Systemic risk in derivative markets: A graph-theory analysis

analysis

Delphine Lautier, Fran k Raynaud

Abstra t

This arti le investigates the time evolution of the integration of derivative

markets through the graph-theory. We fo us on three ategories of underlying

assets: energy and agri ultural produ ts, aswell asnan ial assets. Integration

is seen as a ne essary ondition for systemi risk to appear. In order to fully

understand it, this phenomenon is omprehended through a three-dimensional

approa h: observation time, spatial relationships and term stru tures. Su h

an approa h indeed gives the possibility to investigate pri es sho ks appearing

and propagating in the physi al as well as in the paper markets. In order to

understand the underlying prin iples and dynami behavior of our system, we

sele t spe i tools of statisti al physi s. We rst useminimum spanning trees

as a way to lter the information ontained in the graph. We then study the

topology of the ltered networks in order, rst to see how they are organized,

se ond to quantify the degree of randomness in this organization. Lastly, the

time dependent properties of the trees are examined. On an e onomi point

of view, the emerging taxonomy is meaningful, whi h is a key justi ation of

the use of our methodology. Moreover, we observe an in reasing integration

of the markets through time, as well as a dominan e of spatial over maturity

integration.

1 Introdu tion

This arti le examines the integration of derivative markets, with parti ular emphasis

on ommodityprodu ts. Integrationis onsideredtobeane essary onditioninorder

forsystemi risktoappear. Con ernsabout su hriskhavere ently grown innan ial

more and more integrated, both as regards ea h other and as regards other markets.

For some months now, u tuations in the pri es of energy ommodities have often

been invoked to explain those of soft ommodities likesoy, orn or wheat. Moreover,

sin e ommoditiesare nowadays onsideredas anew lass of assets, they are used by

portfoliomanagersfordiversi ationpurposes. Partofthepri eu tuationsre orded

in ommodity markets mightthereforebeexplainedby externalevents likethe fall in

sto k pri es orin interest rates.

Finan ial literature has looked into su h questions in various ways: herding

behav-ior, o- integration te hniques, spatial integration, et . These studies only take into

a ount two dimensions of integration: spa e and observation time, or maturity and

observation time. The time analysis of the relationships linking dierent spot pri es

ofa ommoditybeing simultaneously tradedinseveral nan ialpla eshas todowith

thespatialdimension ofintegration. When thefo us ispla edonhowthe term

stru -ture of futures pri es (i.e. on a spe i date the relationship linking, several futures

ontra ts with dierent deliverydates) evolvesover time itisthe maturitydimension

ofintegration whi h is examined.

Whilethesestudies onrmthatitishighlylikelythatintegrationisprogressing,none

of themtried to study spatial and maturity evolutions simultaneously. Su h an

anal-ysis is however ru ial as it gives a omplete understanding of systemi risk, namely

the possibility that a pri es sho k o urring ona spe i asset's physi al market an

spread, not only through its own futures market, but also onto other physi al and /

orpapermarkets, and vi eversa.

Takingintoa ounttreedimensionsrequiresanunderstandingofthebehaviorof

om-plexevolvingsystemsand olle tingahuge volume ofdata. This explainswhy weuse

re ent methods originated from statisti al physi s. Many theoreti al and numeri al

tools have been developed re ently in order to investigate the behavior of dynami

omplex systems. Moreover, sin e the pioneer work of [Mantegna, 1999℄ physi ists

startedtoapply the graph-theorytonan ialmarkets. These studiesare summarized

inthe next se tionof this arti le.

In this literature we hoose several measures whi h we found relevant for studying

integration. Firstlyweuseminimumspanningtrees(MST)asaway tolterthe

infor-mationgiven by the orrelation matrix of pri e returns. Minimum spanning trees are

graphsbuilt onthe basis of orrelationstransformed intodistan es. In our ase, they

ii) while onstru ting a MST, we dramati ally ompress the amount of information,

whi h isappre iable whenworking whi h ahuge amountof data. Se ondly, we study

the topology of the ltered networks. The organization between the dierent nodes

of the graph produ es interesting information. After a simple visualization of the

MST,weuse the allometri oe ientsmethodtodeterminewhether anetworkis

to-tallyorganized,totallyrandom,orissituatedsomewherebetweenthesetwoextremes.

Thirdly, given the time dependen y ofthe MST, we study their evolutionover time.

Our rst main result liesin the e onomi meaningfulness of the emerging taxonomy.

In the spatial as well as in the 3-D analyses, the trees are organized into sub-trees

whi h orrespond to the dierent se tors of a tivity under examination: energy

om-modities, agri ultural produ ts, and nan ial assets. Meanwhile, the analysis of the

maturity dimension provides eviden e of a hain-like organization of the trees. The

hierar hi alorderingof thefutures ontra tsbasedontheirdeliverydates anbeseen

asthe mirrorimageofthe Samuelsonee t. Su hintuitiveresultsare veryimportant,

asthey are akey justi ationfor the use of our methodology.

A se ond important result lies in the stru ture of integration that is observed. The

identi ation of the entral node of the tree is very important in su h a study, as,

if a sho k emerges at this entral node, it will have a more important impa t than

anywhere else. Moreover, if integration is a prerequisite for systemati risk to o ur,

thenthe entralvertexisatthe heartofall on erns. Theempiri alstudy showsthat,

in the spatial as well as in the 3-D analysis, between 1998 and 2009, among the 14

dierent underlyingassets under examination, rudeoil has a entralposition.

A third result on erns the evolution of integration over time. In line with previous

studies, we found that spatial integration progresses. We reinfor e this on lusion

with the analysis of the maturity dimension. Several measures indeed lead us to the

on lusion that both spatial and maturity dimensions tend to be more integrated.

Moreover, integration is stronger for the maturity dimension. This result is rather

intuitive: arbitrage operations are far more di ult with physi al assets than with

derivative ontra ts. Last but not least, the analysis of the robustness of the trees'

topology over time shows that during a risis, the trees shrink topologi ally, on the

spatialas wellas onthe maturity dimensions,thus ree ting anintensi ationof the

system's integration. At the same time, they be ome less stable: the topology of the

tion in nan e and physi s. Se tion3 presents the data. In se tion 4, we present the

methodology adopted in the study and the empiri al results. Se tion 5 is devoted to

on lusionsand poli yimpli ations.

2 Preliminary studies

Inthis se tion,we givea briefoverview on the literature onintegration.

The nan ialliterature has investigated the question of integration through dierent

ways. As early as 1990, [Pindy k and Rotenberg, 1990℄ began to study the herding

behavior of investors on ommodity derivative markets. Their seminal work shows

that the persistent tenden y of ommodity pri es to move together an not be

to-tallyexplained by the ommon ee ts of ination,ex hange rates, interest rates and

other ma ro-e onomi variables. This arti lehas inspired several other resear hes on

o-movement. Yet, in this kind of work the identi ation of the relevant e onomi

variables is tri ky. This ould explain why empiri al tests do not really su eed in

on ludingthat there is herdingbehavior in ommodity markets.

Fo usingonspatialintegration,[Jumah and Karbuz, 1999℄initiatedanotherapproa h

tothe systemi risk in ommodity markets. Su h astudy is entered on the

relation-ships between the pri es of raw materialsnegotiated indierent pla es. The authors

initiated several works on spatial integration, based on the methodology of the

o-integration. The empiri al tests show that ommodity markets are more and more

spatially integrated. In the same vein, [Buyuksahin etal., 2008℄ examined the links

between sto k and ommoditymarkets. They were howevernot able to on lude that

the former havean inuen e onthe latter.

Integrationhas alsoatemporaldimension, inthesenseof thepreferredhabitattheory

([Modiglianiand Stut h, 1966℄). In [Lautier, 2005℄, the author studied the

segmen-tation of the term stru ture of ommodity pri es and examined the propagation of

sho ks along the pri es urve, on the rude oil petroleum markets. She showed that

temporalintegration progresses troughtime.

Instatisti alphysi s,theminimumspanningtreeisheavilyusedinordertounderstand

theevolutionof omplexsystems,espe iallynan ialassets. Otherlteringpro edures

have however been used by dierentauthors (see for example [Marsili, 2002℄).

[Bonnanoet al.,2004℄, the authors use this orrelation based method in order to

ex-amine sto ks portfolios and nan ial indexes at dierent time horizons. They also

apply this method in order to falsify widespread markets models, on the basis of a

omparison between the topologi al properties of networks asso iated with real and

arti ial markets. The ltering approa h of the MST an also be used in order to

onstru ta orrelationbased lassi ationofrelevante onomi entities su hasbanks

orhedgefunds, asin[Mi eli and Susinno, 2003℄. Asfaras ommoditiesare on erned,

[Sie zkaand Holyst,2009℄ re ently proposed a study of ommodities lusteringbased

onMST. They found eviden e of a marketsyn hronization. However, their database

ontains ommodities hara terized by lowtransa tion volumes,whi h an introdu e

noiseinthe orrelationmatrix. Lastlythe robustnessovertimeofthe minimum

span-ningtree's hara teristi shasalsobeenexaminedinaseriesofstudies(seeforexample

[Kullmannet al.,2002℄ and [Onnelaet al.,2003b℄).

3 Data

Insert Figures(1)about here.

For our empiri al study, we sele ted futures markets orresponding to three se tors,

namely energy, agri ulture and nan ial assets. On the basis of the Futures Industry

Asso iation's monthly volume reports, we retained those ontra ts hara terized by

the largest transa tion volumes, over a long time period. We used two databases,

Datastreamand Reuters, in order to olle t settlement pri es ona daily basis. With

Datastream the time series had to be rearranged in order to re onstitute daily term

stru tures of futures pri es. Figure (1) summarizes the main hara teristi s of this

database.

Withsu hadatabase, oneofthe di ulties omesfromthefa tthatpri es urvesare

shorterat the beginningof the period. Indeed, over time, the maturities of ontra ts

usuallyrise onaderivativemarket. Thegrowth inthetransa tionvolumesofexisting

ontra ts results in the introdu tion of new delivery dates. Thus, in order to have

ontinuous time series, we had to remove some maturities from the database. On e

this sele tion had been arried out, our database still ontained more than

655, 000

In order to study the integration of derivative markets, we rely on the graph-theory.

Among the dierent tools provided by this method, we sele ted those allowing us to

analyze market integration using a three-dimensional approa h. We rst de ided to

representour pri essystem bystudyingthe orrelationofpri ereturns. Having

trans-formedthese orrelationsintodistan es, wewereable todrawafully onne ted graph

ofthe pri es system,where thenodesof the graphrepresentthe time series offutures

pri es. In order to lter the information ontained in the graph, we rely on spe i

graphs: minimum spanning trees (MST). The method used for the identi ation of

the MST ispresented in the rst part of this se tion. We then study the topologyof

the trees (se ond part) and their dynami behavior (thirdpart).

4.1 Minimum spanning trees: a orrelation-based method

The rst step towards the analysis of market integration was in our ase the

ompu-tationof the syn hronous orrelation oe ients of pri e returns. In order to use the

graph theory, we needed to quantify the distan e between the elements under

exam-ination. We thus extra ted a metri distan e from the orrelation matrix. We were

thenableto onstru tgraphs. Lastly,weusedalteringpro edureinordertoidentify

the MST [Mantegna, 1999℄. Su h a tree an be dened asthe one providingthe best

arrangementof the network's dierent points.

