chapter 45 a dynamic modeling of stock prices and optimal ... · a dynamic modeling of stock prices...

Chapter 45A Dynamic Modeling of Stock Pricesand Optimal Decision Making UsingMVP Theory

Ramin Rajabioun and Ashkan Rahimi-Kian

Abstract In this paper, first a precise mathematical model is obtained for fourcompeting or cooperating companies’ stock prices and then the optimal buy/sellsignals are ascertained for five different agents which are trading in a virtual marketand are trying to maximize their wealth over 1 trading year period. The model isso that gives a good prediction of the next 30th day stock prices. The companiesused in this modeling are all chosen from Boston Stock Market. Genetic Program-ming (GP) is used to produce the predictive mathematical model. The interactionamong companies and the effect imposed by each of five agents on future stockprices are also considered in our modeling. Namely, we have chosen eight compa-nies in order that there is some kind of interrelation among them. Comparison ofthe GP models with Artificial Neural Networks (ANN) and Neuro-Fuzzy Networks(trained by the LoLiMoT algorithm) shows the superior potential of GP in predic-tion. Using these models; five players, each with a specific strategy and all with onecommon goal (wealth maximization), start to trade in a virtual market. We have alsorelaxed the short-sales constraint in our work. Each of the agents has a differentobjective function and all are going to maximize themselves. We have used Parti-cle Swarm Optimization (PSO) as an evolutionary optimization method for wealthmaximization.

Keywords Stock market model � price prediction � Genetic Programming � mean-variance portfolio selection � Particle Swarm Optimization (PSO)

R. Rajabioun (B)School of Electrical and Computer Engineering, Control and Intelligent Processing Center ofExcellence, University of Tehran, Tehran, IranE-mail: [email protected]

S.I. Ao and L. Gelman (Eds.), Advances in Electrical Engineering and ComputationalScience, Lecture Notes in Electrical Engineering.c� Springer Science + Business Media B.V. 2009

525

526 R. Rajabioun, A. Rahimi-Kian

45.1 Introduction

Forecasting the change in market prices and making correct decisions is one of themost principal needs of anyone who economical environments concerns him. Timeseries are the most common methods used in price prediction [1–3]. But the pre-dominant defect of these methods is that they use only the history of a company’sprice to do a prediction. Recently, there has been growing attention to the modelsthat concern the interaction among companies in modeling and the use of game the-ory [4–6] in decision making because of providing more realistic models. Because ofcomplexity of the mutual effects of each company on the others, methods like Artifi-cial Neural Networks (ANN), Neuro-Fuzzy Networks and State Space (SS) modelsare used more often for the stock price modeling. In references [7–10] NeuralNetwork is used to model the stock market and make prediction. In reference [8],Genetic algorithm (GA) is incorporated to improve the learning and generalizabilityof ANNs for stock market prediction. In reference [11] the difference between theprice and the moving average, highest and lowest prices is used as inputs for one-day-ahead price prediction. More over, volume of transactions, market indicatorsand macro economic data are also considered as input variables [12]. There are alsosome studies being performed on the fluctuations and the correlations in stock pricechanges in physics communities, using the concepts and methods in physics [13,14].In reference [15] a neuro-genetic stock prediction system is introduced, which usesgenetic algorithm to select a set of informative input features for a recurrent neuralnetwork. In references [16, 17], the neuro-genetic hybrids for stock prediction areproposed. The genetic algorithm is used to optimize the weights of ANN.

Producing the right buy/sell signals are also important for those who trade inthe stock markets. In reference [18], two simple and popular trading rules includingmoving average and trading range breakout are tested in the Chilean stock market.Their results were compared with the buy-and-hold strategy, and both trading rulesproduced extra returns compared to the buy-and-hold strategy.

Genetic Programming (GP) is a symbolic optimization technique, developed byKoza [19]. It is an evolutionary computational technique based on the so-called“tree representation”. This representation is extremely flexible because trees canrepresent computer programs, mathematical equations, or complete models of pro-cess systems [20]. In reference [21] GP is used to produce a one-day-ahead modelto predict stock prices. This model is tested for a fifty consecutive trading days ofsix stocks and has yielded relatively high returns on investment.

In this paper, we use the GP to find the best mathematical models for the fourcompanies’ stock prices under study. Our GP models are able to predict these stockprices for up to the next 30 days with acceptable prediction errors in the market.Because, the GP is a well known algorithm we will not present it in details. However,reference [22] provides a good review of the GP algorithm.

