# Modelling foreign exchange

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).

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 $$\mu=r^d-r^f$$

• Thank You! I had the Ito's lemma part wrong. Dec 20, 2019 at 20:16