Skip to main content
SpringerLink
Log in

A variational inequality arising from European option pricing with transaction costs

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper we present a method which can transform a variational inequality with gradient constraints into a usual two obstacles problem in one dimensional case. The prototype of the problem is a parabolic variational inequality with the constraints of two first order differential inequalities arising from a two-dimensional model of European call option pricing with transaction costs. We obtain the monotonicity and smoothness of two free boundaries.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Davis M H A, Panas V G, Zariphopoulou T. European option pricing with transaction costs. SIAM J Control Optim, 3(2): 470–493 (1993)

    Article  MathSciNet  Google Scholar 

  2. Rodrigues J F, Santos L. A parabolic quasi-variational inequality arising in a superconductivity model. Ann Sc Norm Super Pisa Cl Sci, 29(4): 153–169 (2000)

    MATH  MathSciNet  Google Scholar 

  3. Santos L. A parabolic variational inequality with non-constant gradient constraint. In: Li T T, Ling L W, Rodrigues J F, eds. Proceedings of the Luso-Chinese Symposium on Nonlinear Evolutionary Equations and their Applications. Singapore: World Scientific, 1999, 203–212

    Google Scholar 

  4. Zhu H. Characterization of variational inequalities in singular stochastic control. Dissertation for the Doctoral Degree. Providence, RI: Brown University, 1992

    Google Scholar 

  5. Dai M, Yi F. Finite-horizon optimal investment with transaction costs: a parabolic double obstacle problem. Preprint, 2007

  6. Blanchet A, Dolbeault J, Monneau R. On the one-dimensional parabolic obstacle problem with variable coefficients. arXiv:math/0410330, 2005

  7. Benes V E, Shepp L A, Witsenhaussen H S. Some solvable stochastic control problems. Stochastics, 4(1): 39–83 (1980)

    MATH  MathSciNet  Google Scholar 

  8. Karatzas I, Shreve S E. Connection between optimal stopping and stochastic control I: monotone follower problems. SIAM J Control Optim, 22: 856–877 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  9. Soner H M, Shreve S E. A free boundary problem related to singular stochastic control: the parabolic case. Comm PDE, 16: 373–424 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Oleinik O A, Radkevich E V. Second Order Equations with Nonnegative Characteristic Form. Providence, RI: American Mathematical Society, 1973

    Google Scholar 

  11. Friedman A. Variational Principle and Free Boundary Problems. New York: John Wiley and Sons, 1982

    Google Scholar 

  12. Ladyzenskaja O A, Solonnikov V A, Ural’ceva N N. Linear and Quasi-linear Equations of Parabolic Type. Providence, RI: American Mathematical Socety, 1968

    Google Scholar 

  13. Friedman A. Parabolic variational inequalities in one space dimension and smoothness of the free boundary. J Funct Anal, 18: 151–176 (1975)

    Article  MATH  Google Scholar 

  14. Black F, Scholes M. The pricing of options and corporate liabilities. J Political Economy, 81: 637–659 (1973)

    Article  Google Scholar 

  15. Davis M H A, Norman A R. Portfolio selection with transaction costs. Math Oper Res, 15: 676–713 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    FaHuai Yi & Zhou Yang

Authors
  1. FaHuai Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhou Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to FaHuai Yi.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10671075), the National Natural Science Foundation of Guangdong Province (Grant No. 5005930), and the University Special Research Fund for PhD Program (Grant No. 20060574002)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yi, F., Yang, Z. A variational inequality arising from European option pricing with transaction costs. Sci. China Ser. A-Math. 51, 935–954 (2008). https://doi.org/10.1007/s11425-007-0175-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-007-0175-4

Keywords

MSC(2000)

Search

Navigation