Tracking errors from discrete hedging in exponential L\'evy models

TL;DR

This paper investigates the errors from discrete hedging in exponential Lévy models, comparing quadratic and delta hedging, and relates convergence rates to the process's Blumenthal-Getoor index.

Contribution

It establishes the rate at which discrete hedging errors vanish and links these rates to the underlying Lévy process characteristics, especially for discontinuous pay-offs.

Findings

Quadratic hedging errors decrease at a quantifiable rate with increased readjustments.

Delta hedging can incur large errors for discontinuous pay-offs.

Convergence rates depend explicitly on the Blumenthal-Getoor index of the Lévy process.

Abstract

We analyze the errors arising from discrete readjustment of the hedging portfolio when hedging options in exponential Levy models, and establish the rate at which the expected squared error goes to zero when the readjustment frequency increases. We compare the quadratic hedging strategy with the common market practice of delta hedging, and show that for discontinuous option pay-offs the latter strategy may suffer from very large discretization errors. For options with discontinuous pay-offs, the convergence rate depends on the underlying Levy process, and we give an explicit relation between the rate and the Blumenthal-Getoor index of the process.