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High order compact finite difference schemes for a nonlinear Black-Scholes equation

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Author Info

  • Bertram Düring

    ()
    (Department of Mathematics and Informatics, University of Mainz)

  • Michel Fournié

    (Laboratoire MIP, Université Paul Sabatier, Toulouse)

  • Ansgar Jüngel

    ()
    (Department of Mathematics and Informatics, University of Mainz)

Abstract

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more e±cient than the considered classical schemes.

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Bibliographic Info

Paper provided by Center of Finance and Econometrics, University of Konstanz in its series CoFE Discussion Paper with number 01-07.

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Length: 16 pages
Date of creation: Sep 2001
Date of revision:
Handle: RePEc:knz:cofedp:0107

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Related research

Keywords: Option pricing; transaction costs; parabolic equations; compact finite difference discretizations;

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References

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  1. Hayne E. Leland., 1984. "Option Pricing and Replication with Transactions Costs," Research Program in Finance Working Papers 144, University of California at Berkeley.
  2. A. E. Whalley & P. Wilmott, 1997. "An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 307-324.
  3. P. A. Forsyth & K. R. Vetzal & R. Zvan, 1999. "A finite element approach to the pricing of discrete lookbacks with stochastic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(2), pages 87-106.
  4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  5. Gerard Gennotte and Hayne Leland., 1989. "Market Liquidity, Hedging and Crashes," Research Program in Finance Working Papers RPF-184, University of California at Berkeley.
  6. (*), Thaleia Zariphopoulou & George M. Constantinides, 1999. "Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences," Finance and Stochastics, Springer, vol. 3(3), pages 345-369.
  7. Jarrow, Robert A., 1992. "Market Manipulation, Bubbles, Corners, and Short Squeezes," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 27(03), pages 311-336, September.
  8. K. Ronnie Sircar & George Papanicolaou, 1998. "General Black-Scholes models accounting for increased market volatility from hedging strategies," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(1), pages 45-82.
  9. E. Platen & M. Schweizer, 1997. "On Feedback Effects from Hedging Derivatives," SFB 373 Discussion Papers 1997,83, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  10. Bernard Bensaid & Jean-Philippe Lesne & Henri Pagès & José Scheinkman, 1992. "Derivative Asset Pricing With Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 63-86.
  11. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  12. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141.
  13. Boyle, Phelim P & Vorst, Ton, 1992. " Option Replication in Discrete Time with Transaction Costs," Journal of Finance, American Finance Association, vol. 47(1), pages 271-93, March.
  14. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
  15. Avellaneda Marco & ParaS Antonio, 1994. "Dynamic hedging portfolios for derivative securities in the presence of large transaction costs," Applied Mathematical Finance, Taylor & Francis Journals, vol. 1(2), pages 165-194.
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