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Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation

  • 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)

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

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Paper provided by Center of Finance and Econometrics, University of Konstanz in its series CoFE Discussion Paper with number 04-02.

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Length: 16 pages
Date of creation: Jan 2004
Date of revision:
Handle: RePEc:knz:cofedp:0402
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  1. Gennotte, Gerard & Leland, Hayne, 1990. "Market Liquidity, Hedging, and Crashes," American Economic Review, American Economic Association, vol. 80(5), pages 999-1021, December.
  2. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  3. 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.
  4. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141.
  5. George M. Constantinides & Thaleia Zariphopoulou, . "Bounds on Prices of Contingent Claims in an Intertemporal Economy with Proportional Transaction Costs and General Preferences," CRSP working papers 347, Center for Research in Security Prices, Graduate School of Business, University of Chicago.
  6. 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.
  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. 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.
  9. 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.
  10. 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.
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