Skip to main content
SpringerLink
Log in

Fast and accurate pricing of barrier options under Lévy processes

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

We suggest two new fast and accurate methods, the fast Wiener–Hopf (FWH) method and the iterative Wiener–Hopf (IWH) method, for pricing barrier options for a wide class of Lévy processes. Both methods use the Wiener–Hopf factorization and the fast Fourier transform algorithm. We demonstrate the accuracy and fast convergence of both methods using Monte Carlo simulations and an accurate finite difference scheme, compare our results with those obtained by the Cont–Voltchkova method, and explain the differences in prices near the barrier.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abate, J., Whitt, W.: A unified framework for numerically inverting Laplace transforms. INFORMS J. Comput. 18, 408–421 (2006)

    Article  MathSciNet  Google Scholar 

  2. Albanese, C., Kuznetsov, A.: Discretization schemes for subordinated processes. Working paper (2003). http://www.level3finance.com/poisson.pdf

  3. Amin, K.: Jump-diffusion option valuation in discrete time. J. Finance 48, 1833–1863 (1993)

    Article  Google Scholar 

  4. Alili, L., Kyprianou, A.: Some remarks on first passage of Lévy processes, the American put option and pasting principles. Ann. Appl. Probab. 15, 2062–2080 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Avram, F., Chan, T., Usabel, M.: On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr’s approximation for American puts. Stoch. Proc. Appl. 100, 75–107 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Asmussen, A., Avram, F., Pistorius, M.R.: Russian and American put options under exponential phase-type Lévy models. Stoch. Proc. Appl. 109, 79–111 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Barndorff-Nielsen, O.E.: Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Barndorff-Nielsen, O.E., Levendorskiǐ, S.: Feller processes of normal inverse Gaussian type. Quant. Finance 1, 318–331 (2001)

    Article  MathSciNet  Google Scholar 

  9. Boyarchenko, M.: Carr’s randomization for finite-lived barrier options: proof of convergence. Working paper (2008). http://ssrn.com/abstract=1275666

  10. Boyarchenko, M.: Discontinuity of value functions of certain options with barriers. Working paper (2009). http://ssrn.com/abstract=1322285

  11. Boyarchenko, N., Levendorskiǐ, S.: On errors and bias of Fourier transform methods in quadratic term structure models. Int. J. Theor. Appl. Finance 10, 273–306 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Boyarchenko, S.I., Levendorskiǐ, S.Z.: Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Finance 3, 549–552 (2000)

    Article  MATH  Google Scholar 

  13. Boyarchenko, S.I., Levendorskiǐ, S.Z.: Perpetual American options under Lévy processes. SIAM J. Control Optim. 40, 1663–1696 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Boyarchenko, S.I., Levendorskiǐ, S.Z.: Non-Gaussian Merton–Black–Scholes Theory. World Scientific, New Jersey (2002)

    MATH  Google Scholar 

  15. Boyarchenko, S.I., Levendorskiǐ, S.Z.: Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Probab. 12, 1261–1298 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Boyarchenko, S.I., Levendorskiǐ, S.Z.: American options: the EPV pricing model. Ann. Finance 1, 267–292 (2005)

    Article  Google Scholar 

  17. Boyarchenko, S.I., Levendorskiǐ, S.Z.: American options in Lévy models with stochastic interest rates. Comput. Finance 12(4) (2009). http://ssrn.com/abstract=1015409

  18. Boyarchenko, S.I., Levendorskiǐ, S.Z.: Pricing American options in regime-switching models. SIAM J. Control Optim. 48, 1353–1376 (2009)

    Article  MathSciNet  Google Scholar 

  19. Broadie, M., Detemple, J.: Option pricing: valuation models and applications. Manag. Sci. 50, 1145–1177 (2004)

    Article  Google Scholar 

  20. Carr, P., Faguet, D.: Fast accurate valuation of American options. Working paper, Cornell University, Ithaca (1994)

  21. Carr, P.: Randomization and the American put. Rev. Financ. Stud. 11, 597–626 (1998)

    Article  Google Scholar 

  22. Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–332 (2002)

    Article  Google Scholar 

  23. Carr, P., Hirsa, A.: Why be backward? Risk 26, 103–107 (2003)

    Google Scholar 

  24. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Press, London (2004)

    MATH  Google Scholar 

  25. Cont, R., Voltchkova, E.: A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43, 1596–1626 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Eberlein, E., Keller, U.: Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995)

    Article  MATH  Google Scholar 

  27. Eberlein, E., Keller, U., Prause, K.: New insights into smile, mispricing and value at risk: the hyperbolic model. J. Bus. 71, 371–406 (1998)

    Article  Google Scholar 

  28. Eskin, G.I.: Boundary Problems for Elliptic Pseudo-Differential Equations. Nauka, Moscow (1973). (Transl. of Mathematical Monographs, vol. 52. Amer. Math. Soc., Providence (1980))

    Google Scholar 

  29. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2003)

    Google Scholar 

  30. Hirsa, A., Madan, D.B.: Pricing American options under variance gamma. J. Comput. Finance 7(2), 63–80 (2003)

    Google Scholar 

  31. Hörmander, L.: Analysis of Partial Differential Operators, vol. III. Springer, Berlin (1985)

    Google Scholar 

  32. Koponen, I.: Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)

    Article  Google Scholar 

  33. Kou, S.G.: A jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101 (2002)

    Article  Google Scholar 

  34. Kou, S., Wang, H.: First passage times of a jump diffusion process. Adv. Appl. Probab. 35, 504–531 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kudryavtsev, O.E., Levendorskiǐ, S.Z.: Pricing of first touch digitals under normal inverse Gaussian processes. Int. J. Theor. Appl. Finance 9, 915–949 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kyprianou, A.E., Pistorius, M.R.: Perpetual options and Canadization through fluctuation theory. Ann. Appl. Probab. 13, 1077–1098 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Levendorskiǐ, S.Z.: Pricing of the American put under Lévy processes. Int. J. Theor. Appl. Finance 7, 303–335 (2004)

    Article  MathSciNet  Google Scholar 

  38. Levendorskiǐ, S., Kudryavtsev, O., Zherder, V.: The relative efficiency of numerical methods for pricing American options under Lévy processes. J. Comput. Finance 9(2), 69–97 (2005)

    Google Scholar 

  39. Lipton, A.: Assets with jumps. Risk Mag. 15, 149–153 (2002)

    Google Scholar 

  40. Longstaff, F., Schwartz, E.: Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14, 113–147 (2001)

    Article  Google Scholar 

  41. Lukacs, E.: Characteristic Functions. Charles Griffin & Company limited, London (1960)

    MATH  Google Scholar 

  42. Madan, D.B., Carr, P., Chang, E.C.: The variance gamma process and option pricing. Eur. Finance Rev. 2, 79–105 (1998)

    Article  MATH  Google Scholar 

  43. Madan, D.B., Yor, M.: Representing the CGMY and Meixner processes as time changed Brownian motions. J. Comput. Finance 12(1), 27–47 (2008)

    Google Scholar 

  44. Matache, A.-M., Nitsche, P.-A., Schwab, C.: Wavelet Galerkin pricing of American options on Lévy driven assets. Quant. Finance 5, 403–424 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  45. Metwally, S., Atiya, A.: Using Brownian bridge for fast simulation of jump-diffusion processes and barrier options. J. Deriv. 10, 43–54 (2002)

    Article  Google Scholar 

  46. Poirot, J., Tankov, P.: Monte Carlo option pricing for tempered stable (CGMY) processes. Asia Pac. Financ. Mark. 13, 327–344 (2006)

    Article  MATH  Google Scholar 

  47. Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  48. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  49. Sepp, A.: Analytical pricing of double-barrier options under a double-exponential jump diffusion process: applications of Laplace transform. Int. J. Theor. Appl. Finance 7, 151–175 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  50. Wang, I.R., Wan, J.W.I., Forsyth, P.A.: Robust numerical valuation of European and American options under the CGMY process. J. Comput. Finance 10(4), 31–69 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, Russian Customs Academy Rostov Branch, Budennovskiy 20, Rostov-on-Don, 344002, Russia

    Oleg Kudryavtsev

  2. Department of Mathematics, The University of Leicester, University Road, Leicester, LE1 7RH, UK

    Sergei Levendorskiǐ

Authors
  1. Oleg Kudryavtsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergei Levendorskiǐ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleg Kudryavtsev.

Additional information

The first author is supported, in part, by grant RFBR 09-01-00781.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kudryavtsev, O., Levendorskiǐ, S. Fast and accurate pricing of barrier options under Lévy processes. Finance Stoch 13, 531–562 (2009). https://doi.org/10.1007/s00780-009-0103-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-009-0103-2

Keywords

Mathematics Subject Classification (2000)

JEL Classification

Search

Navigation