The market price of jump risk for delivery periods: pricing of electricity swaps with geometric averaging

Kempe , Annika; Schmeck, Ma en Diane
A icle — Published Ve sion
The ma ke p ice o jump isk o deli e y pe iods: p icing
o elec ici y swaps wi h geome ic a e aging
Ma hema ics and Financial Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Kempe , Annika; Schmeck, Ma en Diane (2025) : The ma ke p ice o jump isk o
deli e y pe iods: p icing o elec ici y swaps wi h geome ic a e aging, Ma hema ics and Financial
Economics, ISSN 1862-9660, Sp inge , Be lin, Heidelbe g, Vol. 19, Iss. 2, pp. 293-327,
h ps://doi.o g/10.1007/s11579-025-00383-5
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Ma hema ics and Financial Economics (2025) 19:293–327
h ps://doi.o g/10.1007/s11579-025-00383-5
The ma ke p ice o jump isk o deli e y pe iods: p icing o
elec ici y swaps wi h geome ic a e aging
Annika Kempe 1·Ma en Diane Schmeck1
Recei ed: 4 Janua y 2024 / Accep ed: 19 Janua y 2025 / Published online: 10 Ap il 2025
© The Au ho (s) 2025
Abs ac
In his pape , we ex end he ma ke p ice o isk o deli e y pe iods (MPDP) o elec ici y
swap con ac s by in oducing a dimension o jump isk. As in oduced by Kempe e al.
[30], he MPDP a ises h ough he use o geome ic a e aging while p icing elec ici y swaps
in a geome ic amewo k. We adjus he wo k by Kempe e al. [30] in wo di ec ions: Fi s ,
we examine a Me on ype model aking jumps in o accoun . Second, we ans e he model
o he physical measu e by implemen ing mean- e e ing beha io . We compa e swap p ices
esul ing om he a i hme ic (app oxima ed) a e age o he geome ic weigh ed a e age.
Unde he physical measu e, we disco e a decomposi ion o he swap’s ma ke p ice o isk
in o he ins an aneous ma ke p ice o isk and he MPDP.
Keywo ds Elec ici y swaps ·Deli e y pe iod ·MPDP o di usion and jump isk ·
Mean- e e sion ·Jumps ·Samuelson e ec ·Seasonali y
JEL Classi ica ion G130 ·Q400
1 In oduc ion
Wi h he u n o he millennium, p icing de i a i es on elec ici y has become impo an
h ough he libe aliza ion o ene gy ma ke s. Nowadays, new challenges appea due o he
ansi ion o a clima e neu al ene gy sys em: Elec ici y gene a ed om enewable ene gy
sou ces, like wind and sola ene gy, clea ly depends on he wea he condi ions o he sea-
son. Consequen ly, a ising sha e o enewable ene gy induces s onge in e mi ency and
seasonali y e ec s in luencing especially deli e y-dependen p icing e ec s. In elec ici y
ma ke s, such deli e y-dependen u u es con ac s a e he mos impo an de i a i es. They
deli e he unde lying o e a pe iod o ime since elec ici y is no s o able on a la ge scale.
We would like o hank Ch is a Cuchie o o he ui ul commen s and sugges ions. Financial suppo om
he Deu sche Fo schungsgemeinscha (DFG, Ge man Resea ch Founda ion) - SFB 1283/2 2021 - 3172102
26 is g a e ully acknowledged.
BMa en Diane Schmeck
[email p o ec ed]
Annika Kempe
annika_kempe @web.de
1Cen e o Ma hema ical Economics (IMW) a Biele eld Uni e si y, Biele eld, Ge many
123
294 Ma hema ics and Financial Economics (2025) 19:293–327
We he e o e call hem elec ici y swaps. The dependence on he deli e y ime a ec s he
p ice dynamics, he p icing measu e, and he swap’s ma ke p ice o isk o deli e y pe iods
(MPDP) in oduced by Kempe e al. [30]. In his pape , we p o ide an ex ension o he
MPDP. To do so, we adjus he model o a Me on ype model aking jumps in o accoun .
In addi ion, unde he physical measu e, we iden i y a decomposi ion o he swap’s ma ke
p ice o isk in o he ins an aneous ma ke p ice o isk and he MPDP.
The deli e y pe iod is a unique ea u e o elec ici y ma ke s ha di e s om o he com-
modi ies such as oil, gas, o co n. In ac , i plays a c ucial ole in he p icing o elec ici y
swaps. Following he ma ke model app oach, he elec ici y swap p ice esul s om a e -
aging an ins an aneous s eam o u u es wi h espec o he deli e y ime. This app oach
goes back o he amous model by Hea h e al. [23]. I was i s ly connec ed o ene gy- ela ed
de i a i es by Clewlow and S ickland [15] and o elec ici y de i a i es by Bje ksund e al.
[9] ollowed by a ow o wo ks (see, e.g., Koekebakke and Ollma [34], Ben h and Koeke-
bakke [5], Bje ksund e al. [9], Ben h e al. [1], and Kempe e al. [30] o geome ic se ings,
Hinde ks e al. [25] o a s uc u al model and Cuchie o e al. [17] measu e- alued p ocesses).
One s eam o li e a u e in es iga es spo based p ice dynamics and de i e elec ici y u u es
based on he spo p ice e e ing o he day ahead ma ke (see, e.g., Ca ea and Figue oa,
[13], Ca ea and Villaplana [14], Esc ibano e al. [19]). In his pape , we ocus on a HJM- ype
app oach modelling he u u es ma ke di ec ly. Tha is we conside so-called a omic swap
con ac s inducing a deli e y pe iod o a mon h, ha a e used o p ice o e lapping swap
con ac s deli e ing o example o e a qua e o a yea . We e e o Ben h e al. [1]and
Ben h and Koekebakke [5] and Kempe e al. [30] o a cons uc ion o o e lapping swap
con ac s based on a omic swaps.
The deli e y pe iod can be inco po a ed in di e en ways o a e aging. We dis inguish
be ween h ee ypes o a e aging: A i hme ic, app oxima ed, and geome ic a e aging. A i h-
me ic a e aging is heclassicalway oimplemen heswap’sdeli e ype iodandiscon enien
o a i hme ic p ice dynamics. In pa icula , con inuous a i hme ic a e aging is applied by
Ben h e al. [3], Ben h and Koekebakke [5] Ben h e al. [1], Ben h e al. [4], Ben h e al. [7],
Kleisinge -Yu e al. [32], Koekebakke and Ollma [34], and La ini e al. [35], among o he s.
Fo disc e e a i hme ic a e aging, we e e o Lucia and Schwa z [37] and Bu ge e al. [12].
Ins ead, a i hme ic a e aging o geome ic p ice dynamics is poo ly sui ed since he esul ing
swap p ice dynamics a e nei he geome ic no Ma ko ian. I equi es, e.g., an app oxima ion
o he swap p ice ola ili y in oduced by Bje ksund e al. [9] whene e we wan o conside
ac able swap p ice dynamics (see also Ben h e al. [1], Ben h and Koekebakke [5]). We
call his p ocedu e app oxima ed a e aging.Geome ic a e aging, ins ead, does no equi e
any app oxima ions whene e he p ice dynamics a e o geome ic ype and lead o sui able
geome ic dynamics (see Kempe e al. [30]). Hence, he geome ic a e age is ailo -made
o ela i e g ow h a e models. Ne e heless, he geome ic a e age does no p ese e he
ma ingale p ope y. This issue is ackled by Kempe e al. [30] using a measu e change wi h
hei MPDP. Usually, nega i e p ices a e no obse able in he da a o he u u es p ices,
such ha we s ick o a geome ic se ing and compa e he la e a e aging p ocedu es while
adjus ing he MPDP o a Me on ype model.
Bo h pape s, Kempe e al. [30] and Bje ksund e al. [9], in es iga e he modeling o he
deli e y pe iod explici ly h ough a con inuous weigh ed a e aging app oach o geome ic
u u es p ices. Bo h app oaches lead o Ma ko ian and geome ic swap p ice dynamics. We
discuss simila i ies and di e ences be ween hese app oaches and in oduce a numé ai e
caused by he di e en a e aging echniques in Sec .2. In line wi h he ma ke model
app oach, we base he a e aging p ocedu e on a con inuous s eam o u u es con ac s ha
is a ma ingale unde he u u es isk-neu al measu e Q. As he u u es ha e ins an aneous
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Ma hema ics and Financial Economics (2025) 19:293–327 295
Fig. 1 Measu e changes be ween he physical measu e P, he ins an aneous isk-neu al measu e Q,and
he swap’s p icing measu e 
Qas well as hei connec ions wi h he swap’s ma ke p ices o isk PQ, he
ins an aneous ma ke p ice o isk P
Q, and he MPDP deno ed by Q
Q
deli e y, we e e o Qas he ins an aneous isk neu al measu e. The esul ing swap p ice
dynamics based on geome ic a e aging a e no a ma ingale unde Q.We hende ine he
MPDP o di usion and jump isk and a new p icing measu e 
Q, which can hus be used o
p ice de i a i es on he swap. We may e e o 
Qas he “swap’s” isk-neu al measu e since
he swap p ice is a 
Q-ma ingale wi hou any app oxima ions.
