Restricted Export Flexibility and Risk Management with Options and Futures

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* — Q_d) ≤ H*. Given risk aversion and Pf (Q* — Q_d) ≤ 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 ≤ P_d/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

Pf Qf — H* < 0        for S < Pd/Pf,

Pf (Q: — Qd) — H* > 0 for S > Pd/Pf.

¹¹Π_φ is continuous at S = P_d/Pf but not differentiable.

¹²Like I l_i, Π* is continuous at S = Pd/Pf but not differentiable.

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