Parameter Estimation for Constantinides-Ingersoll Model from Discrete Observations

Parameter Estimation for Constantinides-Ingersoll

Model from Discrete Observations

WEI Chao (魏 超)1, SHU Hui-sheng (舒慧生)2*

【摘 要】Abstract: The parameter estimation problem for an economic model called Constantinides-Ingersoll model is investigated based on discrete observations. Euler-Maruyama scheme and iterative method are applied to getting the joint conditional probability density function. The maximum likelihood technique is employed for obtaining the parameter estimators and the explicit expressions of the estimation error are given. The strong consistency properties of the estimators are proved by using the law of large numbers for martingales and the strong law of large numbers. The asymptotic normality of the estimation error for the diffusion parameter is obtained with the help of the strong law of large numbers and central-limit theorem. The simulation for the absolute error between estimators and true values is given and the hypothesis testing is made to verify the effectiveness of the estimators.

【期刊名称】东华大学学报（英文版） 【年(卷),期】2016(033)002 【总页数】5

【关键词】Key words: diffusion process; maximum likelihood estimation(MLE);

discrete

observation;

consistency;

asymptotic

normality; hypothesis testing CLC number: O211.6; O211.9 Document code: A

Article ID: 1672-5220(2016)02- 0183- 05 Introduction

Stochastic phenomena are widespread in nature and many of them can be described by a system of ordinary differential equations which are perturbed by random disturbances. Stochastic model has come to play a key role in many branches of science and industry where more and more people has encountered stochastic processes. Stochastic processes are widely used for model building in social, physical, engineering and life sciences. Recently, stochastic processes are essential to many modern finance theories and have been widely used to model the behavior of main variables such as the instantaneous short-term interest rates, asset prices, asset returns and their volatilities. Statistical inference for stochastic processes is very important in terms of theories and applications in model building. In engineering practice, part or all of the parameters in stochastic systems are always unknown, but the observed values are known. Therefore, parameter estimation has become a serious problem needed to be solved based on observed values. Recently, some methods have been investigated to solve the estimation problem and some of them are wide used such as least-

squares estimation[1-3], maximum likelihood estimation (MLE)[4-6] and Bayesian estimation[7-9]. Nevertheless, a problem occurs in the process of estimating the parameters, which is that the observed data are discretely sampled in time but the model is in continuous time. Generally, there are two methods for solving this problem. One is discretizing the process with Euler-Maruyama scheme[10-11], and the other is approximating the likelihood function[6, 12]. Moreover, Chang and Chen[13] has given the approximate transition densities based on series expansions. Uchida and Yoshida[14] has considered the adaptive maximum likelihood type estimation of both drift and diffusion parameters for an ergodic diffusion process. When the process is observed partially, extended Kalman filter and local linearization filter have been applied[15-16].

Many economic models are described by stochastic differential equations, especially the short-term interestrates. The short-term interest rate is one of the most fundamental and important prices determined in financial markets and more models have been put forward to explain its behavior than for other issues in finance[17-21]. The widely used Cox-Ingersoll-Ross (CIR) pricing formula is a short-term interest rate model and has been studied by many authors[22-23]. However, the drift coefficients in CIR model are linear. The drift and diffusion coefficients in Constantinides-Ingersoll model considered in