CVaR and VaR for a portfolio of derivatives

The use of derivatives can lead to higher yields and lower funding costs. In addition, derivatives are indispensable tools for risk management. We analyze the derivative portfolio hedging problems based on value at risk (VaR) and conditional value at risk (CVaR). As an alternative to VaR, CVaR is a more attractive risk measure since it is coherent.; We show that VaR and CVaR derivative portfolio optimization problems are often ill-posed and the resulting optimal portfolios frequently incur large transaction and management costs. For example, the VaR and CVaR minimizations based on delta-gamma approximations of the derivative values typically have an infinite number of solutions. In addition, we illustrate that the optimal portfolios may perform poorly under slight model error thereby showing the importance of sensitivity testing of portfolio hedging performance with respect to model error. A CVaR optimization model including a proportional cost is proposed to produce optimal portfolios with fewer instruments and smaller transaction costs with similar expected returns and a small compromise in risk. The optimal portfolios under a suitable cost consideration perform much more robustly with respect to model error.; We discuss computational issues for large scale CVaR optimization problems and propose a new method based on a smoothing technique which solves a simulation based CVaR optimization problem much more efficiently than the standard linear programming methods. A comparison is drawn between the smoothing-based method and the linear programming approach for solving the CVaR optimization problem for various kinds of portfolios.; Since CVaR computation for large derivative portfolios can be very computationally expensive, we show how to use variance reduction techniques, including importance and stratified sampling, to significantly reduce the computational time. We also show how these techniques can be used to hedge a portfolio of derivatives with stocks and compare the performance of these optimal portfolios to the corresponding delta hedged portfolios.; Finally, we discuss the CVaR optimization problem in the multiperiod framework. Since the brute-force multiperiod simulation results in a tree with exponential growth, we select a much smaller tree and use interpolation to approximate the problem. We solve for the optimal portfolios by minimizing the CVaR. of the remaining risk and make a comparison with the portfolios obtained from the multiperiod quadratic risk minimization problem.