Mathematical Methods of Operations Research

, Volume 63, Issue 1, pp 123–150 | Cite as

Portfolio problems stopping at first hitting time with application to default risk

  • Holger KraftEmail author
  • Mogens Steffensen
Original Article


In this paper a portfolio problem is considered where trading in the risky asset is stopped if a state process hits a predefined barrier. This state process need not to be perfectly correlated with the risky asset. We give a representation result for the value function and provide a verification theorem. As an application, we explicitly solve the problem by assuming that the state process is an arithmetic Brownian motion. Then the result is used as a starting point to solve and analyze a portfolio problem with default risk modeled by the Black-Cox approach. Finally, we discuss how our results can be applied to a portfolio problem with stochastic interest rates and default risk modeled by the approach of Briys and de Varenne.


Optimal consumption and investment Random time horizon Feynman-Kac representation Barrier options 

JEL Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Black F, Cox JC (1976) Valuing corporate securities: some effects of bond indenture provisions. J Finance 31:351–367CrossRefGoogle Scholar
  2. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81: 637–654CrossRefGoogle Scholar
  3. Blanchet-Scaillet C, El Karoui N, Jeanblanc M, Martellini L (2003) Optimal investment and consumption decisions when time-horizon is uncertain. Preprint 168, Université d’Évry Val d’EssonneGoogle Scholar
  4. Bouchard B, Pham H (2004) Wealth-path dependent utility maximization in incomplete markets. Finance Stoch 8:579–603zbMATHMathSciNetGoogle Scholar
  5. Briys E, de Varenne F (1997) Valuing risky fixed rate debt: an extension. J Financ Quant Anal 32:239–248CrossRefGoogle Scholar
  6. Friedman A (1975) Stochastic differential equations and applications. vol 1. Academic Press, New YorkzbMATHGoogle Scholar
  7. Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. 2nd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. Karatzas I, Wang H (2000) Utility maximization with discretionary stopping. SIAM J Control Optim 39:306–329zbMATHCrossRefMathSciNetGoogle Scholar
  9. Korn R (1997) Optimal portfolios. World Scientific, SingaporezbMATHGoogle Scholar
  10. Korn R, Kraft H (2001) A stochastic control approach to portfolio problems with stochastic interest rates. SIAM J Control Optim 40:1250–1269zbMATHCrossRefMathSciNetGoogle Scholar
  11. Korn R, Kraft H (2003) Optimal portfolios with defaultable securities – a firm value approach. Int J Theor Appl Finance 6:793–819zbMATHCrossRefMathSciNetGoogle Scholar
  12. Kraft H (2003) Elasticity approach to portfolio optimization. Math Methods Oper Res 58: 159–182zbMATHCrossRefMathSciNetGoogle Scholar
  13. Merton RC (1969) Lifetime portfolio selection under uncertainty: the continuous case. Rev Econ Stat 51:247–257CrossRefGoogle Scholar
  14. Merton RC (1971) Optimal consumption and portfolio rules in a continuous-time model. J Econ Theory 3:373–413, Erratum: ebenda 6 (1973), 213–214CrossRefMathSciNetGoogle Scholar
  15. Merton RC (1974) On the pricing of corporate debt: the risk structure of interest rates. J Finance 29:449–479CrossRefGoogle Scholar
  16. Merton RC (1990) Continuous-time finance. Basil Blackwell, CambridgeGoogle Scholar
  17. Zariphopoulou T (1999) Optimal investment and consumption models with non-linear stock dynamics. Math Methods Oper Res 50:271–296zbMATHCrossRefMathSciNetGoogle Scholar
  18. Zariphopoulou T (2001) A solution approach to valuation with unhedgeable risks. Finance Stoch 5:61–82zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations