Modelling electricity forward markets by ambit fields

45 Pages·2011·1.27 MB·English

Modelling electricity forward markets by ambit fields
OleE. Barndorff-Nielsen∗ Fred EspenBenth†
Aarhus University UniversityofOslo&UniversityofAgder
AlmutE. D. Veraart‡
ImperialCollegeLondon
September26,2011
Abstract
Thispaperproposesanewmodellingframeworkforelectricityforwardmarketsbasedonso–
called ambit fields. The new model can capture many of the stylised facts observed in energy
marketsandishighlyanalyticallytractable. Wegiveadetailedaccountontheprobabilisticprop-
ertiesofthenewtypeofmodel,andwediscussmartingaleconditions,optionpricingandchange
ofmeasurewithinthenewmodelclass. Also,wederiveamodelforthetypicallystationaryspot
price,whichisobtainedfromtheforwardmodelthroughalimitingargument.
Keywords: Electricity markets; forward prices; random fields; ambit fields; Levy basis; Samuelson
effect;stochastic volatility.
MSCcodes: 60G10,60G51,60G55,60G57,60G60,91G99.
1 Introduction
This paper introduces a new type of model for electricity forward prices, which is based on ambit
fields and ambit processes. Ambit stochastics constitutes a general probabilistic framework which is
suitablefortempo–spatialmodelling. Ambitprocessesaredefinedasstochasticintegralswithrespect
toamultivariaterandom measure,wheretheintegrand isgivenbyaproductofadeterministic kernel
functionandastochastic volatility fieldandtheintegration iscarriedoutoveranambitsetdescribing
thesphereofinfluenceforthestochastic field.
Due to their very flexible structure, ambit processes have successfully been used for modelling
turbulence in physics and cell growth in biology, see Barndorff-Nielsen & Schmiegel (2004, 2007,
2008a,b,c,2009),VedelJensenetal.(2006). Theaimofthispaperisnowtodevelopanewmodelling
frameworkfor(electricity) forwardmarketsbasedontheambitconcept.
Overthepasttwodecades,themarketsforpowerhavebeenliberalisedinmanyareasintheworld.
The typical electricity market, like for instance the Nordic Nord Pool market or the German EEX
market, organises trade in spot, forward/futures contracts and European options on these. Although
these assets are parallel to other markets, like traditional commodities or stock markets, electricity
∗ThieleCenter, Department ofMathematical Sciences&CREATES,School ofEconomicsandManagement, Aarhus
University,NyMunkegade118,DK-8000AarhusC,Denmark,[email protected]
†Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway, and
FacultyofEconomics,UniversityofAgder,Serviceboks422,N-4604Kristiansand,Norway,[email protected]
‡Department of Mathematics, Imperial College London, 180 Queen’s Gate, SW7 2AZ London, UK,
[email protected]
1
Electronic copy available at: http://ssrn.com/abstract=1938704
1 INTRODUCTION
has its own distinctive features calling for new and more sophisticated stochastic models for risk
managementpurposes, seeBenth,Sˇaltyte˙ Benth&Koekebakker (2008).
Theelectricity spotcannot bestored directly except viareservoirs forhydro–generated power, or
large and expensive batteries. This makes the supply of power very inelastic, and prices may rise
by several magnitudes when demand increases, due to temperature drops, say. Since spot prices are
determined by supply and demand, some form of mean–reversion or stationarity can be observed.
Thespotpriceshavecleardeterministic patternsovertheyear,weekandintra–day. Theliteraturehas
focusedonstochasticmodelsforthespotpricedynamics,whichtakesomeofthevariousstylisedfacts
intoaccount. Recently,averygeneral,yetanalytically tractableclassofmodelshasbeenproposedin
Barndorff-Nielsen et al. (2010), based on Le´vy semistationary processes, which are special cases of
ambitprocesses.
One of the fundamental problems in power market modelling is to understand the formation of
forward prices. Non–storability of the spot makes the usual buy–and–hold hedging arguments break
down, and the notion of convenience yield is not relevant either. There is thus a highly complex
relationship betweenspotandforwards.
A way around this would be to follow the so–called Heath–Jarrow–Morton approach, which has
been introduced in the context of modelling interest rates, see Heath et al. (1992), and model the
forwardpricedynamicsdirectly (rather thanmodelling thespotpriceanddeducing theforwardprice
from the conditional expectation of the spot atdelivery). There are many challenging problems con-
nectedtothiswayofmodelling forwardprices.
