Restricted Export Flexibility and Risk Management with Options and Futures

Pd/Pf\. Since the covariance operator is linear, equation (15) implies Cov[U^,(∏*),S] =
Cov[U^,(H*), S — Pd/Pf] = 0. Using the fact that (S — P_d/Pf ) = max(S — P_d∕Pf, 0) —
max(P_d∕Pf — S, 0) and the linearity again results in Cov[U^,(∏*), max(S^, — Pd/Pf, 0)] =
Cov[U^,(∏*), max(P_d∕Pf — S, 0)]. Using the definition of the covariance operator again
yields

Cov [U^,(∏:), max(Pd/Pf — S, 0)]

= /P‘^/P’ " '(^πj<s )) — e[u^,(∏ :)]}

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

+ /’_/pι {u'(n:(S)) — e[u'(∏:)]}{ — e[max(Pd/Pf — S,0)]} dG(S)

= /P^d/P' {u '(n:⁽S )) — e[u'(∏ :)]}(Pd/P, — s ) dG(S )

= /^Pd/P' {u '(n:(s )) — e[u'(∏ : )]}{(Pd/Pf — s ) — (Pd/Pf — s #)} <jg(s )

+/^Pd/P' {u ' (n:(s)) — e[u'(∏ :)]}(Pd/Pf — s #) dG(s)

< (Pd/Pf — S#) /f {U'(∏J(S)) — e[u'(∏:)]} dG(S)

= (Pd/Pf — S#)1 y' U'(∏((S)) dG(S) [1 — G(Pd/Pf)]

∣_pS U'(∏(S)) dG(S) G(Pd/Pf )!
ʧd/4                            )

= (Pd/Pf — S#) G(PdJP_l) [1 — G(PdJP_l)]

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

where the inequality follows from the fact that {U^,(∏i(S)) — E[U^,(∏:)]} and (Pd/Pf —
S) — (Pd/Pf — S#) = (S# — S) have opposite signs for all S < Pd/Pf. Since the curly
bracketed term in the last line is negative by assumption and (Pd/Pf — S#) is positive,
Cov[U^,(∏:), max(S — Pd/Pf, 0)] is negative for case (b) as well. □

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