Optimal Hedging in Incomplete Markets

ABSTRACT

We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an n-dimensional Brownian motion and an independent Poisson random measure on an n-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of a least expected squared error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.

KEYWORDS:

• Incomplete markets
• pricing kernels
• hedge ratios
• Brownian motion
• Lévy processes
• Lévy measures
• Lévy-Ito processes
• Poisson random measure
• simulations

Acknowledgments

The authors wish to thank D. C. Brody, A. Ciatti, S. Jaimungal and L. Sánchez-Betancourt for helpful discussions. We are also grateful for the helpful comments of an anonymous referee. GB acknowledges support from Timelineapp Tech Ltd, Basildon.

Disclosure statement

No potential conflicts of interest have been reported by the authors.