Hedging Strategy and Electricity Contract Engineering - IFOR

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195
Proof The results follow immediately from Theorem 6.1. Case (i) corresponds
to abundant water and we optimally produce at as low prices as possible, since
a is monotone increasing in b. In case (ii) there is not enough water, even if
we chose the most conservative strategy to produce only at S S4r . In the
general case (iii) the concavity of a in b implies that a convex combination
of two adjacent points is optimal, whereas a being monotone increasing in b
gives the actual _
weights.
Corollary 6.3
Evidently the optimal policy when maximizing the expected profit given a constraint
on the averaged produced water is to choose SK such that
f S dS
WK
Proof Let the distance between the thresholds S4i
S4i 1
go to zero and
consider (6.23) in continuous time, then the results follow 7
immediately from
Theorem 6.1.
Sa S`
Corollary 6.4
The marginal value of water z \ W` 4¡b ] 7
z \ W` ]
b
(i) Is in the trivial case, WK Y where b41 zero
(ii) Is in the general case, where WK
b4i 6
b4i 7 1H for some i, given by
17 aWiS
aWi
, and is constant in the interval c b4i
WK 6
b4i 1 bWi
WK .
bWiS 17 7
Proof In the special case (i) an additional unit of water will not have any value,
since we cannot even utilize the currently available water. The general case
is easily proven by observing that the marginal value of water is given by the
bH slope _
aG of .
Proposition 6.5
The marginal value of water is never exceeding the average price of produced
.
electricity z \ W` ]
z \ W` 4¡b ] 7
z \ W` ]
W`