In this exercise we work with the Black-Scholes setting applied to foreign currency denominated assets. We will see a different use of Girsanov theorem. [for more detail see Musiela and Rutkowski (1997).] Let r,f denote the domestic and the foreign risk-free rates. Let St be the exchange rate, that is, the price of 1 unit of foreign currency in terms of domestic currency. Assume a geometric process for the dynamics of St;
dSt = (r - f)Stdt + σStdWt.
(a) Show that where Wt is Wiener process under probability P.
(b) Is the process a martingale under measure P?
(c) Let P be the probability what does Girsonov theorem imply about the process, Wt - σt,
under P?
(d) shown using Ito formula that
dZt = Zt[(f - r + σ2)dt - σdWt]
where Zt = 1/St.
(e) Under which probability is the process Ztert f eft a martingale?
(f) Can we say that P is the arbitrage-free measure of the foreigneconomy?
