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Wavelet Galerkin pricing of American options on Lévy driven assets. (English) Zbl 1134.91450

Summary: The price of an American-style contract on assets driven by a class of Markov processes containing, in particular, Lévy processes of pure jump type with infinite jump activity is expressed as the solution of a parabolic variational integro-differential inequality (PIDI). A Galerkin discretization in logarithmic price using a wavelet basis is presented. Log-linear complexity in each time-step is achieved by wavelet compression of the moment matrix of the price process’ jump measure and by wavelet preconditioning of the large matrix LCPs at each time-step. Efficiency is demonstrated by numerical experiments for pricing American put contracts on various jump-diffusion and pure jump models. Failure of the smooth pasting principle is observed for American put contracts for certain finite variation pure jump price processes.

MSC:

91B28 Finance etc. (MSC2000)
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[1] DOI: 10.1023/A:1011354913068 · Zbl 1274.91398 · doi:10.1023/A:1011354913068
[2] DOI: 10.1016/S0304-4149(02)00104-7 · Zbl 1060.91061 · doi:10.1016/S0304-4149(02)00104-7
[3] DOI: 10.1214/aoap/1075828052 · Zbl 1042.60023 · doi:10.1214/aoap/1075828052
[4] DOI: 10.1098/rspa.1977.0041 · doi:10.1098/rspa.1977.0041
[5] DOI: 10.1007/s007800050032 · Zbl 0894.90011 · doi:10.1007/s007800050032
[6] DOI: 10.1111/1467-9868.00282 · Zbl 0983.60028 · doi:10.1111/1467-9868.00282
[7] Bensoussan A, Impulse Control and Quasi-variational Inequalities (1984)
[8] Bertoin J, Lévy Processes (1996)
[9] Borici A, Computational Methods in Decision-Making, Economy and Finance (2002)
[10] DOI: 10.1142/S0219024900000541 · Zbl 0973.91037 · doi:10.1142/S0219024900000541
[11] Boyarchenko S, Ann. Appl. Probab. 4 pp 1261– (2002)
[12] DOI: 10.1137/S0363012900373987 · Zbl 1025.60021 · doi:10.1137/S0363012900373987
[13] DOI: 10.1142/9789812777485 · doi:10.1142/9789812777485
[14] DOI: 10.1093/rfs/11.3.597 · Zbl 1386.91134 · doi:10.1093/rfs/11.3.597
[15] Carr P, J. Business (2002)
[16] Carr P, J. Comp. Finance 2 pp 61– (1999)
[17] DOI: 10.1214/aoap/1029962753 · Zbl 1054.91033 · doi:10.1214/aoap/1029962753
[18] Cont R, Financial Modelling with Jump Processes (2004)
[19] DOI: 10.1137/0309028 · doi:10.1137/0309028
[20] DOI: 10.1111/1468-0262.00164 · Zbl 1055.91524 · doi:10.1111/1468-0262.00164
[21] DOI: 10.2307/3318570 · Zbl 0849.60042 · doi:10.2307/3318570
[22] DOI: 10.1111/1467-9965.02001 · Zbl 1072.91019 · doi:10.1111/1467-9965.02001
[23] Eberlein E, Lévy Processes: Theory and Applications pp pp. 319–337– (2001)
[25] Eberlein E, J. Business 71 pp 305– (1998) · doi:10.1086/209749
[26] Eskin GI, Boundary Value Problems for Elliptic Pseudodifferential Equations (1981)
[27] Föllmer H, Applied Stochastic Analysis pp pp. 389–414– (1991)
[28] Fouque J-P, Derivatives in Financial Markets with Stochastic Volatility (2000) · Zbl 0954.91025
[29] Glowinski R, Numerical Analysis of Variational Inequalities (1981)
[30] Jacod J, Limit Theorems for Stochastic Processes (2002)
[31] DOI: 10.1007/BF00047211 · Zbl 0714.90004 · doi:10.1007/BF00047211
[32] DOI: 10.1137/0713050 · Zbl 0337.65055 · doi:10.1137/0713050
[33] Karatzas I, Methods of Mathematical Finance (1998) · doi:10.1007/b98840
[34] DOI: 10.1103/PhysRevE.52.1197 · doi:10.1103/PhysRevE.52.1197
[35] Kornhuber R, Numer. Math. 69 pp 167– (1994)
[36] DOI: 10.1287/mnsc.48.8.1086.166 · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166
[37] DOI: 10.1080/14697680400023295 · doi:10.1080/14697680400023295
[38] Levendorskii SL, Working Paper (2005)
[39] DOI: 10.1002/cpa.3160200302 · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[40] DOI: 10.1023/A:1009703431535 · Zbl 0937.91052 · doi:10.1023/A:1009703431535
[41] DOI: 10.1086/296519 · doi:10.1086/296519
[42] DOI: 10.1051/m2an:2004003 · Zbl 1072.60052 · doi:10.1051/m2an:2004003
[43] DOI: 10.2307/3003143 · doi:10.2307/3003143
[44] DOI: 10.1016/0304-405X(76)90022-2 · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2
[45] Musiela M, Martingale Methods in Financial Modelling (1997)
[46] DOI: 10.1137/S0036142901394844 · Zbl 1050.65134 · doi:10.1137/S0036142901394844
[47] Pham H, J. Math. Sys. Estim. Control 8 pp 1– (1998)
[48] Sato K-I, Lévy Processes and Infinitely Divisible Distributions (1999) · Zbl 0973.60001
[49] Schoutens W, Lévy Processes in Finance, Wiley Series in Probability and Statistics (2003) · doi:10.1002/0470870230
[50] DOI: 10.1142/9789812385192 · doi:10.1142/9789812385192
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