Hedging Strategy and Electricity Contract Engineering - IFOR

68 Risk management
If the market is complete then this probability measure Q is unique and a
unique pricing formula that is independent on the utility functions of the
different players, such as the B&S formula, can be derived. The probability
measure Q is then often referred to as the risk neutral probability measure,
since all contingent claims can be perfectly replicated and hence the risk can
be eliminated.
In an incomplete market however the probability measure Q is not unique
and a wide range of arbitrage free prices can be derived, depending on
which measure that is chosen, as shown in for example [EJ97]. This nonuniqueness
can be interpreted as if the utility function of the different players
comes into play. A pricing model that does not involve the special risk aversion
of the different players will therefore probably fail to give a precise assessment.
3.7.2. Equilibrium
An approach that does not make the assumption of complete markets is the
family of equilibrium pricing models.
The Capital Asset Pricing Model (CAPM) was developed by Sharpe [Sha64].
Under the assumptions of normal distributed log-returns he shows that the
equilibrium expected return of a security can be written as a function of the
expected market return, the interest rate and a dependence measure between
the market and the SXc V asset . Merton [Mer73] presented the Intertemporal
Capital Asset Pricing Model (ICAPM), where he showed that if the investment
opportunity set (interest rates, volatilities etc) is not constant, then the equilibrium
model by Sharpe does not hold. He showed that for each stochastic factor
in the opportunity set one additional term is added to the function describing
the equilibrium expected return of a security. The Consumption-based Capital
Asset Pricing Model (CCAPM) was developed by Breeden [Bre79]. The
equilibrium expected return of a security is here given as a function of the
expected market return, the interest rate and a dependence measure between
the consumption and the security return.
Mainly because of the special characteristics of electricity, such as the non-