The e o e, we call Qalso he “ins an aneous” isk-neu al measu e. I is a clea ad an-
age ha he app oxima ed a e age p ese es he ma ingale p ope y o he swap unde he
measu e Q. A decomposi ion o he ma ke p ice o isk o elec ici y swaps a ises when
u ning o he physical measu e P. Figu e1gi es an o e iew o e he connec ions be ween
he di e en measu es P,Q,and
Qand he swap’s ma ke p ice o isk PQ, he ins an aneous
ma ke p ice o isk P
Q, and he MPDP deno ed by Q
Q.
Indeed, he MPDP is igge ed by ypical ea u es o he elec ici y ma ke en e -
ing he swap’s ola ili y. In pa icula , deli e y-dependen e ec s like seasonali ies and
e m-s uc u e e ec s play a c ucial ole. Fanelli and Schmeck [20] empi ically iden i y
seasonali ies in he swap’s deli e y pe iod by conside ing implied ola ili ies o elec ici y
op ions. Renewable ene gy, like wind and sola ene gy, in ensi y especially he seasonal
e ec s men ioned be o e.
An addi ional p ope y o elec ici y and commodi y ma ke s is he Samuelson e ec (see
Samuelson [43]): The close we each he end o he ma u i y, he mo e e ec he ola ili y
has. Ben h and Pa aschi [6] and Jaeck and Lau ie [27] p o ide empi ical e idence o he
Samuelson e ec in he ola ili y e m-s uc u e o elec ici y swaps. I can also be obse ed
in he implied ola ili ies o elec ici y op ions, especially a ou and in he money (see
Kiesel e al. [31]). Kempe e al. [30] cha ac e ize he MPDP o such seasonali ies and
e m-s uc u e e ec s wi hin a s ochas ic ola ili y model h ough he a iance pe uni o
expec a ion o he deli e y-dependen e ec s. We con ibu e o he li e a u e by in es iga ing
he MPDP analy ically, a ec ed by seasonali ies and he Samuelson e ec . Mo eo e , we lay
he ounda ion o he empi ical analysis o he MPDP by speci ying he model unde he
eal wo ld measu e P.
Fu he cha ac e is ics o he obse e d elec ici y swap p ices a e mean- e e sion and
jump beha io . As men ioned by La ini e al. [35] and Kleisinge -Yu e al. [32] among
o he s, mean- e e sion is an impo an p ope y o he elec ici y swap p ices. Koekebakke
and Ollma [34] empi ically alida e ha he sho - e m p ice a ies a ound he long- e m
p ice, which con i ms mean- e e ing beha io . As Ben h e al. [7], we ace he p oblem
o changing a mean- e e ing p ocess o he isk-neu al measu e. We ex end hei measu e
change o he geome ic se ing. We e en p o ide a p oo o s ochas ic ola ili y se ings in
o de o add ess models such as Kempe e al. [30] and Schneide and Ta in [44]. Besides
mean- e e sion, Ben h e al. [7] include jumps as an ou s anding cha ac e is ic o elec ici y
p ices. They conside compound Poisson p ocesses unde he physical measu e in a mean-
e e ing, a i hme ic se ing. While adjus ing he pape by Kempe e al. [30] o jumps, we
123
296 Ma hema ics and Financial Economics (2025) 19:293–327
es ablish he MPDP o jump isk whene e he jump coe icien elies on deli e y-dependen
e ec s.
In his pape , we ollow a so-called Hea h-Ja ow-Mo on app oach o model o wa d
ma ke s, ha iswede ine heswapwi hdeli e ype iodasa e ageo e anin ini edimensional
s eam o u u es wi h ins an aneous deli e y. No e ha hese u u es a e no aded a he
ma ke , only he swaps wi h deli e y pe iod a e aded. In pa icula , i is no possible o
obse e aded quo es o he u u es and i is no possible o see i he spo p ice con e ges o
he u u es i ime app oaches ma u i y o a u u es. In ac , his ela ionship ypically does
no hold ue o he aded swaps ei he due o he deli e y pe iod (see e.g. Ben h e al. [1]).
Ano he app oach o model he o wa d ma ke is o s a wi h a model o he spo and de ine
he swap p ice as condi ional expec a ion o he a e age spo du ing he deli e y pe iod,
whe e he expec a ion is aken unde some p obabili y measu e ha is equi alen o he eal
wo ld measu e P. No e ha i is no necessa y o ha e a ma ingale measu e o he spo :
as he unde lying elec ici y is no s o able on a la ge scale, i has o be consumed once
pu chased. In pa icula , i is no possible o se up buy- and hold-s a egies, ha a e equi ed
in no-a bi age po olios. In his sense, he elec ici y spo is said o be “no adable” (see
e.g. Ben h e al. [1]).
Ou con ibu ion o he li e a u e is wo old: Fi s , we adjus he pape by Kempe e al.
[30] o he jump case unde he ins an aneous isk-neu al measu e leading o an ex ended
cha ac e iza ion o he MPDP ega ding di usion and jump isk. Second, we ans e he
model o he physical measu e P. Unde Pwe compa e he swap p ices esul ing om
geome ic and app oxima ed a e aging as well as hei isk-neu al measu es e ealing he
decomposi ion o he swap’s ma ke p ice o isk in o he ins an aneous ma ke p ice o isk
and he MPDP. Consequen ly, he model lays he ounda ion o empi ical in es iga ions in
he u u e.
Thepape is o ganizedas ollows: Sec .2p esen s he geome ica e agingapp oachunde
he ins an aneous isk-neu al measu e applied o he jump- ype u u es cu e. In addi ion,
i p esen s he MPDP o di usion and jump isk. Sec ion4in oduces he model unde he
physical measu e and iden i ies he decomposi ion o he swap’s ma ke p ice o isk. An
example unde he physical measu e closes he sec ion.
Finally, Sec .5concludes ou main indings.
2 On he MPDP o diffusion and jump isk
We pa icula ly ocus on an elec ici y swap con ac deli e ing 1 MWh o elec ici y du ing
he ag eed deli e y pe iod (τ1,τ
2]. A a ading day ≤τ1be o e he con ac expi es, we
deno e he swap p ice by F( ,τ
1,τ
2)se led such ha he con ac is en e ed a no cos . I can
be in e p e ed as an a e age p ice o ins an aneous deli e y. Mo i a ed by his in e p e a ion,
we conside a u u es con ac wi h p ice ( ,τ) ha s ands o ins an aneous deli e y a
ime τ∈(τ1,τ
2]. No e ha such a con ac does no exis on he ma ke bu i u ns ou o be
use ul o modeling pu poses when conside ing deli e y pe iods (see, e.g., Ben h e al. [7]
and Kempe e al. [30]).
Following he app oach by Hea h e al. [23], we de i e he p ice o an elec ici y swap
con ac based on an ins an aneous u u es p ice model. Mo e p ecisely, we compa e wo
ypes o swap p ices esul ing om geome ic and app oxima ed a e aging. The goal o his
sec ion is o in es iga e he p icing sp ead be ween bo h app oaches in o de o quan i y he
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Ma hema ics and Financial Economics (2025) 19:293–327 297
consequenceso he app oxima ionand hus he e ec o hep ecise geome ica e agingp o-
cedu e. As he p icing sp ead goes along wi h di e en isk-neu al measu es, we addi ionally
in es iga e he dis ance o bo h isk-neu al measu es quan i ied by he MPDP. Mo eo e , we
cha ac e ize he MPDP o speci ic ola ili y unc ions and di e en jump size dis ibu ions.
Be o e doing so, we would like o epea he main ideas o he MPDP in oduced in Kempe
e al. [30].
2.1 The idea o he MPDP
The p ocedu e o he de i a ion o he swap’s ma ingale measu e used in his pape goes
back o Kempe e al. [30] s a ing wi h a u u es con ac wi h ins an aneous deli e y. The
dynamics o he u u es p ice ( ,τ)is gi en by
d ( ,τ)=σ(ω, ,τ) ( ,τ)dWQ
.(2.1)
Kempe e al. [30] in es iga e he gap be ween he swap’s ma ingale measu e associa ed wi h
wo me hodologies o implemen ing he deli e y pe iod: a i hme ic weigh ed a e age wi h
app oxima ion and he geome ic weigh ed a e age wi hou app oxima ion. The geome ic
weigh ed a e age c ea es a d i e m. Mo e p ecisely, he swap p ice dynamics unde Q
esul ing om geome ic a e aging wi hou app oxima ions e ol e as
dF( ,τ
1,τ
2)
F( ,τ
1,τ
2)=−1
2EUσ(ω, ,U)2−EU[σ(ω, ,U)]2d +EU[σ(ω, ,U)]dWQ
,
(2.2)
whe e he swap’s ola ili y is gi en by
EU[σ(ω, ,U)]=τ2
τ1
w(u,τ
1,τ
2)σ(ω, ,u)du .(2.3)
We e e o a mo e de ailed discussion and mo i a ion o he no a ion o Sec .2.3 below.
