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Stochastic Modelling in Energy MarketsSilvia Lavagnini
Stochastic Modelling in Energy Markets From the Spot Price to
Derivative Contracts
Thesis submitted for the degree of Philosophiae Doctor
Department of Mathematics Faculty of Mathematics and Natural
Sciences
2021
© Silvia Lavagnini, 2021
Series of dissertations submitted to the Faculty of Mathematics and
Natural Sciences, University of Oslo No. 2408
ISSN 1501-7710
All rights reserved. No part of this publication may be reproduced
or transmitted, in any form or by any means, without
permission.
Cover: Hanne Baadsgaard Utigard. Print production: Reprosentralen,
University of Oslo.
“Hvis teltet blåser bort, så legg deg med ansiktet ned. Da finner
jeg deg i morra.”
“If the tent blows away, go to bed face down.
Then I will find you tomorrow.”
Lars Monsen
Preface This thesis is submitted in partial fulfilment of the
requirements for the degree of Philosophiae Doctor at the
University of Oslo. The research presented here was mainly
conducted at the University of Oslo under the supervision of
Professor Fred Espen Benth. The thesis is a collection of five
papers presented in chronological order of writing, and preceded by
an introductory chapter that relates them to each other and
provides background information and motivation for the work.
iii
Acknowledgements I first came to Norway in 2016 as a master student
and fell in love with the blue of its sky and the green of its
forests. I then started my Ph.D. in 2017 and in these almost four
years I had the luck to meet many people who have shared part of
this journey with me. I am really grateful for that.
First of all, I want to thank my supervisor, Fred Espen Benth, who
has guided me through the Ph.D., from the first paper submitted
together, to the last one by myself. In these years I found his
door always open and he had answers to all my questions, including
motivational ones.
I want to thank my two co-authors: Luca Di Persio, who is also one
of the main reasons why I came to Norway in the first place, and
Nils Detering, who hosted me for three months at the University of
California in the beautiful Santa Barbara. I have learnt a lot from
these collaborations, and I hope they will continue. I also thank
Salvador Ortiz-Latorre for the continuos job- and non-job-related
discussions, and for allowing me to teach in his course.
I want to acknowledge also the one-and-a-half year of my Ph.D.
spent as a consultant at Statkraft, during which I was surrounded
by very enthusiastic colleagues. I thank Laxman, Morten, Jørn,
Arne, Vigdis, Andrea and Simen, for welcoming me in their
team.
I thank all my friends and colleagues at the Department of
Mathematics: Adilah, Iben, Dennis, Anton, Alise, Fabian, Marc and
Rossana, for the time and lunches spent together. I thank the small
Italian community with Claudio, Matilde, Giovanni, Michele, Elisa,
Luca and Andrea, for being with me in this journey and making me
feel “a bit more at home”. A special thanks goes to Lorenzo, who
has been a valuable friend in the last four years, supporting me in
the hardest moments of this Ph.D. Finally, I thank Moritz, my
overseas friend, who made my stay in Santa Barbara memorable.
I thank my parents and siblings for the continuous support and the
extremely long video-calls in the evenings where I tried to tell
them all about my adventures in Oslo. I thank my friend Chiara,
with whom I shared many doubts and uncertainties, and Gionata, my
lifelong friend, who has never stopped looking after me and caring
for me despite the distance. I thank all my friends in Verona and
those who motivated and pushed me to start this Ph.D. Finally, I
thank Vegard, who has taught me a lot about life, career and how to
survive in the (sometimes) cold Norway, and who has supported me
when I needed it the most.
Silvia Lavagnini Oslo, May 2021
v
Ringraziamenti Sono arrivata per la prima volta in Norvegia nel
2016 come studente, e mi sono innamorata del suo cielo blu e delle
sue foreste verdi. Ho poi iniziato il dottorato nel 2017 e in
questi quasi quattro anni ho avuto la fortuna di incontrare molte
persone che hanno condiviso parte di questo cammino insieme a me.
Sono molto grata per questo.
Prima di tutto, voglio ringraziare il mio relatore, Fred Espen
Benth, che mi ha guidato lungo il dottorato, dal primo articolo
presentato insieme, all’ultimo mio articolo da sola. In questi anni
ho sempre trovato la sua porta aperta e ha avuto una risposta per
tutte le mie domande, comprese quelle motivazionali.
Voglio ringraziare i miei due coautori: Luca Di Persio, che è anche
una delle ragioni principali per le quali sono venuta in Norvegia
la prima volta, e Nils Detering, il quale mi ha ospitato per tre
mesi alla University of California nella bellissima Santa Barbara.
Ho imparato molto da queste collaborazioni, e spero possano
continuare. Ringrazio anche Salvador Ortiz-Latorre per le continue
discussioni di lavoro e non, e per avermi permesso di insegnare nel
suo corso.
Voglio menzionare anche il periodo di un anno e mezzo speso come
consulente in Statkraft, durante il quale sono stata circondata da
colleghi entusiasti. Ringrazio Laxman, Morten, Jørn, Arne, Vigdis,
Andrea e Simen, per avermi accolto nel loro team.
Ringrazio tutti i miei amici e colleghi del Dipartimento di
Matematica: Iben, Dennis, Anton, Alise, Fabian, Marc e Rossana, per
il tempo e i pranzi spesi insieme. Ringrazio la piccola comunità di
italiani con Claudio, Matilde, Giovanni, Michele, Elisa, Luca e
Andrea, per essere stati con me lungo questo percorso e per avermi
fatto sentire “un po’ più a casa”. Un grazie speciale va a Lorenzo,
il quale negli ultimi quattro anni è stato un amico prezioso,
sostenendomi nei momenti più difficili di questo dottorato. Infine,
ringrazio Moritz, il mio amico d’oltremare, per aver reso il mio
soggiorno a Santa Barbara memorabile.
Ringrazio i miei genitori, mia sorella e mio fratello, per il
continuo supporto e le lunghissime video chiamate di sera dove
cercavo di raccontare tutto sulle mie avventure a Oslo. Ringrazio
la mia amica Chiara, con la quale ho condiviso molti dubbi e
incertezze, e Gionata, il mio amico di una vita, che non ha mai
smesso di cercarmi e preoccuparsi per me nonostante la distanza.
Ringrazio tutti i miei amici di Verona e coloro i quali mi hanno
motivato e spinto ad intraprendere questo dottorato. Infine,
ringrazio Vegard, il quale mi ha insegnato molte cose riguardo alla
vita, alla carriera e al come sopravvivere nella (talvolta) fredda
Norvegia, e mi ha sostenuto quando più ne avevo bisogno.
Silvia Lavagnini Oslo, Marzo 2021
vii
Contents
1 Introduction 1 1.1 A survey of electricity markets . . . . . . .
. . . . . . . . . 3 1.2 Stylized facts of electricity spot markets
. . . . . . . . . . 5 1.3 Stochastic modelling of spot prices . . .
. . . . . . . . . . 9 1.4 Derivative contracts . . . . . . . . . .
. . . . . . . . . . . . 18 1.5 Options in the energy markets . . .
. . . . . . . . . . . . . 24 References . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 28
Papers 36
I Stochastic Modelling of Wind Derivatives in Energy Markets 39 I.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 40
I.2 Spot price model . . . . . . . . . . . . . . . . . . . . . . .
41 I.3 Models for wind . . . . . . . . . . . . . . . . . . . . . .
. . 44 I.4 Income for a wind energy company . . . . . . . . . . . .
. 48 I.5 Quanto options . . . . . . . . . . . . . . . . . . . . . .
. . 52 I.6 Conclusions . . . . . . . . . . . . . . . . . . . . . .
. . . . 60 I.A Proof of Proposition I.4.3 . . . . . . . . . . . . .
. . . . . . 61 References . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 63
II Correlators of Polynomial Processes 65 II.1 Introduction . . . .
. . . . . . . . . . . . . . . . . . . . . . 65 II.2 Polynomial
processes . . . . . . . . . . . . . . . . . . . . . 72 II.3
Two-point correlators . . . . . . . . . . . . . . . . . . . . . 75
II.4 Higher-order correlators . . . . . . . . . . . . . . . . . . .
. 89 II.5 The recursions . . . . . . . . . . . . . . . . . . . . .
. . . . 98 II.6 Numerical performances . . . . . . . . . . . . . .
. . . . . 105 II.7 Conclusions . . . . . . . . . . . . . . . . . .
. . . . . . . . 108 II.A Some combinatorial properties . . . . . .
. . . . . . . . . . 110 II.B An important identity . . . . . . . .
. . . . . . . . . . . . . 115 References . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 118
III CARMA Approximations and Estimation 121 III.1 Introduction . .
. . . . . . . . . . . . . . . . . . . . . . . . 122 III.2
Construction and convergence results . . . . . . . . . . . . 124
III.3 CARMA representation and convergence results . . . . . . 129
III.4 Numerical schemes and results . . . . . . . . . . . . . . . .
134 III.A Proofs of the main results . . . . . . . . . . . . . . .
. . . 143 References . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 155
ix
Contents
IV Accuracy of Deep Learning in Calibrating HJM Forward Curves 157
IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
. 158 IV.2 The forward curve dynamics . . . . . . . . . . . . . . .
. . 160 IV.3 Forward contracts with delivery period . . . . . . . .
. . . 165 IV.4 The neural networks approach . . . . . . . . . . . .
. . . . 169 IV.5 The setting for the experiments . . . . . . . . .
. . . . . . 173 IV.6 Implementation and results . . . . . . . . . .
. . . . . . . . 177 IV.7 Conclusions, remarks and further ideas . .
. . . . . . . . . 183 IV.A Proofs of the main results . . . . . . .
. . . . . . . . . . . 186 IV.B The non-injectivity issue . . . . .
. . . . . . . . . . . . . . 192 References . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 194
V Pricing Asian Options with Correlators 199 V.1 Introduction . . .
. . . . . . . . . . . . . . . . . . . . . . . 199 V.2 Payoff
approximation with Hermite polynomials . . . . . . 201 V.3 Pricing
options with correlators . . . . . . . . . . . . . . . 210 V.4
Polynomial processes and correlator formula . . . . . . . . 215 V.5
Numerical examples . . . . . . . . . . . . . . . . . . . . . . 218
V.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
228 V.A Correlator formula . . . . . . . . . . . . . . . . . . . .
. . . 231 References . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 233
x
Introduction
Climate change is a defining issue of our time, with the potential
to rise sea levels, dry out lands and cause flooding. To meet this
challenge and reduce the emissions of greenhouse gasses, a
transformation of our societies is required. In this direction,
there has been in the recent years a huge growth in renewable power
in order to reduce the environmental impact of traditional energy
sources, such as oil, coal and natural gas. The amount of energy
from sources like wind and solar power has indeed been rising
annually. However, the shift towards a greener energy system comes
at a cost: renewable energy sources are highly dependent on weather
factors such as temperature, wind, cloud and precipitation, which
are volatile and often hard to predict.
As a consequence, renewable energy sources are less reliable than
traditional energy sources, and power prices appear extremely
volatile, introducing challenges in terms of financial risk
management. To encourage and hence facilitate the transition
towards green energy sources, both energy producers and consumers
seek derivative products and optimal hedging strategies to manage
their risk exposure. For this reason, the accurate modelling of
energy markets is a key component for the shift towards a greener
economy. One of the main concerns of mathematical finance and of
this thesis is the study of energy markets and financial products
by accurate stochastic models.
We can identify three different segments within the energy markets:
a market for physical spot trading, forward and futures contracts
on the spot with physical or financial settlement over a period,
and an option market with futures contracts as underlying. Thus the
modelling of energy markets can be divided into three tasks, namely
spot price modelling, derivation or modelling of futures, and
option pricing. With this thesis we bring innovative contributions
for a broad range of financial problems which goes from the
modelling of the spot price to the pricing of derivatives. Here are
the main contributions.
• We provide a model for the correlation between wind power
production in a power plant and the spot price of electricity when
a pure jump process is chosen for modelling (at least) one of the
two processes.
• We open for more accurate estimations of the expected value of
certain functions depending on both spot price and wind power
production, such as the income for a wind power plant or the
tailor-made payoff functions of certain options, such as quanto
options in energy markets.
• In the context of jump-diffusion polynomial processes, we develop
a formula for computing correlators, namely cross-moments of the
process at different time points along its path. With this, we
overcome the algebraic complexity
1
1. Introduction
that one should deal with if applying recursively the moment
formula, and we open for analytical treatment of, e.g., pricing
functionals.
• We construct recursive formulas for the generator matrix
associated with a polynomial jump-diffusion process and its
exponential with respect to the polynomial basis of monomials, also
providing the matrix transformations in order to generalize the
result to any other basis of polynomials.
• We open to new pricing approaches for path-dependent options or
in the context of stochastic volatility by using correlators, which
allow for easy numerical implementations and analytical
tractability.
• We derive pricing formulas for discretely sampled arithmetic
Asian options, by combining polynomial approximation with
generalized Hermite polynomials and the correlator formula for
polynomial jump-diffusion processes. This leads in general to
higher accuracies than a standard Monte Carlo method.
• We give a rigorous mathematical justification to the
representation via stochastic differential equations of CARMA
processes, which are used to model weather variables such as
temperature, but also spot prices.
• By constructing a smoother version of a Lèvy process by
stochastic convolution, we also justify mathematically the use of
numerical methods for simulation and estimation by approximating
derivatives with discrete increments, which is essential for
applying CARMA processes to real data sets.
• We construct a new class of state-dependent volatility operators
for modelling forward curves in a commodity market via an
infinite-dimensional Heath- Jarrow-Morton (HJM) approach.
• We present a deterministic and parametric class of volatility
operators that capture the main characteristics of forward
contracts in electricity markets.
• We propose a new machine-learning approach for calibrating HJM
models in infinite dimension by neural networks, making
infinite-dimensional models more tractable for practical
applications.
The rest of this chapter is as follows. In Section 1.1 we propose a
survey on electricity markets and introduce the complex structure
that makes electricity a special commodity, focussing in particular
on the Nordic power market Nord Pool. In Section 1.2 we review some
of the main stylized facts which are shared by most electricity
markets, and that must be taken into account when choosing a model.
In Section 1.3 we address the problem of spot price modelling by
reviewing some of the most common families of models in the
literature and by briefly discussing the contribution of this
thesis in this regards. We then address the problem of derivative
pricing for forward and futures contracts in Section 1.4 and for
options in Section 1.5, by reviewing different possible approaches
and summarizing the contributions this thesis brings into the
field. The five articles composing the thesis are enumerated in
chronological order of writing.
2
1.1 A survey of electricity markets
Starting from the early 1990s, many countries have opted for a
liberalisation of the energy markets, with the intent of promoting
competitiveness and efficiency gains. In a market where prices are
determined by the interaction of supply and demand, technical
innovation is more stimulated, and the liberalisation has indeed
led in the years to efficient investments and to an increased
importance of energy markets all around the world [95].
The liberalisation of the power sector has also created a need for
organized markets at the wholesale level, such as the power
exchanges. These are commonly launched on a private initiative, for
instance, by a combination of generators, distribution companies,
traders and large consumers. Most of the recently developed
European markets are based on this model. Examples are the European
Energy Exchange (EEX) AG in Germany, the European Power Exchange
(EPEX SPOT) SE in France and the Nordic power market Nord Pool AS.
The latter, in particular, was the world’s first international
power exchange, established in 1992 in Norway and later extended to
Sweden (in 1996), Finland (in 1998) and Denmark (in 2000).
Nowadays, it also includes the Baltic states, the UK and some
Central Western European countries.
The role of a power exchange is to match supply and demand of
electricity to determine and announce a clearing price, commonly
known as the spot price. We recall that for financial assets and
most commodities, the term “spot” defines a market for immediate
delivery and financial settlements up to two business days.
However, electricity is not storable and a classical spot market is
not possible in this case, since the schedule must be checked for
feasibility in regards to transmission constraints. This latter
task is a responsibility of the transmission system operator (TSO),
such as Statnett for Norway. For these reasons, there is no actual
spot price in the electricity markets, but rather a day-ahead
price.
The day-ahead price is established in the form of a once-per-day
two-sided auction. Each morning, the players submit their bids for
purchasing or selling a certain volume of electricity for the 24
hours of the following day, namely from midnight to midnight. Then
the power exchange determines the intersection of the supply curve
with the demand curve, and, around noon each day, publishes the
day-ahead price for each hour of the next day.
In the Nordic areas, the day-ahead market is called Elspot and is
organized by Nord Pool. Here, due to transmission constraints, the
member states are divided into bidding areas in order to handle
congestions in the grid. Norway, for example, is divided into five
areas and Sweden into four, see Figure 1.1. These areas can have
balance, deficit or surplus of electricity. Then electricity flows
from areas where the price offered is lower towards areas where
demand is high and the price offered is higher. In case of
congestion, each of these areas has a different area spot price. A
price common to all the Nordic countries, called the system price,
is calculated assuming no physical constraints between zones and
setting capacities to infinity. This is the day-ahead price, namely
the reference price used in most standard financial contracts
traded in the Nordic regions. The area prices differ from the
system price in those hours with limited capacity.
3
1. Introduction
Figure 1.1: Bidding areas for Nord Pool power exchange (figure from
[82]).
Since the day-ahead price is determined each day at noon for
delivery in the 24 hours of the next day, the time interval between
the price is fixed and the actual delivery can be from 12 to 36
hours, hence quite long. Therefore, market participants are exposed
to a certain risk, for example if they are not able to produce or
consume the electricity that they have sold or bought the day
before in the day-ahead market. In order to adjust this exposure,
hourly contracts are traded in the so-called intra-day market. In
the Nordic countries, the intra-day market is called Elbas and it
offers continuous intra-day trading, allowing also for 30 and 15
minutes products on several borders [82].
Since capacity is restricted and supply and demand must balance,
one of the most important roles of the TSOs is to supervise the
physical electricity contracts. In particular, the TSOs are
responsible for organizing the real-time or balancing market, which
consists of short-term upward regulations (increased generation or
reduced consumption) or downward regulations (decreased generation
or increased consumption). Both supply and demand sides specify
their bids, stating load and time period for generation and
consumption. The TSOs then list the bids for each hour in merit
order, namely according to price, and use the merit order to
balance the power system. In case of grid power deficit, the
real-time market price is set at the highest price of the units
called upon from the merit order, while in case of grid power
surplus, to the lowest price.
The operations described are physical electricity contracts since
they are open to players with proper facilities for production or
consumption. However, the increasing importance of energy in
everyday life has made it an attractive asset for speculators such
as, e.g., investment banks, hedge funds and pension funds.
Tailor-made contracts for these new players are financial
electricity contracts, which do not require to have consumption or
production of electricity to participate in the markets. These
contracts are linked to some reference electricity spot price and
are in fact settled in cash, following different rules in
4
Stylized facts of electricity spot markets
the different power exchanges. Since the mid 90s, the market place
for sale of financial derivatives at Nord
Pool was Eltermin, which was renamed to Nasdaq Commodities in 2008
after being acquired by Nasdaq [81]. The products traded at Nasdaq
Commodities are mainly used by producers, retailers and end-users
as risk management tools, and by traders for speculating in future
spot prices. Derivatives include futures, forwards, Electricity
Price Area Differentials (EPADs) and options with forward contracts
as underlying asset. In particular, differently from other
commodity markets, forwards and futures in the electricity and gas
markets deliver over a period of time rather than at a fixed time
date. Since they exchange a floating spot price against a
forward/futures fixed price, they are often called swaps. All the
financial contracts use the system price as the reference
price.
1.2 Stylized facts of electricity spot markets
Despite being labelled as a commodity, electricity is in practice
different from other commodities since it has to be delivered over
a time interval, so that it is often called a flow commodity. This
leads to a market that is profoundly different both in
infrastructures and organization from usual commodity markets. In
fact, electricity is not storable or at least has very limited
storage possibilities. This means that the consumer cannot buy for
storage, and this has two main consequences. On the one hand, the
lack of store-ability produces strong seasonality and possible
spikes in the prices. On the other hand, it results in price
differentials between bidding areas, as for the system price in the
Nord Pool day-ahead market, and between different markets, such as
between Nord Pool and the EEX.
The main focus of this thesis is electricity. However, many of the
models considered can be generalized to other commodity markets
having similar modelling characteristics, such as limited
store-ability of the spot, seasonally dependent prices with spikes
and with the traded assets being the average based forward
contracts. Typical examples are temperature and gas markets [21],
which will be mentioned further in this introductory chapter. We
shall now review some of the main stylized facts that are shared by
most electricity spot markets, and we shall use the Phelix Base
price index from the European Power Exchange (EPEX) for
illustration.
1.2.1 Price spikes
One of the most pronounced features of electricity markets are
unpredictable extreme changes in the spot prices, known as jumps or
spikes, as it can be observed in Figure 1.2. These are mainly a
consequence of the mechanism that determines prices, namely the
interaction between demand and supply curves. Based on marginal
costs of production and response time, the generation units of a
given utility in a certain region are ranked in what is called the
supply stack. As illustrated in Figure 1.3, there is a great
variability in costs of production between
5
60
40
20
0
20
40
60
80
100
Sp ot
p ric
e (E
UR /M
W h)
Figure 1.2: Phelix Base price index from 1 January 2012 to 31
December 2015.
different types of installation, wind, solar and hydro power with a
virtually null cost at one extreme, and gas turbines at the other
end of the scale. Moreover, some installations that are relatively
fuel efficient can be run year-round, such as hydro, nuclear and
coal-fired, while energy-intensive units such as gas-fired turbines
are used for short periods of very high price and demand
[82].
On the other hand, unexpected weather conditions may cause sudden
changes in demand, such as for excessive heating in winter or for
air-conditioning in summer, depending on the regions. Since the
supply stack is typically flat in the low-demand region, the
intersection between demand and supply is not very sensitive to
demand shifts when the demand is low. But when the demand is higher
and a larger fraction of power comes from expensive sources, even a
small increase in consumption can make prices raise substantially.
When the demand drops, the more expensive production sources are no
longer needed and prices rapidly decrease to a normal level.
However, since many factors influence the supply stack, such as
fluctuations of fuel prices or outages of power plants, price
spikes can still happen even in the condition of a stable
consumption.
From the modelling point of view, we can define a price spike as a
large upward or downward movement that surpasses a specific
threshold for a short period of time. However, there is no common
consensus of what this threshold or time interval should be.
Different authors have proposed different approaches, such as fixed
price thresholds [77], fixed log-price change thresholds [24], or
variable log-price change thresholds [37, 39]. Alternatively, one
can filter out the spikes with some filtering techniques, such as
wavelet decomposition [90], or simply decide not to define the
price spikes since the model specification and calibration
algorithm do not require it [24, 37, 71, 78]. The class of
models
6
Stylized facts of electricity spot markets
Figure 1.3: Supply stack for Nord Pool power exchange (figure from
[82]).
usually adopted for price spikes is the class of Lévy processes,
which allows for a large range of different distributions also
including heavy tails [5].
1.2.2 Seasonality
Another important feature of electricity markets is seasonality,
which can be observed both on supply and demand sides [95]. Demand
exhibits seasonal fluctuations mainly due to climate conditions,
such as temperature or number of daylight hours. Seasonality is
also observable weekly and intra-daily [80]: weekly, there is
higher demand in the weekdays and lower demand in the weekends;
intra-daily, there is usually a peak in the morning, when people
wake up and go to work, and a peak in the late afternoon, when
people come back and start making dinner. However, seasonal
variations are also observed on the supply side. For example, hydro
power is strictly related to precipitation and snow melting, while
photovoltaic power depends on radiation intensity and angle.
One validated technique in order to test for seasonal behaviour is
the autocorrelation function (ACF), which measures the correlations
for data values at different time lags [92]. Values for the
autocorrelations significantly different from zero indicate that
the time series is not the outcome of a random phenomenon. Since
autocorrelations for consecutive lags are formally dependent
(namely, if the first element is related to the second and the
second is related to the third, then we can expect that the first
element is also related somehow to the third one), it is common to
differentiate the series with a lag of 1, or to take the returns of
the series. In this way, one gets rid of the first-order
correlation
7
0
20
40
60
80
0
20
40
60
80
100
0
20
40
60
80
100
Sp ot
p ric
e (E
UR /M
W h)
Figure 1.4: Three different seasonal functions fitted to the Phelix
Base price index for the period 1 January 2012—31 December
2015.
and obtains more insight information for other dependencies [37].
Alternatively, one can consider the partial autocorrelation
function (PACF) [64].
The most common approach to model seasonality in the spot price of
electricity is to use sinusoidal functions [78, 85]. This approach
can in fact be applied to any time resolution studied, from the
yearly resolution, to the weekly or even intra-daily resolutions.
In this case, one needs to adjust the period of the sinusoidal
functions to the one required in the modelling. In particular,
there are two possible approaches for including a seasonal
component in the model, namely by a geometric or arithmetic model.
In the first case, the seasonal component appears as a
multiplicative factor in the model, while in the second case it
appears as an additive factor [21].
In Figure 1.4 we show for illustration three different seasonal
functions that have been fitted to the Phelix Base index price time
series (after removing the negative values). Specifically, we
consider the following deterministic functions:
Λ1(t) = a1 + b1t,
Λ2(t) = a2 + b2t+ c2 sin(2πt) + d2 cos(2πt), Λ3(t) = a3 + b3t+ c3
sin(2πt) + d3 cos(2πt) + e3 sin(12 · 2πt) + f3 cos(12 · 2πt),
with a1, b1, a2, b2, c2, d2, a3, b3, c3, d3, e3, f3 real constants
and the time tmeasured in years. Here, Λ1 captures the linear
(trend) effect of time, Λ2 captures both linear trend and the
yearly seasonality, and Λ3 includes also the monthly
seasonality.
1.2.3 Mean reversion
A well-known feature of energy spot prices is mean reversion, which
is the tendency of prices to move towards a mean level. The speed
of mean reversion depends on several factors, such as the kind of
commodity considered and the delivery provisions associated with
the commodity. We can then have short-term mean reversion or
long-term mean reversion. For example, it is common to observe in
the electricity markets sudden price spikes which revert very fast
to
8
Stochastic modelling of spot prices
the previous price level. On the other hand, in other markets (such
as the oil markets) prices can take up to months before reverting
to the mean level.
One of the most common models adopted for incorporating mean
reversion is a Gaussian Ornstein–Uhlenbeck (OU) process, originally
proposed by [91] for interest-rate modelling. The drift term of an
OU process is proportional to the speed of mean reversion (usually
a negative constant parameter) and to the distance between the
current price level and the mean reversion level. Hence, if the
spot price is below the mean reversion level, then the drift will
be positive with the intent of pushing upward the price level. On
the other hand, if the spot price is above the mean reversion
level, the drift will be negative, pushing the price downward. Over
time, this leads to a price path reverting towards a mean level,
with a rate depending on the speed of mean reversion of the
model.
Many extensions for the Gaussian OU process have been considered.
First of all, the mean reversion level in the drift term can be
constant, deterministic or even stochastic. The same holds for the
speed of mean reversion, which can be modelled by a stochastic
process [14]. In [29] the authors propose a potential function in
the current price level, allowing for a continuously varying
mean-reversion rate. It is also common when modelling electricity
prices to replace the Brownian motion driving the noise with a
non-Gaussian process [9, 13, 80] or to consider a jump-diffusion
process [37, 72]. In [61], the authors couple a mean reversion
process with upward and downward jumps, where the jumps direction
depends on the price level.
1.3 Stochastic modelling of spot prices
Spot-price modelling is the key for derivative pricing and risk
management. If the chosen model is inaccurate and does not capture
the main features of the price process, the results are likely to
be unreliable. On the other hand, it is crucial that the model
complexity is not too high, in order to make it applicable and
appealing for everyday use in the trading departments. Obviously,
there is not the model for electricity that is suited for all
electricity markets, since national electricity markets can have
very different patterns depending on which sources of electricity
are employed. However, many of the characteristics described in
Section 1.2 are common factors for the different electricity
markets, so that similar models can be adapted to the interested
markets.
We work on the real interval [0, T ] and with a probability space
(,F ,P) equipped with a filtration {Ft}t≥0. The most famous model
for the spot price dynamics S of a financial asset is the geometric
Brownian motion, which is defined as the exponential of a drifted
Brownian motion B, namely
S(τ) = S(t) exp (µτ + σB(τ)) , τ ≥ t ≥ 0, (1.1)
for µ and σ > 0 constant. In particular, since the Brownian
motion is a process with independent and stationary increments
which are normally distributed, the logarithmic returns logS (τ +
τ)− logS (τ) are consequently independent and stationary, following
a Gaussian distribution.
9
1. Introduction
α = 1.0, σ = 0.1 x = 3.0, σ = 0.1 x = 2.0, α = 1.0
0 2 4 6 8 10 Time
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
2.0
2.2
2.4
2.6
2.8
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
= 0.1 = 0.3 = 0.5
Figure 1.5: Examples of OU process with long-run mean µ =
2.0.
A classical extension for commodity markets is the Schwartz model
[88], which includes the mean reversion feature in the model by
considering
S(τ) = S(t) exp (X(τ)) , (1.2)
where X is an Ornstein–Uhlenbeck (OU) process following the
dynamics
dX(u) = α(µ−X(u))du+ σdB(u), X(t) = x, (1.3)
with α > 0. Specifically, the parameter α is the rate of mean
reversion, µ is the long-run mean and σ is the volatility of the
process. The main difference with the geometric Brownian motion
defined above is that the Schwartz model varies with time. Indeed,
from equation (1.3) one notices that, for each time instant u, if
the value of the process X(u) differs from the long-term mean µ, it
will tend to be pulled back towards µ with speed α. The solution at
time τ ≥ t ≥ 0 of equation (1.3) is given by
X(τ) = xe−α(τ−t) + µ (
1− e−α(τ−t) )
In particular, by calculating mean and variance of X(τ),
namely
E [X(τ)|X(t) = x] = xe−α(τ−t) + µ (
1− e−α(τ−t) ) ,
2α
) ,
we see that for τ →∞, X(τ) tends to a Gaussian distributed random
variable with mean µ (the long-run mean) and variance σ2/2α. In
Figure 1.5 we illustrate the effect of the parameters x, α and σ on
the path of the OU process.
The Schwartz model can be generalized in different ways. In order
to capture a seasonal effect, one can for example replace the
initial value S(t) in equation (1.2) with a deterministic function
Λ : [0, T ]→ R+ modelling the seasonality of the spot price
process, such as a sinusoidal function. This leads to
S(τ) = Λ(τ) exp (X(τ)) .
Stochastic modelling of spot prices
Moreover, in order to allow the possibility of jumps in the model,
one can include non-Gaussian innovations by adding a Lévy process L
to equation (1.3) and obtaining a so-called jump-diffusion model,
namely
dX(u) = α(µ−X(u))du+ σdB(u) + dL(u), X(t) = x. (1.4)
Alternatively, one can simply replace in equation (1.3) the
Brownian motion B with a Lévy process L, namely
dX(u) = α(µ−X(u))du+ σdL(u), X(t) = x. (1.5)
Further, the constant coefficients α, µ and σ can be replaced by
real-valued continuous functions defined on the interval [0, T
].
As we discussed in Section 1.2.1, sudden jumps in the spot price
level can happen for different reasons and at different time
scales. We might have rare sudden jumps turning back to the mean
level very quickly, or more restrained and slower mean-reverting
variations. This shows the necessity of a model that allows for
different speeds of mean reversion and incorporates a mixture of
jumps and diffusional behaviours of the prices. One possibility is
to consider a series of Gaussian and non-Gaussian OU processes,
namely a model of the form
S(τ) = Λ(τ) exp
m∑ i=1
Yj(τ)
, (1.6)
where the Yj ’s follow dynamics of the form of equation (1.5),
while the Xi’s follow dynamics like the one in equation (1.3) but
with the innovation term σdB(u) possibly replaced by a sum of p
potentially dependent Brownian motions, namely
∑p k=1 σikdBk(u). The idea behind this general formulation is that,
for
example, spikes can be modelled by an OU process driven by a Lévy
process with low frequency and fast mean reversion, while normal
price variations can be modelled by a slower mean-reverting process
driven by a Brownian motion [21].
Several models studied in the literature fall into this general
formulation. One of the most famous is the two-factor model
proposed by [78] for electricity prices in the Nordic power
exchange. Their model corresponds to having m = 2, n = 0 and p = 2
in equation (1.6), namely
S(τ) = Λ(τ) exp (X1(τ) +X2(τ)) dX1(u) = −α1X1(u)du+ σ1dB1(u) dX2(u)
= µ2du+ σ2
( ρdB1(u) +
√ 1− ρ2dB2(u)
) , for −1 ≤ ρ ≤ 1. In particular, the fact that α2 = 0 implies
that X2 is not stationary. Here the idea is to model the long-term
equilibrium price with X2 and the short-term mean-reverting
component with X1. This model was extended
11
1. Introduction
by [93] with m = 2, n = 1 and p = 2 by S(τ) = Λ(τ) exp (X1(τ)
+X2(τ) + Y1(τ)) dX1(u) = (µ1 − α1X1(u)) du+ σ1dB1(u) dX2(u) = (µ2 −
α2X2(u)) du+ σ2
( ρdB1(u) +
.
Here, in particular, X1 + Y1 models the short-term variations,
while X2 the long-term level, which, contrary to the model of [78],
is also a mean reversion process. Other extensions are in [46,
63].
In all the models introduced above, the noise term is defined as a
Brownian motion or a Lévy process rescaled by a constant volatility
σ. However, a constant volatility is often not enough to model the
variability of the spot price. A possible solution is to replace σ
with a real-valued continuous function, which may be deterministic
or stochastic. In the latter case we talk about stochastic
volatility models. Here the idea is to replace σ with a stochastic
process which may be modelled by a second dynamics, such as with a
mean-reverting process, or it may be a function of the current
state of the spot price itself, in which case we talk about
state-dependent models. Two of the most famous stochastic
volatility models are the Heston model [66] and the GARCH model
[28], where the spot price is defined by{
dS(u) = µS(u)du+ √ σ(u)S(u)dB1(u)
dσ(u) = α(θ − σ(u))du+ ξσ(u)βdB2(u)
for β = 1 2 in the Heston model and β = 1 in the GARCH model,
respectively.
Here B1 and B2 are correlated Brownian motions, µ is the rate of
return of the asset price, θ the long-run average price variance, α
the rate of mean-reversion for the volatility and ξ the volatility
of the volatility (“vol of vol”). A typical example of
state-dependent stochastic volatility model is the constant
elasticity of variance (CEV) model [40], where the spot price is
defined by the dynamics
dS(u) = µS(u)du+ σS(u)γdB(u),
with γ and σ positive constants, so that there is no need to
introduce a second dynamics for the volatility process. Stochastic
volatility models, such as the Heston, GARCH and CEV models, can be
combined with the Schwartz model (1.2), as for example in [11, 60].
Remark 1.3.1. Equation (1.6) describes a geometric model where the
logarithm of the spot price is given by the sum of the (logarithm)
seasonal component and the innovation terms. Instead, directly
defining the spot price in additive form yields an arithmetic
model, namely
S(τ) = Λ(τ) + m∑ i=1
Xi(τ) + n∑ j=1
2012-01-01 2012-12-27 2013-12-22 2014-12-17 2015-12-12 Time
60
40
20
0
20
40
60
80
100
Sp ot
p ric
e (E
UR /M
W h)
Negative values
Figure 1.6: Phelix Base price index from 1 January 2012 to 31
December 2015.
The main difference is that an arithmetic model allows in general
for negative prices. In a normal market, this does not make sense
since it would mean that the buyer of a commodity receives money
rather than paying. For this reason, arithmetic models are not very
common for spot price of commodities. However, the phenomenon of
negative prices is not completely unknown within electricity
markets [12, 78]. In Figure 1.6, for example, we observe eight
negative values (highlighted with a red star) for the Phelix Base
price index in the period 1 January 2012—31 December 2015. We can
explain negative prices by thinking that it may be more costly for
a power producer to switch off the plant than to pay someone to use
the surplus of electricity caused by higher supply than demand. In
[13] the authors propose a class of arithmetic models that have
zero probability of negative prices, by considering the seasonal
function Λ(τ) in equation (1.7) as the lower bound to which the
processes Xi’s and Yi’s revert.
1.3.1 Polynomial processes
A class of processes that has recently gained particular attention
within finance is the class of polynomial processes, which we shall
now introduce. A jump-diffusion process is called polynomial if its
extended generator maps any polynomial function to a polynomial
function of equal or lower degree [43]. As a consequence, the
expectation of any polynomial of the future state of the process,
conditioned on the information up to the current state, is given by
a polynomial of the current state. More specifically, the
conditional moments can be calculated in closed form without any
knowledge of the probability distribution nor of the
13
1. Introduction
characteristic function of the process. Examples of polynomial
processes are the Ornstein–Uhlenbeck processes, exponential Lévy
processes and affine processes.
To put this in mathematical terms, let Poln(R) be the space of all
polynomials of degree less than or equal to n on R, and h0(x),
h1(x), h2(x), . . . be orthogonal polynomial functions with values
in R such that {h0(x), h1(x), . . . , hn(x)} forms a basis for
Poln(R). We introduce the vector valued function
Hn : R −→ Rn+1, Hn(x) = (h0(x), h1(x), . . . , hn(x))>,
with > the transpose operator, so that every polynomial function
p ∈ Poln(R) can be represented by p(x) = ~p>nHn(x) =
Hn(x)>~pn where ~pn = (p0, p1, . . . , pn)> ∈ Rn+1 is the
vector of coordinates with respect to the chosen basis.
For B a standard one-dimensional Brownian motion and N(dt, dz) a
compensated Poisson random measure with compensator `(dz)dt, we
introduce a polynomial jump-diffusion process by the stochastic
differential equation (SDE)
dY (u) = b(Y (u))du+ σ(Y (u))dB(u) + ∫ R δ(Y (u−), z)N(du,
dz),
where Y (t−) denotes the left-limit, and
b(x) = b0 + b1x, σ2(x) = σ0 + σ1x+ σ2x 2, δ(x, z) = δ0(z) + δ1(z)x,
(1.8)
for b0, b1, σ0, σ1, σ2 ∈ R, and δ0, δ1 : R→ R integrable functions
with respect to the Lévy measure ` [53]. For every bounded function
f ∈ C2(R), the extended generator associated with the process Y is
given by
Gf(x) = b(x)f ′(x) + 1 2σ
2(x)f ′′(x)+
(f(x+ δ(x, z))− f(x)− f ′(x)δ(x, z)) `(dz). (1.9)
By definition of polynomial process, for every finite m ≥ 0, the
generator G in equation (1.9) maps Polm(R) to itself [45]. In the
one-dimensional setting, this is guaranteed by condition (1.8).
Moreover, for every n ≥ 1 we can introduce the so-called generator
matrix Gn ∈ R(n+1)×(n+1) associated with the process Y by
GHn(x) = GnHn(x),
which is the linear representation of the action of the extended
generator G on the chosen basis vector of polynomials Hn(x).
By [45, Theorem 2.7], it is then possible to prove the moment
formula for a polynomial process Y , stating that
E [p(Y (T ))| Ft] = ~p>n e Gn(T−t)Hn(Y (t)), 0 ≤ t ≤ T,
(1.10)
where p ∈ Poln(R) and ~pn ∈ Rn+1 is its vector of coefficients with
respect to the chosen basis. This tells us that E [p(Y (T ))| Ft]
is a polynomial function in Y (t)
14
Stochastic modelling of spot prices
for every p ∈ Poln(R). In particular, it means that the conditional
moments of a polynomial process can be found in closed form up to
the exponential of the generator matrix Gn. For this reason,
polynomial processes find application in finance and option
pricing. In the literature, we find examples in stochastic
volatility models [1, 3, 52], stochastic portfolio theory [44] and
option pricing [2, 53]. For polynomial processes in the electricity
markets, we mention [94] for one- and two-factor models for the
spot price, [75] for the modelling of long-term electricity
contracts with delivery period, and [10] for the pricing of
derivative products such as forward, futures and options via
polynomial approximations.
In Paper II we extend the moment formula (1.10) in order to compute
correlators. For a polynomial jump-diffusion process Y ,
correlators are defined as the cross-moments of the process at
different time points along its path, namely
E [pm (Y (s0)) pm−1 (Y (s1)) · · · · · p0 (Y (sm))| Ft] ,
(1.11)
where pk ∈ Polnk(R), k = 0, . . . ,m, and t < s0 < s1 < ·
· · < sm < T < ∞. In particular, we refer to expectations
such as the one in equation (1.11) as (m + 1)-point correlators,
and let n := max {n0, . . . , nm}. These expectations can arise in
financial pricing, such as for path-dependent options or in the
context of stochastic volatility models. We also notice that for m
= 0 equation (1.11) corresponds to the moment formula (1.10).
In principle, the expectation in equation (1.11) can be computed by
iterating the moment formula. For the two-point correlator case (m
= 1), one can apply the tower rule for Ft ⊆ Fs0 to get
E [p1 (Y (s0)) p0 (Y (s1))| Ft] = E [p1 (Y (s0)) q0 (Y (s0); s1 −
s0)| Ft] , (1.12)
where q0 (Y (s0); s1 − s0) = E [p0 (Y (s1))| Fs0 ] is the
polynomial obtained by applying the moment formula to p0(x).
Another application of the moment formula, this time to p1(x)q0(x;
s1 − s0), produces an expression for the conditional expectation
(1.12). This can be continued for larger values of m. There are
however two major issues with this procedure. First, the degree of
the polynomials involved increases, which causes a blow-up in
computational complexity. Second, performing the calculations is
non-trivial because of the algebraic complexity of manipulating the
expressions involved. In Paper II we primarily make headway on the
second issue. In particular, we develop a number of results that
enable us to get a handle on, and ultimately implement
algorithmically, the required calculations. Specifically, we focus
on the vector basis of monomials Hn(x) := (1, x, x2, . . . ,
xn)> and obtain all our results by working with this basis.
Change-of-basis formulas are then provided in order to generalize
the setting.
By representing the two polynomial functions p0(x) and p1(x) in
terms of Hn(x) and by taking the product of the two polynomials we
get an object of the form Hn(x)Hn(x)>. Hence we need to deal
with a matrix of functions instead of a vector of functions as for
m = 0. By considering the vectorization of Hn(x)Hn(x)>, namely
the column vector obtained by stacking the columns of the matrix
Hn(x)Hn(x)> into a vector, we move the problem to a
framework
15
1. Introduction
similar to the case m = 0. However, the vectorization of
Hn(x)Hn(x)> contains redundant terms, namely repeated powers of
x. Despite a suitable generator matrix can in principle be
constructed for the cause, in order to generalise to higher order
correlators, we must get rid of the redundant terms. In other
words, we need a transformation that from Hn(x)Hn(x)> returns
H2n(x). Once done that, the generator matrix to consider is simply
G2n. We achieve this by introducing the so-called L-eliminating
matrix and L-duplicating matrix. The framework is then extended by
suitable recursive relations to m ≥ 0.
The strength of our framework lies in multiple facts. First of all,
we provide an explicit formula for correlators that involves only
linear combinations of the matrix exponential of the original
generator matrix Gn. This means that, assuming these exponential
matrices to be exact, we have a formula for computing correlators
that in practice is exact. Our approach outperforms the alternative
solution by iterating the moment formula because we are able to
write things fully explicitly, while this is not straightforward in
the other case. Having closed formulas is an advantage for example
for those applications that require to differentiate, such as for
computing the Greeks. Moreover, numerical experiments show also
that our approach outperforms a Monte Carlo-simulation
algorithm.
1.3.2 CARMA processes
Another class of processes that finds application in energy markets
and, more in general, for modelling weather variables such as
temperature, is the class of continuous-time autoregressive moving
average processes, CARMA in short. These are the continuous-time
version of ARMA models, which are used to model discrete time
series and are built up on two different parts: a part of
autoregression and a part of moving average. In discrete time, an
autoregression (AR) of order p specifies that the output variable
depends linearly on its own p previous values plus an innovation
term. A moving average (MA) of order q < p specifies that the
output variable depends linearly on the current and q past values
of the innovation process.
The AR and MA parts are described by introducing the so-called
characteristic polynomials Φ ∈ Polp(R) and Ψ ∈ Polq(R). Then, for S
the back-shift operator acting on a discrete time series Xt by SkXt
= Xt−k, the AR and MA models are defined respectively by
Φ(S)Xt = c+ εt and Xt = Ψ(S)εt,
for c a constant and εt the white noise terms. These latter ones
are generally assumed to be i.i.d. sampled from a centred Gaussian
distribution. However, this assumption can be relaxed to allow for
other kinds of innovation. Combining the two parts, one obtains the
ARMA model by
Φ(S)Xt = c+ Ψ(S)εt.
To find appropriate values for the orders p and q in the ARMA
model, one usually plots the partial autocorrelation functions for
an estimate of p, and the
16
Stochastic modelling of spot prices
autocorrelation functions for an estimate of q. After choosing p
and q, ARMA models can be fitted, for example, by least squares
regression or maximum likelihood estimation techniques [31].
The idea of ARMA models is brought in continuous time with CARMA
models, where the back-shift operator is replaced by the
differential operator D = d
dt , with DkX(t) denoting the k-th derivative of the process X(t).
Considering the innovation to be modelled by a Lèvy process L, a
CARMA process of order (p, q) is formally defined through the
stochastic differential equation
P (D)X(t) = Q(D)DL(t), (1.13)
where P ∈ Polp(R) and Q ∈ Polq(R) are the characteristic
polynomials of the CARMA process [30]. Since a Lévy process has in
general no differentiable paths, the representation via SDE is only
formal, and the CARMA process is interpreted by means of its
state-space representation{
dY(t) = AY(t)dt+ epdL(t) X(t) = b>Y(t)
where A ∈ Rp×p contains the coefficients of the polynomial P , b ∈
Rp contains the coefficients of the polynomial Q and ep is the p-th
unit vector. If L is any second order Lévy process, X(t) can be
explicitly expressed by
X(t) = b>eA(t−s)Y(s) + b> ∫ t
s
eA(t−u)epdL(u),
from which we observe that a CAR(1) process corresponds to an
Ornstein– Uhlenbeck process (in this case the matrix A is simply A
= (−a1)).
A possible approach in order to estimate CARMA processes from
discrete- time observed time series is by first focussing on their
discrete-time counterparts. It is indeed possible to find a
relation between the coefficients of a CARMA(p, q) process and an
ARMA(r, s) process with 0 ≤ r < s ≤ p. Then one first estimates
the ARMA process and in a second step recovers the CARMA parameters
through this relation [20, 32, 87]. In particular, simulations and
estimation of CARMA processes rely on numerical schemes, such as
the Euler scheme, which approximate derivatives with discrete
increments with respect to a time step > 0. Their convergence
for approaching zero is also a well-studied topic. However, this
opens up for a discrepancy since it is not mathematically clear
what happens when the time step approaches zero as, in general, the
derivatives of a Lévy process are not well defined.
In Paper III we give a mathematical justification to the
representation via SDE for a CARMA process, and, consequently, to
the discrete approximations used for simulating and estimating the
continuous-time dynamics of CARMA processes. This is done by
replacing the Lèvy process L driving the noise with a continuously
differentiable process Lε obtained by stochastic convolution and
converging to L in L2 ([0, T ]× ) with rate 1. An Lε continuously
differentiable solves the differentiability issue only for CAR(p)
processes in which no more than
17
1. Introduction
the first derivative of the driving noise process is required, but
we extend the approach to CARMA(p, q) processes of every order q ≥
0 by the result presented in [32, Proposition 2]. We thus replace
the Lèvy process L with Lε in the SDE defining the CARMA process X,
obtaining a new process Xε which converges to the original CARMA in
L2 ([0, T ]× ) with rate 2.
Since the ε-approximation of the Lévy process justifies the use of
an Euler scheme for every time step , we then present a possible
approach for simulating Lε starting from a simulated path of the
Lévy process L, and then Xε starting from Lε. For a fixed time
interval , this gives a new noise process L
ε and a third process X
ε , for which we study the convergence. The analysis highlights
that the two parameters and ε must be chosen accurately. Finally,
the calibration problem is also addressed by numerical
experiments.
Both ARMA and CARMA models are a convenient parametric family of
stationary processes exhibiting a wide range of autocorrelation
functions which can be used to model the empirical autocorrelations
observed, for example, in financial time-series analysis. Moreover,
letting the innovation process be modelled by a Lévy process allows
for a rich class of marginal distributions with possibly heavy
tails, which are often observed in finance. The interest in
continuous-time models is due to their use in modelling irregularly
spaced data or in financial applications such as option pricing. We
have examples in stochastic volatility modelling applied to
exchange rates [9, 33] and in electricity markets [15, 23, 58, 83,
84]. Finally, CARMA processes find application in modelling weather
variables, such as temperature and wind speed [20, 21]. Given the
significant application of CARMA processes, with Paper III we
justify their mathematical formulation and application in a
rigorous way. Since the new process Lε admits derivatives, this may
also open for further applications in the context of sensitivity
analysis with respect to paths of the driving-noise process.
1.4 Derivative contracts
A derivative is a contract whose value depends on the performance
of an underlying product, such as an asset, an interest rate or
another derivative. Derivatives are mainly used for hedging the
risk coming from the underlying process, by entering a contract
whose value moves in the opposite direction of the underlying
position. They are also used for speculation and for insurance
purposes, such as in the case of weather derivatives (for which it
is not possible to trade in the underlying). We might distinguish
between two main categories of derivatives, namely forward and
futures contracts on one side and options on the other. We clarify
the difference with an anecdote.
The oldest example of derivative is attested by Aristotle and is
dated back to 2400 years ago [42]. Aristotle tells the story of a
poor philosopher named Thales who forecasts a very successful olive
crop for the upcoming season. He then visits one by one the owners
of the olive presses and reserves each press for his exclusive use
in the harvest period. With the bumper crop that comes, the demand
for olive presses is high, and Thales can charge whatever he
pleases
18
Derivative contracts
for those, gaining a lot of money. Despite Aristotle does not give
many details about the contract terms between Thales and the owners
of the olive presses, we can imagine two different alternatives.
One possibility is that the contract gives Thales the right but not
the obligation to use the olive presses. If he does use them, then
he would have to pay the rest of the rent. If he does not, the
owners of the presses can simply keep the deposit and rent the
presses to someone else. In this case we would call the contract an
option. However, if the contract requires Thales to pay for the
olive presses during the harvest time regardless of whether he
rents them or not, we would call it a forward or futures.
These two alternatives make a very big difference for Thales.
Indeed, in the case of a poor harvesting season, with an option
contract Thales would only lose the money of the deposit, but with
a forward- or future-style contract he would be forced to rent the
olive presses, losing money. From this point of view, the option
contract seems preferable. However, the owners of the olive presses
will probably set a higher rent and require a higher deposit to
cover the possibility that the presses might not be used during
harvest time. We then see that the two alternative contracts make a
difference also for the owners of the presses. In particular, a
forward- or future-style contract would allow them to hedge their
risk, since entering such a contract would assure them a fixed
income (the rent from Thales) regardless of whether the season is
outstanding or terrible.
1.4.1 Forward contracts
As we already mentioned, the value of a derivative depends on the
performance of the underlying product. The key for pricing forwards
is then an accurate model for the underlying spot process. Let S be
the spot price and assume to enter a forward contract delivering
the spot at time τ and with price f(t, τ) at time t ≤ τ . The
payoff of this position at the delivery time τ is then S(τ)− f(t,
τ). Letting r > 0 be the constant risk-free interest rate and Q
a pricing measure (being usually the risk-neutral probability
measure, see Section 1.4.3 below), the value at time t of the
forward contract is given by the present expected value of its
payoff under the measure Q [48]. Since the forward contract is
entered at no cost, this value must be equal to zero, namely
e−r(τ−t)EQ [S(τ)− f(t, τ)| Ft] = 0,
where Ft is the filtration containing all market information up to
time t. In particular, the forward price f(t, τ) must be adapted to
Ft because it is set at time t, hence it cannot contain more
information than what given by Ft. Then
f(t, τ) = EQ [S(τ)| Ft] . (1.14)
This yields an arbitrage-free dynamics for the forward price
process t 7→ f(t, τ), since this process is a martingale under Q.
Remark 1.4.1. We point out that both forward and futures contracts
involve the agreement between two parties to buy and sell an asset
at a specified price by a certain date. However, a forward contract
is a customized arrangement made
19
1. Introduction
over-the-counter (OTC) that settles just once at the end of the
agreement. On the other hand, a futures contract has standardized
terms and is traded on an exchange. Moreover, in the case of a
futures contract, prices are settled on a daily basis until the end
of the contract. However, since throughout the thesis we assume
constant risk-free interest rate, futures prices and forward prices
with common maturity coincide [41].
1.4.2 Swap contracts
We know that in energy markets such as for power and gas, the
underlying commodity is not delivered at a fixed time, but rather
over a period of time, called the delivery period. In this case,
the buyer of an electricity forward receives power during the
delivery period (financially or physically), in exchange of paying
a fixed price per MWh. For this reason, we shall call this kind of
contracts with the name of swaps, to avoid confusing them with the
forward contracts with fixed delivery date described above. We
denote by F (t, T1, T2) the price of a swap at time t for a
contract with delivery period [T1, T2], for t ≤ T1 ≤ T2. Then the
payoff at time t for a continuous flow of electricity is
∫ T2 T1 e−r(u−t) (S(u)− F (t, T1, T2)) du.
Similarly to fixed-time forwards, the contract is entered at no
cost, hence we get the relation
e−rtEQ
Ft ]
= 0,
where it is reasonable to assume F (t, T1, T2) to be Ft-adapted,
since the swap price is settled at time t based on the information
available up to this time. Then
F (t, T1, T2) = EQ
[∫ T2
T1
Ft ] . (1.15)
With a similar argument, if the settlement takes place financially
at the end of the delivery period T2 instead of continuously during
the delivery period, then
F (t, T1, T2) = EQ
[∫ T2
T1
Ft ] . (1.16)
To generalize equation (1.15) and (1.16), we introduce the weight
function
w(u, s, t) := w(u)∫ t s w(v)dv
with w(u) := {
e−ru
1 , (1.17)
where w(u) = e−ru if the contract is settled continuously during
the delivery period, and w(u) = 1 if the contract is settled at the
end of the delivery period. Then the link between a swap contract
and the underlying spot is
F (t, T1, T2) = EQ
[∫ T2
T1
Moreover, by commuting the conditional expectation with the
Lebesgue integration in equation (1.18), we also obtain that
F (t, T1, T2) = ∫ T2
T1
w(u, T1, T2)f(t, u)du, (1.19)
which links the price of a swap contract with the fixed-time
forward price.
1.4.3 The risk-neutral measure
So far, we have mentioned the risk-neutral measure Q without giving
any information about it. We recall that in a market with liquidly
traded spot, the discounted spot price is a martingale under Q, so
that there is a well-known relation between a forward contract and
the underlying spot, which is obtained by a buy-and-hold strategy
argument. However, this breaks down in the electricity market,
because electricity cannot be stored, hence it cannot be bought and
kept in a portfolio. Since the spot is not tradable in the usual
sense and the bank account is trivially a martingale under any
equivalent martingale measure after discounting, in equation (1.14)
the choice for Q is open to any equivalent martingale measure. For
this reason, we do not have a unique price based on the arbitrage
argument, but we need additional criteria to pin down the choice of
Q.
One possible approach is to consider a parametric class of
risk-neutral measures that somehow preserves the peculiar
properties of the underlying model. This is the case of the
Girsanov transform for Brownian motions [26] and the Esscher
transform for general Lévy processes [49]. Both these transforms
introduce a parametric change in the drift of the spot. Moreover,
the Girsanov transform preserves the normality of the distribution
of the Brownian motion. Similarly, the Esscher transform preserves
the distributional properties of the jump process. More precisely,
jumps frequency and sizes change, so that the characteristic of
each jump process is altered, but the independent increment
property is preserved [22, 69].
Another possible approach is the so-called rational-expectation
hypothesis assuming Q = P and implying f(t, τ) = E [S(τ)| Ft] .
However, it is reasonable to think that the producer of a commodity
would like to hedge its revenues by selling forwards, hence
accepting a discount on the expected spot price. This means that
f(t, τ) < E [S(τ)| Ft] and leads to the concept of risk
premium:
RP (t, τ) := f(t, τ)− E [S(τ)| Ft] .
The risk premium measures in some sense the difference between
risk-neutral and real-market price predictions, and describes the
fact that hedgers are willing to pay for getting rid of the spot
price risk. Its value is usually negative when the market is in
normal backwardation, but evidence for a term structure in the risk
premium for power markets has been detected in several studies [38,
47].
21
1.4.4 The HJM approach
What is often adopted as an alternative in energy markets, where
the buy- and-hold strategy cannot be applied, is the direct
modelling of the tradable forward prices. This approach comes from
interest-rate theory, introduced by Heath, Jarrow and Morton, so
that it is referred to as the Heath-Jarrow-Morton (HJM) approach
[65]. In the interest-rate setting, the original idea was to model
the entire forward-rate curve directly because short-rate models
are not always flexible enough to be calibrated to the observed
initial term structure [51]. Later the approach has been
transferred to other markets, such as for commodity forwards [16,
35, 39, 76], and other products, such as call options [36,
73].
For forward contracts with fixed settlement time, applying the HJM
approach is straightforward. The general idea is to state the
dynamics of the forward price directly under the risk-neutral
measure, this being the convenient measure if the purpose is to
price options. However, in electricity, gas or weather markets, the
commodity is delivered over a period and applying the HJM approach
for swap contracts is not straightforward. The main issue is that
there exist many different contracts with different delivery
periods, some of them being overlapping. For example, it is
possible to invest in a quarterly contract, but also on three
monthly contracts covering the same period. Similarly, it is
possible to invest in a yearly contract, but also on the
corresponding four quarterly contracts. When modelling these
contracts, arbitrage-free conditions must be satisfied.
If F (t, T1, T2) is the swap price of a contract with delivery
period [T1, T2] and F (t, tk, tk+1), for k = 1, . . . , n, are n
contracts delivering over the intervals [tk, tk+1] with T1 = t1
< t2 < · · · < tn+1 = T2, the no-arbitrage relation is
like
F (t, T1, T2) = n∑ k=1
wkF (t, tk, tk+1), for wk := ∫ tk+1 tk
w(v)dv∫ T2 T1 w(v)dv
,
with w introduced in equation (1.17). To make this hold for
arbitrary delivery periods, we then consider tk := T1 + (k − 1)
with := (T2 − T1)/n, and by letting n→∞, we obtain the continuous
version of such no-arbitrage condition
F (t, T1, T2) = ∫ T2
T1
w(u, T1, T2)f(t, u)du, (1.20)
with w in equation (1.17) and f(t, u) = F (t, u, u). Equation
(1.20) implies that any swap model valid for arbitrary delivery
periods [T1, T2] must come from a forward dynamics. However, in
[21] the authors show that for most interesting cases, such as for
a geometric Brownian motion, this is not satisfied.
The alternative approach is to construct models for swaps directly
based on models for forwards, despite these last ones not being
traded products in the electricity and gas markets. Following the
HJM approach for forward prices, by equation (1.19) one then
obtains suitable models for swaps satisfying the no-arbitrage
relation (1.20). However, since forward prices are not observed in
the market, it comes the question on how to estimate the forward
dynamics
22
Derivative contracts
starting from swaps observed data. A possible approach in this
sense is by applying a smoothing technique, namely by constructing
a smooth forward curve starting from the discrete swap curve
observed in the market [17, 54].
The HJM approach leaves the open question on which dynamics to
choose for the forward curve. The standard choice for a forward
model in energy markets is the lognormal, or geometric Brownian
motion dynamics [7, 16, 39] also with multi-factors [25, 76]. As a
consequence of the different structure with respect to usual
commodity markets, for electricity and gas markets a high degree of
idiosyncratic risk across different maturities is observed [4, 55,
76]. A Principal Component Analysis on the Nord Pool AS forward
contracts shows that 75% of the forward price variation can be
explained by two factors, while this number is closer to 95% in
other markets such as interest rates [76]. A second study reveals
also that more than ten factors are needed to explain 95% of the
volatility [21]. This points out the necessity of modelling forward
curves by a high-dimensional, possibly infinite-dimensional, noise
process.
In Paper IV we model forward contracts in a commodity market by an
infinite-dimensional and state-dependent HJM model. In particular,
an infinite- dimensional setting requires to model forward prices
by the stochastic dynamics of the function-valued stochastic
process f(t, ·). Following [18], we work in the so-called Filipovi
space, which is a separable Hilbert space first introduced for
interest-rate modelling [50]. In our setting, the
Hilbert-space-valued Wiener process driving the forward curve takes
values in L2(O), O being some Borel subset of R (possibly R
itself). Hence the volatility operator must smoothen elements in
L2(O) to elements of the Filipovi space. For this purpose we
introduce a new class of state-dependent volatility operators that
are integral operators with respect to some suitably chosen kernel
function.
We then focus on the pricing of European-style options written on
swaps. Because of the complexity of the model (due to the
state-dependent volatility and the infinite-dimensional setting),
it is in general not possible to derive closed pricing formulas. To
avoid time consuming numerical methods which render calibration
almost impossible, in Paper IV we propose a machine-learning
approach: we adapt the strategy presented in [68] and train a
neural network which approximates option prices as a function of
the HJM model parameters. This step is costly but off-line, meaning
that no market data is used for training. For calibration to market
data, the trained neural network is then used in an on- line
optimization routine. To ensure that the model reflects the current
market situation, it is sufficient to run the calibration step
regularly. This evaluation is fast since neural networks are
deterministic functions in the input parameters.
For the numerical experiments, we restrict to a setting with
deterministic volatility which allows for analytic pricing formulas
to be used in benchmarking the neural-network-based pricing map
without additional error due to, e.g., Monte Carlo simulations. The
results are promising as indeed the neural network shows high
degree of accuracy when approximating the pricing map. However, in
the calibration step, the neural network might fail to recover the
true parameters. In the specified model, several parameter vectors
lead to similar prices for the training set, making it difficult to
recover the true parameters. Moreover,
23
1. Introduction
the trained neural network itself can be non-injective in the input
parameters, and this may make the original meaning of the
parameters get lost in the approximation step. Nevertheless, the
level of accuracy achieved for the prices after calibration shows
that neural networks may indeed be a promising tool to make
infinite-dimensional models more tractable.
1.5 Options in the energy markets
Option contracts constitute another class of derivatives
particularly relevant in energy markets and option pricing is one
of the main concerns of mathematical finance. An option is a
contract that gives its owner the right, but not the obligation, to
buy or sell an underlying asset or instrument at a specified strike
price prior to or on a specified date, depending on the form of the
option. We can indeed distinguish between two main option styles:
American options can be exercised at any trading day prior to
expiration, while European options can only be exercised at expiry.
Some options are traded on the exchanges. This is usually the case
of standardized plain vanilla options, such as European call and
put written on futures. But there exists also a big variety of
products traded over-the-counter (OTC), some of them highly exotic,
which allow to hedge or speculate on different events in both spot
and forward markets.
The French mathematician Louis Bachelier is considered the founder
of mathematical finance and the father of modern option pricing
theory since in 1900 in his PhD thesis Théorie de la Spéculation
[8] he introduced the use of Brownian motion for valuing stock
options. However, mathematical finance emerged as a discipline only
in the 1970s, with the theories on option pricing of the three
economists Fischer Black, Myron Scholes [27] and Robert Merton
[79]. The Black–Scholes formula is one of the most famous results
within mathematical finance and gives a theoretical estimate for
the price of European-style options, under the assumption that the
underlying spot price follows a geometric Brownian motion with
constant drift and volatility.
