Explicit F\"ollmer-Schweizer decomposition and discrete-time hedging in exponential L\'evy models

Abstract

In a financial market driven by an exponential L\'evy process, an explicit representation is shown for the F\"ollmer-Schweizer decomposition of European type options, implying a closed-form expression of the corresponding local risk-minimizing strategies. Using a jump-adjusted approximation scheme, the error caused by discretising the local risk-minimizing strategies is investigated in dependence of properties of the L\'evy measure, the regularity of the pay-off function and the chosen random discretisation times. The rate of this error as the number of expected discretisation times increases is measured in weighted BMO spaces, implying also $L_p$-estimates. Moreover, the effect of a change of measure satisfying a reverse H\"older inequality is addressed.Comment: 28 page