A Stock Index Futures Price Prediction Approach Based on the MULTI-GARCH-LSTM Mixed Model

Abstract

As a type of financial derivative, the price fluctuation of futures is influenced by a multitude of factors, including macroeconomic conditions, policy changes, and market sentiment. The interaction of these factors makes the future trend become complex and difficult to predict. However, for investors, the ability to accurately predict the future trend of stock index futures price is directly related to the correctness of investment decisions and investment returns. Therefore, predicting the stock index futures market remains a leading and critical issue in the field of finance. To improve the accuracy of predicting stock index futures price, this paper introduces an innovative forecasting method by combining the strengths of Long Short-Term Memory (LSTM) networks and various Generalized Autoregressive Conditional Heteroskedasticity (GARCH)-family models namely, MULTI-GARCH-LSTM. This integrated approach is specifically designed to tackle the challenges posed by the nonstationary and nonlinear characteristics of stock index futures price series. This synergy not only enhances the model’s ability to capture a wide range of market behaviors but also significantly improves the precision of future price predictions, catering to the intricate nature of financial time series data. Initially, we extract insights into the volatility characteristics, such as the aggregation of volatility in futures closing prices, by formulating a model from the GARCH family. Subsequently, the LSTM model decodes the complex nonlinear relationships inherent in the futures price series and incorporates assimilated volatility characteristics to predict future prices. The efficacy of this model is validated by applying it to an authentic dataset of gold futures. The empirical findings demonstrate that the performance of our proposed MULTI-GARCH-LSTM hybrid model consistently surpasses that of the individual models, thereby confirming the model’s effectiveness and superior predictive capability.

1. Introduction

The quest to accurately measure and model price risk within the stock market has been a pivotal area of research in financial econometrics. Ederington and Lee notably utilized the standard deviation of returns as a metric for assessing the stock market’s price risk [1], marking an early attempt to quantify volatility in a straightforward manner. However, traditional regression models, which presupposed homoscedasticity (constant variance) in error terms, encountered limitations when applied to financial time series data. In response to these challenges, Engle introduced the Autoregressive Conditional Heteroskedasticity (ARCH) model, which represented a significant advancement by allowing for the modeling of time-varying volatility [2], thus acknowledging and capturing the dynamic nature of financial data volatility. However, the ARCH model faced its own set of limitations, particularly concerning parameter estimation and the need for an extensive number of parameters in cases of high volatility persistence. To address these issues, Bollerslev proposed the GARCH model, which extended the ARCH framework by incorporating lagged conditional variances into the model [3]. Thus, the model reduces the number of parameters necessary to effectively capture volatility clustering—a prevalent phenomenon in financial time series characterized by periods of heightened volatility congregating in close succession. This innovation not only enhanced the efficiency of volatility modeling but also broadened the applicability of econometric analysis in financial market research.

For the past few years, traditional econometric and statistical models such as Autoregressive Integrated Moving Average (ARIMA) [4] and GARCH [5,6] have been used to predict future prices. Lim revealed that symmetric and asymmetric GARCH models have different performances in different time frames [7]. Silvennoinen proposed a new multivariate GARCH model with a time-varying conditional correlation structure [8]. However, such models often overlook the non-stationary and non-linear characteristics of the stock index futures market during the modeling process, thereby hindering the accurate capture of the long-term dependence in the series [9].

The advent of artificial intelligence has ushered in a new era for financial market analysis, with machine learning and deep learning methodologies increasingly being adopted to address the complexities inherent in financial time series forecasting [10]. Deep learning models have gained widespread popularity in financial time series forecasting, primarily due to their exceptional capability to capture complex nonlinear relationships. This attribute is particularly advantageous in overcoming the limitations often encountered with traditional econometric models in the realm of price forecasting tasks. By harnessing the power of deep learning, analysts and investors can more effectively discern and predict intricate market dynamics that are not readily apparent through conventional modeling approaches. Notable examples include the Support Vector Machine (SVM) [11] and the Gradient Boosting Decision Tree (GBDT) [12], both of which have demonstrated efficacy in futures price prediction. Saud and Shakya conducted a comparative analysis of the predictive capabilities of three prominent deep learning models—Recurrent Neural Network (RNN), LSTM, and Gated Recurrent Unit (GRU)—utilizing Nepal Stock Exchange (NEPSE) bank stock as a case study. Their findings demonstrated that the GRU model outperformed its counterparts in terms of accuracy in stock price prediction [13]. Similarly, the LSTM network has been specifically lauded for its superior performance in predicting futures prices, outpacing other methodologies in terms of prediction accuracy [14,15].

However, LSTM models exhibit shortcomings in predicting financial time series, particularly due to their ineffectiveness at capturing the volatility clustering effect. This limitation can result in predictions that are less than optimal. Volatility clustering refers to the aggregation of volatility in financial time series over time, i.e., volatility is clustered over time. Consequently, a combinatorial model approach is introduced for futures price forecasting. This method amalgamates the strengths of various forecasting models, integrating them to effectively capture both linear and nonlinear patterns within the data. By combining different models, not only is there an improvement in the accuracy of the forecasts, but it also provides enhanced interpretability. This dual benefit arises from the combined models’ ability to leverage the diverse capabilities of individual models, offering a more comprehensive and nuanced understanding of market trends and behaviors. For instance, Wang’s integration of ARIMA and GARCH models exemplifies how combining models could significantly improve futures price forecasting accuracy [16]. Kim et al. proposed a hybrid LSTM-GARCH model specifically designed to forecast stock market volatility [17]. Mahajan et al. developed a hybrid model integrating RNN and GARCH methodologies to predict the volatility of the National Stock Exchange Fifty of India (NIFTY 50) index [18]. Arashi et al. used the Autoregressive Moving Average (ARMA)-GARCH model to forecast the National Association of Securities Dealers Automated Quotations (NASDAQ) stock exchange stock index [19]. Kristjanpoller et al. proposed an Artificial Neural Network (ANN) model combined with a GARCH model, known as the ANN-GARCH model to forecast the price volatility of bitcoin [20]. Wu et al. proposed a hybrid model that combines the Convolutional Neural Network (CNN) and LSTM models to predict ten stocks in the U.S. and Taiwan [21]. Zolfaghari et al. similarly proposed a hybrid LSTM-GARCH model, focusing again on stock market volatility for two major indexes in the U.S. stock market including the Dow Jones Industrial Average (DJIA) and Nasdaq Composite (IXIC) [22]. Buyuksahin and Ertekin introduced an ARIMA-ANN hybrid approach to improve the forecasting accuracy of time series data including three well-known benchmark datasets and one public dataset: Wolf’s sunspot data, Canadian lynx data, British pound/US dollar exchange rate data, and the publicly available electricity price of Turkey Intraday Market [23]. Xu and Qin suggested a novel combination of ARIMA and regression tree models for interval-valued time series, expanding the scope of traditional forecasting techniques. It was tested on the Standard & Poor’s 500 (S&P 500) index series, and the results show that the hybrid model has better performance than the individual models [24]. The results show that the hybrid model has better performance than the individual models For an overview of these related results, we refer to the subject monographs [25,26,27,28,29,30] and references within.

Motivated by the challenges and advancements observed in the field, this study introduces a hybrid approach that amalgamates the strengths of both deep learning and econometric approaches. The proposed model, named the MULTI-GARCH-LSTM mixed model is specifically designed to harness the deep learning capabilities of LSTM for understanding complex data patterns and the robust econometric modeling of GARCH to effectively capture volatility clustering phenomena inherent to financial datasets. We first build a GARCH-family model to capture the volatility clustering effect of financial data by calculating volatilities. The GARCH-family model is initially employed to delineate various volatility characteristics, encompassing volatility aggregation and leverage effects prevalent within the futures market. Following this, both the volatility and price series are assimilated as inputs for the LSTM model. By leveraging the self-learning prowess of LSTM, this model skillfully navigates the intricate correlations that exist between price information and volatility feature data in the input series. This strategic approach enables a more nuanced capture of the volatility clustering effect inherent in financial datasets, thereby substantially enhancing the predictive accuracy of futures prices.

The main contributions of this paper are as follows:

-

This paper proposes an improved LSTM-GARCH model that integrates the GARCH family of models to more effectively capture the volatility clustering effects of financial data. The model employs LSTM to identify temporal patterns and GARCH to address conditional volatility, offering an integrated framework that enhances price forecasting and effectively captures volatility clustering effects in financial data, thus contributing to the study of financial time series forecasting.

-

An empirical experiment was conducted on gold futures issued by the Shanghai Futures Exchange (SHFE) to demonstrate the effectiveness and robustness of the proposed model. The results indicate that the proposed model surpasses benchmark models, including the traditional GARCH, LSTM, and GRU models, in terms of prediction accuracy and stability, thereby providing a scientific basis for rational decision-making by investors and the rational use of risk control measures by managers to stabilize the market.

This paper is tightly organized to ensure maximum clarity and comprehensive subject coverage. In Section 2, we embark upon an in-depth exploration where we not only reveal the precise data sets that underpin our investigation but also intricately delineate the innovative prediction models developed in the course of this research. This segment provides readers with a thorough understanding of the empirical foundation and methodological advancements proposed herein. Section 3 methodically charts out the experimental protocol adopted, which is substantiated by an exhaustive examination of the empirical outcomes derived from this study. It presents a step-by-step process, accompanied by a rigorous analysis of the results to underscore the reliability and validity of our findings. Finally, Section 4 serves as the synthesis and culmination point of the paper. It encapsulates the key discoveries and the broader implications stemming from our research endeavor. In doing so, it offers a concise yet comprehensive summary of the study’s core contributions, while providing thoughtful reflections on the overall scope and significance of the work carried out. This section concludes with valuable insights into the potential impact and future directions suggested by our findings.

