Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes

Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes

Abstract

This paper introduces the class of volatility modulated Lévy-driven Volterra () processes and their important subclass of Lévy semistationary () processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.

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0 \volume19 \issue3 2013 \firstpage803 \lastpage845 \doi10.3150/12-BEJ476 \newremarkexExample \newremarkremRemark

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Modelling energy spot prices by LSS processes

{aug}

1]\fnmsOle E. \snmBarndorff-Nielsen\thanksref1label=e1]oebn@imf.au.dk, 2]\fnmsFred Espen \snmBenth\thanksref2label=e2]fredb@math.uio.no and 3]\fnmsAlmut E. D. \snmVeraart\thanksref3label=e3]a.veraart@imperial.ac.uk\corref

energy markets \kwdforward price \kwdgeneralised hyperbolic distribution \kwdLévy semistationary process \kwdSamuelson effect \kwdspot price \kwdstochastic integration \kwdstochastic volatility \kwdvolatility modulated Lévy-driven Volterra process

1 Introduction

Energy markets have been liberalised worldwide in the last two decades. Since then we have witnessed the increasing importance of such commodity markets which organise the trade and supply of energy such as electricity, oil, gas and coal. Closely related markets include also temperature and carbon markets. There is no doubt that such markets will play a vital role in the future given that the global demand for energy is constantly increasing. The main products traded on energy markets are spot prices, futures and forward contracts and options written on them. Recently, there has been an increasing research interest in the question of how such energy prices can be modelled mathematically. In this paper, we will focus on modelling energy spot prices, which include day-ahead as well as real-time prices.

Traditional spot price models typically allow for mean-reversion to reflect the fact that spot prices are determined as equilibrium prices between supply and demand. In particular, they are commonly based on a Gaussian Ornstein–Uhlenbeck (OU) process, see Schwartz Schwartz1997 (), or more generally, on weighted sums of OU processes with different levels of mean-reversion, see, for example, Benth, Kallsen and Meyer-Brandis BenthKallsenMeyerBrandis2007 () and Klüppelberg, Meyer-Brandis and Schmidt KMBS2010 (). In such a modelling framework, the mean-reversion is modelled directly or physically, by claiming that the price change is (negatively) proportional to the current price. In this paper, we interpret the mean-reversion often found in commodity markets in a weak sense meaning that prices typically concentrate around a mean-level for demand and supply reasons. In order to account for such a weak form mean-reversion, we suggest to use a modelling framework which allows to model spot prices (after seasonal adjustment) directly in stationarity. This paper proposes to use the class of volatility modulated Lévy-driven Volterra () processes as the building block for energy spot price models. In particular, the subclass of so-called Lévy semistationary () processes turns out to be of high practical relevance. Our main innovation lies in the fact that we propose a modelling framework for energy spot prices which (1) allows to model deseasonalised energy spot prices directly in stationarity, (2) comprises stochastic volatility, (3) accounts for the possibility of jumps and spikes, (4) features great flexibility in terms of modelling the autocorrelation structure of spot prices and of describing the so-called Samuelson effect, which refers to the finding that the volatility of a forward contract typically increases towards maturity.

We show that the new class of processes is analytically tractable, and we will give a detailed account of the theoretical properties of such processes. Furthermore, we derive explicit expressions for the forward prices implied by our new spot price model. In addition, we will see that our new modelling framework encompasses many classical models such as those based on the Schwartz one-factor mean-reversion model, see Schwartz Schwartz1997 (), and the wider class of continuous-time autoregressive moving-average (CARMA) processes. In that sense, it can also be regarded as a unifying modelling approach for the most commonly used models for energy spot prices. However, the class of processes is much wider and directly allows to model the key special features of energy spot prices and, in particular, the stochastic volatility component.

The remaining part of the paper is structured as follows. We start by introducing the class of processes in Section 2. Next, we formulate both a geometric and an arithmetic spot price model class in Section 3 and describe how our new models embed many of the traditional models used in the recent literature. In Section 4, we derive the forward price dynamics of the models and consider questions like affinity of the forward price with respect to the underlying spot. Section 5 contains an empirical example, where we study electricity spot prices from the European Energy Exchange (EEX). Finally, Section 6 concludes, and the Appendix contains the proofs of the main results.

