Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 64

2.1.3 The Schwartz Model

The stochastic differential equation in the real word is:

d S t ¼ kðl ln S t ÞS t d t þ rS t d W t ð 3 Þ

where S t is the price of the commodity at time t ; k is the reversion rate, r is the
instantaneous volatility and d W t stands for the increment to a standardWiener process.

2.1.4 The Ornstein-Uhlenbeck or O-U Process

The stochastic differential equation in the real word is:

d S t ¼ kðS m S t Þ d t þ r d W t

ð 4 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 of k . Moreover, r is the instantaneous volatility, and
d W t stands for the increment to a standard Wiener process.

2.2 Risk Premium

Let k be the market price of risk. The Risk Premium (RP) is de ned as the dif-

ference between the quotation at time t of a future with maturity T and the spot
price S expected for that time T , i.e. RP

ðt ; TÞ ¼ Fðt ; TÞE t ðS T Þ :

2.3 Equivalent Martingale Measure or Risk-Neutral Measure

In a complete market where there are no opportunities for arbitrage there is a
valuation method based on incorporating the market risk and using the new
probability distribution, which is the risk-neutral measure.
Basically, this means subtracting the market price of risk from the stochastic
differential equation in order to obtain the performance under the equivalent
Martingale measure. Assets can then be valued by discounting them at the riskless
rate. This is equivalent to discounting the expected real-world value with a rate that
is the sum of the riskless rate plus a RP. However, the risk-neutral measure is much
easier to use, since the parameters of the corresponding stochastic process are
relatively easier to estimate. 5 It is important to stress that using risk-neutral

5

For instance by using the enormous amount of information provided by futures markets.