Restricted Export Flexibility and Risk Management with Options and Futures

Therefore, ∏* attains a unique minimum at S = P_d/Pf. (See Figure 4.) Since U''(∙) < 0,
U'(Щ) reaches a unique maximum at S = P_d/Pf. Thus, there exists a unique S > P_d/Pf
defined by U^,(Π*(S)) = E[U^,(Π*(S))]. Similarly, there exists a unique S# < P_d∕Pf
defined by U'(Π^(S#)) = E[U'(∏χ(S^,))].

First, consider case (a), defined by E[U'(∏_:)| S ≥ P_d∕Pf] not exceeding E[U'(Π_:)|S ≤
Pd/Pf], where E[∙∣∙] is the conditional expectation operator. By the definition of the
covariance operator, one can write

Cov [U'(∏:), max(S - Pd/Pf, 0)]

= /Pd^/P’ {u'(∏'(S)) - e[u'(∏:)]}{ - e[max(S - Pd/Pf,0)]} dG(S)

⁺ ∣S_m U '(^πxs )) - ^e[^u '(∏ : )]}

×{(S - Pd/Pf ) - e[ max(S - Pd/Pf, 0)]} dG(S)

= /Sp, {U'(∏(S)) - E[U'(∏:)]}(S - Pd/Pf) dG(S)

= /Sp, {U'(∏*(S)) - E[U'(∏Z)]}{(S - Pd/Pf) - (S - Pd/Pf)} dG(S)

+1Sp, {U'(∏'(S)) - E[U'(∏Z)] }(S - Pd/Pf) dG(S)

< (S - Pd/Pf) jS_/p {U'(∏Z(S)) - E[U'(∏Z)]} dG(S)

= (S - Pd/Pf)1 /S U'(∏Z(S)) dG(S) G(Pd/Pf)

I ^jPd∕p_f     `-

- Is^Fd'^F' U'(∏:(S)) dG(S) [1 - G(Pd/Pf )]1

= (S - Pd/Pf ) G(Pd/Pf ) [1 - G(Pd/Pf )]

× 1e[u '(∏ : )∣ S ≥ Pd/Pf ] - e[u '(∏ :)| S ≤ Pd/Pf ] 1,

where the inequality follows from the fact that {U'(∏X(S)) -E[U'(∏:)]} and (S-Pd/Pf ) -

(S - Pd/Pf ) = (S - S) have opposite signs for all S > Pd/Pf. The curly bracketed
term in the last line is non-positive by assumption. Since (S - Pd/Pf ), G(P_d/Pf ) and
[1 - G(P_d/Pf )] are all positive, Cov[U'(Щ), max(S - Pd/Pf, 0)] is negative for case (a).

__ _ __ ___{__}___. , . ~. _ , . _ ___{__}___. , . ~.

Now, consider case (b), in which E[U'(∏£)| S ≥ P_d/Pf] is greater than E[U'(∏:) ∣S ≤

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