Title: Minimizing CVaR and VaR for Portfolios of Derivatives

      Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are the most frequently used risk measures in current risk management practice. As an alternative to VaR, CVaR is attractive since it is a coherent risk measure and a convex function of the portfolio instrument holdings. We analyze the problem of computing the optimal VaR and CVaR portfolios. In particular, we illustrate that VaR and CVaR minimization problems for a portfolio of derivatives are typically ill-posed. For example, the VaR and CVaR minimizations based on delta-gamma approximations of the derivative values typically have an infinite number of solutions.
       We propose to include cost as an additional preference criterion for the CVaR optimization problem. We demonstrate that, with the addition of a proportional cost, it is possible to compute an optimal CVaR investment portfolio of derivatives with significantly fewer instruments and comparable CVaR and VaR. A computational method based on a smoothing technique is proposed to solve a simulation based CVaR optimization problem efficiently. Comparison is made with the standard interior point method and simplex method for solving the simulation based CVaR optimization problem.
      This work is in collaboration with S. Alexander and T.F. Coleman