Mathematical and Statistical Methods for Actuarial Sciences and Finance

26 C. Bencivenga, G. Sargenti, and R.L. D’Ecclesia

10

15

20

25

30

35

40

45

2002 2003 2004 2005 2006 2007

Brent crude oil

0

100

200

300

400

500

600

2002 2003 2004 2005 2006 2007

EEX elect

0

10

20

30

40

50

60

70

80

90

2002 2003 2004 2005 2006 2007

NBP UK

Fig. 1.Crude oil, natural gas and electricity prices, 2001–2007

i= 1 ,...,25,N=60, where

σi,N=

√

√√

√^1

N− 1

∑iN

τ=(i− 1 )N+ 1

(

ln

Sτ

Sτ− 1

−E

(

ln

Sτ

Sτ− 1

)) 2

.

The oil price volatility swings between 21% and 53%, confirming the non-stationarity
of the data. The same non-stationarity characterises the data of natural gas, fluctuating
between 65% and 330%. Electricity prices, as expected, were far more volatile than
oil and gas prices,^6 with a range of quarterly volatility which swings between around
277% and 868%.
A preliminary analysis is going to be performed on the stationarity of the time
series. In line with most of the recentliterature we transform the original series in
logs. First we test the order of integration of a time series using the Augmented
Dickey-Fuller (ADF) type regression:

yt=α 0 +α 1 t+γyt− 1 +

∑k

j= 1

βjyt−j+t (1)

(^6) Seasonality and mean reversion are common features in commodity pricedynamics; in
addition a jump component has to be included when describing electricity prices.