  We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang
2014) pertaining to the existence of a shadow price process for portfolio
optimisation under proportional transaction costs. There, we established a
positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$
satisfying the condition $(NUPBR)$ of "no unbounded profit with bounded risk".
This condition requires that $S$ is a semimartingale and therefore is too
restrictive for applications to models driven by fractional Brownian motion. In
the present paper, we derive the same conclusion under the weaker condition
$(TWC)$ of "two way crossing", which does not require $S$ to be a
semimartingale. Using a recent result of R.~Peyre, this allows us to show the
existence of a shadow price for exponential fractional Brownian motion and
$all$ utility functions defined on the positive half-line having reasonable
asymptotic elasticity. Prime examples of such utilities are logarithmic or
power utility.
