Matache, Ana-Maria; Petersdorff, Tobias Von; Schwab, Christoph
Fast deterministic pricing of options on Lévy driven assets
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 1 , p. 37-71
Zbl 1072.60052 | MR 2073930 | 2 citations dans Numdam
doi : 10.1051/m2an:2004003
URL stable : http://www.numdam.org/item?id=M2AN_2004__38_1_37_0

Classification:  65N30,  60J75
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) t u+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 θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for 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(Nlog(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.

Bibliographie

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1978). MR 450957 | 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

[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

[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

[6] A. Bensoussan and J.-L. Lions, Impulse control and quasi-variational inequalities. Gauthier-Villars, Paris (1984). MR 756234

[7] J. Bertoin, Lévy processes. Cambridge University Press (1996). MR 1406564 | 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

[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. 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

[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. 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).

[15] R. Cont and P. Tankov, Financial modelling with jump processes. Chapman and Hall/CRC Press (2003). MR 2042661 | Zbl 1052.91043

[16] F. Delbaen and W. Schachermayer, The variance-optimal martingale measure for continuous processes. Bernoulli 2 (1996) 81-105. 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

[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

[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

[20] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987). MR 959133 | 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

[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

[23] I. Karatzas and S.E. Shreve, Methods of Mathematical Finance. Springer-Verlag (1999). MR 1640352 | Zbl 0941.91032

[24] G. Kou, A jump diffusion model for option pricing. Mange. Sci. 48 (2002) 1086-1101.

[25] D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall (1997). Zbl pre05181830

[26] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer-Verlag, Berlin (1972). 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

[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

[31] R.C. Merton, Option pricing when the underlying stocks are discontinuous. J. Financ. Econ. 5 (1976) 125-144. Zbl 1131.91344

[32] 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. Zbl 0991.60045

[33] 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

[34] 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.

[35] 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

[36] P. Protter, Stochastic Integration and Differential Equations. Springer-Verlag (1990). MR 1037262 | Zbl 0694.60047

[37] S. Raible, Lévy processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis Albert-Ludwigs-Universität Freiburg i.Br. (2000). Zbl 0966.60044

[38] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999). MR 1739520 | Zbl 0973.60001

[39] 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

[40] W. Schoutens, Lévy Processes in Finance. Wiley Ser. Probab. Stat., Wiley Publ. (2003).

[41] T. Von Petersdorff and C. Schwab, Wavelet-discretizations of parabolic integro-differential equations. SIAM J. Numer. Anal. 41 (2003) 159-180. Zbl 1050.65134

[42] 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

[43] X. Zhang, Analyse Numerique des Options Américaines dans un Modèle de Diffusion avec Sauts. Ph.D. thesis, École Normale des Ponts et Chaussées (1994).