# Change of numeraire in options with currency exchange features

FV of an EUR denominated option under "COP" risk measure is given by: $$V_t^{COP} = D^{COP} \mathbb{E}_t^{COP} \left[X_T(S_T -K)^+\right]$$ where $X_T$ is the exchange rate COP/EUR.

Pricing the option in EUR risk neutral measure mandates us to write the RHS above as (Girsanov's theorm): $$D^{COP} \mathbb{E}_t^{EUR}\left [\frac{d\mathbb{Q}^{COP}}{d\mathbb{Q}^{EUR}}|_t X_T (S_T-K)^+\right]$$ Where$\frac{d\mathbb{Q}^{COP}}{d\mathbb{Q}^{EUR}}|_t$ is the Radon Nikodym derivative.

How can we argue or derive that Radon Nikodym derivative in our case is given by: $\frac{d\mathbb{Q}^{COP}}{d\mathbb{Q}^{EUR}} |_t = \frac{X_t D^{EUR}}{X_T D^{COP}}$?

• Could you confirm the following? (1) $S_T$ and $K$ are expressed in EUR (2) Interest rates are deterministic (3) $D^{COP}$ is the discount factor for the COP currency Jun 19, 2018 at 9:06
• @byouness: Yes to all. Jun 19, 2018 at 9:58
• Ok, thanks. I answered your question below. Please let me know if anything remains unclear for you. Jun 19, 2018 at 9:59

### 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.

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:

1. Converting the payoff (which is in EUR) to COP at $T$ and then discounting in COP from $T$ to $t$;
2. Discounting the payoff from $T$ to $t$ in EUR and then converting the discounted value at $t$ from EUR to COP.
• Welcome. If the answer is what you were looking for, then could you please accept it so that it's easier to find for ppl who have the same question? Thanks. Jun 19, 2018 at 10:21
• The numéraire $M_t$ is the money market account denominated in EUR (expressed in the COP currency), while the numéraire $N_t$ is the money market account denominated in COP (expressed in the COP currency). Jun 19, 2018 at 11:48