The modeling is done for four companies in Boston Stock Market [23]. Selectedcompanies include: Advanced Micro Devices (AMD), Ericsson (ERIC), Sony(SNE), Philips (PHG), International Business Machines (IBM), Intel Corporation(INTC), Microsoft (MSFT) and Nokia (NOK). These companies are assumed to

45 A Dynamic Modeling of Stock Prices and Optimal Decision 527

have a relationship like competition or cooperation and so their stock prices couldaffect on each other. Letters allocated in parentheses are the symbols using whichone can access the price data of each company. We use the price history of theseeight companies as inputs to predict our four objective companies’ prices including:Ericsson (ERIC), International Business Machines (IBM), Sony (SNE) and Philips(PHG). Obtained four models precision is compared with two traditional methods:(1) Multi Layer Perceptron (MLP) and (2) Neuro-Fuzzy Network trained by LocallyLinear Model Tree (LoLiMoT) method.

After modeling the four companies’ stock prices, we create five agents who tradein a virtual market in order to maximize their wealth. These agents (players) will buyor sell their in hand stocks according to their uniquely defined strategies. Each playerhas a unique objective function. The Buy/Sell actions of each player are obtained soas to maximize its objective function in each trading period. The maximization isdone using the Particle Swarm Optimization (PSO) method [24].

At rest of the paper, in Section 45.2, modeling and prediction is discussed.Section 45.3 demonstrates the virtual stock market and argues its constraints andpresumptions. Then in Section 45.4 the results of our simulations are shown andfinally the conclusion is done in Section 45.5.

45.2 Modeling and Prediction

As stated earlier, our primary goal is to obtain a predictive model that is able topredict the future stock prices precisely. The companies that we are going to pre-dict their stocks include: Ericsson (ERIC), International Business Machines (IBM),Sony (SNE) and Philips (PHG). We presume that these companies have some kindof interrelations with four other companies including: Advanced Micro Devices(AMD), Intel Corporation (INTC), Microsoft (MSFT) and Nokia (NOK). So wedownloaded these eight companies’ price data from the Boston Stock Market [23].The downloaded data encompasses some information like: daily opening price, dailyclosing price, daily highest price, daily lowest price and exchange volume. In thispaper, we predict the average of daily opening and closing prices. Our data set con-tains sampled price data for the interval of 2001/07/08 to 2006/03/11. The criterionused to evaluate the models is the Normalized Mean Square Error (NMSE), whichis defined as follows:

NMSE D

nPiD1

.yi � Oyi /2

nPiD1

y2i

(45.1)

where yi and Oyi are the original and predicted price values, respectively. Figure 45.1shows the NMSE values for the train data set (the first 70%) using GP, ANN(MLP) and Neuro-Fuzzy networks (trained using LoLiMoT). Figure 45.2 depictsthis comparison for the test data set (the last 30%).


IBM ERIC SNE PHG0

0.002

0.004

0.006

0.008

0.01

0.012

NM

SE

MLP

NEUROFUZZY

GP

NMSE for train data

Fig. 45.1 The prediction error (NMSE) for all companies (using train data)

IBM ERIC SNE PHG0

0.1

0.2

0.3

0.4

0.5NMSE for test data

NM

SE

MLP

NEUROFUZZY

GP

Fig. 45.2 The prediction error (NMSE) for all companies (using test data)

The GP-based stock price models were initialized with some functions and ter-minals. The terminals included random number generators together with integersfrom 1 to 16. The functions included: fC;�;�;�; log x; ex; xyg. The populationsizes were set to 600 except for ERIC, which was set to 400. Meanwhile the numberof iterations was set to 80. As it can be seen from Figs. 45.1 and 45.2, the GP-based price model prediction errors are acceptable for the training data set and lessthan both of the MLP and Neuro-Fuzzy models for test data set. The only draw-back of the GP algorithm is its time-consuming modeling characteristics, which isacceptable comparing to its precise modeling.

Until now we have modeled the interactions of eight different companies thataffect the future price values. But due to the fact that buyers/sellers also affect futurestock prices of the companies, it is essential to include such interactions in the mod-eling. Therefore, we augment a new term to our price models in order to includethe effects of the market players’ actions (buy/sell weights) into the future price


Table 45.1 The PSO model parameters

Parameter range [�1, 1]Maximum optimization iterations each day 200Population size 180Acceleration constant 1 2Acceleration constant 2 2

changes. Since there are not much available data on how the buy/sell volumes of themarket players affect the future prices, we decided to add a new term to show theseeffects in our price prediction models as follows:

augmented term D � �W � a � Price vector (45.2)

where:

γ : is a weighting coefficient that regulates the impact of the augmented term onthe models. When γ is large the augmented term makes the model deviate fromthe trend of the time-series market historical data.W: is a weight vector that its elements show each company’s stock trade impacton future prices. The elements of this vector are between 0 and 1.a: is the action vector of all players. Its elements are between -1 and 1 that showthe buy/sell rates of the stocks.Price vector: contains the current stock price values in the market.

