Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 63

Technical uncertainty 4 is not correlated with the general movements of the
economy, and is endogenous to the decision-making process. Examples include the
amount of oil contained in an oil eld and the wind load at a particular location. This

type of uncertainty can be reduced to some extent (and at some expense) by
conducting studies and setting up pilot plants, but it cannot be eliminated altogether.

2.1 Stochastic Process

Some simple stochastic processes are presented below. They are analyzed in greater
depth in the Appendix. They all have a differential equation with a deterministic
part and a stochastic part. There is one version for the real world and another for the
risk-neutral world, where the market price of risk is deduced from the drift.
These stochastic processes which model the behavior of commodities serve to
value real assets when their NPV depends on one or more commodities via their
prices.

2.1.1 Geometric Brownian Motion

Geometric Brownian Motion (GBM) is a continuous time stochastic process that is
widely used in nance to model stock prices via two components: a deterministic

trend and a random element. It is presented here as a basic model prior to the
development of more complex models of energy commodity behavior.
The stochastic differential equation in the real word is:

d S t ¼ aS t d t þ rS t d W t ð 1 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.

2.1.2 Inhomogeneous Geometric Brownian Motion

The stochastic differential equation in the real word is:

d S t ¼ kðS m S t Þ d t þ rS t d W t ð 2 Þ

where S t is the price of the commodity at time t ; k is the reversion rate, S m is the
expected price toward 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.

4 By contrast with economic uncertainty, technical uncertainty refers to the uncertainty of
technical parameters, e.g., recoverable reserves.