We propose a new alternative method to estimate the parameters in one-factor mean reversion processes based on the maximum likelihood technique. This approach makes use of Euler-Maruyama scheme to approximate the continuous-time model and build a new process discretized. The closed formulas for the estimators are obtained. Using simulated data series, we compare the results obtained with the results published by other authors. 1. Introduction Different applications of stochastic differential equations have attracted the attention of many researchers in relation to the theoretical advances needed for modeling and simulation of the dynamic behavior of the main variables affecting the evolution of some phenomena. Applications in fields of economics, finance, physics, insurance, and others have been drivers of these developments. Particularly, through diffusion processes have been carried out modeling interest rates and commodity prices, where they used in their modeling processes mean reversion. Some trends of these studies are based on some a priori knowledge to determine the structure of the model to be used, while other alternatives are part of a general structure, and statistical procedures through the particular structure are determined to represent the phenomenon. Whichever approach is used, in general, in the specification of the model, it is necessary to estimate parameters to achieve a proper fit of the model. Although some parameter estimation methods have been extensively studied in diffusion processes, this has a serious problem, which is that the specification of the models is done in continuous time but the data that is available for adjusting the parameters are discretely sampled in time. A common solution to this problem is to discretize the continuous-time model based on Euler-Maruyama scheme and perform procedures on the theoretical formulation resulting in discrete time. From this scheme alternatives can be proposed for parameter estimation approach among which stands out the maximum likelihood estimation. Following the classification proposed by Yu and Phillips [1], different parameter estimation procedures can be classified globally into three categories according to how to proceed from the model formulation: discretization of continuous process and estimation of the parameters with the resulting discretized process, estimation of the transition probability function for the continuous-time model, and independent estimations of the trend and volatility components by kernel methods. Thus, among the methods that are based on the
References
[1]
J. Yu and P. C. B. Phillips, “A Gaussian approach for continuous time models of the short-term interest rate,” The Econometrics Journal, vol. 4, no. 2, pp. 210–224, 2001.
[2]
K. Nowman, “Gaussian estimation of single-factor continuous time models of the term structure of interest rates,” The Journal of Finance, vol. 52, pp. 1695–1703, 1997.
[3]
J. Yu, “Bias in the estimation of the mean reversion parameter in continuous time models,” Journal of Econometrics, vol. 169, no. 1, pp. 114–122, 2012.
[4]
G. B. Durham and A. R. Gallant, “Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes,” Journal of Business & Economic Statistics, vol. 20, no. 3, pp. 297–338, 2002.
[5]
L. Valdivieso, W. Schoutens, and F. Tuerlinckx, “Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type,” Statistical Inference for Stochastic Processes, vol. 12, no. 1, pp. 1–19, 2009.
[6]
O. Elerian, S. Chib, and N. Shephard, “Likelihood inference for discretely observed nonlinear diffusions,” Econometrica, vol. 69, no. 4, pp. 959–993, 2001.
[7]
A. R. Pedersen, “Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes,” Bernoulli, vol. 1, no. 3, pp. 257–279, 1995.
[8]
T. Koulis and A. Thavaneswaran, “Inference for interest rate models using Milstein's approximation,” The Journal of Mathematical Finance, vol. 3, pp. 110–118, 2013.
[9]
G. J. Jiang and J. L. Knight, “A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model,” Econometric Theory, vol. 13, no. 5, pp. 615–645, 1997.
[10]
D. Florens-Zmirou, “On estimating the diffusion coefficient from discrete observations,” Journal of Applied Probability, vol. 30, no. 4, pp. 790–804, 1993.
[11]
O. Vasicek, “An equilibrium characterization of the term structure,” Journal of Financial Economics, vol. 5, pp. 177–186, 1977.
[12]
M. Brennan and E. Schwartz, “Analyzing convertible bonds,” Journal of Financial and Quantitative Analysis, vol. 15, pp. 907–929, 1980.
[13]
J. Cox, J. Ingersoll, and S. Ross, “An analysis of variable rate loan contracts,” The Journal of Finance, vol. 35, pp. 389–403, 1980.
