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 [2010], 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

(1)

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 σ_{1} is unchanged, while the effective volatility is
given by

The parameters for the simulations are

σ∞ = 0.11

w∞ = 0.18

τ = 1 day

θ_{1} = 810^{-7}

The cross term returns use a lag of 480 steps on the 3 minutes grid,
corresponding to Δt =24 hours.

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.

References

Gilles Zumbach. Volatility conditional on price trends. QuantitativeFinance, 10:431–442, 2010.