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Fast deterministic pricing of options on Lévy driven assets. (English) Zbl 1072.60052
Summary: Arbitrage-free prices $$u$$ of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) $$\partial_tu+{\mathcal A}[u]=0$$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the $$\theta$$-scheme in time and a wavelet Galerkin method with $$N$$ degrees of freedom in log-price space. The dense matrix for $$\mathcal A$$ can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for $$M$$ time steps is bounded by $$O(MN(\log(N))^2)$$ operations and $$O(N\log(N))$$ memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black-Scholes equation. Computational examples for various Lévy price processes are presented.

##### MSC:
 60H30 Applications of stochastic analysis (to PDEs, etc.) 60G51 Processes with independent increments; Lévy processes 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J75 Jump processes (MSC2010) 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65T60 Numerical methods for wavelets 91G20 Derivative securities (option pricing, hedging, etc.)
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##### References:
 [1] R.A. Adams , Sobolev Spaces . Academic Press, New York ( 1978 ). MR 450957 | Zbl 1098.46001 · Zbl 1098.46001 [2] H. Amann , Linear and Quasilinear Parabolic Problems , Vol. I: Abstract Linear Theory, Monographs Math. Birkhäuser, Basel 89 ( 1995 ). MR 1345385 | Zbl 0819.35001 · Zbl 0819.35001 [3] O.E. Barndorff-Nielsen , Exponentially decreasing distributions for the logarithm of particle size . Proc. Roy. Soc. London A 353 ( 1977 ) 401 - 419 . [4] O.E. Barndorff-Nielsen , Normal inverse Gaussian distributions and stochastic volatility modelling . Scand. J. Statis. 24 ( 1997 ) 1 - 14 . Zbl 0934.62109 · Zbl 0934.62109 [5] O.E. Barndorff-Nielsen and N. Shepard , Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics . J. Roy. Stat. Soc. B 63 ( 2001 ) 167 - 241 . Zbl 0983.60028 · Zbl 0983.60028 [6] A. Bensoussan and J.-L. Lions , Impulse control and quasi-variational inequalities . Gauthier-Villars, Paris ( 1984 ). MR 756234 · Zbl 0373.49004 [7] J. Bertoin , Lévy processes . Cambridge University Press ( 1996 ). MR 1406564 | Zbl 0861.60003 · Zbl 0861.60003 [8] F. Black and M. Scholes , The Pricing of Options and Corporate Liabilities . J. Political Economy 81 ( 1973 ) 637 - 654 . Zbl 1092.91524 · Zbl 1092.91524 [9] S. Boyarchenko and S. Levendorski , Barrier options and touch-and-out options under regular Lévy processes of exponential type . Ann. Appl. Probab. 12 ( 2002 ) 1261 - 1298 . Article | Zbl 1015.60036 · Zbl 1015.60036 [10] S. Boyarchenko and S. Levendorski , Option pricing for truncated Lévy processes . Int. J. Theor. Appl. Finance 3 ( 2000 ) 549 - 552 . Zbl 0973.91037 · Zbl 0973.91037 [11] P. Carr and D. Madan , Option valuation using the FFT . J. Comp. Finance 2 ( 1999 ) 61 - 73 . [12] P. Carr , H. Geman , D.B. Madan and M. Yor , The fine structure of asset returns: an empirical investigation . J. Business 75 ( 2002 ) 305 - 332 . [13] T. Chan , Pricing contingent claims on stocks driven by Lévy processes . Ann. Appl. Probab. 9 ( 1999 ) 504 - 528 . Article | Zbl 1054.91033 · Zbl 1054.91033 [14] A. Cohen , Wavelet methods for operator equations , P.G. Ciarlet and J.L. Lions Eds., Elsevier, Amsterdam, Handb. Numer. Anal. VII ( 2000 ). · Zbl 0976.65124 [15] R. Cont and P. Tankov , Financial modelling with jump processes . Chapman and Hall/CRC Press ( 2003 ). MR 2042661 | Zbl 1052.91043 · Zbl 1052.91043 [16] F. Delbaen and W. Schachermayer , The variance-optimal martingale measure for continuous processes . Bernoulli 2 ( 1996 ) 81 - 105 . Article | Zbl 0849.60042 · Zbl 0849.60042 [17] F. Delbaen , P. Grandits , T. Rheinländer , D. Samperi , M. Schweizer and C. Stricker , Exponential hedging and entropic penalties . Math. Finance 12 ( 2002 ) 99 - 123 . Zbl 1072.91019 · Zbl 1072.91019 [18] E. Eberlein , Application of generalized hyperbolic Lévy motions to finance , in Lévy Processes: Theory and Applications, O.E. Barndorff-Nielsen, T. Mikosch and S. Resnick Eds., Birkhäuser ( 2001 ) 319 - 337 . Zbl 0982.60045 · Zbl 0982.60045 [19] H. Föllmer and M. Schweizer , Hedging of contingent claims under incomplete information , in Applied Stochastic Analysis, M.H.A. Davis and R.J. Elliot Eds., Gordon and Breach New York ( 1991 ) 389 - 414 . Zbl 