Deux ontributions en gestion des risques de mati�eres premi�eresSteve OHANA19 septembre 2006

L'es alier de la s ien e est l'�e helle de Ja ob, il ne s'a h�eve qu'aux pieds de DieuAlbert Einstein

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Remer iementsCe travail doit beau oup tout d'abord �a ma dire tri e de th�ese, le professeur Helyette Geman, qui,tout au long de es quatre ann�ees, m'a prodigu�e ave bienveillan e ses onseils et ses intuitions.C'est �a son onta t que j'ai appris e qu'�etait v�eritablement le m�etier de la re her he. Je la remer- ie in�niment pour m'avoir appris (parfois dans la douleur...) �a onstruire et �a r�ediger un arti les ienti�que. Ce manus rit n'aurait sans doute pas �et�e le meme sans son regard attentif et exigeant.Cette th�ese est �egalement le fruit d'un partenariat de trois ans ave la Dire tion de la Re her hede Gaz de Fran e, qui a permis de donner �a mon travail une dimension appliqu�ee. Mes pens�eesvont en parti ulier �a Olivier Bardou, �a Isabelle Garreau, �a David Game, et �a Guillaume Leroy, queje remer ie haleureusement pour leur soutien. Je remer ie C�eline Jerusalem pour son aide ainsique Christian De-La�orest, Damien Reboul-Salze, Florent Bergeret, Christophe Barrerra-Esteve,Gr�egory Benmenzer, Solenne Gueydan, Marion La ombe, Pierre-Laurent Lu ille, et Jeanne Reypour leur a ueil et leur en adrement s ienti�que.Je salue le professeur Ni ole El Karoui, qui m'a fourni l'essentiel de ma formation th�eorique en�nan e, le professeur Dann Lanneuville, qui m'a aid�e �a former mon projet de th�ese, ainsi que lesprofesseurs Guy Cohen, Pierre Carpentier, C�e ile Kharoubi, Steven Shreve, Marija Ili , AbrahamLioui, Paul Kleindorfer, Mi hel Crouhy, et Fr�ed�eri Bonnans, qui m'ont g�en�ereusement onsa r�eiii

leur temps et m'ont apport�e des onnaissan es pr�e ieuses.Ces quatre ann�ees de re her he ont �et�e une exp�erien e humaine passionnante mais �egalement diÆ- ile, o�u l'euphorie laissait pla e parfois au doute et au d�e ouragement. Je veux i i rendre hommageaux personnes dont la on�an e et l'amiti�e m'ont aid�e �a surmonter les �epreuves qui jalonnent la vied'un do torant.Je remer ie en premier lieu ma femme Nathalie pour son amour et son soutien quotidien ; la on�-an e tranquille, la joie de vivre ommuni ative, et l'�energie bienveillante qu'elle d�egage ont �et�e desmoteurs indispensables pour mener �a bien ette th�ese. Ce travail lui est personnellement d�edi�e.Je voudrais remer ier �egalement mes parents pour les valeurs d'humilit�e, d'honnetet�e, et de ques-tionnement qu'ils m'ont transmises; la re her he �etant avant tout une forme d'ouverture d'esprit,de sens ritique, et de quete spirituelle, ils ont �et�e sans au un doute mes premiers guides dans monpar ours de her heur.Je remer ie haleureusement mon fr�ere Jean-Ja ques Ohana, pour l'enthousiasme qu'il a toujoursmanifest�e pour mes "trouvailles", aussi modestes soient-elles, pour sa uriosit�e insatiable et sag�en�erosit�e, qui font qu'il a toujours �et�e pour moi un mod�ele et un guide.Je remer ie mes beaux-parents Marie-Claire et Fabien Belhassen pour leur bienveillan e et la on-�an e qu'ils ont toujours montr�ee pour mon travail.Je voudrais �egalement saluer Mauri e Petrover, �eminent professeur de m�ede ine, qui s'est int�eress�etr�es tot �a mon travail, et m'a a ompagn�e ave �e oute, a�e tion, p�edagogie, et bienveillan e dansma d�e ouverte de la re her he.Il est impossible de ne pas �evoquer i i mes amis et fr�eres Emmanuel Farhi, Emmanuel Goldsztejn,Benjamin Petrover, Ari�e Bibas et Joseph Amar, dont l'importan e dans ma vie est si grande.En�n, je salue mon ousin Mi hael Ohana et mes amis Emmanuel-Juste Duits, Muriel Darmon,iv

Alexandre Espinoza, Harold Hauzy, Emilie Brunel, Aur�elia Crouhy, Pas al L�evy-Garboua, Patri ket Emmanuelle Hayat, Fran ois-Charles S apula, Benjamin Kunstler, et Ygal Levy, qui ont ha un ontribu�e, par leur soutien et leur int�eret pour mes re her hes, �a la on r�etisation de e travail.

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ContentsRemer iements iiiIntrodu tion xi1 Time- onsisten y in managing a ommodity portfolio : a dynami risk measureapproa h 11.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 A omparison of dynami risk obje tives . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Stati risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Risk measures asso iated to a stream of ash ows . . . . . . . . . . . . . . . 61.2.3 Time onsisten y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.4 Risk and substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 The retailer's portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 De omposition results in two parti ular ases . . . . . . . . . . . . . . . . . . 181.3.3 The retailer problem in an in omplete/illiquid market . . . . . . . . . . . . . 201.3.4 A on avity property for Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22vii

1.3.5 Ji as the arbitrage pri e of the portfolio in omplete markets . . . . . . . . . 221.3.6 A model for the evolution of the forward urve and demand . . . . . . . . . . 241.4 Numeri al results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4.1 Expression of the retailer's problem on an event tree . . . . . . . . . . . . . . 261.4.2 Building the event tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.3 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4.4 E�e t of optimal strategies on the �nal and minimal wealths . . . . . . . . . 321.4.5 Portfolio value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.5 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.6 Annex: proof of the onvexity result . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.7 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 A new dependen e model for ommodity forward urves; appli ation to the USnatural gas and oil markets 472.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2 The e onomi relations between oil and natural gas in the US . . . . . . . . . . . . . 522.2.1 Dependen e through the demand . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.2 Dependen e through the supply . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3 Empiri al observation of the dependen e between oil and gas forward urves in the US 602.3.1 Data des ription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3.2 De omposition of daily forward urve moves into short term and long-termsho ks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3.3 Slope and level: two state variables for the shape of the forward urve . . . . 712.3.4 De�nition of lo al and global dependan e stru tures . . . . . . . . . . . . . . 81viii

2.3.5 Analysis of lo al dependen e stru ture . . . . . . . . . . . . . . . . . . . . . . 812.3.6 Analysis of global dependen e stru ture . . . . . . . . . . . . . . . . . . . . . 902.4 A new dependen e model for ommodity forward urves . . . . . . . . . . . . . . . . 972.4.1 Formulation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.4.2 Calibration of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.4.3 Stability of the orre tion me hanisms . . . . . . . . . . . . . . . . . . . . . . 1252.4.4 Evolution of the orrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312.5 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342.6 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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Introdu tionLa gestion des risques �nan iers asso i�es aux portefeuilles de mati�eres premi�eres est un probl�emed'une grande a tualit�e, dans un ontexte de forte volatilit�e des mar h�es de ommodit�es interna-tionaux et de d�er�egulation des mar h�es europ�eens de l'�ele tri it�e et du gaz.Cette th�ese apporte deux ontributions ind�ependantes dans e domaine.Le premier hapitre a trait �a la gestion et �a la valorisation dynamique des portefeuilles de ommodit�es. Les produ teurs, a heteurs et n�ego iants de mati�eres premi�eres sont expos�es �a la fois�a un risque de mar h�e, de par la volatilit�e des prix des mati�eres premi�eres, et �a des risques "nontrad�es" sur le mar h�e, dont les plus importants sont le risque volumique, li�e �a l'in ertitude surleur propre onsommation ou sur elles de leurs lients1, les risques de ontrepartie2, les risquesphysiques3, et les risques politiques4. Ces risques non trad�es et la liquidit�e restreinte et in ertainedes mar h�es de ommodit�es r�eent un ontexte de mar h�e in omplet qui distingue les mar h�es de1Les d�eriv�ees limatiques ont notamment pour vo ation la ouverture �nan i�ere des risques volumiques, mais �etantdonn�ee leur tr�es faible liquidit�e, on peut onsid�erer le risque volumique omme un risque non trad�e sur le mar h�e2La faillite du g�eant des mar h�es de l'�energie qu'�etait Enron a a ru la sensibilit�e des a teurs �a e type de risques;il est �a noter que l'apparition des mar h�es de futures de mati�eres premi�eres a �et�e notamment motiv�ee par e risque3Un exemple est l'in endie qui a rendu indisponible le sto kage anglais de Rough en f�evrier 20064Les r�e entes mena es de rupture de l'approvisionnement europ�een en gaz r�e�ees par la rise diplomatique russo-ukrainienne en sont une bonne illustration xi

mati�eres premi�eres des mar h�es taux et a tions. Ainsi, meme si les a tifs physiques omposantles portefeuilles de mati�eres premi�eres ( ontrats d'approvisionnement exible ("swing options"), apa it�es de sto kage, unit�es de produ tion/transformation...) peuvent etre �e onomiquement as-simil�es �a des options "r�eelles" �e rites sur des sous-ja ents spot ou futures de ommodit�es, ils nepeuvent etre valoris�es �a l'aide des prin ipes d'arbitrage, omme le sont les options portant sur dessous-ja ents a tions ou taux d'int�eret. De plus, l'univers de mar h�e in omplet r�ee des synergiesentre les a tifs d'un portefeuille de mati�eres premi�eres qui interdisent la d�e omposition de la gestionet de la valorisation d'un tel portefeuille en somme d'options r�eelles ind�ependantes. Les mouve-ments r�e ents de rappro hement entre di��erentes ompagnies europ�eennes de gaz et d'�ele tri it�e5ont d'ailleurs �et�e en partie motiv�es par les synergies existant entre leurs di��erents portefeuillesd'a tifs physiques.Pour prendre en ompte es synergies dans la valorisation et la gestion d'un portefeuille de mati�erespremi�eres, il est n�e essaire d'adopter une appro he globale mod�elisant l'intera tion entre les di��erentsa tifs physiques et ontrats omposant e portefeuille.Il existe un nombre important de travaux de re her he op�erationnelle adoptant ette d�emar heglobale. Cependant, la plupart d'entre eux d�eveloppent une vision statique de l'�evaluation duportefeuille: seule la valeur initiale du portefeuille est onsid�er�ee et optimis�ee. Or, la gestion d'unportefeuille est un pro essus dynamique qui implique des d�e ision s�equentielles (utilisation dessto ks, nominations sur les ontrats d'approvisionnement, interventions sur les mar h�es �a terme...),lors desquelles le gestionnaire doit remettre �a jour son rit�ere d'�evaluation, en tenant ompte du5Une inquantaine de fusion/a quisitions entre ompagnies d'�energie ont eu lieu en Europe entre 1998 et 2003,parmi lesquelles Gas Natural-Endesa, EON-Ruhrgas, EDF-Edison, et le mouvement ontinue aujourd'hui ave l'annon e des intentions d'OPA hostiles de EON sur Endesa, d'Enel sur Suez puis l'annon e du possible rappro hementSuez-Gaz de Fran e xii

nouvel �etat du portefeuille et de l'information qui est devenue disponible. Dans le adre d'une�evaluation statique, le planning de d�e ision futures qui est d�e id�e �a la date t ne tient pas omptede la remise �a jour du rit�ere d'�evaluation qui sera n�e essairement op�er�ee �a la date t+ 1. On om-prend d�es lors que l'utilisation d'un tel rit�ere d'�evaluation dans un probl�eme de gestion dynamiquepeut onduire �a mal anti iper la valeur et le risque futurs du portefeuille et �a regretter ex-post desd�e isions pass�ees, e que les �e onomistes appellent l'in onsistan e dynamique.L'obje tif du premier hapitre est d'introduire et d'exp�erimenter sur un exemple on ret un nou-veau rit�ere d'�evaluation dynamique pour les portefeuilles de mati�eres premi�eres, dans le ontexted'un mar h�e partiellement liquide et en pr�esen e d'un risque volumique. Ce rit�ere, onstruit demani�ere r�e ursive �a partir du futur, permet la onsistan e inter-temporelle des d�e isions de gestion.D'autre part, par e qu'il d�epend de deux param�etres fa ilement interpr�etables, l'un ontrolant lar�egularit�e temporelle des ash- ows, l'autre leur dispersion al�eatoire, e rit�ere permet au gestion-naire de trouver le ompromis id�eal entre ri hesse �nale esp�er�ee, risque sur la ri hesse �nale etrisque de tr�esorerie au ours de l'horizon6. En�n, notre appro he est �a la fois un outil de gestion etde valorisation d'un portefeuille; elle rend notamment possible l'�evaluation des a tifs au sein d'unportefeuille, ave des appli ations potentielles importantes en terme de s�ele tion de portefeuille( ession/a quisition/ren�ego iation d'a tifs...).Le deuxi�eme hapitre de ette th�ese a pour obje tif de proposer un mod�ele d'�evolution on-jointe de deux ourbes �a terme de mati�eres premi�eres. De nombreux portefeuilles de mati�eres6Pour d�e�nir le risque sur la ri hesse �nale ou sur la tr�esorerie, on fera appel au on ept de Conditional Value atRisk xiii

premi�eres sont en e�et �a l'interfa e entre plusieurs mar h�es de ommodit�es7. La valeur �nan i�ereet la gestion de es portefeuilles "multi- ommodit�es" ne d�ependent pas seulement des prix spot maisde l'ensemble des ourbes �a terme des di��erentes mati�eres premi�eres impliqu�ees. Par exemple, lavaleur d'une entrale de produ tion d'�ele tri it�e �a partir de gaz d�epend du spread entre les ourbes�a terme du gaz et de l'�ele tri it�e sur la dur�ee de vie de la entrale: le pri ing et la ouvertured'un tel a tif reposeront don sur un mod�ele d'�evolution onjointe des ourbes �a terme du gazet de l'�ele tri it�e. Un deuxi�eme exemple est elui du d�etenteur d'un ontrat de livraison de gazindex�e sur le prix spot du p�etrole qui, pour s�e uriser sa marge, souhaitera au moment "opportun"prendre une position "short" sur le mar h�e �a terme du gaz et une position "long" sur le mar h�e �aterme du p�etrole. Un troisi�eme as mettant en jeu les orr�elations entre ourbes �a terme est eluid'un hedge-fund d�esirant onstruire une strat�egie "long/short" sur les futures de deux mati�erespremi�eres, en pariant sur le retour �a des relations �e onomiques de "long terme" liant historique-ment les prix �a terme8. Sur es trois exemples, on onstate que la onnaissan e des lois d'�evolution onjointe de plusieurs ourbes �a terme des mati�eres premi�eres inter-d�ependantes est essentielle. Or,s'il existe une abondante litt�erature sur la mod�elisation des orr�elations entre les prix de mati�erespremi�eres, elle- i on erne prin ipalement l'intera tion entre quelques points parti uliers sur unememe ourbe �a terme (par exemple, le prix spot et le prix �rst-month) ou sur deux ourbes �aterme (par exemple, les deux prix �rst-month). Jusqu'�a pr�esent, le probl�eme des d�ependan es en-7On peut penser par exemple �a des portefeuilles ontenant des ontrats de gaz index�es sur le prix du p�etrole, desusines de transformation de mati�eres premi�eres omme des raÆneries, des entrales de produ tion de gaz fon tionnantau harbon, au �oul, ou au gaz, des usines de fabri ation de m�etaux ou de papier onsommatri es d'�energie...8Il est important de souligner i i l'instabilit�e temporelle de ertaines relations de long terme entre mati�erespremi�eres: par exemple, la relation entre les prix du rude oil et les prix des produits raÆn�es (aussi appel�ee " ra kspread") s'est montr�ee parti uli�erement instable es derni�eres ann�eesxiv

tre l'ensemble de deux ourbes �a terme n'a pas �et�e en ore trait�e dans la litt�erature. Le deuxi�eme hapitre de ette th�ese se propose d'�elaborer un mod�ele d'�evolution onjointe des ourbes �a terme dedeux mati�eres premi�eres inter-d�ependantes. Notre mod�ele int�egre �a la fois les d�ependan es globaleset lo ales entre deux ourbes �a terme de mati�eres premi�eres. Les mouvements journaliers d'une ourbe �a terme sont d�e ompos�es en un ho ourt terme a�e tant les premi�eres maturit�es et un ho long terme, repr�esentant une translation globale des prix �a terme. Cette d�e omposition desd�eformations se traduit par une d�e omposition de la forme d'une ourbe �a terme omme sommed'une omposante d�eterministe saisonni�ere, d'une omposante al�eatoire de ourt terme (la "pente"),et d'une omposante al�eatoire de long terme ("le niveau"). Les d�ependan es globales on ernentles relations de long terme existant entre pentes et niveaux des deux ourbes �a terme, tandis que lesd�ependan es lo ales d�e rivent les relations entre les ho s ourt et long terme journaliers des deux ourbes. Comme dans un mod�ele de oint�egration lassique, les relations de long terme apparais-sent dans notre mod�ele �a travers une prime de risque dans l'�evolution des prix �a terme: l'originalit�epar rapport �a un mod�ele de oint�egration lassique est la stru ture par terme des primes de risque,qui, dans notre mod�ele, est ompatible ave l'absen e d'arbitrage et est la somme d'une partie" ourt terme", d�ependant de l'�e art �a la relation d'�equilibre sur les pentes, et d'une partie longterme, d�ependant de l'�e art �a la relation de long terme sur les niveaux. Le mod�ele de d�ependan eest appliqu�e aux mar h�es am�eri ains du gaz et du p�etrole ( rude oil et heating oil) de Janvier1999 �a O tobre 2004. Con ernant la stru ture de d�ependan e lo ale, nous mettons en �eviden e desrelations de ausalit�e entre les ho s journaliers des ourbes �a terme gaz et p�etrole, une volatilit�esto hastique pour l'ensemble des ho s, une volatilit�e saisonni�ere pour les ho s ourt terme du gazet du heating oil, des orr�elations positives entre les o-mouvements journaliers des ourbes �a termegaz et p�etrole. Con ernant la stru ture de d�ependan e globale, nous mettons en lumi�ere l'existen exv

d'une forte relation de long terme forte entre les niveaux des ourbes �a terme gaz et p�etrole (ave deux ruptures intervenant au d�ebut de l'ann�ee 2000 et au milieu de l'ann�ee 2003), et une relationde long terme moins signi� ative entre les pentes des ourbes �a terme. En�n, une �etude portant surla stabilit�e temporelle du mod�ele de d�ependan e r�ev�ele que les m�e anismes de orre tion d'erreurrelatifs aux d�ependan es globales se sont renfor �es depuis 2002 et que les orr�elations entre lesmouvements journaliers des ourbes �a terme gaz et p�etrole ont onnu un trend as endant sur lap�eriode 1999-2004.

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Chapter 1Time- onsisten y in managing a ommodity portfolio : a dynami riskmeasure approa h1We onsider the problem of the manager of a storable ommodity (e.g. hydro, natural gas, oal)portfolio fa ing demand risk while having a ess to storage fa ilities and illiquid spot and forwardmarkets. In this setting, we emphasize that a dynami ally onsistent way of managing risk overtime must be introdu ed. In parti ular, we demonstrate the temporal in onsisten y of stati riskobje tives based on �nal wealth and advo ate the use of a new lass of re ursive risk measuressu h as those suggested by Epstein and Zin (1989) and Wang (2000) for portfolio optimization andvaluation. This type of risk measures not only provide time- onsistent de ision plannings but allow1This hapter is a slightly di�erent version of an arti le with the same title written with Pr Geman; I thankGuillaume Leroy, David Game, Olivier Bardou and Jean-Ja que Ohana for their support and helpful suggestions; Ia knowledge also Stanley Zin and Paul Kleindorfer for their omments whi h helped me to improve this paper1

the portfolio manager to ontrol independently the o urren e of ash- ows a ross time and a rossrandom states of nature. We illustrate the dis ussion in an empiri al se tion where the trade-o�between �nal wealth risk and bankrupt y risk at an intermediate date is analyzed and the synergybetween the physi al assets omposing a ommodity portfolio is assessed.

