Under the risk neutral measure, suppose the dynamics of the domestic interest rate r_d and foreign interest rate r_f follow the mean-reversion processes:

Let _ρdf and _ρfX denote the correlation coefficient between the domestic and foreign interest rates, and between the foreign interest rate and exchange rate, respectively. Taking the notional to be unity and assuming that the interest payment to be paid at T₁ is known at time t, show that the sum of the present values at time t of all future interest payments based on the foreign LIBORs in a differential swap is given by (Wei, 1994; Chang, Chung and Yu, 2002)

where B_d (t, T_i) and B_f (t, T_i) are the T_i-maturity domestic and foreign bond discount prices, respectively, and

Using financial arguments, explain why _ρdf and _ρfX enter into the pricing formula, and the differential swap would be worth something even if the domestic and foreign forward rates are the same and evolve with perfect correlation.

Transcribed Image Text:

draka (Mara) dt + oddZa drf = kf (ufrf) dt+of dZf, and the exchange rate X follows XP X = (ra-rf) dt+ox dZx.