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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)

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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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Publisher 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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  1. Gennotte, Gerard & Leland, Hayne, 1990. "Market Liquidity, Hedging, and Crashes," American Economic Review, American Economic Association, vol. 80(5), pages 999-1021, December. [Downloadable!] (restricted)
    Other versions:
  2. Leland, Hayne E, 1985. " Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December. [Downloadable!] (restricted)
    Other versions:
  3. E. Platen & M. Schweizer, . "On Feedback Effects from Hedging Derivatives," Sonderforschungsbereich 373 1997-83, Humboldt Universitaet Berlin.
  4. 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.
    Other versions:
  5. 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 and Francis Journals, vol. 6(2), pages 87-106, June. [Downloadable!] (restricted)
  6. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring. [Downloadable!] (restricted)
  7. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141. [Downloadable!] (restricted)
  8. 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. [Downloadable!]
  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. [Downloadable!] (restricted)
  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. [Downloadable!] (restricted)
  11. K. Ronnie Sircar, George Papanicolaou, 1998. "General Black-Scholes models accounting for increased market volatility from hedging strategies," Applied Mathematical Finance, Taylor and Francis Journals, vol. 5(1), pages 45-82, March. [Downloadable!] (restricted)
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