Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 83

by larger amounts as the maturity period increases. This behavior can be used to
identify whether a commodity is a good candidate for modeling with GBM.
In the real world:

E t ðS T Þ ¼ S t e aðTtÞ

ð 33 Þ

So the RP looks like this:

ðt ; TÞ ¼ Fðt ; TÞE t ðS T Þ ¼ S t e aðTtÞ ½ e kðTtÞ

RP 1 ð 34 Þ

Thus:
(a) if k ¼ 0 then Fðt ; TÞ ¼ E t ðS T Þ and RP ðt ; TÞ ¼ 0 In this case the future is an

:

unbiased estimator of the expected spot price.

(b) if k [ 0 then RP ðt ; TÞ \ 0 : In this case the future is a downward-biased esti-

mator of the spot price.

(c) if k \ 0 then RP ðt ; TÞ [ 0 In this case the future is an upward-biased

:

estimator.

The sign of the RP and the sign of the market price of risk are exactly opposite.
Moreover, assuming a market price of risk that is constantly proportional to the spot
price, the RP tends to zero as the time t approaches the maturity T . This means that
the future is a good estimator for close-at-hand maturity times, because even though
it is biased the bias may be slight. The same cannot be said of more remote maturity
times, where the bias may be signi cant.

Observe that the following is also valid from two futures contracts with matu-
rities T 1 and T 2 onwards:

Fðt ; T 2 Þ ¼ Fðt ; T 1 Þ t e ðakÞðT 2 T 1 Þ

ð 35 Þ

In some cases, this may make it possible to do away with the use of the spot
price S t , which may sometimes not be observable.

A.2 Inhomogeneous Geometric Brownian Motion Model

This is a mean reverting model that has the following stochastic differential equation:

d S t ¼ kðS m S t Þ d t þ rS t d W t ð 36 Þ

where S t is the price of the commodity at time t ; k is the reversion rate, S m is the
expected price to which the value of the commodity tends in the long term, r is the
instantaneous volatility and d W t stands for the increment to a standard Wiener
process.