For FX time series, it is found empirically that trending prices or drifting prices influence the subsequent volatility differently. When the price path shows a distinct trend (up or down), the subsequent volatility tends to be larger. In the opposite case where the price path is drifting sideway, the subsequent volatility tends to be smaller. This effect has been analysed empricaly for FX data in Zumbach , and various models proposed. The possible intuitive explanation for the effect is that a clear price change induces traders to modify their positions (hence creating volatility) whereas still prices lead them to keep their positions.
The trend versus drift shape for the past price path can be measured by a simple product of two non-overlapping returns. In equation, the form
is positive when both returns have the same sign, and negative otherwise. In order to include the effect in a process, this product should be included in the volatility feed-back equation. This can be done in many ways, and the overall strategy is to go along the core structure of a process.
The present page introduce a GARCH(1,1) process with a trend, while more sophisticated variations are given in the related pages. For the GARTCH(1,1) proces, the historical volatility σ1 is unchanged, while the effective volatility is given by
The parameters for the simulations are
The innovations have a Student distribution with 3.3 degree of freedom. The simulation time corresponds to 200 years with a time increment δt = 3 minutes.