Hedging Strategy and Electricity Contract Engineering - IFOR

R
S x GaRKV is convex and continuously differentiable with respect to R , and U d S xV
Fd
is given as the minimization Fd S of GaRKV x with respect R to
Furthermore the R
that minimizes Fd S x GaRKV equals Red S xV
M
3.5 The need for a new risk measure 63
max E lS x G YV
x†ˆ‡
n
d S xV s.t. U C,
(3.3)
where the expected profit is maximized, subject to a constraint C on the risk,
measured as CVaR with a confidence level of c .
The function U d S xV is unfortunately in general not easy to handle, especially
not when involved in an optimization problem. This problem can however be
avoided by introducing the similar function
1
1 c
E lS x G Y V R
Fd S x GaRKV
G
and by sharpening the assumptions needed to define VaR and CVaR, and also
require that the distribution of Y does not depend on x and that lS x G YV is continuous
in x. Under these assumptions [RU00] have shown the following two
theorems
Theorem 3.4
min
Fd S x G?RKV .
U d S xV
j †m‡
left endpoint of argmin
Fd S x GaRbV .
Red S xV
†ˆ‡ j
The Fd S function GaRKV x cannot only be used to measure CVaR, it can also be
used to manage a portfolio in a CVaR perspective, as seen in Theorem 3.5.
Theorem 3.5
U d S xV Minimizing with respect to x X is equivalent to Fd S minimizing G?RKV x
over S all G@RbV x X in the sense that
min
X U d S xV min
x†
Fd S x GaRbV
H xg j@L † X`‰‡