Hedging Strategy and Electricity Contract Engineering - IFOR

3.4 Traditional risk management models 55
lS Let G YV x denote the loss function of a decision variable x
n , which we
can see as a portfolio, and a random vector Y
m representing the future
values of stochastic variables of interest, such as spot price or demand. The
density of Y is denoted by f Y .
We assume lS that G V x is measurable and let S F G V x for each x X denote the
distribution function for the lS loss G YV x , S F GaRbV x S P lS Y G x V RbV Y , where
X is a subset of
n and can be interpreted as the set of available portfolios.
At a given confidence c S 0G 1V level , which in risk management typically
could be 0.99, c the -percentile of the loss distribution is called VaR.
Definition 3.1
VaR with confidence level c
of the loss associated with a decision x is the value
Red inf R F S x GaRKV c .
In Figure 3.1 the 95% percentile of the profit and loss distribution V aR 95% ,
together with CV aR 95% , which will be defined soon, are illustrated.
VaR attempts to provide a meaningful answer to the important question
How much are we likely to lose?
but actually only states that we are c % certain that we will not lose more than
V aRd in the given time horizon.
There are in the industry three traditional methods to calculate VaR; the variance/covariance
method, historical simulation and Monte Carlo simulation.
Further the by Embrechts et al. [EKM97] proposed extreme value theory approach
has increased much attention from the industry in the last years.
Variance/covariance method In this approach the risk is calculated analytically
based on the statistical properties of the risk factors. The assumption
here is that the returns of the positions are multi normal distributed, why only
the first two moments are of interest. This analytical method is therefore called