Skip to main content Accessibility help
×
×
Home

Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case

  • Michael Schröder

Abstract

In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case
      Available formats
      ×

Copyright

Corresponding author

Postal address: Keplerstrasse 30, D-69469 Weinheim (Bergstrasse), Germany.

References

Hide All
Aase Nielsen, J. and Sandmann, K. (1995). Equity linked life insurance. Insurance Math. Econom. 16, 225253.
Aase Nielsen, J. and Sandmann, K. (1996). Uniqueness of the fair premium for equity-linked life insurance contracts. Geneva Papers Risk Insurance Theory 21, 65102.
Aase Nielsen, J. and Sandmann, K. (2002). The fair premium of an equity-linked life and pension insurance. In Advances in Finance and Stochastics, eds Schönbucher, P. and Sandmann, K., Springer, Heidelberg, pp. 218255.
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, Birkhäuser, Boston, MA, pp. 283318.
Bauer, H. (1996). Probability Theory. De Gruyter, Berlin.
Doetsch, G. (1971). Handbuch der Laplacetransformation I. Birkhäuser, Basel.
Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press.
Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407428.
Eberlein, E. (2001). Application of generalized hyperbolic Lévy motions to finance. In Lévy Processes, Birkhäuser, Boston, MA, pp. 319336.
Erdélyi, A. et al. (1981). Higher Transcendental Functions, Vol. II. Krieger, Malabar, FL.
Hämmerlin, G. and Hoffmann, K. H. (1989). Numerische Mathematik. Springer, Heidelberg.
Hull, J. C. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42, 281300.
Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Springer, Heidelberg.
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.
LIFFE (2004). Exchange contract no. 59. One month Euro overnight index average (EONIA) indexed contract. The London International Financial Futures and Options Exchange. Available at http://www.euronext.com/trader/contractspecifications/derivative/wide/0,5786,1732_627725,00.html?euronextCode=EON-LON-FUT.
Musiela, M. and Rutkowski, M. (1997). Martingale Methods in Financial Modelling. Springer, New York.
Nicolato, E. and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13, 445466.
Prause, K. (1999). The generalized hyperbolic model. , Universität Freiburg.
Raible, S. (2000). Lévy processes in finance. , Universität Freiburg.
Sansone, G. (1991). Orthogonal Functions. Dover, New York.
Schröder, M. (2005a). Laguerre series in contingent claim valuation, with applications to Asian options. Math. Finance 15, 491531.
Schröder, M. (2005b). Continuous time methods in the study of discretely sampled functionals of Lévy processes, II: the case of exponential Lévy processes. Working paper.
Schröder, M. (2005c). Continuous time methods in the study of discretely sampled functionals of Lévy processes, III: stochastic volatility models of OU type. Working paper.
Schröder, M. (2006a). On ladder height densities and Laguerre series in the study of stochastic functionals. I. Basic methods and results. Adv. Appl. Prob. 38, 969994.
Schröder, M. (2006b). On ladder height densities and Laguerre series in the study of stochastic functionals. II. Exponential functionals of Brownian motion and Asian option values. Adv. Appl. Prob. 38, 9951027.
Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions. Princeton University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed