Financial prediction and trading strategies using neurofuzzy approaches

520 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 28, NO. 4, AUGUST 1998

Financial Prediction and TradingStrategies Using Neurofuzzy Approaches

Konstantinos N. Pantazopoulos, Lefteri H. Tsoukalas,Member, IEEE,Nikolaos G. Bourbakis,Fellow, IEEE, Michael J. Brun, and Elias N. Houstis

Abstract—Neurofuzzy approaches for predicting financial timeseries are investigated and shown to perform well in the con-text of various trading strategies involving stocks and options.The horizon of prediction is typically a few days and tradingstrategies are examined using historical data. Two methodologiesare presented wherein neural predictors are used to anticipatethe general behavior of financial indexes (moving up, down, orstaying constant) in the context of stocks and options trading.The methodologies are tested with actual financial data and showconsiderable promise as a decision making and planning tool.

Index Terms—Financial engineering prediction, fuzzy, neuralnetworks, neurofuzzy.

I. INTRODUCTION

T HE prediction of financial market indicators is a topicof considerable practical interest and, if successful, may

involve substantial pecuniary rewards. Neural networks havebeen used for several years in the selection of investmentsbecause of their ability to identify patterns of behavior that arenot readily observable. Much of this work has been proprietaryfor the obvious reason that the users want to take advantage ofthe insight into the market that they gained through the use ofneural network technology. Thus, conventional wisdom has itthat if one succeeds in developing a methodology that correctlypredicts the ups and downs of various market indicators sheor he is unlikely to seek publication of results; conversely, ifsomething is published it must really not “make money.” Wetake both views as indicative of the fact that the ultimate testof value for financial prediction lies with the market. Be thatas it may, financial markets, and economic systems in general,present fascinating opportunities for studying complexity andtesting the limits of methodological advances and computa-tional technologies [2]. Unlike the natural systems of physicsand chemistry, financial systems involve the actual interplay ofdecisions and actions taken by millions of individual investorsand institutions on a global scale.

Although participants in the market are constantly involvedin making decisions based on their predictions or anticipations

Manuscript received June 25, 1996; revised November 8, 1997. This workwas supported in part by INEL University Research Consortium, managed byLockheed Martin Idaho Technologies Co. for the U.S. Department of Energyunder Contract DE-AC07-94ID3223. This work was also supported by theNSF under Grants 9123502-CDA, 92022536-CCR, and 620-92-J-0069; byARPA under Grant DAAH04-94-G-0010; and by the Purdue Special InitiativeFellowship and Purdue Research Foundation.

The authors are with the Department of Computer Science, Purdue Univer-sity, West Lafayette, IN 47907-1398 USA (e-mail: [email protected]).

Publisher Item Identifier S 1083-4419(98)04983-8.

of market phenomena, there is considerable disagreementamongst experts as to what degree, if any, financial time seriesare predictable and how to predict them. Some researchershave identified the presence of chaos in financial indicators,implying that financial systems are characterized by non-repetitive and nonpredictable fluctuations arising through theinterplay of a system’s participating agents and its relation toother systems [16]. Others have argued for certain underlyingregularities and patterns and hence for a more predictablemarket system [12], [15]. The “efficient market hypothesis,”currently the most widely held view of market behavior, statesthat no investment “system” or technical strategy can yieldconsistently average returns exceeding the average returns ofthe market as a whole. Any information that would yield extrareturns spreads rapidly and is “priced out” within seconds ofits introduction.

Even though nearly everybody agrees on the complex andnonlinear nature of economic systems, there is skepticism as towhether new approaches to nonlinear modeling, such as neuralnetworks, can improve economic and financial forecasts. Someresearchers claim that neural networks may not offer any majorimprovement over conventional linear forecasting approaches[5], [11]. One possible explanation may be that, while em-pirical studies in the natural sciences are characterized bylarge data sets, often numbering in the tens of thousands, datasets in economic applications usually consist of less than afew thousand observations. Consequently, computational pro-cedures designed in the former context may not be appropriatein the latter [17]. In addition, there is a great variety ofneural computing paradigms, involving various architectures,learning rates, etc., and hence, precise and informative compar-isons may be difficult to make. Lapedes and Farber, amongstothers, have offered interesting and convincing evidence thatin the context of time series prediction, neural networks canvery accurately model the nonlinearities involved [10]. Inrecent years, an increasing number of research in the emergingand promising field of financial engineering is incorporatingneurofuzzy approaches [18], [20].

