Worst-Case Portfolio Optimization in a Market with Bubbles

International Journal of Theoretical and Applied Finance, Vol. 19, No. 2, 1650009 (36 pages), 2016.

Posted: 4 Sep 2013 Last revised: 3 May 2016

See all articles by Christoph Belak

Christoph Belak

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften

Sören Christensen

Gothenburg University

Olaf Menkens

Dublin City University - School of Mathematical Sciences

Date Written: December 17, 2014

Abstract

We study a portfolio optimization problem in a financial market which is under the threat of crashes. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime switching model for the warnings and we make no assumptions about the distribution of the timing and the size of the crashes. Instead, we assume that the investor takes a worst-case perspective towards their impacts. That is, for each admissible trading strategy, we determine the crash scenario (including the no-crash scenario) which minimizes the investor’s expected utility from terminal wealth and then find the trading strategy which maximizes her utility in the worst-case scenario. We characterize the optimal value functions by a system of HJB equations and derive a coupled system of ordinary differential equations for the optimal strategies. We conclude with numerical examples.

Keywords: optimal investment, market crashes, worst-case scenario, regime switching, financial bubbles

JEL Classification: G11

Suggested Citation

Belak, Christoph and Christensen, Sören and Menkens, Olaf, Worst-Case Portfolio Optimization in a Market with Bubbles (December 17, 2014). International Journal of Theoretical and Applied Finance, Vol. 19, No. 2, 1650009 (36 pages), 2016., Available at SSRN: https://ssrn.com/abstract=2319913 or http://dx.doi.org/10.2139/ssrn.2319913

Christoph Belak (Contact Author)

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften ( email )

Institut fur Mathematik, Sekr. MA 7-1
Strasse des 17. Juni 136
Berlin, 10623
Germany

Sören Christensen

Gothenburg University ( email )

Göteborg, 41296
Sweden

Olaf Menkens

Dublin City University - School of Mathematical Sciences ( email )

Dublin
Ireland

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