Fast deterministic pricing of options on Lévy driven assets

Ana-Maria Matache; Tobias von Petersdorff; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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(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.

How to cite

top

Matache, Ana-Maria, von Petersdorff, Tobias, and Schwab, Christoph. "Fast deterministic pricing of options on Lévy driven assets." ESAIM: Mathematical Modelling and Numerical Analysis 38.1 (2010): 37-71. <http://eudml.org/doc/194208>.

@article{Matache2010,
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 θ-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(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. },
author = {Matache, Ana-Maria, von Petersdorff, Tobias, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
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},
month = {3},
number = {1},
pages = {37-71},
publisher = {EDP Sciences},
title = {Fast deterministic pricing of options on Lévy driven assets},
url = {http://eudml.org/doc/194208},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Matache, Ana-Maria
AU - von Petersdorff, Tobias
AU - Schwab, Christoph
TI - Fast deterministic pricing of options on Lévy driven assets
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
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 θ-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(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.
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/194208
ER -

References

top
  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 A353 (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. B63 (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 Economy81 (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. Finance3 (2000) 549-552.  
  11. P. Carr and D. Madan, Option valuation using the FFT. J. Comp. Finance2 (1999) 61–73.  
  12. P. Carr, H. Geman, D.B. Madan and M. Yor, The fine structure of asset returns: an empirical investigation. J. Business75 (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. Bernoulli2 (1996) 81–105.  
  17. F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer and C. Stricker, Exponential hedging and entropic penalties. Math. Finance12 (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. Business63 (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.  URIhttp://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,  URIhttp://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. Bernoulli7 (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. Paris333 (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 NI03013-CPD, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK (2003), , ESAIM: M2AN38 (2004) 93–127.  URIhttp://www.newton.cam.ac.uk/preprints2003.html
  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.