Abstract.
We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Lévy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lévy process and its stochastic exponential are investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: January 2003
Mathematics Subject Classification:
Primary: 60F05, 60G51, 60H30, 91B28; secondary: 60E07, 91B70
JEL Classification:
C22, G11, D81
We would like to thank Jan Kallsen and Ralf Korn for discussions and valuable remarks on a previous version of our paper. The second author would like to thank the participants of the Conference on Lévy Processes at Aarhus University in January 2002 for stimulating remarks. In particular, a discussion with Jan Rosinski on gamma processes has provided more insight into the approximation of the variance gamma model.
Rights and permissions
About this article
Cite this article
Emmer, S., Klüppelberg, C. Optimal portfolios when stock prices follow an exponential Lévy process. Finance and Stochastics 8, 17–44 (2004). https://doi.org/10.1007/s00780-003-0105-4
Issue Date:
DOI: https://doi.org/10.1007/s00780-003-0105-4