Abstract
In recent years more realistic stochastic models for price movements in financial markets have been developed by replacing the classical Brownian motion by Lévy processes. Among these generalized hyperbolic Lévy processes turned out to provide an excellent fit to observed market data. We review the most important facts of the generating distributions and give a detailed derivation of its limits including some well known distributions which also have been applied to financial data. All these are subclasses of extended generalized Ń-convolutions. This fact allows the explicit calculation of characteristic triplets and the construction of uan convergent triangular schemes, whereas classical multinomial approximations are shown to fail in this context. The mixing generalized inverse Gaussian distributions are also considered.
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References
M. Abramowitz and I.A. StegunHandbook of Mathematical Functions(5th ed.), Dover, New York, 1968.
S. Asmussen and J. RoshiskiApproximations of small jumps of Lévy processes with a view towards simulationJ. Appl. Probab., 38 (2001), 482–493.
O.E. Barndorff-Nielsen and P. BlæsildHyperbolic distributions and ramifications: contributions to the theory and applicationsin: C. Taillie, G. Patil and B. Baldessari, Eds., Statistical Distributions in Scientific Work, Vol. 4, (Reidel, Dordrecht) (1981), 19–44.
O.E. Barndorff-NielsenExponentially decreasing distributions for the logarithm of particle sizeProc. Roy. Soc. London A, 353 (1977), 401–419.
O.E. Barndorff-NielsenProcesses of normal inverse Gaussian typeFinance Stoch., 2 (1998), 41–68.
O.E. Barndorff-Nielsen and C. HalgreenInfinite divisibility of the hyperbolic and generalized inverse Gaussian distributionsZ. Wahrscheinlichkeitstheorie verw. Geb., 38 (1977), 309–312.
O.E. Barndorff-Nielsen, J.L. Jensen and M. SorensenSome stationary processes in discrete and continuous timeAdv. Appl. Probab., 30 (1998), 989–1007.
P. BlæsildThe two-dimensional hyperbolic distribution and related distributions with an application to Johannsen’s bean dataBiometrika, 68 (1981), 251–263.
L. BondessonOn simulation from infinitely divisible distributionsAdv. Appl. Prob., 14 (1982), 855–869.
P.P. Carr, H. Geman, D.B. Madan and M. YorThe fine structure of asset returns: an empirical investigationJ. Business, 75 (2002), 305–332.
J.C. Cox, S.A. Ross and M. RubinsteinOption pricing: a simplified approachJ. Fin. Econ., 7 (1979), 229–264.
E. EberleinApplication of generalized hyperbolic Lévy motions to financein: O.E. Barndorff-Nielsen, T. Mikosch and S. Resnick, Eds., Lévy processes: theory and applications (Birkhäuser, Boston) (2001), 319–337.
E. Eberlein and U. KellerHyperbolic distributions in financeBernoulli, 1 (1995), 281–299.
E. Eberlein, U. Keller and K. PrauseNew insights into smile mispricing and value at risk: the hyperbolic modelJ. Business, 71 (1998), 371–405.
E. Eberlein and K. PrauseThe generalized hyperbolic model: financial derivatives and risk measuresin: H. Geman, D. Madan, S. Pliska and T. Vorst, Eds., Mathematical Finance — Bachelier Congress 2000, Springer, Berlin, 2001, 245–267.
E. Eberlein, J. Kallsen and J. KristenRisk management based on stochastic volatilityJ. Risk, 5 (2003), 19–44.
W. FellerAn Introduction to Probability Theory and Its ApplicationsVol. II (2nd ed.), Wiley, New York, 1971.
B.V. GnedenkoOn limit theorems for a random number of random variablesLecture Notes in Math.1021(1983), 167–176.
E. GrosswaldThe Student-t distribution of any degree of freedom is infinitely divisibleZ. Wahrscheinlichkeitstheorie verw. Geb.36(1976), 103–109.
C. HalgreenSelf-decomposability of the generalized inverse Gaussian and hyperbolic distributionsZ.Wahrscheinlichkeitstheorie verw. Geb., 47 (1979), 13–17.
P.R. HalmosMeasure TheorySpringer, New York, 1974.
J. Jacod and A.N. ShiryaevLimit Theorems for Stochastic ProcessesSpringer, Berlin, 1987.
N.L. Johnson, S. Kotz and N. BalakrishnanContinuous Univariate DistributionsVol. 2 (2nd ed.), Wiley, New York, 1995.
B. JorgensenStatistical Properties of the Generalized Inverse Gaussian DistributionLecture Notes in Statistics, 9 (1982), Springer, New York.
M. LoèveProbability Theory I (4thed.), Springer, New York, 1977.
E. LukacsCharacteristic Functions(2nd ed.), Griffin, London, 1970.
D.B. Madan, P.P. Carr and E.C. ChangThe variance gamma process and option pricingEurop. Finance Rev., 2 (1998), 79–105.
S.T. Rachev and L. Rüschendorf.Models for option pricesTheory Probab. Appl.39(1994), 120–152.
A. Rejman, A. Weron and R. WeronOption pricing proposals under the generalized hyperbolic modelComm Statist. Stochastic Models, 13 (1997), 867–885.
T.H. RydbergThe normal inverse Gaussian Lévy process: simulation and approximationComm. Statist. Stochastic Models13(1997), 887–910.
O. ThorinOn the infinite divisibility of the Pareto distributionScand. Act. J., (1977), 31–40.
O. ThorinOn the infinite divisibility of the lognormal distributionScand. Act. J., (1977), 121–148.
O. Thorin, Anextension of the notion of a generalized 1-convolutionScand. Act. J., (1978), 141–149.
S.J. WolfeOn moments of infinitely divisible distribution functionsAnn. Math. Stat., 42 (1971), 2036–2043.
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Eberlein, E., v. Hammerstein, E.A. (2004). Generalized Hyperbolic and Inverse Gaussian Distributions: Limiting Cases and Approximation of Processes. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability, vol 58. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7943-9_15
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DOI: https://doi.org/10.1007/978-3-0348-7943-9_15
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