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Modelling by Lévy Processess for Financial Econometrics

  • Ole E. Barndorff-Nielsen
  • Neil Shephard

Abstract

This paper reviews some recent work in which Lévy processes are used to model and analyse time series from financial econometrics. A main feature of the paper is the use of posi- tive Ornstein-Uhlenbeck-type (OU-type) processes inside stochastic volatility processes. The basic probability theory associated with such models is discussed in some detail.

Keywords

Stochastic Volatility Stochastic Volatility Model Inverse Gaussian Distribution Cumulant Generate Function Levy Process 
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.

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References

  1. [1]
    S. Asmussen (1998), Stochastic simulation with a view towards stochastic processes, Lecture Notes 2, MaPhySto, Aarhus University.Google Scholar
  2. [2]
    S. Asmussen and J. Rosiński (2000), Approximation of small jumps of Lévy processes with a view towards simulation, Research Report 2000–2, MaPhySto, Aarhus University.Google Scholar
  3. [3]
    S. Bar-Lev, D. Bshouty, and G. Letac (1992), Natural exponential families and self-decomposability, Statistics and Probability Letters, 13, 147–152.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    O. E. Barndorff-Nielsen (1986), Sand, wind and statistics, Acta Mechanica, 64, 1–18.CrossRefGoogle Scholar
  5. [5]
    O. E. Barndorff-Nielsen (1998a), Probability and statistics: Selfdecomposability, finance and turbulence, in L. Acccardi and C. C. Heyde, eds., Probability Towards 2000: Proceedings of a Symposium held 2–5 October 1995 at Columbia University, Springer-Verlag, New York, 47–57.Google Scholar
  6. [6]
    O. E. Barndorff-Nielsen (1998b). Processes of normal inverse Gaussian type. Finance and Stochastics, 2, 41–68.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    O. E. Barndorff-Nielsen (1999), Superposition of Ornstein-Uhlenbeck type processes, Theory of Probability and Its Applications, to appear.Google Scholar
  8. [8]
    O. E. Barndorff-Nielsen (2000), Lévy processes and Lévy random fields: From a stochastic modelling perspective, in preparation.Google Scholar
  9. [9]
    O. E. Barndorff-Nielsen, J. L. Jensen, and M. Sørensen (1998), Some stationary processes in discrete and continuous time. Advances in Applied Probability, 30, 989–1007.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    O. E. Barndorff-Nielsen and M. Leonenko (2000), Non-Gaussian scenarios in Burgers turbulence, in preparation.Google Scholar
  11. [11]
    O. E. Barndorff-Nielsen and V. Pérez-Abreu (1999), Stationary and selfsimilar processes driven by Lévy processes, Stochastic Processes and Their Applications, 84, 357–369.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    O. E. Barndorff-Nielsen and K. Prause (2001), Apparent scaling, Finance and Stochastics, 5,103–113.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    O. E. Barndorff-Nielsen and N. Shephard (2001), Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, Journal of the Royal Statistical Society, B63, to appear.Google Scholar
  14. [14]
    J. Bertoin (1996), Lévy Processes, Cambridge University Press, Cambridge.Google Scholar
  15. [15]
    J. Bertoin (2000), Some properties of Burgers turbulence with white or stable noise initial data, in O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds., Lévy Processes: Theory and Applications, Birkhäuser, Boston, 2001 (this volume), 267–279.Google Scholar
  16. [16]
    F. Black (1976), Studies of stock price volatility changes, Proceedings of the Business and Economic Statistics Section, American Statistical Association, 177–181.Google Scholar
  17. [17]
    T. Bollerslev, R. F. Engle, and D. B. Nelson (1994), ARCH models, in R. F. Engle and D. McFadden, eds., The Handbook of Econometrics, Vol. 4, North-Holland, Amsterdam, 2959–3038.Google Scholar
  18. [18]
    L. Bondesson (1982), On simulation from infinitely divisible distributions, Advances in Applied Probability, 14, 855–869.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    A. Brix (1998), Spatial and spatio-temporal models for weed abundance, Ph.D. thesis, Department of Mathematics and Physics, Royal Veterinary and Agricultural University, Copenhagen.Google Scholar
  20. [20]
    J. Y. Campbell, A. W. Lo, and A. C. MacKinlay (1997), The Econometrics of Financial Markets, Princeton University Press, Princeton, NJ.MATHGoogle Scholar
  21. [21]
    T. Chan (1999), Pricing contingent claims on stocks driven by Lévy processes, Annals of Statistics, 27, 504–528.Google Scholar
  22. [22]
    R. Cont, M. Potters, and J. P. Bouchaud (1997), Scaling in stock market data: Stable laws and beyond. in B. Dubrulle, F. Graner, and D. Sornette, eds., Scale Invariance and Beyond: Proceedings of the CNRS Workshop on Scale Invariance, Les Houches, 75–85.Google Scholar
