Least Square Regression Methods for Bermudan Derivatives and Systems of Functions

  • Shigeo KusuokaEmail author
  • Yusuke Morimoto
Part of the Advances in Mathematical Economics book series (MATHECON, volume 19)


Least square regression methods are Monte Carlo methods to solve non-linear problems related to Markov processes and are widely used in practice. In these methods, first we choose a system of functions to approximate value functions. So one of questions on these methods is what kinds of systems of functions one has to take to get a good approximation. In the present paper, we will discuss on this problem.


Computational finance Option pricing Malliavin calculus Least square regression methods 


  1. 1.
    Bally V, Pagés G (2003) A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9:1003–1049MathSciNetCrossRefGoogle Scholar
  2. 2.
    Belomestny D (2011) Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates. Financ Stoch 15:655–683MathSciNetCrossRefGoogle Scholar
  3. 3.
    Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Lecture notes in mathematics, vol 580. Springer, Berlin/New YorkGoogle Scholar
  4. 4.
    Clement E, Lamberton D, Protter P (2002) An analysis of a least squares regression algorithm for American option pricing. Financ Stoch 6:449–471MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gobet E, Lemor J-P, Warin X (2005) A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann Appl Probab 15:2172–2202MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kusuoka S, Morimoto Y (2014) Stochastic mesh methods for Hörmander type diffusion processes. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 18. Springer, Tokyo Heidelberg New York Dordrecht London, pp 61–99Google Scholar
  7. 7.
    Kusuoka S, Stroock DW (1985) Applications of Malliavin calculus II. J Fac Sci Univ Tokyo Sect IA Math 32:1–76MathSciNetGoogle Scholar
  8. 8.
    Longstaff F, Schwartz E (2001) Valuing American options by simulation: a simple least-squares approach. Rev Financ Stud 14:113–147CrossRefGoogle Scholar
  9. 9.
    Tsitsiklis J, Van Roy B (1999) Regression methods for pricing complex American style options. IEEE Trans Neural Netw 12:694–703CrossRefGoogle Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  2. 2.Bank of Tokyo Mitsubishi UFJTokyoJapan

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