Robust One-Period Option Hedging

The paper considers robust optimization to cope with uncertainty about the stock return process in one period option hedging problems. The robust approach relates portfolio choice to uncertainty, making more cautious hedges when uncertainty is high. We represent uncertainty by a set of plausible expected returns of the underlying stocks and show that for this set the robust problem is a second order cone program that can be solved efficiently. We apply the approach to find an optimal portfolio to hedge an index option. Portfolio selection concerns the allocation of wealth to assets such that return is maximized and risk is minimized. The best known mathematical model for portfolio selection is the Markowitz (1952) model. The Markowitz model measures return by the expected value of the random portfolio return and risk by the variance of the portfolio return. The mathematical model is a quadratic programming model. A good reference on portfolio optimization is Zenios (1993). Critics have shown that portfolio optimization is very sensitive to the parameters of the model, in particular to the expected return. The numerical values for the parameters are output from economic models, possibly combined with subjective beliefs. These models are estimated from noisy data and as such subject to statistical and judgemental error. This leads to small inaccuracies in the parameter values. The problem is that small deviations in the parameter values lead to large changes in the optimal portfolio. In classical portfolio optimization the uncertainty in the parameters is not considered and estimates are passed on to the optimization tool as being oracle prophecies. This is also the case for the Markowitz model: the parameter values for expected return and risk are treated as certain. However the parameter values are uncertain and ignoring this leads to a poor actual performance (see e.g. Jobson and Korkie (1981), Michaud (1998, 1989) and Jorion (1986)). Various approaches have been proposed to work around this sensitivity. Statistical literature reports improved parameter estimators; Aı̈t-Sahalia and Brandt (2001) consider direct estimation of portfolio weights; Jagannathan and Ma (2003) impose supplementary ’wrong’ constraints in the optimization model to preclude extreme portfolios (but also opportunities); Horst, Roon and Werker (2002) adjust the coefficient of risk aversion to incorporate estimation uncertainty. 1991 Mathematics Subject Classification. 90C15, 90C20, 90C90, 49M29. JEL codes: C61, G11.