Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 74

Table 4 RBOB gasoline
parameters

Parameter Value

kS m
kþk

99.5614

k þ k

0.5814

c 4.7210

/ 0.2329

This leads to the estimation of two additional parameters (Table 4 ).
The accuracy of the estimate is substantially improved when seasonality is
included. Whenever there is seasonality its impact should be considered, but in
some cases de-seasonalized series should be worked with. Observe how the esti-
mate of the long-term equilibrium point changes.

4 The Case of ICE EUA Futures

This section gives an example of the calculation of the parameters of a GBM
process using quotes for ICE EUA Futures.
Figure 7 shows the ICE EUA Futures quotations for 08/23/2013. As can be seen,
these prices behave in a way compatible with GBM-type modeling, where:

FðT 1 ; T 2 Þ ¼ S t e ðakÞðT 2 T 1 Þ

ð 14 Þ

given that this model implies exponential growth in the price of the futures contract
as the maturity period increases.
It can clearly be seen in Fig. 7 that the GBM estimate is a better approximation
in this case (it is closer to the actual quotation) than the one provided by an IGBM
model with these data. Using a GBM model has signi

cant implications in terms of
volatility and expected value. Volatility is increasing over time in the risk-neutral
world, which signi

cantly affects the calculation of the value of options:

Var ðS T Þ ¼ S t e 2 ðakÞðTtÞ ½e r 2

ðTtÞ

1 ð 15 Þ

When volatility is high the value of the option increases signi cantly.

In the real world, it suf ces to set k ¼ 0, which gives:

e r 2 ðS T Þ ¼ S t e 2 aðTtÞ ½

ðTtÞ

Var 1 ð 16 Þ

For cases in which T is very close to t the expression obtained is T ¼ t þ D t

Var ðS tþDt ÞS t r 2

D t

ð 17 Þ