A note on the suboptimality of path-dependent pay-offs in Lévy markets

Cox & Leland (2000) 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 ones. In this note we provide a novel and simple proof for the Cox & Leland result and we will extend it to general Lévy markets in case 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.