Before revision:  It turns out that, for \mbox{%DIFAUXCMD \cite{BrenierMartingale}\hspace{0pt}%DIFAUXCMD 's construction, the change of numeraire can be viewed as a mirror coupling for positive martingales, while for \mbox{%DIFAUXCMD \cite{HobsonKlimmek2013} }\hspace{0pt}%DIFAUXCMD it }  }\hspace{0pt}%DIFAUXCMD } exchanges forward start straddles of type I and type II giving also that the optimal transport plan in the subhedging problems is the same for both types of options.
After revision:  It turns out that, for \mbox{%DIFAUXCMD \cite{BrenierMartingale}\hspace{0pt}%DIFAUXCMD 's construction, the change of numeraire can be viewed as a mirror coupling for positive martingales, while for \mbox{%DIFAUXCMD \cite{HobsonKlimmek2013} }\hspace{0pt}%DIFAUXCMD it } applied to \mbox{%DIFAUXCMD \cite{HobsonKlimmek2013 }\hspace{0pt}%DIFAUXCMD } exchanges forward start straddles of type I and type II giving also that the optimal transport plan in the subhedging problems is the same for both types of options.