We present a methodology to model electricity price dynamics by applying the interest rate theory toolkit. We construct the electricity market following [16] and applying the Heath, Jarrow and Morton ([7]) model. The electricity returns forward curve evolution using the Regime Switching Volatility is the instrument chosen to reflect into a simulating model the natural seasonality of electricity prices. The model calibration and the volatility parameters estimation allow to simulate in a realistic way the future electricity prices.

Silvana Musti

Viviana Fanelli

English

Research Papers in Economics

 
Dipartimento di Scienze Economiche, Matematiche e Statistiche 
 
Università degli Studi di Foggia 
____________________________________________________________________ 
 
 
Electricity Price Modelling with a Regime 
Switching Volatility 
 
Viviana Fanelli e Silvana Musti 
 
 
Quaderno n. 06/2010 
 
 
 
 
 
 
 
 
 
 
 
 
“Esemplare fuori commercio per il deposito legale agli effetti della legge 15 aprile 2004 n. 106” 
Quaderno riprodotto dal  Dipartimento di Scienze Economiche, Matematiche e Statistiche 
nel mese di giugno 2010 e depositato ai sensi di legge. 
Authors only are responsible for the content of this reprint. 
_______________________________________________________________________________ 
Dipartimento di Scienze Economiche, Matematiche e Statistiche, Largo Papa Giovanni Paolo II, 1, 
71121 Foggia (Italy), Phone +39 0881-75.37.30, Fax +39 0881-77.56.16 
 Electricity Price Modelling with a
Regime Switching Volatility
Viviana Fanelli, Silvana Musti
Department of Economics, Mathematics and Statistics
University of Foggia, Italy
May 2010
Abstract
We present a methodology to model electricity price dynamics by applying
the interest rate theory toolkit. We construct the electricity market following
[16] and applying the Heath, Jarrow and Morton ([7]) model. The electricity
returns forward curve evolution using the Regime Switching Volatility is the
instrument chosen to reﬂect into a simulating model the natural seasonality of
electricity prices. The model calibration and the volatility parameters estimation
allow to simulate in a realistic way the future electricity prices.
1 Introduction
The deregulation process of the electricity markets started in US and in the main
Western countries brought a new dynamic in the market. The electricity market,
once monopolistic, becomes a competitive market where power prices are derived
by the interaction of supply and demand. This new contest, joined with the physical
characteristics of electrical power, has generated new price patterns, never seen before,
neither in ﬁnancial markets, nor in commodity markets. The crucial feature of this
commodity is the non-storability: except hydroelectric power, the electricity has to be
generated exactly when it is requested for consumption. The o er is therefore com-
pletely inelastic to price changes and prices show extremely high volatility and sudden
consistent changes in their levels, called ”spikes”. The market participants, both pro-
ducers and consumers, are dramatically exposed to uncertainties in electrical power
prices: risk management techniques becomes fundamental instruments to quantify
and mitigate the price risk associated with uncertain prices. Therefore, researchers
and practitioners have focused their attention on studying electricity price evolution
setting up mathematical models able to capture the main features of price behavior
allowing both the derivatives pricing and the risk hedging. Electricity is called a ﬂow
commodity: all contracts guarantee the delivery of an established amount of power (1
MWh) continuously over a speciﬁc future time period (1 hour, 1 month, 1 quarter, 1
year). Electricity is traded in an auction system for standardized contracts which can
be settled with physical delivery or simply ﬁnancially. The energy market is made of
two segments: a market for spot trading and a derivative market. In the spot market,
1every day hourly contracts for each of the 24 hours of the coming day are traded
(Day-Ahead Market with physical delivery). Besides, in the derivative market, power
Forward contracts and Future contracts are settled both ﬁnancially or with physi-
cal delivery: the tradable contracts in the Forward Electricity market have monthly,
quarterly and yearly delivery period. Power Forward and Future contracts are the
main derivative product traded on the electricity markets: European and Asian Op-
tions with Power Forward as underlying have been launched in the last few years
in the more developed Electricity Markets (Nord Pool, Scandinavia and NYMEX,
New York) but they do not seem to be attracting much attention. The literature on
stochastic models for electricity and other ﬂow commodity has developed rapidly in
the last few years. Basically, two lines of research focus on electricity price modelling:
the traditional way has concentrated on modelling the stochastic process of the spot
price, by adapting and modifying the approach followed for other energy commodity.
Lucia and Schwartz ([13]) model the natural logarithm of the spot price by assuming
a mean reverting process estimated using spot price data in the Nordic market. The
price evolution of a futures contract is than determined by applying expected value
under an appropriate martingale measure equivalent to the objective one. Other au-
thors ([15] and [5]) suggest a two factor model, in order to take into account the
inﬂuence on the spot price given both by a short term and a long term source of
randomness. The introduction of a jump di usion process appears the natural way
to account for spikes ([5], [3]), even if market incompleteness is introduced. Geman
and Roncoroni [6] suggest an exponential mean reversion model with state dependent
jumps. Huisman ([10]) and De Jong ([4]), among others, suggested Markovian regime
switching models characterized by the occurrence of stable and turbulent periods.
The major disadvantage of the spot price models is that forward prices are given
endogenously from the spot price dynamics. Therefore, the obtained dynamics of for-
ward prices is most of the times not consistent with the market observed prices. The
natural research answer is the whole forward curve modelling. This line of research
refers to the modelling framework of Heath, Jarrow and Morton ([7]) and, using only
few stochastic factors and the initial price curve as given, models futures prices un-
der some equivalent martingale measure in a no-arbitrage environment. Clewlow and
Strickland ([3]) have been the ﬁrst researchers to introduce this approach to the en-
ergy market. Bjerksund et al. ([1]), and Koekebakker et al. ([12]) model a continuum
of instantaneous-delivery forward contracts under risk neutral probability measure.
However, various issues about the physical characteristics of the electricity and the
existence and interpretation of risk neutral measure remain neglected. One question
comprehends all of them: is it possible to derive the pricing approach from Interest
Rate Modelling? There are di erent reasons justifying a negative answer: ﬁrst of all
the non-storability of electrical power cannot be neglected using a pricing method-
ology in which the underlying is assumed storable. Besides, hedging is impossible if
short position are not allowed and the electricity market is far away from being com-
plete, so it is not guaranteed the existence of a risk neutral measure. Koekebakker et
al. ([12]) give an interesting interpretation of the electricity market and demonstrates
that it is possible to create a market framework where it is guaranteed the existence of
a risk neutral measure, the risk neutral measure has an economic interpretation and
prices can be obtained with the instruments used in the ﬁxed income markets, such
as interest rate models. The simple and at the same time innovative consideration
2is that even if energy cannot be stored, it has to be produced. And the producer
can put himself in the condition of having the ability to produce electricity, creating
a sort of ”electricity storability”. From this point of view, the true underlying of
electricity contracts becomes the physical ability to produce power. According to this
perspective, the electricity market becomes more complex and has to be considered as
composed of both power electricity and agreements on power production capacities.
The market reaches the equilibrium and determines the price process for all tradable
assets both physical (production capacity agreements) and ﬁnancial (future electric-
ity prices), S 
t . There is a measure Q, equivalent to the market measure P, such
that the equilibrium asset prices are given by their future revenues, expected with
respect to Q. By using the production portfolios, it is possible to replicate any future
contracts. By considering the market in this more general perspective, electricity
cannot be stored, but a hedging is still possible by production capacity investments.
It makes sense to apply interest rate models because it is possible to consider the
market as composed by inﬁnitely many future contracts with instantaneous delivery
period. The equilibrium in the production capacity market guarantees the existence
of a measure Q, such that electricity asset prices are given by the expected value of
future revenues of certain production portfolios. The equilibrium concept represents
a way to price all contracts by the same measure ”chosen by the market”. The asset
dynamics is therefore described directly under Q.
2 Model setting
In this paper we construct the electricity market following [16] and applying the
Heath, Jarrow and Morton ([7]) model with Regime Switching Volatility. The Elec-
tricity Market (hereafter EM) is given on the time horizon [0,T]: we model the price
dynamics for every t   [0,T] of a futures contract supplying 1 MWh at a time     t.
The stochastic process of prices, (Ft( ))t [0,T] is a positive valued adapted stochastic
process realized on a complete ﬁltered probability space
 
 ,F,P,(Ft)t [0,T]
 
. At time
t = 0 we observe prices (F 
0( ))  [0,T], for all future delivery time. According to the
above mentioned hypothesis, we assume that in the model the following axioms are
valid:
(a) (Ft( ))  [0,T] is almost surely continuous for each     [0,T];
(b) there is a risk neutral measure QF equivalent to P such that for each     [0,T],
(Ft( ))  [0,T] is a QF-martingale;
(c) forward prices start at observed values F0( )=( F 
0 ( ))  [0,T];
(d) terminal prices (Ft(t))t [0,T] form a continuous spot price process.
A currency change is the key instrument making this model similar to the ﬁxed income
market model: the power forward prices at time t, Ft( ) is expressed in units of power
prices just in front of delivery at time t, Ft( ). Under this new currency, power prices
behave like zero coupon bonds, in the sense that their value converges to 1 when t
coincides with  :
pt( ): =
Ft( )
Ft(t)
,    [0,T],t   [0, ].
3

Electricity Price Modelling with a Regime Switching Volatility

http://www.dsems.unifg.it/q062010_abstract.pdf

Electricity Price Modelling with a Regime Switching Volatility

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics