Portfolio selection problem with Value-at-Risk constraints under non-extensive statistical mechanics

The optimal portfolio selection problem is a major issue in the financial field in which the process of asset prices is usually modeled by a Wiener process. That is, the return distribution of the asset is normal. However, several empirical results have shown that the return distribution of the asset has the characteristics of fat tails and aiguilles and is not normal. In this work, we propose an optimal portfolio selection model with a Value-at-Risk (VaR) constraint in which the process of asset prices is modeled by the non-extensive statistical mechanics instead of the classical Wiener process. The model can describe the characteristics of fat tails and aiguilles of returns. Using the dynamic programming principle, we derive a Hamilton–Jacobi–Bellman (HJB) equation. Then, employing the Lagrange multiplier method, we obtain closed-form solutions for the case of logarithmic utility. Moreover, the empirical results show that the price process can more accurately fit the empirical distribution of returns than the familiar Wiener process. In addition, as the time increases, the constraint becomes binding. That is, to control the risk the agent reduces the proportion of the wealth invested in the risky asset. Furthermore, at the same confidence level, the agent reduces the proportion of the wealth invested in the risky asset more quickly under our model than under the model based on the Wiener process. This can give investors a good decision-making reference.