Stationary-increment Student and variance-gamma processes

Abstract

A continuous-time model with stationary increments for asset price {P t } 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−rt P t } a familiar martingale. We then specify the activity time process, {T t }, for which {T t − t} is asymptotically self-similar and {τ t }, with τ t = T t − T t−1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) X t and a model for {X t } 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.