2. Materials and Methods

2.1. Dataset

To verify the reliability and validity of the MULTI-LSTM-GARCH mixed model, we gathered historical data on gold futures prices from SHFE. The data utilized in this study consist of daily trading prices of the SHFE composite index futures for the period spanning 1 January 2013, to 30 September 2023. All the data were collected from the official SHFE website, including daily opening and closing prices, trading volume, open interest, highest and lowest prices, turnover, price changes, and other pertinent information. The dataset comprises 2608 data points, encompassing a broad range of market activity. To facilitate the training and validation of our MULTI-LSTM-GARCH model, we divided the dataset into training and testing sets at an 8:2 ratio, with 80% used for model training and the remaining 20% for testing. This division resulted in a training set containing 2460 data points and a testing set with 616 data points. This ratio was chosen to ensure that the model was trained on a sufficiently large dataset while also allowing for a robust evaluation of the model’s performance on unseen data.

2.2. Symbol Definition

Table 1. Symbol Definition.

${\mathit{i}}_{\mathit{t}}$	The input gate
$f_t$	The forget gate
$o_t$	The output gate
$x_t$	The current input
$h_t$	The output from the hidden layer
$c_t$	The cell state at time t
$y_t$	The closing price of the futures at time t
$\mu$	The mean of the futures price
${\epsilon }_t$	The error term at time t
${\sigma }_t$	The conditional standard deviation at time t
$z_t$	The standardized residual at time t
${\alpha }_0$	The variance of the error term
${\alpha }_i$	The coefficient of the error term
${\beta }_i$	The coefficient of the variance term
${\gamma }_i$	The coefficient of the asymmetric term
p	The order of the variance term
q	The order of the error term
${\mu }_t$	The conditional mean of the futures price
$D(0,1)$	A zero-mean unit variance distribution
$\omega$	The coefficient of the APARCH model
$\delta$	The parameter of the APARCH model

provides a clear and concise reference for the symbols used throughout the paper, ensuring consistency and ease of understanding for the reader:

2.3. Methods

2.3.1. LSTM

LSTM networks introduced by Hochreiter and Schmidhuber in 1997 [31], represent a significant evolution within the domain of Recurrent Neural Networks (RNNs), which first introduced by Rumelhart et al. in 1986, are a class of artificial neural networks uniquely tailored for processing sequential data [32]. By design, RNNs are adept at capturing temporal dependencies, positioning them as ideal candidates for time series forecasting tasks.

Despite their strengths, traditional RNNs encounter limitations in modeling long-term dependencies within sequential data, primarily due to the vanishing and exploding gradient problems during the training phase. These challenges significantly hinder the network’s ability to learn and retain information over extended sequences, thus constraining their applicability in tasks requiring long-term memory.

In response to these limitations, LSTM networks were specifically engineered to address the shortcomings of traditional RNNs by excelling at capturing long-term temporal relationships within time series data. The innovative architecture of LSTMs incorporates unique structures such as input gates, output gates, and forget gates, which collectively enhance the network’s capacity to learn and remember information over prolonged periods. A distinctive feature of LSTM networks is the inclusion of a state unit, represented by c, which plays a crucial role in the network’s ability to preserve information across time intervals.

The ability of LSTM networks to mitigate the vanishing and exploding gradient problems and to maintain information over extended durations renders them exceptionally suited for a wide array of applications. These range from speech recognition to the nuanced domain of financial time series analysis, where the capacity to recognize and remember long-term patterns is invaluable. Figure 1 delineates the intricate structure of an LSTM network, showcasing its comprehensive design tailored for advanced sequential data processing tasks.

In LSTM structure:

-

$i_t$ represents the input gate. This gate controls the extent to which new input data are allowed to flow into the cell state.

-

$f_t$ is the forget gate. It determines how much of the historical data (from the cell’s state) should be retained or discarded.

-

$o_t$ is the output gate, regulating the data that obtains output from the LSTM cell at any given moment.

The network’s current input is represented by $x_t$, while $h_{t-1}$ signifies the output from LSTM’s hidden layer at the previous time step. Similarly, $c_{t-1}$ refers to the cell state at the previous moment. LSTM’s output at time t is indicated by $h_t$ and $c_t$.

These components work together within LSTM, allowing it to effectively process and learn from time series data, particularly in capturing and remembering patterns over long periods. This capability makes LSTMs highly suitable for a range of applications, from speech recognition to financial time series analysis, Figure 1 shows the structure of LSTM.

2.3.2. GARCH Family

GARCH is a popular time series model for modeling volatility clustering, which is widely used in financial market forecasting. Originating as an extension of the ARCH model, the GARCH framework introduces a mechanism to incorporate the variance of the error term from previous periods, thereby enhancing the model’s capacity to capture the persistence of volatility over time. The foundational equations of the GARCH model are delineated as follows:

$$y_t=\mu +{\epsilon }_t,$$

(1)

$${\epsilon }_t={\sigma }_t·z_t,$$

(2)

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^q{\alpha }_i{\epsilon }_{t-i}^2+\sum _{i=1}^p{\beta }_i{\sigma }_{t-i}^2.$$

(3)

Fundamentally, the GARCH model presupposes that the time series’ error term follows a normal distribution. This assumption underpins the model’s capacity to forecast volatility by capturing the clustering of volatility episodes over time. The model’s parameters include ${\alpha }_0$, representing the variance of the error term; ${\alpha }_i$, the coefficient of the error term; ${\beta }_i$, the coefficient of the variance term; and p and q, denoting the orders of the variance and error terms, respectively. It is important to note that the GARCH model is univariate, limiting its application to the volatility prediction of individual time series rather than multiple concurrently.

The GARCH family model is an extension of the GARCH model, which includes the Exponential GARCH (EGARCH) model, Glosten–Jagannathan-Runkle GARCH (GJR-GARCH) model and Asymmetric Power Autoregressive Conditional Heteroskedasticity (APARCH) model. These models are designed to address the limitations of the standard GARCH model, offering enhanced capabilities in capturing the complex dynamics of financial time series data. The EGARCH model is distinguished by its ability to model the asymmetric effects of volatility, a feature particularly relevant in financial markets where positive and negative shocks have disproportionately different impacts on volatility levels. The GJR-GARCH model further extends the GARCH framework by integrating an asymmetric term, enhancing the model’s sensitivity to the skewness of volatility reactions to market movements. Similarly, the APARCH model incorporates a leverage term, enabling it to more accurately capture the leverage effect—a phenomenon where negative market returns lead to higher subsequent volatility compared to positive returns.

These extensions of the GARCH model enrich the analytical toolbox available for financial market forecasting, allowing for a more nuanced and comprehensive understanding of volatility dynamics in financial time series. Each variant brings a unique perspective to volatility modeling, from capturing asymmetric responses to market shocks to accommodating the leverage effect, thereby offering a more detailed and sophisticated approach to volatility prediction in financial econometrics.

2.3.3. EGARCH

The EGARCH model represents a seminal breakthrough in the realm of econometrics, serving as an enhanced derivative of the standard GARCH model. By incorporating an asymmetric component, the EGARCH framework significantly bolsters the model’s ability to accurately capture and reflect the differing impacts of market shocks on volatility dynamics. It was first proposed by Nelson in 1991 [33], and was designed to capture the asymmetric volatility effects observed in financial time series. The model’s formulation is articulated through the following equations:

$$r_t=X_t\beta +{\epsilon }_t,$$

(4)

$$ln{\sigma }_t^2={\alpha }_0+\sum _{i=1}^p{\alpha }_i\left(\left|{\epsilon }_{t-i}\right|-{\gamma }_i{\epsilon }_{t-i}\right)+\sum _{j=1}^q{\beta }_{j}ln{\sigma }_{t-j}^2.$$

(5)

Within this framework, ${\alpha }_0$ denotes the variance of the error term, ${\alpha }_i$ the coefficient of the error term, ${\beta }_j$ the coefficient of the variance term, and p and q are the orders of the variance and error terms, respectively. The EGARCH model is inherently univariate, implying its applicability to the volatility prediction of a singular time series, as opposed to the simultaneous forecasting of multiple series.

A hallmark of the EGARCH model is its asymmetric term, represented by the absolute value of the lagged residual, ${\epsilon }_{t-i}$. This inclusion is instrumental in modeling the asymmetric response of volatility to market shocks, wherein positive and negative shocks exert distinct influences on future volatility. The parameters ${\alpha }_i$ and $\gamma$ are crucial in determining the asymmetry’s magnitude and direction, offering a multifaceted depiction of volatility dynamics.

By integrating the asymmetric term, the EGARCH model affords a more precise portrayal of the time-varying volatility characteristic of financial markets. This enhancement allows for a nuanced understanding of how different types of market shocks—positive or negative—affect volatility, thereby providing a comprehensive tool for analyzing and forecasting the volatility patterns in financial data.

