Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-23T06:20:17.353Z Has data issue: false hasContentIssue false
  • English
  • Français

On Optimal Terminal Wealth Problems with Random Trading Times and Drawdown Constraints

Part of: Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  22 February 2016

Ulrich Rieder  and
Marc Wittlinger
Ulrich Rieder*
Affiliation:
Ulm University
Marc Wittlinger*
Affiliation:
Ulm University
*
∗ Postal address: Institut für Optimierung und Operations Research, Ulm University, Helmholtzstrasse 18, 89081 Ulm, Germany. Email address: ulrich.rieder@uni-ulm.de
∗∗ Postal address: Institute of Mathematical Finance, Ulm University, Helmholtzstrasse 18, 89081 Ulm, Germany. Email address: marc.wittlinger@uni-ulm.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider an investment problem where observing and trading are only possible at random times. In addition, we introduce drawdown constraints which require that the investor's wealth does not fall under a prior fixed percentage of its running maximum. The financial market consists of a riskless bond and a stock which is driven by a Lévy process. Moreover, a general utility function is assumed. In this setting we solve the investment problem using a related limsup Markov decision process. We show that the value function can be characterized as the unique fixed point of the Bellman equation and verify the existence of an optimal stationary policy. Under some mild assumptions the value function can be approximated by the value function of a contracting Markov decision process. We are able to use Howard's policy improvement algorithm for computing the value function as well as an optimal policy. These results are illustrated in a numerical example.

Keywords

Portfolio optimizationilliquid marketrandom trading timedrawdown constraintlimsup Markov decision processHoward's policy improvement algorithmLévy process

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G51: Processes with independent increments; L'evy processes 90C40: Markov and semi-Markov decision processes
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 121 - 138
Copyright
© Applied Probability Trust 

References

Applebaum, D. (2009). Lévy Processes and Stochastic Calculus (Camb. Stud. Adv. Math. 116), 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Bäuerle, N. and Rieder, U. (2009). MDP algorithms for portfolio optimization problems in pure Jump markets. Finance Stoch. 13, 591–611.CrossRefGoogle Scholar
Bäuerle, N. and Rieder, U. (2011). Markov Decision Processes with Applications to Finance. Springer, Heidelberg.CrossRefGoogle Scholar
Cvitanic, J. and Karatzas, I. (1994). On portfolio optimization under ‘drawdown’ constraints. In Constraints (IMA Lecture Notes Math. Appl. 65), Springer, New York, pp. 77–88.Google Scholar
Elie, R. (2008). Finite time Merton strategy under drawdown constraint: a viscosity solution approach. Appl. Math. Optim. 58, 411–431.CrossRefGoogle Scholar
Elie, R. and Touzi, N. (2008). Optimal lifetime consumption and investment under a drawdown constraint. Finance Stoch. 12, 299–330.CrossRefGoogle Scholar
Gassiat, P., Pham, H. and Sîrbu, M. (2011). Optimal investment on finite horizon with random discrete order flow in illiquid markets. Internat. J. Theoret. Appl. Finance 14, 17–40.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer, London.CrossRefGoogle Scholar
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Finance Rev. 2, 79–105.CrossRefGoogle Scholar
Matsumoto, K. (2006). Optimal portfolio of low liquid assets with a log-utility function. Finance Stoch. 10, 121–145.CrossRefGoogle Scholar
Pham, H. and Tankov, P. (2008). A model of optimal consumption under liquidity risk with random trading times. Math. Finance 18, 613–627.CrossRefGoogle Scholar
Rogers, L.-C.-G. and Zane, O. (2002). A simple model of liquidity effects. In Advances in Finance and Stochastics, Springer, Berlin, pp. 161–176.CrossRefGoogle Scholar
Wittlinger, M. S. (2011). Terminal wealth problems in illiquid markets under a drawdown constraint. , Ulm University. Available at http://vts.uni-ulm.de/docs/2012/7821/vts_7821_11297.pdf.Google Scholar