Journal of Probability and Statistics

Volume 2015, Article ID 626020, 17 pages

http://dx.doi.org/10.1155/2015/626020

## An Ambit Stochastic Approach to Pricing Electricity Forward Contracts: The Case of the German Energy Market

Department of Computer Science, University of Verona, Strada le Grazie 15, 37134 Verona, Italy

Received 30 June 2015; Revised 27 September 2015; Accepted 30 September 2015

Academic Editor: Ricardas Zitikis

Copyright © 2015 Luca Di Persio and Isacco Perin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose an ambit stochastic model to study the electricity forward prices. We provide a detailed analysis of the probabilistic properties of such model, discussing the related martingale conditions and deriving concrete implementation of it for the related underlying spot price. The latter is obtained from the forward model through a limiting argument. Furthermore, we show, also providing a concrete example, that a proper specification of these models is able to effectively forecast prices of forward contracts written on the European Energy Exchange (EEX) AG, or German Energy Exchange, market.

#### 1. Introduction

In the last two decades, energy markets have been liberalised in many areas in the world and this has led to the creation of completely new markets as in the case of the Nordic Nord Pool market, the European Energy Exchange (EEX) market, the Italian GME, where GME stands for* Gestore dei Mercati Energetici*, market, and so forth. The latter phenomenon has been underlined by an increasing interest in trading within such commodity frameworks, particularly with respect to electricity, oil, gas, and coal exchange. There is no doubt that such markets will play a vital role in the future given the constant expansion of global demand for energy. From the financial point of view, standard products traded on energy markets are spot prices, forward and futures contracts, and options written on them.

In the present work, we focus our attention on economics aspect linked to the electricity commodity. Our choice is mainly due to the fact that electricity cannot be stored; therefore, it has some peculiar characteristics if compared to other commodities like oil or gas. In particular, as soon as electricity has been produced, it has to be delivered to the grid. This leads to very atypical price patterns, such as large price spikes, strong short-term volatility, and negative electricity prices for short, but not zero, periods of time. The latter is the case if the supply considerably exceeds the demand. A previous type of price behaviours suggests both practitioners and academicians to develop* ad hoc* techniques for this particular framework, hence trying to get finer approaches to electricity trading than for traditional financial markets.

Recent literature has studied, in particular, a class of stochastic models for the electricity price dynamics based on* ambit processes* and* ambit fields* (see, e.g., [1, 2]). Ambit processes are defined as stochastic integrals with respect to a multivariate random measure, where the integrand is given by a product of a deterministic* kernel function* and a* stochastic volatility field* and the integration is carried out over a specific region in space-time, called* ambit set*. This kind of stochastic process was first developed in the study of turbulence, but, because of their flexible structure, they have been applied in a heterogeneous set of areas (see, e.g., [3, 4]) and, between them, in the financial framework to model dynamic processes (see, e.g., [5]).

The wide applicability of ambit processes is due to their flexibility and analytical tractability. As an example, if we consider the energy markets, ambit processes easily incorporate* leptokurtic behaviour in returns*, stochastic volatility,* leverage effects*, and the observed* Samuelson effect* in the volatility, namely, the fact that when the time to maturity approaches zero, the volatility of the forward price increases and converges to the volatility of the underlying spot price, provided the forward price converges to the spot price.

In what follows, we develop a class of models for electricity forward prices which will be based on ambit processes and ambit fields. In addition, we show that a proper specification of these models efficiently forecasts the price of the German* monthly peak forward contracts* within the EEX market scenario.

Although many stylized facts concerning energy markets can be incorporated in an ambit framework, one may question whether ambit processes are not in fact too general to be a good building block for financial models. In fact, we always have to find the right tradeoff between flexibility, effectiveness, and concrete application of the proposed model, particularly in the financial framework where decisions have often to be taken rather quickly. Concerning the latter point of view, one has to consider that ambit processes, at least in their most general definition, do not satisfy the martingale property, a fact that can be considered as a particularly relevant drawback within the* traditional* approach to financial modeling. Nevertheless, we underline that the martingale framework can be recovered assuming appropriate conditions to hold, paying the price to work with a restricted class of ambit processes. Anyway, it is not obvious that we should adopt the martingale framework if our aim is to model electricity forward contracts. In particular, lines of empirical evidence show that electricity prices are not martingales even at the spot price level, but this does not imply any arbitrage because electricity cannot be stored. The last point is of particular relevance since it ensures that our analysis can be carried out using ambit models, without being worried about arbitrage opportunities. We would also like to mention recent results (see, e.g., [6, 7]), which show that subclasses of nonsemimartingales can be used to model financial assets without necessarily giving rise to arbitrage opportunities in markets exhibiting market frictions, such as, for example, transaction costs.

The paper is organized as follows. In Section 2, we briefly review the key definitions and concepts on the basis of both ambit fields analysis and ambit processes; in Section 3, we introduce the modeling framework for electricity forward markets, and we study its key properties highlighting the most relevant model specifications; then, we present suitable conditions under which the ambit processes adopted in the presented models are martingales and we also derive a model for the underlying spot price exploiting the forward model through a limiting argument; finally, in Section 4, we concretely apply the approach proposed in Section 3 to forecast the price of a particular forward contract written within the German EEX market.

#### 2. Ambit Stochastic Approach

The concepts of ambit fields and ambit processes form the building blocks of the model for the electricity forward price that we will present in Section 3. Therefore, in this section, we briefly review the ideas behind ambit stochastic approaches, as they have been introduced by Barndorff-Nielsen and Schmiegel in [3] and then further discussed in [4, 8]; we also refer to [9] and the references therein for further details.

##### 2.1. The General Framework

The general background setting for ambit process and ambit field consists of a stochastic field , considered in space-time domain , a curve , which also belongs to , and the values of the field along such a curve; namely, ; then it follows that is a stochastic process. In most applications, the space is chosen to be , for , or . Moreover, the stochastic field is supposed to be generated by innovations in space-time and the values are assumed to depend only on those innovations that occur prior to or at time . More precisely, at each point , only the innovations in some subset of , where , determine the value of . We refer to as the ambit set associated with and to and as an ambit field and an ambit process, respectively.

Without further assumptions, nothing interesting can be said about the field and the process ; therefore, we have to specify a suitable mathematical structure in the next subsection; nevertheless, without being rigorous, is defined as a stochastic integral plus a smooth term and the integrand in the stochastic integral consists of a deterministic kernel times a positive random variate which is taken to embody the volatility or* intermittency* of the field . We also underline that the volatility field, denoted by , is given as an ambit field and a central issue is what can be learned about from observation of or .

##### 2.2. Ambit Fields and Ambit Processes

In ambit stochastic approach, an essential role is played by the* Lévy bases*, which we give the definition of in what follows mainly following the extensive and detailed discussion provided by [10, 11]. In addition, the classical concept of subordination of Lévy processes is generalized to subordination of Lévy bases or* extended subordination*, by introducing the so-called* meta-times* in [10].

Let be a* Borel set* in and let be the* Borel **-algebra* on . We denote by the sub--algebra of consisting of the elements of bounded with respect to the* Lebesgue measure *; that is,

*Definition 1 (Lévy basis and homogeneous Lévy basis). *A family of random vectors in is an *-valued Lévy basis on * if the following three properties are satisfied: (i)The law of is* infinitely divisible* for all .(ii)For any sequence of disjoint elements of , one has that the random variables are* independent*.(iii) For any sequence of disjoint elements of which satisfy , one has that where the convergence on the right hand side is intended almost surely.
If a Lévy basis has a* stationary* law, then it is called a* homogeneous Lévy basis*.

*Remark 2 (about Lévy bases). *Properties (ii) and (iii) define an* independently scattered random measure*; hence, the class of Lévy bases is a subclass of independently scattered random measures.

We are now in position to state the definitions of ambit field and ambit process.

*Definition 3 (ambit field and ambit process). *An* ambit field * is a random field such that provided the integrals exist over ambit sets and , where is a constant, and are deterministic functions, and are stochastic fields, and is a Lévy basis. Moreover, we define the corresponding* ambit process * as the stochastic process, defined on a given probability space , given by the evaluation of the field along a curve ; namely, where and .

*Remark 4 (particular cases of ambit processes). *Let us specify a particular class of ambit fields that is of particular interest in many applications, which is specified as follows: where the ambit sets and are taken to be homogeneous and* nonanticipative*; that is, is of the form , where only involves negative time coordinates, similar to . In fact, this specific framework allows modeling stationary random fields, since in (4) turns out to be stationary when the fields and are stationary and is independent of the driving Lévy basis . Furthermore, other potentially important features of a random field, such as, for example, isotropy or skewness, can be easily modeled with (4) via an appropriate choice of the deterministic kernels and and the stochastic fields and .

*Remark 5 (stochastic integration). *In order to build relevant models based on ambit fields, we need a suitable integration theory. In [12], Rajput and Rosinski have proposed an integration theory for appropriate deterministic integrands with respect to Lévy bases. However, in the context of ambit fields, we need to integrate stochastic integrands. To solve the latter issue, we can count on an alternative integration concept due to Walsh. Such an idea works even if the integrand depends on the particular Lévy basis chosen (see [13]). Unfortunately, the Walsh approach can be used to define ambit fields only when the driving Lévy basis is square-integrable. In the recent work [14], the ideas of Walsh, Rajput, and Rosinski have been combined to propose an integration concept for random integrands and general Lévy bases. Such an approach relies on [15], which is an earlier work by Bichteler and Jacod. We refer to [16] as a complete reference for the comparison of various integration concepts.

#### 3. Modeling Electricity Forward Price by Ambit Fields

Due to their structure, we can use ambit fields to capture many characteristics of energy markets such as strong seasonal patterns, very pronounced* volatility clusters*, high spikes/jumps, and the Samuelson effect. For these reasons, we review a general model, presented in [5], for electricity forward prices which is based on ambit fields.

##### 3.1. A Model for Forward Prices

Let be a probability space, a Lévy basis, a stochastic field for , and . In addition, we consider the filtration defined by where and where denotes the -null sets of . Note that is right continuous by construction. Using an ambit field, we model the forward price as a stochastic process for each fixed ; namely, where denotes the current time, denotes the volatility of the forward market as a whole, is an ambit set, and is the kernel function. A complete model description is given specifying an ambit set , a kernel function , a stochastic volatility field , and a Lévy basis . Latter choices are often based on qualitative analysis of the markets we want to study as well as concerning the analytical tractability of the resulting model. However, in general, we would like to be well defined in the sense of Walsh (see [13]) and stationary in time. In order to have integrability in the sense of Walsh, we assume that the following conditions hold: (i)The Lévy basis is square-integrable and has zero-mean.(ii) The stochastic volatility field is adapted to the filtration and independent of the Lévy basis .(iii) The kernel function is nonnegative and such that for .(iv) The convolution is integrable with respect to .

Moreover, in order to ensure that is stationary in time, we assume that the following conditions hold:(v) The stochastic volatility field is stationary in .(vi) The ambit set is such that where .

Throughout this section, we suppose that is defined as in (7) and satisfies conditions (i)–(vi).

*Remark 6 (when is a function of ). *The aforementioned conditions on ensure having stationarity in time; however, as soon as we replace by a function of , that is, , is not in general stationary any more. For instance, if we want to reflect the fact that the forward price depends also on the time to maturity, let us indicate it by ; placing , we lose the stationarity of , with respect to time.

*Remark 7 (about ). *If we consider (7), we have that the forward price at time is given by where in the right hand side in (9) we perform integration with respect to a random measure. It follows that the forward price at time 0 is the* value* of the random variable given in (9), contrary to most other models where is considered as a deterministic quantity equal to the observed price.

Under previous conditions (i)–(vi), the ambit field specification provided by (7) is highly analytically tractable and its* conditional cumulant function* is given by the following proposition.

Proposition 8 (conditional cumulant function of ). *Let be a homogeneous Lévy basis; hence, its control measure is proportional to the Lebesgue measure and one can assume, without loss of generality, that the proportionality constant equals 1. Then, the cumulant function of , conditioned to , is given by where is the Lévy seed associated with ; see Subsection 3.2 of [5] for a rigorous definition of Lévy seed associated with a Lévy basis. Moreover, if is Gaussian, then *

*Proof. *The proposition is an immediate consequence of Proposition in [12].

##### 3.2. Examples of Model Specifications

The forward model based on an ambit field has a very general structure; consequently, it is possible to concretely specify exploiting several choices a peculiarity which is highly useful in many applications. There are three components of the model to specify, namely, the Lévy basis , the kernel function , and the volatility . In the following, we present some examples of specification for the* parameters * and , while we discuss volatility modulation in the next subsection.

###### 3.2.1. Specification of the Lévy Basis

Our model is based on the zero-mean and square-integrable assumptions; hence, we can choose any infinitely divisible distribution satisfying these two hypotheses. A natural choice is to consider a Gaussian Lévy basis which implies a smooth random field. Alternatively, we can take a* normal inverse Gaussian* Lévy basis. We could need to relax the zero-mean assumption for the Lévy basis, in order to choose, for example, a* Gamma* or an* inverse Gaussian* Lévy basis.

###### 3.2.2. Specification of the Kernel Function

The kernel function plays a key role in our setting since(i)the kernel function completely determines the space-time* autocorrelation structure* of a zero-mean ambit field (see Section 3.4);(ii)the kernel function characterises the Samuelson effect as we will see in Theorem 28;(iii)the kernel function determines whether the forward price is a martingale or not (see Theorem 19 and Corollary 21).

Recall that the kernel is a function of the three variables , and , where refers to the temporal component and refer to the spatial dimension. A rather natural approach for specifying a kernel function is to assume a factorisation property that we will present by two different realizations each of which has its own relevance in a specific framework. First, we study factorisation into a temporal and a spatial kernel. In particular, we assume that the kernel function factorises as follows: for suitable functions and representing the temporal part and the spatial part, respectively. Equation (12) allows us to study specifications of and separately. In empirical work, it will be particularly interesting to focus in more detail on the question of how to model the spatial kernel function , which determines the correlation between various forward contracts. Even if we are allowed to choose similar or identical types of functions for both the temporal and the spatial dimension, nevertheless, we will see in Section 3.6 that particular choices for lead to a rather natural relation between forward and implied spot prices. Let us briefly study an example which is included in the presented modeling framework.

*Example 9 (exponential kernel function). *Let be a homogeneous symmetric normal inverse Gaussian Lévy basis ; namely, it has Lévy seed with density where denotes the modified Bessel function of the second kind, while , . Then, If the kernel function factorises as in (12), and , then and the latter integral can be computed explicitly for suitable kernel functions, for example, for and for . Then, and, taking as in Remark 6, we have Therefore, the cumulant function is given by for , and the integral in (18) can be easily computed.

Alternative factorisation of the kernel function is given as follows: for suitable functions and . Although the latter does not look very natural at first sight, it is very important since it naturally includes cases where is removed, for example, choosing and ; in fact, in this case, we have . The latter case is crucial when we want to formulate martingale conditions for the forward price (see Section 3.5). Let us consider two particular realizations for (19).

*Example 10 (exponential ). *Miming the Ornstein-Uhlenbeck case, for , we can set hence taking from factorisation (19).

*Example 11 (hyperbolic kernel function). *In [17], the authors propose a model for electricity forward prices with a Lévy basis obtained from a standard Brownian motion and kernel function given by where , , and are positive constants. The authors argued that the Samuelson effect in electricity markets is much steeper than in other commodity markets, defending the choice of a hyperbolic function rather than an exponential one. Note that we can obtain this kernel function from factorisation (19) with and when .

##### 3.3. Space-Time Stochastic Volatility

Although a variety of purely temporal stochastic volatility models can be found in the literature, suitable space-time stochastic volatility models still need to be developed. Until now, volatility modulation within the framework of an ambit field can be achieved by four complementary methods: by introducing a stochastic integrand, by extended subordination, by* probability mixing*, or by* Lévy mixing*.

###### 3.3.1. Volatility Modulation via a Stochastic Integrand

This method has already been suggested in the initial definition of ambit fields (see (2)). In that case, there are essentially two approaches that can be used to construct a stochastic volatility field. In fact, we can specify the stochastic field directly as a random field, for example, as another ambit field, or we can start with a purely temporal or purely spatial stochastic volatility process, respectively, and then suitably generalise it to a random field. In the following, we will present examples of both types of construction. First, we focus on the modeling approach where we directly specify a random field for the volatility field. A natural starting point for modeling the volatility is to combine kernel smoothing of a homogeneous Lévy basis with (nonlinear) transformation to ensure positivity. For instance, let where is a homogeneous Lévy basis independent of , is an integrable kernel function satisfying for , and is a continuous, nonnegative function. Clearly, the kernel function determines the space-time autocorrelation structure of the volatility field .

*Remark 12 (about stochastic volatility). *Note that defined by (22) is stationary in the temporal dimension and if we have for some suitable function , then the stochastic volatility is both stationary in time and homogeneous in space.

*Example 13 ( distribution). *Consider stochastic volatility as in (22), let be a standard normal Lévy basis, and let . Then, is clearly positive and has a pointwise distribution with one degree of freedom.

In what follows, we show how to construct a stochastic volatility field by extending a purely temporal stochastic process to a random field. Note that our objective is to construct a stochastic volatility field which is stationary, at least in the temporal dimension. There are many possibilities to reach the latter goal, but in the following we decide to focus our attention on a particularly relevant one, namely, the* Ornstein-Uhlenbeck type volatility field* (). The choice of an Ornstein-Uhlenbeck process as the stationary base component is motivated by its analytical tractability and also because it tends to perform well in practice, at least in the purely temporal case. Without loss of generality, we restrict our attention to the case ; that is, we consider one spatial dimension. In order to present this type of volatility fields, we have to define the* Lévy supraprocesses*.

*Definition 14 (Lévy supraprocess generated by ). *Suppose that is a stationary, positive, and infinitely divisible process on . Let be a family of stationary processes such that has independent increments and for each the cumulant function of is given by where with for an arbitrary signed measure on . Then, for any fixed , is a Lévy process called* Lévy supraprocess generated by *.

*Definition 15 (). *Suppose that is a positive Ornstein-Uhlenbeck process with rate parameter and such that it is generated by a Lévy subordinator ; that is, A stochastic volatility field on is an if it is defined as where is the spatial rate parameter and is a family of Lévy supraprocesses.

Note that, in the above construction, we start from an Ornstein-Uhlenbeck process in time, , and the spatial structure is then introduced through two steps. First, we multiply the process with an exponential weight dependent on a spatial component, which reaches its maximum for , decaying as it moves away from the purely temporal case. Second, we add an integral which is similar to an Ornstein-Uhlenbeck process in the spatial variable . However, although the stochastic volatility field is stationary in time, note here that the integration starts from rather than from ; hence, the resulting component is not stationary in the spatial variable , even if this could be changed without affecting the structure if such a property is required in a particular application. Finally, note that the process is in general not predictable, which is disadvantageous given that we want to construct stochastic integrals in the sense of Walsh. However, if we choose as an Ornstein-Uhlenbeck process, we obtain a predictable stochastic volatility process.

*Example 16 (covariance function of when is an Ornstein-Uhlenbeck process). *Suppose that is an Ornstein-Uhlenbeck process with rate parameter and generated by a Lévy process . Then, using the notations in Definition 15, we have Furthermore, if and , then for fixed and the covariance function of is

###### 3.3.2. Volatility Modulation via Extended Subordination

An alternative way of volatility modulation is by means of extended subordination. In this modulation, the volatility is incorporated in the modeling through a meta-time associated with an absolutely continuous measure on ; namely, where A natural choice of meta-time is , where is the mapping given by , while The above construction of a meta-time in a space-time model is very general, since it allows defining a variety of models for the random field , leading to new model specifications. For instance, one could model by an or any other model discussed previously. It follows that this way to modulate the volatility is more general than the previous one.

###### 3.3.3. Volatility Modulation via Probability Mixing and Lévy Mixing

Volatility modulation can also be obtained through a probability mixing approach which is realized defining new distributions by randomising a parameter from a given parametric distribution, as in the following classical example.

*Example 17 (normal variance-mean mixture distribution). *Consider as base modelassuming that the corresponding Lévy basis is homogeneous and Gaussian; namely, the corresponding Lévy seed is given by with and . Then, we use probability mixing supposing that is random. Hence, the conditional law of the Lévy seed is given by . Due to the scaling property of the Gaussian distribution, such a model can be represented as in (2); therefore, in this particular case, probability mixing and volatility modulation via a stationary stochastic integrand produce the same result. Such a construction falls into the class of* normal variance-mean mixtures*.

