Notations
- $S_T$ and $K$ are expressed in EUR;
- $D^{CCY}(t,T) = \frac{\beta^{CCY}_t}{\beta^{CCY}_T}$ where $\beta^{CCY}$ is the money market account in currency $CCY$). In other words, it is the (stochastic) discount factor from $t$ to $T$ in the currency $CCY$;
- $X_t$ is the value of 1 EUR in COP.
Answer
The expression of the Radon-Nikodym derivative follows from the numéraire change formula. If $N$ and $M$ are two numéraires with corresponding measures $\mathbb{Q}^N$ and $\mathbb{Q}^M$, then:
$$\frac{d\mathbb{Q}^{N}}{d\mathbb{Q}^{M}}|_t = \frac{N_T}{M_T} \frac{M_t}{N_t}$$
Here, $N_t = \beta^{COP}_t$, while $M_t = \beta^{EUR}_t X_t$.
It follows that:
$$\frac{d\mathbb{Q}^{COP}}{d\mathbb{Q}^{EUR}}|_t = \frac{\beta^{EUR}_t X_t}{\beta^{EUR}_T X_T} \frac{\beta^{COP}_T}{\beta^{COP}_t} = \frac{D^{EUR}(t,T)}{D^{COP}(t,T)} \frac{X_t}{X_T}$$
Leading to the following expression for the option price in COP:
$$\begin{aligned}
V_t^{COP} & = \mathbb{E}^{COP}_t \left[ D^{COP}(t,T) X_T (S_T - K)^+ \right] \\
& = X_t \mathbb{E}^{EUR}_t \left[ D^{EUR}(t,T) (S_T - K)^+ \right]
\end{aligned}$$
Pratically speaking, what this expresses is that these two things are the same:
- Converting the payoff (which is in EUR) to COP at $T$ and then discounting in COP from $T$ to $t$;
- Discounting the payoff from $T$ to $t$ in EUR and then converting the discounted value at $t$ from EUR to COP.
