revenue is unaffected by the exchange rate. For exchange rates above this level, additional
output is exported such that the domestic currency value of marginal revenue linearly
increases in the exchange rate with slope Pf .
To hedge its exchange rate risk exposure, the firm can trade currency futures as well
as currency call and put options on the delivery of the domestic currency per unit of
the foreign currency. Of course, the firm makes its risk management decision at date 0,
i.e. before the realized exchange rate is known. Since the payoffs of any combination
of futures, call options and put options can be replicated by any two of these three
financial instruments using put-call parity (see, e.g., Sercu and Uppal, 1995), one of them
is redundant. Without loss of generality, we restrict the firm to use currency futures and
currency call options. Let F be the futures price and H be the number of currency futures
sold by the firm. In addition, let C denote the premium of a call option with strike price
K and Z denote the number of currency call options written by the firm. For simplicity,
K is chosen to be equal to Pd/Pf .5 The currency derivatives markets are competitive
such that F and C are not affected by the firm’s positions in these markets.
Taking the optimal sales allocation rule described above as given, the firm’s domestic
currency profits at date 1, denoted by Π, can be written as6
~ ~ ~. Γ _. . ~ . . 1 . . . .
Π = jR+(F - S}H + C - max(S - Pd∕Pf, 0)] Z - c(Q), (3)
where RR is defined in equation (2). The firm’s decision problem at date 0 is to choose an
output level, Q, and a hedge portfolio, (H, Z), so as to maximize the expected utility of
its domestic currency profits:
where E[∙] is the expectation operator, ∏ is defined in equation (3) and U(∏) is a von
Neumann-Morgenstern utility function defined over the firm’s domestic currency profits.
The firm is risk averse, U,(∏) > 0 and U"(∏) < 0.
max E
Q,H,Z
[U (∏)]
(4)
5In practice, it is relatively easy to trade in currency options with any strike price since the ma jority
of currency options is traded in the over-the-counter markets where products are not standardized.
6For simplicity, we assume an interest rate of zero such that costs c(Q) can simply be subtracted at
date 1.