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Free boundary and optimal stopping problems for American Asian options

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Abstract

We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.

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References

  1. Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance theory. In: Rogers, L., Talay, D. (eds.) Numerical Methods in Finance, pp. 1–21. Publ. Newton Inst. Cambridge Univ. Press, Cambridge (1997)

    Google Scholar 

  2. Barraquand, J., Pudet, T.: Pricing of American path-dependent contingent claims. Math. Finance 6, 17–51 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barucci, E., Polidoro, S., Vespri, V.: Some results on partial differential equations and Asian options. Math. Models Methods Appl. Sci. 11, 475–497 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ben-Ameur, H., Breton, M., L’Ecuyer, P.: A dynamic programming procedure for pricing American-style Asian options. Manag. Sci. 48, 625–643 (2002)

    Article  Google Scholar 

  5. Bensoussan, A.: On the theory of option pricing. Acta Appl. Math. 2, 139–158 (1984)

    MATH  MathSciNet  Google Scholar 

  6. Bensoussan, A., Lions, J.-L.: Applications of Variational Inequalities in Stochastic Control. Studies in Mathematics and its Applications, vol. 12. North-Holland, Amsterdam (1982). Translated from the French

    MATH  Google Scholar 

  7. Bramanti, M., Cerutti, M.C., Manfredini, M.: L p estimates for some ultraparabolic operators with discontinuous coefficients. J. Math. Anal. Appl. 200, 332–354 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dai, M., Kwok, Y.K.: Characterization of optimal stopping regions of American Asian and lookback options. Math. Finance 16, 63–82 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Di Francesco, M., Pascucci, A.: On a class of degenerate parabolic equations of Kolmogorov type. AMRX Appl. Math. Res. Express 2005, 77–116 (2005)

    Article  MathSciNet  Google Scholar 

  10. Di Francesco, M., Pascucci, A., Polidoro, S.: The obstacle problem for a class of hypoelliptic ultraparabolic equations. Preprint AMS Acta (available on-line at http://www.dm.unibo.it/matecofin) (2007)

  11. Foschi, P., Pascucci, A.: Path dependent volatility. Preprint AMS Acta (available on-line at http://www.dm.unibo.it/matecofin) (2007)

  12. Friedman, A.: Stochastic Differential Equations and Applications, vol. 1. Probability and Mathematical Statistics, vol. 28. Academic [Harcourt Brace Jovanovich Publishers], New York (1975)

    MATH  Google Scholar 

  13. Friedman, A.: Variational Principles and Free-Boundary Problems. Pure and Applied Mathematics. Wiley, New York (1982)

    MATH  Google Scholar 

  14. Fu, M.C., Wu, R.: Optimal exercise policies and simulation-based valuation for American-Asian options. Oper. Res. 51, 52–66 (2003)

    Article  MathSciNet  Google Scholar 

  15. Hansen, A.T., Jørgensen, P.L.: Analytical valuation of American-style Asian options. Manag. Sci. 46, 1116–1136 (2000)

    Article  Google Scholar 

  16. Hobson, D.G., Rogers, L.C.G.: Complete models with stochastic volatility. Math. Finance 8, 27–48 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  18. Huang, K., Thulasiram, R.K.: Parallel algorithm for pricing American Asian options with multi-dimensional assets. In: 19th Annual International Symposium on High Performance Computing Systems and Applications, 15–18 May 2005, Guelph, Ontario, Canada, pp. 177–185. IEEE Computer Society (2005)

  19. Ingersoll, J.E.: Theory of Financial Decision Making. Blackwell, Oxford (1987)

    Google Scholar 

  20. Jaillet, P., Lamberton, D., Lapeyre, B.: Variational inequalities and the pricing of American options. Acta Appl. Math. 21, 263–289 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jiang, L., Dai, M.: Convergence of binomial tree methods for European/American path-dependent options. SIAM J. Numer. Anal. 42, 1094–1109 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Karatzas, I.: On the pricing of American options. Appl. Math. Optim. 17, 37–60 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)

    MATH  Google Scholar 

  24. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Classics in Applied Mathematics, vol. 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). Reprint of the 1980 original

    MATH  Google Scholar 

  25. Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino 52, 29–63 (1994)

    MATH  MathSciNet  Google Scholar 

  26. Manfredini, M., Polidoro, S.: Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1, 651–675 (1998)

    MATH  MathSciNet  Google Scholar 

  27. Marcozzi, M.D.: On the valuation of Asian options by variational methods.. SIAM J. Sci. Comput. 24, 1124–1140 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Meyer, G.H.: On pricing American and Asian options with PDE methods. Acta Math. Univ. Comenian. (N.S.) 70, 153–165 (2000)

    MathSciNet  Google Scholar 

  29. Parrott, K., Clarke, N.: A parallel solution of early exercise Asian options with stochastic volatility. In: Greenwich, M.C., Lai, C.-H., Bjørstad, P.E., Widlund, O. (eds.) Proceedings of the Eleventh Domain Decomposition Conference, pp. 439–446 (1998)

  30. Rogers, L., Shi, Z.: The value of an Asian option. J. Appl. Probab. 32, 1077–1088 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Stroock, D.W., Varadhan, S.: Diffusion processes. I: Existence and uniqueness. Commun. Pure Appl. Math. 22, 345–400 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  32. van Moerbeke, P.: On optimal stopping and free boundary problems. Arch. Ration. Mech. Anal. 60, 101–148 (1975/76)

    Google Scholar 

  33. Wu, L., Kwok, Y.K., Yu, H.: Asian options with the American early exercise feature. Int. J. Theor. Appl. Finance 2, 101–111 (1999)

    Article  MathSciNet  Google Scholar 

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  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Andrea Pascucci

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  1. Andrea Pascucci
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Correspondence to Andrea Pascucci.

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Pascucci, A. Free boundary and optimal stopping problems for American Asian options. Finance Stoch 12, 21–41 (2008). https://doi.org/10.1007/s00780-007-0051-7

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