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

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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.

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References

  1. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Political Econ. 81, 637–659 (1973)

    Article  Google Scholar 

  2. Leland, H.E.: Option pricing and replication with transaction costs. J. Finance 40, 1283–1301 (1985)

    Article  Google Scholar 

  3. Bensaid, B., Lesne, J., Pages, H., Scheinkmann, J.: Derivative asset pricing with transaction costs. Math. Finance 2(2), 63–86 (1992)

    Article  MATH  Google Scholar 

  4. Edirisinghe, C., Naik, V., Uppal, R.: Optimal replication of options with transaction costs and trading restrictions. J. Financ. Quant. Anal. 28, 117–138 (1993)

    Article  Google Scholar 

  5. Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Future Mark. 8, 222–239 (1989)

    Google Scholar 

  6. Davis, M.H.A., Panas, V.G., Zariphopoulou, T.: European option pricing with transaction costs. SIAM J. Control Optim. 31(2), 470–493 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Damgaard, A.: Computation of reservation prices of options with proportional transaction costs. J. Econ. Dyn. Control 30, 415–444 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Monoyios, M.: Option pricing with transaction costs using a Markov chain approximation. J. Econ. Dyn. Control 28, 889–913 (2004)

    Article  MathSciNet  Google Scholar 

  9. Zakamouline, V.: European option pricing and hedging with both fixed and proportional transaction costs. J. Econ. Dyn. Control 30, 1–25 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Damgaard, A.: Utility based option evaluation with proportional transaction costs. J. Econ. Dyn. Control 27, 667–700 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Clewlow, L., Hodge, S.: Optimal delta-hedging under transaction costs. J. Econ. Dyn. Control 21, 1353–1376 (1997)

    Article  MATH  Google Scholar 

  12. Crandall, M.G., Lions, P.L.: Viscosity solution of Hamilton-Jacobi-equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lions, P.L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 1. Commun. Partial Differ. Equ. 8, 1101–1174 (1983)

    Article  MATH  Google Scholar 

  14. Lions, P.L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 2. Commun. Partial Differ. Equ. 8, 1229–1276 (1983)

    Article  MATH  Google Scholar 

  15. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)

    MATH  Google Scholar 

  17. Soner, H.M.: Optimal control with state-space constraints. SIAM J. Control Optim. 24, 552–561 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. Katsoulakis, M.: Viscosity solutions of second order fully nonlinear elliptic equations with state constrains. Indiana Univ. Math. J. 43, 493–518 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Davis, M.H.A., Zariphopoulou, T.: American options and transaction fees. In: Davis, M.H.A., Duffie, D., Fleming, W.A., Shreve, S.E. (eds.) Mathematical Finance, pp. 47–61. Springer, New York (1995)

    Google Scholar 

  20. Rubinov, A.M., Yang, X.Q.: Lagrange-Type Functions in Constrained Non-Convex Optimization. Kluwer Academic Publishers, Dordrecht (2003)

    MATH  Google Scholar 

  21. Wang, S., Yang, X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129, 227–257 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Soner, H.M.: Controlled Markov processes, viscosity solutions and applications to mathematical finance. In: Capuzzo Dolcetta, I.C., Lions, P.L. (eds.) Viscosity Solutions and Applications. Springer, New York (1995)

    Google Scholar 

  23. Zatiphopoulou, T.: Investment-consumption models with transaction fees and Markov-chain parameters. SIAM J. Control Optim. 30, 613–636 (1992)

    Article  MathSciNet  Google Scholar 

  24. Bardi, M., Capuzzo-Dolcetta, J.: Optimal Control and Viscosity Solution of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997)

    Book  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics & Statistics, University of Western Australia, 35 Stirling Hwy, Crawley, WA, 6009, Australia

    W. Li & S. Wang

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  1. W. Li
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  2. S. Wang
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Correspondence to S. Wang.

Additional information

Communicated by X.Q. Yang.

We thank Professor C.J. Goh for several constructive suggestions.

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Li, W., Wang, S. Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs. J Optim Theory Appl 143, 279–293 (2009). https://doi.org/10.1007/s10957-009-9559-7

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  • DOI: https://doi.org/10.1007/s10957-009-9559-7

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