Abstract: In the modified Black-Scholes financial market, this paper investigates continuous-time portfolio optimization problems based on CVaR and CCaR risk measures. The expressions of the optimal investment strategies and the efficient frontiers are obtained in closed form. The optimal investment strategies exhibit two-fund separation theorem which includes the riskless asset and equal-volatility portfolio. If the risk exposure of the risky assets is given by the investors in advance, then the optimal investment strategy can be determined by an estimated volatility matrix, without knowing the parameters of expected return. Hence, the sensitive problem brought by the traditional optimal portfolio weight in estimating the expected returns is solved efficiently. Moreover, the optimal investment strategy and the efficient frontier of the mean-variance model are derived by using our proposed method. Finally, the empirical analysis shows that our optimal portfolio weights are stable over time and obtain a significantly better performance than the equal-weight investment strategy and the minimum variance investment strategy, both of which are not influenced by the estimation of the expected returns.

Key words: portfolio optimization, downside risk measure, modified Black-Scholes financial market, investment strategy, efficient frontier