Environmental Engineering Reference
In-Depth Information
Table 7
Crude Oil (CL) and Fuel Oil (HO) parameters
Parameter
CL
HO
S
0
106.42
130.158
kS
m
kþk
79.5157
94.4471
k þ k
0.5583
0.1556
0.2840
0.2305
σ
This valuation assumes that production is always
3
of a barrel of Gasoline and
1
3
of a barrel of fuel oil from one barrel of crude oil. This is the case if the plant is
operational at all times and the crack spread is always positive.
In practise things are not that simple: there are also
xed costs and other variable
costs. In this case, a number of MonteCarlo simulations of the stochastic process
could be run, with the maximum between the crack spread minus the variable costs
and zero being obtained in each case and with the results being discounted at the
riskless rate. This serves to determine the average current value, from which the
amount of the current
xed must be discounted.
(E)
Stochastic process simulation
This example involves a MonteCarlo simulation. The aim is to demonstrate that
the goodness of the simulation can be checked by checking the resulting volatilities
and correlations against originals from the stochastic model.
Two correlated stochastic processes are simulated below using the Table
7
parameters:
Initially we have:
e
ðk
i
þk
i
Þ
D
t
D
p
e
t
k
i
S
i
m
k
i
k
i
S
i
m
k
i
D
S
t
¼
S
i
0
0
þ r
i
S
t
þ k
i
þ
ð
26
Þ
þ k
i
where
i 2
f g
Along with the correlation
CL, HO
ρ
CL,HO
= 0.7222.
The correlated samples are obtained by applying the following:
e
CL
t
e
CL
¼
q
1
e
HO
t
e
CL
q
CL
;
HO
þ
e
HO
q
2
CL
;
HO
¼
ð
Þ
ð
27
Þ
where e
CL
and e
HO
are uncorrelated random samples.
A simulation is conducted at 25 years with 252 steps per year for both crude oil
(CL) and fuel oil (HO). It can be observed that the volatilities estimated from the
simulation would be very similar to the model parameters. These results are shown
in Table
8
.
The estimated correlation is 0.7229. Volatility estimations and correlation can
therefore be regarded with con
dence.
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