In this context, it is important to note that probability mixing does not generally lead to infinitely divisible distributions. In order to overcome this problem, we present the concept of Lévy mixing. Let be a* factorisable* Lévy basis on with* characteristic quadruplet * (see Subsection of [10] for a rigorous definition of characteristic quadruplet of a Lévy basis and factorisable Lévy basis). Suppose that depends on the parameter , where denotes the space of the parameter . Then, the* Lévy measure* of is given by . Let be a measure on and define where we assume that then there exists a Lévy basis which has as its Lévy measure. We define as the* Lévy basis obtained by Lévy mixing of ** with the measure *. Let us study a concrete example of Lévy mixing.

*Example 18 (superposition of Ornstein-Uhlenbeck type process). *Let us consider the example of superposition of Ornstein-Uhlenbeck type process. Let be a subordinator without drift and with Lévy measure , and consider an Ornstein-Uhlenbeck process for . By a simple calculation, we obtain the following expression for the cumulant function of : for , where is a mixture of with the Lebesgue measure. Lévy mixing can be carried out with respect to the parameter ; namely, where is a measure on satisfyingTaking to be the Lévy basis with Lévy measure and defining the superposition of Ornstein-Uhlenbeck type process with respect to by where , we have that the cumulant function of is given by hence showing that superposition of Ornstein-Uhlenbeck type process can be obtained from an Ornstein-Uhlenbeck process through Lévy mixing.

##### 3.4. Autocorrelation and Cross-Correlation

We would like to underline that the proposed approach allows modeling the entire forward curve. Hence, it is interesting to study the correlation structure between various forward contracts implied by the presented modeling framework. Starting from the definition of covariance function of an ambit field (see Section of [5]), we deduce the* autocorrelation function* of a forward contract . More precisely, assuming that the Lévy basis is homogeneous, and taking , , , and , , we have that the ambit set defined in (8) can be expressed as ; then we define the autocorrelation function as Note that the correlation structure is principally determined by three factors, which are the intersection of the corresponding ambit sets, the kernel function, and the autocorrelation structure of the stochastic volatility field. Furthermore, we could model various commodity forward contracts, such as electricity and natural gas futures, simultaneously. It follows that we can specify different ambit sets, kernel functions, stochastic volatility fields, and Lévy bases in order to obtain a rather flexible correlation structure. In such a situation, it becomes even more clear how flexible the ambit setting is. The details of these multivariate extensions can be found in Section of [5].

##### 3.5. Martingale Condition

In what follows, we give sufficient conditions for the presented model to be a martingale, even if such property may be too restrictive if we are interested in modeling electricity forward contracts. In fact, there are at least two arguments that support the choice of more general classes of stochastic processes than martingales. First, in the energy context, it might not be as crucial that is a martingale as it is in the context of modeling interest rates. In fact, in many emerging electricity markets, a seller may not be able to find any buyer to get rid of a forward; oppositely, it may also happen that there are no sellers when one wants to buy a contract. Hence, the illiquidity prevents possible arbitrage opportunities from being exercised and we need a martingale condition only in the absence of arbitrage setting (see [18]). Second, independently of the particular structure of energy markets, the recent literature in mathematical finance (see, e.g., [6, 18]) has highlighted that some classes of nonsemimartingales, in particular stochastic processes with conditional full support, do not necessarily give rise to arbitrage opportunities reflecting more realistic characteristics of the market. In the null-spatial setting, Pakkanen proves that* Brownian semistationary processes* have in fact conditional full support (see [7]). In future research, it will be interesting to study extensions of the latter result to more general ambit fields. However, the question of establishing martingale conditions for ambit fields is interesting in order to point out a subset of the class we are considering, which can be exploited to model general forward prices, hence without restricting ourselves to the electricity case. In what follows, we formulate the martingale condition for ambit fields more general than those defined by (7), where the ambit set is chosen as in (8), also showing that such condition simplifies in the modeling framework described in Section 3.1.

