Consider an economy involving three currencies. For i = 1, 2, 3, let
denote the value at time t of a zero coupon bond that pays a unit amount at T in currency i. Further, for i, j = 1, 2, 3, i = j, let
denote the value at time t in currency i of one unit of currency j, and let
denote the corresponding forward FX rate at time t for maturity at T. Recall that
Suppose that a complete arbitrage-free model has been specified for the economy for the finite time interval
denote the EMM corresponding to numeraire
(i) Show that the time-zero value, stated in currency 2, of a call option that pays
in currency 2 at time T can be expressed as
(ii) Suppose we are given the prices for all strikes of call options on
and
If the model is calibrated to these prices explain, using the expression in (i), what these prices tell us about the distributional information needed to calculate call prices on
(iii) Let W∗ be a two-dimensional Brownian motion on the probability space (Ω, F,Q1) and let
be the augmented natural filtration generated by W∗. Now suppose that under Q1
the processes
and
satisfy
Where
and a are positive constants and
If Q2
denotes the EMM corresponding to numeraire
show that, under Q2,
Where
is an
Brownian motion and
For this model derive an expression for
the time-zero value of the vanilla FX call option described in (i).