Earlier work by the authors has described neurofuzzy tech-niques employed in stock arbitrage and financial time seriesanticipatory characterization [6], [14]. This work is expandedin Sections III and IV and new results are presented involvingneurofuzzy predictions in the context of two related butdifferent trading strategies, the first based on stocks, andthe second based on options. A stock is a certificate ofownership representing one or more shares of a corporation’s

1083–4419/98$10.00 1998 IEEE

PANTAZOPOULOSet al.: FINANCIAL PREDICTION AND TRADING STRATEGIES 521

equity. Usually stocks are listed, i.e., they are traded in thefinancial markets and world’s stock exchanges. Stock tradingis the activity of buying, selling, or sorting (selling withoutowning) the stock of a publicly traded corporation. An optionis a financial instrument, a form of contract between twoparties, which is also traded in the financial markets andorganized exchanges, much like common stocks, commodities,currencies, etc. The basic difference between options andentities such as stocks, is that options are derivative products,i.e., they are based on a primary asset such as a corporation’sstock. Options can also be based on aggregate metrics suchas market indexes. Market indexes are aggregate measures(weighted averages) of stocks. The price of a market indexfluctuates over time, representing the changes in the prices ofthe individual stocks. The price of the various market indexesis continuously calculated and quoted by various services.

In the case of the option trading, we discuss strategiesinvolving options on the S&P 500 index, i.e., options thatare derived from the S&P 500, although our results can beextended for strategies based on other options in a straightfor-ward manner. The option trading strategy is based on neuralpredictors of the volatility of the S&P 500 which predictup,down or same type of movements of the index. Based onthe superior knowledge provided by the volatility predictions,a profitable trading strategy that involves options is madepossible. The model presented here is a fuzzy recurrent neuralnetwork with one hidden layer that analyzes the stream ofdaily prices of the S&P 500. A neurofuzzy methodology isemployed to guide simulated trades in options based on thelevel of the Standard & Poor’s 500 (S&P 500) index. Wefind that if we use a standard option pricing model (we usethe Black & Scholes modelfound in [4]), our trades wouldyield returns more than 150 times greater than a portfolio fullyinvested directly in the S&P 500 index. It would not be in linewith the efficient market hypothesis, however, to expect suchresults from actual investment using this technique to last forlong periods of time if used in a large scale. The promise ofhigh returns would raise the price of the options until normalrates of return were again reached.

In the case of the stock trading strategy, a neural predictoris designed based on the assumption that “the price of astock should generally change proportionately with (or against)certain market indexes, e.g., the Dow Jones Industrial Average(DJIA), or the S&P 500.” The neural network, after analyzingthese market variables, predicts the true price for a particularstock , and decides whether it is underpriced or overpriced.Although the overall market might swing up one day, anddown the next, the neural network model should be right whenthe market stays constant; the up and down swings shouldoffset each other on the other days. Under this assumption,a neurofuzzy model is developed, reflecting the fact thatsome fluctuations in market indexes might be more importantwith respect to the price of stock than others. A neuralnetwork is employed, where each neuron’s running averageerror from its previous predictions can be used to calculate areliability index for the neuron. At the last layer, the networkmakes a fuzzy or disproportionate weighted average of all theneuron’s predictions based on a Gaussian activation function

of the output neuron’s average error. The network uses severalmarket variables and traces them based on a first order gradientof their running averages. The model presented here, is afuzzy neural network with two hidden layers which analyzeconsecutive sets of market variables.

The rest of the paper is organized in four sections. Section IIdiscusses the neurofuzzy approaches pertinent to the timeseries prediction problem and their practical ramifications; fur-thermore, it provides an introduction to the prediction problemas it is encountered in financial engineering applications and abrief survey of some neural network models used in the markettoday. Section III discusses the neurofuzzy methodology forstock trading strategies and Section IV presents the applicationof a neurofuzzy methodology to trading strategies of optionsbased on predictions of volatility. Section V discusses theresults of the neurofuzzy approach taken.

II. NEUROFUZZY APPROACHES ANDFINANCIAL PREDICTION

Neurofuzzy approaches represent an integration of neuralnetworks and fuzzy logic that have capabilities beyond eitherof these technologies individually (see [21]). In applicationsof financial time series prediction, establishing crisp criteriafor decision making is rather difficult to achieve; neurofuzzyapproaches are shown to be capable of providing improveddecision support in situations where crisp estimates are eithermeaningless or unavailable.

Although about 80% of all neural network applicationsutilize networks in which there are no feedbacks from theoutput of one layer to the inputs of the same layer or earlierlayers of neurons, there are situations (e.g., when dynamicbehavior is involved) where it is advantageous to use feedback.When the output of a neuron is fed back into a neuron in anearlier layer, the output of that neuron is a function of both theinputs from the previous layer at timeand its own output thatexisted at an earlier time, i.e., at time where is thetime for one cycle of calculation. Whereas feedforward neuralnetworks appear to have no memory since the output at anyinstant is dependent entirely on the inputs and the weights atthat instant, neural networks that contain such feedback, calledrecurrent neural networks, exhibit characteristics similar toshort-term memory because the output of the network dependson both current and prior inputs.

Neural networks can be used to predict future values ina time series based on current and historical values. Suchpredictions are, in a sense, a form of inferential measurement.Because of their prediction capabilities, there has been an ex-traordinary amount of interest in the use of neural networks topredict stock market behavior [18]. This popularity continuesin spite of the fact that the predictions of neural networkscannot be explained or verified. Perhaps the main reasons forthe continuing popularity in the field are that neural networksdo not require a parametric system model and that they arerelatively insensitive to unusual data patterns.

