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Worst-Case Value-at-Risk of Non-Linear Portfolios

Steve Zymler (sz02***at***doc.ic.ac.uk)
Daniel Kuhn (dkuhn***at***doc.ic.ac.uk)
Berc Rustem (br***at***doc.ic.ac.uk)

Abstract: 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 non-convex) 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.

Keywords: Value-at-Risk, Derivatives, Robust Optimization, Second-Order Cone Programming, Semidefinite Programming

Category 1: Robust Optimization

Category 2: Applications -- OR and Management Sciences (Finance and Economics )

Citation: Working paper, Department of Computing, Imperial College London, June 2011

Download: [Postscript][PDF]

Entry Submitted: 08/18/2009
Entry Accepted: 08/18/2009
Entry Last Modified: 06/22/2012

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