2.3.4. GJR-GARCH

The GJR-GARCH model extends the traditional GARCH framework by incorporating an asymmetric term, which was first proposed by Glosten et al. in 1993 [34]. This model is designed to capture the leverage effect, a phenomenon where negative financial shocks have a more pronounced and lasting impact on volatility compared to positive shocks. The GJR-GARCH model’s formulation integrates an asymmetric term, thereby capturing the distinct impacts of market shocks on volatility. The model is formally articulated through the following equations:

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^m({\alpha }_i+{\gamma }_i{N}_{t-i})a_{t-i}^2+\sum _{j=1}^s{\beta }_j{\sigma }_{t-j}^2,$$

(6)

$$\begin{array}{c}\hfill N_{t-i}=\left\{\begin{array}{cc}1,\hfill & a_{t-i}<0,\hfill \\ 0,\hfill & a_{t-i}\ge 0.\hfill \end{array}\right.\end{array}$$

(7)

In this framework, ${\alpha }_0$ signifies the variance of the error term, ${\alpha }_i$ is the coefficient of the error term, ${\beta }_j$ is the coefficient of the variance term, and ${\gamma }_i$ is the coefficient of the asymmetric term. The parameters p and q denote the orders of the variance and error terms, respectively, underscoring the model’s capacity to adapt to various temporal structures of volatility. It is essential to note that the GJR-GARCH model is univariate, designed solely for the volatility prediction of individual time series and not suitable for forecasting the volatility across multiple series concurrently.

The inclusion of the asymmetric term $N_{t-i}$ is a distinctive feature of the GJR-GARCH model, allowing for a nuanced understanding of how volatility responds to different market conditions. This model acknowledges the empirical observation that negative financial shocks—indicated by $a_{t-i}<0$—exert a more pronounced and lasting influence on volatility levels than positive shocks. Such flexibility in modeling volatility dynamics makes the GJR-GARCH model an invaluable tool in the precise characterization of financial time series data, enhancing the predictive accuracy and utility of econometric analyses in financial markets. Through this asymmetric lens, the GJR-GARCH model provides a more comprehensive and accurate depiction of volatility’s behavior, reflecting the complex and often skewed nature of financial market fluctuations.

2.3.5. APARCH

The APARCH model is a variant of the GARCH model that incorporates an asymmetric term to capture the leverage effect. This model was first proposed by Ding et al. in 1993 [35], and designed to address the limitations of traditional GARCH models in capturing the asymmetric response of volatility to market shocks. The APARCH model’s formulation is articulated through the following equations:

$$y_t={\mu }_t+{\alpha }_t,$$

(8)

$${\alpha }_t={\sigma }_t{ϵ}_t,$$

(9)

$${ϵ}_t\sim D(0,1),$$

(10)

$${\sigma }_t^{\delta }=\omega +\sum _{i=1}^m{\alpha }_i\left(\right|{\alpha }_{t-i}{|-{\gamma }_i{\alpha }_{t-i})}^{\delta }+\sum _{j=1}^s{\beta }_j{\sigma }_{t-j}^{\delta }.$$

(11)

In this framework, ${\mu }_t$ represents the conditional mean of the futures price, $D(0,1)$ denotes a zero-mean unit variance distribution, $\omega$, ${\alpha }_i$, ${\gamma }_i$, and ${\beta }_j$ are the coefficients that satisfy certain regularity conditions to ensure the positivity of the volatility. The model encompasses several other models, with the standard GARCH model emerging when $\delta =2$ and ${\gamma }_j=0$. The TGARCH model is obtained when $\delta =2$, albeit with a slightly different form. The volatility equation directly utilizes the volatility ${\sigma }_t$ and the new information ${\alpha }_t$ when $\delta =1$, as opposed to their squares. This feature of the APARCH model is particularly useful when conducting hypothesis tests on volatility assumptions, offering a more nuanced and accurate depiction of volatility dynamics in financial time series data.

2.3.6. MULTI-GARCH-LSTM Mixed Model

The GARCH models excel in capturing the volatility clustering phenomenon characteristic of financial data, while LSTM models adeptly discern the complex nonlinear relationships inherent in such data. The fusion of the GARCH and LSTM models harnesses these strengths, promising to significantly enhance the predictive accuracy of financial time series by addressing both volatility clustering and non-linearity.

Initially, the GARCH model quantifies the volatility clustering features of financial datasets. This process involves a preliminary smoothness test of the price series, followed by the establishment of a mean model. Subsequent examination of the mean model’s residuals for an ARCH effect dictates the feasibility of constructing the GARCH model.

Subsequently, the identified volatility clustering characteristics, alongside the financial data, are fed into the LSTM model. This model’s deep learning capabilities enable the intricate learning of the nonlinear relationships within the financial data, thereby aiming to predict future data points with heightened accuracy. Figure 2 delineates the precise LSTM modeling process, underscoring the model’s efficacy in learning nonlinear relationships and enhancing prediction accuracy.

3. Results

3.1. Experimental Setup

In the data preprocessing phase, the raw data underwent cleaning and transformation to prepare it for analysis. This process was crucial to ensure the data’s reliability and usefulness. It involved addressing and rectifying missing values, eliminating any outliers that could skew the results, and normalizing the data. There are days in the data where data such as trading volume are not provided, and we remove the anomalous data of empty trading volume in order to avoid the negative impact on model training due to missing values and outliers. Normalization is a key step in data preprocessing, especially in financial data analysis, as it helps to standardize the range of the data, making it more comparable and easier to work with in predictive modeling. We normalized the data by scaling all the eigenvalues to the same range on a scale of 0 to 1. This normalization process is essential to ensure uniformity and consistency across the dataset, it can help in stabilizing the gradient descent optimization process, thereby facilitating the subsequent modeling and analysis phases.

