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A regression-based smoothing spline Monte Carlo algorithm for pricing American options in discrete time

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Abstract

American options in discrete time can be priced by solving optimal stopping problems. This can be done by computing so-called continuation values, which we represent as regression functions defined recursively by using the continuation values of the next time step. We use Monte Carlo to generate data, and then we apply smoothing spline regression estimates to estimate the continuation values from these data. All parameters of the estimate are chosen data dependent. We present results concerning consistency and the estimates’ rate of convergence.

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References

  • Chow, Y.S., Robbins, H., Siegmund, D.: Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston (1971)

    MATH  Google Scholar 

  • Cover, T.: Rates of convergence for nearest neighbor procedures. In: Proceedings of the Hawaii International Conference on System Sciences, Honolulu, HI, pp. 413–415 (1968)

  • Devroye, L.: Necessary and sufficient conditions for the almost everywhere convergence of nearest neighbor regression function estimates. Z. Wahrscheinlichkeitstheor. Verw. Geb. 61, 467–481 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Dippon, J., Winter, S.: Smoothing spline regression estimates for randomly right censored data. Preprint, University of Stuttgart (2006)

  • Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO. Anal. Numér. 10, 5–12 (1976)

    MathSciNet  Google Scholar 

  • Egloff, D.: Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15, 1–37 (2005)

    Article  MathSciNet  Google Scholar 

  • Egloff, D., Kohler, M., Todorovic, N.: A dynamic look-ahead Monte Carlo algorithm for pricing American options. Ann. Appl. Probab. 17, 1138–1171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Eubank, R.L.: Spline Smoothing and Nonparametric Regression. Dekker, New York (1988)

    MATH  Google Scholar 

  • Franke, J., Diagne, M.: Estimating market risk with neural networks. Stat. Decis. 24, 233–253 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, Berlin (2004)

    MATH  Google Scholar 

  • Glasserman, P., Yu, B.: Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14, 1–30 (2004)

    Article  MathSciNet  Google Scholar 

  • Györfi, L., Kohler, M., Krzyżak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. Springer, Berlin (2002)

    MATH  Google Scholar 

  • Kohler, M.: Nonparametric regression with additional measurement errors in the dependent variable. J. Stat. Plan. Inference 136, 3339–3361 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Kohler, M., Krzyżak, A.: Nonparametric regression estimation using penalized least squares. IEEE Trans. Inf. Theory 47, 3054–3058 (2001)

    Article  MATH  Google Scholar 

  • Kohler, M., Krzyżak, A., Schäfer, D.: Application of structural risk minimization to multivariate smoothing spline regression estimates. Bernoulli 8, 475–489 (2002)

    MathSciNet  MATH  Google Scholar 

  • Kohler, M., Kul, S., Mathé, K.: Least squares estimates for random design regression. Preprint, University of Stuttgart (2004)

  • Kohler, M., Krzyżak, A., Todorovic, N.: Pricing of high-dimensional American options by neural networks (2006, submitted)

  • Longstaff, F.A., Schwartz, E.S.: Valuing American options by simulation: a simple least-square approach. Rev. Financ. Stud. 14, 113–147 (2001)

    Article  Google Scholar 

  • Shiryayev, A.N.: Optimal Stopping Rules. Applications of Mathematics. Springer, Berlin (1978)

    MATH  Google Scholar 

  • Stone, C.J.: Optimal global rates of convergence for nonparametric regression. Ann. Stat. 10, 1040–1053 (1982)

    Article  MATH  Google Scholar 

  • Tsitsiklis, J.N., Van Roy, B.: Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Trans. Automat. Contr. 44, 1840–1851 (1999)

    Article  MATH  Google Scholar 

  • van de Geer, S.: A new approach to least squares estimation, with applications. Ann. Stat. 15, 587–602 (1987)

    Article  MATH  Google Scholar 

  • van de Geer, S.: Regression analysis and empirical processes. CWI-tract 45, Centre for Mathematics and Computer Science, Amsterdam (1988)

  • van de Geer, S.: Estimating a regression function. Ann. Stat. 18, 907–924 (1990)

    Article  MATH  Google Scholar 

  • van de Geer, S.: Empirical Processes in M-Estimation. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  • Wahba, G.: Spline Models for Observational Data. SIAM, Philadelphia (1990)

    MATH  Google Scholar 

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  1. Department of Mathematics, Darmstadt University of Technology, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Michael Kohler

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  1. Michael Kohler
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Correspondence to Michael Kohler.

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Kohler, M. A regression-based smoothing spline Monte Carlo algorithm for pricing American options in discrete time. AStA 92, 153–178 (2008). https://doi.org/10.1007/s10182-008-0067-0

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  • DOI: https://doi.org/10.1007/s10182-008-0067-0

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