Maximum likelihood estimation of the Cox-Ingersoll-Ross model using particle filters. (English) Zbl 1231.91485
Summary: This paper shows how to build in a computationally efficient way a maximum simulated likelihood procedure to estimate the Cox-Ingersoll-Ross model from multivariate time series. The advantage of this estimator is that it takes into account the exact likelihood function while avoiding the huge computational burden associated with MCMC methods and without the ad hoc assumption that certain bond yields are measured without error. The proposed methodology is implemented and tested on simulated data. For realistic parameter values the estimator seems to have good small sample properties, compared to the popular quasi maximum likelihood approach, even using moderate simulation sizes. The effect of simulation errors does not seem to undermine the estimation procedure.
MSC:
| 91G70 | Statistical methods; risk measures |
| 91G30 | Interest rates, asset pricing, etc. (stochastic models) |
| 65C05 | Monte Carlo methods |
| 62M20 | Inference from stochastic processes and prediction |
Software:
GenocopReferences:
| [1] | Bakshi S. B., Chen Z. (1997) An alternative valuation model for contingent claims. Journal of Financial Economics 44: 123–165 · doi:10.1016/S0304-405X(96)00009-8 |
| [2] | Ball C. A., Tourous W. N. (1996) Unit roots and the estimation of interest rate dynamics. Journal of Empirical Finance 3: 215–238 · doi:10.1016/0927-5398(95)00018-6 |
| [3] | Brigo, D., & Mercurio, F. (2000). The CIR++ model and other deterministic-shift extensions of short rate models. In Proceedings of the 4th Columbia-JAFEE conference for mathematical finance and financial engineering (pp. 563–584), Tokyo, 16–17 December. |
| [4] | Chiarella C., Hung H., Tô T.-D. (2009) The volatility structure of the fixed income market under the HJM framework: A nonlinear filtering approach. Computational Statistics and Data Analysis 53: 2075–2088 · Zbl 1453.62069 · doi:10.1016/j.csda.2008.07.036 |
| [5] | Chiarella C., Tô T.-D. (2006) The multifactor nature of the volatility of futures markets. Computational Economics 27: 163–183 · Zbl 1153.91475 · doi:10.1007/s10614-006-9023-9 |
| [6] | Cox J. C., Ingersoll J. E., Ross S. (1985) A theory of the term structure of interest rates. Econometrica 53: 385–407 · Zbl 1274.91447 · doi:10.2307/1911242 |
| [7] | Dai Q., Singleton K. J. (2000) Specification analysis of affine term structure models. Journal of Finance 55: 1943–1978 · doi:10.1111/0022-1082.00278 |
| [8] | de Jong F. (2000) Time-series and cross-section information in affine term structure models. Journal of Business and Economic Statistics 18: 300–314 · doi:10.2307/1392263 |
| [9] | de Jong F., Santa-Clara P. (1999) The dynamics of the forward interest rate curve: A formulation with state variables. Journal of Financial and Quantitative Analysis 34: 131–157 · doi:10.2307/2676249 |
| [10] | Doucet A., de Freitas N., Gordon N. (2001) An introduction to sequential Monte Carlo methods. In: Doucet A., de Freitas N., Gordon N. (eds) Sequential Monte Carlo methods in practice. Springer, New York, pp 3–14 · Zbl 1056.93576 |
| [11] | Doucet A., Godsill S., Andrieu C. (2000) On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing 10: 197–208 · doi:10.1023/A:1008935410038 |
| [12] | Doucet A., Tadić V. B. (2003) Parameter estimation in general state-space models using particle methods. Annals of the Institute of Mathematical Statistics 55: 409–422 · Zbl 1047.62084 |
| [13] | Duan J. C., Simonato J. G. (1999) Estimating and testing exponential-affine term structure models by Kalman filter. Review of Quantitative Finance and Accounting 13: 102–127 · doi:10.1023/A:1008304625054 |
| [14] | Duffee G. R. (1999) Estimating the price of default risk. Review of Financial Studies 12: 197–226 · doi:10.1093/rfs/12.1.197 |
| [15] | Duffee, G. R., & Stanton, R. (2004). Estimation of dynamic term structure models. Working Paper, University of Berkeley. |
| [16] | Durham G. B., Gallant A. R. (2002) Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. Journal of Business and Economic Statistics 20: 325–327 · doi:10.1198/073500102288618397 |
| [17] | Geyer A. L. J., Pichler S. (1999) A state-space approach to estimate and test multifactor Cox–Ingersoll–Ross models of the term structure. Journal of Financial Research 23: 107–130 |
| [18] | Godsill S. J., Doucet A., West M. (2004) Monte Carlo smoothing for nonlinear time series. Journal of the American Statistical Association 99: 156–168 · Zbl 1089.62517 · doi:10.1198/016214504000000151 |
| [19] | Harvey A. C., De Rossi G. (2006) Signal extraction. In: Mills T. C., Patterson K. (eds) Palgrave handbook of econometrics: Volume 1 econometric theory. Palgrave Macmillan, Basingstoke |
| [20] | Hull J., White A. (1990) Pricing interest rate derivative securities. Review of Financial Studies 3: 573–592 · Zbl 1386.91152 · doi:10.1093/rfs/3.4.573 |
| [21] | Hürzeler M., Künsch H. R. (2001) Approximating and maximising the likelihood for a general state-space model. In: Doucet A., de Freitas N., Gordon N. (eds) Sequential Monte Carlo methods in practice. Springer, New York, pp 159–176 |
| [22] | Kitagawa G. (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics 5: 1–25 · doi:10.2307/1390750 |
| [23] | Lamoureux C., Witte H. D. (2002) Empirical analysis of the yield curve: The information in the data viewed through the window of Cox, Ingersoll and Ross. Journal of Finance 57: 1479–1520 · doi:10.1111/1540-6261.00467 |
| [24] | Longstaff F. A., Schwartz S. (1992) Interest rate volatility and the term structure: A two factor general equilibrium model. Journal of Finance 47: 1259–1282 · doi:10.2307/2328939 |
| [25] | Lund, J. (1997). Econometric analysis of continuous-time arbitrage-free models of the term structure of interest rates. Unpublished Working Paper, Aarhus School of Business. |
| [26] | Michalewicz Z. (1999) Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, Berlin · Zbl 0818.68017 |
| [27] | Pitt, M. (2002). Smooth particle filters for likelihood evaluation and maximisation. Warwick Economic Research Papers, Department of Economics, 651. |
| [28] | Pitt M., Shephard N. (1999) Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association 94: 590–599 · Zbl 1072.62639 · doi:10.2307/2670179 |
| [29] | Takahashi A., Sato S. (2001) Monte Carlo filtering approach for estimating the term structure of interest rates. Annals of the Institute of Statistical Mathematics 53: 50–62 · Zbl 0995.62105 · doi:10.1023/A:1017964304055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
