Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 85

Observe also that for a very long-term future the expected real-world value is
precisely S m :

EðS 1 Þ ¼ S m

and

kS m
k þ k

Fðt ; 1Þ ¼

Therefore,

ðt ; 1Þ ¼ kS m
k þ k

RP

If this model is fully representative of the actual behavior of a futures market
long-term quotations provide information on kS m

kþk , which is linked to the MPR such

that when k ¼

; Fðt ; 1Þ ¼ S m is obtained, and if the reversion rate is very high
then the RP is low.
Consider two contracts T 1 and T 2 . The following is obtained:

0

e ðkþkÞðT 1 tÞ

kS m
k þ k þ

kS m
k þ k

Fðt ; T 1 Þ ¼ S t

e ðkþkÞðT 2 tÞ

kS m
k þ k þ

kS m
k þ k

Fðt ; T 2 Þ ¼ S t

It can be deduced that

e ðkþkÞðT 2 T 1 Þ

kS m
k þ k þ

kS m
k þ k

Fðt ; T 2 Þ ¼ Fðt ; T 1 Þ ð 44 Þ

The GBM model is a particular case of this model when S m ¼ 0 and a ¼ k for

this case:

Fðt ; TÞ GBM ¼ S t e ðakÞðTtÞ

ð 45 Þ

which is the same result as obtained above.