In this post, we consider stochastic clocks whose instantaneous rate of activity follows a square root process. We present all intermediate steps in the derivation of the characteristic function of the corresponding time change. The corresponding steps closely resemble those in the derivation of the zero-coupon bond price in the Cox et al. (1985) model for the short-term interest rate.
After introducing the general idea of stochastic clocks in the previous post, we now consider the case where the instantaneous rate of activity follows a non-Gaussian Ornstein-Uhlenbeck process. The major reference is the paper by Barndorff-Nielsen and Shephard (2001b), see also Barndorff-Nielsen and Shephard (2001a). In this post, we discuss the general stochastic setup that applies to all such models that are driven by an almost surely non-decreasing Lévy processes. For the moment, we leave the particular dynamics of this subordinator unspecified but we will provide examples in future posts.
This is the first post in a series on asset price dynamics that are subject to a stochastic clock. The main idea is to not model the volatility as a stochastic process but instead make the rate at which time progresses randomly. In this post, we outline the basic idea and define the terminology.