The purpose of this thesis is to study the hedging of financial derivatives, using the so-called local risk-minimizing strategy, which is a popular quadratic hedging strategy in incomplete markets. The local risk minimization aims at perfectly replicating the derivative. However such strategies cannot be self-financing in general, and therefore allowing for a cost. Then a good strategy should have minimal cost. The problem of finding the local risk minimizing strategy is in this thesis tackled by two methods, via a change of measure using the minimal martingale measure and via backward stochastic differential equations. The financial market model studied is driven by a time-changed Lévy noise, where the time change is independent of the Lévy process. In this thesis two different information flows are considered. Both filtrations are naturally linked to the noise. Information flow G is the large filtration containing information about the future, which captures all statistical properties of the noise. While the smaller information flow F is a more realistic information flow from a financial modeling perspective.