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Smooth Upper Bounds For The Price Function Of American Style Options

Author

Listed:
  • LOUIS BHIM

    (Optiver Asia Pacific, 39 Hunter St., Sydney, NSW 2000, Australia)

  • REIICHIRO KAWAI

    (School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia)

Abstract

We introduce a new approach for systematically obtaining smooth deterministic upper bounds for the price function of American style options. These bounding functions are characterized by sufficient conditions, under which the bounds may be infimized. In a single implementation, the proposed approach obtains explicit bounds in the form of piecewise polynomial functions, which bound the price function from above over the whole problem domain both in time and state. As a consequence, these global bounds store a continuum of information in the form of a finite number of polynomial coefficients. The proposed approach achieves these bounds, free from statistical error, without recourse to sample path simulation, without truncating the naturally unbounded domain that arises in this problem, and without discretizing the time and state variables. Throughout the paper, we demonstrate the effectiveness of the proposed method in obtaining tight upper bounds for American style option prices in a variety of market models and with various payoff structures, such as the multivariate Black Scholes and Heston stochastic volatility models and the American put and butterfly payoff structures. We also discuss extensions of the proposed methodology to perpetual American style options and frameworks in which the underlying asset contains jumps.

Suggested Citation

  • Louis Bhim & Reiichiro Kawai, 2018. "Smooth Upper Bounds For The Price Function Of American Style Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(01), pages 1-38, February.
    • Handle: RePEc:wsi:ijtafx:v:21:y:2018:i:01:n:s0219024918500097
    DOI: 10.1142/S0219024918500097
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    References listed on IDEAS

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    1. Carriere, Jacques F., 1996. "Valuation of the early-exercise price for options using simulations and nonparametric regression," Insurance: Mathematics and Economics, Elsevier, vol. 19(1), pages 19-30, December.
    2. Nan Chen & Paul Glasserman, 2007. "Additive and multiplicative duals for American option pricing," Finance and Stochastics, Springer, vol. 11(2), pages 153-179, April.
    3. Sam Howison & Christoph Reisinger & Jan Hendrik Witte, 2010. "The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options," Papers 1008.0836, arXiv.org, revised May 2013.
    4. Sören Christensen, 2014. "A Method For Pricing American Options Using Semi-Infinite Linear Programming," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 156-172, January.
    5. Egorova, Yana, 2017. "Инвестирование Денежных Средств В Условиях Экономического Кризиса В 2017 Году," MPRA Paper 77648, University Library of Munich, Germany.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    7. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, vol. 50(9), pages 1222-1234, September.
    8. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    9. M. A. H. Dempster & J. P. Hutton, 1999. "Pricing American Stock Options by Linear Programming," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 229-254, July.
    10. Carl Chiarella & Adam Kucera & Andrew Ziogas, 2004. "A Survey of the Integral Representation of American Option Prices," Research Paper Series 118, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    12. Rafael Company & Vera Egorova & Lucas J'odar & Fazlollah Soleymani, 2017. "Computing stable numerical solutions for multidimensional American option pricing problems: a semi-discretization approach," Papers 1701.08545, arXiv.org.
    13. Kolodko A. & Schoenmakers J., 2004. "Upper Bounds for Bermudan Style Derivatives," Monte Carlo Methods and Applications, De Gruyter, vol. 10(3-4), pages 331-343, December.
    14. Martin B. Haugh & Leonid Kogan, 2004. "Pricing American Options: A Duality Approach," Operations Research, INFORMS, vol. 52(2), pages 258-270, April.
    15. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
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