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 Pd/Pf ) = max(S Pd∕Pf, 0)
max(Pd∕Pf S, 0) and the linearity again results in Cov[U,(∏*), max(S, Pd/Pf, 0)] =
Cov[U
,(∏*), max(Pd∕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)

= /Pd/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(PdJPl) [1 G(PdJPl)]

×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 (P
d/Pf S#) is positive,
Cov[U
,(∏:), max(S Pd/Pf, 0)] is negative for case (b) as well.

22



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