Firstly, standard models for the forward dynamics generally depend on the current time and the
time to maturity. However, power market trades in contracts which deliver power over a delivery
period, introducing a new dimension in the modelling. Hence comprehensive forward price models
should be functions of both time to and length of delivery, which calls for random field models in
time and space. Furthermore, since the market trades in contracts with overlapping delivery peri-
ods,specificno–arbitrage conditionsmustbesatisfiedwhichessentiallyputsrestrictionsonthespace
structure of the field. So far, the literature is not very rich on modelling power forward prices ap-
plying the Heath–Jarrow–Morton approach, presumably due to the lack of analytical tractability and
empiricalknowledgeofthepriceevolution.
Empirical studies, see Frestad et al. (2010), have shown that the logarithmic returns of forward
prices are non–normally distributed, with clear signs of (semi-) heavy tails. Also, a principal com-
ponent analysis by Koekebakker & Ollmar (2005) indicates a high degree of idiosyncratic risk in
power forward markets. This strongly points towards random field models which, in addition, allow
forstochastic volatility. Moreover, thestructure determining theinterdependencies betweendifferent
contracts isbyfar not properly understood. Someempirical studies, seeAndresen etal. (2010), sug-
gest that the correlations between contracts are decreasing with time to maturity, whereas the exact
formofthisdecayisnotknown. Buthowtotake‘lengthofdelivery’ intoaccount inmodelling these
interdependencies has been an open question. A first approach on how to tackle these problems will
bepresented laterinthispaper.
Ambit processes provide a flexible class of random field models, where one has a high degree
of flexibility in modelling complex dependencies. These may be probabilistic coming from a driv-
ing Levy basis and the stochastic volatility, or functional from a specification of an ambit set or the
deterministic kernelfunction.
Ourfocuswillbeonambitprocesseswhicharestationaryintime. Assuch,ourmodellingframe-
workdiffersfromthetraditionalmodels,wherestationary processes are(ifatall)reachedbylimiting
arguments. Modelling directly in stationarity seems in fact to be quite natural in various applica-
tions and is e.g. done in physics in the context of modelling turbulence, see e.g. Barndorff-Nielsen
&Schmiegel (2007, 2009). Hereweshow that such anapproach hasstrong potential infinance, too,
when we are concerned with modelling commodity markets. In particular, we will argue that en-
ergyspotpricesaretypicallywell–described bystationaryprocesses, seee.g.Barndorff-Nielsenetal.
2
Electronic copy available at: http://ssrn.com/abstract=1938704
1 INTRODUCTION
(2010) for a detailed discussion on that aspect, and in order to achieve stationarity in the spot price
it makes sense to model the corresponding forward price also in stationarity. The precise relation
betweenthespotandtheforwardpricewillbeestablished laterinthepaper.
Duetotheirgeneralstructure,ambitprocesseseasilyincorporateleptokurticbehaviourinreturns,
stochastic volatility and leverage effects and the observed Samuelson effect in the volatility. Note
thattheSamuelson effect, seeSamuelson(1965), referstothefindingthat, whenthetimetomaturity
approacheszero,thevolatilityoftheforwardincreasesandconvergestothevolatilityoftheunderlying
spotprice(provided theforwardpriceconverges tothespotprice).
Althoughmanystylisedfactsofenergymarketscaneasilybeincorporatedinanambitframework,
one may question whether ambit processes are not in fact too general to be a good building block
for financial models. In particular, one property — the martingale property — is often violated by
general ambit processes. However, wecan and willformulate conditions which ensure that an ambit
process is in fact a martingale. So, if we wish to stay within the martingale framework, we can do
so by using a restricted subclass of ambit processes. On the other hand, in modelling terms, it is
actually not so obvious whether we should stay within the martingale framework if our aim is to
model electricity forward contracts. Given the illiquidity of electricity markets, it cannot be taken
forgrantedthatarbitrage opportunities arisingfromforwardpricesoutside themartingaleframework
can be exercised. Also, we know from recent results in the mathematical finance literature, see e.g.
Guasoni et al. (2008), Pakkanen (2011), that subclasses of non–(semi)–martingales can be used to
model financial assets without necessarily giving rise to arbitrage opportunities in markets which
exhibitmarketfrictions, suchase.g.transaction costs.
Next, wewillnot workwiththemostgeneral class ofambitprocesses since wearemainly inter-
estedinthetime–stationary caseasmentionedbefore.
Last but not least wewill show that the ambit framework can shed some light on the connection
between electricity spot andforward prices. Understanding theinterdependencies between these two
assets is crucial in many applications, e.g. in the hedging of exotic derivatives on the spot using
forwards. Atypical exampleinelectricity marketsisso–called user–time contracts, givingtheholder
therighttobuyspotatagivenpriceonapredefinednumberofhoursinayear,say.
The outline for the remaining part of the paper is as follows. Section 2 gives an overview of
the standard models used for forward markets. Section 3 reviews basic traits of the theory of ambit
fieldsandprocesses. InSection4,weintroduce thenewmodelling frameworkforelectricity forward
markets, study its key properties and highlight the most relevant model specifications. In Section 5,
we show how some of the traditional models for forward prices relate to ambit processes. Section 6
presents the martingale conditions for our new model and discusses option pricing. Moreover, since
we do the modelling under the risk neutral measure, we discuss how a change of measure can be
carried out to get back to the physical probability measure, see Section 7. Next we show what kind
of spot model is implied by our new model for the forward price, and we discuss that, under certain
conditions, the implied spot price process equals in law a Le´vy semistationary process, see Section
8. Inordertogetalsoavisual impression ofthenewmodelsforthetermstructure offorwardprices,
we present a simulation algorithm for ambit fields in Section 9 and highlight the main theoretical
properties of the modelling framework graphically. Section 10 deals with extensions of our new
modelling framework: While wemainly focus onarithmetic models forforward prices inthis paper,
wediscuss brieflyhowgeometricmodelscanbeconstructed. Also,wegiveanoutlook onhowambit
field based models can be used to jointly model time and period of delivery. Finally, Section 11
concludes and Appendix A contains the proofs of our main results and some technical results on the
correlation structureofthenewclassofmodelsandextensions tothemultivariate framework.
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2 OVERVIEWONAPPROACHESTOMODELLINGFORWARDPRICES
2 Overview on approaches to modelling forward prices
Before introducing ambit fields, let us review the exisiting literature on direct modelling of forward
prices in commodity markets, i.e. the approach where one is not starting out with a specification of
theunderlying spotdynamics.
Although commodity markets have very distinct features, most models for energy forward con-
tractshavebeeninspiredbyinstantaneous forwardratesmodelsinthetheoryforthetermstructureof
interestrates,seeKoekebakker&Ollmar(2005)foranoverviewonthesimilaritiesbetweenelectricity
forwardmarketsandinterest rates.
Hence, in order to get an overview on modelling concepts which have been developed in the
context of the term structure of interest rates, but which can also be used inthe context ofelectricity
markets, wewill now review these examples from the interest rate literature. However, later we will
argue that, in order to account for the particular stylised facts of power markets, there is a case for
leaving these models behind and focusing instead on ambit fields as a natural class for describing
energyforwardmarkets.
Throughout the paper, we denote by t R the current time, by T 0 the time of maturity of a
∈ ≥
givenforwardcontract,andbyx = T tthecorresponding timetomaturity. WeuseF (T)todenote
t
−
theprice ofaforward contract attimetwithtimeofmaturity T. Likewise, weusef fortheforward
priceattimetwithtimetomaturityx = T t,whenweworkwiththeMusielaparameterisation, i.e.
−
wedefinef by
f (x) = f (T t) =F (T).
t t t
−
2.1 Multi–factormodels
Motivated bytheclassical Heathetal.(1992) framework, thedynamics oftheforward rateunder the
riskneutralmeasurecanbemodelledby
n
(i) (i)
df (x) = σ (x)dW , for t 0,
t t t ≥
i=1
X
forn IN andwhereW(i) areindependent standard Brownianmotions andσ(i)(x)areindependent
∈
positive stochastic volatility processes for i = 1,...,n. The advantage of using these multi–factor
models is that they are to a high degree analytically tractable. Extensions to allow for jumps in such
models have also been studied in detail in the literature. However, a principal component analysis
by Koekebakker &Ollmar(2005) has indicated that weneed infact many factors (large n)to model
electricity forward prices. Hence it is natural to study extensions to infinite factor models which are
alsocalledrandomfieldmodels.
2.2 Random field models forthe dynamics offorwardrates
Inorder toovercome theshortcomings ofthe multifactor models, Kennedy (1994) has pioneered the
approach ofusing random fieldmodels, in some cases called stochastic string models, formodelling
the term structure of interest rates. Random field models have a continuum of state variables (in our
case forward prices for all maturities) and, hence, are also called infinite factor models, but they are
typically very parsimonious in the sense that they do not require many parameters. Note that finite–
factormodelscanbeaccommodated byrandom fieldmodelsasdegenerate cases.
