Environmental Engineering Reference
In-Depth Information
With the closest futures value, for 8/23/2013, and estimating the parameters in
such a way as to minimize the sum of the squares of the differences between the
actual future and the estimated
gure for each date,
the following values are
5583.
10
This calculation is based on the
future with the closest maturity (Oct 13), for the date of which the price is $106.42/
barrel. As can be seen in Fig.
3
, the
kS
m
obtained:
kþk
¼
79
:
5157 and
k þ k
¼
0
:
t is very close using the values from the two
parameters.
From Fig.
3
it is easy to determine that the behavior of this commodity can be
modeled via a mean reverting stochastic process.
11
It is also possible to use the model in [
26
], where the equation for the future is:
FðT
1
;
T
2
Þ
¼e
½e
kðT
2
T
1
Þ
ln
S
t
þð
1
e
kðT
2
T
1
Þ
ÞX
m
þ
r
4
k
ð
1
e
2
kðT
2
T
1
Þ
ð
13
Þ
The results for this case are shown in Fig.
4
. The
t is very similar to that of the
IGBM case:
The
kS
m
gures obtained in this case are as
follows:
kþk
¼
79
5157 and
:
k þ k
¼
5583. Modeling here would also be via a mean reverting process. No
seasonality is detected in this case.
0
:
3.2 The Case of NY Harbor ULSD Futures
This section gives an example of the calculation of the parameters of a mean
reverting process using quotes for NY Harbor ULSD Futures.
Figure
5
shows the actual and estimated values on 8/23/2013
12
for ULSD Futures.
In this case the values of the estimated parameters are as follows:
kS
m
kþk
¼
94
4771
:
and
k þ k
¼
0
1556. No clear seasonal behavior is detected.
:
3.3 The Case of RBOB Gasoline Futures
This section gives an example of the calculation of the parameters of the seasonal,
mean reverting process using quotes for RBOB Gasoline Futures.
Figure
6
shows the actual and estimated values on 8/23/2013
13
for Gasoline
Futures.
10
Quotations from several days could also have been used, but for the sake of illustration the
simplest method is preferred here.
11
The statistical test lies outside the scope of this chapter.
12
Quotes are converted to $/barrel by multiplying the original $/gallon
gures by 42.
13
Here also the quotes are converted to $/barrel by multiplying the original $/gallon
gures by 42.
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