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On the convergence of the quasi-regression method: polynomial chaos and regularity

Part of: Mathematical finance Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  22 June 2017

Je Guk Kim
Je Guk Kim*
Affiliation:
SungKyunKwan University
*
* Postal address: SKK Business School, SungKyunKwan University, 25-2, Sungkyunkwan-ro, Jongno-gu, Seoul 03063, Republic of Korea. Email address: [email protected]
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Abstract

We present an analysis of convergence of a quasi-regression Monte Carlo method proposed by Glasserman and Yu (2004). We show that the method surely converges to the true price of an American option even under multiple underlyings via polynomial chaos expansion and weaker conditions than those used in Glasserman and Yu (2004). Further, we show the number of simulation paths grows exponentially in the number of basis functions to obtain convergence in implementing the method. Finally, we propose a rate of convergence considering regularity of value functions.

Keywords

American optionMonte Carlopolynomial chaosregularityquasi-regression

MSC classification

Primary: 91G60: Numerical methods (including Monte Carlo methods)
Secondary: 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 424 - 443
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, 9th edn. Springer, New York. Google Scholar
[2] Carriere, J. F. (1996). Valuation of the early-exercise price for options using simulations and nonparametric regression. Insurance Math. Econom. 19, 19–30. CrossRefGoogle Scholar
[3] 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. CrossRefGoogle Scholar
[4] Egloff, D. (2005). Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Prob. 15, 1396–1432. Google Scholar
[5] Egloff, D., Kohler, M. and Todorovic, N. (2007). A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Prob. 17, 1138–1171. Google Scholar
[6] Ernst, O. G., Mugler, H. I., Starkoff, H. J. and Ullmann, E. (2012). On the convergence of generalized polynomial chaos expansions. ESIM Math. Modelling Numer. Anal. 46, 317–339. Google Scholar
[7] Gerhold, S. (2011). The Longstaff-Schwartz algorithm for Levy models: results on fast and slow convergence. Ann. Appl. Prob. 21, 589–608. CrossRefGoogle Scholar
[8] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer, New York. CrossRefGoogle Scholar
[9] Glasserman, P. and Yu, B. (2004). Number of paths versus number of basis functions in American option pricing. Ann. Appl. Prob. 14, 2090–2119. Google Scholar
[10] Guo, B.-Y. (1999). Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68, 1067–1078. CrossRefGoogle Scholar
[11] Lars, L.-C. (2002). L p -norms of Hermite polynomials and an extremal problem on Wiener chaos. Ark. Mat. 40, 133–144. Google Scholar
[12] Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-squares approach. Rev. Financial Studies 14, 113–147. Google Scholar
[13] Owen, A. B. (2000). Assessing linearity in high dimensions. Ann. Statist. 28, 1–19. CrossRefGoogle Scholar
[14] Stentoft, L. (2004). Convergence of the least squares Monte Carlo approach to American option valuation. Manag. Sci. 50, 1193–1203. Google Scholar
[15] Tsitsiklis, J. N. and Van Roy, B. (2001). Regression methods for pricing complex American-style options. IEEE Trans. Neural Networks 12, 694–703. Google Scholar
[16] Xu, C.-L. and Guo, B.-Y. (2003). Hermite spectral and pseudospectral methods for nonlinear partial differential equations. Comput. Appl. Math. 22, 167–193. Google Scholar
[17] Zanger, D. Z. (2013). Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing. Finance Stoch. 17, 503–534. Google Scholar