Let us consider a European-style call option written on the spot
price S with strike price K > 0 and exercise time T ≥ t ≥ 0. The
value at time t, denoted by Π(t), of the option contract is given
by the present expected value of its payoff under the measure Q.
Since the payoff function of a call option is π(x) = max(x−K, 0),
we then get
Π(t) = EQ
Ft] , (1.21)
for r ≥ 0 the risk-free interest rate. If we assume S to be
modelled by a geometric Brownian motion as in equation (1.1), the
price Π(t) can be computed. However, one needs first to define the
risk-neutral measure Q.
From the Girsanov theorem, the idea is to introduce a new process
B(τ) := B(τ) + (µ− r + 1
2σ 2) τσ . The risk-neutral probability Q is then the
probability
measure such that B is a Brownian motion under Q. Hence B(T ) has
Gaussian distribution with mean 0 and variance T under Q [70, 97].
By computing the
24
expectation in equation (1.21), one obtains the Black–Scholes
formula
Π(t) = S(t)Φ(d1)−Ke−r(T−t)Φ(d2),
where
,
and Φ denotes the cumulative distribution function of a standard
Gaussian random variable. Even if the Black–Scholes formula is
widely employed as a useful approximation to reality, one must
understand its limitations, such as the fact that, as discussed in
Section 1.3, spot prices do not usually follow log-normal dynamics
and non-constant volatility is often required to model volatility
changes. However, since it is easy to calculate, the Black–Scholes
formula is widely used in practice – for example, to compute the
implied volatility, which is the volatility implied by option
prices observed in the market [70].
1.5.1 Quanto options
Several studies reveal the existence of a negative correlation
between renewable electricity sources and electricity prices: large
amounts of wind and solar production in the market have the effect
to lower the spot price [67, 89]. It is then important to include
this negative correlation in the model. To motivate this fact, we
consider the case of a wind energy company. If suddenly the wind
intensity is stronger than expected, then the company faces a
surplus of production which must be sold. However, a surplus of
power in the market causes a decrease in the electricity price.
This means that the company is exposed to two different but
correlated kinds of risk. A direct volumetric risk due to the
strong wind and high production, and an indirect price risk due to
the drop in electricity prices. Modelling the correlation between
spot price and wind/solar power production is then important in
order to forecast possible losses and, more importantly, to hedge
against these kinds of risk with tailor-made contracts.
Quanto options are getting popular in this sense: since they take
into account the correlation between energy consumption and certain
weather conditions, they enable price and weather risks to be
controlled at the same time. The label quanto options refers
traditionally to a class of derivatives allowing the investor to be
exposed to price movements in a foreign asset without the
corresponding exchange rate risk [59]. These have typically a
call–put payoff structure. However, the same term is used for a
type of energy options that are different from the currency ones
since the payoff structure is similar to the product of call–put
options, enabling to hedge exposure to the joint price and
volumetric risk.
The literature related to quanto options in energy markets is not
extensive. In [34], a sophisticated parameter-intensive bivariate
model for the joint dynamics of energy prices and temperature is
proposes, incorporating seasonality in means and variances, long
memory, autoregressive patterns and dynamic correlations. Due to
the complexity of the model, the authors apply a
simulation-based
25
1. Introduction
approach to calculate prices. In [19], using a HJM approach, quanto
options prices are derived analytically under the assumption of
log-normal distributions for the involved processes. This allows
for fast implementations and explicit derivations of Delta-hedging
and cross-Gamma hedging parameters. However, price and power
production show different dynamics, hence they might have
univariate marginal distributions from different families, making
it challenging to select a suitable bivariate density. In [86], the
authors propose a copula model for the joint behaviour of prices
and wind power production, which allows for arbitrary marginal
distributions.
In Paper I we propose a new approach to model the correlation
between electricity spot price and power production from a wind
power plant. In particular, we propose to model the logarithm of
the spot price of electricity with a normal inverse Gaussian (NIG)
process and wind speed and wind power production with two Gaussian
Ornstein–Uhlenbeck processes. In order to model the correlation
between spot price and wind power production, we face then the
problem of modelling the correlation between a pure jump process
and a continuous path process. For this, we consider the
approximation for Lévy processes proposed by [6]. The basic idea is
to replace the small jumps of the NIG process not exceeding ε in
absolute value by an appropriate scaled Brownian term Bε, while the
remaining big jumps are modelled by a compound Poisson process Cε.
This allows to consider a linear correlation between the Brownian
motion driving the dynamics of the wind power production and Bε,
letting the process Cε be the independent jump component.
We apply the model to estimate the income from a wind power plant
as the expected value of the discounted product of power production
with spot price, and to price a quanto option in energy markets.
For both these tasks, modelling properly the correlation between
spot price and power production is critical in order to get
valuable results to be used for management choices and risk
hedging. In the case of quanto options, the double-hedging property
implies that the payoff function depends on two underlying assets.
For this, we introduce a payoff function dependent on an energy
price index and an index of power production, for which the
correlation must be taken into account.
1.5.2 Path-dependent options
Path-dependent options are a recurrent example in the energy
markets, and there exist many different types. Among the most
common, we mention Asian options which were traded at the Nord Pool
power exchange in the 1990s. Their payoff is determined by the
average underlying spot price over some agreed time period. In
particular, there exist arithmetic and geometric Asian options,
discretely or continuously sampled, average price or average
strike. Arithmetic and geometric Asian options differ from the way
the average is computed, namely
S(t, T ) = 1 T − t
∫ T
t
1 T − t
Options in the energy markets
respectively. However, the average can be also computed discretely
instead of continuously. This leads to
S(t, T ) = 1 N
1 N
logS(ti) ) ,
for t = t1 < · · · < tN = T . The quantity S(t, T ) can then
be used as the average price or average strike. For a call-payoff
function, this means to have
max(S(t, T )−K) or max(S(T )− S(t, T )).
Due to the averaging, Asian options reduce the volatility inherent
in the option. Another example of path-dependent options is given
by temperature
derivatives, such as the ones traded at the Chicago Mercantile
Exchange (CME). Here it is possible to find futures contracts on
weekly, monthly and seasonal temperatures, and European call and
put options on these futures. In this case, we talk about average
options because they are based on some average-temperature index,
such as the heating-degree days (HDD) measuring the accumulated
degrees when temperature is below 18°C, the cooling-degree days
(CDD) corresponding to the accumulated degrees when temperature is
above 18°C, or the cumulative average temperature (CAT)
corresponding to the daily average temperature. As for Asian
options, the payoff of temperature derivatives is path-dependent.
This means that their payoff does not only depend on the final
value of the underlying process, but on the entire path in the
agreed time interval.
Because of the dependence on the entire path, Asian options are
more challenging to price, also in relation to a more or less
sophisticated model for the underlying process. For example, if S
is considered to be modelled by a geometric Brownian motion, then
it is possible to get a Black–Scholes-type pricing formula. Other
possible approaches include, e.g., Monte Carlo simulations [74],
Laplace transform [62], Fourier transform [56] or the approximation
of the average distribution by fitting integer moments [57].
Recently, a new approach for pricing Asian options has been
considered in relation to polynomial processes as introduced in
Section 1.3.1 and orthogonal polynomials [53, 96].
In Paper V we adopt a similar approach and we price discretely
sampled arithmetic Asian options by orthogonal polynomials in the
context of polynomial jump-diffusion processes. More precisely, we
construct the polynomial expansion for the call payoff function by
generalized Hermite polynomials. These are orthogonal polynomials,
with parameters a, b ∈ R, b > 0, defined for n ≥ 0 by
qa,bn (x) := (−1)nω−1 a,b(x) d
n
( − (x− a)2
X(T ) = 1 m+ 1
27
1. Introduction
with m ≥ 0, we basically obtain an infinite sum of polynomial
functions in X(T ), for which we must compute the expected value.
By the multinomial theorem, we rewrite the terms of this sum as
linear combinations of correlator-type terms as introduced in
equation (1.11), and we compute expectations of this form by the
closed formula for correlators developed in Paper II.
The procedure gives an exact formula for pricing Asian options.
However, the infinite summation must be truncated for numerical
purposes, obtaining an approximation of the true price that depends
on the truncation number N and the two parameters a and b. We study
analytically the behaviour of the approximation error in relation
to these three parameters, which is confirmed by numerical
examples. We also compare the results with a Monte-Carlo-simulation
approach and use the analytical price as benchmark whenever
possible.
Numerical experiments show that the Hermite series can reach much
higher accuracies than Monte Carlo. In particular, the parameter b
influences the speed of convergence of the series and is strongly
related to the standard deviation of the underlying process Y .
However, numerical instabilities are observed mainly due to the
intrinsic exploding nature of polynomial functions of high degree.
In particular, the bigger is the initial value of the process Y ,
the higher is the value of its moments or correlators. High initial
values coupled with high-order powers create numerical
instabilities and possibly prevent the series from
converging.
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[2] Ackerer, D. and Filipovi, D. “Option pricing with orthogonal
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(2020), pp. 47–84.
[3] Ackerer, D., Filipovi, D., and Pulido, S. “The Jacobi
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no. 3 (2018), pp. 667–700.
[4] Andresen, A., Koekebakker, S., and Westgaard, S. “Modeling
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[5] Applebaum, D. Lévy processes and stochastic calculus. Cambridge
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