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Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model

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Abstract

We study the viscosity solutions of integro-differential Hamilton–Jacobi–Bellman equations of degenerate parabolic type. These equations are from the pricing problem for the European passport options in a jump-diffusion model. The passport option is a call option on a trading account. We discuss the mathematical model for pricing problem. We prove the comparison principle, uniqueness and convexity preserving for the viscosity solutions of related pricing equations.

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References

  1. Hyer, T., Lipton-Lifschitz, A., Pugachevsky, D.: Passport to success. Risk 10(9), 127–131 (1997)

    Google Scholar 

  2. Andersen, L., Andreasen, J., Brotherton-Ratcliffe, R.: The passport option. J. Comput. Finance 1(3), 15–36 (1998)

    Google Scholar 

  3. Ahn, H., Penaud, A., Wilmott, P.: Various passport options and their valuation. Appl. Math. Finance 6(4), 275–292 (1998)

    Article  Google Scholar 

  4. Penaud, A., Wilmott, P., Ahn, H.: Exotic passport options. Asia-Pac. Financ. Mark. 6(2), 171–182 (1999)

    Article  MATH  Google Scholar 

  5. Nagayama, I.: Pricing of passport option. J. Math. Sci. Univ. Tokyo 5(4), 747–785 (1998)

    MATH  MathSciNet  Google Scholar 

  6. Henderson, V., Hobson, D.: Local time, coupling and the passport option. Finance Stoch. 4(1), 69–80 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Shreve, S.E., Vecer, J.: Options on a traded account: vacation calls vacation puts and passport options. Finance Stoch. 4(3), 255–274 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chan, S.-S.: The valuation of American passport options. Working Paper, University of Wisconsin-Madison (1999)

  9. Hendersen, V., Hobson, D.: Passport options with stochastic volatility. Appl. Math. Finance 8(2), 97–119 (2001)

    Article  Google Scholar 

  10. Henderson, V., Hobson, D., Kentwell, G.: A new class of commodity hedging strategies: a passport options approach. Int. J. Theor. Appl. Finance 5(3), 255–278 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Vecer, J.: A new PDE approach for pricing arithmetic average Asian options. J. Comput. Finance 4(4), 105–113 (2001)

    Google Scholar 

  12. Ahn, H., Dewynne, J., Hua, P., Penaud, A., Wilmott, P.: The end-of-the-year bonus: how to optimally reward a trader. Int. J. Theor. Appl. Finance 5(2), 279–306 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cox, J.C., Ross, S.A.: The valuation of options for alternative stochastic processes. J. Financ. Econ. 3(2), 145–166 (1976)

    Article  Google Scholar 

  14. Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1), 125–144 (1976)

    Article  MATH  Google Scholar 

  15. Aase, K.K.: Contingent claims valuation when the security price is a combination of an Itô process and a random point process. Stoch. Process. Appl. 28(2), 185–220 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Naik, V., Lee, M.: General equilibrium pricing of options on the market portfolio with discontinuous returns. Rev. Financ. Stud. 3(4), 493–521 (1990)

    Article  Google Scholar 

  17. Yang, C., Jiang, L., Bian, B.: Free boundary and American options in a jump-diffusion model. Euro. Jnl of Applied Mathematics. 17(1), 95–127 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. 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, vol. 1660, pp. 134–185. Springer, Berlin (1997)

    Chapter  Google Scholar 

  21. Soner, H.M.: Stochastic optimal control in finance. Working paper, Princeton University (2004)

  22. Roßberg, T.: A survey: stochastic control of jump-diffusion-processes. Working paper (2000)

  23. Alvarez, O., Lasry, J.-M., Lions, P.-L.: Convexity viscosity solutions and state constraints. J. Math. Pures Appl. 76(3), 265–288 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Feldman, M., McCuan, J.: Constructing convex solutions via Perron’s method. Ann. Univ. Ferrara 53(1), 65–94 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Giga, Y., Goto, S., Ishii, H., Sato, M.H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443–470 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pham, H.: Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Syst. Estim. Control 8, 1–27 (1998)

    MathSciNet  Google Scholar 

  27. Benth, F.E., Karlsen, H.K., Reikvam, K.: A semilinear Black and Scholes partial differential equation for valuing American options. Finance Stoch. 7(3), 277–298 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Topper, J.: A finite element implementation of passport options. Master’s Thesis, University of Oxford (2003)

  29. Hung, P.Y.: The methods of Monte Carlo and finite difference apply to the passport options. Master’s Thesis, Providence University (2004)

  30. Pooley, D.: Numerical methods for nonlinear equations in option pricing. PhD Thesis, University of Waterloo (2003)

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Acknowledgements

This work was supported in part by the Research Program of Shanghai Normal University (No. SK201211), the Major Project of Shanghai Municipal Education Commission (No. 13ZZ107) and Shanghai Normal University Leading Academic Discipline Project (No. DZW912). The authors would like to thank the reviewers for their very helpful comments and suggestions.

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Authors and Affiliations

  1. Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, P.R. China

    Yang Wang & Jizhou Zhang

  2. Department of Mathematics, Tongji University, Shanghai, 200092, P.R. China

    Baojun Bian

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  1. Yang Wang
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  2. Baojun Bian
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Correspondence to Yang Wang.

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Wang, Y., Bian, B. & Zhang, J. Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model. J Optim Theory Appl 161, 122–144 (2014). https://doi.org/10.1007/s10957-013-0382-9

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