Restricted Export Flexibility and Risk Management with Options and Futures

Proposition 1 (Separation) When currency call options⁸ with strike price P_d/Pf are
available, the restricted export flexible firm’s optimal output, Q*, is implicitly given by
c'(Q*) = Pd + PfC. Thus, Q* depends neither on the preferences of the firm nor on the
distribution of the exchange rate.

Since the derivation of c^z(Q*) = Pd + Pf C does not involve equation (6), i.e. the use
of currency futures, the above separation result holds even when H ≡ 0. The intuition
behind Proposition 1 can be explained using the thin dashed line in Figure 2 which exhibits
the firm’s marginal revenue with respect to the exchange rate as given by Pd + Pf max(S —
P_d/Pf, 0). Since the shape of this function exactly mirrors the shape of the call option’s
payoff (plus a constant), call options with strike price P_d/Pf span the firm’s exchange rate
exposure. Thus, the production decision is based on the market price for this exposure
as given by the call premium C . The optimal production decision is to equate marginal
costs with deterministic marginal revenue. If the condition of Proposition 1 is violated,
the firm can make a riskless profit. If, for example, output is less than Q*, then increasing
output and selling the associated exchange rate exposure by writing a call option on Pf
units of foreign currency results in a deterministic profit of Pd + PfC — c^z(Q) > 0. Hence,
the degree of risk aversion and the distribution of the exchange rate cannot affect the
optimal production decision.

A natural question to ask in the context of restricted export flexibility is whether the
tightness of the restrictions affects the firm’s optimal output. The following statement is
a direct consequence of Proposition 1.

Corollary 1 When currency call options with strike price P_d/Pf are available, the re-
stricted export flexible firm’s optimal output, Q*, is not affected by the tightness of the
restrictions arising from Qd and Qf .

This result is a direct implication of the optimality condition, c^z(Q*) = Pd + PfC,
which is unaffected by Qd and Qf . Since marginal revenue is independent of Qd and Qf ,

⁸ Due to put-call parity, the availability of currency futures and put options yields the same result.

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