Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 82

Gourieroux and Jasiak [ 14 ] examine the estimation of the parameters of certain
stochastic differential equations.
The characteristics of four stochastic processes are described below: a GBM
model and three mean reverting models.

A.1 Geometric Brownian Motion Model

This model is not widely used in modeling the behavior of commodity prices
(which in turn can determine the value of investment in energy assets) because it is
considered that the prices of these commodities tend to show mean reverting
behavior, especially when futures market quotations are analyzed.
In the real world, the model is as follows:

d S t ¼ aS t d t þ rS t d W t ð 28 Þ

where S t is the price of the commodity at time t ; a is the drift in the real world, r is
the instantaneous volatility and d W t stands for the increment to a standard Wiener

process.
The risk-neutral version of the model is:

d S t ¼ ða kÞS t d t þ rS d W t ð 29 Þ

where kS t is the market price of risk. 15
It holds that a k ¼ r d with r being the riskless rate and d the convenience
yield, so the following alternate expression can also be used:

d S t ¼ ðr dÞS t d t þ rS d W t ð 30 Þ

If X ¼ ln S ;

'

and Ito s lemma is applied, then the following is obtained:

d t þ r d W t

r
2

d X t ¼ a k ð 31 Þ

In this case it the value of a future with maturity T at time t is obtained from the
following equation:

Fðt ; TÞ ¼ S t e ðakÞðTtÞ ¼ S t e ðrdÞðTtÞ ð 32 Þ

Equation ( 32 ) shows that a commodity with GBM-type behavior in the spot
price should exhibit quotations on the futures market that increase in absolute value

15

Kolos and Ronn [ 18 ] estimate the market price of risk for energy markets.