Environmental Engineering Reference
In-Depth Information
the 3:2:1 crack spread in the series considered. However, if an Ornstein-Uhlenbeck
(O-U) process is used, there is no r multiplied by S t , which means that the possibility
of negative values
is admitted.
In this case,
the estimated volatility is
r OU
1499, i.e. 17 times greater. Thus, the estimated volatility to be applied in a
stochastic process depends on what process is chosen.
¼
29
:
3 Commodities Futures Markets
Futures markets provide one of the biggest sources of information for the valuation
of investments in energy assets. Since counterparty risk is eliminated, the quotes on
such markets are prices in which there is no uncertainty. Thus, the value of
for
instance
a barrel of oil for delivery in 2 years time can be discounted at the risk-
free interest rate (risk-neutral valuation). This section provides some simple
examples using prices from various markets: Light Sweet Crude Oil futures (WTI),
NY Harbor ULSD futures, RBOB Gasoline futures, and ICE EUA Futures. The
examples include mean reverting processes and GBM, along with cases with and
without seasonality.
Futures markets provide information on the values of the parameters in the
deterministic part of the differential equation, always in a risk-neutral world. As in
the GBM case, the value of a k can be estimated easily and accurately. The same
goes for
kS m
kþk and k þ k in the case of the IGBM model. In the case of mean
reversion, these markets provide information on the long-term equilibrium value
and the rate at which the expected price moves toward that value in the risk-neutral
world.
Commodities markets are studied in [ 10 , 12 ].
3.1 The Case of Light Sweet Crude Oil (WTI) Futures
An example is given below of the calculation of the parameters of a mean reverting
process using quotes for Light Sweet Crude Oil (WTI) Futures.
In the case of the IGBM model, the parameters of the following equation are
estimated using data from the futures market.
e ðkþkÞðT 2 T 1 Þ
kS m
k þ k þ
kS m
k þ k
FðT 1 ; T 2 Þ ¼
FðT 1 ; T 1 Þ
ð 12 Þ
This equation can be used to obtain kS m
kþk and k þ k for either an IGBM process or
an O-U process. The difference between the two lies in the volatility estimate to be
used for the model but this does not affect the above formula.
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