Modelling foreign exchange

Question

I am currently stuck in an assignment in mathematics on the modelling of foreign exchanges. Any help is welcome!

Consider a model for the exchange rate between two countries $f$ (foreign) and $d$ (domestic), and denote by $e_t$ the price in domestic currency of one unit of the foreign currency. Assume that the short rate in the two countries is constant and denote it by $r^d$ respectively by $r^f$. $B_{t}^{d} = e^{r^{d}t} (B^{f}_{t} = e^{r^{f}t})$ denote the price of the domestic (foreign) savings account. The exchange-rate dynamics under $P$ is given by $de_{t} = \mu e_{t}dt + \sigma e_{t}dW_{t}$.

Denote by $Q^{d}$ the domestic martingale measure, that is the martingale measure that corresponds to the numeraire $B^{d}$. Show that under $Q^{d}$ the drift of $e_{t}$ is equal to $r^{d} −r^{f}$. Hint: consider the drift of the asset $S^{f}_{t} := e_{t}B^{f}_{t}$ (the value of the foreign savings account denominated in domestic currency).

Answer 1

Define the process $M_t=\frac{S_t^f}{B_t^d}$.

$S_t^f$ is an asset denominated in the domestic currency, therefore the process $M_t$ is a martingale under the $Q^d$ measure ( ratio between an asset and the numeraire)

Use Ito's lemma,

$dM_t=\frac{dS_t^f}{B_t^d}+S_t^fd\left(\frac{1}{B_t^d}\right)$

$dS_t^f=B_t^fde_t+dB_t^fe_t$

$dM_t=\frac{B_t^fde_t+dB_t^fe_t}{B_t^d}+S_t^fd\left(\frac{1}{B_t^d}\right)$

We have $$dB_t^d=r^dB_t^ddt$$ $$d\left(\frac{1}{B_t^d}\right)=-r^dB_t^ddt$$ $$dB_t^f=r^fB_t^fdt$$

Substituting all these equation to the sde of $M_t$, we have

$$d M_t=M_t\left(\mu+r^f-r^d\right)dt+...$$ Finally,we know that the drift of $M_t$ is zero because it is a martingale, therefore $$\mu=r^d-r^f$$