A Nash bargaining solution for a multi period competitive portfolio optimization problem: Co-evolutionary approach

Abstract This study focuses on proposing a Nash bargaining model to solve a novel multi-period competitive portfolio optimization problem for large investors in the stock market who want to maximize their terminal wealth while taking into account competitors' profits. In this study, a Competitive Portfolio Model (CPM) is developed in accordance with the Cournot competition principle for a static, non-cooperative, and non-zero-sum game with complete information. Transaction costs, risk-free assets and cash are also included to match real-world conditions. Also, three criteria including the average value at risk, the mean absolute semi-deviation, and entropy are considered to control the investment risk in the model. Moreover, due to common constraints between players (free floating shares of risky assets) in stock markets, this study falls into the category of Generalized Nash Equilibrium Problems (GNEP). Therefore, to overcome the problem, a Cooperative Co-evolutionary Algorithm (CCA) based on Particle Swarm Optimization (PSO) is customized and used. A few experimental tests and a numerical example with descriptive analytics (using real data from two large mutual funds who invest in the Iranian Stock Exchange market) are used to evaluate the feasibility of the proposed model and the efficiency of the design algorithm. After solving the model, for each time period, investors' trading strategies (trading signals and stock volume in their portfolio) are determined. The results show that the volume of transactions due to the market power of an investor has a significant effect on the terminal wealth of competitors. Also, the results of sensitivity analysis show that profit-making is inversely related to the degree of risk aversion of investors.

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