Abstract
We review the design and analysis of multiresolution (wavelet) methods for the numerical solution of the Kolmogorov equations arising, among others, in financial engineering when Lévy and Feller or additive processes are used to model the dynamics of the risky assets. In particular, the Dirichlet and free boundary problems connected to barrier and American style contracts are specified and solution algorithms based on wavelet representations of the Feller processes’ Dirichlet forms are presented. Feller processes with generators that give rise to Sobolev spaces of variable differentiation order (corresponding to a state-dependent jump intensity) are considered. A copula construction for the systematic construction of parametric multivariate Feller-Lévy processes from univariate ones is presented and the domains of the generators of the resulting multivariate Feller-Lévy processes is identified. New multiresolution norm equivalences in such Sobolev spaces allow for wavelet compression of the matrix representations of the Dirichlet forms. Implementational aspects, in particular the regularization of the process’ Dirichlet form and the singularity-free, fast numerical evaluation of moments of the Dirichlet form with respect to piecewise linear, continuous biorthogonal wavelet bases are addressed. Monte Carlo path simulation techniques for such processes by FFT and symbol localization are outlined. Numerical experiments illustrate multilevel preconditioning of the moment matrices for several exotic contracts as well as for Feller-Lévy processes with variable order jump intensities. Model sensitivity of Lévy models embedded into Feller classes is studied numerically for several types of plain vanilla, barrier and exotic contracts.
Keywords
- Dirichlet forms
- Feller processes
- Kolmogorov equations
- Lévy processes
- Matrix compression
- Option pricing
- Sato processes
- Wavelet discretization
AMS Subject Classification 2000
Primary: 45K05, 60J75, 65N30
Secondary: 45M60
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Amadori, A.: Obstacle problem for nonlinear integrodifferential equations arising in option pricing. Ric. Mat. 56, 1–17 (2007)
Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, New York (2009)
Baiocchi, C.: Discretization of evolution variational inequalities. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds.) Partial Differential Equations and the Calculus of Variations, vol. I of Progr. Nonlinear Differential Equations Appl., pp. 59–92. Birkhäuser, MA (1989)
Barndorff-Nielsen, O.: Processes of normal inverse Gaussian type. Finance Stochast. 2, 41–68 (1998)
Barndorff-Nielsen, O., Levendorskii, S.: Feller processes of normal inverse Gaussian type. Quant. Finance 1, 318–331 (2001)
Bentata, A., Cont, R.: Matching marginal distributions of a semimartingale with a Markov process. Tech. rep., arXive. http://arxiv.org/pdf/0910.3992 (2009)
Benth, F., Nazarova, A., Wobben, M.: The pricing of derivatives under bounded stochastic processes. Tech. rep., University of Oslo (2010)
Berg, C., Forst, G.: Non-symmetric translation invariant Dirichlet forms. Invent. Math. 21, 199–212 (1973)
Bertoin, J.: Lévy Processes. Cambridge University Press, New York (1996)
Bervoets, F., Fang, F., Lord, R., Oosterlee, C.: A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes. SIAM J. Sci. Comput. 30, 1678–1705 (2008)
Beuchler, S., Schneider, R., Schwab, C.: Multiresolution weighted norm equivalences and applications. Numer. Math. 98, 67–97 (2004)
Bony, J., Courrege, P., Priouret, P.: Semi-groupes de feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier Grenoble 18(2), 369–521 (1968)
Böttcher, B.: Construction of time-inhomogeneous Markov processes via evolution equations using pseudo-differential operators. J. Lond. Math. Soc. 78(2), 605–621 (2008)
Böttcher, B., Schilling, R.: Approximation of Feller processes by Markov chains with Lévy increments. Stoch. Dyn. 9, 71–80 (2009)
Böttcher, B., Schnurr, A.: The Euler scheme for Feller processes. Tech. rep., TU Dresden (2009)
Boyarchenko, S., Levendorskii, S.: Non-Gaussian Merton-Black-Scholes Theory. World Scientific Publishing, NJ (2002)
Brézis, H.: Un problème d’evolution avec contraintes unilatérales dépendant du temps. C. R. Acad. Sci. Paris Sér. A–B 274, A310–A312 (1972)
Briani, M., La Chioma, C., Natalini, R.: Convergence of numerical schemes for viscosity solutions of integro-differential degenerate parabolic problems arising in financial theory. Numer. Math. 98, 607–646 (2004)
Carr, P.: Local variance gamma. Tech. rep., Private communication (2009)
Carr, P., Geman, H., Madan, D., Yor, M.: From local volatility to local Lévy models. Quant. Finance 4, 581–588 (2004)
Carr, P., Madan, D.: Option pricing and the fast Fourier transform. J. Comput. Finance 4, 61–73 (1999)
Chernov, A., von Petersdorff, T., Schwab, C.: Exponential convergence of hp quadrature for integral operators with Gevrey kernels. Tech. Rep. 03, SAM, ETH. http://www.sam.math.ethz.ch/reports/2009/03 (2009)
Cohen, A.: Numerical Analysis of Wavelet Methods, vol. 32. North Holland-Elsevier, Amsterdam (2003)
Cooley, J., Tukey, J.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297–301 (1965)
Courrège, P.: Sur la forme intégro-differentielle des opérateurs de c k ∞ dans c satisfaisant du principe du maximum, sém. Théorie du Potentiel 38, 38 (1965)
Cox, A., Hobson, D., Obłój, J.: Time-homogeneous diffusion with a given marginal at a random time. Tech. Rep. http://arxiv.org/pdf/0912.1719v1, arXive (2009)
Crandall, M., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Crandall, M., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)
Dahmen, W., Schneider, R.: Wavelets with complementary boundary conditions, function spaces on the cube. Results Math. 34, 255–293 (1998)
Dappa, H.: Quasiradiale Fouriermultiplikatoren. Ph.D. thesis, University of Darmstadt (1982)
Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998)
Delbaen, F., Shirakawa, H.: A note on option pricing for the constant elasticity of variance model. Asia Pac. Financ. Market. 9, 85–99 (2002)
Dijkema, T.: Adaptive tensor product wavelet methods for solving PDEs. Ph.D. thesis, University of Utrecht (2009)
Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)
Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 343–1376 (2000)
Eberlein, E., Glau, K., Papapantoleon, A.: Analysis of Fourier transform valuation formulas and applications. Appl. Math. Finance 17(3), 211–240 (2009)
Eberlein, E., Papapantoleon, A., Shiryaev, A.N.: On the duality principle in option pricing: Semimartingale setting. Finance Stoch. 12, 265–292 (2008)
Farkas, W., Reich, N., Schwab, C.: Anisotropic stable Lévy Copula processes-analytical and numerical aspects. Math. Models Meth. Appl. Sci. 17, 1405–1443 (2007)
Frigo, M., Johnson, S.: The design and implementation of FFTW3. Proc. IEEE 93(2), 216–231 (2005)
Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical analysis of variational inequalities, vol. 8 of Studies in Mathematics and its Applications. North-Holland, Amsterdam (1981)
Gyöngy, I.: Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theor. Relat. Field. 71, 501–516 (1986)
Harbrecht, H., Stevenson, R.: Wavelets with patchwise cancellation properties. Math. Comp. 75, 1871–1889 (2006)
Hilber, N.: Stabilized wavelet method for pricing in high dimensional stochastic volatility models. Ph.D. thesis, ETH Dissertation No. 18176. http://e-collection.ethbib.ethz.ch/view/eth:41687 (2009)
Hilber, N., Schwab, C., Winter, C.: Variational sensitivity analysis of parametric Markovian market models. Banach Center Publications 83, 85–106 (2008)
Hoh, W.: Pseudodifferential operators generating Markov processes. Habilitationsschrift, University of Bielefeld (1998)
Hoh, W.: Pseudodifferential operators with negative definite symbols of variable order. Rev. Math. Iberoamericana 16, 219–241 (2000)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam (1981)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for variational inequalities of the first kind. M2AN Math. Model. Numer. Anal. 37, 41–62 (2003)
Ito, K., Kunisch, K.: Parabolic variational inequalities: The Lagrange multiplier approach. J. Mat. Pures Appl. 85, 415–449 (2005)
Jackson, K., Jaimungal, S., Surkov, V.: Fourier space time-stepping for option pricing with Lévy models. Tech. rep., SSRN. http://ssrn.com/abstract=1020209 (2007)
Jacob, N.: Pseudo-Differential Operators and Markov Processes vol. 1: Fourier Analysis and Semigroups. Imperial College Press, London (2001)
Jacob, N.: Pseudo-Differential Operators and Markov Processes vol. 2: Generators and their Potential Theory. Imperial College Press, London (2002)
Jacob, N.: Pseudo-Differential Operators and Markov Processes vol. 3: Markov Processes and Applications. Imperial College Press, London (2005)
Jacob, N., Schilling, R.: Lévy-type processes and pseudo differential operators. In: Barndorff-Nielsen, O., Mikosch, T., Resnick, S. (eds.) Lévy Processes: Theory and Applications, pp. 139–167. Birkhäuser, MA (2001)
Jacod, J., Shiriyaev, A.: Limit Theorems for Stochastic Processes. Springer, Heidelberg (2003)
Kallsen, J., Tankov, P.: Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivariate Anal. 97, 1551–1572 (2006)
Keller-Ressel, M., Schachermayer, W., Teichmann, J.: Affine processes are regular. Tech. rep., arXive. http://arxiv.org/pdf/0906.3392v1 (2009)
Kikuchi, K., Negoro, A.: On Markov process generated by pseudodifferential operator of variable order. Osaka J. Math. 34, 319–335 (1997)
