Skip to main content
SpringerLink
Log in

Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.

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

Similar content being viewed by others

References

  1. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)

    Article  Google Scholar 

  2. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Florescu, I., Mariani, M.C.: Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market. Electron. J. Differ. Equ. 2010(62), 1–10 (2010)

    MathSciNet  Google Scholar 

  4. Florescu, I., Viens, F.: Stochastic volatility: option pricing using a multinomial recombining tree. Appl. Math. Finance 15(2), 151–181 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hagan, P., Kumar, D., Lesniewski, A., Woodward, D.: Managing smile risk. Wilmott Magazine (2002)

  6. Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11(3), 215–260 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hoggard, T., Whalley, A.E., Wilmott, P.: Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Res. 7, 21 (1994)

    Google Scholar 

  8. Hull, J.C., White, A.D.: The pricing of options on assets with stochastic volatilities. J. Finance 42(2), 281–300 (1987)

    Article  Google Scholar 

  9. Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Amer. Math. Soc., Providence (2008)

    MATH  Google Scholar 

  10. Leland, H.E.: Option pricing and replication with transaction costs. J. Finance 40, 1283–1301 (1985)

    Article  Google Scholar 

  11. Mariani, M.C., SenGupta, I.: Nonlinear problems modeling stochastic volatility and transaction costs. Quant. Finance (2011, to appear)

  12. Mariani, M.C., Florescu, I., Beccar Varela, M.P., Ncheuguim, E.: Long correlations and Levy models applied to the study of memory effects in high frequency (tick) data. Physica A 388(8), 1659–1664 (2009)

    Article  Google Scholar 

  13. Mariani, M.C., Florescu, I., Beccar Varela, M.P., Ncheuguim, E.: Study of memory effects in international market indices. Physica A 389(8), 1653–1664 (2010)

    Article  Google Scholar 

  14. SenGupta, I.: Differential operator related to the generalized superradiance integral equation. J. Math. Anal. Appl. 369, 101–111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. SenGupta, I.: Spectral analysis for a three-dimensional superradiance problem. J. Math. Anal. Appl. 375, 762–776 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wiggins, J.B.: Option values under stochastic volatility: Theory and empirical estimates. J. Financ. Econ. 19(2), 351–372 (1987)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, USA

    Maria C. Mariani, Indranil SenGupta & Pavel Bezdek

Authors
  1. Maria C. Mariani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Indranil SenGupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pavel Bezdek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria C. Mariani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mariani, M.C., SenGupta, I. & Bezdek, P. Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility. Acta Appl Math 118, 203–220 (2012). https://doi.org/10.1007/s10440-012-9685-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-012-9685-3

Keywords

Search

Navigation