Pricing in Electricity Markets: A Mean
Reverting Jump Diffusion
Model
with
Seasonali ty
ÁLVARO CARTEA & MARCELO G.
FIGUEROA
Birkbeck College. University
of
London, London
WCl
E 7 HX, UK
ABSTRACT
This paper presents a mean-reverting jump diffusion model for the electricity spot
price and derives
(he
corresponding forward price
in
closed~form.
Based
on
historical spot data
and forward data from England and Wales the model
is
calibrated and months, quarters, and
seasons-ahead forward surfaces are presented.
KEY
WORDS:
Energy derivatives, e1ectricity, forward curve, forward surfaces
1.
Introduction
One
of
the key aspects
of
a competitive market
is
deregulation. In most electricity
markets this has, however, only occurred recently, Prior to this, price variations were
often minimal
and
heavily controIled by regulators, In England and Wales in particular,
prices were set by the Electricity Pool, where, due to centralization and inflexible
arrangements, prices failed to reflect falling costs and competition, Deregulation carne in
the recent introduction on
27
March
2001
of
NETA
(New Electricity Trade
Arrangement), removing price controls
and
openly encouraging competition.
Price variations have increased significantly as a consequence
of
the introduction
of
competition, encouraging the pricing
of
a new breed
of
energy-based financial
products
to
hedge the inherent risk,
both
physical
and
financial, in this market.
Most
of
the current transactions
of
instruments in the electricity markets are carried
out
through
bilateral contracts ahead
of
time although electricity
is
also traded on
forward
and
futures markets
and
through
power exchanges.
One
of
the most striking differences
that
singles
out
electricity markets
is
that
electricity
is
very difficult
or
too expensive to store, hence markets must be kept in
balance
on
a second-by-second basis. In England
and
Wales, this
is
done by the
Correspondence Address: Álvaro Cartea, Birkbeck College, University
ofLondon,
Malet Street, Bloomsbury,
London
WCIE
7HX,
UK.
Email: a.cartea@bbk.ac.uk
1
314
A.
Cartea and M.G. Figueroa
National
Grid
Company which operates a balancing mechanism to ensure system
security.l Moreover, a1though power markets may have certain similarities with
other markets, they present intrinsic characteristics which distinguish them. Two
distinctive fea tu res are present in energy markets in general,
and
are very evident
in
electricity markets in particular: the mean reverting nature
of
spot prices and the
existence
of
jumps
or
spikes in the prices.
In stock markets, prices are allowed to evolve 'freely',
but
this
is
not true for
electricity prices; these will generally gravitate
around
the cost
of
production. Under
abnormal market conditions, price spreads are observed in the short run,
but
in
the
long run supply will be adjusted and prices will move towards the level dictated by
the cost
of
production. This adjustment can be captured by mean-reverting
processes, which in turn may be combined with jumps to account for the observed
spikes.
Therefore, to price energy derivatives it
is
essential that the most important
characteristics
of
the evolution
of
the spot,
and
consequently the forward, are
captured. Several approaches may be taken, gene rally falling into two
c1asses
of
models: spot-based models
and
forward-based models. Spot models are appealing
since they tend to be quite tractable
and
also allow for a good mathematical
description
of
the problem in question. Significant contributions have been made by
Schwartz (1997), for instance, where the
author
introduces
an
Ornstein-Uhlenbeck
type
of
model which accounts for the mean reversion
of
prices. Lucía
and
Schwartz
(2002) extend the range
of
these models to two-factor models which incorporate a
deterministic seasonal component. On the other
hand
forward-based models have
been used largely in the N ord Pool Market
of
the Scandinavian countries. These rely
heavily, however, on a large
data
set, which
is
a limiting constraint in the case
of
England
and
Wales. Finally, it must also be pointed
out
that
the choice
of
model
may sometimes be driven by what kind
of
information
is
required.
For
example,
pricing interruptible contracts would require a spot-based model while pricing Asian
options
on
a basket
of
electricity monthly and seasonal forwards calls for forward-
based models.
The spot models described by Schwartz (1997)
and
Lucia
and
Schwartz (2002)
capture the mean reverting nature
of
electricity prices,
but
they fail to account for the
huge and non-negligible observed spikes in the market. A natural extension
is
then to
incorporate a
jump
component
in
the model. This
c1ass
of
jump-diffusion models
was first introduced by Merton (2001) to model equity dynamics. Applying these
jump-diffusion-type models in electricity
is
attractive since solutions for the pricing
of
European options are available in c1osed-form. Nevertheless, it fails to
incorporate both mean reversion
and
jump
diffusion
at
the same time. Clewlow
et al. (2001) describe
an
extension to Merton's model which accounts for
both
the
mean reversion
and
the jumps
but
they do not provide a c1osed-form solution for the
forward. A similar model to the one
we
present, although
not
specific to the analysis
of
electricity spot prices, has be en analysed in Benth et
al.
(2003).
The main contribution
of
this paper
is
twofold. First,
we
present a model that
captures the most important characteristics
of
electricity spot prices such as mean
reversion, jumps
and
seasonality
and
calibrate the parameters to the England
and
Wales market. Second, since
we
are able to calculate
an
expression for the forward
curve in c1osed-form
and
recognizing the lack
of
sufficient
data
for robust parameter
2
Pricing
in
Electricity Markets
315
estimation, we estimate the model parameters exploiting the fact
that
we
can use
both
historical spot
data
and
current forward prices (using the closed-form
expression for the forward).2
The remaining
of
this
paper
is
structured as follows. In Section 2 we present
data
analysis
to
support
the use
of
a model which incorporates
both
mean reversion
and
jumps.
In
Section 3 we present details
of
the spot model
and
derive in closed-form the
expression for the forward curve.
In
Section 4 we discuss the calibration
of
the model
to
data
from England
and
Wales. In Section 5 we present forward surfaces reflecting
the months, quarters
and
seasons-ahead prices. Section 6 concludes.
2.
Data Analysis
For
over three decades most equity models have tried
to
'fix' the main drawback
from assuming Gaussian returns. A clear example
is
the wealth
of
literature
that
deals with stochastic volatility models, jump-diffusion
and
more recently, the use
of
Lévy processes. One
of
the main reasons
to
adopt
these alternative models
is
that
Gaussian
shocks attach very little probability
to
large movements in the underlying
that
are,
on
the contrary, frequently observed in financial markets.
In
this section we
will see
that
in electricity spot markets assuming Gaussian shocks to explain the
evolutÍon
of
the spot dynamics
is
even a
poorer
assumption
than
in equity markets.
Electricity markets exhibit their own intrinsic complexities. There
is
a strong
evidence
of
mean reversion
and
of
spikes in spot prices, which in general are much
more
pronounced
than
in stock markets.
The
former can be observed by simple
inspection
of
the
data
in
both
markets. Figure 1 shows daily closes
of
the
FTSE
1 00
index from 2/01/90
to
18/06/04.
The
nature
of
the
price
path
can be seen as a
combination
of
a deterministic trend together with
random
shocks.
In
contrast,
Figure
2 shows
that
for electricity spot prices in England
and
Wales there
is
a strong
mean reversion.
3
This is, prices tend to oscillate
or
revert
around
a mean level, with
extraordinary periods
of
volatility. These extraordinary periods
of
high volatility are
reflected in the characteristic spikes observed in these markets.
Normality Tests
In
the Black-Scholes model prices are assumed
to
be log-normally distributed, which
is
equivalent
to
saying
that
the returns
of
the prices have a Gaussian
or
Normal
distribution.
4
Although fat tails are observed in
data
from stock markets, indicating
the probability
of
rare events being more frequent
than
predicted by a
Normal
distribution, models based
on
this assumption have been largely used as a
benchmark, albeit modified in
order
to
account for fat tails.
For
electricity though, the
departure
from Normality
is
more extreme. Figure 3
shows a
Normality
test for the electricity spot price from 2/04/01 to 3/03/04.
If
the
returns were indeed Normally distributed the
graph
would be a straight
lineo
We can
clearly observe this
is
not
the case, as evidenced from the fat tails.
For
instance,
corresponding
to
a probability
of
0.003
we
have returns which are higher
than
-0.5;
instead
if
the
data
were perfectly
Normally
distributed, the
dotted
lines suggests the
probability
of
such returns should be virtually zero.
3
316
A.
Cartea
and
M.
G.
Figueroa
6500
6000
5500
(/)
5000
(J.)
-
o
5-
4500
(J.)
(/)
o
u
4000
3500
3000
2500
2000
500
1000 1500
2000
days
Figure 1. FTSElOO daily closes
[rom
2/01/90
to
18/06/04
Deseasonalization
2500 3000 3500
One important assumption
of
the Black-Scholes model
is
that
returns are assumed
to be independently distributed. This can be easily evaluated with
an
autocorrelation
test.
If
the
data
were
in
fact independently distributed, the correlation coefficient
would be close to zero. A strong level
of
autocorrelation
is
evident in electricity
markets, as can be
se
en from Figure
4.
As explained for instance in Pindyck and
Rubinfield (1998) the evidence
of
autocorrelation manifests an underlying
seasonality. Furthermore, the lag
of
days between highly correlated points in the
series reveals the nature
of
the seasonality.
In
this case,
we
may observe
that
the
returns show significant correlation every 7 days (there
is
data
for weekend s also);
which suggests sorne intra-week seasonality.
In
order to estimate the parameters
of
the model,
we
strip the returns from this
seasonality. Although there are several ways
of
deseasonalizing the data,
we
follow a
common approach which
is
to subtract the mean
of
every day across the series
according to
(1)
where
Rr
is
the defined deseasonalized return
at
time t,
rt
the return
at
time t
and
rd
is
the corresponding mean (throughout the series)
of
the particular
4
Pricing
in
Electricity
Markets
317
110
100
90
80
?
~
70
:!:
~
Q)
60
Q
.~
o
50
c.
'"
40
30
20
10
, 100 200 300 400
500
600 700
2/04/01
days
800
900 1000
!
3/03/04
Figure
2.
Averaged daily prices in England
and
Wales from 2/04/01 to 3/03/04
day
rt
represents. Figure 5 shows the autocorrelation test performed
on
the
deseasonalized returns.
As
expected, the strong autocorrelation
is
no longer
evidenced.
Jumps
As seen from the Normality test, the existence
of
fat tails suggest the probability
of
rare events occurring
is
actually much higher than predicted by a
Gaussian distribution.
By
simple inspection
of
Figure 2
we
can easily be
convinced
that
the spikes in electricity
data
cannot
be captured by simple
Gaussian shocks.
We
extract the jumps from the original series
of
returns by writing a numerical
algorithm
that
filters returns with absolute values greater
than
three times the
standard deviation
of
the returns
of
the series
at
that
specific iteration.
5
On
the
second iteration, the standard deviation
of
the remaining series (stripped from
the first filtered returns)
is
again calculated; those returns which are now greater than
three times this last standard deviation are filtered again. The process
is
repeated
until
no
further returns can be filtered. This algorithm allows
us
to estimate the
cumulative frequency
of
jumps
and
other statistical information
of
relevance for
calibrating the model.
6
The relevance
of
the jumps in the electricity market
is
further demonstrated by
comparing Figure 6 to Figure
3;
where
we
can clearly observe
that
after stripping the
returns from the jumps, the Normality test improves notoriously.
5