4.1.1 The orrelation matrix

In order to measure the similarities in the syn hronous time evolution of the futures

ontra ts,webuiltamatrixof orrelation oe ients. Thelatteraredenedasfollows:

ρ

ij

(t) =

hr

i

r

j

i − hr

i

i hr

j

i

q

hr

2

i

i − hr

i

i

2

r

2

j

 − hr

j

i

2

,

(1)

When fo using on the spatial dimension,

i

and

j

stand for the nearby futures pri es

of pairs of assets, like rude oil or orn. In the absen e of reliable spot data, we

approximate the spot pri es with the nearest futures pri es. When fo using on the

maturitydimension, they stand forpairs of deliverydates. They are amixof the two

stands for pri e returns, with

r

i

= (ln F

i

(t) − ln F

i

(t − ∆t)) /∆t

, where

F

i

(t)

is the

settlement pri e of the futures ontra t at

t

.

∆t

is the time window, and

h.i

denotes

the statisti alaverage performedother time,on the tradingdays of the study period.

Foragiventimeperiodandagivensetofdata,wethus omputedthematrixof

N ×N

orrelation oe ients

C

, for all the pairs

ij

.

C

is symmetri with

ρ

ij

when

i = j

.

Thus, is hara terizedby

N (N − 1) /2

oe ients.

4.1.2 From orrelations to distan es

In order to use the graph-theory, we needed to introdu e a metri . The orrelation

oe ient

ρ

ij

indeed annotbe usedas adistan e

d

ij

between

i

and

j

be auseit does

not fulllthe three axioms that dene a metri [Gower, 1966℄:

• d

ij

= 0

if and only if

i = j

,

• d

ij

= d

ji

• d

ij

≤ d

ik

+ d

kj

Ametri

d

ij

an however be extra ted fromthe orrelation oe ients through anon

lineartransformation. Su h a metri is dened asfollows:

d

ij

=

q

(2 (1 − ρ

ij

)).

(2)

A distan e matrix

D

was thus extra ted from the orrelation matrix

C

a ording to

Equation(2).

C

and

D

are both

N × N

dimensional. Whereasthe oe ients

ρ

ij

an

be positive for orrelated returns ornegative for anti- orrelated returns, the distan e

d

ij

representing the distan e between pri e returnsis always positive.

4.1.3 From full onne ted graphs to Minimum Spanning Trees (MST)

Agraph givesa representation of pairwiserelationshipswithina olle tionof dis rete

entities. A simple onne ted graphrepresents allthe possible onne tions between

N

points under examination with

N − 1

links (edges). Ea h point of the graph

onsti-tutes a node (vertex). The graph an be weighted in order to represent the dierent

intensities of the links and / or nodes. Su h weights an represent the distan es

be-tween the nodes.

there are a lotof paths spanning a graph. For aweighted graph,the minimum

span-ningtree (MST) isthe one spanning all thenodes ofthe graph, withoutloops. It has

less weight thanany othertree. Its links are a subset of thoseof the initialgraph.

Through a lteringpro edure (the informationspa e is redu ed from

N(N − 1)/2

to

N − 1

),the MST reveals the mostrelevant onne tionsof ea h elementofthe system.

In our study, they provide for the shortest path linking allnodes. Thus, they an be

seen as a way of revealing the underlying me hanisms of systemi risk: the minimal

spanning tree is indeedthe easiest path for the transmission of apri es sho k.

4.2 Topology of the Minimum Spanning Trees

The rst information given by a minimum spanning tree is the kind of arrangement

found between the verti es. Therefore, the rst step in studying MST lies in their

visualization. We then use the allometri oe ients method in order to determine

whetheraMST istotallyorganized,totallyrandom,orissituatedsomewherebetween

these two extreme topologies. In this part of the study, we onsider the whole time

periodas asingle windowand thus perform a stati analysis.

4.2.1 Visualization and des ription of the MST

Thevisualizationof thetreesis averyimportantstep,asitaddresses the

meaningful-nessofthe taxonomythat emergesfromthe system. Before goingfurther,letusmake

two remarks: rst, we are onsidering links between markets and/or delivery dates

belongingtothe MST.Thus, ifarelationshipbetweentwomarkets ormaturities does

not appear in the tree, this does not mean that this relation does not exist. It just

does not orrespond to a minimal distan e. Se ond, our results naturally depend on

the natureand numberof markets hosen for the study.

In what follows, we will use the term se tor in order to des ribe the grouping of

underlyingassets,whereas wewillretain the term luster inorder todes ribe, for a

spe i market, the groupingof delivery dates in the maturity dimension.

Insert Figure(5) about here.

Figure(5)presents theMSTobtained forthe spatialandforthematuritydimensions.

As far as the spatial dimension is on erned, the MST looks like a star. In Figure

nan ialassets (mainlyonthe right). Moreover, the most onne tednode inthe graph

isEuropean rude oil (LLC),whi h makesit the best andidate for the transmission

ofpri eu tuationsinthe tree (a tually,the same ouldhavebeen saidfor Ameri an

rudeoil (NLC),as the distan e between these produ ts is very short). Last but not

least,theenergy se torseemsthe mostintegrated, asthe distan esbetweenthe nodes

are short. The link between the energy and agri ulturalprodu ts passes through soy

oil (CBO). This is interesting, as the latter an be used for fuel. The link between

ommodities and nan ialassets passes through gold (NGC), whi h an be seen as a

ommodity but also asa reserve of value. The only surprise omes fromthe S

&

P

500

(ISM),whi his more orrelated tosoy oil(CBO) than toother nan ialassets.

Su hanorganizationleadstospe i on lusions regardingsystemi risk. Letus

sup-posethat apri essho krea hesinterestrates(IED).Thestar-likeorganizationofthe

treedoesnot enableustodeterminewhether this sho k omesfromtheenergy orthe

agri ulturalse tors. Thingsare totallydierent inthe maturity dimension.

In this ase, it was not possible to give an illustration for ea h tree, as the database

gathers

14

futures ontra ts. We thus retained a representative tree, that of Brent

rude (LLC). The latter is illustrated by Figure (5)-b. The MST is linear and the

maturities are regularlyordered fromthe rst tothe last deliverydates.

Insert Figure(6) about here.

