Fast deterministic pricing of options on Lévy driven assets

Matache, Ana-Maria; Petersdorff, Tobias Von; Schwab, Christoph

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 37-71.

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) ${\partial }_{t}u+𝒜\left[u\right]=0$. 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 $𝒜$ 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 $O\left(MN{(log\left(N\right))}^2\right)$ operations and $O(Nlog(N\left)\right)$ 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.

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

[2] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory, Monographs Math. Birkhäuser, Basel 89 (1995). | MR | Zbl

[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

[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

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

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

[8] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities. J. Political Economy 81 (1973) 637-654. | Zbl

[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

[10] S. Boyarchenko and S. Levendorski, Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Finance 3 (2000) 549-552. | Zbl

[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

[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 | Zbl

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

[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

[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

[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

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

[21] P. Jaillet, D. Lamberton and B. Lapeyre, Variational inequalities and the pricing of American options. Acta Appl. Math. 21 (1990) 263-289. | Zbl

[22] R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357-1368. | Zbl

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

[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). | Zbl

[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

[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

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

[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

[33] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. Springer-Verlag, New York 44 (1983). | MR | Zbl

[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

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

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

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

[39] D. Schötzau and C. Schwab, $hp$-discontinuous Galerkin time-stepping for parabolic problems. C.R. Acad. Sci. Paris 333 (2001) 1121-1126. | Zbl

[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

[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

[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).