Semi-Static and Sparse Variance-Optimal Hedging

TL;DR

This paper develops methods for semi-static, variance-optimal hedging strategies that combine dynamic and static positions, including sparse strategies selecting few assets, with practical formulas and numerical examples.

Contribution

It introduces a general framework for semi-static variance-optimal hedging, including sparse strategies and tractable Fourier-based formulas under the Heston model.

Findings

Derived explicit formulas for hedging strategies and errors.

Demonstrated sparse hedging for variance swaps using European options.

Validated methods with numerical examples.

Abstract

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.