  In the theory of extreme values of Gaussian processes, many results are
expressed in terms of the Pickands constant $\mathcal{H}_{\alpha}$. This
constant depends on the local self-similarity exponent $\alpha$ of the process,
i.e. locally it is a fractional Brownian motion (fBm) of Hurst index
$H=\alpha/2$. Despite its importance, only two values of the Pickands constant
are known: ${\cal H}_1 =1$ and ${\cal H}_2=1/\sqrt{\pi}$. Here, we extend the
recent perturbative approach to fBm to include drift terms. This allows us to
investigate the Pickands constant $\mathcal{H}_{\alpha}$ around standard
Brownian motion ($\alpha =1$) and to derive the new exact result
$\mathcal{H}_{\alpha}=1 - (\alpha-1) \gamma_{\rm E} + \mathcal{O}\!\left(
\alpha-1\right)^{2}$.