Typically, the input of a single time series into a neuralnetwork is made as shown in Fig. 1. The fluctuating variableis sampled at an appropriate rate to avoid aliasing, andsequential samples are introduced into the input layer in a


Fig. 1. Recurrent neural network for time-series prediction.

Fig. 2. Different neural networks predict different features of a time series.

manner similar to that used in a transverse filter. At everytime increment, a new sample value is introduced into therightmost input neuron, and a sample value in the leftmostinput neuron is discarded. The main difference compared tothe transverse filter is that the sample preceding those in theinput is introduced into the single output neuron as the desiredoutput. In this way, the network is trained to predict the valueof the time series one time increment ahead, based on thepreviously sampled values. The network can be trained topredict more than one time increment ahead, but the accuracyof the prediction decreases when predictions are further intothe future. Since such systems are often used in real time, orwith data from historic records, the amount of training datais usually very large.

Although it is possible to predict multiple outputs, it is bestto predict only one value because the network minimizes thesquare error with respect to all neurons in the output layer.Minimizing square error with respect to a single output givesa more precise result. If multiple time predictions are needed,individual networks should be used for each prediction in themanner shown in Fig. 2 [21].

Generally, large-scale deterministic components, such astrends and seasonal variations, should be eliminated from

inputs. The reason is that the network will attempt to learnthe trend and use it in the prediction. This may be appropriateif the number of input neurons is sufficient for the input datato span a complete cycle (e.g., an annual cycle). If trends areimportant, they can be removed and then added back in later.This allows the network to concentrate on the important detailsnecessary for an accurate prediction.

The standard method of removing a trend is to use aleast squares fit of the data to a straight line, althoughnonlinear fitting may be appropriate in some cases (e.g., cyclicfluctuations). An alternate method of removing trends andseasonal variations is to pass the data through a high-pass filterwith a low cutoff frequency. There are alternative techniques inwhich a low pass filter is used to leave only the slowly varyingtrend which then are subtracted from the original signal, withthe difference being the value sent to the neural network inputlayer.

One of the interesting variations of the above technique forprediction is to use differences between successive sample val-ues as inputs to the neural network. This effectively eliminatesconstant trends and slowly changing trends by converting themto a constant offset. Even seasonal trends are usually removedin this way. Using differences in predicting is generally usefulin stock price predictions, especially if the difference is scaledrelatively to the total price of the stock, which is effectivelythe percent price change.

The vast majority of financial and economic activity in-volves the analysis and prediction of numerous variables.Variables that are encountered in this context are either leadingor trailing economic and financial indicators of some behavioror pattern, as well as prices and indexes. In the case of leadingvariables, the primary reason for the need of prediction andforecasting is that it can be used to take preemptive actions.More than any other context, the financial markets offer abold example of how a reliable prediction can be capitalizedupon. Consequently, it is natural for financial engineers to seekfor the best possible prediction systems. Finance theoreticians


argue for the long standing “efficient market hypothesis” asa basis of denouncing any technique that attempts to finduseful information for the future behavior of stock prices byusing historical data. However, the assumptions underlyingthis hypothesis are in many cases unrealistic. Studies indicatingthat some form of the hypothesis is indeed observed inthe financial markets, as well as others, which concludeto the contrary, add to the confusion. Irrespective of thetheoretical foundations, technical analysis is used widely asa tool facilitating various financial activities. Among the mostchallenging financial activities that can make direct use ofprediction systems is portfolio management and trading.

Portfolio management systems using neural networks havebeen developed by a number of people and reported to achievegood results. Wilson, for example, reports on a hybrid systembased on several price prediction models including technical,adaptive, and statistical models [22]. The first neural networklayer used by Wilson’s system selects which of these models isworking best for each stock followed. Then the selected modelmakes a recommendation on whether to buy, sell, or hold aparticular stock. The final layer of the system decides whichstocks to buy or sell, depending on the maximum risk that theportfolio is allowed to undertake, as it is measured by thebetacorrelation coefficient of the Capital Asset Pricing Model [3].Although the system’s performance is better than S&P 500, theresults show a great volatility in the value of the portfolio overthe 250 weeks test period, including a dip below the S&P 500about halfway through the tested time frame. Nonetheless, theresults are quite impressive, and the concept of dynamicallymanaging the portfolio’s risk factor very promising.

Lucid Capital Network has reported on some of theirpreliminary tests, on neural network models and applicationsto the futures market [8]. Their study amplifies the importanceand effectiveness of some sound preprocessing techniques; itrecommends normalizing the data so as to provide the networkstrictly with day-to-day changes, rather than the actual pricesof equities. In addition it recommends a “holiday management”processing: filling gaps in the database or the holiday intervalswith the previous day’s data. The concern over holiday gaps isa significant source of noise for most neural network platforms.

Lowe has developed a feedforward analog neural networkfor portfolio management [13]. The marketing analysis isbased on the fundamental assumption that the market isnot totally efficient, i.e., some stock prices are not “true.”Additionally, and to some degree contrarily, this approachincorporates “Sharpe’s theory” [19] as well, whose formulasare based upon the assumption that the market is alwaysin equilibrium. Lowe’s model was tested in both U.S. andEuropean markets. Although the test seems very short (lessthan a year), Lowe’s portfolio has earned an impressive15% return, even during the volatile bear market where hisdata was extracted. Ahmad and Fatmi [1] have designeda “quadric neural network” to predict financial time series.Their neural network is based on the Gabor–Kolmogorovpolynomial model. The fundamental idea is that all statisticaland stochastic functions must be studied within the frameworkof self-defined logical, statistical and physical concepts. Theyprovide formulas that systematically lead to the development

Fig. 3. Distribution of thedMV:

of several functions, linear and nonlinear, which ultimatelypredict the financial time series. The model analyzes all theavailable data derived from a single stream of a stock’s dayto day prices. Much of these derived data is arrived at byvarying the time frame used by the network, and dynamicallyupdating the network’s weights through a feedback functionof the least mean square algorithm. Although this approachof systematic formulas is clearly defined and the network wasshown to converge for a hypothetical series, it appears thatsubstantial testing would be required to demonstrate in practiceKolmogorov’s theory and its fundamental idea of accuratelypredicting a series simply by analyzing its single stream ofhistorical pricing.

III. A N EUROFUZZY METHODOLOGY

FOR STOCK TRADING STRATEGIES

In this section, a feedforward neural network model (FFNM)used for stock trading is presented. Each neuron in the networkuses a number of expressions to calculate a prediction. In ad-dition, the FFNM measures its accumulated error and providesa reliability factor which is used in updating its parameters.

A. Preprocessing of the FFNM Input

The input to the FFNM is a set of financial time seriesdescribing a set of market variablesand a stock The market variables can be either daily pricesor running averages for predefined time frames. The number ofdifferent running averages used is a parameter of the systemdenoted with NumA. The first step is the calculation of therelative change of the input, so that different market variablesand the stock can be compared in terms of relative growth [9].This transformation is described by

where, denotes the market variable numberat timeand the logarithmic relative change of Thefunction is used (any base would work) so that thevalues will be evenly distributed as shown Fig. 3.

A ten-fold increase for example, followedby 1/10-fold increase would average to a

or a one-fold increase (i.e., no change).

B. Calculation of the Market Variable’sIdeal Weight (Sensitivity)

The prediction of the FFNM is based on the assumption thatthe price of stock is related to the various market variables.The neural networks are trained to learn this relationship and


Fig. 4. Sensitivity of stockS with respect to a market variableMVi:

use it in establishing movements of the stock price that areout of line. Each market variable has a certain sensitivitywith respect to stock . This sensitivity, or weight, denotedwith , indicates whether moves with or against andto what degree. Equation (1) describes this relation

(1)

From (1) we see that Thusindicates the sensitivity of to stock as shown in Fig. 4.

C. Learning Rate (BETA)

Before the neuron for a particular market variablecalcu-lates its “prediction,” its weight is revised by taking a weightedaverage of its previous weight and the ideal weight (i.e., theteacher variable) which should have been used in the previousstep. Equation (2) indicates the dependence of the weightedaverage on the learning rateBETA, i.e.,

(2)

where is the previous (old) weight and is the idealweight for that step. Since different learning rates vary withthe magnitude of a market variable’s previous performance,severalBETA values are used independently. Thus, a singlemarket variable can provide short term predictions as well aslonger term ones. Each weight corresponds to a particularand with a particular learning rate Equation (3) shows theway BETA is calculated where and NumLdenotes the total number of different learning rates used

(3)

From (3) we see that the smaller the learning rate, thelesser the importance given to historical performance.

D. The Activation Function

At each iteration (each day in this model), the FFNMcalculates components from the overall network prediction,where is the number of market variables used. The outputlayer consists of output neurons. In order to obtain asingle prediction, all these components are averaged based oneach neuron’s reliability index. The reliability index, , isa Gaussian activation function of the average absolute error

, as shown in Fig. 5, and it is described by (4)

(4)

Since is effectively normalized by the weight average,any arbitrary value, e.g., 10, will work. , however, defines

Fig. 5. Gaussian activation functions.

TABLE IMARKET INDEXES USED BY THE FFNM

DJIA Dow Jones Industrial Average (30 companies)DJTA Dow Jones Transportation Average (20 companies)DJUA Dow Jones Utility Average (15 companies)DJ65 Dow Jones 65 Stock CompositeDJFI Dow Jones Futures Index (Commodities)S&P 100 Standard & Poor’s 100 IndexS&P EI Standard & Poor’s Electronic InstrumentsS&P SC Standard & Poor’s SemiconductorsS&P BE Standard & Poor’s Computer SystemsS&P500 Standard & Poor’s 500 Index

the steepness of the curve, and thus defines how much moreimportant a component, e.g., with 1% average error, is thananother component with a 10% average error. If a 20-fold willbe chosen more important (for this example) a isused.

Market variables (the ones used here are market indexes) arethe primary input data that the FFNM analyzes to gauge the“true” value of stock . Thirteen market variables are selectedas network input. These variables are among the most commonmarket indexes which are affected and affect the price of thestock which we use in the evaluation of the performance ofthe FFNM, the common stock of IBM. Some of the indexesused are shown in Table I.

