An extension of the Clark–Ocone formula under benchmark measure for Lévy processes

Abstract

The classical Clark–Ocone theorem states that any random variable can be represented as

where denotes the conditional expectation, is a Brownian motion with canonical filtration and D denotes the Malliavin derivative in the direction of W. Since many applications in financial mathematics require representation of random variables with respect to risk neutral martingale measure, an equivalent martingale measure version of this theorem was stated by Karatzas and Ocone (Stoch. Stoch. Rep. 34 (1991), 187–220). In this paper, we extend these results to be valid for square integrable pure jump Lévy processes with no drift and for square integrable Itô–Lévy processes using Malliavin calculus and white noise analysis. This extension might be useful for some applications in finance. As an application of our result, we calculate explicitly the closest hedge strategy for the digital option whose pay-off, , is square integrable and the stock price is driven by a Lévy process.

Keywords:

• Clark–Ocone formula
• change of measure
• Lévy processes
• Malliavin calculus
• white noise analysis

2000 Mathematics Subject Classification::

• Primary 60H07, 60H40
• Secondary 60G20

Acknowledgements

The author wishes to express her thanks to Prof. Bernt Øksendal and Prof. Giulia Di Nunno for fruitful suggestions and comments. The author would also like to thank the anonymous referee for helpful suggestions.