in the literature as *real-world pricing formula*, see Corollary 9.1.3 in [@bookplaten10]. Note that benchmarked risk minimisation, described in [@duplaten14], gives generally. In case $\Lambda^i$ is a true martingale we obtain, by changing in from the real world probability measure ${\mathbb{P}}$ to the equivalent risk neutral probability measure $\mathbb{Q}^i$, the risk neutral pricing formula $$\begin{aligned}
\label{eq:realWorldEqualsRiskNeutral}
\begin{aligned}
{\mathfrak{V}}^i(t)&=\mathbb{E}\left[\left.\frac{B^i(t)}{B^i(T)}\frac{B^i(T)}{B^i(t)}\frac{D^i(t)}{D^i(T)}{\mathcal{H}}^i(T)\right|{\mathcal{F}}_t\right]\\
&=\mathbb{E}\left[\left.\frac{\Lambda^i(T)}{\Lambda^i(t)}\frac{B^i(t)}{B^i(T)}{\mathcal{H}}^i(T)\right|{\mathcal{F}}_t\right]={\mathbb{E}^{{\mathbb{Q}}^i}\left[\left.\frac{B^i(t)}{B^i(T)}{\mathcal{H}}^i(T)\right|{\mathcal{F}}_t\right]}.
\end{aligned}\end{aligned}$$ This shows that in this case the real-world pricing formula generalises the classical risk-neutral valuation formula and the Radon-Nikodym derivative for the respective risk neutral probability measure is given by . In general, due to the supermartingale property of benchmarked price processes in the case when $\Lambda^i$ is a strict supermartingale, a formally obtained risk neutral price is greater than or equal to the real world price, see [@duplaten14].\
Later on we consider the pricing of zero coupon bonds. According to , the minimal possible price $P^i(t,T)$ at time $t$ of a zero coupon bond, denominated in the $i$-th currency and paying at maturity $T$ one unit of the $i$-th currency, is given by the formula $$\begin{aligned}
\label{zerocouponbond}
P^i(t,T)=D^i(t)\mathbb{E}\left[\left.\frac{1}{D^i(T)}\right|{\mathcal{F}}_t\right].\end{aligned}$$

All benchmarked non-negative price processes are supermartingales. Therefore, as already mentioned, the fair price process of a contingent claim is the 