Changing Numéraires for Currency-Converted Option Payoffs

# Change of numeraire in options with currency exchange features


# 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}} $?

## Answer by byouness (score 2, accepted)

https://quant.stackexchange.com/a/40387

#### 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.