E. FFNM Simulator and Parameter Settings

The FFNM simulator is built using recurrent neural net-works with two hidden layers. The parametersand NumT(size of historical sample) are set accordingly to the numberof market variables and number of daily prices available. Thevalue of NumA can be adjusted as necessary. For example,in order to use only daily prices of market variablesNumAis set equal to one. If then a second dimensionis introduced, which reflects the number of different runningaverages that the system is tracking. When each individualneuron updates its parameters (“learns”) after each pass, ittakes a weighted average of the old parameters and the newones. TheBETAvalue defines how to proportionately averagethe two values. While it is advantageous for some neurons tolearn quickly, it is more advantageous for other nodes to learn


Fig. 6. Using the FFNM with IBM stock, reinvesting dividends, generatedan annualized return over the test period of 20.9%. The parameters settingswereNumA = 1; NumL = 3; NumV = 13; andNumT equal to thetrading days for the given period.

in a slower pace. The number of different learning rates tobe used is defined byNumL. The approach implemented hereis to take a combination of these neurons (introducing a thirddimension to the neural network) by allowing each neuron tohave its own reliability index. The parameterNumL defineshow many of these combinations to follow for each particularneuron. The overall network has neuronswith NumTpasses through them. These neurons all feed intothe last layer, a single output neuron, that proportionatelyaverages (using the reliability index) all of the individualpredictions to come up with a single prediction.

In evaluating the FFNM neural network simulator the timeseries describing the market variables and the stock in the sixyear period (1988–1994) are used. To evaluate the predictivecapabilities of the FFNM model, the investment strategystarts with $1, and every day invests this amount plus allthe available cash balance of the portfolio. This strategy isa “dividend and capital gains reinvestment strategy” whichaccurately reflects a realistic trading practice. The overallperformance of the network is evaluated based on two ob-jectives: the rate of return, and the volatility of its day today performance. To evaluate both, a simple and objectiveapproach is to consider the daily cash balance of a portfoliothat starts with an initial balance of $1 and subsequently tradesthe IBM stock using the FFNM. Fig. 6 illustrates the balanceof the test portfolio for IBM with the FFNM parameters set to

and equal tothe trading days for the six-year period.

IV. A N EUROFUZZY METHODOLOGY FOR

PREDICTION-BASED OPTION TRADING STRATEGIES

In this section a neurofuzzy volatility prediction system thatanticipates the changes in the volatility of the returns of theS&P 500 financial time series is presented. The prediction ofthe neurofuzzy system provides trading recommendations thatare used to implement short-term option trading strategies. Theinput to the prediction system is the daily close value of theS&P 500 index. The time series used is shown in Fig. 7.

A. Preprocessing of the Time Series Input

The S&P 500 time series is processed before used by thevolatility prediction system, as follows. Letbe the S&P 500 daily closing values. The logarithm of thesedata points is Hence, the difference of successivedata points is

where it is assumed that The successive difference ofthe data points approximates the relative returns of the S&P500. Denoting by the time frame, the time series isfurther transformed as

wherestd is the standard deviation, andThe result approximates the historic volatility of the returnsfor a period of NumA days. The time series is scaled by

in order to facilitate neural network processingand training. The result of the preprocessing is a modifiedtime series, shown in Fig. 8.

The modified time series is partitioned into three sets: thetraining set, the test set, and the “out-of-sample” set which isleft completely untouched. The training set is drawn from thelast 6000 elements of the time series with a sampling rate oftwo. The test set is formed from the remaining points. The“out-of-sample” set is formed from the first 10 000 elementsof the time series. The available time series data span a periodfrom 1928 to 1993. The characteristics of the volatility ofthe market have substantially changed in the last 20 yearsof this period primarily due to the introduction of optionproducts. Thus, the characteristics of importance to be learnedand utilized by the proposed neurofuzzy system manifestthemselves in a much bolder manner in the later period.Consequently, training using data from this period was deemedmore appropriate. The fact the simulation results are obtainedusing a portion of the time series that in the actual time lineoccurs in the past should not pose a problem since that periodis as random as any other period in the market.

The training set is further partitioned into three subsets:the up, the same, and thedown, of NumA-sized vectors. Theup set contains the input vectors that precede an increaseof the volatility of the index returns in excess of Thesameset contains input vectors that precede absolute changeless than, or equal, to and thedown set contains inputvectors that precede a decrease in volatility of more than

The three different subsets of the training set are usedto train three corresponding neural networks, with one hiddenrecurrent sigmoidal layer and one linear output layer. The threeindividual neural networks,fNNup, fNNdown, and fNNsame,are trained to recognize patterns that precede the followingmovements in volatility changes: a) thefNNup predicts ahigher than increase, b) thefNNsamepredicts changeof less than in either direction, and c) thefNNdownpredicts a reduction of or more. The training of the threenetworks was done for 15 000 epochs on the training dataand an error threshold (sum square error) of 10. The three


Fig. 7. The Standards and Poor’s S&P 500 time series from 1928 to 1993.

