A Note on the Suboptimality of Path-Dependent Pay-Offs in Lévy Markets

Abstract

Cox and Leland used techniques from the field of stochastic control theory to show that, in the particular case of a Brownian motion for the asset log-returns, risk-averse decision makers with a fixed investment horizon prefer path-independent pay-offs over path-dependent pay-offs. In this note we provide a novel and simple proof for the Cox and Leland result and we will extend it to general Lévy markets where pricing is based on the Esscher transform (exponential tilting). It is also shown that, in these markets, optimal path-independent pay-offs are increasing with the underlying final asset value. We provide examples that allow explicit verification of our theoretical findings and also show that the inefficiency cost of path-dependent pay-offs can be significant. Our results indicate that path-dependent investment pay-offs, the use of which is widespread in financial markets, do not offer good value from the investor's point of view.

Key Words:

• Path-dependent pay-offs
• Lévy markets

Acknowledgements

Steven Vanduffel is grateful to Fortis for financial support (Fortis Chair in Financial and Actuarial Risk Management). He would also like to thank Tim Mortimer from Future Value Consultants who gave him the idea of researching this topic at a Risk Magazine conference in Brussels in 1996. The authors thank two anonymous referees for their careful reading and suggestions that helped improve the paper significantly.