Kennedy (1994) proposed to model the forward rate by acentered, continuous Gaussian random
field plus a continuous deterministic drift. Furthermore he specified a certain structure of the co-
variance function of the random field which ensured that it had independent increments in the time
4
2 OVERVIEWONAPPROACHESTOMODELLINGFORWARDPRICES
direction t (but not necessarily in the time to maturity direction x). Such models include as spe-
cial cases the classical Heath et al. (1992) model when both the drift and the volatility functions are
deterministic and also two–parameter models, such as models based on Brownian sheets. Kennedy
(1994) derived suitable drift conditions which ensure the martingale properties of the corresponding
discounted zerocouponbonds.
In a later article, Kennedy (1997) revisited the continuous Gaussian random field models and he
showed thatthestructure ofthecovariance function ofsuch models canbespecified explicitly ifone
assumes a Markov property. Adding an additional stationarity condition, the correlation structure of
suchprocessesisalreadyverylimitedandKennedy(1997)provedthat,infact,underastrongMarkov
andstationarity assumption theGaussianfieldisnecessarily described byjustthreeparameters.
TheGaussianassumptionwasrelaxedlaterandGoldstein(2000)presentedatermstructuremodel
based on non–Gaussian random fields. Such models incorporate in particular conditional volatil-
ity models, i.e. models which allow for more flexible (i.e. stochastic) behaviour of the (conditional)
volatilities of the innovations to forward rates (in the traditional Kennedy approach such variances
were just constant functions of maturity), and, hence, are particularly relevant for empirical applica-
tions. Also, Goldstein (2000) points out thatone isinterested inverysmooth random fieldmodels in
the context of modelling the term structure of interest rates. Such a smoothness (e.g. in the time to
maturity direction) can be achieved by using integrated random fields, e.g. he proposes to integrate
over an Ornstein–Uhlenbeck process. Goldstein (2000) derived drift conditions for the absence of
arbitrage forsuchgeneralnon–Gaussian randomfieldmodels.
Whilesuchmodelsarequitegeneraland,hence,appealing inpractice, Kimmel(2004) pointsout
that themodels defined byGoldstein (2000) aregenerally specified assolutions toaset ofstochastic
differential equations, where it is difficult to prove the existence and uniqueness of solutions. The
Goldstein (2000) models and many other conditional volatility random field models are in fact com-
plex and often infinite dimensional processes, which lack the key property of the Gaussian random
fieldmodelsintroducedbyKennedy(1994): thattheindividualforwardratesarelowdimensionaldif-
fusion processes. Thelatterproperty isinfact important formodelestimation and derivative pricing.
Hence,Kimmel(2004)proposesanewapproachtorandomfieldmodelswhichallowsforconditional
volatilityandwhichpreservesthekeypropertyoftheKennedy(1994)classofmodels: theclassofla-
tentvariable termstructuremodels. Heprovesthatsuchmodelsensurethattheforwardratesandthe
latentvariables (whicharemodelledasajointdiffusion)followjointlyafinitedimensional diffusion.
A different approach to generalising the Kennedy (1994) framework is proposed by Albeverio
et al. (2004). They suggest to replace the Gaussian random field in the Kennedy (1994) model by a
(purejump)Le´vyfield. Specialcasesofsuchmodelsaree.g.thePoissonandtheGammasheet.
Finally, another approach for modelling forward rates has been proposed by Santa-Clara & Sor-
nette (2001) who build their model on stochastic string shocks. We will review that class of models
later in more detail since it is related (and under some assumptions even a special case) of the new
modellingframeworkwepresentinthispaper.
2.3 Intuitivedescription ofanambitfield basedmodel forforwardprices
Afterwehavereviewed thetraditional models fortheterm structure ofinterest rates, whichare(par-
tially)alsousedformodellingforwardpricesofcommodities,wewishtogiveanintuitivedescription
ofthenewframeworkwepropose inthispaperbeforewepresentallthemathematical details.
As in the aforementioned models, we also propose to use a random field to account for the two
temporal dimensions of current time and time to maturity. However, the main difference of our new
modelling framework compared to the traditional ones is that we model the forward price directly.
This direct modelling approach is in fact twofold: First, we model the forward prices directly rather
thanthespotprice,whichisinlinewiththeHeathetal.(1992)framework. Second,wedonotspecify
the dynamics of the forward price as the solution of an evolution equation, but we specify a random
5
2 OVERVIEWONAPPROACHESTOMODELLINGFORWARDPRICES
field, an ambit field, which explicitly describes the forward price. In particular, we propose to use
randomfieldsgivenbystochastic integrals oftype
h(ξ,s,x,t)σ (ξ)L(dξ,ds), (1)
s
ZAt(x)
asabuilding blockformodelling f (x). Anatural choice forL—motivated bytheuseofLe´vypro-
t
cesses intheone–dimensional framework — istheclass ofLe´vybases, whichareinfinitely divisible
random measures as described in more detail below. Here the integrand is given by the product of a
deterministic kernelfunction handarandom fieldσ describing thestochastic volatility.