2 Preliminaries

Throughout this paper, we suppose that we have given a probability space with a filtration satisfying the ‘usual conditions,’ see Karatzas and Shreve KS (), Definition I.2.25.

2.1 The driving Lévy process

Let denote a càdlàg Lévy process with Lévy–Khinchine representation for , and

for , and the Lévy measure satisfying and . We denote the corresponding characteristic triplet by . In a next step, we extend the definition of the Lévy process to a process defined on the entire real line, by taking an independent copy of , which we denote by and we define for . Throughout the paper denotes such a two-sided Lévy process.

2.2 Volatility modulated Lévy-driven Volterra processes

The class of volatility modulated Lévy-driven Volterra () processes, introduced by Barndorff-Nielsen and Schmiegel BNSchmiegel2008 (), has the form

(1)

where is a constant, is the two-sided Lévy process defined above, are measurable deterministic functions with for , and and are càdlàg stochastic processes which are (throughout the paper) assumed to be independent of . In addition, we assume that is positive. Note that such a process generalises the class of convoluted subordinators defined in Bender and Marquardt BenderMarquardt2009 () to allow for stochastic volatility.

A very important subclass of processes is the new class of Lévy semistationary () processes: We choose two functions such that and with whenever , then an process is given by

(2)

Note that the name Lévy semistationary processes has been derived from the fact that the process  is stationary as soon as and are stationary. In the case that is a two-sided Brownian motion, we call such processes Brownian semistationary () processes, which have recently been introduced by Barndorff-Nielsen and Schmiegel BNSch09 () in the context of modelling turbulence in physics.

The class of processes can be considered as the natural analogue for (semi-) stationary processes of Lévy semimartingales (), given by

{rem*}

The class of processes can be embedded into the class of ambit fields, see Barndorf-Nilsen and Schmiegel BNSch04 (), BNSch07a (), Barndorff-Nielsen, Benth and Veraart BNBV2009 (), BNBV2009Forward ().

Also, it is possible to define and processes for singular kernel functions and , respectively; a function (or ) defined as above is said to be singular if (or ) does not exist or is not finite.

2.3 Integrability conditions

In order to simplify the exposition, we will focus on the stochastic integral in the definition of an (and of an ) process only. That is, throughout the rest of the paper, let

(3)

In this paper, we use the stochastic integration concept described in Basse-O’Connor, Graversen and Pedersen BassePedersen2010 () where a stochastic integration theory on , rather than on compact intervals as in the classical framework, is presented. Throughout the paper, we assume that the filtration is such that is a Lévy process with respect to , see Basse-O’Connor, Graversen and Pedersen BassePedersen2010 (), Section 4, for details.

Let denote the Lévy triplet of associated with a truncation function . According to Basse-O’Connor, Graversen and Pedersen BassePedersen2010 (), Corollary 4.1, for the process with is integrable with respect to if and only if is -predictable and the following conditions hold almost surely:

(4)

When we plug in , we immediately obtain the corresponding integrability conditions for the process. {ex} In the case of a Gaussian Ornstein–Uhlenbeck process, that is, when for and , then the integrability conditions above are clearly satisfied, since we have

Square integrability

For many financial applications, it is natural to restrict the attention to models where the variance is finite, and we focus therefore on Lévy processes with finite second moment. Note that the integrability conditions above do not ensure square-integrability of even if has finite second moment. But substitute the first condition in (2.3) with the stronger condition

(5)

then is square integrable. Clearly, is constant in case of stationarity. For the Lebesgue integral part, we need

(6)

According to the Cauchy–Schwarz inequality, we find

for any constant . Thus, a sufficient condition for (6) to hold is that there exists an such that

which simplifies to

(7)

in the case. Given a model for and , these conditions are simple to verify. Let us consider an example. {ex} In Example 2.3, we showed that for the kernel function and in the case of constant volatility, the conditions (2.3) are satisfied. Next, suppose that there is stochastic volatility, which is defined by the Barndorff-Nielsen and Shephard BNS2001a () stochastic volatility model, that is , for , and a subordinator . Suppose now that has cumulant function for a Lévy measure supported on the positive real axis, and that has finite expectation. In this case, we have that for all . Thus, both (5) and (6) are satisfied (the latter can be seen after using the sufficient conditions), and we find that is a square-integrable stochastic process.

