Fred Espen Benth

Stochastics: An International Journal of Probability and Stochastic Processes Doi: https://doi.org/10.1080/17442508.2024.2424867

The HEat modulated Infinite DImensional Heston (HEIDIH) is introduced as a concrete case of the general framework of infinite dimensional Heston stochastic volatility models of (F.E. Benth, I.C. Simonsen '18) for the pricing of forward contracts. It is supported by a comprehensive numerical analysis. The model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process. This process is a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process, the solution to a stochastic heat equation on the real half-line. The advection and heat equations are driven by independent space-time Gaussian processes which are white in time and coloured in space, with the latter covariance structure expressed by two different kernels. First, a class of weight-stationary kernels are given, under which regularity results for the HEIDIH model in fractional Sobolev spaces are formulated. In particular, the class includes weighted Matérn kernels. Second, numerical approximation of the model is considered. An error decomposition formula, pointwise in space and time, for a finite-difference scheme is proven. For a special case, essentially sharp convergence rates are obtained when this is combined with a fully discrete finite element approximation of the stochastic heat equation. The analysis takes into account a localization error, a pointwise-in-space finite element discretization error and an error stemming from the noise being sampled pointwise in space. The rates obtained in the analysis are higher than what would be obtained using a standard Sobolev embedding technique. Numerical simulations illustrate the results.

Applied Energy 356 Doi: https://doi.org/10.1016/j.apenergy.2023.122338

In this article, we investigate the mismatch of renewable electricity production to demand and how flexibility options enabling spatial and temporal smoothing can reduce risks of variability. As a case study we pick a simplified (partial) 2-region representation of the Norwegian electricity system and focus on wind power. We represent regional electricity production and demand through two stochastic processes: the wind capacity factors are modelled as a two-dimensional Ornstein–Uhlenbeck process and electricity demand consists of realistic base load and temperature-induced load coming from a deseasonalised autoregressive process. We validate these processes, that we have trained on historical data, through Monte Carlo simulations allowing us to generate many statistically representative weather years. For the investigated realisations (weather years) we study deviations of production from demand under different wind capacities, and introduce different scenarios where flexibility options like storage and transmission are available. Our analysis shows that simulated loss values are reduced significantly by cooperation between regions and either mode of flexibility. Combining storage and transmission leads to even more synergies and helps to stabilise production levels and thus reduces likelihoods of inadequacy of renewable power systems.

Stochastics: An International Journal of Probability and Stochastic Processes 97(1) p. 55-80 Doi: https://doi.org/10.1080/17442508.2024.2422122

The relationship between parabolic stochastic partial differential equations and autoregressive moving average (ARMA) time series on the real line is established. This is done in light of semigroup theory, under which the parabolic stochastic partial differential equation admits an Ornstein–Uhlenbeck process in Hilbert space. Hilbert-valued AR(1) (or, ARH(1) for short) processes are shown to naturally appear from sampled Ornstein–Uhlenbeck processes. An error representation of approximating AR(1) time series for evaluated ARH(1) processes is derived for the time dimension. Further, by proper projections of ARH(1) processes into Hilbertian subspaces a spatial error representation is derived for evaluation of such projections. The result shows convergence for approximating ARMA times series with increasing spatial dimensions. A numerical example demonstrates our theoretical results for the stochastic heat equation. The results provide a functional data analysis approach to ARH(1) processes.

Electronic Journal of Probability (EJP) 29 p. 1-41 Doi: https://doi.org/10.1214/24-EJP1182

We propose a novel approach to polynomial processes, which allows us to analyse such processes on general state spaces. We do not need to specify polynomials explicitly but can work with a general sequence of graded vector spaces of functions on the state space. Elements of these graded vector spaces form the polynomials. By introducing a sequence of vector space complements, we obtain the sets of monomials. The basic tool of our analysis is the polynomial action operator, which is a semigroup of operators mapping conditional expected values of polynomials acting on a polynomial process to polynomials of the same or lower grade. We study abstract polynomial processes under algebraic and topological assumptions on the polynomial actions, and establish an affine drift structure. Moreover, we characterize the covariance structure under similar but slightly stronger conditions. A crucial part in our analysis is the use of the (algebraic or topological) dual of the monomials of grade one, which serves as a linearization of the state space of the polynomial process. We provide several examples of polynomial processes that do not fall into the classical setting but are polynomial processes according to our definition and can be analyzed with the tools we provide here.

Quarterly Journal of the Royal Meteorological Society 150 p. 2712-2726 Doi: https://doi.org/10.1002/qj.4731

There are sparse opportunities for direct measurement of upper stratospheric winds, yet improving their representation in subseasonal-to-seasonal prediction models can have significant benefits. There is solid evidence from previous research that global atmospheric infrasound waves are sensitive to stratospheric dynamics. However, there is a lack of results providing a direct mapping between infrasound recordings and polar-cap upper stratospheric winds. The global International Monitoring System (IMS), which monitors compliance with the Comprehensive Nuclear-Test-Ban Treaty, includes ground-based stations that can be used to characterize the infrasound soundscape continuously. In this study, multi-station IMS infrasound data were utilized along with a machine-learning supported stochastic model, Delay-SDE-net, to demonstrate how a near-real-time estimate of the polar-cap averaged zonal wind at 1-hPa pressure level can be found from infrasound data. The infrasound was filtered to a temporal low-frequency regime dominated by microbaroms, which are ambient-noise infrasonic waves continuously radiated into the atmosphere from nonlinear interaction between counter-propagating ocean surface waves. Delay-SDE-net was trained on 5 years (2014–2018) of infrasound data from three stations and the ERA5 reanalysis 1-hPa polar-cap averaged zonal wind. Using infrasound in 2019–2020 for validation, we demonstrate a prediction of the polar-cap averaged zonal wind, with an error standard deviation of around 12 m·s compared with ERA5. These findings highlight the potential of using infrasound data for near-real-time measurements of upper stratospheric dynamics. A long-term goal is to improve high-top atmospheric model accuracy, which can have significant implications for weather and climate prediction.

Journal of Computational Biology 31(8) p. 727-741 Doi: https://doi.org/10.1089/cmb.2023.0414

Finance and Stochastics 28 p. 1117-1146 Doi: https://doi.org/10.1007/s00780-024-00542-4

In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

Finance and Stochastics 28(4) p. 1035-1076 Doi: https://doi.org/10.1007/s00780-024-00546-0

We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

The Annals of Applied Probability 34(2) p. 2208-2242 Doi: https://doi.org/10.1214/23-AAP2019

Stochastic Processes and their Applications 162 p. 299-337 Doi: https://doi.org/10.1016/j.spa.2023.05.003

Energy Economics 117 Doi: https://doi.org/10.1016/j.eneco.2022.106421

We propose a novel stochastic time series model able to explain the stylized features of daily irradiation level data in 5 cities in Germany. The model is suitable for applications to risk management of photovoltaic power production in renewable energy markets. The suggested dynamics is a low-order autoregressive time series with seasonal level given by an atmospheric clear-sky model. Moreover, we detect a skewness property in the residuals which we explain by a winter–summer regime switch. The stochastic variance is modeled by a seasonally varying GARCH-dynamics. The winter and summer standardized residuals are proposed to be a Gaussian mixture model to capture the bimodal distributions. We estimate the model on the observed data, and perform a validation study. An application to energy markets studying the production at risk for a PV-producer is presented.

IMA Journal of Management Mathematics 34(1) p. 187-220 Doi: https://doi.org/10.1093/imaman/dpab032

Springer

Applied Stochastic Models in Business and Industry Doi: https://doi.org/10.1002/asmb.2815

Abstract This paper is concerned with managing risk exposure to temperature using weather derivatives. We consider hedging temperature risk using so‐called HDD‐ and CDD‐index futures, which are instruments written on temperatures in specific locations over specific time periods. The temperatures are modelled as continuous‐time autoregressive (CARMA) processes and pricing of the hedging instrument is done under an equivalent pricing measure. We develop hedging strategies for locations, cutoff temperatures, and time periods different to the ones in the traded contracts, allowing for more flexibility in the hedging application. The dynamic hedging strategies are expressed explicitly by the term structure of the volatility. We also provide numerical case studies with temperatures following a CAR(3)‐process to illustrate the temporal behaviour of the hedge under different scenarios.

Finance and Stochastics 28 p. 81-121 Doi: https://doi.org/10.1007/s00780-023-00520-2

Energies 16(15) Doi: https://doi.org/10.3390/en16155757

The purpose of this paper is to present and discuss pedagogical frameworks and approaches to developing, delivering, and evaluating a new interdisciplinary course within the domain of energy informatics at both Master’s and PhD levels. This study is needed because many papers on sustainable energy engineering education concentrate on course content but provide very little information on the pedagogical methods employed to deliver that content. The proposed new course is called “smart energy and power systems modelling” and is aimed at discussing how mathematical optimization, in the context of computer science, can contribute to more effectively managing smart energy and power systems. Different pedagogical frameworks are discussed and adapted for the specific domain of energy informatics. An ASSURE model coupled with Bloom’s taxonomy is presented for the design of the course and identification of learning objectives; self-regulated learning strategies are discussed to enhance the learning process; a novel model called GPD (Gaussian Progression of Difficulty) for lecture planning was proposed; a teaching-research nexus is discussed for the course planning and enhancement. Adopting qualitative analyses and an inductive approach, this paper offers a thorough reflection on the strengths and weaknesses of the new course, together with improvement possibilities based on fieldwork and direct experience with the students and colleagues. Opportunities and challenges of interdisciplinary teaching are presented in light of real-world experience, with a particular focus on the interaction between mathematics and computer science to study the specific application of energy and power systems.

Communications in statistics. Case studies, data analysis and applications. 9(3) p. 321-349 Doi: https://doi.org/10.1080/23737484.2023.2217137

Energy Economics 118 Doi: https://doi.org/10.1016/j.eneco.2022.106496

We suggest a new methodology for designing robust energy systems. For this, we investigate so-called near-optimal solutions to energy system optimisation models; solutions whose objective values deviate only marginally from the optimum. Using a refined method for obtaining explicit geometric descriptions of these near-optimal feasible spaces, we find designs that are as robust as possible to perturbations. This contributes to the ongoing debate on how to define and work with robustness in energy systems modelling. We apply our methods in an investigation using multiple decades of weather data. For the first time, we run a capacity expansion model of the European power system (one node per country) with a three-hourly temporal resolution and 41 years of weather data. While an optimisation with 41 weather years is at the limits of computational feasibility, we use the near-optimal feasible spaces of single years to gain an understanding of the design space over the full time period. Specifically, we intersect all near-optimal feasible spaces for the individual years in order to get designs that are likely to be feasible over the entire time period. We find significant potential for investment flexibility, and verify the feasibility of these designs by simulating the resulting dispatch problem with four decades of weather data. They are characterised by a shift towards more onshore wind and solar power, while emitting more than 50% less CO2 than a cost-optimal solution over that period. Our work builds on recent developments in the field, including techniques such as Modelling to Generate Alternatives (MGA) and Modelling All Alternatives (MAA), and provides new insights into the geometry of near-optimal feasible spaces and the importance of multi-decade weather variability for energy systems design. We also provide an effective way of working with a multi-decade time frame in a highly parallelised manner. Our implementation is open-sourced, adaptable and is based on PyPSA-Eur.

Annals of Mathematics and Artificial Intelligence 91 p. 75-103 Doi: https://doi.org/10.1007/s10472-022-09824-z

We propose a neural network architecture in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fréchet space X into a Banach space Y. The approximation results are generalising the well known universal approximation theorem for continuous functions from Rn to R, where approximation is done with (multilayer) neural networks Cybenko (1989) Math. Cont. Signals Syst. 2, 303–314 and Hornik et al. (1989) Neural Netw., 2, 359–366 and Funahashi (1989) Neural Netw., 2, 183–192 and Leshno (1993) Neural Netw., 6, 861–867. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Several examples are given of such activation functions. We show furthermore that our neural networks on infinite dimensional spaces can be projected down to finite dimensional subspaces with any desirable accuracy, thus obtaining approximating networks that are easy to implement and allow for fast computation and fitting. The resulting neural network architecture is therefore applicable for prediction tasks based on functional data.

Stochastic Processes and their Applications 145 p. 241-268 Doi: https://doi.org/10.1016/j.spa.2021.12.011

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert–Schmidt norm. In addition, we determine convergence rates for common stochastic volatility models in Hilbert spaces.

Stochastics: An International Journal of Probability and Stochastic Processes p. 1-23 Doi: https://doi.org/10.1080/17442508.2021.2019738

We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

Energy Economics 114 Doi: https://doi.org/10.1016/j.eneco.2022.106300

We analyse how carbon emissions are reduced utilizing the new NordLink cable between Norway and Germany, taking policy plans of platform electrification in Norway into account. Building a stochastic model based on existing data for Norwegian and German electricity production and demand (Open Power system data, 2020) to account for uncertainty, we found that the cable can be used as an effective way to reduce German emissions by exporting German renewable surplus production on windy days to Norway and importing from Norway other days. We found that such flow of power will lead to a possible reduction of the countries’ total emissions by over 4.1 million tons CO annually. Based on a mean reduction of German CO emissions of about 710 g/kWh by electricity imported from Norway, we found that electrification of the Norwegian oil and gas extraction is inferior to an increased export to Germany in terms of total emission reduction, as electrification only gives a reduction of about 425 g/kWh according to numbers from the Norwegian Oil Directorate (Oljedirektoratet, 2020). A similar calculation assuming a realistic full electrification of the platforms only gave a reduction of 2.4 million tons CO annually. In the long term, we conclude that combining both approaches makes the best use of Norwegian surplus energy production in view of carbon emission reduction.

Infinite Dimensional Analysis Quantum Probability and Related Topics 25(2) p. 1-35 Doi: https://doi.org/10.1142/S0219025722500072

We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida–Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space.

Dependence Modeling 10(1) p. 22-28 Doi: https://doi.org/10.1515/demo-2022-0103

Abstract Copulas are appealing tools in multivariate probability theory and statistics. Nevertheless, the transfer of this concept to infinite dimensions entails some nontrivial topological and functional analytic issues, making a deeper theoretical understanding indispensable toward applications. In this short work, we transfer the well-known property of compactness of the set of copulas in finite dimensions to the infinite-dimensional framework. As an application, we prove Sklar’s theorem in infinite dimensions via a topological argument and the notion of inverse systems.

Scandinavian Journal of Statistics Doi: https://doi.org/10.1111/sjos.12559

SIAM Journal on Financial Mathematics 12(4) p. 1374-1415 Doi: https://doi.org/10.1137/21M141556X

In the setting of one-dimensional polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for polynomial processes. The developed framework can, for example, be applied in financial pricing, such as for path-dependent options and in a stochastic volatility models context. In applications to options, having closed and compact formulations is attractive for sensitivity analysis and risk management, since Greeks can be derived explicitly.

Modern Stochastics: Theory and Applications (MSTA) 8(3) p. 349-371 Doi: https://doi.org/10.15559/21-VMSTA178

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

Digital Finance 3 p. 209-248 Doi: https://doi.org/10.1007/s42521-021-00030-w

We price European-style options written on forward contracts in a commodity market, which we model with an infinite-dimensional Heath–Jarrow–Morton (HJM) approach. For this purpose, we introduce a new class of state-dependent volatility operators that map the square integrable noise into the Filipović space of forward curves. For calibration, we specify a fully parametrized version of our model and train a neural network to approximate the true option price as a function of the model parameters. This neural network can then be used to calibrate the HJM parameters based on observed option prices. We conduct a numerical case study based on artificially generated option prices in a deterministic volatility setting. In this setting, we derive closed pricing formulas, allowing us to benchmark the neural network based calibration approach. We also study calibration in illiquid markets with a large bid-ask spread. The experiments reveal a high degree of accuracy in recovering the prices after calibration, even if the original meaning of the model parameters is partly lost in the approximation step.

Infinite Dimensional Analysis Quantum Probability and Related Topics 24(2) Doi: https://doi.org/10.1142/S0219025721500144

International Journal of Theoretical and Applied Finance 24(6 & 7) Doi: https://doi.org/10.1142/S0219024921500345

In this paper, we introduce a dynamical model for the time evolution of probability density functions incorporating uncertainty in the parameters. The uncertainty follows stochastic processes, thereby defining a new class of stochastic processes with values in the space of probability densities. The purpose is to quantify uncertainty that can be used for probabilistic forecasting. Starting from a set of traded prices of equity indices, we do some empirical studies. We apply our dynamic probabilistic forecasting to option pricing, where our proposed notion of model uncertainty reduces to uncertainty on future volatility. A distribution of option prices follows, reflecting the uncertainty on the distribution of the underlying prices. We associate measures of model uncertainty of prices in the sense of Cont.

Quantitative finance (Print) 21(1) p. 165-183 Doi: https://doi.org/10.1080/14697688.2020.1804606

With the introduction of the exchange-traded German wind power futures, opportunities for German wind power producers to hedge their volumetric risk are present. We propose two continuous-time multivariate models for wind power utilization at different wind sites, and discuss the properties and estimation procedures for the models. Applying the models to wind index data for wind sites in Germany and the underlying wind index of exchange-traded wind power futures contracts, the estimation results of both models suggest that they capture key statistical features of the data. We show how these models can be used to find optimal hedging strategies using exchange-traded wind power futures for the owner of a portfolio of so-called tailor-made wind power futures. Both in-sample and out-of-sample hedging scenarios are considered, and, in both cases, significant variance reductions are achieved. Additionally, the risk premium of the German wind power futures is analysed, leading to an indication of the risk premium of tailor-made wind power futures.

Electronic Journal of Probability (EJP) 26 Doi: https://doi.org/10.1214/21-EJP683

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from [18] to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al.

Mathematics 9(2) Doi: https://doi.org/10.3390/math9020124

Operating in energy and commodity markets require a management of risk using derivative products such as forward and futures, as well as options on these. Many of the popular stochastic models for spot dynamics and weather variables developed from empirical studies in commodity and energy markets belong to the class of polynomial jump diffusion processes. We derive a tailor-made framework for efficient polynomial approximation of the main derivatives encountered in commodity and energy markets, encompassing a wide range of arithmetic and geometric models. Our analysis accounts for seasonality effects, delivery periods of forwards and exotic temperature forwards where the underlying “spot” is a nonlinear function of the temperature. We also include in our derivations risk management products such as spread, Asian and quanto options.

Stochastics: An International Journal of Probability and Stochastic Processes Doi: https://doi.org/10.1080/17442508.2020.1802458

We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space B. It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of B being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.

Journal of Commodity Markets p. 1-13 Doi: https://doi.org/10.1016/j.jcomm.2019.100122

We analyse forward prices observed at the Fishpool market, and propose a two-factor continuous-time stochastic process for modelling the time dynamics. The data analysis reveals that the two factors can be assumed to be a non-stationary compound Poisson process and a stationary continuous-time autoregressive dynamics, describing the bumps observed in the forward curves. We use the model to analyse the risk premium in the forward markets, and find a negative premium in the long end of the market which is in line with the theory of normal backwardation. However, contracts with short time to maturity have a risk premium with randomly changing sign, pointing towards a hedging pressure also induced by the demand-side of the market.

International Journal of Theoretical and Applied Finance 23(4) Doi: https://doi.org/10.1142/S0219024920500272

Quantitative finance (Print) 20(9) p. 1441-1456 Doi: https://doi.org/10.1080/14697688.2020.1733059

The liberalization of energy markets worldwide during recent decades has introduced severe implications for the price formation in these markets. Especially within the European day-ahead electricity markets, increased physical connections between different market areas and a joint effort on optimizing the aggregate social welfare have led to highly connected markets. Consequently, observing the exact same hourly day-ahead prices for two or more interconnected electricity markets in Europe happens frequently. This affects the modelling of such prices and in turn the valuation of derivatives written on prices from these market areas. In this paper, we propose a joint model for day-ahead electricity prices in interconnected markets composed of a combination of transformed Ornstein–Uhlenbeck processes. We discuss the properties of the model and propose an estimation procedure based on filtering techniques. Furthermore, the properties of the model reveal that analytical prices are attainable for, e.g., forwards and spread options.

IEEE Transactions on Power Systems 35(2) p. 1085-1098 Doi: https://doi.org/10.1109/TPWRS.2019.2938423

Successful trading in electricity markets relies on the market actor's ability to accurately forecast the electricity price. The fundamental electricity price models use market information, provided by various price drivers, including the residual that contains a risk premium. In the past, researchers investigating risk premium focused primarily on daily spot price levels, ignoring the intraday information hindering the accurate risk premium determination. This paper presents a new KGB Method for modelling of risk premium, based on “ex-ante” approach focused on a yearly product. The method involves a novel KGB Model and its linearized formulation, the KGB Linear Model, which enables capturing the influence of renewable energy sources on risk premium. The four key drivers of the KGB Linear Model were used providing an insight into the influence of RES generation on risk premium evolution. The method was tested on historical data from the German electricity market. The results for the 2010-2014 period reveal overall influence of PV production share on risk premium is greater than that of wind production share, both increasing the risk premium due to their variability and uncertainty. Using the KGB Method, market actors can forecast risk premium using information readily available to them.

Risks 8(1) Doi: https://doi.org/10.3390/risks8010008

Spot option prices, forwards and options on forwards relevant for the commodity markets are computed when the underlying process S is modelled as an exponential of a process ξ with memory as, e.g., a Volterra equation driven by a Lévy process. Moreover, the interest rate and a risk premium ρ representing storage costs, illiquidity, convenience yield or insurance costs, are assumed to be stochastic. When the interest rate is deterministic and the risk premium is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity, the process (ξ;ρ) has an affine structure under the pricing measure Q and an explicit expression for the option price is derived in terms of the Fourier transform of the payoff function.

Mathematics and Financial Economics 13(4) p. 543-577 Doi: https://doi.org/10.1007/s11579-019-00237-x

Journal of Mathematical Analysis and Applications 476(1) p. 196-214 Doi: https://doi.org/10.1016/j.jmaa.2018.12.055

Two stationary and non-negative processes that are based on continuous-time autoregressive moving average (CARMA) processes are discussed. First, we consider a generalization of Cox–Ingersoll–Ross (CIR) processes. Next, we consider CARMA processes driven by compound Poisson processes with exponential jumps which are generalizations of Ornstein–Uhlenbeck (OU) processes driven by the same noise. The way in which the two processes generalize CIR and OU processes and the relation between them will be discussed. Furthermore, the stationary distribution, the autocorrelation function, and pricing of zero-coupon bonds are considered.

Stochastic Analysis and Applications 36(4) p. 733-750 Doi: https://doi.org/10.1080/07362994.2018.1461566

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.

Applied Mathematical Finance 25(1) p. 36-65 Doi: https://doi.org/10.1080/1350486X.2018.1438904

The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.

Stochastic Processes and their Applications 128(2) p. 461-486 Doi: https://doi.org/10.1016/j.spa.2017.05.005

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.

Mathematics and Financial Economics 13(1) p. 87-114 Doi: https://doi.org/10.1007/s11579-018-0221-8

We develop cointegration for multivariate continuous-time stochastic processes, both in finite and infinite dimension. Our definition and analysis are based on factor processes and operators mapping to the space of prices and cointegration. The focus is on commodity markets, where both spot and forward prices are analysed in the context of cointegration. We provide many examples which include the most used continuous-time pricing models, including forward curve models in the Heath-Jarrow-Morton paradigm in Hilbert space.

Transportation Research Part E: Logistics and Transportation Review 113 p. 194-221 Doi: https://doi.org/10.1016/j.tre.2017.10.014

In this paper, we propose a new multivariate model for the dynamics of regional ocean freight rates. We show that a cointegrated system of regional spot freight rates can be decomposed into a common non-stationary market factor and stationary regional deviations. The resulting integrated CAR process is new to the literature. By interpreting the common market factor as the global arithmetic average of the regional rates, both the market factor and the regional deviations are observable which simplifies the calibration of the model. Moreover, forward contracts on the market factor can be traded in the Forward Freight Agreement (FFA) market. We calibrate the model to historical spot rate processes and illustrate the term structures of volatility and correlation between the regional prices and the market factor. Our model is an important contribution towards improved modelling and hedging of regional price risk when derivative market liquidity is concentrated in a single global benchmark.

Finance and Stochastics 22(2) p. 327-366 Doi: https://doi.org/10.1007/s00780-018-0355-9

In this paper, we show how to approximate Heath–Jarrow–Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have a finite-dimensional state space. Moreover, we recover a closed-form representation of the forward price dynamics in the approximation models and derive the rate of convergence to the true dynamics uniformly over an interval of time to maturity under certain additional smoothness conditions. In the Markovian case, we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.

Risks 6(2) Doi: https://doi.org/10.3390/risks6020056

We model the logarithm of the spot price of electricity with a normal inverse Gaussian (NIG) process and the wind speed and wind power production with two Ornstein–Uhlenbeck processes. In order to reproduce the correlation between the spot price and the wind power production, namely between a pure jump process and a continuous path process, respectively, we replace the small jumps of the NIG process by a Brownian term. We then apply our models to two different problems: first, to study from the stochastic point of view the income from a wind power plant, as the expected value of the product between the electricity spot price and the amount of energy produced; then, to construct and price a European put-type quanto option in the wind energy markets that allows the buyer to hedge against low prices and low wind power production in the plant. Calibration of the proposed models and related price formulas is also provided, according to specific datasets.

Computation and Combinatorics in Dynamics, Stochastics and Control p. 297-320 Doi: https://doi.org/10.1007/978-3-030-01593-0_11

Springer Nature

Journal of Energy Markets 10(1) p. 1-25 Doi: https://doi.org/10.21314/jem.2017.157

Stochastics: An International Journal of Probability and Stochastic Processes 89(1) p. 311-347 Doi: https://doi.org/10.1080/17442508.2016.1177057

We lift ambit fields to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fields can be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation. This stochastic partial differential equation is formulated on a suitable Hilbert space of functions on the positive real line with values in the state space of the Hambit field. We provide an explicit construction of such a space. Finally, we apply this interpretation of Hambit fields to develop a finite difference scheme, for which we prove convergence under some Lipschitz conditions. This research was first published in Schochastics: An International Journal of Probability and Stochastic Processes. © Taylor & Francis.

Real Options in Energy and Commodity Markets p. 63-115 Doi: https://doi.org/10.1142/9789813149410_0003

We conduct an empirical investigation of the logreturns of the futures prices of the European Union Emission Trading System emission certificates traded in the Nord Pool market between 2005 and 2013. We observe heaviness, skewness, and high kurtosis of these logreturns. We thus propose modeling the futures logprices using the Barndorff-Nielsen and Shephard (BNS) or the Heston stochastic volatility models.We carry out an empirical comparison between the performances of these models and investigate their stationary autocorrelation structure. In particular, as a consequence of allowing for skewness in the Heston model, we find analytical expressions for the autocorrelation function of the logreturns and their squares. Our analysis indicates the presence of short-range dependence in the observed futures logprice returns. We conclude that the BNS model better describes the empirical features of the observed futures prices than the Heston model. Our findings have relevance for the real option modeling of fossil-fueled power plants when considering emission costs.

Energy Economics 68 p. 283-302 Doi: https://doi.org/10.1016/j.eneco.2017.10.008

The recent price coupling of many European electricity markets has triggered a fundamental change in the interaction of day-ahead prices, challenging additionally the modeling of the joint behavior of prices in interconnected markets. In this paper we propose a regime-switching AR–GARCH copula to model pairs of day-ahead electricity prices in coupled European markets. While capturing key stylized facts empirically substantiated in the literature, this model easily allows us to 1) deviate from the assumption of normal margins and 2) include a more detailed description of the dependence between prices. We base our empirical study on four pairs of prices, namely Germany–France, Germany–Netherlands, Netherlands–Belgium and Germany–Western Denmark. We find that the marginal dynamics are better described by the flexible skew t distribution than the benchmark normal distribution. Also, we find significant evidence of tail dependence in all pairs of interconnected areas we consider. As a first application of the proposed model, we consider the pricing of financial transmission rights, and highlight how the choice of marginal distributions and copula impacts prices. As a second application we consider the forecasting of tail quantiles, and evaluate the out-of-sample performance of competing models.

Journal of Energy Markets 10(2) p. 1-39 Doi: https://doi.org/10.21314/JEM.2017.159

We propose and investigate a valuation model for the income of selling tradeable green certificates (TGCs) in the Swedish–Norwegian market, formulated as a singular stochastic control problem. Our model takes into account the production rate of renewable energy from a “typical” plant, the price of TGCs and the cumulative amount of certificates sold.We assume that the production rate has a dynamics given by an exponential Ornstein–Uhlenbeck process, and the logarithmic TGC price has a dynamics given by a Lévy process. For this class of dynamics, we find optimal decision rules for the state variables and a closed-form solution to the control problem. A case study of ICAP prices and wind production data from Denmark backs up our model choice and shows the relevance of this pricing approach.

Journal of Energy Markets 10(3) p. 1-33 Doi: https://doi.org/10.21314/JEM.2017.164

In recent years, renewable energy has gained importance in producing power in many markets. The aim of this article is to model photovoltaic (PV) production for three transmission operators in Germany. PV power can only be generated during sun hours and the cloud cover will determine its overall production. Therefore, we propose a model that takes into account the sun intensity as a seasonal function. We model the deseasonalized data by an autoregressive process to capture the stochastic dynamics in the data. We present two applications based on our suggested model. First, we build a relationship between electricity spot prices and PV production where the higher the volume of PV production, the lower the power prices. As a further application, we discuss virtual power plant derivatives and energy quanto options. This is a submitted version of an article which will be published in the Journal of Energy Markets. © Incisive Media

Journal of Banking & Finance 95 p. 203-216 Doi: https://doi.org/10.1016/j.jbankfin.2017.03.018

Stochastic models for forward electricity prices are of great relevance nowadays, given the major structural changes in the market due to the increase of renewable energy in the production mix. In this study, we derive a spatio-temporal dynamical model based on the Heath-Jarrow-Morton (HJM) approach under the Musiela parametrization, which ensures an arbitrage-free model for electricity forward prices. The model is fitted to a unique data set of historical price forward curves. As a particular feature of the model, we disentangle the temporal from spatial (maturity) effects on the dynamics of forward prices, and shed light on the statistical properties of risk premia, of the noise volatility term structure and of the spatio-temporal noise correlation structures. We find that the short-term risk premia oscillates around zero, but becomes negative in the long run. We identify the Samuelson effect in the volatility term structure and volatility bumps, explained by market fundamentals. Furthermore we find evidence for coloured noise and correlated residuals, which we model by a Hilbert space-valued normal inverse Gaussian Lévy process with a suitable covariance functional. (Best Energy Paper Award, ECOMFIN 2016, Paris)

Advanced Modelling in Mathematical Finance p. 477-496 Doi: https://doi.org/10.1007/978-3-319-45875-5_20

The Fascination of Probability, Statistics and their Applications. In honour of Ole E. Barndorff-Nielsen p. 153-189 Doi: https://doi.org/10.1007/978-3-319-25826-3_8

Springer Science+Business Media B.V.

International Journal of Theoretical and Applied Finance 19(1) Doi: https://doi.org/10.1142/S0219024916500023

Bernoulli 22(3) p. 1383-1430 Doi: https://doi.org/10.3150/15-BEJ696

We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an Itô formula. Moreover, we derive an alternative definition using a random-field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.

Maritime Economics & Logistics 18(4) p. 391-413 Doi: https://doi.org/10.1057/mel.2015.22

Bernoulli 22(2) p. 774-793 Doi: https://doi.org/10.3150/14-BEJ675

International Journal of Theoretical and Applied Finance 18(6) Doi: https://doi.org/10.1142/S0219024915500387

For a commodity spot price dynamics given by an Ornstein–Uhlenbeck (OU) process with Barndorff-Nielsen and Shephard stochastic volatility, we price forwards using a class of pricing measures that simultaneously allow for change of level and speed in the mean reversion of both the price and the volatility. The risk premium is derived in the case of arithmetic and geometric spot price processes, and it is demonstrated that we can provide flexible shapes that are typically observed in energy markets. In particular, our pricing measure preserves the affine model structure and decomposes into a price and volatility risk premium. In the geometric spot price model, we need to resort to a detailed analysis of a system of Riccati equations, for which we show existence and uniqueness of solution and asymptotic properties that explain the possible risk premium profiles. Among the typical shapes, the risk premium allows for a stochastic change of sign, and can attain positive values in the short end of the forward market and negative in the long end. Preprint of an article published in International Journal of Theoretical and Applied Finance 2015 © World Scientific Publishing Company http://www.worldscientific.com/worldscinet/ijtaf

International Journal of Theoretical and Applied Finance 18(2) Doi: https://doi.org/10.1142/S0219024915500107

Finance and Stochastics 19(4) p. 849-889 Doi: https://doi.org/10.1007/s00780-015-0270-2

We solve the problem of pricing and hedging Asian-style options on energy with a quadratic risk criterion when trading in the underlying future is restricted. Liquid trading in the future is only possible up to the start of a so-called delivery period. After the start of the delivery period, the hedge positions cannot be adjusted any more until maturity. This reflects the trading situation at the Nordic energy market Nord Pool, for example. We show that there exists a unique solution to this combined continuous–discrete quadratic hedging problem if the future price process is a special semimartingale with bounded mean–variance tradeoff. Additionally, under the assumption that the future price process is a local martingale, the hedge positions before the averaging period are inherited from the market specification without trading restriction. As an application, we consider three models and derive their quadratic hedge positions in explicit form: a simple Black–Scholes model with time-dependent volatility, the stochastic volatility model of Barndorff-Nielsen and Shephard, and an exponential additive model. Based on an exponential spot price model driven by two NIG Lévy processes, we determine an exponential additive model for the future price by moment matching techniques. We calculate hedge positions and determine the quadratic hedging error in a simulation study. The final publication is available at Springer via http://dx.doi.org/10.1007/s00780-015-0270-2

Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management p. 801-825 Doi: https://doi.org/10.1002/9781119011590.ch17

IMA Journal of Management Mathematics 26(3) p. 273-297 Doi: https://doi.org/10.1093/imaman/dpu001

SIAM Journal on Financial Mathematics 6(1) p. 825-869 Doi: https://doi.org/10.1137/15100268X

Based on forward curves modelled as Hilbert-space valued processes, we analyze the pricing of various options relevant in energy markets. In particular, we connect empirical evidence about energy forward prices known from the literature to propose stochastic models. Forward prices can be represented as linear functions on a Hilbert space, and options can thus be viewed as derivatives on the whole curve. The value of these options are computed under various specifications, in addition to their deltas. In a second part, cross-commodity models are investigated, leading to a study of square integrable random variables with values in a two-dimensional Hilbert space. We analyze the covariance operator and representations of such variables, as well as presenting applications to the pricing of spread and energy quanto options. © 2015, Society for Industrial and Applied Mathematics

Journal of Energy Markets 18(4) p. 69-92 Doi: https://doi.org/10.21314/jem.2015.133

Stochastics: An International Journal of Probability and Stochastic Processes 87(3) p. 458-476 Doi: https://doi.org/10.1080/17442508.2014.966826

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].

Energy Economics 52 p. 104-117 Doi: https://doi.org/10.1016/j.eneco.2015.09.009

We analyze cointegration in commodity markets, and propose a parametric class of pricing measures which preserves cointegration for forward prices with fixed time to maturity. We present explicit expressions for the term structure of volatility and correlation in the context of our spot price models based on continuous-time autoregressive moving average dynamics for the stationary components. The term structures have many interesting shapes, and we provide some empirical evidence from refined oil future prices at NYMEX defending our modeling idea. Motivated from these results, we present a cointegrated forward price dynamics using the Heath–Jarrow–Morton approach. In this setting, the concept of cointegration is extended to what we call cointegration in the limit, which is an asymptotic form of the notion. The Margrabe formula for spread option prices is shown to hold, with an explicit plug-in volatility. We present several numerical examples showing that cointegration leads to significantly cheaper spread options compared to the complete market case, where cointegration disappears with respect to the pricing measure.

Fields Institute Communications 74 p. 109-148 Doi: https://doi.org/10.1007/978-1-4939-2733-3_5

Journal of Energy Markets 8(1) p. 1-35 Doi: https://doi.org/10.21314/jem.2015.130

Applied Mathematical Finance 22(1) p. 28-62 Doi: https://doi.org/10.1080/1350486X.2014.948708

SIAM Journal on Financial Mathematics 5 p. 685-728 Doi: https://doi.org/10.1137/13093604X

In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein–Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump L´evy process and models the characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricing measure is given by the Esscher transform that preserves the probabilistic structure of the driving L´evy processes while changing the levels of mean reversion. Using this choice one can generate stochastic risk premiums (in geometric spot models) but with (deterministically) changing sign. In this paper we introduce a pricing change of measure, which is an extension of the Esscher transform. With this new change of measure we can also slow down the speed of mean reversion and generate stochastic risk premiums with stochastic nonconstant sign, even in arithmetic spot models. In particular, we can generate risk profiles with positive values in the short end of the forward curve and negative values in the long end. Finally, our pricing measure allows us to have a stationary spot dynamics while still having randomly fluctuating forward prices for contracts far from maturity. © 2014 Society for Industrial and Applied Mathematics

Infinite Dimensional Analysis Quantum Probability and Related Topics 17(2) Doi: https://doi.org/10.1142/S0219025714500118

This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space of Potthoff–Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model. Preprint of an article published in Infinite Dimensional Analysis Quantum Probability and Related Topics. 2014, 17 (2), DOI: http://dx.doi.org/10.1142/S0219025714500118 © World Scientific Publishing Company http://www.worldscientific.com/worldscinet/idaqp

Stochastics: An International Journal of Probability and Stochastic Processes 86(6) p. 932-966 Doi: https://doi.org/10.1080/17442508.2014.895359

In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.

Stochastic Processes and their Applications 124(1) p. 812-847 Doi: https://doi.org/10.1016/j.spa.2013.09.007

Communications in Mathematics and Statistics 2(1) p. 47-106 Doi: https://doi.org/10.1007/s40304-014-0030-1

SIAM Journal on Financial Mathematics 5(1) p. 71-98 Doi: https://doi.org/10.1137/130905320

The present paper discusses simulation of Lévy semistationary (LSS) processes in the context of power markets. A disadvantage of applying numerical integration to obtain trajectories of LSS processes is that such a scheme is not iterative. We address this problem by introducing and analyzing a Fourier simulation scheme for obtaining trajectories of these processes in an iterative manner. Furthermore, we demonstrate that our proposed scheme is well suited for simulation of a wide range of LSS processes, including, in particular, LSS processes indexed by a kernel function which is steep close to the origin. Finally, we put our simulation scheme to work for simulating the price of path-dependent options to demonstrate the advantages of the proposed Fourier simulation scheme. © 2014, Society for Industrial and Applied Mathematics

Energy Pricing Models : Recent Advances, Methods, and Tools p. 223-268 Doi: https://doi.org/10.1007/978-1-137-37027-3_8

Published by Palgrave Macmillan. Reproduced with permission of Palgrave Macmillan. This extract is taken from the author's original manuscript and has not been edited. The definitive, published, version of record is available here: http://www.palgrave.com/page/detail/energy-pricing-models-marcel-prokopczuk/?isb=9781137377340

Springer Science+Business Media B.V.

Quantitative Energy Finance: Modeling, pricing and hedging in energy and commodity markets p. 285-305 Doi: https://doi.org/10.1007/978-1-4614-7248-3_11

Lecture notes in Energy 54 p. 233-260 Doi: https://doi.org/10.1007/978-3-642-55382-0_10

Finance and Stochastics 18(2) p. 407-430 Doi: https://doi.org/10.1007/s00780-013-0224-5

We develop a general approach to portfolio optimization in futures markets. Following the Heath–Jarrow–Morton (HJM) approach, we model the entire futures price curve at once as a solution of a stochastic partial differential equation. We also develop a general formalism to handle portfolios of futures contracts. In the portfolio optimization problem, the agent invests in futures contracts and a risk-free asset, and her objective is to maximize the utility from final wealth. In order to capture self-consistent futures price dynamics, we study a class of futures price curve models which admit a finite-dimensional realization. More precisely, we establish conditions under which the futures price dynamics can be realized in finite dimensions. Using the finite-dimensional realization, we derive a finite-dimensional form of the portfolio optimization problem and study its solution. We also give an economic interpretation of the coordinate process driving the finite-dimensional realization.

Energy Economics 44 p. 392-406 Doi: https://doi.org/10.1016/j.eneco.2014.03.020

Journal of Energy Markets 7(2) p. 35-69 Doi: https://doi.org/10.21314/jem.2014.114

International Journal of Theoretical and Applied Finance 17(2) Doi: https://doi.org/10.1142/S0219024914500083

In this paper, we present a multi-factor continuous-time autoregressive moving-average (CARMA) model for the short and forward interest rates. This model is able to present an adequate statistical description of the short and forward rate dynamics. We show that this is a tractable term structure model and provides closed-form solutions to bond prices, yields, bond option prices, and the term structure of forward rate volatility. We demonstrate the capabilities of our model by calibrating it to a panel of spot rates and the empirical volatility of forward rates simultaneously, making the model consistent with both the spot rate dynamics and forward rate volatility structure.

Recent advances in financial engineering - Proceedings of the International Workshop on Fininance 2012 p. 1-24 Doi: https://doi.org/10.1142/9789814571647_0001

The Interrelationship Between Financial and Energy Markets p. 233-260 Doi: https://doi.org/10.1007/978-3-642-55382-0

In this paper we derive power futures prices from a two-factor spot model being a generalization of the classical Schwartz–Smith commodity dynamics. We include non-Gaussian effects by introducing Lévy processes as the stochastic drivers, and estimate the model to data observed at the European Electricity Exchange in Germany. The spot and futures price models are fitted jointly, including the market price of risk parameterized from an Esscher transform. We apply this model to price call and put options on power futures. It is argued theoretically that the pricing measure for options may be different to the pricing measure of futures from spot in power markets due to the non-storability of the electricity spot. Empirical evidence pointing to this fact is found from option prices observed at the European Electricity Exchange. The definitive version is available at springerlink.com

Advances in Applied Probability 46(3) p. 719-745 Doi: https://doi.org/10.1239/aap/1409319557

In this paper we propose a new modelling framework for electricity futures markets based on so-called ambit fields. The new model can capture many of the stylised facts observed in electricity futures and is highly analytically tractable. We discuss martingale conditions, option pricing, and change of measure within the new model class. Also, we study the corresponding model for the spot price, which is implied by the new futures model, and show that, under certain regularity conditions, the implied spot price can be represented in law as a volatility modulated Volterra process. © Applied Probability Trust 2014

Seminar on Stochastic Analysis, Random Fields and Applications VII - Centro Stefano Franscini, Ascona, May 2011 p. 261-284 Doi: https://doi.org/10.1007/978-3-0348-0545-2_14

Mathematical Modeling with Multidisciplinary Applications p. 257-284 Doi: https://doi.org/10.1002/9781118462706.ch11

SIAM Journal on Scientific Computing 35(5) p. A2207-A2224 Doi: https://doi.org/10.1137/110851080

Advances in Applied Probability 45(2) p. 572-594 Doi: https://doi.org/10.1239/aap/1370870130

Energy Economics 36 p. 55-77 Doi: https://doi.org/10.1016/j.eneco.2012.12.001

Due to the non-storability of electricity and the resulting lack of arbitrage-based arguments to price electricity forward contracts, a significant time-varying risk premium is exhibited. Using EEX data during the introduction of emission certificates and the German “Atom Moratorium” we show that a significant part of the risk premium in electricity forwards is due to different information sets in spot and forward markets. In order to show the existence of the resulting information premium and to analyse its size we design an empirical method based on techniques relating to enlargement of filtrations and the structure of Hilbert spaces.

Advances in Applied Probability 45(2) p. 545-571 Doi: https://doi.org/10.1239/aap/1370870129

Stochastics: An International Journal of Probability and Stochastic Processes 85(5) p. 833-858 Doi: https://doi.org/10.1080/17442508.2012.665056

Stochastics: An International Journal of Probability and Stochastic Processes 85(6) p. 1015-1039 Doi: https://doi.org/10.1080/17442508.2012.736994

World Scientific

Lecture notes in mathematics 2081 p. 109-167 Doi: https://doi.org/10.1007/978-3-319-00413-6_2

We give a short introduction to energy markets, describing how they function and what products are traded. Next we survey some of the popular models that have been proposed in the literature. We extend the analysis of one of these models to include for stochastic volatility effects. In particular, we analyse a mean reverting stochastic spot price dynamics with a stochastic mean level modelled as an Ornstein–Uhlenbeck process. We include in this dynamics a stochastic volatility model of the Barndorff-Nielsen and Shephard type. Some properties of the dynamics are derived and discussed in relation to energy markets. Moreover, we derive a semi-analytical expression for the forward price based on such a spot dynamics. In the last part of these lecture notes we consider a cross-commodity spot price model including jumps. A Margrabe formula for options on the spread is derived, along with an analysis of the dependency risk under an Esscher measure transform. An empirical example demonstrates that the Esscher transform may increase the tail dependency in the bivariate jump part of the spot model. The final publication is available at Springer

Energy Economics 40 p. 259-268 Doi: https://doi.org/10.1016/j.eneco.2013.07.007

Bernoulli 19(3) p. 803-845 Doi: https://doi.org/10.3150/12-BEJ476

Operations Research 60(6) p. 1373-1388 Doi: https://doi.org/10.1287/opre.1120.1112

Review of Development Finance 2(1) p. 22-31 Doi: https://doi.org/10.1016/j.rdf.2012.01.004

The aim of this paper is to study pricing of weather insurance contracts based on temperature indices. Three different pricing methods are analysed: the classical burn approach, index modelling and temperature modelling. We take the data from Malaysia as our empirical case. Our results show that there is a significant difference between the burn and index pricing approaches on one hand, and the temperature modelling method on the other. The latter approach is pricing the insurance contract using a seasonal autoregressive time series model for daily temperature variations, and thus provides a precise probabilistic model for the fine structure of temperature evolution. We complement our pricing analysis by an investigation of the profit/loss distribution from the contract, in the perspective of both the insured and the insurer.

SIAM Journal on Financial Mathematics 3 p. 639-666 Doi: https://doi.org/10.1137/100818261

Energy Economics 34 p. 1589-1616 Doi: https://doi.org/10.1016/j.eneco.2011.11.012

Energy Economics 34(2) p. 592-602 Doi: https://doi.org/10.1016/j.eneco.2011.09.012

In this paper we present a stochastic model for daily average temperature. The model contains seasonality, a low-order autoregressive component and a variance describing the heteroskedastic residuals. The model is estimated on daily average temperature records from Stockholm (Sweden). By comparing the proposed model with the popular model of Campbell and Diebold (2005), we point out some important issues to be addressed when modelling the temperature for application in weather derivatives market.

Stochastic Analysis and Applications 30(1) p. 20-43 Doi: https://doi.org/10.1080/07362994.2012.628906

In power markets one frequently encounters a risk premium being positive in the short end of the forward curve and negative in the long end. Economically it has been argued that the positive premium is reflecting retailers aversion for spike risk, wheras in the long end of the forward curve, the hedging pressure kicks in as in other commodity markets. Mathematically, forward prices are expressed as risk-neutral expectations of the spot at delivery. We apply the Esscher transform on power spot models based on mean-reverting processes driven by independent increment (time-inhomogeneous Lévy) processes. It is shown that the Esscher transform is yielding a change of mean-reversion level. Moreover, we show that an Esscher transform together with jumps occuring seasonally may explain the occurence of a positive risk premium in the short end. This is demonstrated both mathematically and by a numerical example for a two-factor spot model being relevant for electricity markets.

Journal of Energy Markets 4(4) p. 3-28 Doi: https://doi.org/10.21314/jem.2011.065

This is the pre-peer reviewed version of the following article: Benth, Fred Espen; Lempa, Jukka; Nilssen, Trygve Kastberg, On the optimal exercise of swing options in electricity markets, Journal of Energy Markets. 2012, 4 (4), 3-28, which has been published in final form at http://www.risk.net/journal-of-energy-markets/technical-paper/2160760/on-optimal-exercise-swing-options-electricity-markets.

Journal of Energy Markets 5(4) p. 3-31 Doi: https://doi.org/10.21314/jem.2012.080

Journal of Forecasting 30(3) p. 355-376 Doi: https://doi.org/10.1002/for.1179

International Journal of Stochastic Analysis Doi: https://doi.org/10.1155/2011/576791

We propose a continuous-time autoregressive model for the temperature dynamics with volatility being the product of a seasonal function and a stochastic process. We use the Barndorff-Nielsen and Shephard model for the stochastic volatility. The proposed temperature dynamics is flexible enough to model temperature data accurately, and at the same time being analytically tractable. Futures prices for commonly traded contracts at the Chicago Mercantile Exchange on indices like cooling- and heating-degree days and cumulative average temperatures are computed, as well as option prices on them.

Quantitative finance (Print) 11(3) p. 407-421 Doi: https://doi.org/10.1080/14697688.2010.481629

Statistical Tools for Finance and Insurance p. 163-199 Doi: https://doi.org/10.1007/978-3-642-18062-0_5

Statistical research report (Universitetet i Oslo. Matematisk institut

The aim of this paper is to study pricing of weather insurance contracts based on temperature indices. Three different pricing methods are analysed: the classical burn approach, index modelling and temperature modelling. We take the data from Malaysia as our empirical case. Our results show that there is a significant difference between the burn and index pricing approaches on one hand, and the temperature modelling method on the other. The latter approach is pricing the insurance contract using a seasonal autoregressive time series model for daily temperature variations, and thus provides a precise probabilistic model for the fine structure of temperature evolution. We complement our pricing analysis by an investigation of the profit/loss distribution from the contract, in the perspective of both the insured and the insurer.

Communications on Stochastic Analysis 5(2) p. 285-307 Doi: https://doi.org/10.31390/cosa.5.2.03

Applied Mathematical Finance 18(2) p. 93-117 Doi: https://doi.org/10.1080/13504861003722385

Advanced Mathematical Methods for Finance p. 35-74 Doi: https://doi.org/10.2139/ssrn.1597697

Ambit processes are general stochastic processes based on stochastic integrals with respect to Lévy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection between ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Lévy noise analysis.

Mathematical Finance 21(4) p. 595-625 Doi: https://doi.org/10.1111/j.1467-9965.2010.00445.x

We consider the non-Gaussian stochastic volatility model of Barndorff-Nielsen and Shephard for the exponential mean-reversion model of Schwartz proposed for commodity spot prices. We analyze the properties of the stochastic dynamics, and show in particular that the log-spot prices possess a stationary distribution defined as a normal variance-mixture model. Furthermore, the stochastic volatility model allows for explicit forward prices, which may produce a hump structure inherited from the mean-reversion of the stochastic volatility. Although the spot price dynamics has continuous paths, the forward prices will have a jump dynamics, where jumps occur according to changes in the volatility process. We compare with the popular Heston stochastic volatility dynamics, and show that the Barndorff-Nielsen and Shephard model provides a more flexible framework in describing commodity spot prices. An empirical example on UK spot data is included.

QUANTUM PROBABILITY AND INFINITE DIMENSIONAL ANALYSIS - Proceedings of the 29th Conference p. 153-184

International Journal of Applied Mathematics and Statistics 16(M10) p. 11-37

Stochastics: An International Journal of Probability and Stochastic Processes 82(3) p. 291-313 Doi: https://doi.org/10.1080/17442501003629554

Journal of Applied Statistics 37(6) p. 893-909 Doi: https://doi.org/10.1080/02664760902914490

Preprint series (Universitetet i Oslo. Matematisk institutt)

AIP Conference Proceedings 1281 p. 531-534 Doi: https://doi.org/10.1063/1.3498530

Alternative Investments and Strategies p. 93-122

Energy Economics 32(5) p. 1034-1043 Doi: https://doi.org/10.1016/j.eneco.2010.01.005

Energy Journal 31(2) p. 53-86

We analyze the daily returns of Nordic electricity swaps and identify significant risk premia in the short end of the market. On average, long positions in this part of the swap market yield negative returns. The daily returns are distinctively non-normal in terms of tail-fatness, but we find little evidence of asymmetry. We investigate if the flexible four-parameter class of normal inverse Gaussian (NIG) distributions can capture the observed stylized facts and find that this class of distributions offers a remarkably improved fit relative to the normal distribution. We also compare the fit with that of the four-parameter class of stable distributions; the NIG law outperforms the stable law in the vast majority of cases. Thus, the NIG family of distributions, which allows for stochastic dynamics in terms of Levy processes that are suitable for pricing derivatives and Value-at-Risk measurements, is a serious candidate for modeling term structure dynamics in the Nordic electricity market.

International Journal of Theoretical and Applied Finance 12(1) p. 63-82 Doi: https://doi.org/10.1142/S0219024909005117

Journal of Energy Markets 2(3) p. 111-140

Energy Economics 31(1) p. 16-24

Daily average wind speeds are dynamically modelled by a continuous-time autoregressive model with seasonal mean and volatility. Futures prices based on an index of aggregated wind speeds are derived, and it is shown that the Samuelson effect breaks down. The volatility of these futures will decrease when approaching maturity, an effect which is explained by the memory in higher-order autoregressive models. (C) 2008 Elsevier B.V. All rights reserved.

Stochastic Analysis and Applications 27(5) p. 875-896 Doi: https://doi.org/10.1080/07362990903136413

Preprint series (Universitetet i Oslo. Matematisk institutt)

International Journal of Theoretical and Applied Finance 12(4) p. 491-506

Energy Economics 31(1) p. 16-24 Doi: https://doi.org/10.1016/j.eneco.2008.09.009

Preprint series (Universitetet i Oslo. Matematisk institutt)

The main objective of this work is to construct optimal temperature futures from available market-traded contracts to hedge spatial risk. Temperature dynamics are modelled by a stochastic differential equation with spatial dependence. Optimal positions in market-traded futures minimizing the variance are calculated. Examples with numerical simulations based on a fast algorithm for the generation of random fields are presented.

Journal of Banking & Finance 32(10) p. 2006-2021 Doi: https://doi.org/10.1016/j.jbankfin.2007.12.022

International Journal of Theoretical and Applied Finance

Journal of Banking & Finance Journal of Banking &(Volum 32 (10)) p. 2006-2021

World Scientific

Energy Economics 30(3) p. 1116-1157 Doi: https://doi.org/10.1016/j.eneco.2007.06.005

World Scientific

Energy Economics 30(3) p. 1116-1157

We discuss the modeling of electricity contracts traded in many deregulated power markets. These forward/futures type contracts deliver (either physically or financially) electricity over a specified time period, and is frequently referred to as swaps since they in effect represent an exchange of fixed for floating electricity price. We propose to use the Heath-Jarrow-Morton approach to model swap prices since the notion of a spot price is not easily defined in these markets. For general stochastic dynamical models, we connect the spot price, the instantaneous-delivery forward price and the swap price, and analyze two different ways to apply the Heath-Jarrow-Morton approach to swap pricing: Either one specifies a dynamics for the non-existing instantaneous-delivery forwards and derives the implied swap dynamics, or one models directly on the swaps. The former is shown to lead to quite complicated stochastic models for the swap price, even when the forward dynamics is simple. The latter has some theoretical problems due to a no-arbitrage condition that has to be satisfied for swaps with overlapping delivery periods. To overcome this problem, a practical modeling approach is analyzed. The market is supposed only to consist of non-overlapping swaps, and these are modelled directly. A thorough empirical study is performed using data collected from Nord Pool. Our investigations demonstrate that it is possible to state reasonable models for the swap price dynamics which is analytically tractable for risk management and option pricing purposes, however, this is an area of further research. (c) 2007 Elsevier B.V. All rights reserved.

Applied Mathematical Finance 14(2) p. 153-169 Doi: https://doi.org/10.1080/13504860600725031

Journal of Derivatives 15(1) p. 52-66

Journal of Applied Statistics 34(7) p. 823-841 Doi: https://doi.org/10.1080/02664760701511398

Applied Mathematical Finance 14(2) p. 153-169

Quantitative finance (Print) 7(5) p. 553-561 Doi: https://doi.org/10.1080/14697680601155334

Applied Mathematical Finance 14(4) p. 347-363 Doi: https://doi.org/10.1080/13504860601170609

Springer

Scandinavian Journal of Statistics 34 p. 746-767 Doi: https://doi.org/10.1111/j.1467-9469.2007.00564.x

Quantitative finance (Print) 7(5) p. 553-561

We propose an Ornstein-Uhlenbeck process with seasonal volatility to model the time dynamics of daily average temperatures. The model is fitted to approximately 45 years of daily observations recorded in Stockholm, one of the European cities for which there is a trade in weather futures and options on the Chicago Mercantile Exchange. Explicit pricing dynamics for futures contracts written on the number of heating/cooling degree-days (so-called HDD/CDD futures) and the cumulative average daily temperature (so-called CAT futures) are calculated, along with a discussion on how to evaluate call and put options with these futures as underlying.

Scandinavian Journal of Statistics 34(4) p. 746-767

Journal of Derivatives 15(1) p. 52-66

Applied Mathematical Finance 14(4) p. 347-363

International Journal of Theoretical and Applied Finance 9(6) p. 843-867

Studies in Nonlinear Dynamics & Econometrics 10(3)

Stochastics and Stochastics Reports 77(2) p. 109-137

Stochastic Analysis and Applications 23

Finance and Stochastics 9(4) p. 563-575 Doi: https://doi.org/10.1007/s00780-005-0161-z

Applied Mathematical Finance 12(1) p. 53-85

Interfaces and free boundaries (Print) 6 p. 379-404

Applied Mathematics and Optimization 49 p. 27-41

International Journal of Theoretical and Applied Finance 7(2) p. 177-192

Stochastics and Stochastics Reports 76(3) p. 191-211

Mathematical Finance 13(2) p. 215-244

International Journal of Theoretical and Applied Finance 6(8) p. 865-884

International Journal of Theoretical and Applied Finance 6(8) p. 865-884

Applied Mathematical Finance 10(4) p. 303-324

Applied Mathematical Finance 10(4) p. 325-336 Doi: https://doi.org/10.1080/1350486032000160777

Arbitrage theory is used to price forward (futures) contracts in energy markets, where the underlying assets are non‐tradeable. The method is based on the so‐called ‘fitting of the yield curve’ technique from interest rate theory. The spot price dynamics of Schwartz is generalized to multidimensional correlated stochastic processes with Wiener and Lévy noise. Findings are illustrated with examples from oil and electricity markets.

Finance and Stochastics 7(3) p. 277-298

Stochastic Analysis and Applications 20(6) p. 1191-1223

Finance and Stochastics 7(3) p. 277-298

Mathematical Finance 13(2) p. 215-244

Mathematical Finance 13(1) p. 55-72 Doi: https://doi.org/10.1111/1467-9965.t01-1-00005

In a market driven by a Lévy martingale, we consider a claim ;. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ;: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-Haussmann-Ocone theorem.

Stochastics and Stochastics Reports 74 p. 517-569

Journal of the London Mathematical Society 66(2) p. 1-13

Stochastics and Stochastics Reports 74(3-4) p. 517-569

International Journal of Theoretical and Applied Finance 4(5) p. 711-732

Finance and Stochastics 5(4) p. 447-467

Finance and Stochastics 5(3) p. 275-303

Stochastic Analysis and Applications 19(3) p. 329-341

International Journal of Theoretical and Applied Finance 4(5) p. 711-731

Mathematical Geology 33(6) p. 719-744 Doi: https://doi.org/10.1023/1011078716252

Advances in Applied Probability 32 p. 779-799

Potential Analysis 12(4) p. 385-401

Potential Analysis 12(4) p. 385-401

Statistics and Probability Letters 49 p. 345-354 Doi: https://doi.org/10.1016/S0167-7152(00)00067-5

The Metropolis Adjusted Langevin Algorithm (MALA) samples from complex multivariate densities π. The proposal density is based on a discretized version of a Langevin diffusion, and is well defined only for continuously differentiable densities π. We propose a modified MALA algorithm when this condition is not fulfilled or when π has very rapid variations. The algorithm is illustrated on the Strauss model, for which two different classes of smoothing are proposed. In these examples smoothing gives advantages in terms of reduced asymptotic variance.

Infinite Dimensional Analysis Quantum Probability and Related Topics 2(3) p. 381-396

Ukjent p. 333-338

Letters in Mathematical Physics 43 p. 267-278

Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications 51 p. 215-242

Stochastics and Stochastics Reports 63 p. 313-326

Potential Analysis 8 p. 179-193

Potential Analysis 6 p. 127-148

Journal of Functional Analysis 146(2) p. 382-415

Stochastics and Stochastics Reports 58 p. 349-367

Ukjent

Ukjent

Stochastics and Stochastics Reports 1994(51)

Stochastics and Stochastics Reports 51 p. 293-299

Stochastics and Stochastics Reports 51 p. 195-216

Ukjent 8