Finance and Stochastics

, Volume 15, Issue 4, pp 655–683 | Cite as

Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates



The problem of pricing Bermudan options using simulations and nonparametric regression is considered. We derive optimal nonasymptotic bounds for the low biased estimate based on a suboptimal stopping rule constructed from some estimates of the optimal continuation values. These estimates may be of different nature, local or global, with the only requirement being that the deviations of these estimates from the true continuation values can be uniformly bounded in probability. As an illustration, we discuss a class of local polynomial estimates which, under some regularity conditions, yield continuation values estimates possessing the required property.


Bermudan options Nonparametric regression Boundary condition Suboptimal stopping rules 

Mathematics Subject Classification (2000)

62G08 65C05 60G40 

JEL Classification

G10 G12 G13 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersen, L.: A simple approach to the pricing of Bermudan swaptions in the multi-factor LIBOR market model. J. Comput. Finance 3, 5–32 (2000) Google Scholar
  2. 2.
    Audibert, J.-Y., Tsybakov, A.: Fast learning rates for plug-in classifiers. Ann. Stat. 35, 608–633 (2007) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Belomestny, D., Milstein, G.N., Spokoiny, V.: Regression methods in pricing American and Bermudan options using consumption processes. Quant. Finance 9, 315–327 (2006) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Broadie, M., Glasserman, P.: Pricing American-style securities using simulation. J. Econ. Dyn. Control 21, 1323–1352 (1997) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Carrière, J.: Valuation of early-exercise price of options using simulations and nonparametric regression. Insur. Math. Econ. 19, 19–30 (1996) CrossRefMATHGoogle Scholar
  6. 6.
    Clément, E., Lamberton, D., Protter, P.: An analysis of a least squares regression algorithm for American option pricing. Finance Stoch. 6, 449–471 (2002) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Devroye, L., Györfi, L., Lugosi, G.: A Probabilistic Theory of Pattern Recognition. Springer, Berlin (1996) MATHGoogle Scholar
  8. 8.
    Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999) CrossRefMATHGoogle Scholar
  9. 9.
    Egloff, D.: Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15, 1396–1432 (2005) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Egloff, D., Kohler, M., Todorovic, N.: A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Probab. 17, 1138–1171 (2007) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964) MATHGoogle Scholar
  12. 12.
    Giné, E., Guillou, A.: On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals. Ann. Inst. Henri Poincaré B, Probab. Stat. 37, 503–522 (2001) CrossRefMATHGoogle Scholar
  13. 13.
    Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, Berlin (2004) MATHGoogle Scholar
  14. 14.
    Glasserman, P., Yu, B.: Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14, 2090–2119 (2004) MATHMathSciNetGoogle Scholar
  15. 15.
    Kohler, M., Krzyżak, A., Todorovic, N.: Pricing of high-dimensional American options by neural networks. Math. Finance 20, 383–410 (2010) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. Transl. Math. Monogr., vol. 23. Amer. Math. Soc., Providence (1968) Google Scholar
  17. 17.
    Longstaff, F., Schwartz, E.: Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14, 113–147 (2001) CrossRefGoogle Scholar
  18. 18.
    Mammen, E., Tsybakov, A.: Smooth discrimination analysis. Ann. Stat. 27, 1808–1829 (1999) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991) MATHGoogle Scholar
  20. 20.
    Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, 28–76 (1994) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Tsitsiklis, J., Van Roy, B.: Regression methods for pricing complex American style options. IEEE Trans. Neural Netw. 12, 694–703 (1999) CrossRefGoogle Scholar
  22. 22.
    Van Roy, B.: On regression-based stopping times. Discrete Event Dyn. Syst. 20, 307–324 (2009) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

Personalised recommendations