Plane Thermoelastic Waves in Infinite Half-Space Caused Operational Research in Engineering Sciences: Theory and Applications Vol. 5, Issue 1, 2022, pp. 185-204 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta040422196m * Corresponding author. saimamustafa28@gmail.com (S. Mustafa), arfajutt60@gmail.com (A. A. Bajwa), shafqat905@e.gzhu.edu.cn (S. Iqbal), A NEW FUZZY GARCH MODEL TO FORECAST STOCK MARKET TECHNICAL ANALYSIS Saima Mustafa 1, Arfa Amjad Bajwa 1, Shafqat Iqbal 2* 1 Department of Mathematics and Statistics, PMAS Arid Agriculture University, Rawalpindi, Pakistan 2 School of Economics and Statistics, Guangzhou University, Guangzhou, China Received: 15 January 2022 Accepted: 14 March 2022 First online: 04 April 2022 Original scientific paper Abstract: Decision making process in stock trading is a complex one. Stock market is a key factor of monetary markets and signs of economic growth. In some circumstances, traditional forecasting methods cannot contract with determining and sometimes data consist of uncertain and imprecise properties which are not handled by quantitative models. In order to achieve the main objective, accuracy and efficiency of time series forecasting, we move towards the fuzzy time series modeling. Fuzzy time series is different from other time series as it is represented in linguistics values rather than a numeric value. The Fuzzy set theory includes many types of membership functions. In this study, we will utilize the Fuzzy approach and trapezoidal membership function to develop the fuzzy generalized auto regression conditional heteroscedasticity (FGARCH) model by using the fuzzy least square techniques to forecasting stock exchange market prices. The experimental results show that the proposed forecasting system can accurately forecast stock prices. The accuracy measures RMSE, MAD, MAPE, MSE, and Theil-U-Statistics have values of 18.17, 15.65, 2.339, 301.998, and 0.003212, respectively, which confirmed that the proposed system is considered to be useful for forecasting the stock index prices, which outperforms conventional GARCH models. Key words: Fuzzy time series, Membership function, trapezoidal fuzzy approach, GARCH model, Forecasting. 1. Introduction Forecasting is a significant feature in economics, commerce, various branches of science and marketing. It is a technique that predicts the future behavior of output on the basis of present and past output of yield and past trends. The economy of a nation to a great extent relies on upon capital business sector on upon capital S. Mustafa et al./Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 185-204 186 business sector, forecasting of stock market and their drifts are important factor in attaining significant gains in financial market. In capital and derivative pricing, investment plans, fund distribution and risk control processes, the accurately computation and prediction of financial- volatility plays a vital role (Franke & Westerhoff, 2011; Haugom, Langeland, Molnár, & Westgaard, 2014; A. Y. Huang, 2011), also fuzzy-Garh models for forecasting financial volatility (Hung, 2011a, 2011b; Maciel, Gomide, & Ballini, 2016). The stock price has deep impact in financial event of the country and large-scale economics approach. However, predicting and forecasting the stocks trading, prices and movement is not an easy task because of the serious impact of full-scale financial variable, including general monetary condition, political interference, financial specialist’s decision, sudden and unexpected change in security exchanges. Apart from the statistical models that have been used to understand and forecast variations in the stock market, a lot of attention has also been shifted to the applications of various soft computing application. There are different time series models proposed by the different researchers. Due to appropriateness and efficiency Fuzzy time series models are used in different studies (Bisht, Joshi, & Kumar, 2018; Iqbal & Zhang, 2020; Yu, 2005). Fuzzy set theory, provides an authoritative framework to handle with vague or ambiguous problems and can express linguistic values and human subjective decisions of natural language, (Zadeh, 1965). Fuzzy time series was first presented by (Song & Chissom, 1993, 1994). Furthermore, many fuzzy time series models were developed by researchers using different theories (Chen & Tanuwijaya, 2011; Egrioglu, Bas, Yolcu, & Chen, 2020; Hassan et al., 2020; Iqbal, Zhang, Arif, Hassan, & Ahmad, 2020; Lu, Chen, Pedrycz, Liu, & Yang, 2015; Wang, Lei, Fan, & Wang, 2016; Xiao, Gong, & Zou, 2009). Some analysts developed FTS forecsting models using probabilistic fuzzy set theory and reported significant results (Gupta & Kumar, 2019; W.-J. Huang, Zhang, & Li, 2012). Some fuzzy forecasting models in the environment of intuitionistic fuzzy set theory with equal length intervals are developed by (Abhishekh, Gautam, & Singh, 2018),(Bas, Yolcu, & Egrioglu, 2021) and also some work with unequal length intervals introduced by (Lei, Lei, & Fan, 2016) and (Iqbal & Zhang, 2020). In Addition, a novel method to forecast time series data was introduced by (Soto, Melin, & Castillo, 2018), using ensembles of IT2FNN models with fuzzy integrator optimization. There also some studies in which fuzzy based forecasting techniques are compared with classical models like ARIMA (Iqbal, Zhang, Arif, Wang, & Dicu, 2018). Technical analysis is a tool to predict future stock value developments by analyzing the past succession of stock costs. The generalized autoregressive conditional heteroscedasticity (GARCH) model is one of the famous econometric models used to estimates the volatility in financial market, stock markets. GARCH model is an econometric model, to describe an appropriate approach to estimate the in-monetarist markets volatility in monetarist markets, (Engle, 1982). GARCH models are useful across an extensive range of applications, also they do have boundaries as this model is only part of a solution. Although these models are usually applied to return series, financial decisions are rarely based solely on expected returns and volatilities. These models are parametric specifications that operate best under relatively stable market conditions. GARCH is explicitly designed to model time-varying conditional variances, Generalized Auto-Regressive Conditional Heteroscedasticity models often failed to capture highly irregular phenomena, including wild market fluctuations (e.g., crashes and subsequent A New Fuzzy Grach Model to forecast Stock Market Technical Analysis 187 rebounds), and other highly unanticipated events that can lead to significant structural change. GARCH models often fail to fully capture the fat tails distribution observed in asset return series. A fat-tailed distribution is a probability distribution that has the property, along with the other heavy-tailed distributions, that its revelations excess skewness or kurtosis. This comparison is often made relative to the normal distribution, or to the exponential distribution. Heteroscedasticity explains some of the fat tail behavior, but typically not all of it. Fat tail distributions, such as student-t, have been applied in GARCH modeling, but often the choice of distribution is a matter of trial and error. For this purpose, fuzzy model is proposed known as Fuzzy Generalized Auto-Regressive Conditional Heteroscedasticity (FGARCH) model in this paper. Although several fuzzy GARCH models based on different statistical and machine learning approaches are developed, such as (Hung, 2009, 2011a; Popov & Bykhanov, 2005), and (Maciel et al., 2016), but our proposed Fuzzy Generalized Auto-Regressive Conditional Heteroscedasticity (FGARCH) model is the best option because it is useful in investment on assets returns but also operates best under wide market fluctuation. In this paper, a new fuzzy model is proposed known as Fuzzy Generalized Auto- Regressive Conditional Heteroscedasticity (FGARCH) with fuzzy least square techniques and fuzzy trapezoidal approach. The motivation to use trapezoidal membership function is that it outperforms the different types of membership functions when it comes to develop a fuzzy-model for decision making and applicable to real-world applications. The proposed fuzzy model is the best option because it is useful in investment on assets returns but also operates best under wide market fluctuation. The objectives of the current study are explained as: (i) to estimate the unknown parameter by using the Generalized Auto-Regressive Conditional Heteroscedasticity and forecasting fuzzy models, (ii) to articulate the fuzzy model by using the fuzzy least square technique, (iii) to evaluate the comparison between forecast produced from classical model and proposed fuzzy model and also select the best performance model from them. The remaining paper comprises in the following stages. First section describes the introduction part. Second section briefly explains the earlier work done by the researchers in classical and fuzzy forecasting model. In third section, briefly described the methodology of the classical econometric model “Generalized Auto- Regressive Conditional Heteroscedasticity (GARCH)” and fuzzy model “Proposed Fuzzy Generalized Auto-Regressive Conditional Heteroscedasticity (FGARCH)”by using fuzzy least square method. This section also comprises concept of limitation in Generalized Auto-Regressive Conditional Heteroscedasticity (GARCH), perceptive to move towards fuzzy model. Fourth section included the results obtained from classical and proposed models with comparing the efficiency of the both models by using different endorsements. 2. Basic Theories 2.1. Fuzzy Set A fuzzy set Z˜ in the universe of information U can be defined as a set of ordered pairs and it can be represented mathematically as S. Mustafa et al./Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 185-204 188 (1) Here (x) is degree of membership of x, which assumes values in the range from 0 to 1, i.e., (x) ∈ [0,1]. 2.2. Trapezoidal membership function Trapezoidal membership function is described using the following equation T = (2) Where, x represents real value within the universe of discourse. a, b, c, d represent a x- coordinates of the four heads of trapezoidal and values should validate the following condition a