All our numerical experiments are carried out on a PC with Intel Core i7-10875H CPU at 2.30 GHz and 16 GB of physical memory. The PC runs Python Version: 3.8.10 on a Windows 10 Education 64-bit operating system, and the analysis was primarily conducted using Python, a widely recognized programming language in the data science community due to its extensive libraries and frameworks that support machine learning and time series analysis. Specifically, we utilized the following Python libraries:

• NumPy and Pandas: They were instrumental in data manipulation and numerical computations.
• TensorFlow and Keras: These libraries were crucial for building and training the LSTM components of our model due to their comprehensive support for deep learning algorithms.
• Arch: This library was used for implementing various GARCH models, including the standard GARCH, EGARCH, and GJR-GARCH, providing robust tools for volatility modeling.
• Matplotlib: This library was employed for data visualization, enabling the creation of insightful plots and graphs to facilitate a deeper understanding of the data.
• Scikit-learn: This library was utilized for model evaluation and performance metrics, offering a comprehensive suite of tools for assessing the model’s predictive accuracy.

3.2. Evaluation Metrics

To rigorously assess the performance of the MULTI-LSTM-GARCH model, we deploy a suite of evaluation metrics, including the Coefficient of Determination ($R^2$), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). These metrics are instrumental in providing a multifaceted evaluation of the model’s prediction accuracy and overall effectiveness:

$R^2$ quantifies the proportion of the variance in the dependent variable that is predictable from the independent variables. This metric serves as a critical measure of the model’s explanatory power, offering insights into the model’s ability to capture the underlying patterns in the data.

$$R^2=1-\frac{{\sum }_{i=1}^n{(y_i-{\hat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}.$$

(12)

MSE quantifies the average squared deviation between predicted and actual values, serving as a critical measure of the model’s precision. By squaring the errors, MSE emphasizes larger discrepancies, offering insight into the model’s performance under varied scenarios.

$$\mathrm{MSE}=\frac{1}{n}\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2.$$

(13)

RMSE, the square root of MSE, scales the errors back to their original units, facilitating an intuitive understanding of the model’s prediction error magnitude. RMSE is particularly useful for identifying how significantly the predicted values deviate from the actual values on average, providing a straightforward interpretation of the model’s accuracy.

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}.$$

(14)

MAE calculates the average absolute difference between the predicted and actual values, disregarding the direction of the errors. This metric offers a direct measure of the average prediction error magnitude, ensuring an unbiased assessment of the model’s predictive capacity.

$$\mathrm{MAE}=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\hat{y}}_i\right|.$$

(15)

MAPE quantifies the percentage difference between the predicted and actual values, offering a relative measure of the model’s prediction accuracy. By normalizing the errors based on the actual values, MAPE provides a standardized assessment of the model’s performance, facilitating comparisons across different datasets and models.

$$\mathrm{MAPE}=\frac{1}{n}\sum _{i=1}^n\left|\frac{y_i-{\hat{y}}_i}{y_i}\right|\times 100\%.$$

(16)

These metrics collectively enable a comprehensive analysis of the MULTI-LSTM-GARCH model’s predictive reliability and accuracy, ensuring a robust evaluation framework. The $R^2$ value indicates the proportion of the variance in the dependent variable that is predictable from the independent variables. A higher $R^2$ value signifies that the model is more effective in capturing market trends or shocks. It indicates that the model can assimilate and reflect changes in market conditions, such as sudden price movements or volatility, within its predictions. The MSE and RMSE measure the average squared difference between the predicted and actual values, with lower values indicating a more accurate model. High values of MSE and RMSE indicate poor model performance, possibly suggesting that the model struggles to adapt to rapid changes in volatility or fails to capture long-term market trends effectively. The MAE measures the average absolute difference between the predicted and actual values, providing a more robust measure of prediction accuracy. By leveraging these evaluation criteria, we can ascertain the model’s efficacy in capturing the dynamic and complex patterns inherent in stock index futures prices, ultimately validating the model’s utility in financial forecasting endeavors.

3.3. Experimental Design

To evaluate the efficacy of the MULTI-LSTM-GARCH mixed model, we conduct a comparative analysis against a range of baseline models, including the LSTM model combined with each of the GARCH-family models (EGARCH, GJR-GARCH, and APARCH) independently, as well as their various combinations, which are standard in the realm of stock index futures price prediction. This evaluation was aimed to compare with baseline models to highlight the enhanced predictive capability of the combined approach. This comprehensive analysis provides a detailed understanding of the MULTI-LSTM-GARCH model’s predictive accuracy and robustness, offering valuable insights into the model’s performance across varied scenarios.

