Asymptotic Power Utility-Based Pricing and Hedging

TL;DR

This paper develops an alternative method for power utility-based pricing and hedging that avoids additional state variables by using semimartingale characteristics, with applications to exponential Lévy and stochastic volatility models.

Contribution

It introduces a new representation for utility-based pricing and hedging that simplifies computations by using the original numeraire, extending previous mean-variance hedging approaches.

Findings

Effective application to exponential Lévy processes.

Extension to stochastic volatility models.

Simplified computation of utility-based prices.

Abstract

Kramkov and Sirbu (2006, 2007) have shown that first-order approximations of power utility-based prices and hedging strategies can be computed by solving a mean-variance hedging problem under a specific equivalent martingale measure and relative to a suitable numeraire. In order to avoid the introduction of an additional state variable necessitated by the change of numeraire, we propose an alternative representation in terms of the original numeraire. More specifically, we characterize the relevant quantities using semimartingale characteristics similarly as in Cerny and Kallsen (2007) for mean-variance hedging. These results are illustrated by applying them to exponential L\'evy processes and stochastic volatility models of Barndorff-Nielsen and Shephard type.