A novel improved trigonometric neural network algorithm for solving price-dividend functions of continuous time one-dimensional asset-pricing models

Abstract

Asset pricing model is the pillar of modern financial market price theory. It is of great practical and theoretical significance to solve the equilibrium price-dividend function of the asset pricing model. To solve the asset pricing model, this paper develops a novel neural network method called improved trigonometric neural network, which consists of three parts: the improved trigonometric function, the initial-condition extreme learning machine algorithm and the reduction algorithm. First, the equilibrium price-dividend function is described as a stochastic differential equation and this function is translated into a second-order ordinary differential equation equivalently. Second, the improved trigonometric neural network is used to solve the differential equation with initial conditions, where the improved trigonometric function is used to serve as the activation function. The reduction algorithm is proposed to obtain a simpler network structure and a faster computing speed. Third, numerical experiments of the improved trigonometric neural network show that the numerical solution of the price-dividend function will be obtained precisely, quickly and feasibly. Compared with several methods to solve asset pricing model, the improved trigonometric neural network has the highest accuracy and fastest speed.

Introduction

Capital asset pricing model plays an important role in modern financial theory. This model is one of the most famous and important models in finance, which is widely used in capital cost budget [1], [2], [3], asset valuation [4], [5], resource allocation [6], [7], [8] and so on. It is of great practical significance and value to study the asset pricing model and its related calculation problems.

In recent decades, asset pricing theory has become one of the fastest developing fields in financial theory. One-dimensional asset-pricing model is a branch of the asset pricing model, which can be divided into continuous time model and discrete time model. The key problems of both models are to solve equilibrium price-dividend functions. In a continuous time model, the equilibrium price-dividend function can be translated into an ordinary differential equation (ODE). The ODE has analytic solutions when the stochastic discount factor (SDF) has a linear structure or the coefficients for the ODE are analytic [9]. In a discrete time model, the equilibrium price-dividend function can be translated into an integral equation (IE). By using higher order polynomial approximations [10], [11], [12], numerical solutions of many discrete time models can be obtained no matter whether the SDF is linear. However, there are two difficulties with discrete time models. One difficulty is the high cost of calculation since the coefficients have to be calculated simultaneously [9]. Another difficulty is the IE of the model must be true within a range for the state variable [13]. By using analytic methods to continuous time asset-pricing models, the second difficulty can be avoided. However, the running time of the analytic methods is longer and the operation cost is higher compared with machine learning methods. Therefore, in order to overcome the first difficulty, machine learning methods are applied to continuous time asset-pricing models.

This paper attempts to use machine learning methods to solve continuous time asset-pricing models so as to overcome the above two difficulties. In continuous time asset-pricing models, the equity premium literature indicates that an important property of non-linear SDF is to capture the dynamic behavior of the stock premium [14], [15], [16]. Therefore, a continuous time asset-pricing model with nonlinear SDF is considered in the paper. Specifically, the Campbell and Cochrane asset-pricing model (CC model) is considered, which is a continuous time one-dimensional asset-pricing model with non-linear analytic SDF and dividend process [17]. Solving this model is equivalent to solving the equilibrium price-dividend function, which is described as a stochastic differential equation (SDE). Furthermore, the equilibrium price-dividend function is translated equivalently into a second-order ordinary differential equation (ODE) with initial values, i.e. an initial value problem (IVP). Once the IVP is solved, the CC model is solved. In this paper, the extreme learning machine (ELM) method is applied to solve the IVP of the CC model.

