## 5 Conclusions and Future Perspectives  
In this study, we propose the PSO Deep-ConvLSTM futures price spreads prediction model, aiming to offer a superior predictive tool to aid investors in devising efficient arbitrage strategies within futures markets. The model's efficacy was tested using real historical futures data and assessed by juxtaposing the data with alternate time series models. The empirical observations revealed the PSO Deep-ConvLSTM model's distinct precision and superiority in both single-step and multistep predictions for futures price spread forecasting, particularly its augmented capacity to apprehend the nonlinear attributes embedded in the data. Additionally, we employed multiple data sets to further enhance the model's confidence stability and generalization ability. The model exhibited comprehensive analytical competencies, considering multiple market indicators, thereby affirming its reliability in handling the complexities and volatility inherent in futures markets.  
The model's supremacy is reflected in two primary facets. Initially, the application of PSO methodology enhances the weight initialization and precision of parameters within the ConvLSTM network model, thus augmenting the objectivity of parameter selection. Moreover, its multidimensional and multistep predictive capacities pave the way for constructing arbitrage strategies based on forecast accuracy and comprehensiveness, elements of paramount importance to financial market participants.  
To conclude, while the PSO Deep-ConvLSTM model has demonstrated significant performance potential, opportunities for amplification remain. These can include the integration of strategic statistical combinations and the addition of more futures market traits. Additionally, extending the model's application to other financial markets might further quantify its generalizability and adaptability. Anticipated future research may build upon this groundwork, proferring an expanded range of applications within the financial sector, thus presenting an array of profound possibilities.  
Figure 11: Forecasting performance of various models for eight-step-ahead prediction of indicator values.  
Figure 10: Forecasting performance of various models for four-step-ahead prediction of indicator values.  
Figure 9: Forecasting performance of various models for one-step-ahead prediction of indicator values.  
**Author Contributions:** Conceptualization, S.T.; methodology, S.T.; software, S.T., Z.Z. and S.C.; validation, S.T., P.Q. and B.Y.; formal analysis, P.Q., investigation, S.T. and M.Z.; resources, S.T.; data curation, S.T.; writing--original draft preparation, S.T.; writing--review and editing, S.T.; visualization, S.T.; supervision, S.T. and P.Q.; project administration, S.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.