The Annals of Applied Probability

Maximizing the probability of a perfect hedge

Jakša Cvitanić and Gennady Spivak

Full-text: Open access

Abstract

In the framework of continuous-time, Itô processes models for financial markets, we study the problem of maximizing the probability of an agent's wealth at time T being no less than the value C of a contingent claim with expiration time T. The solution to the problem has been known in the context of complete markets and recently also for incomplete markets; we rederive the complete markets solution using a powerful and simple duality method, developed in utility maximization literature. We then show how to modify this approach to solve the problem in a market with partial information, the one in which we have only a prior distribution on the vector of return rates of the risky assets. Finally, the same problem is solved in markets in which the wealth process of the agent has a nonlinear drift. These include the case of different borrowing and lending rates, as well as "large investor" models. We also provide a number of explicitly solved examples.

Article information

Source
Ann. Appl. Probab. Volume 9, Number 4 (1999), 1303-1328.

Dates
First available: 21 August 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1029962873

Mathematical Reviews number (MathSciNet)
MR1728563

Digital Object Identifier
doi:10.1214/aoap/1029962873

Zentralblatt MATH identifier
0966.91042

Subjects
Primary: 90A09 90A46
Secondary: 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Hedging partial information large investor margin requirements

Citation

Spivak, Gennady; Cvitanić, Jakša. Maximizing the probability of a perfect hedge. The Annals of Applied Probability 9 (1999), no. 4, 1303--1328. doi:10.1214/aoap/1029962873. http://projecteuclid.org/euclid.aoap/1029962873.


Export citation

References

  • Browne, S. (1996). Reaching goal by a deadline: continuous-time active portfolio management. Adv. Appl. Probab. To appear.
  • Browne, S. and Whitt, W. (1996). Portfolio choice and the Bayesian Kelly criterion. Adv. Appl. Probab. 28 1145-1176.
  • Cox, J. and Huang, C. F. (1989). Optimal consumption and portfolio policies when asset prices folllow a diffusion process. J. Econom. Theory 49 33-83.
  • Cuoco, D. and Cvitani´c, J. (1998). Optimal consumption choices for a large investor. J. Econom. Dy nam. Control 22 401-436. Cvitani´c, J. (1997a). Nonlinear financial markets: hedging and portfolio optimization. In Mathematics of Derivative Securities (M. A. H. Dempster and S. R. Pliska, eds.) 227-254 Cambridge Univ. Press. Cvitani´c, J. (1997b). Optimal trading under constraints. In Financial Mathematics. Lecture Notes in Math. 1656 123-190. Springer, Berlin.
  • Cvitani´c, J. (1998). Minimizing expected loss of hedging in incomplete and constrained markets. Preprint.
  • Cvitani´c, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2 767-818.
  • Cvitani´c, J. and Karatzas, I. (1998). On dy namic measures of risk. Finance and Stochastics. To appear.
  • Cvitani´c, J., Karatzas, I. and Soner, H. M. (1998). Backward stochastic differential equations with constraints on the gains-process. Ann. Probab. 26 1522-1551.
  • El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71.
  • F ¨ollmer, H. and Leukert, P. (1998). Quantile hedging. Finance and Stochastics. To appear.
  • He, H. and Pearson, N. (1991). Consumption and portfolio policies with incomplete markets and short-sale constraints: the infinite-dimensional case. J. Econom. Theory 54 259-304.
  • Heath, D. (1993). A continuous-time version of Kulldorff's result. Unpublished manuscript.
  • Karatzas, I. (1996). Lectures on the Mathematics of Finance. Amer. Math. Soc., Providence, RI.
  • Karatzas, I. (1997). Adaptive control of a diffusion to a goal and a parabolic monge-ampere-ty pe equation. Asian J. Math. 1 324-341.
  • 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 Optim. 25 1557-1586.
  • Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G. L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29 702-730.
  • Karatzas, I., Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Karatzas, I. and Zhao, X. (1998). Bayesian Adaptive Portfolio Optimization. Preprint, Columbia Univ.
  • Kulldorff, M. (1993). Optimal control of a favorable game with a time-limit. SIAM J. Control Optim. 31 52-69.
  • Lakner, P. (1994). Utility maximization with partial information. Stochastic Process. Appl. 56 247-273.
  • Pliska, S. R. (1986). A stochastic calculus model of continuous trading: optimal portfolios. Math. Oper. Res. 11 371-382.
  • Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, New York.
  • Xu, G.-L. and Shreve, S. (1992). A duality method for optimal consumption and investment under short-selling prohibition II. Constant market coefficients. Ann. Appl. Probab. 2 314-328.