0738.90007 · Zbl 0738.90007 [20] J. Jacod and A.N. Shiryaev , Limit Theorems for Stochastic Processes . Springer-Verlag, Berlin ( 1987 ). MR 959133 | Zbl 0635.60021 · Zbl 0635.60021 [21] P. Jaillet , D. Lamberton and B. Lapeyre , Variational inequalities and the pricing of American options . Acta Appl. Math. 21 ( 1990 ) 263 - 289 . Zbl 0714.90004 · Zbl 0714.90004 [22] R. Kangro and R. Nicolaides , Far field boundary conditions for Black-Scholes equations . SIAM J. Numer. Anal. 38 ( 2000 ) 1357 - 1368 . Zbl 0990.35013 · Zbl 0990.35013 [23] I. Karatzas and S.E. Shreve , Methods of Mathematical Finance . Springer-Verlag ( 1999 ). MR 1640352 | Zbl 0941.91032 · Zbl 0941.91032 [24] G. Kou , A jump diffusion model for option pricing . Mange. Sci. 48 ( 2002 ) 1086 - 1101 . · Zbl 1216.91039 [25] D. Lamberton and B. Lapeyre , Introduction to Stochastic Calculus Applied to Finance . Chapman & Hall ( 1997 ). Zbl pre05181830 · Zbl 1167.60001 [26] J.L. Lions and E. Magenes , Non-homogeneous boundary value problems and applications . Springer-Verlag, Berlin ( 1972 ). Zbl 0223.35039 · Zbl 0223.35039 [27] D.B. Madan and E. Seneta , The variance gamma (V.G.) model for share market returns . J. Business 63 ( 1990 ) 511 - 524 . [28] D.B. Madan , P. Carr and E. Chang , The variance gamma process and option pricing . Eur. Finance Rev. 2 ( 1998 ) 79 - 105 . Zbl 0937.91052 · Zbl 0937.91052 [29] A.M. Matache , T. von Petersdorff and C. Schwab , Fast deterministic pricing of options on Lévy driven assets . Report 2002 - 11 , Seminar for Applied Mathematics, ETH Zürich. http://www.sam.math.ethz.ch/reports/details/include.shtml?2002/2002-11.html [30] A.M. Matache , P.A. Nitsche and C. Schwab , Wavelet Galerkin pricing of American options on Lévy driven assets . Research Report 2003 - 06 , Seminar for Applied Mathematics, ETH Zürich, http://www.sam.math.ethz.ch/reports/details/include.shtml?2003/2003-06.html Zbl 1134.91450 · Zbl 1134.91450 [30] R.C. Merton , Option pricing when the underlying stocks are discontinuous . J. Financ. Econ. 5 ( 1976 ) 125 - 144 . Zbl 1131.91344 · Zbl 1131.91344 [31] D. Nualart and W. Schoutens , Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance . Bernoulli 7 ( 2001 ) 761 - 776 . Article | Zbl 0991.60045 · Zbl 0991.60045 [32] A. Pazy , Semigroups of linear operators and applications to partial differential equations . Appl. Math. Sci. Springer-Verlag, New York 44 ( 1983 ). MR 710486 | Zbl 0516.47023 · Zbl 0516.47023 [33] T. von Petersdorff and C. Schwab , Fully discrete multiscale Galerkin BEM , in Multiresolution Analysis and Partial Differential Equations, W. Dahmen, P. Kurdila and P. Oswald Eds., Academic Press, New York, Wavelet Anal. Appl. 6 ( 1997 ) 287 - 346 . [34] K. Prause , The Generalized Hyperbolic Model: Estimation , Financial Derivatives, and Risk Measures. Ph.D. thesis Albert-Ludwigs-Universität Freiburg i.Br. ( 1999 ). Zbl 0944.91026 · Zbl 0944.91026 [35] P. Protter , Stochastic Integration and Differential Equations . Springer-Verlag ( 1990 ). MR 1037262 | Zbl 0694.60047 · Zbl 0694.60047 [36] S. Raible , Lévy processes in Finance: Theory , Numerics, and Empirical Facts. Ph.D. thesis Albert-Ludwigs-Universität Freiburg i.Br. ( 2000 ). Zbl 0966.60044 · Zbl 0966.60044 [37] K.-I. Sato , Lévy Processes and Infinitely Divisible Distributions . Cambridge University Press ( 1999 ). MR 1739520 | Zbl 0973.60001 · Zbl 0973.60001 [38] D. Schötzau and C. Schwab , $$hp$$-discontinuous Galerkin time-stepping for parabolic problems . C.R. Acad. Sci. Paris 333 ( 2001 ) 1121 - 1126 . Zbl 0993.65108 · Zbl 0993.65108 [39] W. Schoutens , Lévy Processes in Finance . Wiley Ser. Probab. Stat., Wiley Publ. ( 2003 ). [40] T. von Petersdorff and C. Schwab , Wavelet-discretizations of parabolic integro-differential equations . SIAM J. Numer. Anal. 41 ( 2003 ) 159 - 180 . Zbl 1050.65134 · Zbl 1050.65134 [41] T. von Petersdorff and C. Schwab , Numerical solution of parabolic equations in high dimensions . Report NI03013-CPD, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK ( 2003 ), http://www.newton.cam.ac.uk/preprints2003.html, ESAIM: M2AN 38 ( 2004 ) 93 - 127 . Numdam | Zbl 1083.65095 · Zbl 1083.65095 [42] X. Zhang , Analyse Numerique des Options Américaines dans un Modèle de Diffusion avec Sauts . Ph.D. thesis, École Normale des Ponts et Chaussées ( 1994 ).
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