Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-21T07:40:37.428Z Has data issue: false hasContentIssue false
  • English
  • Français

An optimal portfolio problem in a defaultable market

Part of: Stochastic systems and control Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

Lijun Bo ,
Yongjin Wang  and
Xuewei Yang
Lijun Bo*
Affiliation:
Xidian University
Yongjin Wang*
Affiliation:
Nankai University
Xuewei Yang*
Affiliation:
Nankai University
*
∗ Postal address: Department of Mathematics, Xidian University, Xi'an 710071, P. R. China.
∗∗ Postal address: School of Business, Nankai University, Tianjin 300071, P. R. China.
∗∗∗ Postal address: School of Mathematical Sciences and TEDA Institute of Computational Finance, Nankai University, Tianjin 300071, P. R. China. Email address: xwyangnk@yahoo.com.cn
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 a portfolio optimization problem in a defaultable market. The investor can dynamically choose a consumption rate and allocate his/her wealth among three financial securities: a defaultable perpetual bond, a default-free risky asset, and a money market account. Both the default risk premium and the default intensity of the defaultable bond are assumed to rely on some stochastic factor which is described by a diffusion process. The goal is to maximize the infinite-horizon expected discounted log utility of consumption. We apply the dynamic programming principle to deduce a Hamilton-Jacobi-Bellman equation. Then an optimal Markov control policy and the optimal value function is explicitly presented in a verification theorem. Finally, a numerical analysis is presented for illustration.

Keywords

Portfolio optimizationdefaultable securitystochastic factorHamilton-Jacobi-Bellman equationsub/super-solution

MSC classification

Primary: 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 689 - 705
Copyright
Copyright © Applied Probability Trust 2010 

References

Biagini, F. and Cretarola, A. (2006). Local risk-minimization for defaultable claims with recovery process. Preprint. LMU University of München and University of Bologna.Google Scholar
Biagini, F. and Cretarola, A. (2007). Quadratic hedging methods for defaultable claims. Appl. Math. Optimization 56, 425–443.CrossRefGoogle Scholar
Biagini, F. and Cretarola, A. (2009). Local risk minimization for defaultable markets. Math. Finance 19, 669–689.CrossRefGoogle Scholar
Bielecki, T. R. and Jang, I. (2006). Portfolio optimization with a defaultable security. Asia-Pacific Financial Markets 13, 113–127.Google Scholar
Bielecki, T. R. and Rutkowski, M. (2002). Credit Risk: Modelling, Valuation and Hedging. Springer, Berlin.Google Scholar
Cox, J. C. and Huang, C.-F. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econom. Theory 49, 33–83.Google Scholar
Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. North Holland, Amsterdam.Google Scholar
Duffie, D. (2001). Dynamic Asset Pricing Theory, 3rd edn. Princeton University Press.Google Scholar
Duffie, D. and Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Rev. Financial Studies 12, 687–720.CrossRefGoogle Scholar
Fleming, W. H. and Hernández-Hernández, D. (2003). An optimal consumption model with stochastic volatility. Finance Stoch. 7, 245–262.CrossRefGoogle Scholar
Fleming, W. H. and Pang, T. (2004). An application of stochastic control theory to financial economics. SIAM J. Control Optimization 43, 502–531.Google Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.Google Scholar
Ishikawa, Y. (2004). Optimal control problem associated with Jump processes. Appl. Math. Optimization 50, 21–65.Google Scholar
Jang, I. (2005). Portfolio optimization with defaultable securities. , University of Illinois at Chicago.Google Scholar
Jarrow, R. A., Lando, D. and Yu, F. (2005). Default risk and diversification: theory and empirical applications. Math. Finance 51, 1–26.Google Scholar
Jin, X. and Hou, Y. (2002). Optimal investment with default risk. FAME research paper no. 46, Switzerland.Google Scholar
Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a ‘small investor’ on a finite horizon. SIAM J. Control Optimization 27, 1557–1586.Google Scholar
Korn, R. and Kraft, H. (2003). Optimal portfolios with defaultable securities: a firm value approach. Internat. J. Theory Appl. Finance 6, 793–819.Google Scholar
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous case. Rev. Econom. Statist. 51, 247–257.CrossRefGoogle Scholar
Merton, R. C. (1971). Optimal consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413.Google Scholar
Merton, R. C. (1992). Continuous-Time Finance. Blackwell, Oxford.Google Scholar
Pang, T. (2006). Stochastic portfolio optimization with log utility. Internat. J. Theory Appl. Finance 9, 869–887.Google Scholar
Pham, H. (2002). Smooth solutions to optimal investment methods with stochastic volatilities and portfolio constraints. Appl. Math. Optimization 46, 55–78.Google Scholar
Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar