Isao Shoji

Comparative study of estimation methods for continuous time stochastic processes

In this paper we investigate the finite sample performances of five estimation methods for a continuous-time stochastic process from discrete observations. Applying these methods to two examples of stochastic differential equations, one with linear drift and state-dependent diffusion coefficients and the other with nonlinear drift and constant diffusion coefficients, Monte Carlo experiments are carried out to evaluate the finite sample performance of each method. The Monte Carlo results indicate that the differences between the methods are large when the discrete- time interval is large. In addition, these differences are noticeable in estimations of the diffusion coefficients.

Searching for an optimal rotation age for forest stand management under stochastic log prices

Due to rapid change in timber prices in the Japanese market most likely affected by imported timber from countries such as the U.S., Canada, and the Nordic countries, the domestic forest managers have been facing a large degree of future price uncertainty. Because of this, it becomes necessary to take the future price uncertainty into account within the forest management framework. In this paper, the continuous time stochastic process, i.e., the geometric Brownian motion, has been used to model the log price process. The binomial option pricing approximation was then applied to value the Sugi (Cryptomeria japonica) and Hinoki (Chamaecyparis obtusa) forested land under stochastic log prices in order to search for an optimal rotation age. Our experiments with the proposed two state stochastic dynamic programming model showed that when the current log price is high enough to cover all costs, an optimal rotation age from the stochastic price and deterministic price models coincides, although the total expected present net value from management activities differs. Also it was shown that as the current log price decreases, an optimal rotation age derived from the stochastic price model becomes longer than that from the deterministic price model. If the current log price further decreases, then forest management will be abandoned, and the forest stand will be converted into alternative uses.

Approximation of continuous time stochastic processes by a local linearization method

This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be O((Δt) 2 ) and O(Δt) in the L p sense respectively, where Δt is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.

A statistical comparison of the short-term interest rate models for Japan, U.S., and Germany

We propose a simple and practical model selection method for continuous time models. We apply the method to several continuous time short-term interest rate models using discrete time series data of Japan, U.S. and Germany. All the models can be easily estimated from discrete observations, and their performances can be evaluated in a uniform statistical framework. The models that allow dependence of volatility on the level of interest rates tend to perform well empirically. The degree of volatility dependence on the interest rate levels seems to be different across the countries. For the German data, we observe that a model with nonlinear drift performs better than the best linear drift model.

Modeling the term structure of interest rates with general short-rate models

Abstract. We propose a model of the term structure of interest rates with generally specified processes of the short-rate. As documented by many studies, it is important to model the short-rate generally to capture its observed behavior. This adequate specification is essential for term structure models to explain yields accurately. With such general conditions, however, term structure models cannot be obtained explicitly. In this paper we derive an analytical model of the term structure by locally approximating a short-rate process with nonlinear drift and diffusion terms. The empirical analysis confirms that the proposed model outperforms the existing affine term structure model.

Approximation of Non-Linear Term Structure Models

“Equilibrium term structure models, like those of Vasicek, or Cox, Ingersoll, and Ross (CIR), start with assumptions about the behavior of the short rate; the full term structure is then obtained by projecting the short rate process forward. To obtain closed form valuation equations, the drift of the short rate is typically assumed to be linear, but empirical evidence suggests that its behavior is actually more complex. Short rates seem to follow a driftless random walk when they are at intermediate levels, but to exhibit mean reversion when the rate becomes very low or very high. However, incorporating a non-linear drift leads to complicated estimation problems. In this article, Takamizawa and Shoji present a linearization technique that allows them to fit a model with non-linear drift more simply. They show that with U.S. interest rate data, their approach fits better in-sample and is more accurate out of sample than a linear drift CIR model.”

Nonparametric estimation of nonlinear dynamics by metric-based local linear approximation

This paper discusses nonparametric estimation of nonlinear dynamical system models by a method of metric-based local linear approximation. We assume no functional form of a given model but estimate it from experimental data by approximating the curve implied by the function by the tangent plane around the neighborhood of a tangent point. To specify an appropriate neighborhood, we prepare a metric defined over the Euclidean space in which the curve exists and then evaluate the closeness to the tangent point according to the distances. The proposed method differs from the first order polynomial modeling in discerning the metric and the weighting function, but the first order polynomial modeling with Gaussian kernels is shown to be a special version of the proposed method. Simulation studies and application to ECG signals show the proposed method is easy to manipulate and has performance comparable to or better than the first order local polynomial modeling.

A note on asymptotic properties of the estimator derived from the Euler method for diffusion processes at discrete times

In this note we investigate asymptotic properties of an estimator, called the Euler estimator, which is obtained by maximizing the likelihood function of the process discretized by the Euler method. By linking the Euler estimator of the coefficients of the drift function of a stochastic differential equation with the least square estimator and the maximum likelihood estimator based on the likelihood ratio approach, it is shown that the three estimators are equivalent. Furthermore, it is also shown that the Euler estimator of a coefficient of the diffusion term has consistency.

On the accuracy of the local linear approximation for the term structure of interest rates

We examine by numerical experiments the accuracy of an analytical approximation for the nonlinear term structure of interest rates, which is obtained by applying the local linear approximation to a generally specified process of the short rate. Under various short-rate models, we compare discount-bond prices computed by the approximation with those calculated by the Monte Carlo method as the benchmark, which shows that deviations are small. Also in this paper, we show that the approximation originally derived in single-factor framework can be easily extended to a multifactor counterpart. We examine the accuracy using an illustrative two-factor model, which also shows the approximation is accurate.

An empirical analysis of the volatility of the Japanese stock price index: a non-parametric approach

This paper presents an empirical analysis of stochastic features of volatility in the Japanese stock price index, or TOPIX, using high-frequency data sampled every 5 min. The process of TOPIX is modeled by a stochastic differential equation with the time-homogeneous drift and diffusion coefficients. To avoid the risk of misspecification for the volatility function, which is defined by the squared diffusion coefficient, the local polynomial model is applied to the data, and then produced the estimates of the volatility function together with their confidence intervals. The result of the estimation suggests that the volatility function shows similar patterns for one period, but drastically changes for another.