Advertisement

Survival probability of LIFFE bond futures via the Mittag-Leffler function

  • Francesco Mainardi
  • Marco Raberto
  • Enrico Scalas
  • Rudolf Gorenflo
Conference paper

Abstract

The tick-by-tick dynamics of financial markets can be modeled by a continuous-time random walk (CTRW), as recently proposed by Scalas et al [16]. Here we point out the consistency of the model with the empirical analysis of the survival probability for certain bond futures (BUND and BTP) traded in 1997 at LIFFE, London. This requires the introduction of the Mittag-Leffler function as interpolating between a stretched exponential at small times and power-law at large times.

Keywords

Master Equation Fractional Derivative Fractional Calculus Delivery Date Caputo Fractional Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bachelier LJB (1900) Theorie de la Speculation. Ann. Ecole Normal Supérieure 17:21–86. Reprinted by Editions Jaques Gabay, Paris, 1995. [English translation with Editor’s notes in [4], pp. 17–78]MathSciNetMATHGoogle Scholar
  2. 2.
    Black F, Scholes M(1973) The pricing of options and corporate liabilities,. Journal of Political Economy 81:637–659.CrossRefGoogle Scholar
  3. 3.
    Bouchaud J-P, Potters M (2000) Theory of Financial Risk: from Statistical Physics to Risk Management. Cambridge Univ. Press, Cambridge.Google Scholar
  4. 4.
    Cootner PH (ed) (1964) The Random Character of Stock Market Prices. MIT Press, Cambridge MA.Google Scholar
  5. 5.
    Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (1955) Higher Transcendental Functions. McGraw-Hill, New York, Vol 3, pp 206–227MATHGoogle Scholar
  6. 6.
    Gorenflo R, Mainardi F (1997) Fractional calculus, integral and differential equations of fractional order. In: Carpinteri A, Mainardi F (eds) Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag, Wien and New York, pp 223–276 [CISM Lecture Notes, Vol. 378] [reprinted in NEWS 010101 http://www.fracalmo.org] Google Scholar
  7. 7.
    Mainardi F, Gorenflo R (2000) On Mittag-Leffler type functions in fractional evolution processes. J. Comput. and Appl. Mathematics 118:283–299MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Mainardi F, Raberto M, Gorenflo R., Scalas E (2000) Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287:468–481ADSCrossRefGoogle Scholar
  9. 9.
    Mandelbrot BB (1963) The variation of certain speculative prices. Journal of Business 36:394–419. [Reprinted in [4], pp. 307–332]CrossRefGoogle Scholar
  10. 10.
    Mantegna RN, Stanley HE (2000) An Introduction to Econophysics. Cambridge University Press, CambridgeGoogle Scholar
  11. 11.
    Merton RC (1990) Continuous Time Finance. Blackwell, Cambridge MAGoogle Scholar
  12. 12.
    Montroll EW, Weiss GH (1965) Random walks on lattices, II. J. Math. Phys. 6:167–181MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Podlubny I (1999) Fractional Differential Equations. Academic Press, San DiegoMATHGoogle Scholar
  14. 14.
    Samko SG, Kilbas AA, Marichev OI (1993) Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam [Translated from the Russian (Nauka i Tekhnika, Minsk, 1987)]MATHGoogle Scholar
  15. 15.
    Samuelson PA (1965) Rational theory of warrant pricing. Industrial Management Review 6:13–31Google Scholar
  16. 16.
    Scalas E, Gorenflo R, Mainardi F (2000) Fractional calculus and continuoustime finance. Physica A 284:376–384MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Japan 2002

Authors and Affiliations

  • Francesco Mainardi
    • 1
  • Marco Raberto
    • 2
  • Enrico Scalas
    • 3
    • 4
  • Rudolf Gorenflo
    • 5
  1. 1.Dipartimento di FisicaUniversità di Bologna and INFN Sezione di BolognaBolognaItaly
  2. 2.Dipartimento di Ingegneria Biofisica ed ElettronicaUniversità di GenovaGenovaItaly
  3. 3.Dipartimento di Scienze e Tecnologie AvanzateUniversità del Piemonte OrientaleAlessandriaItaly
  4. 4.INFN Sezione di TorinoTorinoItaly
  5. 5.Erstes Mathematisches InstitutFreie Universität BerlinBerlinGermany

Personalised recommendations