Annals of Applied Probability

The Longstaff–Schwartz algorithm for Lévy models: Results on fast and slow convergence

Stefan Gerhold

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We investigate the Longstaff–Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090–2119] and Stentoft [Manag. Sci. 50 (2004) 1193–1203] to several Lévy models, in particular the geometric Meixner model. A convenient setting to analyze this convergence problem is provided by the Lévy–Sheffer systems introduced by Schoutens and Teugels.

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Ann. Appl. Probab., Volume 21, Number 2 (2011), 589-608.

First available in Project Euclid: 22 March 2011

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Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Option pricing dynamic programming Monte Carlo regression orthogonal polynomials Lévy–Meixner systems


Gerhold, Stefan. The Longstaff–Schwartz algorithm for Lévy models: Results on fast and slow convergence. Ann. Appl. Probab. 21 (2011), no. 2, 589--608. doi:10.1214/10-AAP704.

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  • [1] Bacinello, A. R., Biffis, E. and Millossovich, P. (2009). Pricing life insurance contracts with early exercise features. J. Comput. Appl. Math. 233 27–35.
  • [2] Belomestny, D., Bender, C. and Schoenmakers, J. (2009). True upper bounds for Bermudan products via non-nested Monte Carlo. Math. Finance 19 53–71.
  • [3] Brigo, D. and Mercurio, F. (2006). Interest Rate Models—Theory and Practice: With Smile, Inflation and Credit, 2nd ed. Springer, Berlin.
  • [4] Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of a least squares regression method for American option pricing. Finance Stoch. 6 449–471.
  • [5] Cont, R., Tankov, P. and Voltchkova, E. (2004). Option pricing models with jumps: Integro-differential equations and inverse problems. In Proceedings of ECCOMAS 2004 (P. Neittaanmäki et al., eds.). Jyväskylä, Finland.
  • [6] de Bruijn, N. G. (1958). Asymptotic Methods in Analysis. Bibliotheca Mathematica 4. North-Holland, Amsterdam.
  • [7] de Jong, R. M. (2002). A note on “Convergence rates and asymptotic normality for series estimators”: Uniform convergence rates by W. K. Newey. J. Econometrics 111 1–9.
  • [8] Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the Gamma process. Astin Bull. 21 177–192.
  • [9] Egloff, D. (2005). Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15 1396–1432.
  • [10] Egloff, D., Kohler, M. and Todorovic, N. (2007). A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Probab. 17 1138–1171.
  • [11] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [12] Fouque, J.-P. and Han, C.-H. (2009). Asymmetric variance reduction for pricing American options. In Mathematical Modelling and Numerical Methods in Finance (A. Bensoussan, P. Ciarlet and Q. Zhang, eds.). Handbook of Numerical Analysis 15 169–187. North-Holland, Amsterdam.
  • [13] Gerber, H. U. and Shiu, E. S. W. (1994). Option pricing by Esscher transforms. Transactions of Society of Actuaries 46 99–191.
  • [14] Glasserman, P. and Yu, B. (2004). Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14 2090–2119.
  • [15] Godefroy, M. (1901). La Fonction Gamma. Gauthier-Villars, Paris.
  • [16] Grigelionis, B. (1999). Processes of Meixner type. Liet. Mat. Rink. 39 40–51.
  • [17] Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: A simple least-squares approach. Rev. Financial Stud. 14 113–148.
  • [18] Meixner, J. (1934). Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. London Math. Soc. 9 6–13.
  • [19] Nabben, R. (1999). Two-sided bounds on the inverses of diagonally dominant tridiagonal matrices. Linear Algebra Appl. 287 289–305. Special issue celebrating the 60th birthday of Ludwig Elsner.
  • [20] Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics 79 147–168.
  • [21] Schoutens, W. (2000). Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics 146. Springer, New York.
  • [22] Schoutens, W. (2002). Meixner processes: Theory and applications in finance. EURANDOM Report 2002-004, EURANDOM, Eindhoven.
  • [23] Schoutens, W. and Teugels, J. L. (1998). Lévy processes, polynomials and martingales. Comm. Statist. Stochastic Models 14 335–349. Special issue in honor of Marcel F. Neuts.
  • [24] Shreve, S. E. (2004). Stochastic Calculus for Finance. II. Continuous-Time Models. Springer, New York.
  • [25] Stentoft, L. (2004). Convergence of the least squares Monte Carlo approach to American option valuation. Manag. Sci. 50 1193–1203.
  • [26] Szegő, G. (1975). Orthogonal Polynomials, 4th ed. American Mathematical Society, Colloquium Publications XXIII. Amer. Math. Soc., Providence, RI.
  • [27] Zeng, J. (1992). Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 1–22.