Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 84

Under the equivalent Martingale measure this can be expressed as:

d S t ¼½ kðS m S t ÞkS t d t þ rS t d W t ð 37 Þ

where the market price of risk (MPR) is proportional to S t :
The expected value at a time T can be calculated as follows:

Eð d S t Þ

d t þðk þ kÞEðS t Þ ¼ kS m

ð 38 Þ

This can be solved by using e ðkþkÞt as an integration factor, which gives:

kS m

k þ k þ C 0 e ðkþkÞt

E Q ðS t Þ ¼ ð 39 Þ

where C 0 is a constant determined as a function of the initial conditions. Since it
must hold that E t ðS t Þ ¼ S t the following is obtained:

kS m
k þ k

C 0 ¼ S t

e ðkþkÞðTtÞ

kS m
k þ k þ

kS m
k þ k

Fðt ; TÞ ¼ S t ð 40 Þ

It is easy to check that formula ( 40 ) complies with the differential equation. It
can also be checked that this solution complies with the differential equation:

1
2 r 2 S 2 F SS þ ½ kðS m S t ÞkS t F S ¼ F T

ð 41 Þ

with the terminal condition Fðt ; tÞ ¼ S t
where the subindexes of F refer to the derivatives corresponding to future
Eq. ( 40 ).
The expected future value of the spot price for the mean reverting model can
easily be obtained from Eq. ( 40 ) by assuming k ¼

0. It works out to:

e kðTtÞ E t ðS T Þ ¼ S m þðS t S m Þ ð 42 Þ

The RP is:

RP ðt ; TÞ ¼ Fðt ; TÞE t ðS T Þ ð 43 Þ

In this case a high reversion rate k reduces the RP.