2

1.1 Introdu tionWe onsider the situation of a retailer, who is engaged in long-term sale ontra ts, owns storagefa ilities and an trade the ommodity in illiquid spot and forward markets. The retailer is fa inga portfolio optimization problem, that translates into de iding at ea h time step whi h quantity toinje t in or withdraw from her storage fa ilities and trade in the spot and forward market, and aportfolio valuation problem, that onsists in assessing the value of the global portfolio and of ea hasset omposing it. The optimization and the valuation take pla e in the ontext of two types ofrisk: the volume risk that arises from the random demand of long-term ustomers and is related toexogenous non traded variables su h as weather, and the pri e risk that is linked to the volatilityof the ommodity pri e.In this in omplete market setting, the value of the retailer's portfolio is not uniquely determinedby arbitrage onsiderations and an integrated portfolio approa h is needed to handle liquidity on-straints.The sto hasti programming literature, on the one hand, has essentially treated situations whereportfolio management is analyzed through a mean-varian e riterion applied to �nal or intermedi-ate wealths, and fully de�ned at the �rst de ision date. In parti ular, the risk re-evaluations arisingat intermediate de ision dates are not taken into a ount, leading to possible on i ts betweende isions taken over time. Examples of this approa h are found in Unger (2002), where a CVaR onstraint on the �nal wealth is addressed through a Monte-Carlo approa h, in Martinez-de-Albenizand Sim hi-Levi (2005), where mean-varian e trade-o�s are onsidered and yield expli it solutionsin a one-step framework, and in Li and Kleindorfer (2004), where the ase of a multi-period VaR onstraint on ash ows is examined.The literature on de ision theory, on the other hand, has paid a deserved attention to the prob-3

lem of dynami hoi e under un ertainty. Originally, it was the problem of dynami onsumptionplanning that was analyzed by e onomists. In a seminal paper, Epstein and Zin (1989) introdu ea set of dynami utilities, de�ned re ursively in a dis rete time setting, and allowing one to sepa-rately a ount for the issue of substitution - ontrolling onsumption over time- and risk aversion- ontrolling onsumption a ross random states of nature. In �nan e, dynami risk measures werere ently introdu ed to a ount for the o urren e of a stream of random ash- ows over time. Ageneral requirement for these risk measures is their time- onsisten y (see e.g., Artzner et al. (2002))be ause, as emphasized by Wang (2000), multi-period risks are reevaluated as new information be- omes available, whi h raises the issue of the ompatibility between onse utive de isions impliedby the risk measure.Our arti le, to our knowledge, is the third attempt after Chen et al. (2004) and Ei hhorn andRomis h (2005) to use dynami risk obje tives in inventory and ontra ts portfolio problems. Ei h-horn and Romis h (2005) use a restri tion of the set of oherent dynami risk measures de�ned byArtzner et al. (2002) to solve an ele tri ity portfolio optimization problem but do not raise theproblem of time onsisten y of optimal strategies. Chen et al. (2004) de�ne their obje tive fun tionas an additive inter-temporal utility of the onsumption pro ess of the portfolio manager. Instead,we hoose the Epstein and Zin (1989) non additive inter-temporal utility obje tive and apply itdire tly to the ash ow pro ess. The impa t of this hange is signi� ant : in our setting, the initialwealth is not a state variable, the only state variables being the inventory level, and the umulativepositions in the forward market for ea h future delivery period; in addition, the retailer's problemappears as a ash- ow stream management one rather than a onsumption planning one; lastly,the exibility of the non additive inter-temporal utility allows the portfolio manager to separately ontrol the distribution of ash ows a ross time periods and a ross states of nature, whi h is not4

allowed by an additive utility obje tive on the onsumption pro ess2.The ontribution of this paper is twofold: i) on the methodologi al side, we de�ne the on ept oftime- onsisten y of optimal strategies, show that the lassi ally used stati risk measures on �nalwealth are not time- onsistent and advo ate the use of re ursive utilities as a time- onsistent and exible measure for portfolio risk management and valuation; ii) on the operational side, we providea tra table framework to dynami ally manage physi al assets under random demand and evolutionof spot and forward ommodity pri es, and show on a numeri al example how the use of re ursiveutilities an help strike a trade-o� between �nal and intermediate wealth risk management andassess the synergy between the physi al assets omposing a ommodity portfolio.The remainder of the paper is organized as follows. In se tion 2, we de�ne the time- onsisten y ofoptimal strategies and ompare two obje tives with respe t to the issues of time- onsisten y, andrisk/substitution preferen es. In se tion 3, we present the retailer's portfolio management problemand provide a pri ing formula and bid/ask pri es for physi al ommodity assets. Se tion 4 presentsa numeri al illustration of the main �ndings. Se tion 5 ontains on luding omments.1.2 A omparison of dynami risk obje tivesThe obje tive of this se tion is to present two examples dynami risk preferen es and assess theirtime- onsisten y properties, whi h we view as an original ontribution of the paper.2Note that our framework redu es to the one of Chen et al.(2004) when substitution preferen es are ignored andwhen CARA utility fun tions are used 5

1.2.1 Stati risk measuresIn the ase of one period settings, a number of stati risk measures have been de�ned to expresspreferen es of risk averse agents (see e.g, Artzner et al. (2000) and Frittelli and Rosazza Gianin(2004)). Mathemati ally, a (stati ) risk measure is a fun tion, here denoted �, asso iating to a ontingent laim X a real number �(X). �(X) represents the pri e that it is a eptable to pay inorder to pur hase X and ��(�X) represents the apital that must be provisioned in order to makea short position in X a eptable.1.2.2 Risk measures asso iated to a stream of ash owsPossible riteria for ash ow streams assessmentDe�ned on a �ltered probability spa e (;F ;P; (Ft)), the dis rete-time sto hasti pro ess G =(Gi)i=1;:::;T , represents a sequen e of random ash ows o urring at times (�i)i=1;:::;T . G is theset of all F�i-adapted ash ow pro esses from i = 1 to i = T . We hoose F�1 = f;;g (G1 isdeterministi ), and F�T = F , so that full information is revealed at date �T .A dynami value measure V = (Vi)i=1;:::;T onsists of mappings Vi : G�! R that asso iate to ea h ash ow pro ess G 2 G and to ea h ! 2 a real number Vi(G;!). The resulting sto hasti pro ess(Vi) is F�i-adapted. Finan ially, it represents the value of the sequen e of ash ows (Gk)k=1;:::;Tor the apital requirement to over the liabilities (�Gk)k=1;:::;T at dates (�k).Let us now propose two ategories of dynami values measures for streams of ash ows:1. The �rst ategory onsists of extensions of stati riteria depending on the wealth a umulated6

