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Multi-asset American options and parallel quantization. (English) Zbl 1273.91457

Summary: We present a parallel implementation of the optimal quantization method on a grid computing. Its purpose is to price instantaneously multidimensional American options. Numerical tests are proceeded with variable number of processors, from 4 to 128. Finally, a spatial extrapolation of Richardson-Romberg is introduced to speed up the convergence rate and stabilize the results.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
91G20 Derivative securities (option pricing, hedging, etc.)
65C50 Other computational problems in probability (MSC2010)
65Y05 Parallel numerical computation
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[1] Bally V, Pagès G (2003) A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9(6):1003-1049 · Zbl 1042.60021 · doi:10.3150/bj/1072215199
[2] Bally V, Pagès G, Printems J (2001) A stochastic quantization method for nonlinear problems. Monte Carlo Methods Appl 7(1-2):21-34 · Zbl 1035.65008
[3] Bally V, Pagès G, Printems J (2005) A quantization tree method for pricing and hedging multi-dimensional American options. Math Financ 15(1):119-168 · Zbl 1127.91023 · doi:10.1111/j.0960-1627.2005.00213.x
[4] Bardou O, Bouthemy S, Pagès G (2009) Optimal quantization for the pricing of swing options. Appl Math Financ 16(2):183-217 · Zbl 1169.91337 · doi:10.1080/13504860802453218
[5] Barraquand J, Martineau D (1995) Numerical valuation of high dimensional multivariate American securities. J Financ Quant Anal 30(3):383-405 · doi:10.2307/2331347
[6] Broadie M, Glasserman P (1997) Pricing American-style securities using simulation. J Econ Dyn Control 21(8-9):1323-1352 · Zbl 0901.90009 · doi:10.1016/S0165-1889(97)00029-8
[7] Fournié É, Lasry JM, Lebouchoux J, Lions PL (2001) Applications of Malliavin calculus to Monte Carlo methods in finance II. Finance Stoch 5:201-236 · Zbl 0973.60061 · doi:10.1007/PL00013529
[8] Fournié É, Lasry JM, Lebouchoux J, Lions PL, Touzi N (1999) Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch 3:391-412 · Zbl 0947.60066 · doi:10.1007/s007800050068
[9] Gersho A, Gray RM (1992) Vector quantization and signal compression. Kluwer, Boston. · Zbl 0782.94001 · doi:10.1007/978-1-4615-3626-0
[10] Graf S, Luschgy H (2000) Foundations of quantization for probability distributions. Lecture notes in mathematics, no 1730. Springer, Berlin · Zbl 0951.60003 · doi:10.1007/BFb0103945
[11] Graf S, Luschgy H, Pagès G (2008) Distortion mismatch in the quantization of probability measures. ESAIM P&S 12:127-153 · Zbl 1196.60062 · doi:10.1051/ps:2007044
[12] Kargin V (2005) Lattice option pricing by multidimensional interpolation. Math Financ 15(4):635-647 · Zbl 1107.91049 · doi:10.1111/j.1467-9965.2005.00254.x
[13] Kieffer J (1982) Exponential rate of convergence for the Lloyd’s method I. IEEE Trans Inf Theory (Special issue on Quantization) 28(2):205-210 · Zbl 0525.94006 · doi:10.1109/TIT.1982.1056482
[14] Lions PL, Régnier H (2001) Calcul des prix et des sensibilités d’une option Américaine par une méthode de Monte Carlo, working paper coc ixis · Zbl 0476.94008
[15] Longstaff FA, Schwartz ES (2001) Valuing American options by simulation: a simple least-squares approach. Rev Financ Stud 14:113-148 · Zbl 1386.91144 · doi:10.1093/rfs/14.1.113
[16] Pagès G (1998) A space quantization method for numerical integration. J Comput Appl Math 89:1-38 · Zbl 0908.65012 · doi:10.1016/S0377-0427(97)00190-8
[17] Pagès G, Printems J (2003) Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Methods Appl 9(2):135-166 · Zbl 1029.65012 · doi:10.1515/156939603322663321
[18] Pagès G, Wilbertz B (2011) Optimal Delaunay and Voronoi quantization schemes for pricing American style options. Carmona R, Del Moral P, Hu P, Oudjane N, Numerical Methods in Finance, Springer New York, Series: Proceeding in Mathematics (to appear 2011-2012) · Zbl 1029.65012
[19] Tsitsiklis JN, Van Roy B (1999) Optimal stopping of markov processes: Hilbert space theory, approximation, algorithms, and an application to pricing high-dimentional financial derivatives. IEEE Trans Automat Contr 44(10):1840-1851 · Zbl 0958.60042 · doi:10.1109/9.793723
[20] Villeneuve S, Zanette A (2002) Parabolic A.D.I. methods for pricing American option on two stocks. Math Oper Res 27(1):121-149 · Zbl 1082.60515 · doi:10.1287/moor.27.1.121.341
[21] Zador PL (1982) Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans Inf Theory (Special issue on Quantization) 28(2):139-149 · Zbl 0476.94008 · doi:10.1109/TIT.1982.1056490
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