Knopova, V., Schilling, R.: Transition density estimates for a class of Lévy and Lévy-type processes. Tech. rep., arXive. http://arxiv.org/abs/0912.1482 (2009)
Lee, R.: Option pricing by transform methods: Extensions, unification, and error control. J. Comput. Finance 7(3), 51–86 (2004)
Leentvaar, C.: Pricing multi-asset options with sparse grids. Ph.D. thesis, TU Delft (2008)
Lewis, A.: A simple option formula for general jump-diffusion and other exponential Lévy processes. Tech. rep., SSRN. http://papers.ssrn.com/sol3/papers. cfm?abstract_id=282110 (2001)
Matache, A.-M., Schwab, C., Wihler, T.P.: Linear complexity solution of parabolic integrodifferential equations. Numer. Math. 104, 69–102 (2004)
Nicolato, E., Venardos, E.: Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Math. Finance 13, 445–466 (2003)
Nochetto, R., Savaré, G., Verdi, C.: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math. 53, 525–589 (2000)
Pham, H.: Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl. Math. Optim. 35, 145–164 (1997)
Pham, H.: Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. J. Math. Syst. Estim. Contr. 8, 1–27 (electronic) (1998)
Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes in C. The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Primbs, M.: Stabile biorthogonale Spline-Waveletbasen auf dem Intervall. Ph.D. thesis, University of Duisburg-Essen (2006)
Reich, N.: Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces. Ph.D. thesis, ETH Dissertation No. 17661. http://e-collection. ethbib.ethz.ch/view/eth:30174 (2008)
Reich, N.: Anisotropic operator symbols arising from multivariate jump processes. J. Integr. Equat. Operat. Theor. 63, 127–150 (2009)
Reich, N.: Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces. ESAIM: M2AN 44(1), 33–73 (2009)
Reich, N., Schwab, C., Winter, C.: On Kolmogorov equations for anisotropic multivariate Lévy processes. Tech. Rep. 03, SAM, ETH. http://www.sam.math.ethz.ch/ reports/2008/03 (2008)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York (1990)
Roman, S.: The formula of Faà di Bruno. Am. Math. Mon. 87, 805–809 (1980)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, Cambridge (1999)
Savaré, G.: Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6, 377–418 (1996)
Sayah, A.: Equation d’Hamilton-Jacobi du premier ordre avec terme intégro-différentiel: Parties i et ii. Comm. PDE 16, 1057–1074 (1991)
Schilling, R.: On Feller processes with sample paths in Besov spaces. Math. Ann. 309, 663–675 (1996)
Schilling, R.: Feller processes generated by pseudo-differential operators: On the Hausdorff dimension of their sample paths. J. Theor. Probab. 11, 303–330 (1998)
Schilling, R.: Growth and Hölder conditions for the sample paths of Feller processes. Probab. Theor. Relat. Field. 112, 565–611 (1998)
Schilling, R.: On the existence of Feller processes with a given generator. Tech. rep., TU Dresden. http://www.math.tu-dresden.de/sto/schilling/papers/existenz (2004)
Schilling, R., Schnurr, A.: The symbol associated with the solution of a stochastic differential equation. Tech. rep., TU Dresden. http://arxiv.org/pdf/0912.1458v1 (2009)
Schneider, R., Reichmann, O., Schwab, C.: Wavelet solution of variable order pseudodifferential equations. Calcolo 47(2), 65–101 (2009)
Schnurr, J.: The symbol of a Markov Semimartingale. Ph.D. thesis, TU Dresden (2009)
Schötzau, D., Schwab, C.: hp-discontinuous Galerkin time-stepping for parabolic problems. C. R. Acad. Sci. Paris 333, 1121–1126 (2001)
Schoutens, W.: Lévy Processes in Finance. Wiley, Chichester (2003)
Schwab, C.: Variable order composite quadrature of singular and nearly singular integrals. Computing 53, 173–194 (1994)
Soner, H.: Optimal control with state-space constraint II. SIAM J. Control Optim. 24, 1110–1122 (1986)
Stroock, D.: Diffusion processes associated with Lévy generators. Wahrscheinlichkeitstheorie verw. Gebiete 32, 209–244 (1975)
Urban, K.: Wavelet Methods for Elliptic Partial Differential Equations. Oxford University Press, Oxford (2009)
Van Loan, C.: Computational Frameworks for the Fast Fourier Transform. SIAM Publications, Philadelphia (1992)
Winter, C.: Wavelet Galerkin schemes for option pricing in multidimensional Lévy models. Ph.D. thesis, ETH Dissertation No. 18221. http://e-collection.ethbib. ethz.ch/view/eth:41555 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Reichmann, O., Schwab, C. (2010). Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing. In: Barndorff-Nielsen, O., Bertoin, J., Jacod, J., Klüppelberg, C. (eds) Lévy Matters I. Lecture Notes in Mathematics(), vol 2001. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14007-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-14007-5_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14006-8
Online ISBN: 978-3-642-14007-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