This addi ional d i e m gi es eason o he exis ence o he MPDP whene e he u u es
ola ili y depends on he deli e y pe iod. In pa icula , he MPDP is de ined by
Q
Q
1:= −1
2
VU[σ(ω, ,U)]
EU[σ(ω, ,U)],(2.4)
a ec ing he B ownian mo ion WQ. The MPDP in oduced by Kempe e al. [30] is essen ial
o change he measu e o he swap’s isk-neu al measu e 
Q. While Kempe e al. [30]base
he de i a ion o he elec ici y swap and he MPDP on a geome ic dynamics wi h s ochas ic
ola ili y in he spi i o Hes on [24], his pape ocuses on a geome ic jump di usion wi h
de e minis ic ola ili y. Mo eo e , his pape in es iga es no only he connec ion be ween
he u u es’ and swap’s isk neu al measu e as in Kempe e al. [30] bu c ea es also a b idge
o he physical measu e.
2.2 The model
Conside a il e ed p obabili y space (, F,(F ) ∈[0,τ],Q), whe e he il a ion sa is ies he
usual condi ions. We i s model he solu ion o a u u es con ac and hen de i e he co e-
sponding dynamics o a oid lacks o exis ence in he p esence o jumps (see Papapan oleon
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298 Ma hema ics and Financial Economics (2025) 19:293–327
[40]). A ime ≤τ, le he loga i hmic p ice p ocess o he u u es con ac be de ined as
ln ( ,τ)=ln (0,τ)+
0
σ(s,τ)dWQ
s+
0
η(s,τ)d
JQ
s−
0
cQ(s,τ)ds ,(2.5)
wi h ini ial non- andom condi ions (0,τ) > 0. Mo eo e , WQis a one-dimensional s an-
da d B ownian mo ion unde Qindependen o he jump p ocess 
JQ. In pa icula , 
JQis
a compound compensa ed jump p ocess de ined h ough he compensa ed Poisson andom
measu e 
NQ(d ,dz)=N(d ,dz)−Q(dz)d :

JQ
=
0R
z
NQ(ds,dz), (2.6)
wi h Lé y measu e Q(dz)=λQG(dz), which is independen o he deli e y ime, and
whe e λQ>0 indica es he jump in ensi y and G(dz) he jump size dis ibu ion. The las
e m in Equa ion (2.5) de ines he compensa o o he loga i hmic e u n unde he cu en
measu e Q:
cQ( ,τ)=1
2σ2( ,τ)+ψQ(iη( ,τ)), (2.7)
whe e ψQ(i )is he in eg and o he Lé y-Khin chine exponen ial de ined h ough he
momen gene a ing unc ion
ψQ( ):= Re z −1− zQ(dz). (2.8)
We assume ha he u u es p ice ola ili y and jump coe icien s, σ( ,τ) and η( ,τ),a e
de e minis ic and ha he u u es p ice ( ,τ) is F -adap ed o ∈[0,τ].We u he
assume ha heysa is ysui able in eg abili yand measu abili ycondi ions(see Assump ion 1
in Appendix A o de ails) o ensu e ha he p ocess in Equa ion (2.5)isaQ-ma ingale, and
ha Equa ion (2.5) gi es he unique solu ion o he p ocess e ol ing as
d ( ,τ)
( −,τ) =σ( ,τ)dWQ
+Reη( ,τ )z−1
NQ(d ,dz). (2.9)
As σ( ,τ) depends on bo h, ading ime and deli e y ime τ, we allow o ola ili y
s uc u es as he Samuelson e ec o seasonali ies in he deli e y ime, which a e add essed
in Examples 3.1 and 3.2.
2.3 Implemen ing he deli e y pe iod
Following he Hea h-Ja ow-Mo on app oach o p ice u u es and swaps in elec ici y ma -
ke s, he swap p ice is usually de ined as he a i hme ic weigh ed a e age o u u es p ices
(see, e.g., Ben h e al. [1], Bje ksund e al. [9], and Ben h e al. [7]):
FA( ,τ
1,τ
2):= τ2
τ1
w(u,τ
1,τ
2) ( ,u)du,(2.10)
o a gene al weigh unc ion
w(u,τ
1,τ
2):= ˆw(u)
τ2
τ1ˆw( )d , o u∈(τ1,τ
2],(2.11)
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Ma hema ics and Financial Economics (2025) 19:293–327 299
whe e ˆw:(τ1,τ
2]→R+
0is he co esponding se lemen unc ion which is de e minis ic,
in eg able and non-nega i e. No e ha wde ines a p obabili y densi y unc ion wi h suppo
on (τ1,τ
2]since i is posi i e and in eg a es o one, ha is τ2
τ1w(u,τ
1,τ
2)du =1. Hence,
we deno e Uas a andom deli e y a iable wi h densi y w(u,τ
1,τ
2)(see also Kempe e al.
[30]). The mos popula example is gi en by a cons an se lemen ype ˆw(u)=1, such ha
he densi y becomes w(u,τ
1,τ
2)=1
τ2−τ1and U∼U((τ1,τ
2])is uni o mly dis ibu ed o e
he deli e y pe iod. This co esponds o a one- ime se lemen . A con inuous se lemen o e
he ime in e al (τ1,τ
2]is co e ed by a con inuous discoun unc ion ˆw(u)=e− u,whe e
is he cons an in e es a e (see, e.g., Ben h e al. [1]).
Thea i hme ica e ageo he u u esp iceasinEqua ion (2.10)leads o ac abledynamics
o he swap as long as one assumes an a i hme ic s uc u e o he u u es p ices as well. This
is based on he ac ha a i hme ic a e aging is ailo -made o absolu e g ow h a e models.
Ne e heless, i one de ines he u u es p ice as a geome ic p ocess as in Equa ion (2.9), one
can show ha he dynamics o he swap p ice FAde ined h ough Equa ion (2.10)a egi en
by
dFA( ,τ
1,τ
2)
FA( −,τ
1,τ
2)=σ( ,τ
2)−τ2
τ1
∂σ
∂u( ,u)w(τ, τ1,τ
2)
w(τ, τ1,u)
FA( ,τ
1,u)
FA( ,τ
1,τ
2)dudWQ
+Reη( ,τ2)z−1−τ2
τ1
∂eη(s,u)z
∂u
w(τ, τ1,τ
2)
w(τ, τ1,u)
FA( ,τ
1,u)
FA( ,τ
1,τ
2)du
NQ(dz,d ),
(2.12)
o any τ∈(τ1,τ
2](see Ben h e al. [1], c . Chap e 6.3.1). Thus, he dynamics o he swap
p ice is nei he a geome ic p ocess no Ma ko ian, which makes i unhandy o u he
analysis. To o e come his issue, Bje ksund e al. [9] sugges an app oxima ion in he se up
wi hou jumps, which we call app oxima ed a e aging since i is he a i hme ic a e age o
app oxima ed loga i hmic e u ns. App oxima ed a e aging main ains he ma ingale p op-
e y meaning ha he swap is a ma ingale whene e is a ma ingale. I we ans e he
app oxima ed a e aging p ocedu e o ou jump se ing, we can de ine he swap p ice p ocess
based on app oxima ed a e aging by
dFa( ,τ
1,τ
2)
Fa( −,τ
1,τ
2):= τ2
τ1
w(u,τ
1,τ
2)d ( ,u)
( −,u)du .(2.13)
Incon as ,geome ic a e aging o igina es om he a i hme ica e ageo loga i hmic e u ns
wi hou any need o app oxima ions. Hence, in line wi h Kempe e al. [30], we de ine he
swap p ice o igina ing om geome ic a e aging by
F( ,τ
1,τ
2):= eτ2
τ1w(u,τ1,τ2)ln ( ,u)du ,(2.14)
(see also Kemna and Vo s [29]). Assume ha he ola ili y and jump coe icien s sa is y
u he in eg abili y condi ions (see Assump ion 2in Appendix A). I u ns ou , ha he
esul ing swap p ice dynamics is a geome ic p ocess wi h a non-ze o d i e m:
Lemma 2.1 (The Swap P ice unde Q) Le Assump ion 2in Appendix A be sa is ied. Unde
he ins an aneous p icing measu e Q, he dynamics o he swap p ice p ocess F(·,τ
1,τ
2),
de ined in Equa ion (2.14), a e gi en by
dF( ,τ
1,τ
2)
F( −,τ
1,τ
2)=E[σ( ,U)]dWQ
+ReE[η( ,U)]z−1
NQ(d ,dz)
−1
2V[σ( ,U)]+E[ψQ(η( ,U))]−ψQ(E[η( ,U)])d ,
(2.15)
123
300 Ma hema ics and Financial Economics (2025) 19:293–327
whe e U deno es he andom deli e y a iable wi h densi y w(u,τ
1,τ
2).
P oo Plugging he in eg al ep esen a ion o he u u es a e p ocess om Equa ion (2.5)
in o Equa ion (2.14)gi esusF( ,τ
1,τ
2)=F(0,τ
1,τ
2)e¯
X( ,τ1,τ2), whe e an applica ion o
he s ochas ic Fubini Theo em (see P o e [42], c . Theo em 65, Chap e IV.6) leads o
¯
X( ,τ
1,τ
2)=
0
Eσ(s,U)dWQ
s
+
0
E[η(s,U)]d
JQ
s−1
2
0
Eσ2(s,U)ds −
0
E[ψQ(η(s,U))]ds .(2.16)
Then, Equa ion (2.15) ollows using I ô’s o mula (see, e.g., Øksendal and Sulem [39]). 
Ha ing p esen ed he h ee p ocedu es o con inuous ime a e aging ha a e used o de i e
he swap om an unde lying u u es cu e, we would like o compa e hem: A i hme ic
a e aging, de ined by Equa ion (2.10), is ac able o a i hme ic u u es cu es, whe eas
app oxima ed a e aging, de ined by Equa ion (2.13), and geome ic a e aging, de ined by
Equa ion (2.14), a e well sui ed o geome ic u u es cu es. In line wi h a se ies o li e a u e
(see Koekebakke and Ollma [34], Ben h and Koekebakke [5], Bje ksund e al. [9], Ben h
e al. [1], and Kempe e al. [30]), we ollow he geome ic app oach. Ou goal h oughou
his pape is o in es iga e he p icing sp ead be ween geome ic and app oxima ed a e aging
analy ically.
3 The MPDP
Al hough he u u es p ice and he app oxima ed Faa e ma ingales unde he p icing
measu e Q, he swap p ice Fis no a Q-ma ingale: Indeed, he swap p ice p ocess unde Q
hasanega i ed i e mconsis ingo wopa sgi enby heswap’s a ianceand hedi e ence
be ween he a e aged Lé y-Khin chine in eg and and he Lé y-Khin chine in eg and o he
a e aged jump coe icien . Hence, using geome ic a e aging leads o a new in e p e a ion
o isk ela ed o he deli e y pe iod as we will analyze in he ollowing.
Analogous o Kempe e al. [30], we de i e he co esponding isk-neu al measu e 
Q
unde which he elec ici y swap p ice Fis a ma ingale. Fo de i ing he swap’s isk-neu al
measu e, we hus de ine he MPDP ex ended o jumps in he ollowing.
De ini ion 3.1 [(The MPDP)] A ime ∈[0,τ
1], hema ke p ice o di usion and jump
isk o deli e y pe iods associa ed o he deli e y pe iod (τ1,τ
2]is de ined by Q
Q:=
(Q
Q
1,
Q
Q
2),whe e
Q
Q
1( ,τ
1,τ
2):= −1
2
V[σ( ,U)]
E[σ( ,U)],(3.1)
Q
Q
2( ,τ
1,τ
2):= −RE[eη( ,U)z]−eE[η( ,U)]zQ(dz)
ReE[η( ,U)]z−1Q(dz).(3.2)
In gene al, he MPDP does no coincide wi h he ma ke p ice o isk. In ac , i is an addi-
ional isk ha has o be aken in o accoun whene e app oxima ed a e aging is conduc ed.
Technically speaking, he MPDP cha ac e izes he dis ance be ween he ma ingale measu e
o he swaps, Fand Fa, esul ing om geome ic and app oxima ed a e aging. In pa icula ,
1 e e s o he addi ional di usion isk, which is measu able and F -adap ed as σ( ,u)is.
123
Ma hema ics and Financial Economics (2025) 19:293–327 307
−1
2E[σ( ,U)]2+R
E[eη( ,U)z]−1−ln E[eη( ,U)z]d .
Taking Eq.2.16 in o accoun , he p icing sp ead is gi en by
Fa( ,τ
1,τ
2)−F( ,τ
1,τ
2)=Fa( ,τ
1,τ
2)(1−D( ,τ
1,τ
2)),
such ha F( ,τ
1,τ
2)=Fa( ,τ
1,τ
2)D( ,τ
1,τ
2),whe e
D( ,τ
1,τ
2)=e¯
X( ,τ1,τ2)−¯
Xa( ,τ1,τ2)=e−1
2
0V[σ(s,U)]ds−
0Rln E[eη(s,U)z]−E[η(s,U)]zN(ds,dz).
(iii)I η( ,u)⊥⊥ u, hen
D( ,τ
1,τ
2)=e−1
2
0V[σ(s,U)]ds .
Since V[σ(·,U)]≥0 by Jensen, i ollows ha D( ,τ
1,τ
2)∈(0,1]and hus F≤Fa.

We conclude ha a i hme ic and in speci ic cases app oxima ed a e aging lead o highe
swap p ices han he geome ic a e age. We would like o s ess ha Din Equa ion (3.26)
is no a ec ed by measu e changes since i is cha ac e ized by a d i componen and a pu e
jump componen exclusi ely (see also Equa ion (3.28)). Mo eo e , no e ha Dcan be seen
as s ochas ic discoun ac o , which can be used o de i e he swap p ice Fgi en Fa.Vice
e sa, conside
Fa( ,τ
1,τ
2)=F( ,τ
1,τ
2)D−1( ,τ
1,τ
2). (3.27)
The exponen ial pa o D−1can be in e p e ed as a p ice (p emium) pe sha e, which we
pay o an imp ecise a e aged swap. Mo eo e , we can see Das he p ice p ocess o a
non-di idend paying asse e ol ing as
dD( ,τ
1,τ
2)
D( −,τ
1,τ
2)=−1
2V[σ( ,U)]d +ReE[η( ,U)]z
E[eη( ,U)z]−1N(d ,dz), (3.28)
such ha we can in e p e Das a numé ai e. I Fais a ma ingale, hen F
Dis also a ma ingale.
I Fis a ma ingale, hen FaDis a ma ingale (see, e.g., Sh e e [46], c . Theo em 9.2.2). We
can hus use i o p ice op ions and o he de i a i es on he swap. In he subsequen sec ion,
we in oduce he model unde i s physical measu e P.
Rema k 3.2 (Deli e y-Dependen In ensi y) Le us conside an adjus ed e sion o he u u es
p ice unde he ins an aneous p icing measu e Qsimila o Equa ion (2.9)gi enby
d ( ,τ)
( −,τ) =σ( ,τ)dWQ
+Reηz−1
Nτ(d ,dz), (3.29)
whe e η∈Rand he compensa ed Poisson andom measu e is de ined by 
Nτ(d ,dz):=
N(d ,dz)−λQ(τ)G(dz)d , wi h a jump in ensi y adjus ed o a de e minis ic, posi i e, and
bounded unc ion o he deli e y ime.
(i)Analogous o Lemma 2.1, he dynamics o he swap p ice, de ined by geome ic
a e aging, a e gi en by
dF( ,τ
1,τ
2)
F( −,τ
1,τ
2)=E[σ( ,U)]dWQ
+Reηz−1N(d ,dz)
−1
2V[σ( ,U)]+E[λQ(U)]Reηz−1G(dz)d ,
(3.30)
123

308 Ma hema ics and Financial Economics (2025) 19:293–327
whe e Uis he andom deli e y a iable wi h densi y w(u,τ
1,τ
2).
(ii)I he ola ili y is independen o deli e y ime, hen he app oxima ed and geome ic
a e age coincide and so hei isk-neu al p icing measu e. Howe e , he esul ing
swap p ice p ocess in Equa ion (3.30) is no a ma ingale unde Qsince he in ensi y
is a ec ed by he a e aging p ocedu e.
(iii)In he case o deli e y-dependen ola ili y, he MPDP om De ini ion 3.1 adjus s
o (Q
Q
1,0). Hence, he MPDP associa ed o he B ownian mo ion s ays he same
and i s second dimension becomes ze o since he jump coe icien ηis independen o
deli e y ime. Howe e , unde he assump ion ha he in ensi y is deli e y-dependen ,
he swap p ice is in gene al no a ma ingale unde 
Q.
(i ) The swap p ice is a 
Q−ma ingale, only i he swap’s jump in ensi y unde 
Qis
gi en by E[λQ(U)], i.e., i 
Nτ1,τ2(d ,dz):= N(d ,dz)−E[λQ(U)]G(dz)d ,isa
compensa ed Poisson andom measu e unde 
Q.
