On the suboptimality of path-dependent pay-offs in Lévy markets

Cox & Leland (2000) use techniques from the field of stochastic control theory to show that in the particular case of a Brownian motion for the asset returns all risk averse decision makers with a fixed investment horizon prefer path-independent pay-offs over path-dependent ones. We will provide a novel and simple proof for the Cox & Leland result and we will extend it to general, not necessarily complete, Lévy markets. It is also shown that in these markets optimal path-independent pay-offs have final values increasing with the underlying asset value. Our results imply that path-dependent investment pay-offs, the use of which is widespread in financial markets, do not appear to offer good value for risk averse decision makers with a fixed investment horizon.