Theorem 19 (martingale condition for general ambit fields). *Let for some and, for fixed , let where is defined as in (8), is a deterministic kernel function, is an adapted, nonnegative random field, and is a Lévy basis satisfying both (i) and (iv). Further, and are assumed to be independent. Then, is a martingale with respect to defined as in (5) if and only if, for all , it holds for some deterministic kernel function .*

*Proof. *Measurability and integrability properties are trivially verified according to what we have seen so far; therefore, we just have to show that for all . Note that, for , we have that . Using the independence property of and and the fact that is a zero-mean process, we find where Without loss of generality, we can assume that . Since is a Lévy basis with zero-mean, we know that and from the* Itô isometry* (see Theorem in [13]) we therefore get that where is the* covariance measure* induced by ; see, for example, [13], for a rigorous definition of covariance measure induced by a Lévy basis. Thus, in order to obtain , we need that, for all , Note that there is only one class of functions which satisfy the above condition, namely, functions of the form for all , and for some deterministic kernel function .

*Remark 20 (Lévy basis with mean different from zero). *If we would like to work with Lévy bases which do not have zero-mean, then the martingale condition has to be extended by an additional drift condition.

Corollary 21 (martingale condition for ). *One gets that is a martingale with respect to if and only if, for all , , the following holds: for some deterministic kernel function ; hence, the latter is a special case of the factorisation given in (19).*

*Proof. *This result follows directly from Theorem 19 considering .

Clearly, the martingale condition is rather strong and it is necessary to check whether there are actually any relevant cases, not excluded by condition (52). Therefore, let us study some examples.

*Example 22 (exponential kernel function). *If we choose to be of the form for some , then the above martingale condition is clearly satisfied. Note that this choice of the kernel function belongs to both factorisation classes (12) and (19).

*Example 23 (factorisation (19)). *Let us consider kernel functions which factorise as in (19); that is, then the choice of the function does not have any impact concerning whether the ambit field is a martingale; indeed such a property is solely determined by the function . It follows that every choice of the form satisfies the martingale condition.

##### 3.6. Derivation of a Spot Model from the Forward One

Under the absence of arbitrage assumption, the forward price of a contract tends to be equal to the spot price of the underlying as the time to maturity tends to zero. In this subsection, we investigate in detail the nature of the spot price model implied by the above modeling framework for the forward price. Note that this study should be understood as a theoretical exercise, since we do not observe convergence of the electricity forward price to the electricity spot price in real markets.

Let be the forward price and let be the spot price of the underlying. When the time to maturity tends to 0, we would have that tends to ; hence, we define the spot price as

Lemma 24 (convergence of to ). *Suppose thatthen in as time to maturity tends to zero, where denotes the risk neutral probability measure.*

*Proof. *The result follows immediately by applying the Itô isometry, as proved by Theorem in [13].

The previous lemma gives us that the forward price will tend continuously in variance to the spot price as time to maturity decreases to zero. Note also that whenever is a stationary field, the condition in the above lemma is translated to convergence of to in .

Using Proposition 8 for the forward price, we can easily derive the conditional cumulant function for the implied spot price.

Proposition 25 (conditional cumulant function of ). *Let be a homogeneous Lévy basis. Then, for defined as in (56), the cumulant function conditioned to is given by where is the Lévy seed associated with .*

*Proof. *This result is an immediate consequence of Proposition 8.

A case of some special interest is given by considering the driving Lévy basis of the ambit field to be a homogeneous Gaussian Lévy basis; in fact, in this case, we get the following result.

Corollary 26 (spot price when is a Gaussian Lévy basis). *Let be as in (56) and let be a homogeneous Gaussian Lévy basis with where is the Lévy seed associated with . Then, the cumulant function of conditioned to is given by If the kernel function can be factorized as in (12), then where implying that is equal in law to where is the standard Brownian motion.*

*Proof. *The result follows by straightforward calculations, exploiting Proposition 25 and the definition of Gaussian Lévy basis.

*Remark 27 (modeling spot price). *Note that in (63) is a Brownian semistationary process, which has been used as a model for energy spot prices in [1].

###### 3.6.1. The Samuelson Effect

Recall that the Samuelson effect describes the empirical fact that the volatility of the forward price increasingly converges to the volatility of the underlying spot price when the time to maturity tends to zero. Here, we show that this effect is naturally included in our modeling framework.

