Risk Minimization and Portfolio Diversification

We consider a capital at risk (CaR) minimization problem in an incomplete market Black-Scholes setting. The optimization problem is studied, given the possibility that a correlation constraint between the wealth process and a financial index is imposed. The optimal portfolio is not unique and it is analytically characterized. In the special case of complete market, the optimal portfolio is unique and is obtained in closed form. The effect of the correlation constraint is also explored; it turns out that this constraint leads to a more diversified portfolio.