between date �i and date �T : Wi;T := TX�=i G�Vi(G;!) = �(Wi;T jF�i) (1.1)In the above equation, � is a one-step risk measure and the notation �(:jF�i) refers to ondi-tioning on the information available at date �i.2. A se ond ategory of riteria (proposed by Epstein and Zin (1989) and Wang (2000)) arere ursively onstru ted from the end of the time period by de�ning:VT (G;!) = GTVi(G;!) = W (Gi; �(Vi+1jF�i)) 8i � T � 1 (1.2)In the above equation, � is a one-step ertainty equivalent3 and the mapping W : R2 !R is alled an aggregator. In this framework, the date �i value is assessed re ursively byaggregation of the urrent ash ow Gi and ertainty equivalent of Vi+1 seen from date �i.An important observation is that the pro ess (Vi) is F�i-adapted.1.2.3 Time onsisten yTime- onsisten y is a property whi h guarantees that preferen es implied by a dynami valuemeasure do not on i t over time.3We adopt Wang's de�nition of the ertainty equivalent, i.e., a stati measure � verifying the monotoni ity property(whi h insures that if a random variable X is larger than Y in every state of the world, then �(X) � �(Y )) andredu ed to the identity on the spa e of onstant random variables.7

Examples of time-in onsisten iesConsider the two ash ow streams A and B, where all transition probabilities are supposed toequal 0:5:����������

�����HHHHH�����HHHHH

3 1(state u)0(state d)

7(state uu)1(state ud)6(state du)1(state dd)A����������

�����HHHHH�����HHHHH

3 2(state u)1(state d)

4(state uu)1(state ud)3(state du)1(state dd)BLet us evaluate stream A using the dynami value measure (1.1) with �(X) = u�1(E [u(X)℄),u(x) = ln(x):V2(A; u) = exp(E (ln(WA2;3ju))) = exp(0:5(ln(8) + ln(2))) = 4; V2(A; d) = exp(E (ln(WA2;3jd))) = p6V1(A) = exp(E (ln(W1;3))) = exp(0:25(ln(11) + ln(5) + ln(9) + ln(4))) = (55 � 36) 14Now evaluate stream B:V2(B; u) = exp(E (ln(WB2;3ju))) = exp(0:5(ln(6)+ln(3))) = p18; V2(B; d) = exp(E (ln(WB2;3jd))) = p8V1(B) = exp(E (ln(WB1;3))) = exp(0:25(ln(9) + ln(6) + ln(7) + ln(5))) = (54 � 35) 14We thus have simultaneously the following inequalities:V2(A; u) < V2(B; u); V2(A; d) < V2(B; d); V1(A) > V1(B)8

As a result, the dynami value measure V de�ned in (1.1) quali�es B as preferable to A in all statesof the world at time 2 and A preferable to B at time 1, hen e its time in onsisten y.Time onsisten y does not hold either if � is a mean-varian e instead of an expe ted utility riterionin equation (1.1). To see this, onsider the two following ash ow streams A (left) and B (right),with transition probabilities being written on top of ea h ar :����������

�����HHHHH0 0 (state u)0 (state d)

1 (state uu)0 (state ud)0

12123414

A����������0 0 (state u)

0 (state d)0.50

1212BLet us evaluate stream A using the dynami value measure (1.1) with �(X) = E (X) � V ar(X):V2(A; u) = E(WA2;3 ju))� V ar(WA2;3ju)) = 34 � (34 � 916) = 916V2(A; d) = E(WA2;3 jd))� V ar(WA2;3jd)) = 0V1(A) = E(WA1;3))� V ar(WA1;3)) = 12 � 34 � (38 � 964) = 964

Now evaluate stream B:V2(B; u) = E (WB2;3 ju))� V ar(WB2;3ju)) = 12V2(B; d) = E (WB2;3 jd)) � V ar(WB2;3jd)) = 0V1(B) = E (WB1;3))� V ar(WB1;3)) = 12 � 12 � (12 � 14 � 116) = 316 = 12649

We thus have simultaneously the following inequalities:V2(A; u) > V2(B; u);V2(A; d) � V2(B; d);V1(A) < V1(B)Let us now formally de�ne the time- onsisten y property:De�nition 1.2.1: The dynami value measure V is intrinsi ly time onsistent iffor A;B 2 G, for t 2 T = f1; :::; Tg, for ! 2 ;8>><>>: A(t; !) � B(t; !)Vt+1(A;!0) � Vt+1(B;!0) 8!0 2 Ht(!) ) Vt(A;!) � Vt(B;!)In the above de�nition, Ht(!) denotes the set of events !0 2 having the same history as !up to time t: intuitively, it is the set of all possible subsequent events after time t bran hing froma given s enario !.Property 1.2.2: If the aggregator W is monotoni , then the dynami value measures of the re ur-sive type (1.2) are intrinsi ly time- onsistentProof : for any t 2 T , for any ! 2 , if A(t; !) � B(t; !) and Vt+1(A;!0) � Vt+1(B;!0) 8!0 2Ht(!),then, by monotoni ity of ertainty equivalents, �(Vt+1(A; :)jFt)(!) � �(Vt+1(B; :)jFt)(!).In turn, by monotoni ity of the aggregator W ,Vt(A;!) =W (A(t; !); �(Vt+1(A; :)jFt)(!)) �W (B(t; !); �(Vt+1(B; :)jFt)(!)) = Vt(B;!).�De�nition of time onsisten y of optimal strategies and omparison of the two riteriaIn the previous se tion, we de�ned an intrinsi time- onsisten y property, related to the evaluationof exogenous streams of random ash- ows. In this se tion, we assume instead that the ash ows depend on de isions that are made at ea h date �i, using the information available at thisdate. De ision at date �i is the result of the optimization of a dynami value measure of the type10