  23. [23]
    E. Csáki, M. Csörgö, Z. Y. Lin, and P. Révész (1991), On infinite series of independent Ornstein-Uhlenbeck processes, Stochastic Processes and Their Applications, 39, 25–44.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    F. X. Diebold and M. Nerlove (1989), The dynamics of exchange rate volatility: A multivariate latent factor ARCH model, Journal of Applied Econometrics, 4, 1–21.CrossRefGoogle Scholar
  25. [25]
    E. Eberlein (2000), Application of generalized hyperbolic Lévy motion to finance, in O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds., Lévy Processes: Theory and Applications, Birkhäuser, Boston, 2001 (this volume), 319–336.Google Scholar
  26. [26]
    E. Eberlein and U. Keller (1995), Hyperbolic distributions in finance, Bernoulli, 1, 281–299.MATHCrossRefGoogle Scholar
  27. [27]
    R. F. Engle and J. R. Russell (1998), Forecasting transaction rates: The autoregressive conditional duration model. Econometrica, 66, 1127–1162.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    T. S. Ferguson and M. J. Klass (1972), A representation of independent increment processes without Gaussian components, Annals of Mathematical Statistics, 43, 1634–1643.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    U. Frisch (1995), Turbulence, Cambridge University Press, Cambridge.MATHGoogle Scholar
  30. [30]
    E. Ghysels, A. C. Harvey, and E. Renault (1996), Stochastic volatility, in C. R. Rao and G. S. Maddala, eds., Statistical Methods in Finance, North-Holland, Amsterdam, 119–191.CrossRefGoogle Scholar
  31. [31]
    C. Halgreen (1979), Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 47, 13–17.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    S. Hodges and A. Carverhill (1993), Quasi mean reversion in an efficient stock market: The characterization of economic equilibria which support the Black-Scholes option pricing, Economic Journal, 103, 395–405.CrossRefGoogle Scholar
  33. [33]
    S. Hodges and M. J. P. Selby (1997), The risk premium in trading equilibria which support the Black-Scholes option pricing, in M. Dempster and S. Pliska, eds., Mathematics of Derivative Securities, Cambridge University Press, Cambridge, 41–53.Google Scholar
  34. [34]
    F. Hubalek and L. Krawczyk (1999), Simple explicit formulae for variance-optimal hedging for processes with stationary independent increments, unpublished paper.Google Scholar
  35. [35]
    J. Jacod and A. N. Shiryaev (1987), Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin.MATHGoogle Scholar
  36. [36]
    Z. J. Jurek and J. D. Mason (1993), Operator-Limit Distributions in Probability Theory, Wiley, New York.MATHGoogle Scholar
  37. [37]
    Z. J. Jurek and W. Vervaat (1983), An integral representation for selfdecomposable Banach space valued random variables, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 62, 247–262.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    M. King, E. Sentana, and S. Wadhwani (1994), Volatility and links between national stock markets, Econometrica, 62, 901–933.MATHCrossRefGoogle Scholar
  39. [39]
    I. Koponen (1995), Analytic approach to the problem of convergence of truncated Lévy flights towards the gaussian stochastic process, Physics Review E, 52, 1197–1199.CrossRefGoogle Scholar
  40. [40]
    S. Levendorskii(2000), Generalized truncated Lévy processes and generalizations of the Black-Scholes formula and equation, with some applications to option pricing, personal communication to O. E. Barndorff-Nielsen.Google Scholar
  41. [41]
    P. Lévy (1937), Theories de L’Addition Aléatories, Gauthier-Villars, Paris.Google Scholar
  42. [42]
    Z. Y. Lin (1995), On large increments of infinite series of Ornstein-Uhlenbeck processes, Stochastic Processes and Their Applications, 60, 161–169.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    M. Loéve (1955), Probability Theory, Van Nostrand, Princeton, NJ.MATHGoogle Scholar
  44. [44]
    R. Mantegna and H. E. Stanley (2000), Introduction to Econophysics: Correlation and Complexity in Finance, Cambridge: Cambridge University Press.Google Scholar
  45. [45]
    M. B. Marcus (1987), ???-Radial Processes and Random Fourier Series, Memoirs of the American Mathematical Society 368, AMS, Providence, RI.Google Scholar
  46. [46]
    D. B. Nelson (1991), Conditional heteroskedasticity in asset pricing: A new approach, Econometrica, 59, 347–370.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    E. Nicolato (1999), A class of stochastic volatility models for the term structure of interest rates, Ph.D. thesis, Department of Mathematics Sciences, Aarhus University.Google Scholar
  48. [48]
    E. Nicolato and E. Venardos (2001), Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type with a leverage effect, in preparation.Google Scholar
  49. [49]
