Stochastic Equilibrium Level of the Underlying Process

The values of $$\overline{S}$$ and $$\overline{X}$$ denote the mean values, i.e. the long-run equilibrium levels, of the price process and the log-price process, respectively. In the preceding chapters, these parameters are held constant, which implies that in the long-run, every shock in the price process is removed due to the mean reversion. In this section, we model the mean of the price process itself as an additional stochastic factor $$\mathfrak{X}$$ , which is equivalent to the assumption that there exist shocks in the price process which persist. We address both a mean-reverting process and a Brownian motion process with (presumably positive) drift as subordinated equilibrium level process. While the former accounts for additional risk in price variations, the latter can represent inflation effects. The combination with jumps in the equilibrium level process as discussed in Sect. 6.3.2 may account for regime shift risk. For instance, the crude oil price behavior in the last few years can be modeled according to either a large value of the drift component $${\mu }_{\mathfrak{X}}$$ or a large upward jump in the equilibrium level process.