Measure-valued processes for energy markets Christa CuchieroLuca Di PersioFrancesco Guida Sara Svaluto-Ferro

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Measure-valued processes for energy markets

Christa Cuchiero∗Luca Di Persio †Francesco Guida ‡

Sara Svaluto-Ferro §

Abstract

We introduce a framework that allows to employ (non-negative) measure-valued

processes for energy market modeling, in particular for electricity and gas futures.

Interpreting the process’ spatial structure as time to maturity, we show how the

Heath-Jarrow-Morton approach (see Heath et al. (1992)) can be translated to this

framework, thus guaranteeing arbitrage free modeling in inﬁnite dimensions. We

derive an analog to the HJM-drift condition and then treat in a Markovian set-

ting existence of non-negative measure-valued diﬀusions that satisfy this condition.

To analyze mathematically convenient classes we build on Cuchiero et al. (2021)

and consider measure-valued polynomial and aﬃne diﬀusions, where we can pre-

cisely specify the diﬀusion part in terms of continuous functions satisfying certain

admissibility conditions. For calibration purposes these functions can then be pa-

rameterized by neural networks yielding measure-valued analogs of neural SPDEs.

By combining Fourier approaches or the moment formula with stochastic gradient

descent methods, this then allows for tractable calibration procedures which we

also test by way of example on market data. We also sketch how measure-valued

processes can be applied in the context of renewable energy production modeling.

Keywords: HJM term structure modeling; energy markets; (neural) measure-valued

processes, polynomial and aﬃne diﬀusions; Dawson-Watanabe type superprocesses

AMS MSC 2020: 91B72, 91B74, 60J68

Contents

1 Introduction 2

1.1 Notation and basic deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . 6

2 The large ﬁnancial market and the no-arbitrage condition 7

∗Vienna University, Department of Statistics and Operations Research, Data Science @ Uni Vienna,

Kolingasse 14-16, 1090 Wien, Austria, christa.cuchiero@univie.ac.at

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

luca.dipersio@univr.it

‡University of Trento and University of Verona, Department of Mathematics, Via Sommarive 14,

38123 Povo, Italy, francesco.guida@unitn.it

§University of Verona, Department of Economics, Via Cantarane 24, 37129 Verona, Italy,

luca.dipersio@univr.it

The ﬁrst author gratefully acknowledges ﬁnancial support through grant Y 1235 of the START-program.

arXiv:2210.09331v1 [q-fin.MF] 17 Oct 2022

3 A Heath-Jarrow-Morton approach for measure-valued processes 8

4 Measure-valued diﬀusions satisfying the HJM condition 14

4.1 Cylindrical functions and polynomials . . . . . . . . . . . . . . . . . . . . 14

4.2 Directionalderivatives ............................ 14

4.3 Diﬀusion-type operators and martingale problems . . . . . . . . . . . . . 15

4.4 Existence of measure-valued diﬀusions satisfying the HJM drift-condition 15

5 Tractable examples of aﬃne and polynomial type 18

5.1 Black-Scholes type measure-valued HJM-models . . . . . . . . . . . . . . 22

5.2 Aﬃne measure-valued HJM-models . . . . . . . . . . . . . . . . . . . . . 25

5.3 Optionpricing................................. 27

5.3.1 Option pricing in Black-Scholes type measure-valued HJM-models 27

5.3.2 Option pricing in aﬃne measure-valued HJM-models . . . . . . . 28

5.4 Calibration to market option prices . . . . . . . . . . . . . . . . . . . . . 30

6 Measure-valued processes for renewable energy production 32

A Auxiliary results 33

B Existence for martingale problems 35

1 Introduction

In this article we show how to employ non-negative measure-valued processes for energy

market modeling, in particular for electricity and gas futures. Before describing in detail

our approach we start by discussing some important features of these markets, following

Benth et al. (2008).

The most liquidly traded products in electricity, gas, and also temperature markets

are future contracts as well as options written on these. These future contracts deliver the

underlying commodity over a certain period rather than at one ﬁxed instance of time. By

their nature, these contracts are often also called swaps, since they represent an exchange

between ﬁxed and variable commodity prices. We shall however employ the terms future

or future contract throughout.

Our focus lies on these future markets as they exhibit higher liquidity than the spot

energy markets. Indeed, even though it is possible to trade power and gas commodities

on the spot market, one usually faces high storage and transaction costs, which in turn

explains the lack of liquidity on the spot market. As a consequence it is potentially

more diﬃcult to recognize and model features or characteristic patterns of the spot price

behavior. Nevertheless stochastic models for the spot prices are widely applied and also

used to derive the dynamics of future prices based on no-arbitrage principles. Indeed,

this is one of the two main approaches how to to set up tractable mathematical models

for future contracts in power markets. The second approach consists in directly modeling

the complete forward curve by applying a Heath-Jarrow-Morton (HJM) type approach

(see Heath et al. (1992) and Benth and Kr¨uhner (2014) in the context of energy markets).