The analysis on the maturity dimension gives rise tothree remarks. Firstly,this

lin-ear topology ree ts the presen e of the Samuelson ee t. In derivative markets, the

movementsin the pri esof the prompt ontra ts are larger thanthe otherones. This

results in a de reasing pattern of volatilities along the pri es urve. Se ondly, this

type of organization impa ts the possible transmission of pri es sho ks. The most

likely path for a sho k is indeed unique and passes through ea h maturity, one after

the other. Thirdly, the short part of the urves are generally less orrelated with the

otherparts. This phenomenon an result frompri es sho ks emerging in the physi al

marketwiththe mostnearby pri e beingthe most ae ted;it ould alsoree tnoises

introdu edon the rst maturity by investors inthe derivative market.

Letusnowturntothethree-dimensionalanalysis. Figure(6)representsthe

3

-Dstati

MST. Itsshape bringsto mindthat observed inthe spatialdimension. However, it is

enhan ed by the presen e of the dierent maturities available for ea h market. The

rudeoil(LLC)andAmeri anheatingoil(NHO)arefoundatthe enter ofthegraph.

They are the three losestnodes ofthe graph. Moreover, the agri ultural se toris no

longerlinked togold. It is nowdire tly linked toAmeri an rude oil(NCL).

It would have been interesting to knowwhi h maturities onne t two markets or

se -tors. E onomi intuitionsuggests twokindsof onne tions: they ouldappear onthe

shortestoronthelongestpartofthe urves. Intherst ase,thepri e'ssystemwould

be essentially driven by underlying assets; in the se ond one, it would be dominated

by derivative markets. However, a loser analysis of the

3

-D trees does not provide

eviden eofeither kindofexpe ted organization. Moreover, theanalysis ofthe tree at

dierentperiodsdoesnotlead tothe on lusionthat thereissomethinglikeapattern

in the way onne tions o ur. Further investigations are thus ne essary in order to

study the links between markets and se tors more pre isely. We oer an initial

re-sponse tothis problem atthe end of this se tion.

4.2.2 Allometri properties of the MST

Star-like trees are symptomati of a random organization, whereas hain-like trees

reveal a strong stru ture. The omputation of the allometri oe ients of the MST

provides a meansof quantifying the degree of randomness inthe tree.

The rst model of the allometri s aling ona spanning tree was developed by

[Banavar etal.,1999℄. The rst step of the pro edure onsists in initializing ea h

node ofthe tree with the value

1

. Thenthe root or entral vertex of the tree must be

identied. Inwhat follows,the rootisdened as thenode havingthe highestnumber

of links atta hed to it. Startingfrom this root,the method onsists in assigning two

oe ients

A

i

and

B

i

toea h node

i

of the tree, where:

A

i

=

X

j

A

j

+ 1 and B

i

=

X

j

B

j

+ A

i

,

(3)

j

stands for all the nodes onne ted to

i

in the MST. The allometri s aling relation

isdened asthe relationship between

A

i

and

B

i

:

B ∼ A

η

,

(4)

η

is the allometri exponent. It represents the degree or randomness of the tree and

standsbetween two extreme values:

1

+

for star-like trees and

2

−

Figure(2) summarizes the allometri propertiesof the MST for ea hdimension. The

leftpanelreprodu esthe dierentexponentsand givesthe error resultingfroma

non-linearregression. Figure(7) gives anillustrationof the allometri oe ients in

3

-D.

Thedashedline orresponds tothebesttwithanexponentequalto

1.85

. Thegure

shows that the oe ients are well des ribed by the powerlawwith anexponent.

Asfarasthespatialdimensionis on erned,theexponentsindi atethatevenifFigure

(5) seems to show a star-like organization, the shape of the MST is rather omplex

andstandsexa tlybetween thetwoasymptoti topologies. Thereisanorderingofthe

tree,whi hiswellillustrated by theagri ulturalse tor,whi hformsaregularbran h.

Within the maturity dimension, the oe ients tend towards their asymptoti value

η = 2

−

. They are however a bit smaller than

2

, due to nite size ee ts (there is a

nite number of maturities). Su h a result is rather intuitive but nevertheless

inter-esting: arbitrage operations on the futures ontra ts related to the same underlying

asset are easyand rapidly undertaken, resultingin aperfe t orderingof the maturity

dates.

Even if the topology of the spatial and 3-D trees seems similar, they are

quantita-tively dierent. Theallometri exponentfor the three-dimensionalishigher: the best

t from our data gives an exponent lose to

1.757

, whi h must be ompared to the

value of

1.493

for the spatial ase. Thus, the topologyof our system, in

3

-D, israther

omplex. Itistheresultoftwodrivingfor es: thestar-likeorganizationindu edbythe

spatialdimensionandthe hain-likeorganizationarisingfromthematuritydimension.

4.3 Dynami al studies

Be ause they are based on orrelation oe ients, our Minimum Spanning Trees are

intrinsi allytime dependent. Therefore, its isne essary to study the time dependent

properties of the graphs. On the basis of the entire graph, rstly we examined the

dynami al properties of the orrelation oe ients, as well as the node's strength,

whi h provides informationonhow far a given node is orrelated to the other nodes.

In order to study the robustness of the topology of the MST, we then omputed the

graph'slength,whi hrevealsthestateofthesystemataspe i time. Lastly,survival

retained a rollingtime window with a size of

∆T = 480

onse utive tradingdays.

4.3.1 Correlation oe ients

In order to examine the time evolution of our system, we investigated the mean

or-relations of the returns and their varian es ([Sie zkaand Holyst, 2009℄). The mean

orrelation

C

T

_(t)

for the orrelation oe ient

ρ

T

ij

in a time window

[t − ∆T, t]

an

bedened as follows:

C

T

_{(t) =}

2

N (N − 1)

X

i<j

ρ

T

ij

(t) ,

(5) The varian e

σ

2

C

(t)

of the mean orrelation isgiven by:

σ

C

2

(t) =

2

N (N − 1)

X

i<j

ρ

T

ij

(t) − C

T

(t)

2

.

(6)

Insert Figures(8)and (9) about here.

Figure(8) represents the mean orrelation and its varian e onthe spatial dimension.

Itshows that themean orrelationof thepri es system in reasesover time,espe ially

after

2007

. The varian e exhibits a similar trend. Moreover, it rea hes its maximum

onthe

09/19/2008

, fourdays after the Lehman Brothers'bankrupt y.

We then examine the maturity dimension. Firstly, we fo us on the statisti al

prop-erties of the orrelation oe ients of two futures ontra ts, represented by Figure

(9). They are very dierent for these ontra ts. The maturities of Brent rude oil

(LLC) are more and more integrated over time: at the end of the period, the mean

orrelation is lose to

1

. Su h a trend does not appear for the eurodollar ontra t

(IED). This is onsistent with the peripheral position of the interest rate market in

the orrelation lands ape. As far as rude oil is on erned, the level of integration

be omes so strong that the varian e de reases and exhibits an anti orrelation with

the mean orrelation. The result was totally dierent in the spatial ase: the mean

orrelationand itsvarian ewhere orrelated([Onnela etal.,2003a℄)alsoobservesu h

apositive orrelation during pri esgrowth and nan ial rises).

Insert Figures(3), (10) about here.

Figure(3)summarizesthestatisti alpropertiesofthe mean orrelationsandvarian es

forthe

14

markets,onthematuritydimension. It onrmsthat, foralmostevery

is quitelow, when ompared with other markets, espe ially for London Natural Gas.

Meanwhile, their mean varian e ishigh.

Mergingspa eandmaturity,inthreedimensions, wealsoobserveanimportantrise in

the mean orrelation and varian e, as shown in Figures(10)-a and (10)-b. Moreover,

these values are orrelated.

4.3.2 Node's strength

The node's strength, al ulated for ea h node

i

, indi ates the loseness of one node

i

tothe others. It isdened as follows:

S

i

=

X

i6=j

1

d

ij

.

(7)

In our ase, the node's strength provides information on the intensity of the

orrela-tions linking a given node to the others. When

S

i

is high, the node is lose to the

others.

Insert Figure(11) about here.

Figure(11) represents the time evolutionof the node's strength forea h node within

the fully onne ted graph, in the spatial dimension. The gure has been separated

into four panels: the energy se tor is at the top, with Ameri an produ ts on the left

and European produ ts onthe right, the agri ultural se toris atthe bottomleft and

nan ialassets are at the bottom right.

Figure 11) prompts the following remarks: at the end of the period, out of all the

assets studied, the two rude oilsand Ameri an heating oil show the greatest node's

strength. Theseare followed by soy oil(CBO),other agri ultural assets,the S

&

P

500

ontra t (ISM),gold(NGC),theeuro dollarex hangerate (CEU)and European gas

oil (LLE). The more distant nodes are those representing the eurodollar (IED) and

naturalgases (NNG and LNG).

Whenthe timeevolutionof this measure is on erned,the se tor shows dierent

pat-terns: the integration movement, hara terizedby anin rease in the node's strength,

emerges earlier for the energy se tor than for the agri ultural se tor. However, it

de reases for energy at the end of the period, whi h is not the ase for agri ultural

in rease afterO tober

2005

. Last but not least, mostof theprodu ts exhibitastrong

in rease, ex ept for natural gases and interest rate ontra ts. Thus, whereas the ore

ofthetree be omesmoreandmore integrated,the peripheralassets donotfollowthis

movement.

Insert Figures(3)and (12) about here.

As far as the maturity dimension is on erned, it was not possible to represent the

node's strength for all futures ontra ts. Moreover, the omputation of mean node's

strength,onallmaturitiesfor ea h ontra t, would leadtothe samekindofresults as

thoseprovided byFigure(3). Therefore, weagainretained,theBrent rudeoil(LLC)

and the eurodollar ontra t (IED) examples. We then hose three delivery dates for

these ontra ts, as shown in Figures (12)-a and (12)-b. The rst maturity is drawn

withane line,the lastmaturitywithawidelineandtheintermediarymaturitywith

amediumwidthline. All the observed nodes'strength growovertime, ex ept forthe

rst eurodollar (IED)maturity. Moreover, in ea h ase, the strongest node isthe one

whi h orresponds to the intermediary maturity, whereas the weakest one represents

the rst maturity.

4.3.3 Normalized tree's length

Letusnowexaminesomeofthepropertiesofthe lteredinformation. Thenormalized

tree's length an be dened as the sum of the lengths of the edges belonging to the

MST:

L (t) =

1

N − 1

X

(i,j)∈M ST

d

ij

,

(8)

where

t

denotes the date of the onstru tion of the tree and

N − 1

is the number

of edges in the MST. The length of a tree is longer as the distan es in rease, and

onsequently when orrelationsarelow. Thus, the morethelengthshortens, themore

integrated the system is.

Insert Figures(13),(14) and (4) about here.

Figure (13)-a represents the dynami behavior of the normalized length of the MST

in its spatial dimension. The general pattern is that the length de reases, whi h

re-e ts the integration of the system. This information onrms what was observed

on the basis of the node's strengths. However we must remember that we are now

From a systemi point of view, this means that a pri es sho k will be less and less

absorbed asitpasses through thetree. Amore indepthexaminationofthe graphalso

shows a very importantde rease between O tober

2006

and O tober

2008

,as well as

signi ant u tuationsinSeptemberand O tober

2008

. Weleavethe analysisof su h

events for future studies.

In the maturity dimension, as integration in reases, the normalized tree's length also

diminishes. This phenomenon is illustrated by Figures(14)-aand -b, whi h represent

theevolutionsre ordedfor theeurodollar ontra t(IED)and forBrent rude (LLC).

Asfar asthe interestrate ontra t is on erned, the tree's lengthrst in reases, then

inmid-

2001

it drops sharply and remains fairly stable after that date. For rude oil,

the de rease is onstant and steady , ex ept for afew surges.

Figure(4) summarizes the main results on erning the tree's length for ea h futures

ontra t. However, it is not easy to ompare the tree's lengths of futures ontra ts

whenthe latterhave adierent number of delivery dates.

4.3.4 Survival ratios

TherobustnessoftheMSTovertimeisexaminedby omputingthesinglestepsurvival

ratio of the links,

S

R

. This quantity refers to the fra tion of edges in the MST, that

survivesbetween two onse utive trading days ([Onnela etal., 2003b℄):

S

R

(t) =

1

N − 1

|E (t) ∩ E (t − 1)| .

(9)

Inthisequation,

E(t)

referstothesetofthetree'sedgesatdatet,

∩

istheinterse tion

operator, and

| . |

gives the number of elements ontained in the set. Under normal

ir umstan es, thetopologyof the trees, between two dates,should beverystable, at

least when of the window lengths parameter

∆T

presents small values. While some

u tuationsof the survival ratiosmight bedue toreal hanges inthe behavior of the

system, it is worth noting that others may simply be due to noise. In this study, we

mostly examine the presen e of trends inthe way these ratios evolve.

Figure (13)-b represents their evolution in the spatial dimension. Most of the time,

this measure remains onstant, with a value greater than

0.9

. Thus, the topology of

possibletoidentify foureventswhere

1/4

of theedgeshas beenshued. Su h aresult

also allsforfurtherinvestigation,asareorganizationofthesystem anbeinterpreted

asthe result of a pri es sho k.

In the maturity dimension, Figures (14)-a and -bexhibit dierent patterns for rude

oil (LLC) and interest rates (IED). As far as rude oil is on erned, while the trees

shrink in the metri sense, the organization of the MST is very stable. Few events

seem to destabilize the edges of the trees, ex ept for the very end of the period, i.e.

fromthe end of

2008

. Again, what happens on the eurodollar is totally dierent. In

mid-

2001

,aroundthetime ofthe internet risis,whenthe lengthofthe treein reases,

the tree also be omes more spa ed out. This sparseness omes with an important

amount of reorganizations, and u tuations in the survival ratio are greater as the

lengthin reases.

Amore ompleteviewofwhathappens inthematuritydimensionisoeredbyFigure

(4). It exhibits the high level of stability of the trees in the way delivery dates are

organized.

Lastly, as far as the

3

-D trees are on erned, the survival ratios do not give any

fur-therinformationthaninthespatialandmaturitydimension. However, amorespe i

analysis ofthese trees, based ona pruningmethod,providessome interesting results.

4.3.5 Pruning the trees

As far as the stability of the trees is on erned, espe ially in

3

-D, when fo using on

thewhole system, itisinteresting todistinguishbetween reorganizationso urring in

aspe i market, between dierentdeliverydatesof the same ontra t,and

reorgani-zationthat hangesthe natureof the linksbetween twomarkets oreven between two

se tors. Equation (9) however givesthe same weight to every kind of reorganization,

whateveritsnature. Thetroubleis,a hangeinintra-maturitylinksdoesnot havethe

samemeaning, fromane onomi point ofview, as amovement ae tingthe

relation-ship between two markets or se tors. As we are interested, atleast initially,instrong

eventsae tingthemarkets,intermarketsandinterse torsreorganizationsseemmore

relevant. Thus, in order to distinguish between these ategories of displa ements,we

what-fromthe analysis. It signies that with pruned trees, the information on the spe i

maturity that is responsible for the onnexion between markets is nolonger relevant.

Su h trees enable usto ompute the survivalratios on the sole basis of market links.

Insert Figures(16)and (17) about here.

Figure(16)-adisplays the survivalratio of the redu ed trees. As observed previously,

the ratiois fairly stable. However, several events ause asigni ant rearrangementof

the tree. This is the ase, for example, for two spe i dates, namely

02/09/04

and

09/16/08

. A brief fo us on these twodates shows that the tree is totally rearranged.

In

2004

, the trees be ome highly linear, the nan ial assets se tor is at the enter of

thegraph,and ommoditiesappearmainlyattheperipheryofthesystem. Conversely,

in

2008

, the tree has a typi al star-like shape showing an organization based on the

dierent se tors studied.

Another interesting hara teristi of the pruned survival ratios is that they provide

informationonthe lengthofperiodsof marketstability. Overthe entireperiodof our

study, we measured the length of time

τ

orresponding to a stability period, and we

omputed the o urren es

N (τ )

of su h periods. Figure (16)-b displays our results.

Itshows that

N (τ )

de reases stronglywith

τ

,with a possible power lawbehavior, as

shown in the log-log s ale inset of Figure (16)-b. There are few stable periods that

last a long time, and mu h more stable periods that last a short time. We need to

rene the former result, but if su h a power law is onrmed, it will mean that the

markets an have stableperiodsof any length.

Finally, another interesting result lies in the analysis of those links whi h are most

frequently responsible for the reorganization of the trees. With fourteen markets,

there are ninety one links in our system. Some of them - twenty six - never appear.

Among the remaining sixty- ve trees, some appear very frequently and, on the

on-trary,othersdisplayveryfewo urren es. Figure(17)reprodu esthesetwo ategories

of links and the frequen y in whi h they appear in the MST. The most robust links

have a frequen y equalto one, whi h means that the links are always present. They

mainly orrespondtothe agri ulturalse tor,with thefollowingpairs: wheatand orn

(CW-CC),soy beans and orn (CS-CC),soy oiland soy beans (CBO-CS). The link

between gold and the euro-dollar ex hange rate (NGC-CEU) is also always present.

Asexpe ted,therelationshipsbetweenthetwo rudeoils(NCL-LLC)are very stable,

nomi point of view, as interest rates are embedded in forward ex hange rates. The

othertailofthe urve ontains ten links hara terizedby afrequen y lowerthan

0.01

.

