exchange rate distribution affect the firm’s optimal production and risk management
decisions. Making fairly priced currency options available to the firm has a positive effect
on production and, consequently, on expected exports and expected domestic sales if the
currency futures market is unbiased.
Appendix
A. Proof of Proposition 4
Partially differentiating the firm’s profits, Π*, as given in equation (3) with Z = 0, with
respect to S yields11
—T = Pf Qf - H* + ∂S max(S - Pd/Pf, 0)Pf (Q* - Qd - Qf ). (17)
∂S ∂S
The remainder of the proof is by contradiction. Inspection of equation (17) reveals that
dn*/dS ≤ 0 if Pf (Q* — Qd) ≤ H*. Given risk aversion and Pf (Q* — Qd) ≤ H*, U,(Π*)
is non-decreasing in S for S > Pd/Pf and strictly increasing for S ≤ Pd/Pf . Hence,
Cov(U,(Π*),S') > 0 for Pf (Q* — Qd) ≤ H*. Thus, equation (15) requires H* < Pf(Q* —
Qd).
Likewise, equation (17) implies ∂Π*∕∂S ≥ 0 if PfQf ≥ H. Thus, given PfQf ≥ H,
U,(Π*) is non-increasing in S for S ≤ Pd/Pf and decreasing in S for S > Pd/Pf such that
the covariance in equation (15) is negative. Hence, (15) implies H* >Pf Qf. °
B. Proof of Proposition 5
We have to show that Cov[U,(∏*), max(S — Pd/Pf, 0)] is negative. Partially differentiating
Π* with respect to S, given the optimal futures position H* as stated in Proposition 4,
results in12
∂ ∏*
∂S
Pf Qf — H* < 0 for S < Pd/Pf,
Pf (Q: — Qd) — H* > 0 for S > Pd/Pf.
11Πφ is continuous at S = Pd/Pf but not differentiable.
12Like I li, Π* is continuous at S = Pd/Pf but not differentiable.
20