4 The eal-wo ld model
A ypical ea u e o elec ici y p ices beyond seasonali ies and he Samuelson e ec is he
mean- e e ing beha io (see, e.g., Ben h e al. [1] and Ben h e al. [7]). In o de o implemen
he d i ea u e, we de i e he u u es unde he physical measu e P. No e ha we will include
mean- e e sion a he u u es and hus he swap’s a e le el. We hen conside he esul ing
ma ke p ices o isk ans e ing o he ins an aneous and he swap’s isk-neu al measu e.
4.1 The swap p ice unde he physical measu e
We now de i e he p ice o a swap con ac ha deli e s one uni o elec ici y du ing he
ixed deli e y pe iod (τ1,τ
2], simila o Sec .2bu now unde he physical measu e P. Hence,
s a ing om he physical measu e P, he loga i hmic u u es p ice p ocess om Equa ion
(2.5), gi en by
ln ( ,τ)=e−
0κ(s)ds ln (0,τ)+
0
e−
κ(q)dqμ( , τ)d
+
0
e−
κ(q)dqσ( ,τ)dWP
+
0
e−
κ(q)dqη( , τ)d
JP
,
(4.1)
whe e WPis a B ownian mo ion unde he physical measu e Pindependen o he compound
compensa ed jump p ocess 
JP. In pa icula , 
JPis de ined h ough he P-compensa ed Pois-
son andom measu e 
NP(d ,dz)=N(d ,dz)−P(dz)d wi h Lé y measu e P(dz)=
λPG(dz) ha is independen o deli e y ime. No e ha λP>0 indica es he jump in ensi y
unde he physical measu e and G(dz)is he jump size dis ibu ion.
In o de o cha ac e ize he u u es p ice in mo e de ail, we in oduce he ollowing lemma.
Lemma 4.1 We assume ha he coe icien s sa is y sui able in eg abili y and measu abili y
condi ions (see Assump ion 4in Appendix A) such ha
Equa ion 4.1 is he unique s ong solu ion o he dynamics
d ( ,τ)
( −,τ) =σ( ,τ)dWP
+Reη( ,τ )z−1
NP(d ,dz)+cP( ,τ,ln ( ,τ))d ,(4.2)
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Ma hema ics and Financial Economics (2025) 19:293–327 309
whe e he d i - e m is cha ac e ized by
cP( ,τ,Y)=μ( ,τ)−κ( )Y+1
2σ( ,τ)
2+ψP(η( ,τ)). (4.3)
Hence, he loga i hmic u u es e ol es as
dln ( ,τ)=(μ( ,τ)−κ( )ln ( ,τ)
)d +σ( ,τ)dWP
+η( ,τ)d
JP
.(4.4)
P oo The unique s ong solu ion ollows om Ben h e al. [1] (c . P oposi ion 3.1). Applying
I o’s o mula leads o he desi ed dynamics (see Øksendal and Sulem [39], c . Theo em 1.16).

No e ha he assump ion behind he model induces a ini e second momen as well as a
ini e momen gene a ing unc ion o he jump size dis ibu ion. In Examples 3.3 o 3.6,we
conside sui able dis ibu ions o hese jump sizes.
Rema k 4.1 In he li e a u e, we some imes ind he applica ion o logno mal dis ibu ed
jump sizes (see, e.g., Bo o ko a and Pe mana [10] and Bo o ko a and Pe mana [11]). This
dis ibu ion, howe e , is no sui able o ou se ing since i s momen gene a ing unc ion
E[eηZ]is no ini e a any posi i e alue η(see, e.g., G ay and Pi s [22], c . Chap e 2.2.6).
Hence, helogno maldis ibu ioncon adic s hein eg abili yassump ioninAssump ion4(i)
unde he physical measu e in Appendix A.
As in he p e ious sec ion, we now de i e he swap p ices esul ing om geome ic and
app oxima ed a e aging.
Lemma 4.2 [The Swap P ice unde P] Le Assump ion 4and 5in Appendix A be sa is ied.
Then,
he swap p ice based on geome ic a e aging e ol es as
dF( ,τ1,τ2)
F( −,τ1,τ2)=E[σ( ,U)]dWP
+ReE[η( ,U)]z−1
NP(d ,dz)+cP( ,τ
1,τ
2,ln F( ,τ
1,τ
2))d ,(4.5)
whe e he d i e m is gi en by
cP( ,τ
1,τ
2,¯
Y)=E[μ( ,U)]−κ( )¯
Y+1
2E[σ( ,U)]2+ψP(E[η( ,U)]). (4.6)
P oo Following he conside a ions in he p e ious sec ion, he swap p ice is de ined by he
geome ic a e age in Equa ion (2.14). Using he in eg al ep esen a ion o Eq.4.4 and he
s ochas ic Fubini heo em (see P o e [42], c . Theo em 65), we can in oduce he dynamics
o he swap’s loga i hmic e u n by
dln F( ,τ
1,τ
2)
=(E[μ( ,U)]−κ( )ln F( ,τ
1,τ
2))d +E[σ( ,U)]dWP
+E[η( ,U)]d
JP
.
(4.7)
An applica ion o I o’s o mula (see Øksendal and Sulem [39], c . Theo em 1.16) yields he
desi ed swap dynamics. 
No e ha he speed o mean- e e sion κ( )has o be independen o he deli e y ime. This
assump ion ensu es ha ln F, in Equa ion (4.7), is again an O ns ein-Uhlenbeck p ocess and
ha he swap’s p ice dynamics in Equa ion (4.5) s ay ac able. This is also in line wi h he
123
310 Ma hema ics and Financial Economics (2025) 19:293–327
indings in Ben h e al. [7] (c . P oposi ion 2.2) and La ini e al. [35]. In pa icula , he mean-
e e ing e ec comp ises he jump componen as well, e en i we implemen i h ough a
measu e change o he B ownian pa . Mo e p ecisely, mean- e e sion connec ed o jumps
co e s indeed a unique ea u e o elec ici y ma ke s known as spikes: Spikes a e la ge jumps
quickly e u ning o he “no mal” le el (see, e.g., Klüppelbe g e al. [33]). They a ise as
elec ici y is no s o able on a la ge scale and since he elec ici y demand is no elas ic (see
Bo o ko a and Schmeck [11]).
Le us now in es iga e he swap p ice unde he physical measu e esul ing om app ox-
ima ed a e aging (see Eq.2.13 in o de o compa e he p icing sp ead be ween bo h
app oaches.
Lemma 4.3 Le Assump ion 4and 5in Appendix A be sa is ied. Then, he swap p ice dynamics
based on app oxima ed a e aging e ol e as
dFa( ,τ
1,τ
2)
Fa( −,τ
1,τ
2)
=E[σ( ,U)]dWP
+RE[eη( ,U)z]−1
NP(d ,dz)+EU[cP( ,U,ln ( ,U))]d ,
(4.8)
wi h EUdeno ing he expec a ion wi h espec o he andom deli e y a iable U ha ing
densi y w(u,τ
1,τ
2).
P oo We use he app oxima ed a e aging me hodology (see Equa ion (2.13)) in o de o
de i e he swap p ice e olu ion and apply he s ochas ic Fubini heo em (see P o e [42], c .
Theo em 65) leading o Equa ion (4.8). 
4.2 The isk p emium
We would like o close his sec ion wi h a discussion o he isk p emium in ou se ing. The
isk p emium usually ep esen s he di e ence be ween he o wa d p ice and he spo p ice
p edic ion a deli e y ime (c . o ins ance Ben h e al. [2]). In he case o swaps, Ben h e al.
[7] ex end he de ini ion o he isk p emium as he di e ence be ween he swap p ice and he
expec ed alue o he spo p ice weigh ed o e he deli e y pe iod. Inspi ed by he ex ended
de ini ion o he isk p emium o swaps by Ben h e al. [7], we conside he isk p emium as
he di e ences be ween he swap p ice and he expec ed alue o he geome ically a e aged
u u esp iceweigh ed o e he deli e ype iod.Addi ionally,we conside he iskp emium as
a p icing sp ead (simila o in Co olla y 3.1) esul ing om di e en p icing me hodologies,
i.e. he di e ence be ween wo p ices unde he same measu e.
Co olla y 4.1 (Risk P emium)
(i)The isk p emium be ween he swap p ice F and F Ais always non-posi i e unde he
physical measu e, i.e.,
RPF,FA( ,τ
1,τ
2):= F( ,τ
1,τ
2)−FA( ,τ
1,τ
2)≤0.(4.9)
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Ma hema ics and Financial Economics (2025) 19:293–327 311
(ii)The isk p emium be ween he swap p ice F and Faunde he physical measu e is
de e mined by
RPF,Fa( ,τ
1,τ
2):= F( ,τ
1,τ
2)−Fa( ,τ
1,τ
2)=Fa( ,τ
1,τ
2)(D( ,τ
1,τ
2)−1),
(4.10)
whe e D( ,τ
1,τ
2)is de ined in Equa ion (3.26). Hence, he dis ance be ween F and
Fais no a ec ed by he measu e change.
(iii)The isk p emium o he geome ic swap p ice be ween he physical and he swap’s
isk neu al measu e is de ined by
RPF( ,τ
1,τ
2):= F( ,τ
1,τ
2)−EP[F(τ1,τ
1,τ
2)|F ](4.11)
and is explici ly de e mined by
RPF( ,τ
1,τ
2)=F( ,τ
1,τ
2)(1−DRP( ,τ
1,τ
2)),(4.12)
whe e
DRP( ,τ
1,τ
2)
:= exp τ1
0e−τ1
sκ( )d E[μ(s,U)]ds +1
2τ1
E[σ(s,U)]2e−2τ1
sκ( )d +1ds
+τ1
ψPE[η(s,U)]e−τ1
sκ( )d +ψ
Q(E[η(s,U)])ds
+
0e−
sκ( )d E[σ(s,U)]dWP
s−
0E[σ(s,U)]dW
Q
s
+
0e−
sκ( )d E[η(s,U)]d
JP
s−
0E[η(s,U)]d
J
Q
s.
(4.13)
P oo (i)Fisalwayssmalle o equal han FAanalogous o Co olla y3.1(i)sinceJensen’s
inequali y also holds unde he physical measu e.
(ii)Analogous o Co olla y 3.1 (ii).
(iii)A s aigh o wa d e alua ion o EP[F(τ1,τ
1,τ
2)|F ]leads o
EP[F(τ1,τ
1,τ
2)|F ]=F( ,τ
1,τ
2)DRP( ,τ
1,τ
2). (4.14)
So ha Equa ion (4.12) di ec ly ollows.

Hence, he sp ead be ween he swap p ices Fand Faunde he physical measu e P
coincides wi h he p icing sp ead om Co olla y 3.1 (ii) unde he ins an aneous isk measu e
Q, as he numé ai e in Equa ion (3.28) is no a ec ed by a change o measu e.
4.3 The swap p ice Funde i s isk-neu al measu e

Q
In o de o de i e he swap’s ma ingale measu e 
Q, we in oduce he swap’s ma ke p ice o
isk o he swap p ice esul ing om geome ic a e aging in he nex de ini ion:
De ini ion 4.1 We de ine he swap’s ma ke p ice o isk by P
Q:= (P
Q
1,
P
Q
2),whe e
P
Q
1( ,τ
1,τ
2):= E[μ( ,U)]−κ( )ln F( ,τ
1,τ
2)+1
2E[σ( ,U)]2
E[σ( ,U)],(4.15)
123
312 Ma hema ics and Financial Economics (2025) 19:293–327
P
Q
2( ,τ
1,τ
2):= 1−R
zP(dz)E[η( ,U)]
ReE[η( ,U)]z−1P(dz).(4.16)
No e ha he ma ke p ice o isk does no en e he jump size dis ibu ion since we es ic
P
Q
2 o depend on ading ime and deli e y pe iod. Hence, he ma ke p ice o jump isk
a ec s he jump in ensi y only.
We ollow he me hodology o Ben h e al. [7] o change he measu e om he physical
measu e P o he swap’s isk-neu al measu e 
Q. The e o e, le π=(π1,π
2)be a p edic able
p ocess sa is ying
Eτ1
0π(s,τ
1,τ
2)2ds<∞.(4.17)
We de ine a new p ocess ZP
Qbeing he unique s ong solu ion o
dZP
Q( ,τ
1,τ
2)=ZP
Q( −,τ
1,τ
2)dH( ,τ
1,τ
2), (4.18)
such ha ZP
Q(0,τ
1,τ
2)=1, whe e
dH( ,τ
1,τ
2)=π1( ,τ
1,τ
2)dWP
+π2( ,τ
1,τ
2)d
JP
.(4.19)
I πjsa is ies Equa ion (4.17), hen His a well-de ined squa e in eg able ma ingale. No e
ha he p ocess ZP
Qis known as he Doléans-Dade exponen ial o H ha is explici ly gi en
by
ZP
Q( ,τ
1,τ
2)=eH( ,τ1,τ2)−1
2
0π1(s,τ1,τ2)2ds 
0<s≤
(1+H(s,τ
1,τ
2))e−H(s,τ1,τ2).
(4.20)
I ZP
Qis a s ic ly posi i e ma ingale, hen we can de ine he equi alen p obabili y measu e

Qby
d
Q
dP=ZP
Q(τ1,τ
1,τ
2), (4.21)
whe e ZP
Q unc ions as he Radon-Nikodym de i a i e. I we u he assume ha
EP[ZP
Q(τ1,τ
1,τ
2)]=1, hen Gi sano ’s heo em (see Øksendal and Sulem [39], c .
Theo em 1.35) s a es o π:= −P
Q ha
W
Q
=WP
+
0
P
Q
1(s,τ
1,τ
2)ds ,(4.22)
is a B ownian mo ion wi h espec o 
Qand

N
Q(d ,dz)=
NP(d ,dz)+P
Q
2( ,τ
1,τ
2)P(dz)d ,(4.23)
is a 
Q-compensa ed Poisson andom measu e o N(·,·).
Unde he abo e assump ions speci ied la e a s aigh o wa d alua ion leads o he
ollowing esul :
P oposi ion 4.1 The swap p ice p ocess F de ined in Equa ion (2.14)is a ma ingale unde

Qgi en by
dF( ,τ
1,τ
2)
F( −,τ
1,τ
2)=E[σ( ,U)]dW
Q
+ReE[η( ,U)]z−1
N
Q(d ,dz). (4.24)
123

Ma hema ics and Financial Economics (2025) 19:293–327 313
We would like o in es iga e he consequences o ou p e ious assump ions.
Rema k 4.2 (i)The Doléans-Dade exponen ial in Equa ion (4.20) is posi i e i
π2(s−)J>−1, i.e., i P
Q
2J<1. Hence, simila o Ben h e al. [7], we need o
assume ha he ma ke p ice o jump isk is bounded and de e minis ic o e he en i e
ime pe iod such ha P
Q
2( ,τ
1,τ
2)z<1 o P-a.e. z∈Rand o each ∈[0,τ
1].
(ii)I ln ,andsolnF, is d i en by a compensa ed Poisson p ocess only, hen he swap’s
ma ke p ice o isk is a ained by P
Q:= (0,
P
Q
2),whe e
P
Q
2( ,τ
1,τ
2):= 1−E[η( ,U)]RzP(dz)
ReE[η( ,U)]z−1P(dz)+E[μ( ,U)]−κ( )ln F( ,τ
1,τ
2)
ReE[η( ,U)]z−1P(dz).
(4.25)
In his se ing, we need o equi e ha κ( )≡0.
No e ha a posi i e local ma ingale is a supe ma ingale. Hence, in o de o p o e
ha he Radon-Nikodym densi y ZP
Qis a ue ma ingale, i is su icien o e i y ha
EP[ZP
Q(τ1,τ
1,τ
2)]=1 is sa is ied, which is p o en in he nex p oposi ion.
P oposi ion 4.2 Unde Assump ion 6in Appendix A, he p ocess ZP
Qde ined by Equa ion
(4.18)is a s ic ly posi i e ue ma ingale.
P oo In Appendix B, we p o e his p oposi ion e en in a s ochas ic ola ili y amewo k. 
4.4 The app oxima ed swap p ice Faunde he ins an aneous isk-neu al measu e
We in oduce he ins an aneous ma ke p ice o isk o he app oxima ed swap p ice in he
nex de ini ion.
De ini ion 4.2 We de ine he ins an aneous ma ke p ice o isk o he app oxima ed swap
by PQ := (PQ
1,
PQ
2),whe e
PQ
1( ,τ
1,τ
2):= E[μ( ,U)]−κ( )ln F( ,τ
1,τ
2)+1
2E[σ2( ,U)]
E[σ( ,U)],(4.26)
PQ
2( ,τ
1,τ
2):= 1−R
zP(dz)E[η( ,U)]
RE[eη( ,U)z]−1P(dz).(4.27)
No e ha we assume ha he ma ke p ice o jump isk a ec s he jump in ensi y only.
The ma ke p ice o isk does no en e he jump size dis ibu ion since we es ic PQ
2 o
depend on ading and deli e y pe iod.
Simila o he las subsec ion, we can de ine he equi alen (ins an aneous) p obabili y
measu e Qby
dQ
dP=ZPQ(τ1,τ
1,τ
2), (4.28)
whe e ZPQ unc ions as he Radon-Nikodym de i a i e cha ac e ized by π:= −PQ.I we
u he assume ha EP[ZPQ(τ1,τ
1,τ
2)]=1, hen Gi sano ’s heo em (see Øksendal and
Sulem [39], c . Theo em 1.35) s a es ha
WQ
=WP
+
0
PQ
1(s,τ
1,τ
2)ds ,(4.29)
123
314 Ma hema ics and Financial Economics (2025) 19:293–327
is a B ownian mo ion wi h espec o Qand

NQ(d ,dz)=
NP(d ,dz)+PQ
2( ,τ
1,τ
2)P(dz)d ,(4.30)
is a Q-compensa ed Poisson andom measu e o N(·,·).
Unde he abo e assump ions a s aigh o wa d alua ion leads o he ollowing esul :
P oposi ion 4.3 The app oxima ed swap p ice p ocess Fade ined in Equa ion (2.13)is a
ma ingale unde Qgi en by
dFa( ,τ
1,τ
2)
Fa( −,τ
1,τ
2)=E[σ( ,U)]dWQ
+RE[eη( ,U)z]−1
NQ(d ,dz). (4.31)
We e e o Sec .4.3 o he consequences o he assump ions made abo e.
4.5 The decomposi ion o he ma ke p ice o isk
F om he p e ious subsec ions, we know he co esponding ma ke p ices o isk o he
swap p ice esul ing om geome ic a e aging (see De ini ion 4.1) and om app oxima ed
a e aging (see De ini ion 4.2). In his subsec ion, we iden i y a clea dis inc ion be ween
bo h ma ke p ices o isk leading o a speci ic decomposi ion ha is s ongly connec ed o
he MPDP.
We now in oduce he decomposi ion o he swap’s ma ke p ice o isk, om De -
ini ion 4.1, which inally connec s he ins an aneous ma ke p ice o isk, speci ied in
De ini ion 4.2, and he MPDP, de ined in De ini ion 3.1. The decomposi ion esul is s a ed
in he nex p oposi ion.
P oposi ion 4.4 The swap’s ma ke p ice o isk, P
Q, esul ing om geome ic a e aging
(see De ini ion 4.1), decomposes in o
P
Q
j( ,τ
1,τ
2)=PQ
j( ,τ
1,τ
2)+¯
Q
Q
j( ,τ
1,τ
2), o j =1,2,(4.32)
whe e PQ
jis speci ied in De ini ion 4.2 and ¯
Q
Q
jde ines he sp ead o di usion and jump
isk. Mo e p ecisely,
¯
Q
Q
1( ,τ
1,τ
2)=−1
2
V[σ( ,U)]
E[σ( ,U)],(4.33)
¯
Q
Q
2( ,τ
1,τ
2)=−E[η( ,U)]R
zG(dz)RE[eη( ,U)z]−eE[η( ,U)]zG(dz)
RE[eη( ,U)z]−1G(dz)ReE[η( ,U)]z−1G(dz),
(4.34)
whe e ¯
Q
Q
2is independen o he jump in ensi y.
P oo The esul is a ained by sub ac ing he swap’s ma ke p ice o isk P
Qde ined in
De ini ion 4.1 om he ins an aneous ma ke p ice o isk PQ de ined in De ini ion 4.2.
Hence, we ound a ep esen a ion o he swap’s ma ke p ice o isk o he swap p ice F,
cha ac e ized by he ins an aneous ma ke p ice o isk o he app oxima ed swap Faand he
sp ead ¯
Q
Q=¯
Q
Q
1,¯
Q
Q
2. We u he in es iga e he sp ead in he nex lemma.
Lemma 4.4 (i)The sp ead o di usion isk, ¯
Q
Q
1( ,τ
1,τ
2), is nega i e o all ading imes
∈[0,τ
1].
123
Ma hema ics and Financial Economics (2025) 19:293–327 315
(ii)I he a e age jump size is posi i e, i.e., i RzG(dz)>0, hen he sp ead o jump isk,
¯
Q
Q
2, is nega i e.
(iii)I he a e age jump size is ze o, i.e., i RzG(dz)=0, hen he sp ead o jump isk,
¯
Q
Q
2,isze o.
(i ) I he ola ili y is independen o he deli e y, i.e., i σ( ,u)⊥⊥ u, hen he sp ead o
di usion isk is ze o, i.e., ¯
Q
Q
1( ,τ
1,τ
2)≡0.
( ) I he jump coe icien is independen o he deli e y, i.e., i η( ,u)⊥⊥ u, hen he sp ead
o jump isk is ze o, i.e., ¯
Q
Q
2( ,τ
1,τ
2)≡0.
P oo The esul s in (i)and (ii) ollow di ec ly om Jensen’s inequali y. The esul s in (iii)
and (i ) ollow om he ac ha he nume a o becomes ze o whene e he deli e y pe iod
disappea s. 
As a esul , whene e he sp ead ¯
Q
Q
jis nega i e o j=1,2, hen he app oxima ed
swap induces mo e isk han he swap p ice based on geome ic a e aging. In pa icula , he
conside ed sp ead has he same p ope ies as he MPDP (see Kempe e al. [30]). Indeed, a
compa ison wi h ou p e ious conside a ions in Sec .2gi es he ollowing insigh s:
Rema k 4.3 (i)The sp ead o di usion isk, ¯
Q
Q
1, coincides wi h he MPDP o di usion
isk, Q
Q
1, in Equa ion (3.1) om Sec .2.
(ii)The sp ead o jump isk, ¯
Q
Q
2, does no coincide wi h he MPDP o jump isk, Q
Q
2,
om Equa ion (3.2) bu wi h Q
Q
2(1−PQ
2). This connec ion occu s na u ally by he
change o measu e.
(iii)The condi ion in Lemma 4.4(iii)holds ue, o example, when jump sizes ollow a e
s anda d no mal dis ibu ion.
Hence, s a ing om he physical measu e, we can ind he swaps ue ma ingale measu e
based on he swap’s ma ke p ice o isk de ined in De ini ion 4.1. I we would like o adjus
al eady exis ing models using he ins an aneous ma ke p ice o isk, we can easily adjus he
model h ough he sp ead de ined in P oposi ion 4.4 ha is s ongly connec ed o he MPDP
de ined in De ini ion 3.1.
4.6 An example based on sho - e m long- e m e olu ion
In his sec ion, we gi e an example in he spi i o he popula sho - e m long- e m models
based on Gibson and Schwa z [21] and Schwa z and Smi h [45]. On he spo le el, hese
models ypicallydi ide hep icee olu ionin oanon-s a iona yGaussiancomponen ,gin ing
he long e m mean o spo p ices. I is in luenced e.g. by poli ical o egula o y decisions.
Fu he mo e, a s a iona y mean e e ing componen desc ibes sho e m p ice luc ua ions
due o imbalances in supply and demand. This mean e e ing sho e m componen leads
o a Samuelson e ec in he u u es p ice dynamics, see e.g. Ben h and Schmeck [8]. Then,
he u u es p ices e ol e acco ding o
d ( ,τ)
( −,τ) =σdWP
+Reη( ,τ )z−1
NP(d ,dz)
+μ−κln ( ,τ)+1
2σ2+ψP(η( ,τ))
d ,(4.35)
123
316 Ma hema ics and Financial Economics (2025) 19:293–327
whe e η( ,τ) =e−(τ− )wi h >0 cap u es he e m s uc u e e ec in he spi i o
Samuelson (c . he ola ili y in Example 3.2), and 
NPis a compund Poisson p ocess. As
we wan o gi e an example unde he physical measu e, we ha e added a d i e m simila
o Equa ion (4.3) and assume cons an d i and mean- e e sion pa ame e s μand κ>0
o simplici y. Assuming a g adual in low o enewables, jumps a e mo e likely downwa d
poin ing as opposed o upwa d poin ing jumps in powe sys ems wi h mo e nuclea , hyd o,
gas, and empe a u e dependen demand (c . Pa aschi e al. [41] and Hinde ks and Wagne
[26]). To cap u e he enewable e ec wi hin he jump size dis ibu ion, we assume o
simplici y a nega i e jump size dis ibu ion cap u ed by he Di ac measu e assigned o a
jump size o -1.
Unde Assump ions 4and 5in Appendix A, he swap p ice dynamics unde he physical
measu e Pbased on geome ic a e aging e ol e acco ding o Lemma 4.2 as
dF( ,τ
1,τ
2)
F( −,τ
1,τ
2)=σdWP
+ReE[η( ,U)]z−1
NP(d ,dz)
+μ−κln F( ,τ
1,τ
2)+1
2σ+ψP(E[η( ,U)])d ,
(4.36)
whe e he swap’s jump coe icien o a cons an se lemen ype unc ion w(u,τ
1,τ
2)=
1
τ2−τ1is gi en by
E[η( ,U)]=1−e−(τ2−τ1)
(τ2−τ1) η( ,τ
1). (4.37)
No e ha he Assump ions 4and 5a e sa is ied, since μ,κ,andσa e cons an , η( ,u)is
de e minis ic and ini e o ∈[0,τ
1]and u∈[τ1,τ
2] o a ini e ading ho izon τ1<∞,
and he second momen o he nega i e exponen ial dis ibu ion as well as he momen
gene a ing unc ion exis . To u n o he swap’s isk neu al measu e 
Q, he swap’s ma ke
p ice o ola ili y and jump isk a e gi en acco ding o De ini ion 4.1 by
P
Q
1( ,τ
1,τ
2):= μ−κln F( ,τ
1,τ
2)+1
2σ2
σ,(4.38)
P
Q
2( ,τ
1,τ
2):= 1+E[η( ,U)]
λ−
JReE[η( ,U)]z−1G(dz).(4.39)
Thinking in he spi i o he sho - e m long- e m amewo k, he Gaussian pa o he
dynamics s ands o long e m e olu ions. Thus, he e 1cap u es e ec s connec ed o he
inco po a ion o he deli e y pe iod ha ha e a long e m cha ac e . On he o he hand, he
Jump componen s ands o sho - e m e olu ions due o imbalances in supply and demand.
