Skip to main content Accessibility help
×
Home

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

  • Je Guk Kim (a1)

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.

Copyright

Corresponding author

* Postal address: SKK Business School, SungKyunKwan University, 25-2, Sungkyunkwan-ro, Jongno-gu, Seoul 03063, Republic of Korea. Email address: jkim74@vols.utk.edu

References

Hide All
[1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, 9th edn. Springer, New York.
[2] Carriere, J. F. (1996). Valuation of the early-exercise price for options using simulations and nonparametric regression. Insurance Math. Econom. 19, 1930.
[3] Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of a least squares regression method for American option pricing. Finance Stoch. 6, 449471.
[4] Egloff, D. (2005). Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Prob. 15, 13961432.
[5] Egloff, D., Kohler, M. and Todorovic, N. (2007). A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Prob. 17, 11381171.
[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, 317339.
[7] Gerhold, S. (2011). The Longstaff-Schwartz algorithm for Levy models: results on fast and slow convergence. Ann. Appl. Prob. 21, 589608.
[8] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer, New York.
[9] Glasserman, P. and Yu, B. (2004). Number of paths versus number of basis functions in American option pricing. Ann. Appl. Prob. 14, 20902119.
[10] Guo, B.-Y. (1999). Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68, 10671078.
[11] Lars, L.-C. (2002). L p -norms of Hermite polynomials and an extremal problem on Wiener chaos. Ark. Mat. 40, 133144.
[12] Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-squares approach. Rev. Financial Studies 14, 113147.
[13] Owen, A. B. (2000). Assessing linearity in high dimensions. Ann. Statist. 28, 119.
[14] Stentoft, L. (2004). Convergence of the least squares Monte Carlo approach to American option valuation. Manag. Sci. 50, 11931203.
[15] Tsitsiklis, J. N. and Van Roy, B. (2001). Regression methods for pricing complex American-style options. IEEE Trans. Neural Networks 12, 694703.
[16] Xu, C.-L. and Guo, B.-Y. (2003). Hermite spectral and pseudospectral methods for nonlinear partial differential equations. Comput. Appl. Math. 22, 167193.
[17] Zanger, D. Z. (2013). Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing. Finance Stoch. 17, 503534.

Keywords

MSC classification

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

  • Je Guk Kim (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed