SIAM Review
Interior point algorithms: theory and analysis
Interior point algorithms: theory and analysis
Mathematics of Operations Research
Operations Research
Constructing Risk Measures from Uncertainty Sets
Operations Research
SIAM Review
Robust solutions of uncertain linear programs
Operations Research Letters
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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 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 nonconvex quadratic functions of the returns of the derivative underliers. These models lead to new worst-case polyhedral VaR WPVaR and worst-case quadratic VaR WQVaR approximations, respectively. WPVaR serves as a VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that---unlike VaR that may discourage diversification---WPVaR and WQVaR are in fact coherent risk measures. We also reveal connections to robust portfolio optimization. This paper was accepted by Dimitris Bertsimas, optimization.