Portfolio optimization problems involving Value-at-Risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are further compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR over all return distributions of the derivative underliers with given first- and second-order moments. The derivative returns are modelled as convex piecewise linear or - by using a delta-gamma approximation - as (possibly non-convex) quadratic functions of the returns of the derivative underliers. These models lead to new Worst-Case Polyhedral VaR (WCPVaR) and Worst-Case Quadratic VaR (WCQVaR) approximations, respectively. WCPVaR is a suitable VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WCQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that WCPVaR and WCQVaR optimization can be formulated as tractable second-order cone and semidefinite programs, respectively, and reveal interesting connections to robust portfolio optimization. Numerical experiments demonstrate the benefits of incorporating non-linear relationships between the asset returns into a worst-case VaR model.
