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.
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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.
References
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[2] Ackerer, D. and Filipovi, D. “Option pricing with orthogonal polynomial expansions”. In: Mathematical Finance vol. 30, no. 1 (2020), pp. 47–84.
[3] Ackerer, D., Filipovi, D., and Pulido, S. “The Jacobi stochastic volatility model”. In: Finance and Stochastics vol. 22, no. 3 (2018), pp. 667–700.
[4] Andresen, A., Koekebakker, S., and Westgaard, S. “Modeling electricity forward prices using the multivariate normal inverse Gaussian distribution”. In: The Journal of Energy Markets vol. 3, no. 3 (2010), pp. 3–25.
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