3 The new model class for energy spot prices

This section presents the new modelling framework for energy spot prices, which is based on processes. As before, for ease of exposition, we will disregard the drift part in the general process for most of our analysis and rather use with

(8)

as the building block for energy spot price, see (1) for the precise definition of all components. Throughout the paper, we assume that the corresponding integrability conditions hold. We can use the process defined in (8) as the building block to define both a geometric and an arithmetic model for the energy spot price. Also, we need to account for trends and seasonal effects. Let denote a bounded and measurable deterministic seasonality and trend function.

In a geometric set up, we define the spot price by

(9)

In such a modelling framework, the deseasonalised, logarithmic spot price is given by a process. Alternatively, one can construct a spot price model which is of arithmetic type. In particular, we define the electricity spot price by

(10)

(Note that the seasonal function in the geometric and the arithmetic model is typically not the same.) For general asset price models, one usually formulates conditions which ensure that prices can only take positive values. We can easily ensure positivity of our arithmetic model by imposing that is a Lévy subordinator and that the kernel function takes only positive values.

3.1 Model properties

Possibility of modelling in stationarity

We have formulated the new spot price model in the general form based on a process to be able to account for non-stationary effects, see, for example, Burger et al. BKMS2003 (), Burger, Graeber and Schindlmayr BGS2007 (). If the empirical data analysis, however, supports the assumption of working under stationarity, then we will restrict ourselves to the analysis of processes with stationary stochastic volatility. As mentioned in the Introduction, traditional models for energy spot prices are typically based on mean-reverting stochastic processes, see, for example, Schwartz Schwartz1997 (), since such a modelling framework reflects the fact that commodity spot prices are equilibrium prices determined by supply and demand. Stationarity can be regarded as a weak form of mean-reversion and is often found in empirical studies on energy spot prices; one such example will be presented in this paper.

The initial value

In order to be able to have a stationary model, the lower integration bound in the definition of the process, and in particular for the process, is chosen to be rather than 0. Clearly, in any real application, we observe data from a starting value onwards, which is traditionally chosen as the observation at time . Hence, while processes are defined on the entire real line, we only define the spot price for . The observed initial value of the spot price at time is assumed to be a realisation of the random variable and , respectively. Such a choice guarantees that the deseasonalised spot price is a stationary process, provided we are in the stationary framework.

The driving Lévy process

Since and processes are driven by a general Lévy process , it is possible to account for price jumps and spikes, which are often observed in electricity markets. At the same time, one can also allow for Brownian motion-driven models, which are very common in, for example, temperature markets, see, for example, Benth, Härdle and Cabrera BenthCabreraHaerdle2009 ().

Stochastic volatility

A key ingredient of our new modelling framework which sets the model apart from many traditional models is the fact that it allows for stochastic volatility. Volatility clusters are often found in energy prices, see, for example, Hikspoors and Jaimungal HJ (), Trolle and Schwartz TrolleSchwartz2009 (), Benth B (), Benth and Vos BenthVos (), Koopman, Ooms and Carnero Koopmanetal2007 (), Veraart and Veraart VV2 (). Therefore, it is important to have a stochastic volatility component, given by , in the model. Note that a very general model for the volatility process would be to choose an process, that is, and

(11)

where denotes a deterministic, positive function and is a Lévy subordinator. In fact, if we want to ensure that the volatility is stationary, we can work with a function of the form , for a deterministic, positive function .

Autocorrelation structure and Samuelson effect

The kernel function (or ) plays a vital role in our model and introduces a flexibility which many traditional models lack: We will see in Section 3.2 that the kernel function – together with the autocorrelation function of the stochastic volatility process – determines the autocorrelation function of the process . Hence our – based models are able to produce various types of autocorrelation functions depending on the choice of the kernel function . It is important to stress here that this can be achieved by using one process only, whereas some traditional models need to introduce a multi-factor structure to obtain a comparable modelling flexibility. Also due to the flexibility in the choice of the kernel function, we can achieve greater flexibility in modelling the shape of the Samuelson effect often observed in forward prices, including the hyperbolic one suggested by Bjerksund, Rasmussen and Stensland BRS () as a reasonable volatility feature in power markets. Note that we obtain the modelling flexibility in terms of the general kernel function here since we specify our model directly through a stochastic integral whereas most of the traditional models are specified through evolutionary equations, which limit the choices of kernel functions associated with solutions to such equations. In that context, we note that a or an process cannot in general be written in form of a stochastic differential equation (due to the non-semimartingale character of the process). In Section 3.3, we will discuss sufficient conditions which ensure that an process is a semimartingale.