[14]
K. Chan, F. Karolyi, F. Longstaff, and A. Sanders, “An empirical comparison of alternative models of short term interest rates,” The Journal of Finance, vol. 47, pp. 1209–1227, 1992.
[15]
A. Lari-Lavassani, A. Sadeghi, and A. Ware, “Mean reverting models for energy option pricing,” Electronic Publications of the International Energy Credit Association, 2001, http://www.ieca.net.
[16]
D. Pilipovic, Energy Risk: Valuing and Managing Energy Derivatives, McGraw-Hill, New York, NY, USA, 2007.
[17]
G. Courtadon, “The Pricing of options on default-free bonds,” Journal of Financial and Quantitative Analysis, vol. 17, pp. 75–100, 1982.
[18]
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, pp. 385–407, 1985.
[19]
K. Dunn and J. McConnell, “Valuation of GNMA Mortgage-backed securities,” The Journal of Finance, vol. 36, no. 3, pp. 599–616, 1981.
[20]
K. Ramaswamy and S. Sundaresan, “The valuation of floating-rate instruments,” Journal of Financial Economics, vol. 17, pp. 251–272, 1986.
[21]
S. Sundaresan, Valuation of Swaps. Working Paper, Columbia University, 1989.
[22]
F. Longstaff, “The valuation of options on yields,” Journal of Financial Economics, vol. 26, no. 1, pp. 97–122, 1990.
[23]
G. Constantinides and J. Ingersoll, “Optimal bond trading with personal taxes,” Journal of Financial Economics, vol. 13, pp. 299–335, 1984.
[24]
G. Jiang and J. Knight, “Finite sample comparison of alternative estimators of Ito diffusion processes: a Monte Carlo study,” The Journal of Computational Finance, vol. 2, no. 3, 1999.
[25]
L. Giet and M. Lubrano, Bayesian Inference in Reducible Stochastic Differential Equations, Document de Travail No. 2004–57, GREQAM University d’Aix-Marseille, 2004.
[26]
A. R. Bergstrom, “Gaussian estimation of structural parameters in higher order continuous time dynamic models,” Econometrica, vol. 51, no. 1, pp. 117–152, 1983.
[27]
A. Bergstrom, “Continuous time stochastic models and issues of aggregation over time,” in Handbook of Econometrics, Z. Griliches and M. D. Intriligator, Eds., vol. 2, Elsevier, Amsterdam, The Netherlands, 1984.
[28]
A. Bergstrom, “The estimation of parameters in nonstationary higher-order continuous-time dynamic models,” Econometric Theory, vol. 1, pp. 369–385, 1985.
[29]
A. Bergstrom, “The estimation of open higher-order continuous time dynamic models with mixed stock and flow data,” Econometric Theory, vol. 2, pp. 350–373, 1986.
[30]
A. Bergstrom, Continuous Time Econometric Modelling, Oxford University Press, Oxford, UK, 1990.
[31]
P. C. B. Phillips and J. Yu, “Corrigendum to ‘A Gaussian approach for continuous time models of short-term interest rates’,” The Econometrics Journal, vol. 14, no. 1, pp. 126–129, 2011.
[32]
M. J. D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives,” The Computer Journal, vol. 7, pp. 155–162, 1964.
[33]
V. S. Huzurbazar, “The likelihood equation, consistency and the maxima of the likelihood function,” Annals of Eugenics, vol. 14, pp. 185–200, 1948.
[34]
A. Kohatsu-Higa and S. Ogawa, “Weak rate of convergence for an Euler scheme of nonlinear SDE's,” Monte Carlo Methods and Applications, vol. 3, no. 4, pp. 327–345, 1997.
[35]
V. Bally and D. Talay, “The law of the Euler scheme for stochastic differential equations: error analysis with Malliavin calculus,” Mathematics and Computers in Simulation, vol. 38, no. 1–3, pp. 35–41, 1995.
[36]
A. Zellner, “Bayesian analysis of regression error terms,” Journal of the American Statistical Association, vol. 70, no. 349, pp. 138–144, 1975.
