International Journal of Differential Equations

Volume 2018, Article ID 4762485, 11 pages

https://doi.org/10.1155/2018/4762485

## Applications of Parameterized Nonlinear Ordinary Differential Equations and Dynamic Systems: An Example of the Taiwan Stock Index

^{1}Department of Mathematical Sciences, National Chengchi University, Taipei 116, Taiwan^{2}Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei 106, Taiwan^{3}Department of Statistics, National Chengchi University, Taipei 116, Taiwan

Correspondence should be addressed to Yong-Shiuan Lee; wt.ude.uccn@10545399

Received 29 June 2017; Revised 24 September 2017; Accepted 25 February 2018; Published 12 April 2018

Academic Editor: Tongxing Li

Copyright © 2018 Meng-Rong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Considering the phenomenon of the mean reversion and the different speeds of stock prices in the bull market and in the bear market, we propose four dynamic models each of which is represented by a parameterized ordinary differential equation in this study. Based on existing studies, the models are in the form of either the logistic growth or the law of Newton’s cooling. We solve the models by dynamic integration and apply them to the daily closing prices of the Taiwan stock index, Taiwan Stock Exchange Capitalization Weighted Stock Index. The empirical study shows that some of the models fit the prices well and the forecasting ability of the best model is acceptable even though the martingale forecasts the prices slightly better. To increase the forecasting ability and to broaden the scope of applications of the dynamic models, we will model the coefficients of the dynamic models in the future. Applying the models to the market without the price limit is also our future work.

#### 1. Introduction

Stock indices draw a lot of attention in the financial field and modelling stock prices is one of the major topics. Nevertheless, in most existing studies, stock price returns are modelled instead of stock prices themselves. It results from the theory of random walks stated and published in many books and theses. Fama’s 1965 article [1] is one of the most commonly known ones. Among massive researches of modelling stock prices, Chen et al. [2] have investigated and modelled the mean reversion of stock prices. They characterized the phenomenon by three dynamic models derived from the concept of Newton’s law of cooling. However, the type of the stock movement other than mean reversion should be included in a more reasonable model. Also, the speed of the convergence to the implied equilibrium should be described if we try to increase the accuracy of the model.

From past experiences, the uptrend and downtrend of the stock price, usually during a bull market and a bear market, respectively, move differently. The uptrend usually goes up fast at first due to good news and then slows down gradually as a result of selling pressure, but the downtrend generally drops steeply. The stock price movement and the logistic growth perform very much alike; nevertheless, stock prices fluctuate more dynamically in the real world. Therefore, modelling this phenomenon is the main goal in the study.

We examined a series of the daily closing prices of Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and treated it as a dynamic system. The only assumption in this study is that the stock prices are related to time. In order to depict the movement with respect to time, we attempt to modify the logistic growth model and bring in the technique of dynamic integration to construct some models. Furthermore, the models describing the mean reversion in the existing study only consist one time related coefficient [2]. We include two time-varying coefficients for constructing new models to increase the accuracy. Hence, we will introduce the nonlinear dynamic models in this study. From this point of view, modelling the stock prices may be more reliable.

In Section 2, we provide the literature review of the logistic growth model and dynamic integration. Section 3 is the methodology of how we build the nonlinear dynamic models. Section 4 consists of the empirical study of applying the dynamic models to TAIEX. In Section 5, we make some conclusions from the empirical study and some suggestions for future works.

#### 2. Literature Review

The logistic growth model was named by the mathematician Verhulst. He solved the logistic equation and called the solution the logistic function [3–5] to express the population growth. The logistic model is continuous in time and can be stated by the differential equation:where is time, is the growth rate, is the carrying capacity, and is the population size. The logistic growth model was first used to describe the population growth in a restricted environment and then extensively applied to all kinds of fields such as Biology, Chemistry, Economics, Demography, and Statistics.

Concerning the topic of macroeconomics, Teräsvirta and Anderson [6] included logistic function to build a smooth transition autoregressive (STAR) model for describing business cycles. Based on their research, González-Rivera [7] applied STGARCH model on stock returns and exchange rates to explore the asymmetric response of conditional variances to positive and negative news. Other than applying logistic function, the nonlinear time series model such as GARCH is applied to explain the nonlinear phenomenon in the behaviors of the stock returns in the existing studies [8, 9]. Besides, there are neural network models applied to model the stock index [10–12]. Guresen et al. [10] list a table of existing studies of artificial neural networks and financial time series models. The exponential law of the stock movement is described and analyzed by Gkranas et al. [13] and Zarikas et al. [14]. However, applying logistic equation to stock prices directly is absent in existing studies.

