Explicit F\"ollmer--Schweizer decomposition and discretization with jump correction in exponential L\'evy models

TL;DR

This paper derives explicit formulas for hedging in exponential Lévy models and analyzes the accuracy of jump-corrected discretization methods, providing convergence rates and error estimates under various conditions.

Contribution

It introduces a closed-form Föllmer--Schweizer decomposition for European options and a jump correction discretization method with proven convergence rates in exponential Lévy models.

Findings

Explicit Föllmer--Schweizer decomposition derived

Discretization error converges at a quantifiable rate

Impact of Lévy measure and payoff regularity on discretization error

Abstract

We investigate two hedging problems in exponential L\'evy models. First, we provide an explicit representation for the F\"ollmer--Schweizer decomposition of European type options under mild conditions, which implies a closed-form expression of the corresponding local risk-minimizing strategies. Secondly, we discretize stochastic integrals driven by an exponential L\'evy process using a jump correction method. The convergence rate of the resulting discretization error as the expected number of discretization times increases is measured in weighted BMO spaces, implying also $L_p$-estimates, $p \in (2, \infty)$. Moreover, the effect of a change of measure satisfying a reverse H\"older inequality is addressed. As an application, the error caused by discretizing the local risk-minimizing strategies is investigated in dependence of properties of the L\'evy measure, the regularity of the payoff function and the chosen random discretization times.