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Stationary-increment Student and variance-gamma processes

Part of: Stochastic processes Distribution theory - Probability Inference from stochastic processes Statistics

Published online by Cambridge University Press:  14 July 2016

Richard Finlay  and
Eugene Seneta
Richard Finlay*
Affiliation:
University of Sydney
Eugene Seneta*
Affiliation:
University of Sydney
*
∗Postal address: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia.
∗Postal address: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia.
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Abstract

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A continuous-time model with stationary increments for asset price {Pt} is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e−rtPt} a familiar martingale. We then specify the activity time process, {Tt}, for which {Tt − t} is asymptotically self-similar and {τt}, with τt = Tt − Tt−1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) Xt and a model for {Xt} which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.

Keywords

Skew-stationary processskewnessmartingalesubordinatorvolatilitylong-range dependencet-distributionvariance-gamma distributionLaplace distributionself-similarityheavy tailminimum χ2 estimationefficient market

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60E99: None of the above, but in this section 62-07: Data analysis 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 441 - 453
Copyright
© Applied Probability Trust 2006 

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