Machine learning algorithms are more and more extensively applied in various fields. Neural networks are one of the most popular algorithms and attract extensive attention in various fields [18], [19], [20]. Cao et al. built a fully deconvolutional neural network and applied it to the problem of single image super-resolution [21]. Xiao et al. proposed several novel nonlinear finite-time recurrent neural networks which can address the problem of motion tracking of robot manipulators [22], [23]. The extreme learning machine is one of neural networks, whose input weights and biases are fixed so that the ELM algorithm has a fast learning speed and a good generalization performance. Hu et al. used the random forest (RF) and the ELM to forecast salinity time series [24] and Mariano et al. applied the ELM to the reverse engineering of gene regulatory networks from expression time series [19]. These methods are suitable for time series instead of solving ordinary differential equations. Liao et al. proposed a hybrid classification approach called ATELMSRC using the ELM and sparse representation classifier [25] and Kim et al. proposed the multimodal sparse hierarchical extreme learning machine to identify Alzheimer’s disease and mild cognitive impairment [26]. Both methods are good for classification. In the financial field, many problems such as asset pricing model and bankruptcy probability can be described as ordinary differential equations (ODEs). Therefore, many researchers proposed various algorithms to solve ODEs and applied them to the above problems. Yang et al. proposed a Legendre improved extreme learning machine (L-IELM) [27] to solve ODEs. Zhou et al. proposed an improved ELM algorithm (I-ELM) to solve the integro-differential equation of ruin probability [28]. In addition, there exist other machine learning methods developed for solving ODEs such as least squares support vector machines (LSSVM) [29] and genetic algorithm (GA) [30]. Most of the above methods assume a trial solution for the ODE to be solved. For example, a trial solution for the ODE of L-IELM is represented using Legendre polynomials, a trial solution used in LSSVM is represented by the RBF kernel function, and the GA assumes a trial solution using the power function.

Inspired by the literature of algorithms to solve ODEs, the authors assume a trial solution represented by a Fourier series after comparing different kinds of orthogonal polynomials. Then, we develop a novel improved trigonometric neural network (ITNN) based on the extreme learning machine. Furthermore, the reduction algorithm for number of neurons is proposed to simplify the network structure since different functions have different distributions of harmonics across the spectrum.

In this paper, a novel improved trigonometric neural network is proposed to solve the IVP of the CC model. The improved trigonometric neural network consists of three parts: the improved trigonometric function based on Fourier series expansion, the initial-condition extreme learning machine (ICELM) algorithm and the reduction algorithm. Firstly, based on Fourier series expansion theory, the improved trigonometric function is proposed to serve as the activation function of the ITNN algorithm. Secondly, the ICELM is proposed to solve the initial value problem, where two initial conditions are added to the network model. Moreover, the reduction algorithm of the number of network neurons is proposed to get a simpler network structure and a higher computing speed. Fig. 1 shows the general framework of this paper. As shown in Fig. 1, the equilibrium price-dividend function of one-dimensional continuous time asset pricing models can be translated into an ordinary differential equation (ODE). The ODE with two initial conditions is called an initial value problem (IVP). Next, the IVP is solved by the ITNN in this paper. With the improved trigonometric function to serve as activation function, ITNN is applied to calculate the numerical solutions, corresponding optimal output weights $\widehat{\beta }$, optimal neurons number $\widehat{N}$ and optimal value $\widehat{l}$. Then, the reduction algorithm is applied to reduce the number of the network neurons under the condition of satisfying the requirement of precision. Next, the numerical solutions of the IVP are obtained feasibly and efficiently. Finally, the price-dividend ratio $p\left(\tau \right)$, i.e., the numerical solutions of IVP, is used to calculate the instantaneous expected return, instantaneous standard deviation for the return on equity and Sharpe ratio.

The rest of the paper is organized as follows. Campbell and Cochrane’s asset-pricing model and the initial value problem of price-dividend function are introduced in Section 2. Section 3 investigates the traditional ELM algorithm. Section 4 illustrates the proposed improved trigonometric neural network algorithm. To verify the feasibility and superiority of the ITNN algorithm, Section 5 carries out some numerical and comparative experiments for solving the price-dividend function of the Campbell and Cochrane asset-pricing model. In Section 6, some conclusions are provided. The main contributions of this paper are summarized as follows:

• The ITNN algorithm is presented to solve the price-dividend function of the Campbell and Cochrane asset-pricing model. The proposed method is able to solve different continuous time asset pricing models with analytic SDF and different initial conditions.

• The ITNN algorithm is used to reproduce all the results in CC model. The comparative results with the analytic method [9] demonstrate that the solutions obtained by the ITNN can accurately represent the behavior of stock returns while the accuracy is higher and time cost is lower than that of the analytic method.

• The improved trigonometric function based on the Fourier series is proposed to serve as the activation function of the ITNN algorithm. Numerical simulations and a comparative study among different orthogonal polynomials verify the high accuracy of the ITNN algorithm with the improved trigonometric function.

• The reduction algorithm of the number of network neurons is used to obtain a simpler network structure and a higher computing speed.

• Comparative studies among traditional methods to solve the IVP such as the ITNN, analytic method [11], L-IELM [27], I-ELM [28], LSSVM [29] and GA [30] are carried out to demonstrate the superiority of the ITNN algorithm.