TO THE FINANCIAL MATHEMATICS
TO THE FINANCIAL MATHEMATICS
OF ENERGY MARKETS
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen
op vrijdag 1 februari 2008 om 12.30 uur
door
Ferry Jaya PERMANA
Magister Sains Matematika,
Institut Teknologi Bandung, Indonesi¨e
Prof. dr. F. M. Dekking
Samenstelling Promotiecommissie:
Rector Magnificus voorzitter
Prof. dr. F. M. Dekking Technische Universiteit Delft, promotor
Dr. S. Borovkova Vrije Universiteit Amsterdam,
Dr. J. A. M. van der Weide Technische Universiteit Delft
Prof. dr. ir. C. W. Oosterlee Technische Universiteit Delft
Prof. dr. H. Geman Birkbeck, University of London, UK
Prof. dr. S. J. Koopman Vrije Universiteit Amsterdam
Dr. S. Schrans Fortis Bank, Brussel, Belgium
Prof. dr. ir. A. W. Heemink Technische Universiteit Delft (reservelid)
Het Stieltjes Instituut heeft bijgedragen in the drukkosten van het proefschrift.
THOMASSTIELTJESINSTITUTE FORMATHEMATICS
ISBN: 978-90-9022713-9
Copyright c 2008 by Ferry Jaya Permana.
All rights reserved. No part of this publication may be reproduced, stored in a re-trieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the author.
my mother, my ’family’, and the memory of my father.
Contents
Preface
1
I
Modelling electricity prices
15
1 Introduction 17
2 Jump-diffusion and L´evy models 19
2.1 Literature review . . . 19
2.1.1 Overview of the available electricity price models . . . 19
2.1.2 Overview of the available jumps and volatility models . . . . 20
2.2 Stochastic models for electricity spot prices . . . 21
2.2.1 Existing jump-diffusion models . . . 21
2.2.2 Model estimation issues . . . 23
2.2.3 The proposed models . . . 24
2.3 Potential jump-diffusion model . . . 26
2.4 Potential L´evy jump models . . . 28
2.4.1 Potential L´evy diffusion . . . 28
2.4.2 The α-stable distribution . . . 30
2.4.3 Two potential L´evy models . . . 31
2.5 Model estimation and application to historical prices . . . 33
2.5.1 Filtering spikes and estimating seasonalities . . . 33
2.5.2 Application of the potential jump-diffusion model . . . 37
2.5.3 Application of the potential L´evy models . . . 41
3 Conclusions and future work 53
II
Modelling implied volatility surfaces
55
1 Introduction 57
2.1 Overview of available models . . . 59
2.1.1 Parametric model . . . 59
2.1.2 Non-parametric model . . . 61
2.2 Semi-parametric model . . . 63
2.3 Application to oil market data . . . 66
2.3.1 Implied volatility surfaces . . . 66
2.3.2 European vs. Asian options . . . 68
3 Conclusions and future work 75
III
Derivatives in energy markets
77
1 Introduction 79 2 Asian options 83 2.1 Geometric average price options . . . 832.2 The existing methods for arithmetic average price option price . . . 85
2.2.1 Vorst method . . . 85
2.2.2 Curran method . . . 87
2.2.3 Log-normal approach (Wakeman method) . . . 88
2.2.4 Reciprocal gamma approach (Milevsky method) . . . 88
2.3 Already Issued Option . . . 90
2.4 Simulation study . . . 91
3 Basket options and implied correlations 95 3.1 The GLN approach . . . 96
3.1.1 Basket Distribution . . . 97
3.1.2 Parameters estimation . . . 100
3.2 Option valuation and the greeks . . . 102
3.3 Other analytical approximations . . . 107
3.4 Simulation study . . . 109
3.5 Application to oil markets . . . 113
4 Asian basket options 119 4.1 The GLN approach . . . 119
4.1.1 The price . . . 120
4.1.2 The greeks . . . 126
4.2 Simulation study . . . 127
4.3 Application to oil markets . . . 129
A Greeks of Asian options (newly issued options) 137
B Estimating the parameters of a shifted log-normal distribution 143
C Greeks of basket options 147
D Closed formulae for vegas of Asian basket options 153
Bibliography
157
Summary
157
Samenvatting
161
Acknowledgements
165
Preface
Commodities surround us everywhere in our daily life. People consume staple foods such as wheat, rice, potato or corn, and also need vegetables, fruits and meats. Cot-ton is used to make clothes. Metals and wood are essential raw material for making tools, machinery or even cars and houses. Moreover, in the modern world, the need for oil, gas or electricity as energy resources is high and still growing. Commodities have major impact on economies of developed and developing countries. As a result, worldwide commodity markets have grown explosively in the past several decades.
Commodities are defined as consumption assets whose scarcity has a major im-pact on the world and country-specific economic development (?). In the old days, commodities were traded in the so-called spot markets (similar to common market places), where buyers and sellers met to make transactions for immediately delivery. The history of “modern” commodity markets has begun in the 16th century, when growers in Japan sold their rice at the time of planting in order to finance the pro-duction. This transaction did the first so-called forward contract. Such contracts did become rapidly popular. In the 18th century, numerous forward contracts on potatoes and other agricultural products took place in the USA. Standardization of such contracts in terms of quantity, quality and delivery date of a commodity has become inevitable. This triggered the establishment of NYCE (New York Cotton Exchange) in 1842 and CBOT (Chicago Board of Trade) in 1848.
In the Netherlands, commodity markets began in the 16th century. Forward contracts on tulip bulbs have been traded in the Netherlands since the early 1600’s. It was not long before many speculators joined in and led the market to a crash, by failing to honor their agreements. Soon afterwards, such transactions were even declared illegal. However, commodity trade in the Netherlands was not affected by this unfortunate incident. The Amsterdam Exchange (Amsterdamse Beurs) is considered the oldest stock exchange in the world. It started trading in the 16th century, when Dutch traders decided to take charge of the spice import from Asia. The prime trading and shipping companies of Holland merged to form one large company called VOC (Verenigde Oostindische Compagnie). In order to finance their ships and equipment, in 1602 the VOC established the Amsterdam Exchange, which is still active now as part of the European Stock Exchange Euronext.
Of all commodities, energy has become the biggest market-traded commodity (although modern commodity markets have roots in the trading of agricultural
ucts). This happened after the deregulation in oil and natural gas industries in the 1980s, followed by deregulation in electricity industry in the 1990s. Before the deregulation, prices were set by the regulator, i.e., governments. Energy prices were relatively stable, but consumers had to pay high premiums for inefficient cost, e.g., complex cross-subsidies from area with surpluses to area with shortages or ineffi-cient technology. Deregulation led to a free market with more competitive prices, but revealed that energy prices are the most volatile among all commodities, which exposed both energy producers and consumers to many financial risks.
Commodity (and especially energy) prices are much more volatile than prices on stocks, bonds or other financial indices. This high volatility is caused by rapid changes in supply and demand due to many reasons. Fluctuations of oil prices often are associated with fluctuations in supply due to political events, e.g., the Gulf war or the terrorist attacks of September 11, 2001. For natural gas and electricity, price fluctuations are mostly led by weather conditions, i.e., fluctuations in demand.
As a consequence of the extreme volatility of energy prices, energy market par-ticipants are susceptible to significant market risk (i.e., the risk associated with the uncertainty about the price). There are several ways to manage the price risk. The simplest way is to buy and store the physical commodity (e.g., the oil) when prices fall and use it when prices increase. However, this is an expensive way of managing the price risk because the storage costs are usually quite high. And for electricity this does not work at all since it is completely non-storable and must be consumed once generated.
