Fast deterministic pricing of options on Lévy driven assets

Mathematical Modelling and Numerical Analysis

Volumn 38, Issue 1, 2004, Pages 37-71

  Matache, Ana Maria ^a   Von Petersdorff, Tobias ^b   Schwab, Christoph ^a  

^a ETH ZURICH   (Switzerland)

^b Bldg 142   (United States)

Author keywords

Galerkin finite element method; GMRES; L vy processes; Markov processes; Matrix compression; Parabolic partial Integro differential equations; Wavelet

Indexed keywords

EID: 1342304982     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2004003     Document Type: Article

References (43)

1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1978).
2. H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory, Monographs Math. Birkhäuser, Basel 89 (1995).
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.
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.
6. A. Bensoussan and J.-L. Lions, Impulse control and quasi-variational inequalities. Gauthier-Villars, Paris (1984).
7. J. Bertoin, Lévy processes. Cambridge University Press (1996).
8. F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities. J. Political Economy 81 (1973) 637-654.
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.
10. S. Boyarchenko and S. Levendorski, Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Finance 3 (2000) 549-552.
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.
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).
16. F. Delbaen and W. Schachermayer, The variance-optimal martingale measure for continuous processes. Bernoulli 2 (1996) 81-105.
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.
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.
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.
20. J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987).
21. P. Jaillet, D. Lamberton and B. Lapeyre, Variational inequalities and the pricing of American options. Acta Appl. Math. 21 (1990) 263-289.
22. R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357-1368.
23. I. Karatzas and S.E. Shreve. Methods of Mathematical Finance. Springer-Verlag (1999).
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).
26. J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer-Verlag, Berlin (1972).
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.
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
31. R.C. Merton, Option pricing when the underlying stocks are discontinuous. J. Financ. Econ. 5 (1976) 125-144.
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.
33. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. Springer-Verlag, New York 44 (1983).
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).
36. P. Protter, Stochastic Integration and Differential Equations. Springer-Verlag (1990).
37. S. Raible, Lévy processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis Albert-Ludwigs-Universität Freiburg i.Br. (2000).
38. K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999).
39. D. Schötzau and C. Schwab, hp-discontinuous Galerkin time-stepping for parabolic problems. C.R. Acad. Sci. Paris 333 (2001) 1121-1126.
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.
42. T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high, dimensions. Report N103013-CPD, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK (2003), http://www.newton.cam.ac.uk/prepririts2003.html, ESAIM: M2AN 38 (2004) 93-127.
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).