The Annals of Applied Probability

Approximating Lévy processes with completely monotone jumps

Daniel Hackmann and Alexey Kuznetsov

Full-text: Open access


Lévy processes with completely monotone jumps appear frequently in various applications of probability. For example, all popular stock price models based on Lévy processes (such as the Variance Gamma, CGMY/KoBoL and Normal Inverse Gaussian) belong to this class. In this paper we continue the work started in [Int. J. Theor. Appl. Finance 13 (2010) 63–91, Quant. Finance 10 (2010) 629–644] and develop a simple yet very efficient method for approximating processes with completely monotone jumps by processes with hyperexponential jumps, the latter being the most convenient class for performing numerical computations. Our approach is based on connecting Lévy processes with completely monotone jumps with several areas of classical analysis, including Padé approximations, Gaussian quadrature and orthogonal polynomials.

Article information

Ann. Appl. Probab., Volume 26, Number 1 (2016), 328-359.

Received: April 2014
Revised: December 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 26C15: Rational functions [See also 14Pxx]

Lévy processes complete monotonicity hyperexponential processes Padé approximation rational interpolation Gaussian quadrature Stieltjes functions Jacobi polynomials


Hackmann, Daniel; Kuznetsov, Alexey. Approximating Lévy processes with completely monotone jumps. Ann. Appl. Probab. 26 (2016), no. 1, 328--359. doi:10.1214/14-AAP1093.

Export citation


  • [1] Allen, G. D., Chui, C. K., Madych, W. R., Narcowich, F. J. and Smith, P. W. (1975). Padé approximation of Stieltjes series. J. Approx. Theory 14 302–316.
  • [2] Bailey, D. H. (1995). A fortran-90 based multiprecision system. ACM Trans. Math. Software 21 379–387.
  • [3] Baker, G. A. Jr. and Graves-Morris, P. (1996). Padé Approximants, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [4] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press, Cambridge.
  • [5] Boyarchenko, M., de Innocentis, M. and Levendorskiĭ, S. (2011). Prices of barrier and first-touch digital options in Lévy-driven models, near barrier. Int. J. Theor. Appl. Finance 14 1045–1090.
  • [6] Cai, N. and Kou, S. (2012). Pricing Asian options under a hyper-exponential jump diffusion model. Oper. Res. 60 64–77.
  • [7] Cai, N. and Kou, S. G. (2011). Option pricing under a mixed-exponential jump diffusion model. Management Science 57 2067–2081.
  • [8] Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. The Journal of Business 75 305–333.
  • [9] Carr, P. and Madan, D. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2 61–73.
  • [10] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall, Boca Raton, FL.
  • [11] Crosby, J., Le Saux, N. and Mijatović, A. (2010). Approximating Lévy processes with a view to option pricing. Int. J. Theor. Appl. Finance 13 63–91.
  • [12] Donoghue, W. F. Jr. (1974). The interpolation of Pick functions. Rocky Mountain J. Math. 4 169–174.
  • [13] Fourati, S. (2012). Explicit solutions of the exit problem for a class of Lévy processes; applications to the pricing of double-barrier options. Stochastic Process. Appl. 122 1034–1067.
  • [14] Gautschi, W. (1970). On the construction of Gaussian quadrature rules from modified moments. Math. Comp. 24 245–260.
  • [15] Golub, G. H. and Welsch, J. H. (1969). Calculation of Gauss quadrature rules. Math. Comp. 23 221–230.
  • [16] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th ed. Elsevier/Academic Press, Amsterdam.
  • [17] Hackmann, D. and Kuznetsov, A. (2014). Asian options and meromorphic Lévy processes. Finance Stoch. 18 825–844.
  • [18] Iserles, A. (1979). A note on Padé approximations and generalized hypergeometric functions. BIT Numerical Mathematics 19 543–545.
  • [19] Jeannin, M. and Pistorius, M. (2010). A transform approach to compute prices and Greeks of barrier options driven by a class of Lévy processes. Quant. Finance 10 629–644.
  • [20] Kalugin, G. A., Jeffrey, D. J., Corless, R. M. and Borwein, P. B. (2012). Stieltjes and other integral representations for functions of Lambert $W$. Integral Transforms Spec. Funct. 23 581–593.
  • [21] Kudryavtsev, O. and Levendorskiĭ, S. (2009). Fast and accurate pricing of barrier options under Lévy processes. Finance Stoch. 13 531–562.
  • [22] Kuznetsov, A. (2010). Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 1801–1830.
  • [23] Kuznetsov, A. (2010). Wiener–Hopf factorization for a family of Lévy processes related to theta functions. J. Appl. Probab. 47 1023–1033.
  • [24] Kuznetsov, A. (2012). On the distribution of exponential functionals for Lévy processes with jumps of rational transform. Stochastic Process. Appl. 122 654–663.
  • [25] Kuznetsov, A., Kyprianou, A. E. and Pardo, J. C. (2012). Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22 1101–1135.
  • [26] Kwaśnicki, M. (2013). Rogers functions and fluctuation theory. Available at arXiv:1312.1866.
  • [27] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd ed. Springer, Heidelberg.
  • [28] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Probab. 45 118–134.
  • [29] Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The Variance Gamma process and option pricing. European Finance Review 2 79–105.
  • [30] Rogers, L. C. G. (1983). Wiener–Hopf factorization of diffusions and Lévy processes. Proc. Lond. Math. Soc. (3) 47 177–191.
  • [31] Schilling, R. L., Song, R. and Vondraček, Z. (2012). Bernstein Functions: Theory and Applications, 2nd ed. De Gruyter Studies in Mathematics 37. de Gruyter, Berlin.
  • [32] Szegö, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI.
  • [33] Weideman, J. A. C. (2005). Padé approximations to the logarithm. I. Derivation via differential equations. Quaest. Math. 28 375–390.