A cheaper way of managing the price risk is using so-called derivatives contracts, or simply derivatives. The simplest derivatives contract is the forward contract we already mentioned, where the purchase (and hence the price) of a commodity is agreed now, but the commodity itself is delivered sometime in the future. A futures contract is very similar to a forward contract, only it is standardized (in terms of quality and quantity of the underlying commodity) and it is traded via an exchange and not directly between two counterparties.
There are myriads of other derivatives in modern financial and commodity mar-kets. One thing they have in common is that they cannot exist without the un-derlying asset (e.g., a commodity) and their price is derived from the price of that underlying asset. Derivatives have become very popular in the past few decades, as they can be a very effective and efficient tool for managing risk, if used wisely. At present, almost all activity in commodity markets takes place in the trading of com-modity derivatives (especially forward and futures contracts) and not commodities themselves. For example, in the case of oil, trading volumes in derivatives markets are nine times larger than those occurring in trading of actual (physical) oil, and this ratio is consistently increasing with the arrival of new financial players.
Derivatives are essentially bets on the prices of the underlying asset. And as one can bet on the direction of a stock or oil price, one can also bet on e.g., what kind of weather it will be this winter in The Netherlands, or whether a certain catastrophic event will take place (e.g., whether a strong hurricane hits Texas this autumn). The realization of this fact has led to the emergence of two new classes of derivatives: weather and catastrophe derivatives.
Weather plays an important role in many industries: in agriculture, weather affects crop harvests. The energy industry is also very susceptible to weather: pro-duction of electricity (especially hydro-electricity) heavily depends on the rainfall. Moreover, severe weather conditions can raise the demand for natural gas and elec-tricity. Tourism, construction and many other industries are also susceptible to risks associated with weather conditions. Nowadays many of these industries can protect their profits from weather risks by buying or selling appropriate weather derivatives. Such derivatives are traded in the US, Europe and Japan, on many exchanges such as Chicago Mercantile Exchange.
Catastrophe derivatives bear more similarities to insurance policies: a holder of a catastrophe derivative will receive a substantial payment if a certain catastrophic event takes place (of course, this holder has to pay a price for such a derivative at the start, just as one has to pay an insurance premium). The big difference with insur-ance, however, is that catastrophe derivatives are not issued by insurance companies (although they use such derivatives very actively now in their daily operations), but are freely traded on exchanges. It is clear who would buy a catastrophe derivative: a construction company, a real estate developer or an insurance company which will have to pay large compensations in case of a catastrophic event. But who would sell such a derivative? Anyone who wants to diversify their portfolio, since occurrence of a catastrophic event (and hence, a cash flow associated with a catastrophe deriva-tive) is almost uncorrelated to movements of financial markets or financial crises. In this respect, catastrophe derivatives are the so-called zero-beta assets, i.e., their prices do not move together with prices of stocks or financial indices.
Another interesting new development in commodity trading has been the intro-duction of the so-called emission trading, or emission markets. On these markets,
the allowances to companies to emit greenhouse gasses such as CO2, are freely traded
as any other commodity. Since the ratification of the Kyoto Protocol in 2006, each participating countrys government sets a limit on the amount of carbon emission that any company can produce. If a company pollutes more than its limit, it has to buy extra emission allowances from a company that pollutes less than its limit. In this way, carbon emission allowances have become a traded commodity. These allowances and derivatives on them are traded on many exchanges worldwide, e.g., on the European Energy Exchange.
The simplest commodity derivatives are forward and futures contracts. Another popular derivatives contract is an option: a right (and not an obligation, as it is the case with forward or futures contract) to buy or sell a certain asset (e.g., a commodity) at a fixed price on a fixed date. This is the so-called plain vanilla, or European option. However, real financial markets (and especially commodity markets) are more like zoos, populated by all kinds of derivatives, which can be much more complicated agreements. Such complicated contracts are called exotic derivatives, and they fascinate finance researchers, just as exotic birds or animals fascinate most people.
Energy options and volatility
The first option trading was legitimized in 1934 in the USA for some agricultural products. For the next several decades, the option trading was virtually inexistent. The main reason for this was the fact that it was not clear what a fair price of an option should be and how the seller of an option contract can manage risks associated with it. Then, in 1973, American economists Fischer Black and Myron S. Scholes published a groundbreaking article, which at once answered all these questions and introduced the famous Black-Scholes formula for the fair price of an option. Also in 1973, another economist Robert C. Merton published a paper expanding the mathematical understanding of the options pricing model of Black and Scholes. In the same year, the Chicago Board Options Exchange was established and began trading standardized call options. The impact of the work by Black, Scholes and Merton was so enormous, that trading in options grew exponentially since then and reached the total volume of 20 trillion US dollars worldwide in 1995 (from practically zero in 1972). In recognition of their groundbreaking work, Merton and Scholes were awarded the 1997 Nobel Prize in Economics for the famous Black-Scholes option price formula; Fisher Black unfortunately was ineligible, having died in 1995. Since then, the option trading has been dramatically growing even further, not only in traded volumes (in 2006 the option trading volume has already reached 270 trillion US dollars), but also in the types of traded options. Options have become a practically standard tool of risk management.
On a more academic note, the work by Black, Scholes and Merton gave rise to an entire new discipline on the border between mathematics and finance, the so-called financial mathematics or financial engineering. It has grown rapidly as a new scientific area, has given a lot of inspiration for researchers and provided many new contributions in both mathematics and finance. The present thesis is also an attempt to contribute to the area of mathematical finance.
There are two classes of derivative pricing models: analytical and numerical approaches. Analytical approaches are particularly attractive for practitioners: they usually provide a closed form expression for a derivative’s price, so it is fast and easy to implement, while numerical approaches are more flexible, but tend to be time consuming.
Most commodity derivatives are exotic, i.e., more complicated than European options. There are inherent reasons for this. For example, most energy delivery contracts are based on the average oil price over a certain period of time. In this case, a so-called Asian option is better suited for risk management purposes: for Asian options, the settlement price depends on the average asset price over a certain time period and not on the asset price on one specific date, as it is the case with European options. Another example arises from the fact that oil is physically traded twice: as a refinery feedstock and as a refinery product. So the portfolio of most energy companies is a basket of several assets. For example, a refinery company must buy crude oil and sell refinery products (e.g., gasoline and heating oil). Hence, its portfolio consists of a so-called short (i.e., negative) position on crude oil and long positions on gasoline and heating oil. A power station is a similar example: it must buy fuel (e.g., natural gas) to produce and sell electricity. A so-called basket option, an option whose payoff depends on the weighted sum of the assets in a portfolio, is an effective risk management tool in this case.
The Black-Scholes model heavily depends on the assumption that the price pro-cess of the asset underlying an option is a Geometric Brownian Motion (GBM). It means that the underlying asset (e.g., stocks, futures) price’s distribution is log-normal and the price log-returns or relative returns are log-normally distributed. Unfor-tunately, the celebrated Black-Scholes formula cannot be directly applied for valuing and managing most exotic options. For example, in cases of Asian or basket op-tions, Black-Scholes approach fails due to the fact that the arithmetic average or the weighted sum of log-normal random variables is not log-normal anymore. Valuation of exotic derivatives (such as Asian or basket options) requires specific tools, usually much more complex than the Black-Scholes formula.
Our main contribution in the area of exotic derivatives is a new method for valuing and hedging general basket options, i.e., options on portfolios containing several assets and also long/short positions. This method uses a generalized of family log-normal distributions, the so-called GLN (Generalized log-normal) distribution, to approximate the basket distribution. It can deal with negative skewness and negative values, typical for general baskets. The GLN approach leads to closed formulae for option’s price and greeks, it allows us to stay within Black-Scholes framework, and can easily be extended to Asian basket options.
op-tion and usually the one that cannot be directly observed. There are two kinds of volatilities: historical and implied. Historical volatility is calculated from the past movement of prices over a specified time period. Technically, it is the annual stan-dard deviation of the log of the changes in the asset price. Although the historical volatility reflects actual (past) market fluctuations, it is based on “stale” data that does not necessarily reflect current market conditions.
The so-called implied volatility is obtained from a liquid option price by inverting the Black-Scholes formula. The implied volatility is generally considered to be the best volatility forecast. It is more responsive to current market conditions because it reflects the perceptions of numerous market participants about the market risk. The Black-Scholes model assumes a constant volatility across all strikes and maturities. It means we would obtain the same implied volatility value if we invert prices of options with different strikes and maturities. In practice, this is not the case. If at-the-money implied volatility is lower than both in- and out-of-at-the-money, then the plot of the implied volatility versus the strike price exhibits the so-called “smile”, a famous phenomenon in derivatives markets. Other shapes of the implied volatility are also possible.The implied volatility can also vary in time to maturity. Then the most convenient way to visualize it is to use a three-dimensional graph, a so-called implied volatility surface, where the implied volatility is plotted as a function of the time to maturity and the strike price.
The implied volatility surface can be modelled in a parametric as well as non parametric way. However, neither approach is suitable for commodity options, due to several reasons. Our contribution in this area is a new approach for modelling the implied volatility surface: a semi-parametric method, which incorporates the simplicity of a parametric method and the flexibility of a non-parametric method. It allows for many realistic volatility functions, is very flexible, fast and easy to implement.
Modelling electricity prices
The great use of electricity began in the 18th century, when people invented elec-tricity appliances: lighting, electric motor, heating, and so on. As the practical use of electricity grew and multiplied, so did the demand for its production. Electricity has fundamentally transformed the way we live and has become one of the most im-portant energy resources. It has been a vital commodity which needs high amounts of capital investments. It is the reason that electricity industry used to be organized as a state-owned monopoly for a long time. The liberalization of electricity markets began in the late 1980s in the UK with the privatization of the UK electricity
in-dustry, followed by many countries. The monopolistic structure was replaced by a deregulated market. However, deregulation in the electricity industry in the 1990s was not as successful as in oil and natural gas industry. The viability and validity of deregulation in the electricity industry was debated after bankruptcy of Enron and some US energy merchants.
Deregulation in the electricity industry led to a free market. The electricity market has become competitive and the electricity price forecast has become an essential input to many risk management and capital budgeting models, for decision making, and for strategy management in a power company. This has created a demand for research in the modelling of electricity prices.
The electricity prices are characterized by seasonalities. Prices increase in the winter (and slightly in the summer) as the demand for electricity is getting higher than in the mid-seasons. This demand is also influenced by activities over a week, higher during weekdays and lower in weekends. While many other commodities exhibit seasonalities, electricity also has unique features: it is non-storable and has to be transported by a transmission network. Severe weather conditions, plant outages or failures in a transmission network cause an unbalanced supply and demand. On the other hand, surge production is limited and very expensive. These features cause price spikes, that is the price rises sharply and falls back shortly afterwards to normal level. Moreover, in electricity trading, a cooperation between the spot market and the transmission system operator is necessary. The spot market should ensure the balancing between supplies and demands, and the transmission system operator should arrange the delivery.
Deregulated spot power markets have been organized mainly in two forms: power exchange (PX), e.g., UKPX (United Kingdom Power Exchange) in London, APX (Amsterdam Power Exchange) and EEX (European Energy Exchange) in Leipzig, and pool markets, e.g., Nordpool in Scandanavia, and NYPOOL (New York In-trastate Access Settlement Pool). In a power exchange (PX), a trade is conducted through bilateral contracts. Bilateral bids specify quantities and prices, trades have to be completed the day before the delivery, so that both market participants and the system operator have enough time to arrange the physical delivery. In a pool market, suppliers make bids to the pool and then the system operator computes the expected demand for each hour of the following day, or both buyers and sellers make bids to the pool and then the system operator builds a demand function analogous to the supply function. Both in a pool or PX, a spot market plays the role of ensur-ing that total generation meets demand. It also is organized as a day-ahead market: by 1:00 p.m., the system operator has finalized the balance of supply and demand for each hour of the following day.
futures (forward) price represents the expected future spot price (discounted at the risk free interest rate). For storable commodities, information about expected future prices also reflects expectations about future supply and demand, and hence, is valuable for making the storage decisions of such commodities. In the general context of commodity markets, under no-arbitrage assumption, the relationship between the spot and the forward price for a commodity is established by a ‘cash and carry’ argument, which is given by:
fT(t) = S(t)e(r−y)(T −t),
where fT(t) denotes the forward price at time t for maturity T , S(t) is the spot price
at time t, r is the continuously compound interest rate, and y is convenience yield on the commodity, which represents benefit of holding the physical commodity over the interval (t, T ). This relationship can be explained as follows: buy commodities in the cash market at the spot price S(t) at time t through a loan, pay the cost of storage, benefit from holding the physical commodity over the interval (t, T ), and realize a cash and carry arbitrage at maturity T . The spot-forward relationship in electricity collapses because it is non-storable (except, partially, for hydroelectricity). It is more reasonable if the argument:
“Forward price = Spot price + Cost of carry” is changed to
“Forward price = Spot price + Risk preminum π(t, T )”,
where π(t, T ) may have different signs for different values of T − t. Furthermore, the
forward price fT(t) does not converge smoothly to the spot (or day ahead) price, e.g.,
due to possible price spikes. So, for electricity we need to model both the dynamics
of the spot price (S(t))t≥0 and the dynamics of the forward price fT(t), 0 ≤ t < T .
Modelling spikes is the most challenging task in modelling electricity spot prices. The standard approach to model daily electricity spot prices or, more often, the logarithm of daily spot prices, is to represent it as a sum of deterministic sea-sonal component and a stochastic process. The classical model is a mean-reversion jump diffusion model, which models the stochastic process as the sum of Ornstein-Uhlenbeck (mean-reversion) and some jump processes. A compound Poisson process is often used to model the jump process. In this case, we model the jump inter-arrival times by exponential random variables and the jump size, for example, by a ran-dom variable having a log-normal distribution. However, such a model has some disadvantages. A mean-reversion process assumes a constant mean-reversion rate. In the electricity price process, that is not the case, since after a jump, the price
will fall back more rapidly to the average price level , a so-called fast relaxation process, than after a normal shock, a so-called standard mean-reversion process. A log-normal distribution used to model the jump size fails to capture the heavy tails in the price and returns distributions, induced by spikes. Another issue is the difficulty to determine which movements in the stochastic component of the price process are part of the continuous dynamics or spikes.
In this thesis, we address most of these modelling issues. We introduce two stochastic models for the electricity price: a potential jump-diffusion and a potential L´evy model. Using the potential jump-diffusion model, the stochastic process is decomposed into a potential driven diffusion and the Poisson jump process. A potential function allows to model the mean-reversion rate as a continuous function whose values depend on the distance from the average price level. It is a more flexible and realistic approach than using a constant mean-reversion rate.
In the L´evy jump model, the stochastic component is decomposed into a drift modelled by a potential function (a so-called potential function based relaxation), and random perturbations modelled by a class of L´evy process, e.g., those with α-stable distribution. The main advantages of the L´evy jump model are that it is not necessary to determine which stochastic price fluctuations are arising from the continuous dynamics, and the α-stable distribution is heavy-tailed enough to capture the heavy tails induced by the spikes/jumps.
Outline of this thesis
This thesis consists of three parts.
Part 1 is concerned with electricity price models. This part is preceded by in-troduction which gives an overview of some existing models and describes model estimation issues in the modelling of electricity prices. We represent the (log) elec-tricity spot price as the sum of a deterministic seasonal component and a stochastic component. We first explain the modelling of the seasonal component, then we in-troduce two models for the stochastic component: a potential jump-diffusion model and a potential L´evy model. In order to test performance of our proposed models, both models are applied to three datasets in the European power markets.
Part 2 is devoted to modeling the implied volatility in energy markets. In this part, we give an overview of parametric and non-parametric methods for modelling the implied volatility surface. We introduce the so-called semi-parametric method. Performance of this method is investigated by applying it to data from the energy markets.
and Asian basket options. We first give an overview of some existing analytical approximation methods to valuing and hedging Asian options. We derive closed formulae for option’s greeks. Performance of these methods is investigated on the basis of simulations. We introduce the GLN approach for general basket option and then we extend it to valuation and hedging of Asian basket options. Performance of the GLN approach is tested on the basis of simulation and applications to data from energy markets. We demonstrate how to calculate the implied correlation using spread option prices from the energy markets.
Main contributions of the thesis
We have introduced and developed several new models in the area of financial math-ematics. Our works mainly motivated by energy markets. However, some of these models, e.g., option pricing models for basket and Asia basket options, are also ap-plicable as risk management tools in other financial or commodity markets. Our main contributions presented in this thesis are:
• The potential jump-diffusion model and potential L´evy model for electricity spot prices
• The semi-parametric model for implied volatility surface
• The GLN (Generalized Log-Normal) approach for valuation and hedging of basket and Asian basket options
Publication details
This thesis is based on several publications, conference proceedings and conference presentations (collaboration with Svetlana Borovkova).
Publications:
1. Modelling Electricity Prices by Potential Jump-Diffusion, Stochastic Finance, 2006, p. 239-264, Springer, ISBN: 0-387-28262-9.
2. A Closed Form Approach to Valuation and Hedging of Basket and Spread Options, The Journal of Derivatives, Vol. 14 No.4 (Summer 2007) , p. 8-24. ISSN: 1074-1240 (joint work with J.A.M van der Weide).
3. Implied Volatility in Oil Markets, 2007, tentatively accepted by Computa-tional Statistics and Data Analysis.
4. Asian Basket Options and Implied Correlations, 2007, ready for submission. 5. Modelling Electricity Prices by the Potential L´evy, 2007, ready for submission
(joint work with I. Pavlyukevich).
Conference proceedings:
1. Implied Volatility in Oil Markets , Proceedings of the 3rd Forecasting Financial Market Conference, June 1-3, 2005, Marseille, France.
2. Average Price Option in Energy Markets, Proceedings of the International Conference on Applied Mathematics , 22-26 August 2005, Institut Teknologi Bandung, Bandung, Indonesia. ISBN: 90-365-2244-7.
3. Empirical Analysis of Analytical Approximation Approaches for Pricing and Hedging Basket Options, Proceedings of the 4th Actuarial and Financial Mathematics Day, February 10, 2006, Brussels, Belgium (joint work with J.A.M van der Weide).
4. Asian Basket Options and Implied Correlation in Oil Markets, Proceedings of the Financial Engineering and Application Conference, September 24-26, 2007, Berkeley, CA, USA, ISBN:978-0-88986-681-2.
Popular publication:
1. Trading Commodities: Derivatives and Risks, Vuurwerk (Relatiemagazine van de Faculteit der Economische Wetenschappen en Bedrijfskunde, Vrije Univer-siteit Amsterdam), 3(6), November 2007.
Note that,
Preface is based on popular publication (1).
Part I: “Modelling electricity prices” is based on publications (1) and (5).
Part II: “Implied volatility in energy markets” is based on publication (3) and con-ference proceeding (1).
Part III, Chapter 2: “Asian options” is based on conference proceeding (2).
Part III, Chapter 3: “Basket options and implied correlations” is based on publica-tions (2) and (4) and conference proceeding (3).
Part III, Chapter 4: “Asian basket options” is based on publication (4) and confe-rence proceeding (4).
Conference presentations:
• Modelling electricity prices by the potential jump-diffusion, International Con-ference on Stochastic Finance 2004, September 26-30, 2004, Lisbon, Portugal. Presentation: Svetlana Borovkova.
• Implied Volatility in Oil Markets , The 3rd Forcasting Financial Market Con-ference 1-3 June 2005, Marseille, France.
Presentation: Ferry J. Permana.
• Average Price Option in Energy Markets, International Conference on Ap-plied Mathematics, August 22-26, 2005, Institut Teknologi Bandung, Ban-dung, Indonesia.
Presentation: Ferry J. Permana.
• Empirical Analysis of Analytical Approximation Approaches for Pricing and Hedging Basket Options, The 4th Actuarial and Financial Mathematics Day, February 10, 2006, Brussels, Belgium.
Presentation: Ferry J. Permana.
• A Closed Form Approach to Valuation and Hedging of Basket and Spread Op-tions, presented in: The 12th International Conference on Computing in Eco-nomics and Finance, June 22-24, 2006, Limasol, Cyprus and The 4th World Congress Bachelier Finance Society 2006, 17-20 August 2006, Tokyo, Japan. Presentation: Svetlana Borovkova.
• Generalized Log-normal Distribution Approach to the Valuation and Hedging Basket Options, Konferensi Nasional Matematika XIII, July 24-27, 2006, Se-marang, Indonesia.
Presentation: Ferry J. Permana.
• Basket Options and Implied Correlations in Oil Markets, January 17-18, 2007, Commodities 2007, Birkbeck College, University of London, UK.
Presentation: Svetlana Borovkova.
• Modelling Electricity Prices by the Potential L´evy , The 15th Annual Sym-posium of Society Nonlinear Dynamics and Econometrics, March 15-16, 2007, Paris, France.
Presentation: Ferry J. Permana.
• Basket and Spread Options and Implied Correlations: A Closed Form Ap-proach, the 25th edition of the Erasmus Finance Day, May 11, 2007, Rotter-dam, The Netherlands.
• A GLN Approach to Valuation and Hedging of Asian Basket Options, SEAMS-GMU International Conference on Mathematics and Its Applications, July 24-27, 2007, Yogyakarta, Indonesia.
Presentation: Ferry J. Permana.
• Asian Basket Options and Implied Correlation in Oil Markets, Financial En-gineering and Application Conference, September 24-26, 2007, Berkeley, CA, USA.
Part I
Modelling electricity prices
Chapter 1
Introduction
The recent liberalization of European electricity markets offers benefits to both providers and consumers. It also presents new modelling, pricing and risk manage-ment challenges to researchers and practitioners involved in energy markets.
Electricity is a flow commodity of strategic importance, characterized by non-storability, inelasticity of demand and large-scale long-term investments. These factors lead to price features rarely seen in other commodity markets: complex seasonal patterns, periods of high volatility and price spikes. These characteristics make traditional modelling and pricing methods of financial analysis unsuitable for electricity. Risks in electricity markets are dramatically more pronounced: prices can increase by more than 1000% within hours, and return to their normal levels shortly afterwards - a behavior usually referred to as spikes.
Modelling the evolution of the spot price is the core of energy market research. The model for the electricity spot price suggested by ? is a mean-reversion diffusion (with seasonally varying mean). This model does not incorporate spikes. However, correctly modelling spikes remains the most difficult part of electricity price mod-elling, because after an upward jump, the price has to be forced back to normal levels. Jump-diffusions, sometimes suggested for modelling prices of financial assets, are now routinely used by many authors for modelling electricity prices (?, ?, ?, ?, ?). In such models, Poisson jumps are added to a mean-reversion diffusion process. Mean-reversion jump-diffusions provide for spike behavior, since the mean-reversion forces the price to return to its normal level. However, a high rate of mean-reversion is needed to force the price back after a jump. Hence, the degree of mean-reversion outside spikes can be grossly misspecified.
So far, the only model successfully separating mean-reversion and spikes is the Markov regime-switching model, introduced by ?. Their model postulates that the price can be in one of three regimes: normal mean-reversion, spike regime and return from a spike to a normal price level. Transitions from one regime to another are governed by a discrete time Markov process, with transition probabilities estimated
from historical data (as well as the model parameters in each regime). A similar approach is taken by ?, where regimes are defined in terms of the price exceeding a certain pre-specified threshold. These regime-switching models allow for a richer structure than the mean-reversion with jumps. However, these model did not find widespread acceptance by practitioners, who are accustomed to diffusion-like models, preferring them to regime-switching models.
Here we introduce a number of new approaches for modelling electricity spot prices: potential diffusion and potential L´evy models. In the potential jump-diffusion model we use a recently developed potential field model (?). This involves a jump-diffusion with drift given by a potential function derived from the process’ invariant distribution. The potential forces the price to return to its average level (driven by a seasonal component) after a jump. The potential function model is close in spirit to a mean-reversion jump-diffusion, with a more general drift term. It is also a generalization of the regime-switching model of ?, which allows the rate of mean-reversion to be different in each regime. The rate of reversion to a mean price level in the potential function is a continuous function of the distance from this level, specified by the model builder. This allows for a richer model structure, while still retaining the jump-diffusion framework. In fact, the mean-reversion jump-diffusion features in our model as a special case.
In the potential L´evy model, we incorporate a potential function as a drift and a class of L´evy process, i.e., those with an α-stable distribution, for modelling the stochastic price fluctuations. The proposed potential function is inspired by the solutions of the non-linear stochastic differential equations introduced by ?. After a jump, the potential function forces the price to return rapidly toward its average level with the higher reversal rate (a so-called fast relaxation process) than the ‘normal’ mean-reversion rate. Here, we suggest a potential function with different rates: a higher rate for the fast relaxation process and a lower rate for the mean-reverting process. An α-stable distribution used for modelling jumps allows to capture the heavy tails induced by jumps. Moreover, disentangling the price movements as a part of continuous dynamics or jump dynamics is not necessary.
Chapter 2
Jump-diffusion and L´
evy models
2.1
Literature review
Research on electricity price modelling is rapidly expanding. Various models for data from power markets have been proposed, for a good review see e.g., ?. We will review several available models for electricity. In the first subsection, we give overview of some recent results on electricity prices modelling. In the next subsection, we give an overview of methods proposed to model jumps and volatility.
2.1.1 Overview of the available electricity price models
Seasonality is a characteristic feature of electricity prices. For the Spanish Electricity Pool, ? used the HEGY weekly seasonal unit root test (introduced by Hylleberg, Engle, Granger and Yoo (1990)) and the CH seasonal stationarity test (introduced by Canova and Hansen (1995)) to show that a model with a unit autoregressive root at zero frequency, as well as unit roots at all of the weekly frequencies, could approximate remarkably well the weekly seasonal stochastic component of the logs of the daily average electricity prices.
? tested several models for California hourly electricity prices, including
mean-reversion, jump-diffusion, ARMAX(1,0), EGARCH and also models incorporating time-dependent jump intensity and weather data. They suggested to take into ac-count mean-reversion, time of the day effects, weekend and weekday effects and seasonal effects as the characteristic features of electricity spot prices. They also notice that volatility clustering and higher order autocorrelation are the two most important features of electricity prices.
? highlighted the importance of regression effects, periodicity, long memory and
volatility in electricity prices. They proposed the periodic seasonal Reg-ARFIMA-GARCH
models for daily electricity spot prices in Nord Pool and three more electricity mar-kets (EEX, APX and Powernext in France). The parameters of such models are jointly estimated by the method of approximate maximum likelihood. Given the fact that the time series of electricity spot prices are usually fat-tailed, the sea-sonal periodic ARFIMA disturbances of such a model are assumed to be Student-t distributed.
Another class of electricity price models are L´evy-based models. An applica-tion of such models yields an advantage: disentangling diffusion and jumps is not necessary. ? introduced generalized hyperbolic distributions which allow exact fit-ting of the returns generated by the model to the empirically observed returns of Nord Pool. Such distributions allow to reproduce the volatility of electricity prices accurately.
A class of L´evy distributions, the so-called α-stable distribution, was proposed by ?. He showed that such a distribution is heavy-tailed enough to fit the log-return of the electricity spot prices from Nord Pool and EEX.
2.1.2 Overview of the available jumps and volatility models
Continuous Brownian motion and jump components are often considered to be in-volved in modelling of financial asset prices, e.g., electricity prices or asset returns. Modelling jumps and disentangling jumps from diffusions is needed to improve fore-cast of realized volatility for asset pricing, portfolio choice or risk management. However, estimating parameters of jump-diffusion models is nontrivial, and the difficulty in estimation often prevents their implementation in applications. This difficulty originates from the problem of determining whether stochastic price fluc-tuations in the underlying process are part of the continuous dynamics, or whether they are part of the jump dynamics. This attracts extensive research on modelling jumps and volatility.
? identified the jump-diffusion process that can accommodate the main features
of the daily S&P500 returns: they should incorporate discrete jumps and stochastic volatility, with return innovations and diffusion volatility strongly negatively corre-lated. In this respect, they extended the class of stochastic volatility diffusions by allowing for Poisson jumps of time-varying intensity in returns. Parameters of such models are estimated by implementing the efficient method of moments (EMM).
A partial generalization of quadratic variation, a so-called bipower variation, introduced by ?, is capable to split up the individual components of the quadratic variation into that due to the continuous part of prices and that due to jumps.
Other approaches are (quasi) maximum likelihood (ML) methods and the gener-alized method of moments (GMM). In the ML method, the density of the
jump-diffusion returns is derived analytically, by means of characteristic functions or Fourier transforms, or approximated by a mixture of normal distributions (see ?, ?). In the generalized method of moments, the first few moments are derived analyti-cally (e.g., see ? who points out the importance of matching higher (up to 4th) order moments), and the sample moments are set equal to the corresponding population moments. Then the resulting equations are solved for the unknown parameters, providing the parameter estimates.
The maximum likelihood method is efficient and hence more attractive than the generalized method of moments. For many models, GMM has been shown to have the lower overall estimation efficiency, when compared to maximum likelihood-based methods (?). The main attraction of the GMM method is its general applicability; even when the expression for the density function is unknown, moment conditions may still be available in analytical form for many models of practical interest.
2.2
Stochastic models for electricity spot prices
2.2.1 Existing jump-diffusion models
As mentioned in the Introduction, electricity spot prices are characterized by com-plex seasonal patterns, periods of high volatility and price spikes. All these features
can be seen in Figure 1, which shows daily electricity spot prices1 on three major
European power exchanges: European Energy Exchange (EEX, Leipzig, Germany), Amsterdam Power Exchange (APX) and UK Power Exchange (UKPX, London). On the y-axis is the price in Euros per Megawatt/hour, and on the x-axis - the number of calendar days since the first day in the analyzed dataset (Jan. 1, 2002 for EEX, Jan. 1, 2001 for APX and March 27, 2001 for UKPX). Note that the price on the UK power exchange is quoted in Euros and not in British pounds - UKPX is managed by the Amsterdam Power Exchange (APX) and prices are quoted in Euros to ensure better coordination with other European power exchanges.
Price spikes dominate the graphs, but seasonal (yearly and weekly) behavior is also visible. Generally, the average price level is higher in winter and summer than during mid-seasons (due to the use of heating and air conditioner), and higher on weekdays than on weekends (due to business activities).
The standard approach is to model the spot price P (t) or, more often, its lo-garithm Y (t) = log P (t) on each day t as the sum of the deterministic seasonal
1
0 100 200 300 400 500 600 700 800 900 0 20 40 60 80 100 120 140 160 180 day Euro/MWh 0 200 400 600 800 1000 1200 0 100 200 300 400 500 600 700 day Euro/MWh 0 500 1000 1500 0 20 40 60 80 100 120 day Euro/MWh
Figure 1: EEX (01.02.02-13.05.04), APX (01.01.01-11.02.04) and UKPX (27.03.01-30.04.05) electricity spot prices.
component s(t) and a stochastic process X(t):
Y (t) = log P (t) = s(t) + X(t). (1)
The seasonal component s(t) usually consists of two periodic functions: with the periods of 1 year and 1 week (sometimes a half-yearly periodic component is added as well).
The specification (1) for the (log) price is used by most researchers and practi-tioners working in energy markets (e.g., ?, ?, ?, ? and many others). The particular specification of the stochastic process X(t) leads to different models. For example, in
? X(t) is modelled as the stationary mean-reversion (Ornstein-Uhlenbeck) process:
dX(t) = −αX(t)dt + σdW (t),
where α is the mean-reversion rate, σ is the volatility and W (t) is standard Brow-nian motion. The general form of a mean-reversion process is given by stochastic differential equation
dX(t) = −α (m − X(t)) dt + σdW (t), (2)
where m is the mean price level.
? model X(t) by the sum of two stochastic processes: a short-term
mean-reversion component and a long-term equilibrium price level, which follows a geo-metric Brownian Motion:
dX(t) = [−αX(t)dt + σdW1(t)] + [µdt + ηdW2(t)],
with µ being the growth rate and η - the second volatility parameter.
In mean-reversion jump-diffusion models (studied in the articles mentioned in the Introduction), X(t) is decomposed into a mean-reversion diffusion and a Poisson
jump component:
dX(t) = −αX(t)dt + σdW (t) + Jdz(t),
where z(t) is the Poisson process with intensity λ and J is a random variable rep-resenting random jump size. The general form of a mean-reversion jump-diffusion process is
dX(t) = −α(m − X(t))dt + σdW (t) + Jdz(t), (3)
The regime-switching model by ? incorporates a regime-switching mechanism in the specification for X(t) (however, it does not take yearly seasonality into account). Three regimes are defined: “normal” mean-reversion, spike initiation and spike re-versal regimes. The transition from one regime to another is given by a Markov transition probability matrix, which reflects the fact that the spike initiation regime is usually followed by spike reversal. The mean-reversion rate in the spike reversal regime is assumed to be much higher than the mean-reversion rate in the “normal” regime. A similar approach is proposed by ?, where regimes are define in terms of the price exceeding a certain pre-specified threshold. Here, the stochastic component X(t) is decomposed into a mean-reversion diffusion and a Poisson jump component:
dX(t) = −α m(t) − X(t−)dt + σdW (t) + h(t−)dJ(t),
where X(t−) is the left-hand limit of the deterministic seasonal function at time t.
The first term insures that any shift away from the trend generates smooth reversion to the standard level m(t) at mean-reversion rate α. The function h has the form
h(X(t)) = (
+1 , if X(t) < τ
−1 , if X(t) ≥ τ ,
where the threshold τ is calibrated to market data. It defines the algebraic effect of the i-th jump in the jump process J(t), which has the form
J(t) = N (t) X i=1
Ji,
where N (t) is a Poisson process with time-varying intensity, Ji are exponentially
distributed. ? show that such a model fits the moments of order 1,2 and 4, but not the skewness.
2.2.2 Model estimation issues
The problem of disentangling diffusion from jumps arises when we model electric-ity prices using jump-diffusion models. If there was a way of disentangling jumps
from the diffusion component, then jumps could be analyzed and jump parameters more accurately estimated. Both ML and GMM methods are known to grossly overestimate the jump frequency and underestimate the mean jump size (?, ?, ?, ?). Moreover, such estimation procedures do not explicitly separate jumps from the dif-fusion component, so graphical methods of jump distribution analysis (histograms, QQ-plots) are inapplicable.
Here, we shall follow an approach of jump filtering (step-wise procedure), similar to the method proposed by ?. We separate price moves that we classify as jumps by applying a simple sequential filtering procedure, and estimate separately the parameters of the diffusion and the jump components. However, it must be stressed that identifying the jumps without any restrictions is always an ad-hoc task by the researcher, possibly introducing sample selection error to the model.
Disentangling jumps from continuous stochastic dynamics is not necessary if we use a L´evy process. Such models have been applied for log-returns of electricity prices (see ?,?). There is an inherent weakness in all estimation procedures for processes given by a general stochastic differential equation and observed at dis-crete time intervals. In all these procedures, the theoretical stationary distribution of the process is matched to the empirical distribution of the observed log-returns. However, the exact distribution of the log-returns, given some time discretization step, is not the same as the stationary distribution of the process, since the dis-cretization of the continuous time stochastic differential equation by e.g., the Euler scheme involves higher order terms. For some specific models (such as Geometric Brownian Motion) the exact distribution of the log-returns can be derived, but for most of the general models this distribution is not known. Then one can only hope that, for a sufficiently small time discretization step, the process’ stationary distri-bution and the log-returns distridistri-bution are close. This is a reasonable assumption and it is supported by the results on weak convergence of numerical solutions for stochastic differential equation (?, ?). Regarding this issue, we will propose a class of L´evy-based models applied for electricity spot prices.
2.2.3 The proposed models
Mean-reversion diffusion models are unrealistic in that they do not accommodate spikes. The main drawback of mean-reversion jump-diffusion models is that the mean-reversion rate is constant, the underlying assumption being that all the shocks affecting the price die out at the same rate, regardless of the shock size. This is in conflict with economic intuition, which suggests that large shocks should die out more rapidly due to powerful forces of supply and demand, while small shocks should slowly revert to the previous price level, due to the adjustment of production costs.
This is also an empirically observed phenomenon: in most markets, the electricity price returns towards the average price level more rapidly from greater excursions, e.g., a jump.
The regime-switching approach of ? allows for a non-constant mean-reversion rate by specifying a number of regimes, each with its own mean-reversion rate (which is still constant within each regime).
For the electricity price it is reasonable to assume a non-constant mean-reversion rate, e.g., one that itself depends on the distance from the mean level. A more general way of modelling this, is to allow the mean-reversion rate to be a continuous function of the distance to the mean price level. For instance, the price return can be a quadratic or exponential function of this distance. By this, we would allow for a continuum of mean-reversion regimes. This can be incorporated into the model by extending the class of possible drift forms. We shall explore this idea, by specifying the drift term by means of a potential function, then combine such a potential function and Poisson jumps into the potential jump-diffusion model.
The distribution of jump sizes is often assumed to be log-normal. It turns out that the log-normal distribution is too light-tailed to properly match the empirical jump size distribution. As a solution to this problem, we suggest a novel approach to jump size modelling, inspired by recent results on L´evy diffusions by ?. In this approach, we combine the potential function with a specific jump size distribution which arises from a L´evy measure on a one-sided infinite interval, bounded away from zero. This distribution (we shall called it the P-I distribution) is more heavy-tailed than e.g., log-normal and it can be easily calibrated to the empirically observed jump characteristics. We shall call it the potential P-I jump-diffusion model.
One of the estimation issues in modelling electricity prices is the difficulty to determine which movements in the price are part of the continuous dynamics, and which are jumps. As a solution for this problem, we propose the so-called poten-tial L´evy model incorporating a potenpoten-tial function as the drift and a class of L´evy distributions, i.e., those with an α-stable distribution, to model the stochastic price fluctuations. In this model, a potential function has different reversal rates. In a fast relaxation process, the potential function forces the price to return more rapidly toward the mean price level than in a standard mean-reversion process. So the re-versal rate in a fast relaxation process is higher than in a standard mean-reversion process. Such a model does not necessarily contain Brownian motion component, since the corresponding L´evy process is responsible for both small and large price moves. In this model, an α-stable distribution captures the heavy tails and provides a reasonable specification of jump frequencies and sizes.
2.3
Potential jump-diffusion model
The potential function approach was introduced in ? for modelling commodity prices that exhibit multiple attraction regions (such as oil prices). The potential function model is a diffusion model with the drift specified in terms of the potential function. The potential function is closely related to the invariant distribution of the process, hence this approach can encompass multimodality in price distributions. Here we shall apply the potential function approach for a different purpose, namely for modelling a varying rate of mean-reversion in the jump-diffusion context.
The original potential function model postulates that the (log) price process
(X(t))t evolves according to the stochastic differential equation:
dX(t) = −U′(X(t))dt + σdW (t), (4)
where U : R −→ R is a twice continuously differentiable function, such that
U (x) −→ ∞ as |x| −→ ∞, and R−∞∞ exp(−2U(x)/σ2)dx < ∞. These conditions
assure that the invariant distribution of the process (X(t))tis the Gibbs distribution
with density
πσ(x) = exp(−2U (x)
σ2 ) (5)
(for proof see e.g. ?).
The relationship (5) means that there is one-to-one correspondence between the invariant distribution of the process and the diffusion’s drift, given by the potential. So characteristics such as multimodality of the price distribution can be incorporated into the model by the proper choice of the potential function.
The potential (together with the volatility σ) can be estimated from historical data by first estimating
Gσ(x) = 2
σ2U (x) = − log(πσ(x))
by
ˆ
Gσ(x) = − log(ˆπ(x)),
where ˆπ is some estimate of the observations’ marginal density (e.g., a kernel
esti-mator or a histogram smoothed by a polynomial or a sum of Gaussian densities). The volatility σ can then be estimated by discretizing the equation (4) by an
Euler scheme with ∆t = 1 day and noting that σ2/2 is the linear regression coefficient
(without intercept) of the increments (Xi+1− Xi) on ( ˆGσ(Xi)).
The model (4) describes a continuous time process. If we observe time series
observed at discrete time points. The observations (Xi) come from the distribution
with density πσ given by (5).
For a small time interval ∆t, the discretization of the equation (4)
∆Xt= Xt+∆t− Xt= −∇U(Xt)∆t + σ∆Wt,
gives an Euler scheme for the numerical solution to the diffusion equation (4). If we assume that the observation interval is small enough, the approximate model motivated by the above Euler scheme for the observed time series can be given by
¯
Xi+1= ¯Xi− ∇U( ¯Xi)h + ǫi. (6)
Here h is the time interval between observations (for power prices h = 1 day) and the
(ǫi) are the increments of the process σWtover the intervals, so they are independent
normally distributed random vectors with independent components having mean 0
and variance hσ2.
If the underlying process evolves according to (4), the exact evolution equation for the time series of observations at discrete intervals is not given by (6), but by an equation with higher order terms. Conversely, if the exact evolution of the
process is given by (6), the density of its invariant distribution is no longer πσ.
However, from the results on weak convergence of numerical solutions for SDE (?, ?) it follows that, under some differentiability conditions on U , for decreasing h,
the invariant distribution of ¯Xi converges weakly to the invariant distribution of
Xt, i.e., to the Gibbs distribution with the density πσ. So for a relatively short
observation interval, we hope that the invariant distribution is not too far away from the Gibbs distribution. However, as noted in the previous section, matching the empirical returns distribution to the stationary distribution of the underlying process remains here a weak point, just as is the case for all models starting with a stochastic differential equation with general diffusion coefficients.
For more detail on the estimation of the potential model and its applications to commodity prices see ?.
In the spirit of mean-reversion jump-diffusion models discussed in the previous section, we model the stochastic component of the log-price X(t) by
dX(t) = −U′(X(t))dt + σdW (t) + dN (t), (7)
where U (x) is a properly chosen potential function, N (t) is the compound Poisson process with intensity λ and increments having e.g., a log-normal distribution. In the absence of jumps, the process is the diffusion with drift given by a potential function. Note that the mean-reversion is incorporated into (7) as a special case, by taking U (x) a quadratic function.
The invariant marginal distribution of the process X(t) evolving as in (7) is no longer the Gibbs distribution (5) (due to the presence of jumps), so the direct estimation of U from the empirical distribution of observations on X(t) is no longer possible. One way to proceed is to assume a certain parametric form for U . For example, a simple way to model a state-dependent reversion rate, is to assume that the potential function is a polynomial of some fixed (and even) degree, higher than 2 (the degree 2 corresponds to the constant mean-reversion rate). This also assures that the conditions on the potential are satisfied. Then all the parameters of the model (7) can be simultaneously estimated from the observed de-seasoned log-prices
(Xi)i by the method of maximum likelihood or the generalized method of moments,
in the spirit of ?. However, also in this case the jump parameters (frequency and mean jump size) may be significantly misspecified.
Here we follow a different route: we apply a jump filtering procedure and esti-mate the diffusion-related parameters (i.e., the potential function and the volatility σ) from the filtered series, by matching the process’ stationary distribution to the empirically observed one, using the relationship (5). The jump intensity and jump size parameters are estimated from the series of observed jumps. We apply the po-tential jump-diffusion model to data from energy power exchanges. The results are given in Section 2.5.2.
