Inference for Interest Rate Models Using Milstein’s Approximation

ABSTRACT

A class of martingale estimating functions based on the first two moments of the observed process provides a convenient framework for estimating the parameters of diffusion processes [1]. In the Bayesian set up, combined estimating functions had been studied for diffusion processes in [2] with filtering applications. However, when the conditional mean and the conditional variance are functions of parameters of interest in a diffusion process model, the basic martingales generating components of quadratic estimating functions are such that one is an absolute continuous function with respect to the other [3, p. 94]. Hence, the combined martingale estimating functions cannot be constructed for continuous-time diffusion processes. In this paper, a general framework for *parameter estimation* of discretely observed interest rate models is developed by using the Milstein approximation and closed form expressions for the information gain are also obtained. The method is used to study the estimates of the parameters for an extended version of the CoxIngersoll-Ross interest rate model.

Cite this paper

T. Koulis and A. Thavaneswaran, "Inference for Interest Rate Models Using Milstein’s Approximation,"*Journal of Mathematical Finance*, Vol. 3 No. 1, 2013, pp. 110-118. doi: 10.4236/jmf.2013.31010.

T. Koulis and A. Thavaneswaran, "Inference for Interest Rate Models Using Milstein’s Approximation,"

References

[1] A. Thavaneswaran and M. E. Thompson, “Optimal Estimation for Semimartingales,” Journal of Applied Probability, Vol. 23, No. 2, 1986, pp. 409-417. doi:10.2307/3214183

[2] A. Thavaneswaran and M. E. Thompson, “A Criterion for Filtering in Semimartingale Models,” Stochastic Processes and Their Applications, Vol. 28, No. 2, 1988, pp. 259-265. doi:10.1016/0304-4149(88)90099-3

[3] C. Heyde, “Quasi-Likelihood and Its Application: A Ge- neral Approach to Optimal Parameter Estimation,” Springer Series in Statistics, Springer, 1997. doi:10.1007/b98823

[4] V. P. Godambe, “The Foundations of Finite Sample Estimation in Stochastic Processes,” Biometrika, Vol. 72, No. 2, 1985, pp. 419-428. doi:10.1093/biomet/72.2.419

[5] J. E. Hutton and P. I. Nelson, “Quasilikelihood Estimation for Semimartingales,” Stochastic Processes and Their Applications, Vol. 22, No. 2, 1986, pp. 245-257. doi:10.1016/0304-4149(86)90004-9

[6] U. V. Naik-Nimbalkar and M. B. Rajarshi, “Filtering and Smoothing via Estimating Functions,” Journal of the American Statistical Association, Vol. 90, No. 429, 1995, pp. 301-306. doi:10.1080/01621459.1995.10476513

[7] A. Paseka, T. Koulis and A. Thavaneswaran, “Interest Rate Models,” Journal of Mathematical Finance, Vol. 2, No. 2, 2012, pp. 141-158. doi:10.4236/jmf.2012.22016

[8] A. Thavaneswaran, Y. Liang and N. Ravishanker, “Inference for Diffusion Processes Using Combined Estimating Functions,” Sri Lankan Journal of Applied Statistics, Vol. 12, 2011, pp. 145-160.

[9] M. Jeong and J. Y. Park, “Asymptotic Theory of Maximum Likelihood Estimator for Diffusion Model,” Working Paper, Indiana University, Bloomington, 2010.

[10] O. Elerian, “A Note on the Existence of a Closed form Conditional Transition Density for the Milstein Scheme,” Economics Discussion Paper 1998-W18, Nuffield College, Oxford, 1998.

[11] B. G. Lindsay, “Using Empirical Partially Bayes Inference for Increased Efficiency,” The Annals of Statistics, Vol. 13, No. 3, 1985, pp. 914-931. doi:10.1214/aos/1176349646

[12] J. C. Cox, J. E. Ingersoll Jr. and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. doi:10.2307/1911242

[13] Y. Ait-Sahalia, “Testing Continuous-Time Models of The Spot Interest Rate,” Review of Financial Studies, Vol. 9, No. 2, 1996, pp. 385-426. doi:10.1093/rfs/9.2.385

[1] A. Thavaneswaran and M. E. Thompson, “Optimal Estimation for Semimartingales,” Journal of Applied Probability, Vol. 23, No. 2, 1986, pp. 409-417. doi:10.2307/3214183

[2] A. Thavaneswaran and M. E. Thompson, “A Criterion for Filtering in Semimartingale Models,” Stochastic Processes and Their Applications, Vol. 28, No. 2, 1988, pp. 259-265. doi:10.1016/0304-4149(88)90099-3

[3] C. Heyde, “Quasi-Likelihood and Its Application: A Ge- neral Approach to Optimal Parameter Estimation,” Springer Series in Statistics, Springer, 1997. doi:10.1007/b98823

[4] V. P. Godambe, “The Foundations of Finite Sample Estimation in Stochastic Processes,” Biometrika, Vol. 72, No. 2, 1985, pp. 419-428. doi:10.1093/biomet/72.2.419

[5] J. E. Hutton and P. I. Nelson, “Quasilikelihood Estimation for Semimartingales,” Stochastic Processes and Their Applications, Vol. 22, No. 2, 1986, pp. 245-257. doi:10.1016/0304-4149(86)90004-9

[6] U. V. Naik-Nimbalkar and M. B. Rajarshi, “Filtering and Smoothing via Estimating Functions,” Journal of the American Statistical Association, Vol. 90, No. 429, 1995, pp. 301-306. doi:10.1080/01621459.1995.10476513

[7] A. Paseka, T. Koulis and A. Thavaneswaran, “Interest Rate Models,” Journal of Mathematical Finance, Vol. 2, No. 2, 2012, pp. 141-158. doi:10.4236/jmf.2012.22016

[8] A. Thavaneswaran, Y. Liang and N. Ravishanker, “Inference for Diffusion Processes Using Combined Estimating Functions,” Sri Lankan Journal of Applied Statistics, Vol. 12, 2011, pp. 145-160.

[9] M. Jeong and J. Y. Park, “Asymptotic Theory of Maximum Likelihood Estimator for Diffusion Model,” Working Paper, Indiana University, Bloomington, 2010.

[10] O. Elerian, “A Note on the Existence of a Closed form Conditional Transition Density for the Milstein Scheme,” Economics Discussion Paper 1998-W18, Nuffield College, Oxford, 1998.

[11] B. G. Lindsay, “Using Empirical Partially Bayes Inference for Increased Efficiency,” The Annals of Statistics, Vol. 13, No. 3, 1985, pp. 914-931. doi:10.1214/aos/1176349646

[12] J. C. Cox, J. E. Ingersoll Jr. and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. doi:10.2307/1911242

[13] Y. Ait-Sahalia, “Testing Continuous-Time Models of The Spot Interest Rate,” Review of Financial Studies, Vol. 9, No. 2, 1996, pp. 385-426. doi:10.1093/rfs/9.2.385