In AxelCode-R/Master-Thesis: Does not matter.

Future Research

The variants of the Particle Swarm Optimization were examined for their efficacy in solving the index tracking problem. The local PSO and the self-adaptive velocity PSO were found to be particularly effective, as they increase the diversity within the swarm and prevent premature convergence in local minima. Additionally, the self-adaptive velocity PSO offers the advantage of reducing the number of hyperparameters, making it capable of solving a wide range of problems without the need for extensive fine-tuning. The implementation of the self-adaptive velocity PSO, as described in [@FaYa2014], was found to be effective, but further research is needed to determine whether it can be improved by combining it with the local variant.

A backtesting study was conducted in the final chapter, evaluating the practical application of the index tracking problem for retail investors. The results were promising, but further evaluation is necessary to confirm the stability of the results, given the path-dependent nature of the portfolios.

Conclusion

This thesis has been concerned with the analysis and testing of the PSO. The focus has been on optimization problems in the field of quantitative portfolio management. In practice, it is relevant to solve these kind of optimization problems with the same complexity as in real life, which is made possible by the PSO. It has been shown that even complex optimization problems can be solved in a stable manner.

It was shown in chapter \@ref(activevspassive) that more and more investors invest in passive managed funds, as they are more stable over time and have lower costs. To ensure competitiveness, it is important that quantitative strategies are automated as much as possible. The use of PSO in solving optimization problems in quantitative portfolio management is particularly relevant in this context, as it allows for the efficient and effective solution of these complex problems.

In chapter \@ref(challenges), various optimization problems commonly encountered in the realm of quantitative investment strategies within passive investment were examined and presented through the use of examples. One of the issues analyzed was the mean-variance portfolio (MVP) problem, the solution of which yields an optimal portfolio in terms of risk and return. Additionally, the topic of index tracking problem (ITP) was introduced, which aims to create a tracking portfolio that closely follows its benchmark. It has been demonstrated that it is beneficial to achieve the ITP by minimizing the mean square tracking error, which was abbreviated as ITP-MSTE.

In chapter \@ref(analyticalsolver), a quadratic optimizer called solve.QP was used to solve the quadratic problems with their constraints in the continuous case of the MVP problem and the ITP-MSTE. The solve.QP was also used as a benchmark to test the quality of the PSO in the following chapter \@ref(spso). It was found that although the PSO requires significantly more computation time, the results are often close to the solve.QP solution. In addition, the standard PSO was explained in detail and illustrated using visualizations and WebApps. It was shown how the penalty method works to take constraints into account during optimization. Furthermore, the convergence behavior of the standard PSO was analyzed, leading to the conclusion that the standard PSO is not a local search algorithm, but can be transformed into a local and also a global search algorithm by applying small modifications. It has been shown that PSO leads to stable results in the chosen financial optimization problems, provided that sufficient computing time is used.

Since in reality a portfolio consists of integer numbers of assets, it was shown that quadratic optimizers and subsequent rounding are not sufficient for discrete problems. For the PSO, it was shown how to deal with such discrete problems and then the discrete ITP-MSTE was solved, yielding stable results. This is particularly important in the context of quantitative portfolio management, as many real-world problems involve discrete variables.

In the next chapter \@ref(psovariants) different variants of the PSO were analyzed and compared with the standard PSO. It was found that the local PSO and the self-adaptive velocity PSO are the best variants to solve the discrete ITP-MSTE. In addition, the created PSO variants were compared with other metaheuristics applied to the same problem. It was found that the local PSO and the self-adaptive velocity PSO are better than the other metaheuristics. Since the self-adaptive velocity PSO has no hyperparameters that need to be adjusted, it was tested extensively in the following chapter.

In the final chapter of this thesis, the self-adaptive velocity PSO algorithm was applied in a practical scenario. A simulation was conducted in which the optimization problem of a private investor who regularly rebalances their portfolio on a monthly basis was considered. To accurately simulate the conditions of a private investor who adjusts their portfolio on a monthly basis, the ITP-MSTE model was designed to consider the transaction costs and the maximum limit of rebalancing allowed in a single month. The self-adaptive velocity PSO was then utilized to solve this optimization problem and was repeatedly executed until no constraints were violated, which was achieved in all cases. The results of this simulation indicate that the model has been solved effectively. However, further extensive testing is required to accurately assess the stability of the proposed model, as the portfolios in the backtests are path dependent.

Overall, it has been demonstrated that PSO can effectively yield stable results in financial optimization problems when given sufficient computational resources. The simplicity of formulating even complex problems is a key advantage of this method. As the financial industry continues to evolve and place increasing emphasis on automation of quantitative strategies, it is likely that optimization techniques like PSO will become increasingly prevalent.

AxelCode-R/Master-Thesis documentation built on Feb. 25, 2023, 7:57 p.m.