    E. A. Novikov (1994), Infinitely divisible distributions in turbulence, Physics Reviews E, 50, R3303–R3305.CrossRefGoogle Scholar
  50. [50]
    J. Pitman and M. Yor (1981), Bessel processes and infinite divisible laws, in D. Williams, ed., Stochastic Integrals, Lecture Notes in Mathematics 851, Springer-Verlag, Berlin, 285–370.CrossRefGoogle Scholar
  51. [51]
    M. K. Pitt and N. Shephard (1999), Time varying covariances: A factor stochastic volatility approach (with discussion), in J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds., Bayesian Statistics 6, Oxford University Press, Oxford, 547–570.Google Scholar
  52. [52]
    P. Protter (1992), Stochastic Integration and Differential Equations, Springer-Verlag, New York.Google Scholar
  53. [53]
    J. Rosiński (1991), On a class of infinitely divisible processes represented as mixtures of Gaussian processes, in S. Cambanis, G. Samorodnitsky, and M. S. Taqqu, eds., Stable Processes and Related Topics, Birkhäuser, Basel, 27–41.CrossRefGoogle Scholar
  54. [54]
    J. Rosiński (2000), Series representations of Lévy processes from the perspective of point processes, in O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds., Lévy Processes: Theory and Applications, Birkhäuser, Boston, 2001 (this volume), 401–415.Google Scholar
  55. [55]
    T. H. Rydberg (1999), Generalized hyperbolic diffusions with applications towards finance, Mathematical Finance, 9, 183–201.MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    T. H. Rydberg and N. Shephard (2000), A modelling framework for the prices and times of trades made on the NYSE, in W. J. Fitzgerald, R. L. Smith, A. T. Walden, and P. C. Young, eds., Nonlinear and Nonstationary Signal Processing, Isaac Newton Institute and Cambridge University Press, Cambridge, to appear.Google Scholar
  57. [57]
    G. Samorodnitsky and M. S. Taqqu (1994), Stable Non-Gaussian Random Processes, Chapman and Hall, New York.MATHGoogle Scholar
  58. [58]
    K. Sato (1980), Class L of multivariate distribution functions and its subclasses, Journal of Multivariate Analysis, 10, 207–232.MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    K. Sato (1982), Absolute continuity of multivariate distributions of class L, Journal of Multivariate Analysis, 12, 89–94.MathSciNetMATHCrossRefGoogle Scholar
  60. [60]
    K. Sato (1999), Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.MATHGoogle Scholar
  61. [61]
    K. Sato and M. Yamazato (1983), Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type, Stochastic Processes and Their Applications, 17, 73–100.MathSciNetCrossRefGoogle Scholar
  62. [62]
    K. Sato and M. Yamazato (1984), Stationary processes of Ornstein-Uhlenbeck type, in K. Itô and J. V. Prokhorov, eds., Probability Theory and Mathematical Statistics: Fourth USSR-Japan Symposium Proceedings, 1982, Lecture Notes in Mathematics 1021, Springer-Verlag, Berlin, 541–551.Google Scholar
  63. [63]
    N. Shephard (1996), Statistical aspects of ARCH and stochastic volatility. in D. R. Cox, D. V. Hinkley, and O. E. Barndorff-Nielson, eds., Time Series Models in Econometrics, Finance and Other Fields, Chapman and Hall, London, 1–67.Google Scholar
  64. [64]
    A. V. Skorohod (1991), Random Processes with Independent Increments Kluwer, Dordrecht.MATHCrossRefGoogle Scholar
  65. [65]
    F. W. Steutel (1970), Preservation of Infinite Divisibility under Mixing and Related Topics, Mathematics Centre Tracts 33, Mathematical Centrum, Amsterdam.MATHGoogle Scholar
  66. [66]
    S. J. Taylor (1994), Modelling stochastic volatility, Mathematical Finance, 4, 183–204.MATHCrossRefGoogle Scholar
  67. [67]
    K. Urbanik (1969), Self-decomposable probability distributions on Rm, Zastosowania Matematyki, 10, 91–97.MathSciNetMATHGoogle Scholar
  68. [68]
    W. Vervaat (1979), On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Advances in Applied Probability, 11, 750–783.MathSciNetMATHCrossRefGoogle Scholar
  69. [69]
    S. Walker, and P. Damien (2000), Representations of Lévy processes without Gaussian components, Biometrika, 87, 477–483.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    J. B. Walsh (1981), A stochastic model of neural response, Advances in Applied Probability, 13,231–281.MathSciNetMATHCrossRefGoogle Scholar
  71. [71]
    S. J. Wolfe (1982), On a continuous analogue of the stochastic difference equation xn = pxn-\ + bn, Stochastic Processes and Their Applications, 12, 301–312.MathSciNetMATHCrossRefGoogle Scholar
  72. [72]
    R. L. Wolpert and K. Ickstadt (1998), Poisson/gamma random field models for spatial statistics, Biometrika, 85, 251–267.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Neil Shephard
    • 2
  1. 1.Centre for Mathematical Physics and Stochastics (MaPhySto)University of AarhusAarhus CDenmark
  2. 2.Nuffield CollegeUniversity of OxfordOxfordUK

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