des ribed above. This optimization not only yields the �rst de ision at that date, but a wholede ision planning for all subsequent stages. The question we pose in this se tion is the following:are optimal plannings onsistent over time?Let us de�ne the problem formally: onsider a ash ow sequen e (Gi)1�i�T , o urring at dates(�i)i�1, depending on de isions (qi)1�i�T and on a multi-dimensional random pro ess (�i)1�i�T :Gi := f(qi; �i). (�i) is assumed to be of the type �i+1 = g(�i; �i+1) for some reasonably behavedfun tion g, and a white noise ve tor pro ess (�i).We introdu e the state variables xi on whi h depend de isions at time �i and denote A(xi) the setof admissible strategies (qk)i�k�T at time �i. We suppose that, after de ision qi is made at time �i,the state xi leads to xi+1 = h(xi; qi; �i+1; �i+1), where h is a deterministi fun tion and (�i) a whitenoise ve tor pro ess possibly orrelated with (�i). We denote (F�i) the �ltration generated by thepro esses (�i; �i); (qi) is supposed to be an (F�i)-adapted pro ess.Lastly, we onsider the following optimization problem, related to a dynami value measure V :Ji(xi) := Max(qk)k�t2A(xi)Vi(G) (1.3)We denote (q�ik (xi))k�i the resulting (F�i)-adapted optimal strategy de ided at date �i4. Thequestion of onsisten y of optimal strategies an be formulated in the following way:Is q�ii+1(xi; �i+1; �i+1) equal to (q�(i+1)i+1 (xi+1)), where xi+1 = h(xi; q�i(xi); �i+1; �i+1)?We now turn to the time onsisten y of optimal strategies derived from the two dynami valuemeasures de�ned above.- First, let us onsider the �nal wealth obje tive de�ned in equation (1.1) with �(X) = u�1(E [u(X)℄),i.e,4We suppose throughout this se tion that all en ountered optimization problems have a unique solution11

Vi(G;!) = u�1 (E(u(Gi +Gi+1 + :::+GT )jF�i)))5:Ji(xi) : = Max(qk)k�i2A(xi)Vi(G)= u�1�Maxqi Max(qk)k�i+1E �i (E �i+1 (u(Gi +Gi+1 + :::+GT )))�= u�1�Maxqi E �i ( Max(qk)k�i+12A(xi+1)E �i+1 (u(Gi +Gi+1 + :::+GT )))�The date �i+1 implied problem Max(qk)k�i+1E �i+1 (u(Gi+Gi+1+ :::+GT ))) di�ers from the one derivedfrom the dynami value measure (Vi), i.e., Max(qk)k�i+1Vi+1 = E �i+1 (u(Gi+1 + Gi+2 + ::: + GT )). As aresult, the optimal strategy de ided at time i di�ers from the optimal strategy exhibited at timei+ 1.Time in onsisten y remains if we use a mean-varian e obje tive instead of an expe ted utility. Inorder to further investigate this issue, let us onsider a sequen e of three ash ows (G1; G2; G3),depending on the (F�i)-adapted pro ess (��i)i=1;2;3 and F�i-measurable de isions (qi)i=1;2;3, andlet us de ompose the varian e of the sum of these ash ows. As usual, we denote V ar�i(X) :=V ar(XjF�i).V ar�1(G1 +G2 +G3) = V ar�1(G2 +G3) = E �1 [(G2 +G3)2℄� [E �1 (G2 +G3)℄2= E �1 [E �2 ((G2 +G3)2)℄� [E �1 (E �2 (G2 +G3))℄2= E �1 [E �2 ((G2 +G3)2)℄� E �1 ([E �2 (G2 +G3)℄2) + E �1 ([E �2 (G2 +G3)℄2)� [E �1 (E �2 (G2 +G3))℄2= E �1 [V ar�2(G2 +G3)℄ + V ar�1(E �2 (G2 +G3)) = E �1 [V ar�2(G3)℄ + V ar�1(G2 + E �2 (G3))The last equality illuminates why total varian e is time in onsistent: the F�1-measurable termV ar�1(G2 + E �2 (G3)) is ontrolled by both de isions q1 and q2, in ontrast to the term G1, whi hdepends only on the de ision q1. This fa t ompromises the existen e of any dynami programming5From now on, we will denote E(X jF�i ) = E�i (X) 12

equation linking optimal strategies at dates �1 and �2:J1(x1) : = Max(qk)k=1;2;32A(x1) fE �1 (G1 +G2 +G3)� V ar�1(G1 +G2 +G3)g= Max(qk)k=1;2;3 fG1(q1)� V ar�1(G2 + E �2 (G3)) + E �1 (E �2 (G2 +G3)� V ar�2(G3))g6= Maxq1 �G1(q1)� V ar�1(G2 + E �2 (G3)) + E �1 ( Max(qk)k=2;32A(x2)E �2 (G2 +G3)� V ar�2(G3))�- We now turn to the dynami value measures des ribed in equation (1.2).As a �rst observation, let us onsider the ase of a linear aggregator W (x; y) = x+ y. The date �iobje tive derived from the value measure Vi de�ned by equation (1.2) is then:Ji(xi) : = Max(qk)k�i2A(xi)Vi(G)= Max(qk)k�i fGi(qi) + ��i(Vi+1)g= Maxqi �Gi(qi) + Max(qk)k�i+12A(xi+1)��i(Vi+1)�The question at this stage is to know whether permuting the operators Max and operator � islegitimate in the last equality, i.e., if the following property holds:Max(qk)k�i+1��i(Vi+1) ?= ��i( Max(qk)k�i+1Vi+1) (1.4)If the permutation is valid, then the optimal strategies will be time- onsistent sin e the date �i+1implied problem Max(qk)k�i+1Vi+1 will oin ide with the optimization problem at stage i+1; otherwise,they will not. 13

Let us try the aggregator W (x; y) = ��1(�(x) + ��(y)) and ertainty equivalent �(X) =u�1(E [u(X)℄), where u and � are in reasing fun tions and � is a positive dis ounting fa tor6:Ji(xi) : = Max(qk)k�i2A(xi)Vi(G) = Max(qk)k�i2A(xi)��1(�(Gi(qi) + ��(��i(Vi+1)))= ��1� Max(qk)k�i2A(xi) f�(Gi(qi)) + ��(��i(Vi+1))g�= ��1�Maxqi ��(Gi(qi)) + ��( Max(qk)k�i+1��i(Vi+1))��The inversion between operators Max and � in the last equality is permitted asMax(qk)k�i+1��i(Vi+1) = Max(qk)k�i+1u�1 (E �i (u(Vi+1))) = u�1�E �i ( Max(qk)k�i+12A(xi+1)u(Vi+1))�= u�1�E �i (u( Max(qk)k�i+12A(xi+1)Vi+1))� = ��i( Max(qk)k�i+12A(xi+1)Vi+1)We an now present a general suÆ ient ondition of time onsisten y for optimal strategies:Property 1.2.3: If there exist non de reasing fun tions a b, , and d and positive numbers �t su hthat Vi(G) = a hfb(Gi(qi)) + �i [E i(d (Vi+1(G))℄gi (1.5)then the dynami value measure (Vi) leads to time- onsistent optimal strategies.For the re ursive value pro ess de�ned by utility fun tions � and u, equation (1.5) holds witha = ��1, b = �, = � Æ u�1, and d = u. In the ase of lassi al expe tation maximization(risk-neutrality), equation (1.5) holds with a = b = = d = Id.1.2.4 Risk and substitutionWe have mentioned earlier that the problem of dynami optimization under un ertainty involvestwo dimensions, one with respe t to the distribution of ash ows a ross states of nature, the other6This parti ular hoi e for the aggregator and the ertainty equivalent was �rst suggested by Epstein and Zin(1989) and later on extended by Wang (2000) to in orporate ambiguity aversion14