A unifying approach for traditional spot price models

As already mentioned above, energy spot prices are typically modelled in stationarity, hence the class of processes is particularly relevant for applications. In the following, we will show that many of the traditional spot price models can be embedded into our process-based framework.

Our new framework nests the stationary version of the classical one-factor Schwartz Schwartz1997 () model studied for oil prices. By letting be a Lévy process with the pure-jump part given as a compound Poisson process, Cartea and Figueroa CF () successfully fitted the Schwartz model to electricity spot prices in the UK market. Benth and Šaltytė Benth BSB-energy () used a normal inverse Gaussian Lévy process to model UK spot gas and Brent crude oil spot prices. Another example which is nested by the class of processes is a model studied in Benth B () in the context of gas markets, where the deseasonalised logarithmic spot price dynamics is assumed to follow a one-factor Schwartz process with stochastic volatility. A more general class of models which is nested is the class of so-called CARMA-processes, which has been successfully used in temperature modelling and weather derivatives pricing, see Benth, Šaltytė Benth and Koekebakker BSBK-temp (), Benth, Härdle and López Cabrera BenthCabreraHaerdle2009 () and Härdle and López Cabrera CabreraHaerdle2009 (), and more recently for electricity prices by García, Klüppelberg and Müller GKM2010 (), Benth et al. BenthKlueppelberVos (). A CARMA process is the continuous-time analogue of an ARMA time series, see Brockwell Brockwell2001a (), Brockwell Brockwell2001b () for definition and details. More precisely, suppose that for nonnegative integers

where and is a -dimensional OU process of the form

(12)

with

Here we use the notation for the -identity matrix, the th coordinate vector (where the first entries are zero and the th entry is 1) and is the transpose of , with and for . In Brockwell Brock (), it is shown that if all the eigenvalues of have negative real parts, then defined as

is the (strictly) stationary solution of (12). Moreover,

(13)

is a process. Hence, specifying in (13), the log-spot price dynamics will be an process, but without stochastic volatility. García, Klüppelberg and Müller GKM2010 () argue for dynamics as an appropriate class of models for the deseasonalised log-spot price at the Singapore New Electricity Market. The innovation process is chosen to be in the class of stable processes. From Benth, Šaltytė Benth and Koekebakker BSBK-temp (), Brownian motion-driven models seem appropriate for modelling daily average temperatures, and are applied for temperature derivatives pricing, including forward price dynamics of various contracts. More recently, the dynamics of wind speeds have been modelled by a Brownian motion-driven model, and applied to wind derivatives pricing, see Benth and Šaltytė Benth BSB-wind () for more details.

Finally note that the arithmetic model based on a superposition of processes nests the non-Gaussian Ornstein–Uhlenbeck model which has recently been proposed for modelling electricity spot prices, see Benth, Kallsen and Meyer-Brandis BenthKallsenMeyerBrandis2007 ().

We emphasis again that, beyond the fact that processes can be regarded as a unifying modelling approach which nest many of the existing spot price models, they also open up for entirely new model specifications, including more general choices of the kernel function (resulting in non-linear models) and the presence of stochastic volatility.

3.2 Second order structure

Next, we study the second order structure of volatility modulated Volterra processes , where , assuming the integrability conditions (2.3) hold and that in addition is square integrable. Let and . Recall that throughout the paper we assume that the stochastic volatility is independent of the driving Lévy process. Note that proofs of the following results are easy and hence omitted.

Proposition 1

The conditional second order structure of is given by

Corollary 1

The conditional second order structure of is given by

The unconditional second order structure of is then given as follows.

Proposition 2

The second order structure of for stationary is given by

where denotes the autocovariance function of , for .

