Fast deterministic pricing of options on Lévy driven assets

Ana-Maria Matache; Tobias Von Petersdorff; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 1, page 37-71
  • ISSN: 0764-583X

Abstract

top
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 ( M N ( 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.

How to cite

top

Matache, Ana-Maria, Petersdorff, Tobias Von, and Schwab, Christoph. "Fast deterministic pricing of options on Lévy driven assets." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.1 (2004): 37-71. <http://eudml.org/doc/245641>.

@article{Matache2004,
abstract = {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) $\partial _t u + \{\mathcal \{A\}\}[u] = 0$. 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.},
author = {Matache, Ana-Maria, Petersdorff, Tobias Von, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {parabolic partial integro-differential equations; Lévy processes; Markov processes; Galerkin finite element method; wavelet; matrix compression; GMRES; arbitrage-free prices; European contracts on risky assets; parabolic partial integro-differential equation; Galerkin method; Lévy price processes},
language = {eng},
number = {1},
pages = {37-71},
publisher = {EDP-Sciences},
title = {Fast deterministic pricing of options on Lévy driven assets},
url = {http://eudml.org/doc/245641},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Matache, Ana-Maria
AU - Petersdorff, Tobias Von
AU - Schwab, Christoph
TI - Fast deterministic pricing of options on Lévy driven assets
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 1
SP - 37
EP - 71
AB - 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) $\partial _t u + {\mathcal {A}}[u] = 0$. 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.
LA - eng
KW - parabolic partial integro-differential equations; Lévy processes; Markov processes; Galerkin finite element method; wavelet; matrix compression; GMRES; arbitrage-free prices; European contracts on risky assets; parabolic partial integro-differential equation; Galerkin method; Lévy price processes
UR - http://eudml.org/doc/245641
ER -

References

top
  1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1978). Zbl1098.46001MR450957
  2. [2] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory, Monographs Math. Birkhäuser, Basel 89 (1995). Zbl0819.35001MR1345385
  3. [3] O.E. Barndorff-Nielsen, Exponentially decreasing distributions for the logarithm of particle size. Proc. Roy. Soc. London A 353 (1977) 401–419. 
  4. [4] O.E. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statis. 24 (1997) 1–14. Zbl0934.62109
  5. [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. Zbl0983.60028
  6. [6] A. Bensoussan and J.-L. Lions, Impulse control and quasi-variational inequalities. Gauthier-Villars, Paris (1984). Zbl0373.49004MR756234
  7. [7] J. Bertoin, Lévy processes. Cambridge University Press (1996). Zbl0861.60003MR1406564
  8. [8] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities. J. Political Economy 81 (1973) 637–654. Zbl1092.91524
  9. [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. Zbl1015.60036
  10. [10] S. Boyarchenko and S. Levendorski, Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Finance 3 (2000) 549-552. Zbl0973.91037
  11. [11] P. Carr and D. Madan, Option valuation using the FFT. J. Comp. Finance 2 (1999) 61–73. 
  12. [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. [13] T. Chan, Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab. 9 (1999) 504–528. Zbl1054.91033
  14. [14] A. Cohen, Wavelet methods for operator equations, P.G. Ciarlet and J.L. Lions Eds., Elsevier, Amsterdam, Handb. Numer. Anal. VII (2000). 
  15. [15] R. Cont and P. Tankov, Financial modelling with jump processes. Chapman and Hall/CRC Press (2003). Zbl1052.91043MR2042661
  16. [16] F. Delbaen and W. Schachermayer, The variance-optimal martingale measure for continuous processes. Bernoulli 2 (1996) 81–105. Zbl0849.60042
  17. [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. Zbl1072.91019
  18. [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. Zbl0982.60045
  19. [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. Zbl0738.90007
  20. [20] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987). Zbl0635.60021MR959133
  21. [21] P. Jaillet, D. Lamberton and B. Lapeyre, Variational inequalities and the pricing of American options. Acta Appl. Math. 21 (1990) 263–289. Zbl0714.90004
  22. [22] R. Kangro and R. Nicolaides, Far field boundary conditions for Black–Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357–1368. Zbl0990.35013
  23. [23] I. Karatzas and S.E. Shreve, Methods of Mathematical Finance. Springer-Verlag (1999). Zbl0941.91032MR1640352
  24. [24] G. Kou, A jump diffusion model for option pricing. Mange. Sci. 48 (2002) 1086–1101. Zbl1216.91039
  25. [25] D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall (1997). Zbl1167.60001
  26. [26] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer-Verlag, Berlin (1972). Zbl0223.35039
  27. [27] D.B. Madan and E. Seneta, The variance gamma (V.G.) model for share market returns. J. Business 63 (1990) 511–524. 
  28. [28] D.B. Madan, P. Carr and E. Chang, The variance gamma process and option pricing. Eur. Finance Rev. 2 (1998) 79–105. Zbl0937.91052
  29. [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 Zbl1072.60052
  30. [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 Zbl1134.91450
  31. [31] R.C. Merton, Option pricing when the underlying stocks are discontinuous. J. Financ. Econ. 5 (1976) 125–144. Zbl1131.91344
  32. [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. Zbl0991.60045
  33. [33] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. Springer-Verlag, New York 44 (1983). Zbl0516.47023MR710486
  34. [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. [35] K. Prause, The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures. Ph.D. thesis Albert-Ludwigs-Universität Freiburg i.Br. (1999). Zbl0944.91026
  36. [36] P. Protter, Stochastic Integration and Differential Equations. Springer-Verlag (1990). Zbl0694.60047MR1037262
  37. [37] S. Raible, Lévy processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis Albert-Ludwigs-Universität Freiburg i.Br. (2000). Zbl0966.60044
  38. [38] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999). Zbl0973.60001MR1739520
  39. [39] D. Schötzau and C. Schwab, h p -discontinuous Galerkin time-stepping for parabolic problems. C.R. Acad. Sci. Paris 333 (2001) 1121–1126. Zbl0993.65108
  40. [40] W. Schoutens, Lévy Processes in Finance. Wiley Ser. Probab. Stat., Wiley Publ. (2003). 
  41. [41] T. von Petersdorff and C. Schwab, Wavelet-discretizations of parabolic integro-differential equations. SIAM J. Numer. Anal. 41 (2003) 159–180. Zbl1050.65134
  42. [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. Zbl1083.65095
  43. [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). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.