Exchange Rate Economics: Theories and Evidence

Target zone models 293
which is clearly negative. The smooth pasting condition of the S curve is in fact
well known from the option pricing literature and in the analysis of irreversible
investment (see,for example,Dumas 1988). The analogy with option pricing
may be illustrated in the following way. Consider a (continuous time) present
value formulation of expression (12.1) (see our discussion of such present value
formulations of the monetary model in Chapter 4):
( ) ∫ 1 ∞
s t = (m + v)e −(1/α)(t−τ) dτ,(12.11)
α
t
where as in equation (4.6) the current spot rate may be viewed as the present
discount value of future realisations of m + v,where α is the discounting factor. By
differentiating (12.11) with respect to t yields the equation (12.1).
If the price of an asset is given by (12.11),where m is held constant at,say,m 0
then the value of this asset would be:
( ) ∫ 1 ∞
˜s t = (m 0 + v)e −(1/α)(t−τ) dτ,(12.12)
α
t
and the actual exchange rate, s,may be regarded as a compound asset,that is,the
sum of the imaginary asset,whose price is determined by (12.12),and the right to
sell the asset at a price s,plus the obligation to sell at the price ¯s on demand. The
derivation of the S curve then becomes the combined price of the two options:
the requirement to sell on demand becomes more important the higher ˜s is,so
the price of the compound asset falls below ˜s at high v. Conversely,the right to
sell the asset at s supports the value of the asset at low v.
So far the assumption about the behaviour of the money supply in defence of
the target zone has not been made explicit. This behaviour can be illustrated in
the following way. Suppose that initially the market is at point 1 on Figure 12.2.
There is then a series of positive shocks to velocity which moves the system to a
point such as 2. However,at this point any further increases in v would have to
be offset by reductions in m so that the exchange rate stays constant as the market
S
2 3
1
4
v
Figure 12.2 Monetary policy and a family of S curves.