over onse utive time periods. The �rst dimension has an e�e t on the �nal wealth distributionwhile the se ond one impa ts the likelihood of bankrupt y within the time period.Dynami value measures de�ned in equations (1.1) are not appropriate to apture the risk atta hedto intermediate ash ows sin e they are based on �nal wealth. By ontrast, re ursive dynami value measures allows one to disentangle randomness and time omponents, via the ertainty equiv-alent � and the aggregator W (respe tively a ounting for the risk aversion and the substitutionpreferen es of the de ision maker). For instan e, in the ase of re ursive dynami value measuresbased on utility fun tions, the on avity of the fun tions u and � leads to the smoothing of ash ows distributions in both dimensions and in turn to a joint ontrol of the �nal wealth risk andbankrupt y risk.Remark: The hoi e u = � in re ursive value measures derived from utility fun tions u and� leads to the lassi al obje tive: Vi(G) = u�1(E �i (PTk=i ��k��iu(Gk))), whi h has been widelyused in onsumption and portfolio hoi e problems in �nan e (e.g., onsumption-based CAPM).Of ourse, this obje tive is time onsistent and aptures both risk aversion and substitution; itsdrawba k is that it does not o�er as mu h exibility as a more general re ursive value measuresin e risk aversion and substitution are represented by the same fun tion u.As a on lusion of this se tion, we an state that re ursive dynami value measures with utilitytype aggregator and ertainty equivalent are satisfa tory in regard to time onsisten y of optimalstrategies and inter-temporal risk management.15

1.3 The retailer's portfolio problem1.3.1 The modelWe adopt a dis rete time setting, with a �nite horizon. The de ision periods are denoted (pi),i = 1; :::; T (typi ally months or quarters). The dates (�i) are de�ning the periods (pi).-date 1 date 2 ... date T�1 �2 �Tperiod 1 period 2 ... period TWe assume from now on that the retailer's portfolio is omposed of one sale ontra t and onestorage reservoir. In addition, the ommodity is supposed to be traded, stored, and onsumedin the same lo ation (in order to avoid transmission osts and onstraints). The problem an berepresented in a stylized diagram:-66??retailerstorage

market lientLmax is the maximal level of storage, the minimal level of storage (at any date) being 0, Linit isthe initial storage level, Lend is the minimal storage level at the end of the horizon. Li represents thestorage level at the end of period pi. Qinji denotes maximal inje tion in period pi, Qdrawi maximalwithdrawal; we suppose there are no inje tion/withdrawal osts nor holding ost. di denotes the lient's random demand in period pi, Ksi is the �xed selling pri e of the ommodity for period i.Only forward ontra ts are onsidered; ash ows due to forward ontra ting are settled at16

maturity of the ontra t and ounter-party risk ignored. We denote by F (i; j) the forward pri eof the ommodity quoted during pi for delivery in period pj7 (j � i) and Si the spot pri e of the ommodity, where Si := F (i; i).Remarks:1. In our model, trading is only authorized at de ision dates2. Even in the ase of illiquid markets, the retailer is assumed to be a pri e-taker, meaning thather trading de isions will have no impa t on market pri esStorage de ision variables orresponding to period pi are subje t to the following onstraints:0 � qinji � Qinji ; 0 � qdrawi � Qdrawi i � 1 (1.6)L0 := Linit; Li+1 = Li + qinji � qdrawi 0 � i � T (1.7)0 � Li � Lmax 8i = 1; :::; T ; LT � Lend (1.8)n(i; j) denotes the net number of forward ontra ts bought during period pi for delivery in periodpj (j � i), the ase i = j being a spot transa tion. N(i; j) represents the total forward position atthe end of period pi for delivery in period pj and satis�es the onditions:N(0; j) := 0 8j � 1; N(i; j) = N(i� 1; j) + n(i; j) 8 1 � i � j (1.9)We model the sequen e of events and de isions in the following way: during period pi, the retailerdis overs the lient's demand and de ides on date �i whi h quantities n(i; j) to buy on the spotand forward market and qinji or qdrawi to inje t in or withdraw from storage, respe ting the physi albalan e of ommodity ows during period pi i.e.,N(i; i) + qdrawi � qinji = di 8 1 � i � T (1.10)7Here, F (i; j) an be onsidered as the average pri e over all the quotation dates belonging to period pi of allforward ontra ts for delivery in period pj 17

Equation (1.10) expresses that market and storage are the two ways to serve demand at period pi.We de�ne the dis rete set of states of nature . Ea h ! 2 represents a realization of the pro ess�i = (di; F (i; j)j�i), i = 1:::T . We denote by (F�i) the �ltration generated by (�i). Throughoutthe paper, we assume the absen e of arbitrage opportunities in the ommodity spot and forwardmarkets. On (;F ;F�i), we de�ne a risk-neutral probability measure P, under whi h forward pri esare martingales8.We de�ne the set A of admissible strategies as:A := n(qi)i�1 = (qdrawi ; qinji ; n(i; j)j�i)i�1 F�i �measurable and verifying onstraints (1.6) to (1.10)o1.3.2 De omposition results in two parti ular asesIn this se tion, it is assumed that there are neither onstraints nor osts asso iated to trading inthe forward market. The risk-free interest rate r is supposed onstant. The goal here is to presenttwo ases where the pri ing issues and management of the portfolio are parti ularly simple:- the �rst ase is the one of a liquid market and deterministi demand- the se ond ase in ludes un ertain demand but assumes risk-neutrality of the retailer, hen e theuse of a riterion of expe ted pro�t maximizationIn both ases, a full de omposition of the portfolio value and management is possible.The total ash ow during period pi is denoted as Gi and may be written as:Gi = diKsi � TXj=i e�r(�j��i)F (i; j)n(i; j) (1.11)8We hoose here to work under a risk-neutral probability measure P to rule out a spe ulative use of the spot andforward markets; indeed, if forward pri es were not martingales under P, the trading de isions implied by our model ould be in uen ed by possible spreads between forward pri es and P-expe ted values of spot pri es, a feature whi his not relevant in the retailer's ontext 18