Initially, we implement the LSTM model with each of the GARCH-family models (EGARCH, GJR-GARCH, and APARCH) independently to assess their individual predictive capabilities. This step serves as a benchmark for evaluating the performance of the combined MULTI-LSTM-GARCH model, offering a baseline for comparison.

Subsequently, in our pursuit to identify the most effective combination for futures price prediction, we integrate the LSTM model with more than one GARCH-family model. This process involves combining the LSTM model with EGARCH, GJR-GARCH, and APARCH models in various configurations, including LSTM-EGARCH, LSTM-GJR-GARCH, LSTM-APARCH, LSTM-EGARCH-GJR-GARCH, LSTM-EGARCH-APARCH, LSTM-GJR-GARCH-APARCH, and the proposed MULTI-LSTM-GARCH model. This systematic exploration of combined models aims to identify the optimal configuration for futures price prediction, offering a nuanced understanding of the interplay between LSTM and GARCH-family models in financial forecasting.

The final stage of our analysis involved a comprehensive comparison of the performances of these models. This process aimed to discern the most optimal model for futures price prediction.

To elucidate the methodology and enhance comprehension,

Table 2. Input Layer Variables of the Combined Models.

Model	EG	GJR-G	AP	LSTM	Explanatory Variable
LSTM-EG	✓			✓	✓
LSTM-GJR		✓		✓	✓
LSTM-AP			✓	✓	✓
LSTM-EGGJR	✓	✓		✓	✓
LSTM-EGAP	✓		✓	✓	✓
LSTM-GJRAP		✓	✓	✓	✓
MULTI-LSTM-GARCH	✓	✓	✓	✓	✓

in the manuscript methodically delineates the input variables for both the standalone and integrated models. The designations “EGARCH”, “GJR-GARCH”, and “APARCH” correspond to parameters sourced from GARCH-family models, subsequently integrated as inputs in the combined model framework. The term “explanatory variable” refers to the daily futures price series, with “✓” indicating the inclusion of the variable in the models—where “EG” indicates EGARCH term, “GJR-G” the GJR-GARCH term, “AP” the aPARCH term, “LSTM” the LSTM term, “LSTM-EG” the LSTM-EGARCH term, “LSTM-GJR” the LSTM-GJR-GARCH term, “LSTM-AP” the LSTM-APARCH term, “LSTM-EGGJR” the LSTM-EGARCH-GJR-GARCH term, “LSTM-EGAP” the LSTM-EGARCH-APARCH term, “LSTM-GJRAP” the LSTM-GJR-GARCH-APARCH term, and “MULTI-LSTM-GARCH” the MULTI-LSTM-GARCH term.

The selection of appropriate parameters is a critical aspect that significantly influences the prediction performance of any model. In this research, we determine the parameters for the GARCH-family models using the maximum likelihood estimation method. This approach ensures the most probable parameters for the model given the observed data. For the LSTM model, we employ the grid search method, a systematic approach to tuning hyperparameters, to identify the most effective parameter settings.

The LSTM model’s architecture features an input layer, hidden layers (with neuron counts set at 128, 128, and 64 for each layer, respectively), and an output layer. The selection of the Adam optimization algorithm (with a learning rate of 0.001) optimizes the model’s efficiency and effectiveness. Training parameters include 100 epochs and a batch size of 32, with dropout regularization (probability of 0.1) to prevent overfitting. The input data comprise the price data of the preceding 49 days to forecast the subsequent day’s price, aiming to capture pertinent trends for precise prediction.

In parallel, we delve into the parameter setting for the GARCH-family models. Utilizing the Akaike Information Criterion (AIC), Schwarz Criterion (SC), and Hannan–Quinn Criterion (HQ), the models EGARCH (1, 1, 0), GJR-GARCH (1, 1, 1), and APARCH (1, 1, 1) under the Generalized Error Distribution (GED) were selected. These models, designed to forecast future prices, derive their characteristic term coefficients to shed light on market volatility dynamics.

This dual-faceted approach, leveraging both LSTM and GARCH-family models with different parameters, aims to furnish precise and reliable predictions in the fluctuating domain of stock index futures pricing. Through this integration, the research seeks to capture both the volatility clustering and nonlinear relationship characteristics of financial data, thereby augmenting the accuracy of financial predictions.

Table 3. The Combine Model Prediction.

