We consider the problem of hedging a contingent claim with a “semistatic” 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 semistatic hedging strategy, i.e., a strategy that only uses a small subset of available hedging assets and discuss parallels to the variable‐selection problem in linear regression. The methods developed are illustrated in an extended numerical example where we compute a sparse semistatic hedge for a variance swap using European options as static hedging assets.

In a financial market model, we consider the variance-optimal semi-static hedging of a given contingent claim, a generalization of the classic variance-optimal hedging. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy, we use a Fourier approach in a general multidimensional semimartingale factor model. As a special case, we recover existing results for variance-optimal hedging in affine stochastic volatility models. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in both the Heston and the 3/2-model, the latter of which is a non-affine stochastic volatility model. * Corresponding author. Mail: Paolo.Di_Tella@tu-dresden.de. † MKR thanks Johannes Muhle-Karbe for early discussions on the idea of "Variance-Optimal Semi-Static Hedging". We acknowledge funding from the German Research Foundation (DFG) under grant ZUK 64 (all authors) and KE 1736/1-1 (MKR, MH) using the orthogonal component L in the GKW decomposition (2.3).Moreover, notice that, if L ∈ H 2 0 is orthogonal to L 2 (S) in the Hilbert space sense, then L is also orthogonal to S, i. e. LS is a martingale starting at zero and, in particular, L, S = 0. Therefore, from (2.3) we can compute the optimal strategy ϑ * by S, H = ϑ * · S, S = · 0