  We consider hedging of a contingent claim by a 'semi-static' strategy
composed of a dynamic position in one asset and static (buy-and-hold) positions
in other assets. We give general representations of the optimal strategy and
the hedging error under the criterion of variance-optimality and provide
tractable formulas using Fourier-integration in case of the Heston model. We
also consider the problem of optimally selecting a sparse semi-static hedging
strategy, i.e. a strategy which only uses a small subset of available hedging
assets. The developed methods are illustrated in an extended numerical example
where we compute a sparse semi-static hedge for a variance swap using European
options as static hedging assets.
