Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 65

valuation does not assume that investors are risk-neutral: in fact they are not. Risk-
neutral valuation uses risk-neutral probabilities.
For this method to be used the market must be complete, because otherwise the
risk-neutral measure is not unique. With incomplete markets, there are project-
speci

c private risks. Contingent claims can be valued by replicating portfolios of
market assets or by using risk-neutral methods when markets are complete. A
replicating portfolio can still be built up when only market uncertainties exist, but if
some values are not traded assets then the market is incomplete. The growing
number of marketplaces and market participants is markedly conducive to the
application of this method. One of the most frequent causes of market incom-
pleteness is price jumps, as found for instance in electricity markets.
The possibility of cover enables futures prices to be used instead of expected
spot prices. This means a shift from the real world with a RP to risk-neutral world
with no RP simply by changing the drift of the stochastic differential equation.
When the market is not complete any of the three following options may be
used:

(a) Assume that the market is complete and use the above method. In this case
reliability will depend on the degree of incompleteness, but the method will
not be completely correct because some market values will be missing.
(b) Assume that market participants are risk-neutral and discount at the riskless
rate, using the actual probabilities. This may not be a good option.
(c) Use dynamic programming with an exogenous discount rate. In this case, the
problem of determining what that discount rate should be arises. This method
is used in some examples by Dixit and Pindyck [ 9 ].

There is also a fourth option, consisting of establishing a utility function to make
calculations, but this method is usually only used in the academic world.

2.4 Convenience Yield

The convenience yield d is the bene t or premium associated with holding an

underlying product or physical good rather than the contract or derivative product.
Users of a consumption asset may obtain a bene t from physically holding the asset

(as inventory) prior to T (maturity) which is not obtained from holding the futures
contract. Such bene

ts include the ability to pro t from temporary shortages, and

the ability to keep a production process running.
The convenience yield is equivalent to a dividend on a share. Convenience
yields are frequently found in commodities, which leads to the trend in the sto-
chastic differential equation that governs the price under the equivalent Martingale
measure being

ðr dÞS t , i.e., the riskless rate minus the convenience yield. Wie
and Zhu [ 29 ] study the convenience yield and the RP in the US natural gas market.