Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 28J / c / Solution

For a Dollar investor, the Euro bank account has Dollar value $e^{r_{E}t}/Y_t$. After Dollar discounting, it is $e^{(r_E-r_D)t}/Y_t$. Its conditional mean growth exponent is $r_E-r_D+{\sigma }^2-\mu$, so the martingale requirement givesWith equal rates, $Y^{-1}$ is a martingale under $Q_D$ by part (a), whereas $Y$ has positive drift ${\sigma }^2$ and is not a martingale when $\sigma \ne 0$. In the degenerate zero-volatility case, both exchange rates are constant and both are martingales; the asserted nonmartingale conclusion implicitly requires a genuinely risky exchange rate.