1994 .

[15]  M. Chesney,et al.  Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model , 1989, Journal of Financial and Quantitative Analysis.

[16]  Daniel B. Nelson,et al.  Filtering and Forecasting with Misspecified Arch Models Ii: Making the Right Forecast with the Wrong Model , 1992 .

[17]  P. Brugière.  Théorème de limite centrale pour un estimateur non paramétrique de la variance d'un processus de diffusion multidimensionnelle , 1993 .

[18]  S. Ross,et al.  The valuation of options for alternative stochastic processes , 1976 .