Wewilldescribe inmoredetailbelow,howstochastic integralsoftype(1)havetobeunderstood.
Note here that we integrate over a set A (x), the ambit set, which can be chosen in many different
t
ways. Wewilldiscussthechoiceofsuchsetslaterinthepaper.
An important motivation for the use of ambit processes is that we wish to work with processes
whicharestationaryintime,i.e.int,ratherthanformulatingamodelwhichconvergestoastationary
process. Hence,weworkwithstochasticintegralsstartingfrom inthetemporaldimension,more
−∞
precisely, we choose ambit sets of the form A (x) = (ξ,s) : < s t,ξ I (s,x) , where
t t
{ −∞ ≤ ∈ }
I (s,x)istypicallyanintervalincludingx,ratherthanintegratingfrom0,whichiswhatthetraditional
t
modelsdowhichareconstructed assolutions ofstochastic partialdifferential equations (SPDEs). (In
fact, many traditional models coming from SPDEs can be included in an ambit framework when
choosingtheambitsetA (x) = [0,t] x ,seeBarndorff-Nielsen, Benth&Veraart(2011)formore
t
×{ }
details.)
Inordertoobtain modelswhicharestationary inthetimecomponent t,butnotnecessarily inthe
time to maturity component x, we assume that the kernel function depends on t and s only through
the difference t s, so having that h is of the form h(ξ,s,x,t) = k(ξ,t s,x), that σ isstationary
− −
intimeandthatA (x)hasacertainstructure, asdescribed below. Thenthespecification (1)takesthe
t
form
k(ξ,t s,x)σ (ξ)L(dξ,ds). (2)
s
−
ZAt(x)
Note that Hikspoors & Jaimungal (2008), Benth (2011) and Barndorff-Nielsen et al. (2010) pro-
vide empirical evidence that spot and forward prices are influenced by a stochastic volatility field σ.
Hereweassume that σ describes the volatility oftheforward market asawhole. Moreprecisely, we
willassumethat thevolatility oftheforward depends onprevious states ofthevolatility both intime
and in space, where the spatial dimension reflects the timeto maturity. Wewillcome back to that in
Section4.2.3.
The general structure of ambit fields makes it possible to allow for general dependencies be-
tween forward contracts. In the electricity market, a forward contract has a close resemblance with
its neighbouring contracts, meaning contracts which are close in maturity. Empirics (by principal
component analysis) suggest that the electricity markets need many factors, see e.g. Koekebakker &
Ollmar (2005), to explain the risk, contrary to interest rate markets where one finds 3–4 sources of
noise as relevant. Since electricity is a non–storable commodity, forward looking information plays
a crucial role in settling forward prices. Different information at different maturities, such as plant
maintenance,weatherforecasts,politicaldecisionsetc.,giverisetoahighdegreeofidiosyncratic risk
in the forward market, see Benth & Meyer-Brandis (2009). These empirical and theoretical findings
justify arandom fieldmodelinelectricity andalsoindicate thatthere isahigh degree ofdependency
around contracts which are close in maturity, but much weaker dependence when maturities are far-
therapart. Thestructure oftheambitfieldandthevolatility fieldwhichwepropose inthispaperwill
allowusto“bundle” contractstogether inaflexiblefashion.
6
3 AMBITFIELDSANDPROCESSES
3 Ambit fields and processes
Thissection reviews the concept ofambit fields andambit processes which form the building blocks
ofournewmodelfortheelectricity forwardprice. Foradetailedaccount onthistopicseeBarndorff-
Nielsen, Benth &Veraart (2011) and Barndorff-Nielsen & Schmiegel (2007). Throughout the paper,
we denote by (Ω, ,P∗) our probability space. Note that we use the notation since we will later
F ∗
refertothisprobability measureasariskneutralprobability measure.