2cap u es deli e y dependend e ec s connec ed o expec a ions on e.g. nega i e spikes in
he unde lying.
Unde Assump ion 6, he swap p ice p ocess Fis a ma ingale unde he isk neu al
measu e 
Qe ol ing as
dF( ,τ
1,τ
2)
F( −,τ
1,τ
2)=σdW
Q
+ReE[η( ,U)]z−1
N
Q(d ,dz), (4.40)
whe e W
Q
and 
N
Q(d ,dz)a e as in Equa ions (4.22)and(4.23). No e ha he Assump ion 6
(i) is sa is ied whene e
0≥2e−E[η( ,u)]+E[η( ,u)]−2,(4.41)
123
Ma hema ics and Financial Economics (2025) 19:293–327 323
+s
0
σνν( )dB
Qn( )2
()
≤4 2
0+E
Qnsup
s∈[0,τ1]s
0
κνθνd 2
+sup
s∈[0,τ1]s
0
(κν+δν1[0,ˆτn]( ))ν( )d 2
+sup
s∈[0,τ1]s
0
σνν( )dB
Qn
2
()
≤4 2
0+4E
Qnτ1
0
κνθνd 2
+4E
Qnτ1
0
(κν+δν1[0,ˆτn]( ))ν( )d 2
+4E
Qnτ1
0
σνν( )dB
Qn
2
()
≤4 2
0+4τ1τ1
0
κ2
νθ2
νd +4τ1(κν+|δν|)2τ1
0
E
Qnsup
s∈[0, ]
ν(s)2d
+4σ2
νE
Qnτ1
0
ν( )d ,
whe e he i s equali y ep esen s he in eg al e sion o ν. Inequali y () esul s om he
Cauchy-Schwa z inequali y o he sum and an applica ion o he iangle inequali y. We
apply Doob’s inequali y o all expec a ions in Inequali y (). In Inequali y (), we apply
he Cauchy-Schwa z inequali y o he i s and second in eg al and apply I o’s isome y o he
las summand. We inish wi h he s ochas ic Fubini o he second in eg al while making he
in eg and e en bigge . No e ha o he las summand, we ha e E
Qnτ1
0ν( )d ≤˜cν⊥⊥ n
since we can ind explici exp essions in Con and Tanko [16] (c . Chap e 15). Se ing
cν:= 4 2
0+16τ2
1κ2
νθ2
ν+16σ2
ν˜cν, hen, by G onwall, we ecei e E
Qnsups∈[0,τ1]|ν(s)|2≤
cνe16(κν+|δν|)2τ2
1=: c2⊥⊥ n.
Nex , we show ha |ln F|2is uni o mly in eg able:
E
Qnsup
s∈[0,τ1]|ln F(s,τ
1,τ
2)|2
=E
Qnsup
s∈[0,τ1]ln F(0,τ
1,τ
2)+s
01−1[0,ˆτn]( )(E[μ( ,U)]−κ( )ln F( ,τ
1,τ
2))d
−s
0
1
2E[σ( ,U)]2ν( )1[0,ˆτn]( )d
+s
0E[σ( ,U)]ν( )dW
Qn
+s
0E[η( ,U)]d
J
Qn
−s
0E[η( ,U)]1−E[η( ,U)]RzP(dz)
ReE[η( ,U)]z−1P(dz)1[0,ˆτn]( )R
zP(dz)d 2
()
≤7ln F(0,τ
1,τ
2)2+E
Qnsup
s∈[0,τ1]s
01−1[0,ˆτn]( )E[μ( ,U)]d 2
123

324 Ma hema ics and Financial Economics (2025) 19:293–327
+E
Qnsup
s∈[0,τ1]s
0
1
2E[σ( ,U)]2ν( )1[0,ˆτn]( )d 2
+E
Qnsup
s∈[0,τ1]s
01−1[0,ˆτn]( )κ( )ln F( ,τ
1,τ
2)d 2
+E
Qnsup
s∈[0,τ1]s
0E[σ( ,U)]ν( )dW
Qn
2+E
Qnsup
s∈[0,τ1]s
0E[η( ,U)]d
J
Qn
2
+E
Qn⎡
⎣sup
s∈[0,τ1]s
0E[η( ,U)]1−E[η( ,U)]RzP(dz)
ReE[η( ,U)]z−1P(dz)1[0,ˆτn]( )R
zP(dz)d 2⎤
⎦
()
≤7ln F(0,τ
1,τ
2)2+4E
Qnτ1
01−1[0,ˆτn]( )E[μ( ,U)]d 2
+4E
Qnτ1
0
1
2E[σ( ,U)]2ν( )1[0,ˆτn]( )d 2
+4E
Qnτ1
01−1[0,ˆτn]( )κ( )ln F( ,τ
1,τ
2)d 2
+4E
Qnτ1
0E[σ( ,U)]ν( )dW
Qn
2+4E
Qnτ1
0E[η( ,U)]d
J
Qn
2
+4E
Qn⎡
⎣τ1
0E[η( ,U)]1−E[η( ,U)]RzP(dz)
ReE[η( ,U)]z−1P(dz)1[0,ˆτn]( )R
zP(dz)d 2⎤
⎦
()
≤7ln F(0,τ
1,τ
2)2+4τ1τ1
0E[μ( ,U)]2d
+4τ1
0E[σ( ,U)]4d E
Qnτ1
0ν( )2d +4τ1
0κ( )2d E
Qnτ1
0ln F( ,τ
1,τ
2)2d 
+4E
Qnτ1
0E[σ( ,U)]ν( )dW
Qn
2+4E
Qnτ1
0E[η( ,U)]d
J
Qn
2
+4τ1
0E[η( ,U)]2d E
Qn
⎡
⎢
⎣τ1
0⎛
⎜
⎝1−E[η( ,U)]2RzP(dz)2
ReE[η( ,U)]z−1P(dz)2⎞
⎟
⎠R
zP(dz)2
d ⎤
⎥
⎦
≤7ln F(0,τ
1,τ
2)2+4τ1τ1
0E[μ( ,U)]2d +4c2τ1τ1
0E[σ( ,U)]4d
+4τ1
0κ( )2d E
Qn
τ1
0sup
s∈[0, ]
ln F(s,τ
1,τ
2)2d 
+4*τ1
0E[σ( ,U)]4d √τ1c2+4τ1
0E[η( ,U)]2R
z2
Qn(dz)d
+4τ1
0E[η( ,U)]2d τ1
0⎛
⎜
⎝R
zP(dz)2
+E[η( ,U)]2RzP(dz)4
ReE[η( ,U)]z−1P(dz)2⎞
⎟
⎠d ,
=: cY+28 τ1
0κ( )2d E
Qnτ1
0sup
s∈[0, ]
ln F(s,τ
1,τ
2)2d .
123
Ma hema ics and Financial Economics (2025) 19:293–327 325
The i s equali y ep esen s he in eg al e sion o ln F. Inequali y () esul s om he
Cauchy-Schwa z inequali y o he sum and an applica ion o he iangle inequali y. We
apply Doob’s inequali y o all expec a ions in Inequali y (). In Inequali y (), we apply
he Cauchy-Schwa z inequali y o he i s h ee in eg als. We inish wi h I ô-Lé y Isome y
(seeØksendalandSulem[39],c .Theo em1.17) o helas summandandanapplica iono he
s ochas ic Fubini heo em o he ou h summand (including ln F) while making he in eg and
e en bigge . By he p e ious conside a ions, we know ha E
Qnτ1
0E[σ( ,U)]2ν( )d ≤
+τ1
0E[σ( ,U)]4d E
Qn+τ1
0ν( )2d ≤+τ1
0E[σ( ,U)]4d √τ1c2is bounded inde-
penden ly o nand ha τ1
0E[σ( ,U)]4d E
Qnτ1
0ν( )2d ≤c2τ1τ1
0E[σ( ,U)]4d
is independen o n. By he choice o cY, an applica ion o G onwall’s inequali y yields
EPnsups∈[0,τ1]|ln F(s,τ
1,τ
2)|2≤cYe28τ1
0κ( )2d =: c3⊥⊥ n,such ha we ha e shown,
ha ZP
Qis indeed a ue ma ingale.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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