  Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed
by the Hurst exponent $H$. It generalises standard Brownian motion
(corresponding to $H=1/2$). We study the probability distribution of the
maximum $m$ of the process and the time $t_{\rm max}$ at which the maximum is
reached. They are encoded in a path integral, which we evaluate perturbatively
around a Brownian, setting $H=1/2 + \varepsilon$. This allows us to derive
analytic results beyond the scaling exponents. Extensive numerical simulations
for different values of $H$ test these analytical predictions and show
excellent agreement, even for large $\varepsilon$.
