FuzzyTree crossover for multi-valued stock valuation☆
Introduction
Stock valuation sums up the activities of estimating the intrinsic value of a business entity. It is important to securities analysis, loan decisions, and leveraged buyout analysis, and so on. Investors could suffer vast losses if they were to make improper decisions based on an inaccurate business value. To determine the value of business, it is necessary to understand the activities disclosed in financial statements. Conventional methods to stock valuation using financial statements are divided into three categories: asset appraisal [11], discounted present value [1], [12], and price/earnings multiples price [5]. According to these valuation methods, using different critical input variables gives rise to different outcomes even for the same stock. This suggests that the value of a business may be multi-valued rather than single-valued. Another drawback of these approaches is their use of pre-determined functions to design valuation models, which try to estimate the value of business from linear functions under specific assumptions and limitations. They always fail to fully capture flexibility and uncertainty.
The various soft computing technologies do provide alternative solutions to financial problems. For example, fuzzy logic is used as a possibility distribution of portfolios [7], [17], [19], or for the credit analysis of loans [5], [8]. Neural networks are used to predict financial distress [2], [3], [6]. One of the evolutionary computational techniques, genetic programming (GP), is applied to a stock market [10], futures or options pricing [9], and a foreign exchange market [4]. However, few studies have used soft computing methods for stock valuation. In this article we apply both fuzzy numbers to manifest multi-valued uncertainty and genetic programming to optimize an effective stock valuation model.
It is well recognized that crossover and selection operators play the major role in terms of generating solutions in GP [15]. A subtree crossover operator usually destroys building blocks (i.e., an effective partial trees), due to randomly and blindly choosing crossover points. Hence, many investigators propose new crossover methods to obtain more effective building blocks by reserving crucial schemata. For example, a hierarchical crossover can be combined with simulated annealing and hill climbing to find correct solutions via shrinking, growth, or internal substitution while preserving syntactic correctness [22]. Depth-dependent crossover accumulates building blocks according to the depth of a node. The depth selection ratio is higher for nodes closer to a root node [20]. A directing crossover reduces the amount of unviable code (bloat) in individuals while searching for a parsimonious solution [21]. It involves the identification of highly fit nodes to be used as crossover points during the operator application. The island model crossover applies a subtree crossover to aborigines and the depth-dependent crossover is for immigrants based on their ages, which demonstrates how long they can survive in the deme [16]. This can integrate many schemata to form a bigger building block of different deme. The dynamic page-based crossover is described in terms of the number of pages for all individuals. Pages are expressed as a number of instructions, which represents the dynamic change for all individuals in the population. These give rise to a succinct solution without penalizing optimization ability [13], [14]. These crossover operators only exchange constant schemata for all individuals in the population. No new genotype is generated even when swapping partial trees in the dedicated population. These are few studies available for exploring the new crossover operator in fuzzy genetic programming except for the subtree.
The objective of the present study is to develop a FuzzyTree crossover for a multi-valued stock valuation model which improves the convergence phenomenon. We generalize crisp expression trees evolved through GP to fuzzy ones by introducing fuzzy numbers and fuzzy arithmetic operators in the trees. FuzzyTree uses a subtree crossover operator if the selected crossover point is an internal node; otherwise, the selected terminal nodes are snipped into pieces and interchanged with each other. This could improve convergence via protecting the building block and increasing the variety of genotypes. Since the stock value is estimated by a fuzzy expression tree which calculates to a fuzzy number, the real stock values become multi-valued. In addition, the resulting trapezoidal fuzzy stock value induces a natural trading strategy which can readily be executed and evaluated.
Section snippets
Fuzzy genetic programming approach
Our FuzzyTree model generalizes the subtree crossover to produce the next generation’s fuzzy GP individual. The fuzzy GP individual is represented by a fuzzy GP tree as shown in Fig. 1, which contains terminal nodes (ni or vi) and internal nodes (fi). Each terminal node is represented by a trapezoidal fuzzy number. The detailed FuzzyTree crossover algorithm is described in Section 2.1. Section 2.2 introduces the encoding for each fuzzy node. The related arithmetic processing of our fuzzy GP
Experimental results
The simulation environment, sample data, and experimental results are described in this section. Our FuzzyTree-based program is written in Borland C++ Builder 6.0.
Conclusions
Stock valuation is important to investment policymakers. The conventional methods, such as asset appraisal, discounted present value, and price/earnings multiples, only use simple estimation functions to evaluate a stock’s value, lack flexibility and give rise to uncertainty. Therefore, we apply a fuzzy number to represent multi-valued uncertainty and GP to optimize the valuation model. However, the conventional crossover method is not available, because each leaf node is a trapezoidal fuzzy
Acknowledgement
We are extremely grateful to two anonymous referees and the editor, Witold Pedrycz, for many useful comments and suggestions on this paper.
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This work was supported by the National Science Council, Taiwan, ROC, under contracts NSC 92-2626-H-238-001.