Change of numeraire in the two-marginals martingale transport problem

Abstract

In this paper, we apply change of numeraire techniques to the optimal transport approach for computing model-free prices of derivatives in a two-period setting. In particular, we consider the optimal transport plan constructed in Hobson and Klimmek (Finance Stoch. 19:189–214, 2015 ) as well as the one introduced in Beiglböck and Juillet (Ann. Probab. 44:42–106, 2016 ) and further studied in Henry-Labordère and Touzi (Finance Stoch. 20:635–668, 2016 ). We show that in the case of positive martingales, a suitable change of numeraire applied to Hobson and Klimmek (Finance Stoch. 19:189–214, 2015 ) exchanges forward start straddles of type I and type II, so that the optimal transport plan in the subhedging problems is the same for both types of options. Moreover, for Henry-Labordère and Touzi’s (Finance Stoch. 20:635–668, 2016 ) construction, the right-monotone transference plan can be viewed as a mirror coupling of its left counterpart under the change of numeraire.

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