As the skewed return distribution is a prominent feature in nonlinear portfolio selection problems which involve derivative assets with nonlinear payoff structures, Value-at-Risk (VaR) is particularly suitable to serve as a risk measure in nonlinear portfolio selection. Unfortunately, the nonlinear portfolio selection formulation using VaR risk measure is in general a computationally intractable optimization problem. We investigate in this paper nonlinear portfolio selection models using approximate parametric Value-at-Risk. More specifically, we use first-order and second-order approximations of VaR for constructing portfolio selection models, and show that the portfolio selection models based on Delta-only, Delta–Gamma-normal and worst-case Delta–Gamma VaR approximations can be reformulated as second-order cone programs, which are polynomially solvable using interior-point methods. Our simulation and empirical results suggest that the model using Delta–Gamma-normal VaR approximation performs the best in terms of a balance between approximation accuracy and computational efficiency.

When almost all underlying assets suddenly lose a certain part of their nominal value in a market crash, the diversification effect of portfolios in a normal market condition no longer works. We integrate the crash risk into portfolio management and investigate performance measures, hedging and optimization of portfolio selection involving derivatives. A suitable convex conic programming framework based on parametric approximation method is proposed to make the problem a tractable one. Simulation analysis and empirical study are performed to test the proposed approach.