The unconditional second order structure of is then given as follows.

Corollary 2

The second order structure of for stationary is given by

where denotes the autocovariance function of , for . Hence, we have

(14)
Corollary 3

If or if has zero autocorrelation, then

The last corollary shows that we get the same autocorrelation function as in the model. From the results above, we clearly see the influence of the general damping function on the correlation structure. A particular choice of , which is interesting in the energy context is studied in the next example. {ex} Consider the case for and , which is motivated from the forward model of Bjerksund, Rasmussen and Stensland BRS (), which we shall return to in Section 4. We have that This ensures integrability of over with respect to any square integrable martingale Lévy process . Furthermore, . Thus,

Observe that since can be written as

it follows that the process is a semimartingale according to the Knight condition, see Knight Kni92 () and also Basse Basse2008El (), Basse and Pedersen BassePedersen2009 (), Basse-O’Connor, Graversen and Pedersen BassePedersen2010 ().

3.3 Semimartingale conditions and absence of arbitrage

We pointed out that the subclass of processes are particularly relevant for modelling energy spot prices since they allow one to model directly in stationarity. Let us focus on this class in more detail. Clearly, an process is in general not a semimartingale. However, we can formulate sufficient conditions on the kernel function and on the stochastic volatility component which ensure the semimartingale property. The sufficient conditions are in line with the conditions formulated for processes in Barndorff-Nielsen and Schmiegel BNSch09 (), see also Barndorff-Nielsen and Basse-O’Connor BNBasse2009 (). Note that the proofs of the following results are provided in the Appendix.

Proposition 3

Let be an process as defined in (2). Suppose the following conditions hold: {longlist}[(iii)]

.

The function values and exist and are finite.

The kernel function is absolutely continuous with square integrable derivative .

The process is square integrable for each .

The process is integrable for each . Then is a semimartingale with representation

(15)

where for and

{ex}

An example of a kernel function which satisfies the above conditions is given by

For , is given by a volatility modulated Ornstein–Uhlenbeck process. In a next step, we are now able to find a representation for the quadratic variation of an process provided the conditions of Proposition 3 are satisfied.

Proposition 4

Let be an process and suppose that the sufficient conditions for to be a semimartingale (as formulated in Proposition 3) hold. Then, the quadratic variation of is given by

Note that the quadratic variation is a prominent measure of accumulated stochastic volatility or intermittency over a certain period of time and, hence, is a key object of interest in many areas of application and, in particular, in finance.

The question of deriving semimartingale conditions for processes is closely linked to the question whether a spot price model based on an process is prone to arbitrage opportunities. In classical financial theory, we usually stick to the semimartingale framework to ensure the absence of arbitrage. Nevertheless one might ask the question whether one could still work with the wider class of processes which are not semimartingales. Here we note that the standard semimartingale assumption in mathematical finance is only valid for tradeable assets in the sense of assets which can be held in a portfolio. Hence, when dealing with, for example, electricity spot prices, this assumption is not valid since electricity is essentially non-storable. Hence, such a spot price cannot be part of any financial portfolio and, therefore, the requirement of being a martingale under some equivalent measure is not necessary.

Guasoni, Rásonyi and Schachermayer GRS2008 () have pointed out that, while in frictionless markets martingale measures play a key role, this is not the case any more in the presence of market imperfections. In fact, in markets with transaction costs, consistent price systems as introduced in Schachermayer Schachermayer2004 () are essential. In such a set-up, even processes which are not semimartingales can ensure that we have no free lunch with vanishing risk in the sense of Delbaen and Schachermayer DelbaenSchachermayer1994 (). It turns out that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs, see Guasoni, Rásonyi and Schachermayer GRS2008 (). It has recently been shown by Pakkanen Pakkanen2010 (), that under certain conditions, a process has conditional full support. This means that such processes can be used in financial applications without necessarily giving rise to arbitrage opportunities.

3.4 Model extensions

Let us briefly point out some model extensions concerning a multi-factor structure, non-stationary effects, multivariate models and alternative methods for incorporating stochastic volatility.

