We construct a process with inverse gamma increments and an asymptotically self-similar limit. This construction supports the use of long-range-dependent t subordinator models for actual financial data as advocated in Heyde and Leonenko (2005), in that it allows for noninteger-valued model parameters, as is found empirically in model estimation from data.

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.

This paper has as its main theme the fitting in practice of the variance-gamma distribution, which allows for skewness, by moment methods. This fitting procedure allows for possible dependence of increments in log returns, while retaining their stationarity. It is intended as a step in a partial synthesis of some ideas of Madan, Carr and Chang (1998) and of Heyde (1999). Standard estimation and hypothesis-testing theory depends on a large sample of observations which are independently as well as identically distributed and consequently may give inappropriate conclusions in the presence of dependence.