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Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs

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  • Lesmana, Donny Citra
  • Wang, Song

Abstract

We propose a penalty method for a finite-dimensional nonlinear complementarity problem (NCP) arising from the discretization of the infinite-dimensional free boundary/obstacle problem governing the valuation of American options under transaction costs. In this method, the NCP is approximated by a system of nonlinear equations containing a power penalty term. We show that the mapping involved in the system is continuous and strongly monotone. Thus, the unique solvability of both the NCP and the penalty equation and the exponential convergence of the solution to the penalty equation to that of the NCP are guaranteed by an existing theory. Numerical results will be presented to demonstrate the convergence rates and usefulness of this penalty method.

Suggested Citation

  • Lesmana, Donny Citra & Wang, Song, 2015. "Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 318-330.
    • Handle: RePEc:eee:apmaco:v:251:y:2015:i:c:p:318-330
    DOI: 10.1016/j.amc.2014.11.060
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    References listed on IDEAS

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    1. S. Wang & X. Q. Yang & K. L. Teo, 2006. "Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation," Journal of Optimization Theory and Applications, Springer, vol. 129(2), pages 227-254, May.
    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.
    3. Damgaard, Anders, 2003. "Utility based option evaluation with proportional transaction costs," Journal of Economic Dynamics and Control, Elsevier, vol. 27(4), pages 667-700, February.
    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 637-654, May-June.
    5. 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.
    6. Pascal Heider, 2010. "Numerical Methods for Non-Linear Black-Scholes Equations," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(1), pages 59-81.
    7. W. Li & S. Wang, 2009. "Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs," Journal of Optimization Theory and Applications, Springer, vol. 143(2), pages 279-293, November.
    8. Boyle, Phelim P & Vorst, Ton, 1992. "Option Replication in Discrete Time with Transaction Costs," Journal of Finance, American Finance Association, vol. 47(1), pages 271-293, March.
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    Cited by:

    1. Al–Zhour, Zeyad & Barfeie, Mahdiar & Soleymani, Fazlollah & Tohidi, Emran, 2019. "A computational method to price with transaction costs under the nonlinear Black–Scholes model," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 291-301.
    2. Jose Cruz & Daniel Sevcovic, 2020. "On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models," Papers 2003.03851, arXiv.org.
    3. Pedro Polvora & Daniel Sevcovic, 2021. "Utility indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation," Papers 2108.12598, arXiv.org.

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