  We investigate the portfolio selection problem against the systemic risk
which is measured by CoVaR. We first demonstrate that the systemic risk of pure
stock portfolios is essentially uncontrollable due to the contagion effect and
the seesaw effect. Next, we prove that it is necessary and sufficient to
introduce options to make the systemic risk controllable by the correlation
hedging and the extreme loss hedging. In addition to systemic risk control, we
show that using options can also enhance return-risk performance. Then, with a
reasonable approximation of the conditional distribution of optioned
portfolios, we show that the portfolio optimization problem can be formulated
as a second-order cone program (SOCP) that allows for efficient computation.
Finally, we carry out comprehensive simulations and empirical tests to
illustrate the theoretical findings and the performance of our method.