3.1 Reviewofthe theory ofambitfields andprocesses
Thegeneral framework for defining anambitprocess isasfollows. LetY = Y (x) withY (x) :=
t t
{ }
Y(x,t) denote a stochastic field in space–time R and let τ (θ) = (x(θ),t(θ)) denote a curve
X ×
in R. The values of the field along the curve are then given by X = Y (x(θ)). Clearly,
θ t(θ)
X ×
X = X denotes a stochastic process. In most applications, the space is chosen to be Rd for
θ
{ } X
d = 1,2 or 3. Further, the stochastic field is assumed to be generated by innovations in space–time
with values Y (x) which are supposed to depend only on innovations that occur prior to or at time
t
t and in general only on a restricted set of the corresponding part of space–time. I.e., at each point
(x,t),thevalueofY (x)isonlydetermined byinnovations insomesubset A (x)of R (where
t t t
X ×
R = ( ,t]),whichwecalltheambitsetassociatedto(x,t). Furthermore,werefertoY andX as
t
−∞
anambitfield andanambitprocess, respectively.
In order to use such general ambit fields in applications, we have to impose some structural as-
sumptions. More precisely, we will define Y (x) as a stochastic integral plus a smooth term, where
t
the integrand in the stochastic integral will consist of a deterministic kernel times a positive random
variatewhichistakentoembodythevolatilityofthefieldY. Moreprecisely, wethinkofambitfields
asbeingoftheform
Y (x)= µ+ h(ξ,s,x,t)σ (ξ)L(dξ,ds)+ q(ξ,s,x,t)a (ξ)dξds, (3)
t s s
ZAt(x) ZDt(x)
whereA (x),andD (x)areambitsets,handqaredeterministicfunctions,σ 0isastochasticfield
t t
≥
referred to as volatility, a is also a stochastic field, and L is a Le´vy basis. Throughout the paper we
willassumethatthevolatility fieldσ isindependent oftheLe´vybasisLformodelling convenience.
Thecorresponding ambitprocessX alongthecurveτ isthengivenby
X = µ+ h(ξ,s,τ(θ))σ (ξ)L(dξ,ds)+ q(ξ,s,τ(θ))a (ξ)dξds, (4)
θ s s
ZA(θ) ZD(θ)
whereA(θ)= A (x(θ))andD(θ)= D (x(θ)).
t(θ) t(θ)
Of particular interest in many applications are ambit processes that are stationary in time and
nonanticipative. Morespecifically, theymaybederivedfromambitfieldsY oftheform
Y (x) = µ+ h(ξ,t s,x)σ (ξ)L(dξ,ds)+ q(ξ,t s,x)a (ξ)dξds. (5)
t s s
− −
ZAt(x) ZDt(x)
HeretheambitsetsA (x)andD (x)aretakentobehomogeneous andnonanticipative i.e. A (x)is
t t t
of the form A (x) = A+(x,t) where A only involves negative time coordinates, and similarly for
t
D (x). Weassumefurtherthath(ξ,u,x) = q(ξ,u,x) = 0foru 0.
t
≤
Due to the structural assumptions we made to define ambit fields, we obtain a class of random
fieldswhichishighly analytically tractable. Inparticular, wecanderivemomentsandthecorrelation
structure explicitly, seetheAppendixA.4fordetailedresults.
In any concrete modelling, one has to specify the various components of the ambit field, and we
dothatforelectricity forwardpricesinSection4.1.
7
3 AMBITFIELDSANDPROCESSES
3.2 Background onLe´vy bases
Let denotetheδ–ringofsubsetsofanarbitrarynon–emptysetS,suchthatthereexistsanincreasing
S
sequence S ofsetsin with S = S,seeRajput&Rosinski(1989). Recallfrome.g.Rajput&
n n n
{ } S ∪
Rosinski (1989), Pedersen (2003), Barndorff-Nielsen (2011) that a Le´vy basis L = L(B),B
{ ∈ S}
defined on a probability space (Ω, ,P) is an independently scattered random measure with Le´vy–
F
Khinchinrepresentation
C v L(B) = log(E(exp(ivL(B))),
{ ‡ }
givenby
1
C v L(B) = iva(B) v2b(B)+ eivr 1 ivrI (r) l(dr,B), (6)
[−1,1]
{ ‡ } − 2 R − −
Z
(cid:0) (cid:1)
where a is a signed measure on , b is a measure on , l(, ) is the generalised Le´vy measure such
S S · ·
thatl(dr,B)isaLe´vymeasureonRforfixedB andameasureon forfixeddr. Withoutlossof
∈ S S
generalitywecanassumethatthegeneralisedLe´vymeasurefactorisesasl(dr,dη) = U(dr,η)µ(dη),
where µ is a measure on . Concretely, wetake µ to be the control measure, see Rajput & Rosinski
S
(1989), definedby
µ(B)= a (B)+b(B)+ min(1,r2)l(dr,B), (7)
| | R
Z
where denotesthetotalvariation. Further, U(dr,η)isaLe´vymeasureforfixedη.
|·|
Notethataandbareabsolutelycontinuouswithrespecttoµandwecanwritea(dη) = a(η)µ(dη),
andb(dη) = b(η)µ(dη).
