Volatility Forecasts of the S&P100 by Evolutionary Programming in a Modified Time Series Data Mining Framework

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1 Volatility Forecasts of the S&P100 by Evolutionary Programming in a Modified Time Series Data Mining Framework Irwin Ma, Université du Québec, Montréal (Québec), acron@aei.ca Tony Wong, Université du Québec, Montréal (Québec), tony.wong@etsmtl.ca Thiagas Sankar, Université du Québec, Montréal (Québec), thiagas.sankar@etsmtl.ca Raymond Siu, Motorola Canada (Québec), raymond.siu@motorola.com Abstract Traditional parametric methods have limited success in estimating and forecasting the volatility of financial securities. Recent advance in evolutionary computation has provided additional tools to conduct data mining effectively. The current work applies the genetic programming in a Time Series Data Mining framework to characterize the S&P100 high frequency data in order to forecast the one step ahead integrated volatility. Results of the experiment have shown to be superior to those derived by the traditional methods. Keywords: Soft Computing, data mining and management, financial volatility forecasting. 1. Introduction The daily volatility is a key variable in the evaluation of financial risk and options. Integrated Volatility (IV) is calculated from the cumulative squared intraday returns of the underlying securities at high frequencies as defined by Anderson et al. [1]. In deriving IV, the daily volatility is converted from a latent variable into an observable one. Traditional financial engineering methods based on parametric models such as the GARCH model family, have limited success in volatility forecasting due to their rigid as well as linear structure [2]. The requirement of distribution assumption further hinders the accuracy of forecasting [3]. Except those based on proprietary methodology, there is still a lack of publicly available and effective method to deal with the non-linearity inherent in the volatility series of financial indices [4, 5]. In this regard, recent development in financial time series analysis could be beneficial to the forecasting problem. Work conducted in [6] has established that by using a genetic algorithm method, the one day ahead moving direction of the S&P100 volatility could be forecasted at an average accuracy of up to 75%. This paper attempts to forecast the numerical values of the volatility by formulating a nonlinear and non parametric approach based on GP in the Time Series Data Mining (TSDM) framework. 2. Literature Review As indicated by Radzikowski [7], modern parametric option-pricing models, where volatility is often the only stochastic variable, were expected by many to: Be well-specified, Consistently outperform other models, Be statistically consistent with underlying asset return dynamics, Provide a statistical theory of option pricing error, and Be elegant and not difficult to estimate But they failed to deliver, as they either are too complex, have poor out-of-sample performance, make unrealistic distribution assumptions, and/or use implausible and/or inconsistent implied parameters. While parametric models provide internal consistency, they do not out-perform simplistic approaches out-of-sample. Even the most complex modern parametric models are imperfect and are outperformed by simple, less general models. Jackwerth and Rubinstein [8] applied series of tests to variety of models and concluded that naïve approaches are consistently

2 the best, stochastic deviation models are next best, then there are deterministic volatility models and finally the traditional parametric models. Prompted by shortcomings of parametric models, new class of methods was created that do not rely on pre-assumed models but instead try to uncover or induce the model/process of computing prices from vast quantities of historic data. Many of them utilize learning methods of Artificial Intelligence. Non-parametric approaches are particularly useful when parametric solution either lead to bias, or are too complex to use, or do not exist at all. Many recent publications attempted to use genetic algorithms/ programming to forecast financial time series such as stocks, indices and options: [9 13]. In search of a relatively more general approach in data mining of financial time series, G. Szpiro [14] came up with a method that permits the discovery of equations of the data-generating process in symbolic form. The genetic algorithm that is described there uses parts of equations as building blocks to breed ever better formulas. Apart from furnishing a deeper understanding of the dynamics of a process, the method also permits global predictions and forecasts. Povinelli [15] formulated the Time Series Data Mining (TSDM) framework that reveals hidden temporal patterns that are characteristic and predictive of time series events. This contrasts with other time series analysis techniques, which characterize and predict all observations. Upon reviewing the above literature, a better idea would be that instead of using one single formula to explain the entire time series, one could use multiple genetic programs to explore sequentially in the search space to obtain an overall estimation represented by a set of formula rules. The genetic programs try to find the best fitting rules based on the input time series. The best formula rules are then combined as proposed by Chen and Stolfo [16] and used to forecast the future IV values. 3. Problem Statement Different patterns, linear or non-linear including the stylized clustering effect of volatility may repeat in different time intervals. This is true when dealing with different types of financial securities or dealing with different historical periods for the same underlying security. By making use of the stylized characteristics of financial volatility, extend the TSDM method with GP to forecast as many events/non-events as practically feasible in the IV time series in order to guide option trading. 4. The Rule-based GP Forecasting Method Data mining is the analysis of data with the goal of uncovering hidden patterns especially those complex relationships in large data sets [15]. In this section, background information regarding Povinelli s TSDM method is introduced. The TSDM methods create a new structure for analyzing time series by adapting concepts from data mining, time series analysis, genetic algorithms, and nonlinear dynamics system [15]. They are designed to predict nonstationary, nonperiodic and irregular time series, and not restricted by the use of predefined templates. More specifically, they help discover hidden temporal structures predictive of sharp movements in time series, using a time-delay embedding process that reconstructs the time series into a phase space that is topologically equivalent to the original system under certain assumptions [15]. The TSDM methods are developed and applicable to make one-step predictions for time series data sets[17]. In order to extract nonstationary temporal patterns, a specific TSDM method could be used to address quasi-stationary temporal patterns, i.e., temporal patterns that are characteristic and predictive of events for a limited time window Q. It is called the Time Series Data Mining evolving temporal pattern (TSDMe2) method, which uses a fixed training window and a single period prediction window. The TSDMe2 method differs from the other TSDM methods in how the observed and testing time series are formed. The TSDMe2 method creates the overlapping observed time series:

3 X j = {x j, t=j,, j+ N }. (1) The testing time series is formed from a single observation: Y j = {x j, t=j,, j+ N }. (2) Where x j is the time series value at time t=j, while N is the size of the window. In characterizing different patterns hidden in the time series, there are two key factors to consider, number of pattern types and size of the patterns (or windows Q). By parsimony, the simplest characterization of events possible is desired, i.e., as small a dimensional phase space Q as possible and as few characterization patterns as necessary. However, the following modifications have been made to the typical TSDM in order to implement the proposed data mining procedure, i) to increase the pattern characterizations by involving as many as 100 different arithmetic expressions to describe a windowed time series; ii) to use a 4-lag recursive memory as the size of the patterns Q. For definitions of some related concepts, refer to [15]. There will be many different patterns in financial time series, linear and nonlinear. A financial series is a dynamic entity which is affected by many variables, economic, financial, politics, psychological, legal, etc. It is philosophically unwise to use one fix model, linear or nonlinear to estimate such a process, let alone forecasting. In using statistical models to estimate time series, wherever abrupt structural changes the model will need to adjust and change its parameters. On the other hand, the application of volatility estimation in option trading deem necessary to extract also the non-event, so that one could capitalize on the time value of the option. In our case, we used 100 different formula rules to match the frequently appearing events and to extract different patterns buried in noise. Therefore, there is a high probability to extract the patterns and further to forecast the one step-ahead activity. Note that in each of the 100 rules used to characterize different patterns, the value of δ could be considered as the margin of accuracy the rules match the points in the window. The patterns are determined by the four previous points to make a prediction for the value at the next time interval. The 4-lag recursive system is used due to the potential weekly seansonality of the IV time series as well as the convenience of weekly option trading. Using 4- lag approach could save time and memory and is particularly useful in dealing with volatility forecasting because of past research showing that most of the information is contained in the most recent lags, resulting the popularity of GARCH(1,1) or other short memory models. The current method involves matching clustering patterns, which include sharp fluctuations in the IV series. To find these temporal patterns the time series is embedded into a reconstructed phase space with a time delay of one and a dimension of four [15]. Once the data is embedded, temporal structures are located using a genetic programming search. Clusters are made of points within a fixed distance of the temporal structures δ. The event characterization function g(t)= x t+1, determines the value given to the prediction made from the clustering using the temporal structures. This value is the IV value for the next time interval. The temporal structures are next ordered by how well each predicts the IV movements. A ranking function is defined as the average value within a temporal structure, and it is used to order the structures for optimization. The optimization is a search to find the best set of temporal structures and is done with GP that finds fitness value parameters that maximize the ranking function f (P) the frequency of the correct guessed patterns. The GP uses a combination of Monte Carlo search for population initialization with a fixed percentage selection, crossover and mutation to find the optimal P*, and a limited number of generations to halt the genetic programming [15, 17] In general, combination can potentially eliminate the predictions which might be generated due to the noise in the data. In the case of the rule learning process, independent trials of the GP can be considered to explore different parts of the search space, thereby learning different types of patterns for prediction. As a result, at a given time some rules generate better predictors than others, thus making them ideal candidates as base predictors for integration to achieve increased predictive accuracy.

