Su, Stephanie See Weng (2023) Stochastic gradient descent approach with standard error and its application to financial portfolio optimization problems. Doctoral thesis, Universiti Tun Hussein Onn Malaysia.

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Abstract

Stochastic optimization in financial portfolio investment is a challenging task. In this thesis, a computational approach is proposed to solve the financial portfolio optimization problems. For this purpose, the stochastic gradient descent (SGD) method is overviewed, and its recent variant, the adaptive moment estimation (Adam) approach, is investigated. Notice that the updating rule in the Adam algorithm consists of the component of the second moment of past gradients, which is also known as the standard deviation of gradients. Hence, in our study, the computational algorithm mainly focuses on the SGD and Adam algorithms, and the standard error (SE) of sampling of the past gradients is added to the updating rule. So, the convergence rate can be fastened with fewer iteration numbers. On this basis, the proposed algorithm is known as the AdamSE algorithm. On the other hand, the application of the SGD, Adam and AdamSE algorithms to financial portfolio optimization models for the Employees Provident Fund (EPF) is examined. Here, a simulated mean-variance model is defined by using the parameters of the expected return and the covariance matrix from the classical mean-variance model, and the performance of algorithms is observed. Then, a mean-value at risk (mean-VaR) model is introduced, and the standard error of sampling of past gradients is associated with the AdamSE algorithm for obtaining different iteration steps toward the optimal solution. Next, a Black-Litterman model is studied, and different types of gradients in the measure of the central tendency of mean, median and mode gradients are employed in the AdamSE algorithm to express the efficiency of the algorithm. Accordingly, through these financial portfolio optimization models, the features of the AdamSE algorithm are demonstrated. Therefore, the efficiency of the proposed algorithm is proven. In conclusion, the practical application of these SGD algorithms to financial portfolio optimization problems is verified

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