The best value for the γ factor obtained to be 0.1. The W-vector was chosen asfollows: W = [0.1 0.05 0.1 0.2 0.2 0.2 0.05 0.1]. The corresponding companies’symbol vector is: (AMD ERIC IBM INTC MSFT NOK PHG SNE).

The augmented term makes it possible for us to see the effect of each player’smarket decision on the stock prices and other players’ wealth (similar to a non-cooperative game).

Our objective in the next section would be to find the best market actions (sell/buyof each stock) of each player so as to maximize its expected objective function(wealth) in the market. Our market simulation studies are done in a Virtual StockMarket and by means of an evolutionary optimization algorithm (PSO). In our simu-lations a common PSO with inertia was used. Table 45.1 shows the parameters usedin the PSO optimization.

45.3 Virtual Stock Market

We assume five players (agents) in the stock market. We also assume that theseplayers have no stocks at the beginning of the market. They just have US$5,000,000and intend to buy stocks that would maximize their expected wealth in the market.The players are free to buy and sell stocks in each round of the market. There are


1,000,000 stocks assumed to be available from each company in our virtual stockmarket (4,000,000 stocks in total). The only limitation imposed by the market isthe maximum number of stocks each player can buy or sell each day. This buy/sellvolume is limited to 1,000 stocks trading per day for each company. This constraintis essential because if there is no limitation the whole stocks might be bought atthe beginning of the trading period by one of the players. This way there will beno chance for other players to buy some of the stocks. Through the augmented termadded to the stock price models we can see the effect of each agent’s action (sell/buystocks) on the future prices and other agents’ wealth in the market.

We assume five players (agents) with different objective functions and differentstrategies in the market, but we assume that all the agents have access to the stockprice models (developed in Section 45.2) symmetrically.

The players’ strategies are as follows:

Strategy of player 1:

This player buys (sells) the maximum number of allowed stocks when the predictionshows an increase (decrease) in next 30 day prices compared to the average pricesof the last 10 days.


This player uses the Mean-Variance Analysis (MVA). He chooses the standard devi-ation of the expected return (rp/ as a measure of risk (�p/. He plots the opportunityset (efficient frontier) for a four-asset portfolio and takes an average risk and for anaverage return each day. A sample opportunity set for a four-asset portfolio is shownin Fig. 45.3.


This player believes in Random Walk Theory. He believes that the stock prices areunpredictable and therefore, he buys and sells stocks randomly.

Fig. 45.3 A sample oppor-tunity set for a four-assetportfolio (dotted line: theopportunity set with A and Bassets only)


0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Mean-Variance Efficient Frontier and Random Portfolios

Risk (Standard Deviation)

Exp

ecte

d R

etu

rn

Working Region of Player 5

Some of Selectable

Points

Working Region of

Player 2

Working

Region of

Player 4

Fig. 45.4 The working regions of players 2, 4 and 5 on the risk-return efficient frontier (the redpoints can be selected in each trading day)


This player acts just like player 2. The only difference is in his risk averse behavior.To reach the minimum risk, in each stage, he selects the buy/sell weights on the kneeof efficient frontier curve.


This player also acts like player 2 with the difference that this player is risk lover.Therefore, in each stage this player selects the buy/sell weights with the maximumrisk and maximum expected return.

The working regions of players 2, 4 and 5 on the risk-return efficient frontier areshown in Fig. 45.4.

These five players buy and sell in the virtual stock market. In the related litera-ture it is usually seen that short-sales are disregarded when optimizing the players’objective functions and the optimization is just done through stock purchases. How-ever, in this paper we have relaxed this constraint and allowed the players to buyand sell their stocks when needed.