A straightforward extension of our model is to study a superposition of processes for the spot price dynamics. That is, we could replace the process by a superposition of factors:

(16)

and where all are defined as in (8) for independent Lévy processes and independent stochastic volatility processes , in both the geometric and the arithmetic model. Such models include the Benth, Kallsen and Meyer-Brandis BenthKallsenMeyerBrandis2007 () model as a special case. A superposition of factors opens up for separate modelling of spikes and other effects. For instance, one could let the first factor account for the spikes, using a Lévy process with big jumps at low frequency, while the function forces the jumps back at a high speed. The next factor(s) could model the “normal” variations of the market, where one observes a slower force of mean-reversion, and high frequent Brownian-like noise, see Veraart and Veraart VV2 () for extensions along these lines. Note that all the results we derive in this paper based on the one factor model can be easily generalised to accommodate for the multi-factor framework. It should be noted that this type of “superposition” is quite different from the concept behind supOU processes as studied in, for example, Barndorff-Nielsen and Stelzer BNStelzer2010b ().

In order to study various energy spot prices simultaneously, one can consider extensions to a multivariate framework along the lines of Barndorff-Nielsen and Stelzer BNStelzer2010 (), BNStelzer2010b (), Veraart and Veraart VV2 ().

In addition, another interesting aspect which we leave for future research is the question of alternative ways of introducing stochastic volatility in processes. So far, we have introduced stochastic volatility by considering a stochastic proportional of the driving Lévy process, that is, we work with a stochastic integral of with respect to . An alternative model specification could be based on a stochastic time change , where . Such models can be constructed in a fashion similar to that of volatility modulated non-Gaussian Ornstein–Uhlenbeck processes introduced in Barndorff-Nielsen and Veraart BNVeraart2011 (). We know that outside the Brownian or stable Lévy framework, stochastic proportional and stochastic time change are not equivalent. Whereas in the first case the jump size is modulated by a volatility term, in the latter case the speed of the process is changed randomly. These two concepts are in fact fundamentally different (except for the special cases pointed out above) and, hence, it will be worth investigating whether a combination of stochastic proportional and stochastic time change might be useful in certain applications.

4 Pricing of forward contracts

In this subsection, we are concerned with the calculation of the forward price at time for contracts maturing at time . We denote by a finite time horizon for the forward market, meaning that all contracts of interest mature before this date. Note that in energy markets, the corresponding commodity typically gets delivered over a delivery period rather than at a fixed point in time. Extensions to such a framework can be dealt with using standard methods, see, for example, Benth, Šaltytė Benth and Koekebakker BSBK-book () for more details.

Let denote the spot price, being either of geometric or arithmetic kind as defined in (9) and (10), respectively, with

where the stochastic volatility is chosen as previously defined in (11). Clearly, the corresponding results for processes can be obtained by choosing . We use the conventional definition of a forward price in incomplete markets, see Duffie duffie (), ensuring the martingale property of ,

(17)

with being an equivalent probability measure to . Here, we suppose that , the space of integrable random variables. In a moment, we shall introduce sufficient conditions for this.

4.1 Change of measure by generalised Esscher transform

In finance, one usually uses equivalent martingale measures , meaning that the equivalent probability measure should turn the discounted price dynamics of the underlying asset into a (local) -martingale. However, as we have already discussed, this restriction is not relevant in, for example, electricity markets since the spot is not tradeable. Thus, we may choose any equivalent probability as pricing measure. In practice, however, one restricts to a parametric class of equivalent probability measures, and the standard choice seems to be given by the Esscher transform, see Benth, Šaltytė Benth and Koekebakker BSBK-book (), Shiryaev Sh (). The Esscher transform naturally extends the Girsanov transform to Lévy processes.

To this end, consider defined as the (generalised) Esscher transform of for a parameter being a Borel measurable function. Following Shiryaev Sh () (or Benth, Šaltytė Benth and Koekebakker BSBK-book (), Barndorff-Nielsen and Shiryaev BNShiryaev2010 ()), is defined via the Radon–Nikodym density process

(18)

for being a real-valued function which is integrable with respect to the Lévy process on , and

(for ) being the log-moment generating function of , assuming that the moment generating function of exists.

