  This paper analyzes a problem of optimal static hedging using derivatives in
incomplete markets. The investor is assumed to have a risk exposure to two
underlying assets. The hedging instruments are vanilla options written on a
single underlying asset. The hedging problem is formulated as a utility
maximization problem whereby the form of the optimal static hedge is
determined. Among our results, a semi-analytical solution for the optimizer is
found through variational methods for exponential, power/logarithmic, and
quadratic utility. When vanilla options are available for each underlying
asset, the optimal solution is related to the fixed points of a Lipschitz map.
In the case of exponential utility, there is only one such fixed point, and
subsequent iterations of the map converge to it.