In the following, we define the objective functions for all players and demonstratetheir optimization process. For players 2, 4 and 5 that the risk values are importantin their decisions, we define their objective function as follows:

Ei D �E.rpi/� .1 � �/�pi

; i D 2; 4; 5 (45.3)

where Ei : is the Expected return of player i , �: is a constant between 0 and 1.In fact, this is a weight that shows the relative importance of the expected return


(E.rp// versus the risk (�p/ of player-i , for � D 1 the risk term disappears formthe objective function of player-i: Ei D E.rpi/. In our market simulation studies, wechose � D f0; 0:5; 1g for players f2, 4, 5g respectively according to their definedrisk behaviors in the market. The players’ objective functions were optimized withrespect to their decision variables (stock sell/buy actions) using the Particle SwarmOptimization method and the results are presented and analyzed in the followingsection.

45.4 The Market Simulation Results

The market simulation results for the five players are presented and analyzed inthis section. Figures 45.5 and 45.6 show the optimal buy/sell actions for players 1and 5 for each company’s stock (ERIC, IBM, PHG and SNE). The optimal buy/sellactions for players 2, 3 and 4 are shown in the appendix. In these figures, the buyactions are positive, sell actions are negative and no-actions are zero.

If the buy action gets +1 (or �1), then the player, should buy (or sell) the max-imum number of stocks allowed for that company. In Fig. 45.7, the wealth of eachplayer is shown for 1 trading year period. The wealth is measured as the values ofthe in-hand stocks added to the cash amount in-hand for each player.

As can be seen from Fig. 45.7, players 1 and 5 have done better than the restof them in terms of the wealth maximization for 1 year stock trading. Figure 45.8shows the expected risk values in each trading day for each player. As we expected,player 1 has the minimum expected risk over the trading period and has alsoobtained the maximum return from the market.

0 100 200 300 400-1

0

1Player 1 Trade weights for each portfolio assets

ERIC

0 100 200 300 400-1

0

1IBM

0 100 200 300 400-1

0

1PHG

0 100 200 300 400-1

0

1

Trading year

SNE

Fig. 45.5 The optimal trading actions of player 1 for all companies’ stocks


0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

-1

0

1Player 5 Trade weights for each portfolio assets

ERIC

-0.5

0

0.5IBM

-1

-1

0

1PHG

0

1

Trading year

SNE


0 100 200 300 4000

0.5

1

1.5

2

2.5x 107 Treasure profile for each player

Tre

as

ure

in

ha

nd

($

)

Trading year

player 1player 2

player 3

player 4

player 5

Fig. 45.7 The wealth of all players for all days of the trading year

Its strategy was to buy/sell maximum stocks with respect to the comparison ofthe predicted future-prices’ trends with those of the moving average 10-day-beforeprices. Since the GP prediction models had small prediction errors for the test data,this player did well in the market by relying on the prediction results.

Among players 2, 4 and 5 (referring to Fig. 45.7), we can see that player 5 withthe maximum risk level has made the most wealth (expected returns) and stands inthe second rank (after player 1) in terms of market returns. Player 3’s strategy wasto buy and sell randomly; by referring to Figs. 45.7 and 45.8, one can see that hisexpected returns are similar to those for player 2 (see the Fig. 45.9, 45.10 and 45.11in the Appendix) but, his expected risks values are more than other players.


0 100 200 300 4000

2040

P1

Risk Values (%)

0 100 200 300 4000

2040

P2

0 100 200 300 4000

2040

P3

0 100 200 300 4000

2040

P4

0 100 200 300 4000

2040

Trading Year

P5

Fig. 45.8 The expected risk values for each player over the 1 trading year (P1 to P5 from top tobottom graphs)

45.5 Conclusion

In this paper, precise price predictive models were obtained for four companies’stocks using the GP. This model incorporated the effects of the players’ actions onthe stock prices and other players’ wealth. After the GP model was verified (usingthe test data), it was used for making sell/buy decisions by five agents that tradedin a virtual stock market. The trading period was considered 1 year for our marketsimulation studies. Five different strategies and objective functions were definedfor the market trading agents (with different risk attitudes). The PSO algorithm wasused to obtain the optimal buy/sell actions for each player in order to maximize theirobjective functions (expected returns). The players’ strategies and their expectedrisk-returns were obtained and analyzed for the 1 year trading period. Our marketsimulation studies showed that the player (P1) was the most successful one in ourvirtual stock market.


Appendix

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

-1

0

1

-1

0

1

-1

0

1

-1

0

1

Player 2 Trade weights for each portfolio assets

ERIC

IBM

PHG

Trading year

SNE


0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

Trading year

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5


-1

0

1

ERIC

IBM

PHG

SNE



Trading year


0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

-1

0

1

-0.5

0

0.5

-1

0

1

-1

0

1

ERIC

IBM

PHG

SNE


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