We shall adapt this second approach and model directly an analog of the forward curve,

however with non-negative measure-valued processes rather than function-valued ones.

In the sequel, we shall omit “non-negative” and only use “measure” for “non-negative

measure”.

There are many mathematical motivations to deal with this kind of processes. In

general, measure-valued processes are often used for modeling dynamical systems for

which the spatial structure plays a signiﬁcant role. In the current setting, time to maturity

takes the role of the spatial structure. This is similar to common forward curve modeling

via stochastic partial diﬀerential equation (SPDE) as for instance in Benth and Kr¨uhner

(2014) or Benth et al. (2021b). The potential advantage of using measure-valued processes

instead of function-valued ones is that many spatial stochastic processes do not fall into

the framework of SPDEs. In addition, it is often easier to establish existence of a measure-

valued process rather than of an analogous SPDE, say in some Hilbert space, which would

for instance correspond to its Lebesgue density, but which does not necessarily exist.

The decisive economic reason for using measure-valued processes in electricity and

gas modeling however lies in the very nature of the future contracts, namely as products

that deliver the underlying commodity always over a certain period instead of one ﬁxed

instance in time. To formulate this mathematically, denote the price of a future with

delivery over the time interval (τ1, τ2] at time 0 ≤t≤τ1by F(t, τ1, τ2). Then, following

Benth et al. (2008), F(t, τ1, τ2) can be written as a weighted integral of instantaneous

forward prices f(t, u) with delivery at one ﬁxed time τ1< u ≤τ2, i.e.

F(t, τ1, τ2) = Zτ2

τ1

w(u, τ1, τ2)f(t, u)du, (1.1)

where w(u, τ1, τ2) denotes some weight function. The crucial motivation for measure-

valued process now comes from the fact that there is no trading with the instantaneous

forwards f(t, u) for obvious reasons. Thus, rather than using f(t, u)du in the expression

of the future prices, we can also use a measure.

An additional motivation stems from the empirically well documented fact that energy

spot prices can jump at predictable times, e.g. due to maintenance works in the power grid

or political decisions which currently inﬂuence these markets substantially. In the context

of such stochastic discontinuities we refer to the pioneering work Fontana et al. (2020)

in the setup of multiple yield curves. To illustrate why these predictable jumps lead to

measure-valued forward processes, consider the simple example where the spot price S

exhibits a jump at the deterministic time t∗. In this case we can write Su=e

Su+J{u≥t∗}

for some jump-size Jwhich is supposed to be Ft∗-measurable1and some process e

Sthat we

suppose for simplicity to be continuous. As the instantaneous forward price is according

to (Benth et al.,2008, equation (1.6)) given by

f(t, u) = EQ[Su|Ft] = EQ[e

Su+J{u≥t∗}|Ft] = EQ[e

Su|Ft] + EQ[J|Ft]{u≥t∗},

for each t < t∗< T and u∈[t, T ], this implies that u7→ f(t, u) is discontinuous at

u=t∗. As forward curve modeling requires to take derivatives with respect to u, the

corresponding SPDE contains a diﬀerential operator and we thus have to deal with dis-

tributional derivatives, leading naturally to measure-valued processes that can be treated

1Here, (Ft)0≤t≤Tdenotes some ﬁltration supporting the spot price process and T > 0 some ﬁnite

time horizon.

with the analysis presented in this paper. We also refer to Assefa and Harms (2022) where

stochastic discontinuities are also used as one motivation for term structure models based

on cylindrical measure-valued processes.

Due to these reasons one of the main goals of this article is to establish a Heath-Jarrow-

Morton (HJM) approach for measure-valued processes. To this end we pass to the Musiela

parametrization and thus consider time to maturity instead of time of maturity.

More precisely, ﬁx a ﬁnite time horizon T, consider a ﬁltered probability space (Ω,F,

(Ft)t∈[0,T ],P), and a measure-valued process (µt)t∈[0,T ]supported on the compact interval

E:= [0, T ]. The measure µtwill play the role of f(t, t+x)dx, meaning that if its Lebesgue

density µt(dx)/dx existed, it would correspond to instantaneous forward prices at time t

with time to maturity x. Then future prices at time tgiven by (1.1) become

F(t, τ1, τ2) = Z(τ1−t,τ2−t]

w(t+x;τ1, τ2)µt(dx), t ∈[0, τ1].

Note that the integration domain does not include the left boundary point τ1−tas we

suppose a delivery over the left-open interval (τ1, τ2]. Observe that the this choice permits

to cumulatively add delivery periods, meaning that,

F(t, τ1, τn) =

n−1

k=1 Z(τk−t,τk+1−t]

w(t+x;τ1, τn)µt(dx), t ∈[0, τ1]

for each τ1<· · · < τn.

The next step is now to specify a suitable no-arbitrage condition. As long as the future

contracts exist for potentially all delivery periods, it is natural to rely on no-arbitrage

conditions for large ﬁnancial markets allowing for a continuum of assets. In this respect

the notion of no asymptotic free lunch with vanishing risk (NAFLVR), as introduced in

Cuchiero et al. (2016b), qualiﬁes as an appropriate and economically meaningful condi-

tion. Mathematically, in our setting, this is equivalent to the existence of an equivalent

(local) martingale measure Q∼Punder which the (discounted) price processes of these

contracts are (local) martingales. Modulo some technical conditions, this can then be

translated to the following HJM-drift condition on the measure-valued process (µt)t∈[0,T ]

(see Theorem 3.1): if there exists an equivalent measure Q∼Psuch that

hφ, µi+Z·

hφ0

s, µsids,

is Q-martingale for all appropriate test functions φ, then NAFLVR holds. Here, the

brackets mean hφ, µi=REφ(x)µ(dx). Note that this is a weak formulation to make sense

out of

“dµt(dx) = d

dxµt(dx)dt +dNt(dx)”,

where Ndenotes a measure-valued local martingale. Indeed, this would correspond

to a (non-existing) strong formulation, well-known from function-valued forward curve

modeling, see e.g. (Benth et al.,2021b, Section 2). Note that even though we here focus

on energy markets, a similar framework can be used for term structure models in interest

rate theory. The HJM-drift condition of course has to be adapted to guarantee that the

bond prices are (local) martingales.

Clearly the HJM-drift condition restricts the choice of the measure-valued process

since the drift part is completely determined, but we are free to specify the martingale

part as long as we do not leave the state space of (non-negative) measures. To establish

existence of measure-valued diﬀusion processes satisfying the drift condition, we rely on

the martingale problem formulation of Ethier and Kurtz (2005) in locally compact and

separable spaces which applies to our setting of measures on a compact set equipped with

the weak-∗-topology. Based on the positive maximum principle we then give suﬃcient

conditions for the existence of such measure-valued diﬀusions (see Theorem 4.3). Addi-

tionally to these requirements we look for tractable speciﬁcations coming from the class

of aﬃne and polynomial processes as introduced in Cuchiero et al. (2021), where we can

precisely specify the diﬀusion part due to suﬃcient and (partly) necessary conditions on

its form. This setup then allows for tractable pricing procedures via the moment formula

well-known from polynomial processes (see e.g. Cuchiero et al. (2012), Filipovi´c and Lars-

son (2016)) and Fourier approaches. This holds in particular true for the aﬃne class as we

obtain explicit solutions for the Riccati PDEs. These pricing methods can then of course

be used for calibration purposes. To this end we parameterize the function valued charac-

teristics of polynomial models as neural networks to get neural measure-valued processes,

an analog to neural SDEs and SPDEs. In Section 5.4 we exemplify such a calibration

for the particular case of an aﬃne measure-valued model which we calibrate to market

call options written on certain future contracts whose prices are obtained from the EEX

German Power data (see https://www.eex.com/en/market-data/power/options).

Apart from modeling energy futures via the described HJM approach, measure-valued

processes also qualify for modeling quantities related to renewable energy production. In

Section 6we exemplify this brieﬂy by considering wind energy markets.

Finally, let us remark that the setting of cylindrical measures considered in Assefa and

Harms (2022) constitutes an interesting alternative approach to term structure modeling.

In contrast to our framework there cylindrical measure-valued processes arising from

SPDEs driven by some cylindrical Brownian motion are used to model the forwards.

Note that being merely a cylindrical measure-valued process is a weaker requirement

than being a measure-valued process. This weakening in turn allows to deﬁne SPDEs via

cylindrical integrals, which would to be possible in the current measure-valued Markov

process setup. One advantage of our approach is that we do not need Lipschitz conditions

on the volatility function (imposed in Assefa and Harms (2022)), which is essential in

order to accommodate aﬃne and polynomial speciﬁcations. Another diﬀerence is that

we consider concrete model speciﬁcations, in particular conditions that guarantee to stay

in the state-space of non-negative measures, while the focus in Assefa and Harms (2022)

lies rather on the abstract framework of cylindrical stochastic integration. Moreover, in

the application to energy contracts, in Assefa and Harms (2022) the time of maturity

parametrization (in contrast to the Musiela parametrization) is considered, which does not

require to specify a HJM drift condition. Note that we could do this as well by specifying

a measure-valued (local) martingale. The disadvantage of this approach is that – in order

to keep the interpretation as forward process – the support of this measure-valued process

should be [t, T ] and thus depends on the running time which is diﬃcult to achieve. We

therefore opted for the HJM approach with time to maturity where the support of the

measure-valued process is constant over time.

The remainder of the paper is organized as follows: in Section 1.1 we introduce

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Measure-valuedprocessesforenergymarketsChristaCuchiero*LucaDiPersioFrancescoGuidaSaraSvaluto-Ferro§AbstractWeintroduceaframeworkthatallowstoemploy(non-negative)measure-valuedprocessesforenergymarketmodeling,inparticularforelectricityandgasfutures.Interpretingtheprocess'spatialstructureastimetomatu...

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Measure-valued processes for energy markets Christa CuchieroLuca Di PersioFrancesco Guida Sara Svaluto-Ferro

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