t \leq \bar T$ and $i,j=1,\ldots,N$.

Given Assumption \[AnsatzFX\], it is immediate to compute via a direct application of the Itô formula the dynamics of all exchange rates and all primary security accounts in all currency denominations.

The exchange rate $S^{i,j}(t)$ under the real world probability measure ${\mathbb{P}}$ evolves according to the dynamics $$\begin{aligned}
\begin{aligned}
\label{exchangerates}
\frac{{\mathrm{d}}S^{i,j}(t)}{S^{i,j}(t)}&= ( r^i(t)-r^j(t)){\mathrm{d}}t+\langle \boldsymbol{\pi}^i(t)-\boldsymbol{\pi}^j(t),\boldsymbol{\pi}^i(t){\mathrm{d}}t+{\mathrm{d}}\boldsymbol{Z}(t)\rangle,\\
  S^{i,j}(0)&=s^{i,j}>0, 
  \end{aligned}\end{aligned}$$ and the generic $j$-th primary security account $B^{i,j}$, in $i$-th currency denomination and under the real world probability measure ${\mathbb{P}}$, evolves according to the dynamics $$\begin{aligned}
\begin{aligned}
\label{eq:primarySecuritySDE}
\frac{{\mathrm{d}}B^{i,j}(t)}{B^{i,j}(t)}&= r^i(t){\mathrm{d}}t+\langle \boldsymbol{\pi}^i(t)-\boldsymbol{\pi}^j(t),\boldsymbol{\pi}^i(t){\mathrm{d}}t+{\mathrm{d}}\boldsymbol{Z}(t)\rangle, \\
  B^{i,j}(0)&=b^{i,j},
  \end{aligned}\end{aligned}$$ for $\ i,j = 1,\ldots,N$ and $t\in[0,\bar T]$.

The Benchmark Approach {#sec:benchmarkApproach}
----------------------

In the present paper we evaluate contingent claims under the *benchmark approach* of [@bookplaten10]. Under this approach, price processes denominated in terms of the GOP are called benchmarked price processes. More precisely, for $i,j=1,\ldots N$, let us introduce the benchmarked price process $$\begin{aligned}
\hat{B}^{j}=\left\{\hat{B}^{j}(t):=\frac{B^{i,j}(t)}{D^i(t)},\ 0\leq t\leq \bar{T}\right\}.\end{aligned}$$ We call $\hat{B}^{j}$ the *benchmarked $j$-th primary security account*. Note that $\hat{B}^{j}$ does not depend on the index $i$ of the currency denomination we started from. Given and , upon an application of the Itô formula, it is immediate to conclude that all benchmarked 