As for applying dynamic system and differential equations to stock prices, Chiang-Lin et al. [15] applied “parabola approximation” and “dynamic integration” proposed in [16] to model the Taiwan stock index, TAIEX, and evaluate its derivative by considering the index as a dynamic system. Li et al. [17] also applied the same methods to model German DAX and tried to develop a risk detecting instrument of approaching financial catastrophe. Chen et al. [2] further combined Newton’s law of cooling and dynamic integration to explain the phenomenon of mean reversion by modelling TAIEX. On the basis of existing researches, we try to transform the logistic model into some models by considering the stock price, the velocity of the stock price, and the stock return as the subject, respectively. We also solve the equations representing the models by dynamic integration. The details of the dynamic models are in the following section.

#### 3. Methodology

In order to characterize the movement of the stock prices, we only assume that the stock price is a function of time and consider it as a dynamic system. Since we suspect that the rise and fall of the stock prices may be described by the logistic growth model, we combine the concept of the logistic growth model and the dynamic integration to build dynamic models. The models are represented by differential equations which are detailed below along with their solutions.

*Model A (dynamic logistic model)*where is the stock price at time . We can rewrite (2) asand hence model A is in the form of the logistic growth model.

Because the stock prices change dynamically in reality, we assume that the coefficients and are constant during a very short time. Then we consider the parameterized differential equation:with given values of two points at time and . Let and the solution is

Model A describes the relationship between the stock price, , and its velocity, . From the economic point of view, there exists an implied equilibrium in the market. All other things being equal, if the stock price is below the equilibrium, it will rise towards the equilibrium price; if the stock price is above the equilibrium, it will decline. The phenomenon is also called the mean reversion. This model characterizes the mean reversion of the stock prices if is negative, is positive and when the expected future stock price is higher than the current stock price. Under the circumstances, the implied equilibrium is at and the stock price moves as how the logistic growth model describes. The convergence of the simulated theoretical stock prices by (5), (6) is plotted in Figures 7 and 8. Otherwise, the movement of the stock prices follows a quadratic differential equation with no constant growth rate [15]. In this situation, the stock prices will not converge to the implied equilibrium.

In addition to model A, we further consider both the velocity, , and the acceleration, , of the stock price to construct a model. In this instance, the model may represent the movement more properly. Hence, we propose model B in the following.

*Model B (dynamic transformed logistic model)*where is the stock price at time .

Assuming the coefficients and are constant during a very short time, we consider the parameterized differential equation with given values of two points at times and :and it can be solved as follows. Let and ; then the solution is

Model B describes the relationship between the velocity, , and the acceleration, , of the stock price. The relationship is in the form of the logistic growth model. When is negative and is positive, the velocity moves in the way that the logistic growth model depicts. That is, the velocity reverses to an implied equilibrium velocity so that the stock price goes up or down with an approximately constant speed. In the case of other combinations of the coefficients ( and ), the velocity diverges and hence the stock prices fluctuate more dynamically.

Other than the velocity, the relative growth rate of the stock price, , which is also known as the return is a meaningful measure in the field of finance. Therefore, we change the velocity and the acceleration in model B into the relative growth rate and its derivative to build model C. Model C provides another kind of differential equation characterizing the stock price movement.

*Model C (dynamic relative growth rate transformed logistic model)*where is the stock price at time and is the relative growth rate of the stock price at time .

Assuming the coefficients and are constant during a very short time, we consider the parameterized differential equation with given values of two points at times and :and it can be solved as follows. Let and the solution is

Model C describes the relationship between the relative growth rate of the stock price, , and its derivative. Since the relative growth rate is the return of the stock price, when is negative and is positive, the return moves in the way that the logistic growth model depicts. That is, the return reverses to a constant implied equilibrium. In the case of other combinations of the coefficients ( and ), the return diverges and the fluctuations of the stock prices are more dynamical but different from what model B describes.

The dynamic models above are built on the work of Chen et al. [2]. But all of them consist of two coefficients instead of one as in their study. Hence we generalize one of their models into a two-coefficient dynamic model and compare it with the three models above. This extended model is stated in the following.

*Model D (dynamic general Newton model)*where is the stock price at time . We assume the coefficients in (15) are constant during a very short time interval and consider the parameterized differential equation:between two given data points at times and . The solution can be found in the article of Chen et al. [2].

As applying these dynamic models, we discretize and convert them into difference equations since the time interval between the two given points at times and is relatively short compared to the interval of the complete data. The discretization and the parameterization make the differential equations solvable. In the empirical study, we fit and forecast the data by the solutions. We also adopt the dynamic forecasting method discussed in [2] to obtain more accurate theoretical values.

The forecasting error is measured by MAPE (Mean Absolute Percentage Error) and RMSPE (Root Mean Square Percentage Error). They are defined as and , where is the theoretical value of the model at time , is the market value at time , and is the length of data points. Lewis [18] suggested that the forecasting ability of a model can be classified as in Table 1.