Hedging Strategy and Electricity Contract Engineering - IFOR

R
R
M
R
R
64 Risk management
where moreover
argmin
argmin
x† X
H xg j@L † X`‰‡
argmin
S xŠ G?R Š V
Fd S x GaRbV
Fd S xŠ G?RbV
xŠ
U d S xV?G‹R Š
†m‡ j
A proof of Theorem 3.4 and 3.5 is given in [RU00]. lS If G x V Y is convex with
respect to x, then (3.2) and (3.3) can with help Fd S of GaRbV x be solved with
convex programming, since both the objective function and the constraint are
then convex.
The integral Fd S in G?RKV x can be approximated by sampling from the probability
distribution of Y . 9 If we sample a collection of vectors y P?P@P
G 1G y J then
S x GaRbV can be approximated by
Fd
S 1
1
c V J
J
j… 1
M P
Fd S x GaRKV
lS x G y jV
If the loss lS function G x V Y is linear in x, the optimization problem involving
CVaR can be solved with linear programming. This is a very nice feature of
CVaR, since linear programming can handle very large problems efficiently.
The lS terms G x y jV Fd S in GaRKV x are not linear, but piecewise linear. This
can however be resolved by replacing these terms by the auxiliary variables z j ,
and imposing the constraints z j x G y jV R , z lS j 0, 1G
P?P@P
G j J. The
optimization problem (3.2) can then be reduced to the linear program
min
1
n g z†ˆ‡
J g j †m‡ H 1I d L J j… 1 z j
x†m‡
s.t. z j x G y jV RŒG z lS j j 1G 0G
1
J
J
J
j… 1 lS x G y jV R,
P?P@P
G J
(3.4)
and (3.3) similarly to
9 There is a whole theory on how to sample and generate random numbers. We have
in this work used the Monte Carlo method, but there are a vast number of variancereduction
techniques, such as Latin hypercube sampling. For more on sampling see, for
example, [Nie92].