Abstract
In modern finance theory, the short-term interest rate is important in characterizing the term structure of interest rates and in pricing interest-rate-contingent-claims. There is some pioneering work in the continuous-time framework, for example by Vasicek (1997) and Cox et al. (1985). A survey of is provided by Chan et al. (1992). Chan et al. (1992) show that a wide variety of well-known one-factor models for short rates can be nested within the following stochastic different equation (SDE):
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ait-Sahalia, Y. (1996) “Testing Continuous-Time Models of the Spot Interest Rate,” Review of Financial Studies, 9 (2): 385–426.
Ait-Sahalia, Y. (1997) “Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approximation Approach,” Econometrica, 70 (1): 223–262.
Ait-Sahalia, Y. (1999) “Transition Densities for Interest Rate and Other Nonlinear Diffusions,” Journal of Finance, 54 (4): 1361–1395.
Andersen, T. G. and Lund, J. (1997) “Estimating Continuous-Time Stochastic Volatility Models of the Short-Term Interest Rate,” Journal of Econometrics, 77 (1): 343–377.
Bollerslev, T. (1986) “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, 31 (3): 307–327.
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994) Time Series Analysis, 3rd edn, Englewood Cliffs, NJ: Prentice Hall.
Box, G. E. P. and Pierce, D. A. (1970) “Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models,” Journal of the American Statistical Association, 65 (332): 1509–1526.
Breiman, L. (1973) Statistics, Boston, Mass.: Houghton Mifflin.
Brenner, R. J., Harjes, R. H., and Kroner, K. F. (1996) “Another Look at Models of the Short-Term Interest Rate,” Journal of Financial and Quantitative Analysis, 31 (1): 85–107.
Chan, K. C, Karolyi, G. A., Longstaff, F. A., and Sanders, A. B. (1992) “An Empirical Comparison of Alternative Models of the Short-Term Interest Rate,” Journal of Finance, 47 (3): 1209–1227.
Cox, J. C, Ingersoll, J. E., and Ross, S. A. (1985) “An Intertemporal General Equilibrium Model of Asset Prices,” Econometrica, 53 (2): 363–384.
Cox, J. C, Ingersoll, J. E. and Ross, S. A. (1985) “A Theory of the Term Structure of Interest Rates,” Econometrica, 53 (2): 385–407.
Dacunha-Castelle, D. and Florens-Zmirou, D. (1986) “Estimation of the Coefficients of a Diffusion from Discrete Observations,” Stochastics, 19 (4): 263–284.
Duffie, D. and Glynn, P. (2004) “Estimation of Continuous-Time Markov Processes Sampled at Random Time Intervals,” Econometrica, 72 (6): 1773–1808.
Durbin, J. (1970) “Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors Are Lagged Dependent Variables,” Biometrika, 38 (3): 410–421.
Elerian, O. (1998) “A Note on the Existence of a Closed Form Conditional Transition Density for the Milstein Scheme,” Nuffield College Economics Working Papers 1998-W18.
Engle, R. F. (1982) “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50 (4): 987–1007.
Frohn, J. (1995) Grundausbildung in Oekonometrie, 2nd edn, Berlin: Walter de Gruyter.
Fuller, W. A. (1996). Introduction to Statistical Time Series, 2nd edn, New York: Wiley.
Gallant, A. R. and Tauchen, G. (1996) “Which Moments to Match?” Econometric Theory, 12 (4): 657–681.
Gallant, A. R. and White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic System, Oxford: Blackwell.
Gourieroux, C, Monfort, A., and Renault, E. (1993) “Indirect Inference,” Journal of Applied Econometrics, 8 (S): S85–S118.
Hansen, L. A. and Scheinkman, J. A. (1995) “Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes,” Econometrica, 63 (4): 767–804.
Karatzas, I. and Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus, New York: Springer.
Koedijk, K. G., Nissen, F. G. J. A., Schotman, P. C. and Wolff, C. C. P. (1997) “The Dynamics of Short-Term Interest Rate Volatility Reconsidered,” European Finance Review, 1 (1): 105–130.
Ljung, G. M. and Box, G. E. P. (1978) “On Measure of Lack of Fit in Time Series Models,” Biometrika, 65 (2): 297–303.
Lo, A. W. (1988) “Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data,” Econometric Theory, 4 (August): 231–247.
Pedersen, A. R. (1995). “A New Approach to Maximum Likelihood Estimation for Stochastic Differential Equations Based on Discrete Observations,” Scandinavian Journal of Statistics, 22 (1): 55–71.
Shoji, I. and Ozaki, T. (1997) “Comparative Study of Estimation Methods for Continuous Time Stochastic Processes,” Journal of Time Series Analysis, 18 (5): 485–506.
Shoji, I. and Ozaki, T. (1998) “Estimation for Nonlinear Stochastic Differential Equations by Local Linearization Method,” Stochastic Analysis and Applications, 16 (4): 733–752.
Thompson, S. B. (2002) “Evaluating the Goodness of Fit of Conditional Distributions, with an Application to Affine Term Structure Models,” Working Paper, Department of Economics, Harvard University.
Vasicek, O. (1977) “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, 5 (1): 177–188.
Editor information
Editors and Affiliations
Copyright information
© 2011 Chih-Ying Hsiao and Willi Semmler
About this chapter
Cite this chapter
Hsiao, CY., Semmler, W. (2011). Continuous and Discrete Time Modeling of Short-Term Interest Rates. In: Gregoriou, G.N., Pascalau, R. (eds) Financial Econometrics Modeling: Derivatives Pricing, Hedge Funds and Term Structure Models. Palgrave Macmillan, London. https://doi.org/10.1057/9780230295209_9
Download citation
DOI: https://doi.org/10.1057/9780230295209_9
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-1-349-32892-5
Online ISBN: 978-0-230-29520-9
eBook Packages: Palgrave Economics & Finance CollectionEconomics and Finance (R0)