On the methods of pricing American options: case study

S.I.: Advances of OR in Commodities and Financial Modelling
  • 151 Downloads

Abstract

In this study, a comparative analysis of numerical and approximation methods for pricing American options is performed. Binomial and finite difference approximations are discussed; furthermore, Roll-Geske-Whaley, Barone-Adesi and Whaley and Bjerksund-Stensland analytical approximations as well as the least-squares Monte Carlo method of Longstaff and Schwartz are presented. Applicability and efficiency in almost all circumstances, numerical solutions of the corresponding free boundary problem is emphasized. Methods used in pricing American options are also compared on dividend and non-dividend paying assets; and their pros and cons are discussed along with numerical experiments.

Keywords

American options Numerical methods Analytical approximations Bounds 

References

  1. Andersen, L., & Broadie, M. (2004). Primal-dual simulation algorithm for pricing multidimensional American options. Management Science, 50, 1222–1234.CrossRefGoogle Scholar
  2. Averbukh, V.: Pricing American options using Monte Carlo simulation. Ph.D. dissertation, Cornell University (1997)Google Scholar
  3. Barone-Adesi, G., & Whaley, R. E. (1987). Efficient analytic approximation of American options values. The Journal of Finance, 42(2), 301–320.CrossRefGoogle Scholar
  4. Barraquand, J., & Martineau, D. (1995). Numerical valuation of high dimensional multivariate American securities. Journal of Financial and Quantitative Analysis, 30, 383–405.CrossRefGoogle Scholar
  5. Bjerksund, P., Stensland, G. (1992). American exchange options and a put-call transformation. Working paper no. 4/92, Institute of Finance and Management Science, Norwegian School of Economics and Business Administration.Google Scholar
  6. Bjerksund, P., & Stensland, G. (1993). Closed-form approximation of American options. Scandinavian Journal of Management, 9, 87–99.CrossRefGoogle Scholar
  7. Bjerksund, P., Stensland, G. (2002). Closed-form valuation of American options. Working Paper NHH.Google Scholar
  8. Black, F. (1975). Fact and fantasy in the use of options. Financial Analysts Journal, 31(4), 36–72.CrossRefGoogle Scholar
  9. Bossaerts, P. (1989). Simulation estimators of optimal early exercise. Working paper, Carnegie-Mellon University.Google Scholar
  10. Boyle, P. P. (1988). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 23, 1–12.CrossRefGoogle Scholar
  11. Brennan, M., & Schwartz, E. (1978). Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis. Journal of Financial and Quantitative Analysis, 13(3), 461–474.CrossRefGoogle Scholar
  12. Carr, P. (1998). Randomization and the American put. Review of Financial Studies, 11, 597–626.CrossRefGoogle Scholar
  13. Carriere, J. (1996). Valuation of early-exercise price of options using simulations and nonparametric regression. Insurance: Mathematics and Economics, 19, 19–30.Google Scholar
  14. Courtadon, G. (1982). A more accurate finite difference approximation for the valuation of options. Journal of Financial and Quantitative Analysis, 17, 697–705.CrossRefGoogle Scholar
  15. Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). On the valuation of American call options on stocks with known dividends. Journal of Financial Economics, 7, 229–264.CrossRefGoogle Scholar
  16. Crank, J., & Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proceedings of the Cambridge Philosophical Society, 43(1), 50–67.CrossRefGoogle Scholar
  17. Cryer, C. W. (1971). The solution of a quadratic programming problem using systematic overrelaxation. SIAM Journal on Control and Optimization, 9(3), 385–392.CrossRefGoogle Scholar
  18. Garcia, D. (1999). A Monte Carlo method for pricing American options. Working paper, University of California, Berkeley.Google Scholar
  19. Geske, R. (1979). The valuation of compound options. Journal of Financial Economics, 7, 63–81.CrossRefGoogle Scholar
  20. Geske, R. (1981). Comments on Whaley’s note. Journal of Financial Economics, 9(2), 213–215.CrossRefGoogle Scholar
  21. Gilli, M., Schumann, E. (2009). Implementing binomial trees. Comisef Working Paper Series.Google Scholar
  22. Haug, E. G., Haug, J., & Lewis, A. (2003). Back to basics: A new approach to the discrete dividend problem. London: Wilmott Magazine.Google Scholar
  23. Haugh, M., & Kogan, L. (2004). Pricing American options: A duality approach. Operations Research, 52, 258–270.CrossRefGoogle Scholar
  24. Ibanez, A., Zapatero, F. (1998). Monte Carlo valuation of American options through computation of the optimal exercise frontier. Working paper, University of Southern California.Google Scholar
  25. Kozpinar, S. (2013). Pricing American options under discrete and continuous time setting. M.Sc. Thesis, Middle East Technical University.Google Scholar
  26. Kwok, Y. K. (2008). Mathematical Models of Financial Derivatives. Berlin: Springer Finance.Google Scholar
  27. Longstaff, F. A., & Schwartz, E. S. (2001). Valuing American options by simulation: A simple least-squares approach. The Society for Financial Studies, 14(1), 113–147.CrossRefGoogle Scholar
  28. Morton, K. W., & Mayers, D. F. (1995). Numerical solutions of partial differential equations. Cambridge: Cambridge University Press.Google Scholar
  29. Protter, P., Clement, E., & Lamberton, D. (2002). An analysis of a least squares regression method for American option pricing. Finance and Stochastics, 6(4), 449–471.CrossRefGoogle Scholar
  30. Rogers, L. (2002). Monte Carlo valuation of American options. Mathematical Finance, 3, 271–286.CrossRefGoogle Scholar
  31. Roll, R. (1977). An analytic valuation formula for unprotected American call options on stocks with known dividends. Journal of Financial Economics, 5, 251–258.CrossRefGoogle Scholar
  32. Schwartz, E. S. (1977). The valuation of warrants: Implementing a new approach. Journal of Financial Economics, 13, 79–93.CrossRefGoogle Scholar
  33. Tilley, J. A. (1993). Valuing American options in a path simulation model. Transactions of the Society of the Actuaries, 45, 83–104.Google Scholar
  34. Tsitsiklis, J., & Van Roy, B. (1999). Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Transactions on Automatic Control, 44, 1840–1851.CrossRefGoogle Scholar
  35. Uğur, Ö. (2009). An introduction to computational finance. London: Imperial College Press.Google Scholar
  36. Whaley, R. E. (1981). On the valuation of American call options on stocks with known dividends. Journal of Financial Economics, 9, 207–211.CrossRefGoogle Scholar
  37. Wu, Z. (2012). Pricing American options using Monte Carlo method. M.Sc. Thesis, University of Oxford.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Applied MathematicsMiddle East Technical UniversityÇankaya, AnkaraTurkey
  2. 2.Department of MathematicsAtilim UniversityIncek Gölbaşı, AnkaraTurkey

Personalised recommendations