Global "black swan" events, such as the 2007-9 global financial crisis and the 2020-23 Covid-19 pandemic, caused significant disruption to financial markets, in turn leading to increased interest in risk diversification. Meanwhile, sparse portfolio selection is critical for reducing transaction costs and improving managerial efficiency. This paper introduces the risk diversification mean-variance (RDMV) model. By analyzing the structural properties of this model, we prove that under mild conditions on the regularization parameter, all Karush-Kuhn-Tucker (KKT) points of the model globally minimize its objective function. We validate this theoretical characterization by comparing our solutions with a global optimum obtained using Gurobi, the global optimization solver. Further, we provide theoretical results on the relationship between the regularization parameter and sparsity, indicating that the RDMV model exhibits a certain degree of sparsity in portfolio weights without explicit sparse regularization terms. To solve the RDMV model efficiently, we introduce a proximal gradient algorithm with guaranteed convergence. Moreover, an adaptive regularization-parameter-setting strategy is designed to balance risk diversification and sparsity in the RDMV model. Finally, empirical analysis demonstrates that the RDMV model outperforms classical mean-variance, sparse portfolio and risk-diversified portfolio models in terms of sparsity, stability and net profit during the global financial crisis and the Covid-19 pandemic.
The signature Wasserstein generative adversarial network (SigWGAN) developed by Ni et al in 2021 achieved great results in time series generation using a recurrent neural network in combination with log signatures as the generator, trained to maximize the similarity between signatures given a static loss function. As an alternative to the static loss function on signatures, we propose using neural networks to minimize the Wasserstein distance between log signatures to reduce the target dimension of the generator and increase performance by introducing a learnable discriminator. We validate our proposed model on synthetic and real-world data across several performance evaluation metrics, showcasing its effectiveness for financial time series generation.
We develop a new algorithm that allows us to compute pairwise-correlation sensitivities in a Monte Carlo framework by modifying only one trajectory at a time, resulting in a significant decrease in Brownian noise, computing time and memory requirements. We apply this algorithm to the case of the risk management of a large portfolio of options on baskets of equities, but the same algorithm can be used for computing correlation sensitivities in any Monte Carlo framework. We show how this idea can also simplify the computation of correlation Greeks in the framework of adjoint algorithmic differentiation.
The Cox-Ingersoll-Ross model is widely used in financial engineering for the pricing of interest rate derivatives. Various Euler-Maruyama-and Milstein-type discretization schemes have been developed to approximate its solution, among which the full truncation Euler-Maruyama method is popular in practice and is known to be p-strongly pound convergent with order 1/2 under certain parameter conditions. In this paper, we investigate the projected Euler-Maruyama method for solving the Cox-Ingersoll-Ross model. By combining the projection technique with the normalized error analysis introduced by Cozma and Reisinger, we establish the p-strong pound convergence of the projected Euler-Maruyama method with order 1/2 over a significantly wide range of parameter settings.