Fig. 8. Modified S&P 500, time series from 1928 to 1993 (after preprocessing).

neural networks,fNNup, fNNdown, and fNNsame, providefuzzy predictions for the change in volatilityfup, fdown, andfsame, respectively. The membership functions of the threefuzzy outputs are triangular.

B. Fuzzy Post-Processing of the Neural Network’s Output

Each network gives three outputs, the left centerand right which represent the membership function ofthe fuzzy prediction. In the training phase the true value isfuzzified as a triangular fuzzy number centered in the truevalue with a support of For the output of a particularnetwork to be meaningful the condition (constraint of shape)

must be satisfied. If it is not, the output is discarded.Recall that each network is trained to recognize a particular

class of input patterns. The following post-processing rules areused to evaluate the performance of the fuzzy neural networkover the test and “out-of-sample” sets.

• Rule 1: If all three outputs do not satisfy the constraintof shape predict nothing.Else

• Rule 2: If all three satisfy the constraintthendiscard theone with the most distorted membership function and takethe fuzzy mean of the other two.Else

• Rule 3: If less than three satisfy the constraintthen takethe fuzzy mean of those two that satisfy the constraint.

In all cases that there is prediction, the prediction is thecentroid of the resulting fuzzy number. In choosing the mostdistorted membership function to discard, in Rule 2, thefollowing criterion is used:


(a)

(b)

(c)

Fig. 9. Comparing the modified S&P 500 time series with the neural network predictions (dashed lines denote predictions, solid lines denote actual timeseries data): (a) The network trained with patterns of upward movement of the index generally overpredicts the time series. (b) The network trained withpatterns of “about the same” movements of the index generally stays close to the index. (c) The network trained with patterns of downwards movementgenerally underpredicts the actual S&P 500 time series.

TABLE IIRELATIVE RMS ERROR OF THENEUROFUZZY PREDICTION

SYSTEM FOR THE TWO DISTINCT SUBSETS OF THETIME SERIES

Test Set “Out-of-Sample” Set0.2864 0.2781

• Compare the total support of the fuzzy number. Removethe one with biggest support. The support is computedas

The relative rms error

where, is the number of samples, the true output, andthe predicted value, is used to study the performance of

the system on the test and “out-of-sample” sets.Table II summarizes the relative rms on the two distinct

subsets of the input time series.The smaller rms error in the “out of sample” set is explained

by the fact that the variability of the time series is much smallerfor this set than that for the test set. Fig. 9 shows the predictionof each of the three neural networks to the financial time seriesfor a snapshot of the “out-of-sample” for and

The values plotted in Fig. 9 are the centroids of theindividual predictions.

C. Trading Strategies Using Options Basedon Volatility Prediction

A financial instrument whose value is an increasing functionof the volatility of returns of a particular asset is an option onthat asset. Thus, option trading offers a convenient and realistic

framework for testing the neurofuzzy volatility predictionsystem. Typical options are eithercalls or puts1. A straddleis an option formed from onecall and oneput option, bothon the same underlying asset, with the same expiration date,and the same strike price, packaged together. The interestedreader can find an in-depth discussion of options and financialderivatives in [7].

Prior to expiration, the value of an option can becomputed as a function of a number of parameters. Accordingto the widely used Black & Scholes option pricing model [4],the value is given as

where

price of the underlying asset, e.g., the S&P 500;current time;dividend yield;volatility of the underlying asset’s returns;strike price;risk-free rate of return;expiration of the option measured in years.

and are constants known in advance,is taken to bethe T-Bill2 rate with matching duration as the duration of theoption, and is approximated by a constant dividend yield.

1A call is a financial contract giving the right, but not the obligation, to thecontract holder tobuy a specified asset (theunderlyingasset) for a specifiedprice (thestrike price) on, or before, a specified date (theexpirationdate). Aput is similar, only it gives the right tosell the underlying asset for the strikeprice on, or before, the expiration date.

2T-Bills are treasury bills, short term U.S. government debt that is consid-ered to be free of risk. An alternative would be to use money market accountyields or yields on certificates of deposit (CD’s). However, it is commonpractice to approximate the short term risk-free rate of return by the rateoffered from T-Bills.


Fig. 10. Payoff of astraddle.

is described as a stochastic process whose value is determinedby the laws of supply and demand. The time to expirationis calculated as years.3 The payoff of an optionis the value of the option at the expiration date. Prior to theexpiration date the option is traded in the financial markets asany other asset (e.g., stock) and its price fluctuates based ona number of economic and other parameters. For example, anS&P 500 Dec’99 650 Calloption or anS&P 500 Dec’99 650Put could be bought today; their value at the expiration willdepend on the actual price of the underlying asset, which inthis case, is the S&P 500 index. Notice that the value of theS&P 500 is not known with certainty, since, it changes as theprices of the stocks contributing to the index (500 stocks ofU.S. corporations from various industrial sectors) change.

The payoff value for acall with strike price is given by

where SP500stands for the S&P 500 price and is theoption value at the expiration date The payoff value for aput is given by

where is the option value at the expiration date. Astraddle, which is onecall and oneput packaged togetheras one option, has a payoff shown in Fig. 10 and given by

The volatility a measure of the variability of the timeseries, is taken to be constant in most practical cases. It isfound either as the volatility of the more recent past of the timeseries (historic volatility), or, as the volatility implied fromother option prices quoted in the financial markets (impliedvolatility). However, the volatility parameter of the optionprice function refers to the future volatility of the underlyingasset time series. A successful trading strategy, which usesa neurofuzzy volatility predictor, can be built based on thisdeficiency, i.e., of using historic or implied volatility to priceoptions. If options are priced today using either of the twoestimates mentioned above (historic or implied volatility),and their price fluctuates according to the unknown futurevolatility, a reliable prediction of future volatility can beof tremendous value. It can be used to identify overpricedor underpriced options and execute appropriate trades. Theoption value is an increasing function of volatility; the higher

3In order to convert in years, we use the number of trading days in a yearand not the actual days. A widely used approximation for the trading days in252 per year [7].

TABLE IIIASSUMPTIONS FOR THEPARAMETERS USED TO PRICE OPTIONS

S The price of the index is taken to be the closing price for theday; i.e., we execute all trades simultaneously at the end of theday.

K The strike price is taken to be the at-the-money value, i.e. sameas the price of the underlying S&P 500 index.

� The volatility that the market uses to price the options is thehistoric volatility for the last 30 days.

r The risk-free interest rate is taken to be the T-Bill rate,approximately 5%.

q The constant dividend yield of the index is taken to beapproximately 2.5%.

(T � t) The duration of the options used is 1 month. This in practicetranslates into using the option with the closest expiration date.

the volatility, the higher the option value and vice-versa.Consequently, a system that would provide an estimate for thefuture volatility trend could be used to build a trading strategy,based on options, that would provide above normal returns. Ifwe know that future volatility is going to be increasing (up),with a reliability of some degree, then we could buy options.Since the options are selling at current volatility estimates,their price will increase according to the predicted upwardmovement of volatility. If the prediction is realized, we standto gain, ceteris paribus, the difference wheredenotes the predicted volatility and the volatility currentlyused by the market. This will be realized if nothing elsechanges in this period.

Intuitively, a prediction of higher volatility implies that theasset price is going to move; it does not however specifyin which direction. Consequently, if, as a reaction to theprediction for higher volatility we buy call options and theasset price moves down, we might lose. On the other handif we buy put options and the asset price goes up we mightalso lose. A “safer” way to react on a prediction for highervolatility would be to buy straddles, since, whichever way theprice of the underlying is going to move we are going to profitassuming that the prediction of higher volatility is realized.

D. Evaluating the Prediction CapabilitiesUsing Option Trading Strategies

In order to evaluate the predictive capabilities of the neu-rofuzzy volatility prediction system, its recommendation isemployed in the context of an option trading strategy thatinvolvesstraddle“at-the-money” options on the S&P 500. Forpricing the options the Black & Scholes pricing model [4] isused.

The current market price of the options, i.e., the priceat which they can be purchased, is computed using theassumptions listed in Table III.

The trading strategy simulation is summarized as follows.

1) Start with an initial amount (cash balance) of $1000.2) At the beginning of each day, mark-to-market (i.e.,

compute the market value) of the portfolio of options.3) Liquidate the portfolio, if it contains holdings other than

cash.4) If the neurofuzzy volatility prediction system recom-

mends “hold” do nothing.


Fig. 11. The returns from investing into the S&P 500 between 1928 and 1968 (the vertical axis indicates the portfolio balance in dollars).

Fig. 12. Returns from investing according to neurofuzzy predictions of the market for the period between 1928–1968 (vertical axis indicates portfoliobalance in millions of dollars).

5) If the volatility prediction system recommends “buy,”purchase as many “at-the-money”straddlesas the cashbalance permits.

6) Go to step 2.

The above trading strategy is simulated on the “out-of-sample” set for approximately 10 000 days. The comparisonis done between a portfolio fully invested in the S&P 500index and the option portfolio managed using the neurofuzzyvolatility prediction system in the same period. The portfoliosand their market value are computed at the beginning of eachtrading day. Returns from investing into the S&P 500 areshown in Fig. 11. Returns from the option trading strategyusing the volatility prediction system for the same period andwith the same initial investment are shown in Fig. 12.

The portfolio based on the S&P 500 had a balance of $3862at the end of the period. The option portfolio, which wasmanaged using the volatility prediction system, had a balanceof $625 120 at the end of the same period. The resultingperformance is substantially better for the option tradingstrategy. The higher volatility exhibited by the daily balancesof the option portfolio is in part expected and explained by the

leverage of options compared to stocks. Each option gives theright to buy or sell, depending on the type of option, 100 sharesof a certain asset. Also, the choice to fully invest the availablecash balance at all times contributes to the balance volatility.The approximately 73% reliability of the neurofuzzy predictorleaves a 27% margin for false predictions. Note that correctprediction of direction for volatility movement says nothingabout the magnitude of the movement. That means the gainsfrom correct predictions and losses from incorrect predictionsvary. A trade, which is made based on a faulty prediction,results in a loss, which in order to be recovered, it requirestwice the increase. For example, a 50% loss must be followedby a 100% profit in order to reach the same balance as theone before the faulty prediction. In general, the above tradingstrategy demonstrates the potential of the methodology usedto predict the volatility movements. In a realistic frameworkit would be combined with more sophisticated managementand hedging activities in order to reduce the variability of thereturns. The success of the option trading strategy is primarilyattributed to the prediction of the future volatility. In all casesthe value of the option portfolio is more than that of the S&P500 portfolio.


