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Option pricing with transaction costs and a nonlinear Black-Scholes equation

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Author Info

  • Halil Mete Soner

    (Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA)

  • Guy Barles

    (FacultÊ des Sciences et Techniques, UniversitÊ de Tours, Parc de Grandmont, F-37200 Tours, France)

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    Abstract

    In a market with transaction costs, generally, there is no nontrivial portfolio that dominates a contingent claim. Therefore, in such a market, preferences have to be introduced in order to evaluate the prices of options. The main goal of this article is to quantify this dependence on preferences in the specific example of a European call option. This is achieved by using the utility function approach of Hodges and Neuberger together with an asymptotic analysis of partial differential equations. We are led to a nonlinear Black-Scholes equation with an adjusted volatility which is a function of the second derivative of the price itself. In this model, our attitude towards risk is summarized in one free parameter a which appears in the nonlinear Black-Scholes equation : we provide an upper bound for the probability of missing the hedge in terms of a and the magnitude of the proportional transaction cost which shows the connections between this parameter a and the risk.

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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 2 (1998)
    Issue (Month): 4 ()
    Pages: 369-397

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    Handle: RePEc:spr:finsto:v:2:y:1998:i:4:p:369-397

    Note: received: March 1996; final version received: May 1997
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    Web page: http://www.springerlink.com/content/101164/

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    Related research

    Keywords: Transaction costs; options; viscosity solutions; dynamic programming;

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    Citations

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    Cited by:
    1. Dylan Possamai & Nizar Touzi & H. Mete Soner, 2012. "Large liquidity expansion of super-hedging costs," Papers 1208.3785, arXiv.org.
    2. Jiatu Cai & Masaaki Fukasawa, 2014. "Asymptotic replication with modified volatility under small transaction costs," Papers 1408.5677, arXiv.org.
    3. Daniel Sevcovic, 2007. "An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation," Papers 0710.5301, arXiv.org.
    4. Dylan Possama\"i & H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs: the multidimensional case," Papers 1212.6275, arXiv.org, revised Jan 2013.
    5. Jan Kallsen & Johannes Muhle-Karbe, 2012. "Option Pricing and Hedging with Small Transaction Costs," Papers 1209.2555, arXiv.org, revised Dec 2012.
    6. Bertram Düring & Michel Fournié & Ansgar Jüngel, 2004. "Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation," CoFE Discussion Paper, Center of Finance and Econometrics, University of Konstanz 04-02, Center of Finance and Econometrics, University of Konstanz.
    7. H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs," Papers 1202.6131, arXiv.org, revised Jun 2013.
    8. Maxim Bichuch, 2011. "Pricing a Contingent Claim Liability with Transaction Costs Using Asymptotic Analysis for Optimal Investment," Papers 1112.3012, arXiv.org.
    9. H. Mete Soner & Umut Cetin & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," LSE Research Online Documents on Economics, London School of Economics and Political Science, LSE Library 28992, London School of Economics and Political Science, LSE Library.
    10. Daniel Sevcovic, 2008. "Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Scholes equations," Papers 0805.0611, arXiv.org.
    11. Olivier Gu\'eant & Jiang Pu & Guillaume Royer, 2013. "Accelerated Share Repurchase: pricing and execution strategy," Papers 1312.5617, arXiv.org, revised Jan 2014.
    12. Bertram Düring & Michel Fournié & Ansgar Jüngel, 2001. "High order compact finite difference schemes for a nonlinear Black-Scholes equation," CoFE Discussion Paper, Center of Finance and Econometrics, University of Konstanz 01-07, Center of Finance and Econometrics, University of Konstanz.
    13. Jouini, Elyès, 2001. "Arbitrage and control problems in finance: A presentation," Economics Papers from University Paris Dauphine 123456789/5590, Paris Dauphine University.
    14. Lai, Tze Leung & Lim, Tiong Wee, 2009. "Option hedging theory under transaction costs," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 33(12), pages 1945-1961, December.
    15. Umut Çetin & H. Soner & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," Finance and Stochastics, Springer, Springer, vol. 14(3), pages 317-341, September.
    16. Valeri Zakamouline, 2004. "A Unified Approach to Portfolio Optimization with Linear Transaction Costs," GE, Growth, Math methods, EconWPA 0404003, EconWPA, revised 21 Apr 2004.
    17. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2013. "Hedging under an expected loss constraint with small transaction costs," Papers 1309.4916, arXiv.org, revised Sep 2014.
    18. Olivier Gu\'eant & Jiang Pu, 2013. "Option pricing and hedging with execution costs and market impact," Papers 1311.4342, arXiv.org, revised Aug 2014.
    19. Ale\v{s} \v{C}ern\'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy models and the time step equivalent of jumps," Papers 1309.7833, arXiv.org, revised Nov 2013.
    20. Golbabai, A. & Ballestra, L.V. & Ahmadian, D., 2013. "Superconvergence of the finite element solutions of the Black–Scholes equation," Finance Research Letters, Elsevier, Elsevier, vol. 10(1), pages 17-26.

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