The lowest values orrespond to the asso iation of interest rates and gas oiland that

ofinterestrates and gold.

5 Con lusions and poli y impli ations

In this arti le, we study the question of systemi risk in energy derivative markets

based on two hoi es. First we fo us on market integration, as it an be seen as a

ne essary ondition for the propagation of a pri es sho k. More spe i ally, we fo us

on the simultaneous orrelations of pri e returns. Se ondly, based on the fa t that

previous studies mainly fo used solely on the spatio-temporal dimension of

integra-tion,we introdu eamaturitydimension analysisand we perform athree-dimensional

analysis.

In the ontext of an empiri al analysis whi h aims to understand the organization

and the dynami behavior of a high dimensional pri e system, the graph-theory has

proven veryuseful. Itledustorepresent ourpri esystemasagraph,wherethenodes

are pri e returns and the links represent the orrelation between these returns. In

the ontext of the graph theory, Minimum Spanning Trees are often used to

under-stand omplex systems. They are parti ularlyinteresting in our ontext, as they are

lterednetworks enabling us to identify the most probable and the shortest path for

the transmissionof a pri es sho k.

The visualization of the MST rst shows a star-like organization of the trees in the

spatialdimension,whereasthematuritydimensionis hara terizedby hain-liketrees.

Thesetwotopologiesmergeinthethree-dimensionalanalysis,butthestar-like

organi-zationstilldominates. Thestar-likeorganizationreprodu esthethreedierentse tors

studied: energy, agri ultureandnan e, and the hain-likestru ture ree ts the

pres-en e of a Samuelson ee t. These intuitive results are very important, as they are a

key justi ationfor the use of our methodology.

The Ameri an and European rude oils are both found at the enter of the graph

and ensure the links with agri ultural produ ts and nan ial assets. Thus the rst

unlessitisabsorbedqui kly, itwillne essarilypassthrough rudeoilbeforespreading

toother energy produ ts and se tors. Moreover, a sho k will have an impa t on the

wholesystem thatwill be allthe greater the loser itis tothe heart of the system.

Another important on lusion is that the level of integration is more important in

the maturity dimension than in the spatial one. On e again, this result is intuitive:

arbitrage operationsare far easier with standardizedfutures ontra ts written onthe

sameunderlyingassetthanwithprodu tsofdierentnaturessu has ornbushelsand

interest rates. The analysis of how this level evolvesover time shows that integration

in reases signi antlyonboth the spatialand maturity dimensions. Su h anin rease

an be observed on the whole pri es system. It is even more evident in the energy

se tor (with the ex eption of the Ameri an and European natural gas markets) as

well as in the agri ultural se tor. The latter is highly integrated at the end of our

period. Lastly, asfar asthe nan ialse tor is on erned,no remarkable trend an be

highlighted. Thus, as time goes on, the heart of the pri e system be omes stronger

whereas where the peripheralassets are found doesnot hange signi antly.

Lastbutnot least,thedynami analysisalsoreveals, byusingsurvivalratios,thatthe

system is fairly stable. This is true, ex ept for spe i events leading to important

re ongurations of the trees and requiring aspe i analysis. We leave these studies

forfuture analyses.

Su h results have very important onsequen es, for regulatory as well as for hedging

and diversi ation purposes. The move towards integration started some time ago

and there is probably no way to stop or refrain it. However, knowledge of its

har-a teristi s is important, as regulation authorities may a t in order to prevent pri es

sho ks fromo urring, espe iallyinpla es where their impa t may beimportant. As

far as diversi ation is on erned, portfolio managers should probably fo us on the

less stablepartsof the graph. The linksinthe treeswhi h hange the mostshouldbe

thebest andidatesfordiversi ationopportunities. Lastly,oneimportant on ernfor

hedgingistheinformation onveyed byfutures pri esand itsmeaning. Thein reasing

integration ofderivative markets isprobablynot a problemforhedgingpurposes,

un-less a pri es sho k appears somewhere in the system. In su h a ase, the information

relatedtothetransmissionpathofthesho kisimportant,aspri esmighttemporarily

be ome irrelevant.

label of the futures ontra t in Datastream, and the olumn entitled "Pla e" indi ates the

geographi lo alizationoftransa tions. The olumn"Maturities"indi atesthelastmaturity

Allometric coefficient

Name

Static coefficient

Dynamical coefficient

IED

_1,927

_{± 0,056}

_1,913

_{± 0,011}

LNG

_1,874

_{± 0,002}

_1,886

_{± 0,059}

LLE

_1,88

_{± 0,003}

_1,943

_{± 0,02}

NNG

_1,75

_{± 0,037}

_1,774

_{± 0,018}

LLC

_1,889

_{± 0,003}

_1,904

_{± 0,095}

NCL

_1,994

_{± 0,045}

_1,906

_{± 0,013}

NGC

_1,732

_{± 0,092}

_1,908

_{± 0,013}

CBO

_1,889

_{± 0,003}

_1,886

_{± 0,032}

CS

_1,848

_{± 0,095}

_1,822

_{± 0,095}

CW

_1,864

_{± 0,13}

_1,761

_{± 0,125}

CC

_1,88

_{± 0,003}

_1,834

_{± 0,024}

Spatial

_1,493

_{± 0,056}

_1,621

_{± 0,024}

3-D

_1,757

_{± 0,023}

_1,85

_{± 0,009}

Figure2: Allometri propertiesofthetrees. Stati anddynami alexponentsforea hfutures

IED

CEU

ISM

CBO

CS

LNG

LLE

NHO

NNG

LLC

NCL

NGC

CC

CW

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

M14

M15

M16

M17

M18

Figure5: Stati minimumspanning trees. Leftpanel: MSTfor thespatial dimension,built

fromthe orrelation oe ientsofpri esreturns, 30/04/01-01/08/09. Right panel: MSTon

the maturity dimension, built from the orrelation oe ients of the Brent rude oil LLC,

M0 M1 M2 M3 M4 M5 M6 M7 M199 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M25 M24 M26 M27 M28 M29 M30 M31 M33 M32 M34 M35 M37 M36 M38 M39 M192 M42 M43 M44 M45 M46 M47 M48 M49 M50 M60 M51 M52 M53 M150 M54 M55 M56 M57 M58 M59 M61 M62 M63 M64 M65 M66 M67 M78 M68 M79 M69 M80 M70 M81 M71 M82 M72 M83 M73 M84 M85 M74 M75 M86 M76 M87 M77 M88 M89 M90 M91 M92 M93 M94 M95 M96 M97 M98 M99 M100 M101 M102 M103 M104 M105 M121 M106 M107 M108 M181 M109 M110 M111 M112 M113 M114 M115 M116 M117 M118 M119 M120 M122 M123 M157 M124 M125 M197 M126 M127 M128 M129 M130 M131 M132 M133 M134 M136 M135 M137 M142 M143 M138 M140 M141 M139 M144 M145 M146 M147 M148 M149 M151 M152 M153 M154 M155 M156 M158 M159 M160 M161 M162 M163 M164 M165 M166 M167 M171 M168 M169 M201 M170 M172 M173 M178 M174 M175 M177 M179 M176 M180 M182 M183 M184 M185 M186 M187 M188 M189 M205 M190 M191 M193 M194 M195 M196 M198 M200 M202 M203 M204 M206 M207 M208 M209 M218 M210 M211 M212 M213 M214 M215 M216 M217 M219

Figure 6: MST for the three-dimensional analysis, 27/06/2000-12/08/2009. The

dier-ent futures ontra ts are represented by the following symbols: empty ir le: IED, point:

ISM,o tagon: LNG, ellipse: LLE, box: NNG, hexagon: LLC,triangle: NCL,house: NHO,

diamond: NGC, inverted triangle: CBO, triple o tagon: CEU, double ir le: CS, double

1

10

100

A

1

10

100

1000

B

Figure7: Allometri propertiesofthetrees.

3

-Ddynami alallometri oe ientsinlog-log

17-01-2004

09-07-2009

t

0,1

0,2

0,3

0,4

C

T

(a)

17-01-2004

09-07-2009

t

0,04

0,06

0,08

σ

2

C

(b)

PSfrag repla ements

Figure8: Correlation oe ientsinthespatial dimension. Figure(a): Mean ofthe

30-10-1995

03-09-2002

08-07-2009

t

0,8

0,85

0,9

0,95

C

T

LLC

IED

(a)

30-10-1995

03-09-2002

08-07-2009

t

0

0,02

0,04

0,06

0,08

σ

_C

(b)

PSfrag repla ements

Figure9: Correlation oe ientsinthematuritydimensionfor theeurodollar IED (dashed

lines)and the Brent rude oil LLC (bla klines). Figure (a): Mean of the orrelation

17-01-2004

09-07-2009

t

0,2

0,3

0,4

C

T

(a)

17-01-2004

09-07-2009

t

0,08

0,1

0,12

0,14

σ

2

T

(b)

PSfrag repla ements

Figure10: Correlation oe ientsinthreedimensions. Figure(a): Meanof the orrelation

17-01-2004

09-07-2009

t

10

12

14

16

S

NNG

NCL

NHO

(a)

17-01-2004

09-07-2009

t

10

12

14

16

S

LNG

LLE

LLC

(b)

17-01-2004

09-07-2009

t

10

12

14

S

CBO

CS

CW

CC

(c)

17-01-2004

09-07-2009

t

9

10

11

12

S

IED

ISM

NGC

CEU

(d)

Figure 11: Nodes strength of the markets in the spatial dimension. Figure (a): Ameri an

energy produ ts. Figure(b): European energy produ ts. Figure( ): Agri ultural produ ts.

16-01-2004

08-07-2009

t

0

100

200

S

(a)

26-07-1998

16-01-2004

08-07-2009

t

0

100

200

S

(b)

Figure 12: Node strength in the maturity dimension, for three maturities. Figure (a):

17-01-2004

09-07-2009

t

0,9

1

1,1

(a)

17-01-2004

09-07-2009

t

0,25

0,5

0,75

1

S

_R

(b)

PSfrag repla ements

L

16-01-2004

08-07-2009

t

0

0,1

0,2

(a)

26-07-1998

16-01-2004

08-07-2009

t

0,1

0,15

0,2

0,25

0

0,5

1

1,5

2

S

_R

0

0,5

1

1,5

2

S

_R

(b)

PSfrag repla ements

L

L

Figure 14: Maturitydimension,normalizedtree'slength andsurvivalratiosforthe eurodollarIED (a)and the

IED

CBO

CEU

CW

LNG

LLE

LLC

NNG

NHO

NGC

NCL

CS

CC

IED

CEU

LNG

LLE

NHO

NNG

NCL

LLC

NGC

CBO

CS

CC

CW

Figure 15: Pruned minimum spanning trees of the events 09/02/2004 (left panel) and

0

50

100

150

200

250

300

τ

0

5

10

15

20

25

30

N(

τ

)

(a)

17-01-2004

09-07-2009

t

0,4

0,6

0,8

1

1,2

S

_R

(a)

1

10

100

10

Figure 16: Properties of pruned trees. Figure (a): Survival ratio. Figure (b): Number of

IED-LLE

IED-NGC

LLC-CS

NNG-CBO

NHO-CS

LNG-NGC

NHO-CW

NCL-CEU

ISM-CEU

NCL-CS

NGC-CW

LLE-CEU

0

0,005

0,01

0,015

(a)

ISM-NNG LLE-LLC NCL-CBO

NNG-LLC

NNG-NCL

NNG-NHO

LNG-LLE

LLE-NHO

NCL-NHO

IED-CEU

LLC-NCL

NGC-CEU

CBO-CS

CS-CC

CW-CC

0,4

0,8

1,2

(b)

Figure 17: Frequen y of links apparition in the pruned minimum spanning trees. Figure

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