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We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang2014) pertaining to the existence of a shadow price process for portfoliooptimisation under proportional transaction costs. There, we established apositive 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 toorestrictive for applications to models driven by fractional Brownian motion. Inthe present paper, we derive the same conclusion under the weaker condition$(TWC)$ of "two way crossing", which does not require $S$ to be asemimartingale. Using a recent result of R.~Peyre, this allows us to show theexistence of a shadow price for exponential fractional Brownian motion and$all$ utility functions defined on the positive half-line having reasonableasymptotic elasticity. Prime examples of such utilities are logarithmic orpower utility.

Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs