Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 86

A.3 The Schwartz Model

The Schwartz model is implemented, but with a different speci cation of risk as kS t

in the differential equation for the spot price under the equivalent Martingale
measure. 16
Start from:

d S t ¼ kðl ln S t ÞS t d t þ rS t d z t ð 46 Þ

the risk-neutral version of which with the modeling of the market price of risk
selected is:

d t þ rS t d z t d S t ¼½ kðl ln S t ÞS t kS t ð 47 Þ

With X t ¼ ln S t the model takes the following form:

d t þ r d z t

r 2
2

d X ¼ kðl XÞk ð 48 Þ

with X m ¼ l

r 2 k

k

2 k

d X t ¼ kðX m X t Þ d t þ r d z t

ð 49 Þ

from which the future equation is obtained as:

e ½e kðTtÞ ln S t þð 1 e kðTtÞ ÞX m þ r 4 k ð 1 e 2 kðTtÞ ¼ e ½

Fðt ; TÞ ¼ ð 50 Þ

and, as can be checked, Fðt ; tÞ ¼ S t .

A.4 The Ornstein-Uhlenbeck (O-U) Process

The differential equation for this stochastic process is:

d S t ¼ kðS m S t Þ d t þ r d W t ; ð 51 Þ

where S t denotes the price at time t . This current value tends to the S m level in the
long term at a reversion rate k . r is the instantaneous volatility, and d W t stands for
the increment to a standard Wiener process.
This model allows S t to take both negative and positive values. The price has a
conditional mean

16

In the original model by Schwartz [ 26 ] this appears as a risk premium for the log of the price.