2.4
Potential L´
evy jump models
2.4.1 Potential L´evy diffusion
? consider the stochastic differential equation
Xtε = x −
Z t
0
U′(Xsε−)ds + εLt, ε > 0 (8)
on a filtered probability space (Ω, F, (Ft)t≥0, P). Here L = Ltis a L´evy process, and
the noise intensity parameter ε is small compared to the other parameters. They study the exit problem of solutions of the non-linear stochastic differential equation (8) from a bounded or unbounded interval which contains the unique asymptotically stable critical point of the deterministic dynamical system
Yt(x) = x −
Z t
0
U′(Ys(x))ds.
Let U be a potential function and L be a L´evy process whose L´evy measure is
ν(dy) = dy
Some conditions on the potential function U must be imposed. First, U has a
‘parabolic’ shape with non-degenerate global minimum at the origin. Second, U′
is at least locally Lipschitz and increases faster than a linear function at ±∞ to
guarantee the existence of a strong unique solution of (8) on R. Last, U ∈ C3 in
some sufficiently large interval containing the origin in order to obtain some fine
small noise approximation of Xε.
The first exit problem for the process Xε from a bounded interval in the small
noise limit ε → 0 is formulated as follows. Let I = [−b, a], a, b > 0 and define the
first exit time from I as
σ(ε) = inf{t ≥ 0 : Xtε∈ [−b, a]}./
? analyze the laws of σ(ε), in particular its mean value, as ε → 0.
The key technique is the decomposition of the L´evy process into ε-dependent
small and large jump components. In mathematical terms, at time t the process Lt
is represented as a sum of two independent processes
Lt= ξtε+ ηεt.
The L´evy measures corresponding to the processes ξε and ηε are
νξε(dy) = ν
1
|y|1+αI{0 ≤ |y| ≤
1 √ε}dy , νηε(dy) = ν 1 |y|1+αI{|y| > 1 √ε}dy .
The process ξεhas an infinite L´evy measure with support−√1
ε,
1
√ε\{0} and makes
infinitely many jumps on any time interval of positive length. The absolute value
of its jumps does not exceed 1/√ε. The L´evy measure of η νηε(.) is finite. The
process ξε is a compound Poisson process with intensity β
ε and jumps distributed
according to the law βενηε(.), where
βε= νηε(R) = Z R\[− 1 √ε,+√1ε] dy |y|1+α = 2 αε α/2.
Denote τk, k ≥ 0, the arrival times of the jumps of ηε with τ0 = 0, Tk = τk− τk−1
the jump inter-arrival times, and Wk= ηετk−η
ε
τk−1 the jump heights of ηε. The three
processes (Tk)k≥1, (Wk)k≥1 and ξε are independent. Furthermore,
P(Tk ≥ u) = Z ∞ u βεexp (−βεu)dee u, u ≥ 0, E(Tk) = 1 βε = α 2ε −α/2→ ∞ as ε → 0,
P(Wk∈ A) = 1 βε Z AI{|y| > 1 √ ε} 1
|y|1+αdy for any Borel set A ⊆ R.
We are going to use these results to model the potential function and jump distribution. We consider the first exit problem of the stochastic component X(t)
from a bounded interval [m − √1
ε, m +
1 √
ε], where m denotes the mean level. In our
case, it is not possible to choose the small noise limit ε → 0. The most realistic way
is to choose the value of √1ε as big as possible, then we consider that a jump has
occured if the value of |X(t) − m| exceeds √1ε.
2.4.2 The α-stable distribution
? introduced a class of L´evy processes with infinite variance, which posses scale
invariance and self-similarity properties (see e.g., ?). The distribution of such a
process is the so-called α-stable distribution, denoted by Sα(σ, β, µ), where α ∈ (0, 2]
is the index of stability, β ∈ [−1, 1] is the skewness parameter, σ ≥ 0 is the scale parameter and µ is the shift parameter (for α 6= 1). The L´evy measure of a random variable X following an α-stable distribution with 0 < α ≤ 2 is
ν(x) = A
xα+1I{x > 0} +
B
|x|α+1I{x < 0},
for some positive constants A and B. The characteristic function of X ∼ Sα(σ, β, µ)
has the form
ΦX(z) =
(
exp− σα|z|α(1 − iβsgn z tanπα
2 ) + iµz
, if α 6= 1
exp− σ|z|(1 + iβπ2sgn z log |z|) + iµz , if α = 1
The probability density and distribution functions of α-stable distributions are
not known in closed form except in the three cases: Gaussian (S2(σ, 0, µ)), Cauchy
(S1(σ, 0, µ)) and L´evy (S1/2(σ, 1, µ)) distributions. Two approaches have been
pro-posed to tackle this problem: Fast Fourier Transform applied to the characteristic function (?) and direct integration (?). Due to the lack of the closed form formulae for the density and distribution functions, the simulation and the parameter estima-tion for α-stable distribuestima-tions are cumbersome. ? introduced a direct method for
simulating random variable X ∼ Sα(1, 1, 0) which is generalized by ?. ? proposed an
algorithm for constructing a standard stable random variable X ∼ Sα(1, β, 0). Some
methods have been proposed for estimating the parameters of an α-stable distribu-tion: by estimating the tail index as proposed by ?, the quantile method proposed by ?, or the maximum likelihood method proposed by ? and ?. We will use the algorithm proposed by ? for simulating α-stable random variables and the quantile method, proposed by ? for estimating the parameters of an α-stable distribution.
2.4.3 Two potential L´evy models
Let (m ± √1
ε) be the chosen upper and lower thresholds for deciding whether the
stochastic price fluctuations are driven by normal shocks or by jumps. We assume
that within a bounded intervalm−√1ε, m+√1εthe stochastic price fluctuations are
driven only by normal shocks. Here, the price process is a standard mean-reversion. When a jump occurs, the price moves (up or down) outside of this interval. Then a potential function forces the price to return rapidly towards the mean-reversion level with a higher reversal rate than the mean-reversion rate. We call it the fast relaxation process.
Regarding the mean-reversion and fast relaxation processes, we define the po-tential function U (x) as a real-valued function having a ‘parabolic’ shape with the non-degenerate global minimum at the mean-reversion (global mean) level m. Its
first derivative U′(x) has the form
U′ x= A x − ms−1+ C0 , x > m + √1ε (f ast relaxation) −γ(x − m) , |x − m| ≤ √1ε (mean − reversion)
−A m − xs−1− C0 , x > m − √1ε (f ast relaxation).
(9)
Note that s ≥ 2, since the ’force’ of fast relaxation is bigger than ’force’ of mean reversion.
Imkeller and Pavlyukevich model assumes that the jump distribution is a sym-metric α-stable L´evy process. Here, we relax this assumption and assume that the jump distribution is not necessarily symmetric. This is a more realistic assumption,
as shown by historical data from energy markets. The process ηε
t makes finitely
many jumps on any finite time interval and has the finite L´evy measure
νεη(dy) = ν m− (−y)1+α1(y<− 1 √ǫ)+ m+ y1+α1(y> 1 √ε) dy,
where m−+m−= 1, m+and m−are the relative frequencies of positive and negative
jumps, respectively. The distribution of the jump inter-arrival times is exponential with intensity βε= νηε(R) = Z R\−1√ ε, 1 √ ε m− (−y)1+α + m+ y1+α dy = 1 αε α/2. (10)
The distribution function of the jump size ηε
t is P (Wk ∈ A) = 1 βε = Z A m− (−y)1+α + m+ y1+α 1{|y|> 1 √ ε} dy. (11)