Model	R2	RMSE	MAE	MSE	MAPE
LSTM-EG	0.954768	6.507605	5.068048	42.348923	0.011894
LSTM-GJR	0.933886	7.867636	6.203575	61.899699	0.014558
LSTM-AP	0.948680	6.931718	5.218503	48.048710	0.012165
LSTM-EGGJR	0.954379	6.535533	5.549370	42.713188	0.013268
LSTM-EGAP	0.974523	4.883969	3.670806	23.853152	0.008654
LSTM-GJRAP	0.977843	4.554671	3.796484	20.745027	0.009467
MULTI-LSTM-GARCH	0.979456	4.385722	3.651949	19.234559	0.008878

and Figure 3 presents the results of the comparative analysis of the MULTI-LSTM-GARCH model, LSTM Model with one or two GARCH-family models. The plot illustrates the predicted values against the actual values, showcasing the model’s predictive accuracy and reliability, the MULTI-LSTM-GARCH model shows a close alignment between the predicted and actual values, underscoring the model’s robust predictive capabilities. The evaluation metrics include the Coefficient of Determination ($R^2$), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). The $R^2$ value of the MULTI-LSTM-GARCH model is 0.979456 in Table 3, which is higher than the LSTM model with single GARCH-family models and the LSTM model with two GARCH-family models. The results underscore the superior predictive accuracy of the MULTI-LSTM-GARCH model, as evidenced by its higher $R^2$ value and lower error metrics compared to the standalone LSTM model and other combined models. The MULTI-LSTM-GARCH model’s enhanced performance is attributed to its ability to capture both the nonlinear relationships and volatility clustering inherent in financial time series data, thereby offering more accurate and reliable predictions.

Figure 4 is a multi-bar chart representing the comparative analysis of forecasting accuracy, focusing on the MSE for each model as specified in the provided table. The chart compares the MSE of different models: LSTM-EG, LSTM-GJR, LSTM-AP, LSTM-EGGJR, LSTM-EGAP, LSTM-GJRAP, and MULTI-LSTM-GARCH. From the chart, it’s evident that the MULTI-LSTM-GARCH model has the highest accuracy, with the lowest MSE value among all models. Compared with the lowest MAE value of the LSTM model with single GARCH-family models, the MAE is reduced from 5.068048 to 3.651949, and compared with the LSTM model with two GARCH-family models, the MAE is reduced from 3.670806 to 3.651949. This indicates that the MULTI-LSTM-GARCH model outperforms the other models in terms of forecasting accuracy, highlighting its superior predictive capabilities in financial time series analysis. The LSTM component’s ability to capture complex temporal relationships in the data was evident in the reduced error metrics. Similarly, the GARCH component’s handling of volatility clustering contributed to the model’s overall predictive strength. This aligns with the analysis indicating a significant improvement in the predictive performance of the model, especially in its ability to capture asymmetric volatility effects, resulting in more accurate forecasts.

In Figure 3, after 2023-07 the actual value is higher than the predicted value, which indicates that the model may not be able to capture the sudden changes in the market or too long time series data may affect the model’s prediction accuracy. The model is a complex model that requires substantial computational resources. Although it used nearly a decade of futures market data, it still can not predict the price accurately after predicting the price for a long time. This indicates that if the model is applied to futures markets with a smaller amount of data, its forecasting results may not be as good.

4. Conclusions

This study underscores the enhanced predictive capabilities afforded by the integration of LSTM and GARCH family models, culminating in the development of the MULTI-LSTM-GARCH model for forecasting stock index futures prices. The model distinguishes itself by adeptly capturing both the intricate short-term dependencies and the nuanced long-term volatility patterns characteristic of the stock index futures market, thereby serving as a pivotal tool for traders and investors.

The superior performance of the MULTI-LSTM-GARCH model is principally attributed to its dual-component structure. The LSTM module excels in identifying complex temporal relationships and nonlinear dynamics within the market data. Simultaneously, the GARCH component elucidates volatility clustering and asymmetric volatility phenomena, enhancing the model’s comprehensive understanding of market behaviors.

Despite its significant advantages, including significant improvements in prediction accuracy, the model is not without limitations. The complexity of the MULTI-LSTM-GARCH model may necessitate extended training durations and substantial computational resources. The requirement for high-quality, extensive datasets, coupled with the model’s computational demands, poses challenges in terms of efficiency and accessibility. Maybe we can migrate the model to a more powerful computing platform to improve the model’s performance or reduce the amount of data used in the model to improve the model’s prediction accuracy.

Although the MULTI-LSTM-GARCH model is trained based on the gold futures trading market data, the ideas and methods of this model can be applied to various stock index futures in the stock index futures market. The model’s efficacy in capturing sudden market changes and adapting to evolving trends is pivotal for its utility in real-world financial applications. However, we must recognize that the model’s performance is contingent upon prevailing market conditions, and its adaptability to market changes is crucial for sustained relevance, so there is no guarantee that it can consistently achieve accuracy in different markets. The dynamic nature of financial markets necessitates continuous model updates and recalibration to maintain predictive accuracy.

The promising results achieved by the MULTI-LSTM-GARCH model pave the way for further explorations into hybrid predictive models within the realm of financial forecasting. This study contributes valuable insights to the financial forecasting field and suggests a promising trajectory for subsequent research efforts aimed at refining and enhancing predictive models for financial markets. The insights gleaned from this research endeavor provide a solid foundation for future studies aimed at exploring and developing sophisticated hybrid models for financial market analysis.