Theorem 28 (nondecreasing of the conditional variance of ). *Assume that the function is monotonically nondecreasing in for every . The conditional variance of the forward price given by is monotonically nondecreasing in , for , where the term is a suitable constant; see Section A.2 of [5] for a rigorous definition of .*

*Proof. *For , we have and, taking , then since is nondecreasing in .

*Remark 29 (Samuelson effect). *The conditional variance of the spot price is given by and from the above theorem it follows that for . Since we can consider as a monotonically increasing sequence bounded from below by , there exists the limit . Under the condition in Lemma 24, this limit is given by ; hence, it equals the conditional variance of spot price, which implies that we have a Samuelson effect.

##### 3.7. Simulation Algorithm

In what follows, we present a possible algorithm, taken from [5], to simulate an ambit field of the following type: A problem that we immediately note is that we perform integration over an ambit set , which is unbounded; hence, we have to truncate the ambit set before performing any computations. Therefore, we introduce the ambit field where and . Obviously, if and , then converges to .

Taking , we construct a grid for by dividing the interval into equidistant intervals of length , where , and by dividing into equidistant intervals of length , where . Next, we simulate an ambit field through the following steps:(1)Simulate the stochastic field on the grid points in order to obtain the values of stochastic volatility for and . In case we have no stochastic volatility, we set , for all and for all .(2) Simulate random variables , where is the Lévy basis computed on for and .(3) Compute the approximated ambit field in this way: We note that the previous algorithm can be used also to simulate stochastic volatility. In the last step of the simulation algorithm, we use the definition of stochastic integral in the sense of Walsh for simple processes, which ensures the convergence of (70) to the actual stochastic integral, when .

#### 4. Empirical Study

In what follows, we exploit results and techniques developed throughout previous sections to give a detailed analysis as well as a forecast analysis concerning forward contracts within the German energy markets, namely, within the framework of the EEX. The latter ones are characterized by various types of forward contracts whose structure mainly depends on the characterizing time slots each of them takes into consideration, the duration of the contract, whether there are some lower or upper bounds in the quantity of electricity exchanged, and so forth. We focus our attention on a widely used forward contract that assures the supply of electricity at a fixed price for a month from 9 a.m. to 8 p.m., but only during working days. The latter type of contract is traded in the* over-the-counter market* and it is called a monthly peak forward contract. Trades for such contracts take place in the three months before the effective supply of requested electricity. In particular, we analyze twelve peak contracts, one for each month of the year, from April 2012 to March 2013.

##### 4.1. Model Specification and Parameters Calibration

Following the approach presented in Section 3.1, we model the forward price as an ambit process for each fixed ; namely, where denotes the current time, is an interval that depends on the peak contract considered, denotes the volatility of the forward market as a whole, is a Lévy basis, and is the kernel function. We also suppose that satisfies conditions (i)–(vi) stated in Section 3.1, and being the maturity time, we model the fact that the forward price depends also on time to maturity ; hence, we use the* spatial* variable to define a second time variable, namely, the* time to maturity* one.

There are three components of the model which we have to specify, namely, the Lévy basis , the kernel function , and the volatility . The choices we made are mainly based on qualitative analysis of the market of interest, but also taking into consideration analytical as well as numerical tractability of the resulting model and, last but not least, the empirical evidence.

###### 4.1.1. Kernel Function

Every demand in general, and electricity consumption in particular, is affected by seasonality, which is also the case for forward prices. In fact, they depend on the period of the year; indeed it is easy to recognize a cyclic behaviour for such values. In order to depict this kind of periodicity, in (71), we consider a kernel function of the following type: where is the price of the forward contract at time taken from the data, takes into account the seasonal influences that affect electricity markets, and models the average price trend over the peak contracts considered, which are relative to one year. Clearly, the periodicity acts solely on the average trend of forward contracts and this is the reason why multiplies only and not also in (72).

Exploiting related time series data and the fact that in winter the demand for electricity is greater than in summer, we choose the following bell-shaped function to model the effect of seasonality on forward prices: where , , , and , while is a constant that depends only on the length of the forecast. Short-term forecasts are less influenced by seasonality than long-term ones; therefore, the parameter takes care of such phenomenon. In particular, we choose , where is the length of the forecast, so that the periodicity function is significant only in medium- or long-term previsions. Moreover, the variable spans the different months of the year; therefore, the integration interval is different for every peak contract considered (see Table 1).