V. CONCLUSIONS

Neural networks, when properly configured, offer superbpattern matching capabilities that can be used for predictingthe ups and downs of financial indicators such as the S&P 500index. The fuzzification of such neural predictions may leadto robust and overall successful trading strategies.

We have presented two cases in which neurofuzzy predictionapproaches are used successfully to predict financial variablesin a way that is meaningful from an investment point ofview, i.e., resulting in profitable trading of stocks and options.However, another interpretation of our results is possible.They could be a challenge to standard pricing models. Thehigh returns our simulations yielded would surely have led inreality to higher asset prices—higher than the standard modelspredict. The relaxation of some market conditions such as the“commissions” paid for transactions, the “bid-ask” spreadsof quoted prices and the tax liabilities (capital gains andincome from investment), if removed, will result in smallerreturns that those reported here. Nevertheless, the models thatwere presented and the realistic strategies in the context ofwhich they were employed, clearly demonstrate the potentialof neurofuzzy modeling in the arena of financial predictionand management. Subsequent studies should consider a stan-dardization of benchmarking for such systems, other than puremarket performance, which will enable cross comparisons ina manner that overcomes the privacy usually accompanyingfinancial systems’ results.

ACKNOWLEDGMENT

The authors are grateful to A. Hobbs for substantial con-tributions to the stock pricing model and to the refereeswhose comments and suggestions materially contributed tothe revision of the paper.

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Konstantinos N. Pantazopoulosreceived the B.E.degree in computer engineering from the Universityof Patras, Patras, Greece, in 1991 and the M.S.degree from Purdue University, West Lafayette, IN,in 1995. He is currently pursuing the Ph.D. degreein computational finance at Purdue University.

From 1991 to 1993 he was a Computer Scientistwith First Informatics S.A., Athens, Greece. Hehas been a part-time Research Assistant at thePurdue University Softlab group since 1993. Hisresearch interests include numerical algorithms and

applications in option pricing systems, computational intelligence in financialapplications, and Internet security issues.

Mr. Pantazopoulos received the Purdue Special Initiative Fellowship in1996 and the Purdue Research Foundation Fellowship in 1997. He is a memberof the ACM and the UPE society.

Lefteri H. Tsoukalas (S’88–M’89) received theB.S. degree in electrical and computer engineeringin 1981, the M.S. degree in nuclear engineeringand computer science in 1983, and the Ph.D. de-gree in 1989, all from the University of Illinois,Champaign-Urbana.

He joined the faculty of Purdue University, WestLafayette, IN, in August 1994. He is now an As-sociate Professor there. He has several years ofexperience in fuzzy and neural computing tech-nologies with over 80 research publications in the

field, including the book, (co-authored with R. E. Uhrig),Fuzzy and NeuralApproaches in Engineering(New York: Wiley, 1997). He has considerableindustrial experience in developing high-performance, massively parallelsystems for on-line monitoring, diagnostics and control applications.

Nikolaos G. Bourbakis (A’80–SM’88–F’96) re-ceived the Ph.D. degree from the University ofPatras, Patras, Greece, in 1983.

In 1991, he joined Binghamton University, Bing-hamton, NY, where he is a Professor with the Elec-trical Engineering Department. His major researchinterests include artificial intelligence, machine vi-sion, robotics, and knowledge-based VLSI design.He is the editor-in-chief of theInternational Journalof Artificial Intelligence Tools.


Michael J. Brun received the Ph.D. degree ineconomics from the University of Illinois, Urbana-Champaign, in 1992.

He taught at Illinois State University, Normal,from 1989 to 1994, and currently teaches economicsat the Slovak Agricultural University, Nitra, Slo-vakia. His current interests are in the dynamics ofeconomic transition and the problems of applyingmacroeconomic theory to the global economy.

Dr. Brun is a member of the American Societyof Cybernetics and the American Economic Asso-ciation.

Elias N. Houstis received the Ph.D. degree inapplied mathematics from Purdue University, WestLafayette, IN, in 1974.

He is a Professor of Computer Science and Direc-tor of the Computational Science and EngineeringProgram at Purdue University. His research inter-ests include parallel computing, neural computing,and computational intelligence for scientific appli-cations. He is currently working on the design of aproblem solving environment called PDELab for ap-plications modeled by partial differential equations

and implemented on a parallel virtual machine environment.Dr. Houstis is a member of the ACM and the International Federation for

Information Processing (IFIP) Working Group 2.5 (Numerical Software).

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Financial prediction and trading strategies using neurofuzzy approaches