Forη ,letL′(η)beaninfinitelydivisible randomvariable suchthat e
∈ S
e
C v L′(η) = log E(exp(ivL′(η)) ,
{ ‡ }
(cid:0) (cid:1)
with
1
C v L′(η) =iva(η) v2b(η)+ eivr 1 ivrI (r) U(dr,η), (8)
[−1,1]
{ ‡ } − 2 R − −
Z
(cid:0) (cid:1)
thenwehave e e
C v L(dη) = C v L′(η) µ(dη). (9)
{ ‡ } { ‡ }
Inthefollowing, wewill(asinBarndorff-Nielsen (2011))refertoL′(η)astheLe´vyseed ofLatη.
IfU(dr,η) does notdepend onη, wecalll andLfactorisable. IfLisfactorisable, with Rn
S ⊂
andifa(η),b(η)donotdependonηandifµisproportionaltotheLebesguemeasure,thenLiscalled
homogeneous. Sointhehomogeneouscase,wehavethatµ(dη) = cleb(dη)foraconstantc. Inorder
tosimpelifyteheexposition wewillthroughout thepaperassumethattheconstant inthehomogeneous
caseisgivenbyc = 1.
3.3 Integration concepts withrespect to a Le´vy basis
Since ambit processes are defined as stochastic integrals with respect to a Le´vy basis, we briefly
review in this section in which sense this stochastic integration should be understood. Throughout
the rest of the paper, we work with stochastic integration with respect to martingale measures as
defined by Walsh (1986), see also Dalang & Quer-Sardanyons (2011) for a review. We will review
thistheoryherebrieflyandrefertoBarndorff-Nielsen, Benth&Veraart(2011)foradetailedoverview
8
3 AMBITFIELDSANDPROCESSES
onintegration concepts withrespect toLe´vybases. Notethattheintegration theory duetoWalshcan
beregardedasItoˆ integration extendedtorandomfields.
In the following we will present the integration theory on a bounded domain and comment later
onhowonecanextendthetheorytothecaseofanunbounded domain.
LetS denoteabounded Borelsetin = Rd forad Nandlet(S, ,leb)denote ameasurable
X ∈ S
space,where denotestheBorelσ–algebra onS andlebistheLebesguemeasure.
S
LetLdenoteaLe´vybasisonS [0,T] (Rd+1)forsomeT > 0. Notethat (Rd+1)refersto
× ∈ B B
theBorelsetsgenerated byRd+1 and ()referstothebounded Borelsetsgenerated byS.
b
B ·
ForanyA (S)and0 t T,wedefine
b
∈ B ≤ ≤
L (A) =L(A,t) = L(A (0,t]).
t
×
HereL ()isameasure–valuedprocess,whichforafixedsetA (S),L (A)isanadditiveprocess
t b t
· ∈ B
inlaw.
Inthefollowing, wewanttouse theL (A)asintegrators asinWalsh(1986). Inorder todothat,
t
weworkunderthesquare–integrability assumption, i.e.:
Assumption(A1): ForeachA (S),wehavethatL (A) L2(Ω, ,P∗).
b t
∈ B ∈ F
Notethat,inparticular, assumption (A1)excludes α–stable Le´vybasesforα < 2.
Remark1. Notethatthesquareintegrabilityassumptionisneededforstudyingcertaindynamicprop-
erties of ambit fields, such as martingale conditions. Otherwise one could work with the integration
conceptintroducedbyRajput&Rosinski(1989)(providedthestochasticvolatilityfieldσisindepen-
dent of the Le´vy basis L), which would in particular also work for the case when L is a stable Le´vy
basis.
Next,wedefinethefiltration by
t
F
= ∞ 0 , where 0 = σ L (A) : A (S),0 < s t , (10)
Ft ∩n=1Ft+1/n Ft { s ∈ Bb ≤ }∨N
andwhere denotestheP–nullsetsof . Notethat isright–continuous byconstruction.
t
N F F
Inthefollowing, wewillunlessotherwisestated, workwithoutlossofgenerality underthezero–
meanassumption onL,i.e.
