is rather strong and may be empirically rejected, see e.g. [@currencyplaten06], the findings in [@martinoplaten13] and the necessary and sufficient conditions of [@ruf15]. Hence, in the present paper, we shall allow each $\Lambda^i$ to be either a true martingale or a strict local martingale. To work in such a generalized setting requires a more general pricing concept than the one provided under the classical risk neutral paradigm. In the following, we will employ the notion of *real world pricing*: a price process ${\mathfrak{V}}^i=\left\{{\mathfrak{V}}^i(t), 0\leq t \leq \bar{T}\right\}$, here denominated in $i$-th currency, is said to be *fair* if, when expressed in units of the GOP $D^i$, forms a ${\mathbb{P}}$-martingale, this means its benchmarked value forms a true ${\mathbb{P}}$-martingale, see Definition 9.1.2 in [@bookplaten10]. For a fixed maturity $T\in [0, \bar{T}]$, we let ${\mathcal{H}}(T)={\mathfrak{V}}(T)$ be an ${\mathcal{F}}_T$-measurable non-negative benchmarked contingent claim, such that when expressed in units of the $i$-th currency denomination as ${\mathcal{H}}^i(T)=D^i(T){\mathcal{H}}(T)$ we have $$\begin{aligned}
\mathbb{E}\left[\left.{\mathcal{H}}(T)\right|{\mathcal{F}}_t\right]=\mathbb{E}\left[\left.\frac{{\mathcal{H}}^i(T)}{D^i(T)}\right|{\mathcal{F}}_t\right]<\infty\end{aligned}$$ for all $0\leq t\leq T\leq \bar{T}$, $i=1,...,N$. The benchmarked fair price ${\hat V}(t)=V^i(t)/D^i(t)$ of this contingent claim is the minimal possible price and given by the following conditional expectation under the real-world probability measure ${\mathbb{P}}$: $$\begin{aligned}
\label{eq:realWorldPricingFormula}
{\hat {\mathfrak{V}}(t)}=\mathbb{E}\left[\left.{\hat {\mathcal{H}}(T)}\right|{\mathcal{F}}_t\right],\end{aligned}$$ which is known 