4 5. Introduction of GP GP was first developed as an extension to Genetic Algorithm (GA). The most important feature of GP is their representation of individual solution structures. Unlike traditional GA s, which usually represents individuals as vectors of fixed length, GP individual is represented as hierarchical composition of tree-like structure with variable length from basic building blocks called functions and terminals. The function set is composed of the statements, operators, and functions available to the GP system. The terminal set is comprised of the inputs and constants to the GP program. Genetic programming possesses no inherent limitations on the types of functions, as long as the closure property is satisfied, that is each function should be able to handle gracefully all values it might receive as inputs. Traditionally, genetic programming uses a generational evolutionary algorithm. In this approach, there exist well-defined and distinct generations. Each generation holds a complete population of individuals. The newer population is created from the genetic operation on the older population and then replaces it either partially or completely [11]. 6. Data Analysis & Results The intraday data of S&P100 index (OEX) between 1987 and Aug is acquired from TickData Inc. of the U. S. Part of the data set X t, the 15-minute high-low prices between Dec. 3, 2001 and Dec. 31, 2002 are taken for training purpose. The second part, e.g. between Jan. 2 and Aug. 29, 2003 will be used to test the validity of the rules. The first 21 days of both sets of data are used to prepare for the 21-day moving average, in order to take the monthly effect into consideration, to de-trend and to improve the forecasting accuracy. The corresponding normalized IV s were then calculated and fed into the GA s to forecast the moving directions and to find the best 100 rules by maximizing the value f [6]. The GP programs are then applied to forecast the IV values at the selected time ranges ahead, e.g., one day ahead. The execution cycle of the generational GP algorithm includes the following steps: 1. Initialize the population. An initial population of 100 is created randomly from the basic building blocks. 2. Evaluate the individual programs in the existing population. A value for fitness, e.g. the absolute difference between the individual and the desired one is assigned to each solution depending on how close it actually is to solving the problem (thus arriving to the answer of the desired problem). 3. Until the new population is fully populated, repeat the following steps: a. Select an individual or individuals in the population using the selection algorithm b. Perform genetic operations (crossover & mutation) on the selected individuals c. Insert the result of the genetic operations into the new population. 4. If the termination criterion is fulfilled, then continue. Otherwise, replace the existing population with the new population and repeat steps Present the best individuals in the population as the output from the algorithm. Parameters used in the current study are listed in the following table: Table 1: GP Configuration Generations 25/50/100 Populations 100 Function set +, -, %, *, sin, exp, sqrt, ln Terminal set {x(t-1), x(t-4)} Fitness diff. btw. calc. and desired Max depth of new individual 6 Max depth of new subtrees for mutation 6

5 Max depth of individuals after crossover 17 Mutation rate 0.05 Generation method (selection) 50 % Note that based on the findings in Neely and Weller s research [18], the fitness of the GP operation in the current investigation is derived from the Mean Absolute Error between the generated individual and the actual IV value. Two separate tests have been conducted on the 2002 training data set by pre-processing them with the GA s [6], one for 500 generations and the other The intermediate results are then passed through GP programs and the final results of percentage accuracy are shown in Table 2. For example, an initial population of 100 rules is generated and 25 generations of GP are performed with a maximum depth of six of new individuals. GP Parameters 2002 data (500 generations GA) 2002 data (1000 generations GA) [25, 100, 6] 72.65, 73.21, , 66.00, [50, 100, 6] 71.46, 75.36, , 69.13, [100, 100, 6] 71.44, 69.60, , 67.49, Table 2. The forecasting accuracy for 2003 IV data based on both 2002 training data sets The training data set was then pre-processed using GA s [6] and 1000 generation GP was implemented to obtain the results as shown in Table 3. GP Parameters 2001/2002 data (based on 1000 generations GA) [25, 100, 6] 78.25, 77.66, [50, 100, 6] 80.20, 79.51, [100, 100, 6] 79.10, 78.86, Table 3. The forecasting accuracy for 2003 IV data based on 2001/2002 training data sets An interesting phenomenon could be observed that the forecasting accuracy in the current tests is not positively related to the number of generations used in either GA or the subsequent GP operations. This may be caused by the early convergence to the local minima in the search process. Further investigation and appropriate search strategy may be necessary to resolve the issue. 7. Conclusions Based on the novel Time Series Data Mining (TSDM) framework and its associated methods, this paper has made use of GP to find optimal temporal pattern clusters that both characterize and predict time series events. The TSDMe2 method was created for discovering multiple temporal pattern clusters in a time series. Additionally, a time series windowing techniques was adapted to allow prediction of non-stationary events. This paper has demonstrated that the modified TSDM framework successfully characterize and predict complex, non-periodic and irregular IV time series. This was done through testing the S&P100 index of different years. The one step ahead forecasting accuracy reaches an average of 74% with standard deviation of 4.6%. Future work includes testing other financial indices over a wider time span, combining the modified TSDM e2 method with other data processing techniques such as wavelet transform in order to optimize the forecast time horizon, using parallel processing techniques to accelerate GP process, etc.

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