Assumption(A2): ForeachA (S),wehavethatE(L (A)) = 0.
b t
∈ B
One can show that under the assumptions (A1) and (A2), L (A) is a (square–integrable) mar-
t
tingale with respect to the filtration ( ) . Note that these two properties together with the
t 0≤t≤T
F
fact that L0(A) = 0 a.s. ensure that (Lt(A))t≥0,A∈B(Rd) is a martingale measure with respect to
( ) inthesenseofWalsh(1986). Furthermore, wehavethefollowingorthogonality property:
t 0≤t≤T
F
If A,B (S) with A B = , then L (A) and L (B) are independent. Martingale measures
b t t
∈ B ∩ ∅
which satisfy such an orthogonality property are referred to as orthogonal martingale measures by
Walsh(1986), seealsoBarndorff-Nielsen, Benth&Veraart(2011)formoredetails.
Forsuchmeasures, Walsh(1986)introduces theircovariance measureQby
Q(A [0,t]) =< L(A) > , (11)
t
×
for A (Rd). Note that Q is a positive measure and is used by Walsh (1986) when defining
∈ B
stochastic integration withrespecttoL.
Walsh (1986) defines stochastic integration in the following way. Let ζ(ξ,s) be an elementary
randomfieldζ(ξ,s),i.e.ithastheform
ζ(ξ,s,ω) = X(ω)I (s)I (ξ), (12)
(a,b] A
9
3 AMBITFIELDSANDPROCESSES
where 0 a < t, a b, X is bounded and –measurable, and A . For such elementary
a
≤ ≤ F ∈ S
functions, thestochastic integralwithrespect toLcanbedefinedas
t
ζ(ξ,s)L(dξ,ds) := X(L (A B) L (A B)) , (13)
t∧b t∧a
∩ − ∩
Z0 ZB
foreveryB . ItturnsoutthatthestochasticintegralbecomesamartingalemeasureitselfinB (for
∈ S
fixed a,b,A). Clearly, the above integral can easily be generalised to allow for integrands given by
simplerandomfields,i.e.finitelinearcombinationsofelementaryrandomfields. Let denotetheset
T
of simple random fieldsand letthe predictable σ–algebra be the σ–algebra generated by . Then
P T
wecall a random field predictable provided it is –measurable. The aim is now to define stochastic
P
integralswithrespecttoLwheretheintegrandisgivenbyapredictable random field.
InordertodothatWalsh(1986)definesanorm onthepredictable randomfieldsζ by
L
k·k
ζ 2 := E ζ2(ξ,s)Q(dξ,ds) , (14)
k kL
"Z[0,T]×S #
which determines the Hilbert space := L2(Ω [0,T] S, ,Q), and he shows that is dense
L
P × × P T
in . Hence, in order to define the stochastic integral of ζ , one can choose an approxi-
L L
P ∈ P
mating sequence ζ such that ζ ζ 0 as n . Clearly, for each A ,
n n n L
{ } ⊂ T k − k → → ∞ ∈ S
ζ (ξ,s)L(dξ,ds)isaCauchysequenceinL2(Ω, ,P),andthusthereexistsalimitwhichis
[0,t]×A n F
definedasthestochastic integralofζ.
R
Then, this stochastic integral is again a martingale measure and satisfies the following Itoˆ–type
isometry:
2
E ζ(ξ,s)L(dξ,ds) = ζ 2 , (15)
  k kL
Z[0,T]×A !
 
see(Walsh1986,Theorem2.5)formoredetails.
Remark2. Inorder touseWalsh–type integration inthecontext ofambitfields,wenotethefollow-
ing:
General ambit sets A (x) are not necessarily bounded. However, the stochastic integration
t
•
conceptreviewedabovecanbeextendedtounboundedambitsetsusingstandardarguments,cf.
Walsh(1986, p.289).
For ambit fields with ambit sets A (x) ( ,t], we define Walsh–type integrals for
t
• ⊂ X × −∞
integrands oftheform
ζ(ξ,s)= ζ(ξ,s,x,t) = I (ξ,s)h(ξ,s,x,t)σ (ξ). (16)
At(x) s
The original Walsh’s integration theory covers integrands which do not depend on the time
•
index t. Clearly, the integrand given in(16) generally exhibits t–dependence due tothe choice
oftheambitsetA (x)andduetothedeterministickernelfunctionh. Inordertoallowfortime
t
dependenceintheintegrand,wecandefinetheintegralsintheWalshsenseforanyfixedt. Note
that in the case of having t–dependence in the integrand, the resulting stochastic integral is, in
general, notamartingalemeasureanymore. WewillcomebacktothisissueinSection6.
In order to ensure that the ambit fields (as defined in (3)) are well–defined (in the Walsh–sense),
throughout therestofthepaper, wewillworkunderthefollowingassumption:
10