Skip to main content
SpringerLink
Log in
Book cover

International Conference on Numerical Analysis and Its Applications

NAA 2012: Numerical Analysis and Its Applications pp 363–370Cite as

Positivity Preserving Numerical Method for Non-linear Black-Scholes Models

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8236))

Abstract

A motivation for studying the nonlinear Black- Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor’s preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. In this work we develop positivity preserving algorithm for solving a large class of non-linear models in mathematical finance on the original (infinite) domain. Numerical examples are discussed.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abe, R., Ishimura, N.: Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem. Proc. Japan Acad. Ser. A 84, 11–14 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agliardi, R., Popivanov, P., Slavova, A.: Nonhypoellipticity and comparisson principle for partial differential equations of Black-Scholes type, Nonlin. Analisys: Real Word Appl. 12, 1429–1436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alshina, E., Kalitkin, N., Panchenko, S.: Numerical solution of boundary value problem in unlimited area. Math. Modelling 14(11), 10–22 (2002) (in Russian)

    Google Scholar 

  4. Ankudinova, J., Ehrhardt, M.: On tne numerical solution of nonlinear Black-Scholes equations. Int. J. of Comp. and Math. with Appl. 56, 779–812 (2008)

    MathSciNet  Google Scholar 

  5. Avellaneda, M., Parás, A.: Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model. Appl. Math. Fin. 3(1), 21–52 (1996)

    Article  MATH  Google Scholar 

  6. Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance and Stochastics 2(4), 369–397 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Company, R., Navarro, E., Pintos, J., Ponsoda, E.: Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Int. J. of Comp. and Math. with Appl. 56, 813–821 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Düring, B., Fournié, M., Jüngel, A.: Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation. ESAIM: M2AN 38(2), 359–369 (2004)

    Article  MATH  Google Scholar 

  9. Faragó, I., Komáromi, N.: Nonnegativity of the numerical solution of parabolic problems, Colloquia matematica societatis János Bolyai. Numer. Meth. 59, 173–179 (1990)

    Google Scholar 

  10. Frey, R., Patie, P.: Risk management for derivatives in illiquid markets: a simulation-study. In: Adv. in Fin. and Stoch., pp. 137–159. Springer, Berlin (2002)

    Chapter  Google Scholar 

  11. Gerisch, A., Griffiths, D.F., Weiner, R., Chaplain, M.A.J.: A positive splitting method for mixed hyperbolic-parabolic systems, Num. Meth. for PDEs 17(2), 152–168 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heider, P.: Numerical Methods for Non-Linear Black-Scholes Equations. Appl. Appl. Math. Fin. 17(1), 59–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, Heidelberg (2003)

    Book  MATH  Google Scholar 

  14. Ishimura, N., Koleva, M.N., Vulkov, L.G.: Numerical solution of a nonlinear evolution equation for the risk preference. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 445–452. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  15. Jandačka, M., Ševčovič, D.: On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J. of Appl. Math. 3, 235–258 (2005)

    Article  Google Scholar 

  16. Knabner, P., Angerman, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer (2003)

    Google Scholar 

  17. Koleva, M.N., Vulkov, L.G.: A kernel-based algorithm for numerical solution of nonlinear pDEs in finance. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2011. LNCS, vol. 7116, pp. 566–573. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  18. Kusmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. of Comp. Phys. 198(1), 131–158 (2004)

    Article  Google Scholar 

  19. Leland, H.E.: Option pricing and replication with transactions costs. J. of Finance 40(5), 1283–1301 (1985)

    Article  Google Scholar 

  20. Lesmana, D., Wang, S.: An upwind finite difference method for a nonlinear Black–Scholes equation governing European option valuation under transaction costs. Applied Math. and Comp. (in press)

    Google Scholar 

  21. LeVeque, R.J.: Numerical Methods for Conservation Laws, Birkhäuser (1992)

    Google Scholar 

  22. MacKinnon, R.J., Carey, G.F.: Positivity-preserving, flux-limited finite-difference and finite-element methods for reactive transport. Numer. Meth. Fluids 41, 151–183 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oosterlee, C.W., Leentvaar, C.W., Huang, X.: Accurate American option pricing by grid stretching and high order finite dfferences, Tech. rep. Delft Institute of Appl. Math., Delft University of Technology, Delft, the Netherlands (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FNSE, University of Rousse, 8 Studentska Str., Rousse, 7017, Bulgaria

    Miglena N. Koleva

Authors
  1. Miglena N. Koleva
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technlogies, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl25A, 1113, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Pázm\’{a}ny P\’{e}ter s\’{e}t\’{a}ny 1/C,, Eötvös Loránd University, 1117, Budapest, Hungary

    István Faragó

  3. University of Rousse, 8 Studentska Str.,, 7017, Rousse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Koleva, M.N. (2013). Positivity Preserving Numerical Method for Non-linear Black-Scholes Models. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-41515-9_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-41514-2

  • Online ISBN: 978-3-642-41515-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation