Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-22T11:14:40.405Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATING THE SKEWNESS IN DISCRETELY OBSERVED LÉVY PROCESSES

Published online by Cambridge University Press:  01 October 2004

Jeannette H.C. Woerner
Jeannette H.C. Woerner
Affiliation:
University of Göttingen
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider models for financial data by Lévy processes, including hyperbolic, normal inverse Gaussian, and Carr, Geman, Madan, and Yor (CGMY) processes. They are given by their Lévy triplet (μ(θ),σ2,eθxg(x)ν(dx)), where μ denotes the drift, σ2 the diffusion, and eθxg(x)ν(dx) the Lévy measure, and the unknown parameter θ models the skewness of the process. We provide local asymptotic normality results and construct efficient estimators for the skewness parameter θ taking into account different discrete sampling schemes.I thank Prof. Dr. L. Rüschendorf for his steady encouragement, the referees for helpful comments, and the German National Scholarship Foundation for financial support.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 5 , October 2004 , pp. 927 - 942
Copyright
© 2004 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Akritas, M.G. & R.A. Johnson (1981) Asymptotic inference in Levy processes of the discontinuous type. Annals of Statistics 9, 604–614.Google Scholar
Bachelier, L. (1900) Theorie de la speculation. Gauthier-Villar.
Barndorff-Nielsen, O.E. (1977) Exponentially decreasing distributions for the logarithm of particle size. In Proceedings of the Royal Society London A, vol. 353, pp. 401–419.
Barndorff-Nielsen, O.E. (1996) Probability and statistics: Selfdecomposability, finance and turbulence. In L. Accardi et al. (ed.), Proceedings of the Conference “Probability towards 2000,” Centre for Applied Probability, Cornell University.
Barndorff-Nielsen, O.E. (1998) Processes of normal inverse Gaussian type. Finance and Stochastics 2, 41–68.Google Scholar
Carr, P., H. Geman, D. B. Madan, & M. Yor (2002) The fine structure of asset returns: An empirical investigation. Journal of Business 75, 305–332.Google Scholar
Eberlein, E. & U. Keller (1995) Hyperbolic distributions in finance. Bernoulli 1, 281–299.Google Scholar
Gnedenko, B.V. & A.N. Kolmogorov (1968) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.
Godambe, V.P. & C.C. Heyde (1987) Quasi-likelihood and optimal estimation. International Statistical Review 55, 231–244.Google Scholar
Hájek, J. (1972) Local asymptotic minimax and admissibility in estimation. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability vol. 1, pp. 175–194.
Heyde, C.C. (1988) Fixed sample and asymptotic optimality for classes of estimating functions. Contemporary Mathematics 80, 241–247.Google Scholar
Janssen, A. (1992) Conditions for local asymptotic normality of exponential families. Statistics and Decisions 10, 173–182.Google Scholar
Jeganathan, P. (1981) On a decomposition of the limit distribution of a sequence of estimators. Sankhya A 43, 26–36.Google Scholar
Jeganathan, P. (1983) Some properties of risk functions in estimation when the limit of the experiment is mixed normal. Sankhya A 45, 66–86.Google Scholar
Keller, U. (1997) Realistic Modelling of Financial Derivatives. Ph.D. thesis, University of Freiburg.
Küchler, U. & M. Sørensen (1997) Exponential Families of Stochastic Processes. Springer-Verlag.
Le Cam, L. (1960) Locally asymptotically normal families of distributions. University of California Publications in Statistics 3, 37–98.Google Scholar
Le Cam, L. & G. Yang (1990) Asymptotics in Statistics. Springer.
Madan, D.B. & E. Senata (1990) The variance gamma (VG) model for share market returns. Journal of Business 63, 511–524.Google Scholar
Prause, K. (1999) The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures. Ph.D. thesis, University of Freiburg.
Rachev, S.T. & S. Mittnik (2000) Stable Paretian Models in Finance. Wiley.
Raible, S. (2000) Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis, University of Freiburg.
Rydberg, T. (1997) The normal inverse Gaussian Levy process: Simulation and approximation. Communications in Statistics, Stochastic Models 13, 887–910.Google Scholar
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Shiryaev, A.N. (1999) Essentials of Stochastic Finance: Facts, Models and Theory. World Scientific.
Skorokhod, A.V. (1957) On the differentiability of measures which correspond to stochastic processes, part I. Processes with independent increments. Theory of Probability and Applications 2, 407–432.Google Scholar
Tucker, H.G. (1962) Absolute continuity of infinitely divisible distributions. Pacific Journal of Mathematics 12, 1125–1129.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge Series on Statistical and Probabilistic Mathematics. Cambridge University Press.
Witting, H. (1985) Mathematische Statistik I. B.G. Teubner.
Woerner, J.H.C. (2001) Statistical Analysis for Discretely Observed Lévy Processes. Ph.D. thesis, University of Freiburg.