Skip to main content
SpringerLink
Log in

Analysis of a finite volume element method for a degenerate parabolic equation in the zero-coupon bond pricing

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

We construct and analyze a stable exponentially fitted numerical scheme for a degenerate parabolic equation in the zero-coupon bond pricing. Introducing weighted Sobolev spaces, we present the Gärding coercivity and the weak maximum principle for the differential solution. The differential problem is discretized by a fitted finite volume element method resolving the degeneration. We derive coercivity of the discrete bilinear form as we also show that the fully discrete system matrix is essentially of positive type which implies the maximum principle for the implicit time stepping. Numerical experiments validate the theoretical results.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Achdou Y, Pironneau O (2005) Computational methods for option pricing. SIAM Ser Front Appl Math

  • Angermann L (2008) Discretization of the Black-Sholes operator with a natural left-hand side boundary condition. Far East J Appl Math 30(1):1–41

    MATH  MathSciNet  Google Scholar 

  • Angermann L, Wang S (2007) Convergence of a fitted finite volume method for the penalized Black-Sholes equation governing European and American option pricing. Numer Math 106:1–40

    Article  MATH  MathSciNet  Google Scholar 

  • Bramble JH, Hubbard BE (1964) New monotone type approximations for elliptic problems. Math Comput 18(87):349–367

    Google Scholar 

  • Bramble JH, Hubbard BE, Thomée V (1969) Convergence estimates for essentially positive type discrete Dirichlet problems. Math Comput 23:695–709

    Article  MATH  Google Scholar 

  • Bonnans J Fr, Tan X (2011) Monotonicity condition for the \(\theta \)-scheme for diffusion equations. INRIA, Raport de Recherche, No 7778

  • Cen Z, Le A (2011) A robust and accurate finite difference method for a generalized Black-Scholes equation. J Comput Appl Math 235:3728–3733

    Article  MATH  MathSciNet  Google Scholar 

  • Chernogorova T, Stehlíková B (2012) A comparison of asymptotic formulae with finite-difference approximations for pricing zero coupon bond. Numer Algor 59:571–588

    Article  MATH  Google Scholar 

  • Chernogorova T, Valkov R (2011) Finite volume difference scheme for a degenerate parabolic equation in the zero-coupon bond pricing. Math Comput Model 54:2659–2671

    Article  MATH  MathSciNet  Google Scholar 

  • Deng ZC, Yu JN, Yang L (2010) An inverse problem arizen in the zero bond pricing. Nonl Anal Real World Appls 11(3):1278–1288

    Google Scholar 

  • Grossmann C, Roos H-G (2007) Numerical treatment of partial differential equations, 3d edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • Gyulov T, Valkov R (2011) Classical and weak solutions for two models in mathematical finance. AIP Conf Proc 1410:195–202

    Article  MathSciNet  Google Scholar 

  • Knabner P, Angermann L (2003) Numerical methods for elliptic and parabolic partial differential equations. Springer, New York

  • Kufner A (1985) Weighted Sobolev spaces. Wiley, New York

    MATH  Google Scholar 

  • Lions J-L, Magenes E (1968) Problèmes aux Limites non Homogènes et Applications, vols. I and II. Dunod, Paris

  • Miller JJH, Wang S (1994) A new non-conforming Petrov-Galerkin finite element method with triangular elements for a singularly perturbed advection-diffusion problem. IMA J Numer Anal 14:257–276

    Article  MATH  MathSciNet  Google Scholar 

  • Mikula K, Ševčovič D, Stehlíková B (2011) Analytical and numerical methods for pricing financial derivatives. Nova Science Publishers Inc., Hauppauge

    Google Scholar 

  • Oleinik OA, Radkevich EV (1973) Second order equations with nonnegative characteristic form. Plenum Press, New York

    Book  Google Scholar 

  • Ramírez-Espinoza GI, Ehrhardt M (2013) Conservative and finite volume methods for the convection-dominated pricing problem. Adv Appl Math Mech 5(6):759–790

    MATH  MathSciNet  Google Scholar 

  • Stampfli J, Goodman V (2001) The Mathematics of Finance, Modeling and Hedgeing. Thomas Learning

  • Valkov R (2014) Fitted finite volume method for a generalized Black-Scholes equation transformed on finite interval. Numer Algor 65:195–220

    Article  MATH  MathSciNet  Google Scholar 

  • Wilmott P, Howison S, Dewynne J (1995) The mathematics of financial derivatives. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Wang S (2004) A novel fitted volume method for the Black-Scholes equation governing option pricing. IMA J Numer Anal 24:699–720

    Article  MathSciNet  Google Scholar 

  • Zhou HJ, Yiu KFC, Li LK (2011) Evaluating American put options on zero-coupon bonds by a penalty method. J Comput Appl Math 235:3921–3931

    Article  MATH  MathSciNet  Google Scholar 

  • Zhu Y-I, Wu X (2004) Derivative securities and difference methods. Springer, Berlin

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The authors are supported by the Sofia University Foundation under Grant No 106/2013. The first author is also supported by the European Social Fund through the Human Resource Development Operational Programme under contract BG051PO001-3.3.06-0052 (2012/2014). The second author is also supported by the Bulgarian National Fund under Project DID 02/37/09.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria

    T. Chernogorova & R. Valkov

Authors
  1. T. Chernogorova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Valkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Valkov.

Additional information

Communicated by Josselin Garnier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chernogorova, T., Valkov, R. Analysis of a finite volume element method for a degenerate parabolic equation in the zero-coupon bond pricing. Comp. Appl. Math. 34, 619–646 (2015). https://doi.org/10.1007/s40314-014-0128-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-014-0128-9

Keywords

Mathematics Subject Classification

Search

Navigation