A special choice is the ‘constant’ measure change, that is, letting

(19)

In this case, if under the measure , has characteristic triplet , where is the drift, is the squared volatility of the continuous martingale part and is the Lévy measure in the Lévy–Khinchine representation, see Shiryaev Sh (), a fairly straightforward calculation shows that, see Shiryaev Sh () again, the Esscher transform preserves the Lévy property of , and the characteristic triplet under the measure on the interval becomes , where

This comes from the simple fact that the logarithmic moment generating function of under is

(20)
{rem*}

It is important to note here that the choice of (as, e.g., in (19)) forces us to choose a starting time since the function will not be integrable with respect to on the unbounded interval . Recall that the only reason why we model from rather than from is the fact that we want to be able to obtain a stationary process under the probability measure . Throughout this section, we choose the starting time to be zero, which is a convenient choice since , and it is also practically reasonable since this can be considered as the time from which we start to observe the process. With such a choice, we do not introduce any risk premium for . In the general case, with a time-dependent parameter function , the characteristic triplet of  under will become time-dependent, and hence the Lévy process property is lost. Instead, will be an independent increment process (sometimes called an additive process). Note that if , a Brownian motion, the Esscher transform is simply a Girsanov change of measure where for and a -Brownian motion .

Similarly, we do a (generalised) Esscher transform of , the subordinator driving the stochastic volatility model, see (11). We define to have the Radon–Nikodym density process

for being a real-valued function which is integrable with respect to on , and being the log-moment generating function of . Since is a subordinator, we obtain

where and denotes the Lévy measure associated with . {rem*} Our discussion above on choosing a starting value applies to the measure transform for the volatility process as well, and hence throughout the paper we will work under the assumption that for . Note in particular, that this assumption implies that under the risk-neutral probability measure, the characteristic triplets of and only change on the time interval . On the interval , we have the same characteristic triplet for and as under . Choosing , with a constant , an Esscher transform will give a characteristic triplet , which thus preserves the subordinator property of under . For the general case, the process will be a time-inhomogeneous subordinator (independent increment process with positive jumps). The log-moment generating function of under the measure is denoted by .

In order to ensure the existence of the (generalised) Esscher transforms, we need some conditions. We need that there exists a constant such that , and where . (Similarly, we must have such a condition for the Lévy measure of the subordinator driving the stochastic volatility, that is, ). Also, we must require that exponential moments of and exist. More precisely, we suppose that parameter functions and of the (generalised) Esscher transform are such that

(21)

The exponential integrability conditions of the Lévy measures of and imply the existence of exponential moments, and thus that the Esscher transforms and are well defined.

We define the probability as the class of pricing measures for deriving forward prices. In this respect, may be referred to as the market price of risk, whereas is the market price of volatility risk. We note that a choice will put more weight to the positive jumps in the price dynamics, and less on the negative, increasing the “risk” for big upward movements in the prices under .

Let us denote by the expectation operator with respect to , and by the expectation with respect to .

Forward price in the geometric case

Suppose that the spot price is defined by the geometric model

where is defined as in (3). In order to have the forward price well defined, we need to ensure that the spot price is integrable with respect to the chosen pricing measure . We discuss this issue in more detail in the following.

We know that is positive and in general not bounded since it is defined via a subordinator. Thus, (for ) is unbounded as well. Supposing that has exponential moments of all orders, we can calculate as follows using iterated expectations conditioning on the filtration generated by the paths of , for :

To have that , the two integrals must be finite. This puts additional restrictions on the choice of and the specifications of and . We note that when applying the Esscher transform, we must require that has exponential moments of all orders, a rather strong restriction on the possible class of driving Lévy processes. In our empirical study, however, we will later see that the empirically relevant cases are either that is a Brownian motion or that is a generalised hyperbolic Lévy process, which possess exponential moments of all orders.

We are now ready to price forwards under the Esscher transform.

Proposition 5

Suppose that . Then, the forward price for is given by

4.2 Change of measure by the Girsanov transform in the Brownian case

As a special case, consider , where is a two-sided standard Brownian motion under . In this case we apply the Girsanov transform rather than the generalised Esscher transform, and it turns out that a rescaling of the transform parameter function by the volatility is convenient for pricing of forwards. To this end, consider the Girsanov transform

(22)

that is, we set for . Supposing that the Novikov condition

holds, we know that is a Brownian motion for under a probability having density process