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Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs

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  • W. Li

    (University of Western Australia)

  • S. Wang

    (University of Western Australia)

Abstract

We present a novel penalty approach to the Hamilton-Jacobi-Bellman (HJB) equation arising from the valuation of European options with proportional transaction costs. We first approximate the HJB equation by a quasilinear 2nd-order partial differential equation containing two linear penalty terms with penalty parameters λ 1 and λ 2 respectively. Then, we show that there exists a unique viscosity solution to the penalized equation. Finally, we prove that, when both λ 1 and λ 2 approach infinity, the viscosity solution to the penalized equation converges to that of the corresponding original HJB equation.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:2:d:10.1007_s10957-009-9559-7
    DOI: 10.1007/s10957-009-9559-7
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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. Monoyios, Michael, 2004. "Option pricing with transaction costs using a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 28(5), pages 889-913, February.
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    4. Clewlow, Les & Hodges, Stewart, 1997. "Optimal delta-hedging under transactions costs," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1353-1376, June.
    5. Edirisinghe, Chanaka & Naik, Vasanttilak & Uppal, Raman, 1993. "Optimal Replication of Options with Transactions Costs and Trading Restrictions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(1), pages 117-138, March.
    6. 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.
    7. Bernard Bensaid & Jean‐Philippe Lesne & Henri Pagès & José Scheinkman, 1992. "Derivative Asset Pricing With Transaction Costs1," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 63-86, April.
    8. 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.
    9. Damgaard, Anders, 2006. "Computation of reservation prices of options with proportional transaction costs," Journal of Economic Dynamics and Control, Elsevier, vol. 30(3), pages 415-444, March.
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    Cited by:

    1. Roy Cerqueti & Daniele Marazzina & Marco Ventura, 2016. "Optimal Investment in Research and Development Under Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 296-309, January.
    2. Wen Li & Song Wang, 2014. "A numerical method for pricing European options with proportional transaction costs," Journal of Global Optimization, Springer, vol. 60(1), pages 59-78, September.
    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.
    4. Song Wang, 2015. "A penalty approach to a discretized double obstacle problem with derivative constraints," Journal of Global Optimization, Springer, vol. 62(4), pages 775-790, August.
    5. Y. Zhou & S. Wang & X. Yang, 2014. "A penalty approximation method for a semilinear parabolic double obstacle problem," Journal of Global Optimization, Springer, vol. 60(3), pages 531-550, November.
    6. Yan, Dong & Lin, Sha & Hu, Zhihao & Yang, Ben-Zhang, 2022. "Pricing American options with stochastic volatility and small nonlinear price impact: A PDE approach," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    7. Roland Herzog & Karl Kunisch & Jörn Sass, 2013. "Primal-dual methods for the computation of trading regions under proportional transaction costs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 101-130, February.
    8. K. Zhang & K. Teo & M. Swartz, 2014. "A Robust Numerical Scheme For Pricing American Options Under Regime Switching Based On Penalty Method," Computational Economics, Springer;Society for Computational Economics, vol. 43(4), pages 463-483, April.
    9. Lu, Xiaoping & Yan, Dong & Zhu, Song-Ping, 2022. "Optimal exercise of American puts with transaction costs under utility maximization," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    10. 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.

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