Remark: Cash ows due to forward trading are in this paper registered at transa tion date anddis ounted from delivery date at the risk free interest rate r. We adopt this unusual rule be ausewe want ash ows at dates �i to depend only on date �i de isions and not on previous ones9,as would be the ase if ash ows from forward transa tion had been registered at delivery date.Sin e interest rates are onsidered deterministi , this representation has no onsequen es on the�nal wealth but may have some on intermediate wealths10.Assuming liquid spot markets, the oupling onstraint (1.10) an be treated as an impli it one andwe fa e a fully de omposable problem, with onstraints only on individual assets.Deriving from (1.9) and (1.10) the volume n(i; i) of spot transa tions, equation (1.11) be omes:Gi = diKsi � n(i; i)Si � TXj=i+1 e�r(�j��i)n(i; j)F (i; j)= qdrawi Si � qinji Si + di(Ksi � Si) +N(i� 1; i)Si � TXj=i+1 e�r(�j��i)n(i; j)F (i; j)In this form, Gi appears like the sum of three omponents:1. qdrawi Si � qinji Si = period pi payo� from the storage fa ility. Storage de isions taken overtime are inter-dependent due to the apa ity onstraints expressed in equation (1.7)2. di(Ksi �Si) = period pi payo� from the sale ontra t devoided of any optionality, whi h is infa t a strip of swaps ex hanging the sale ontra t pri e Ksi for the spot pri e Si. The volumeinvolved at period pi is either �xed (deterministi demand) or random (unknown demand)3. N(i� 1; i)Si �PTj=i+1 e�r(�j��i)n(i; j)F (i; j) = period pi ash ow from forward ontra ts9in a ordan e with the setting de�ned in se tion 1.2.310we thus assume here that the retailer provisions in advan e all the future gains or liabilities at the signature ofa forward ontra t 19

Under this form, the portfolio appears as a ombination of various options written on the ommodityspot pri e while the forward market appears as a way to hedge the spot pri e risk. The abovesplitting of ash ows suggests a de omposition of the portfolio's value. In fa t, the latter will onlybe possible in two parti ular ases:� Portfolio de omposition in a omplete market setting: here, we assume that the demandpro ess (di) is deterministi (e.g., the ontra t sets a �xed volume to be delivered in all futureperiods). Then, the arbitrage pri e of the portfolio is the sum of maximal expe ted ash owsunder the (unique) risk-neutral probability measure; this value is the sum of the arbitragepri es of storage and sale ontra t. In this framework, the obvious strategy for the portfoliomanager onsists in optimizing independently the storage fa ility against the spot marketunder the risk-neutral measure, and hedging spot pri e risk using the forward market.� Portfolio de omposition for a risk-neutral retailer in a liquid market: we assume here that theretailer fa es both demand and pri e risks but is risk-neutral, i.e., she only tries to maximizeher expe ted pro�t. Under the assumption that the physi al measure is a risk-neutral measure,the optimal strategy for the risk-neutral retailer onsists again in optimizing independently thestorage fa ility against the spot market and doing no trade in the forward market. Moreover,under deterministi demand, the optimum of the risk-neutral retailer's obje tive orrespondsto the arbitrage pri e of the portfolio.1.3.3 The retailer problem in an in omplete/illiquid marketIlliquidity is modeled by deterministi volume onstraints on spot and forward trading, of the form:nb(i; i + �) � nmaxb (i; �); ns(i; i + �) � nmaxs (i; �) (1.12)20

where nb(i; j) and ns(i; j) stand for the number of bought and sold forward ontra ts during periodpi for delivery in period pj (with n(i; j) = nb(i; j) � ns(i; j)).We de�ne the set of admissible strategies from state xi = (Li; N(i; :); �i):A(xi) := n(qk)k�i = (qdrawk ; qinjk ; n(k; j)j�k)k�i Fk �measurable verifying admissibility onstraintso(1.13)and the analogous set of illiquid market admissible strategies Aliq(xi). The restri tions of theprevious de ision sets to date �i, de�ning the admissibility sets for de isions qi only, will be denotedby Ai(xi) and Aliqi (xi).We an now formulate the retailer's optimization problem as:Ji(xi) := Max(qk)k�i2Aliq(xi)Vi(G) (1.14)where G is de�ned by (1.11) and Vi(G) by the re ursive equation (1.2), with aggregator W and ertainty equivalent � derived from on ave in reasing fun tions � and u and positive dis ountfa tors (�i): W (x; y) = ��1(�(x) + �i�(y)); �(X) = u�1(E [u(X)℄)We denote su h a dynami value measure as V �;ut (G).The optimal value Ji(xi) satis�es the dynami programming equation:Ji(xi) = ��1( Maxqi2Aliqi (xi) ��(Gi(qi)) + �i� Æ u�1(E i(u(Ji+1(xi+1))))) (1.15)where the state xi+1 is given by the transition equation xi+1 = (Li + qinji � qdrawi ; N(i; :) +n(i; :); g(�i; �i+1)). The existen e of equation (1.15) guarantees the time onsisten y of optimalstrategies, as shown in the previous se tion. 21

1.3.4 A on avity property for JiProposition 1.3.1:Choosing CARA type utilities �(x) = �e��x and u(x) = �e��x su h that 0 < � � �, for all datest, and all states xi su h that Aliqt (xi) 6= ;, the maximization problem:Maxqi2Aliqi (xi)��(Gi(qi)) + �i� Æ u�1(E �i (u(Ji+1(xi+1))))is on ave with respe t to de isions qi. Moreover, the de ision set Aliqi (xi) is onvex. The resultalso holds for � = Id and u of CARA type.The proof is provided in the annex.1.3.5 Ji as the arbitrage pri e of the portfolio in omplete marketsIn this se tion, we show that, in omplete markets, Ji is the arbitrage pri e of the portfolio un-der the two onditions: �(x) = x (no preferen e for smooth versus irregular ash ows in timedimension) and �i = e�r(�i+1��i) (one period dis ount fa tor). These two assumptions will holdthroughout se tion 3.5.Property 1.3.2:For a on ave fun tion u, Ji(xi) = Max(qk)k�i2Aliq(xi)V Id;ui (G) is never greater than the risk-neutralobje tive Jrni (xi) = Max(qk)k�i2Aliq(xi)V Id;Idi (G)Proof : The on avity of u implies that for all random variables X:u�1(E [u(X)℄) � E (X) (1.16)It results, by a simple re ursion, that:8G 2 G; 8i 2 T ; V Id;ui (G) = Gi + �iu�1(E �i (u(V Id;ut+1 ))) � Gt + �iE �i (V Id;Idi+1 ) = V Id;Idi (G)22

and the property holds. �Property 1.3.3: When onditional values Vk+1 omputed at stages k (k = i; ::; T � 1) are nonsto hasti , then V Id;ui is the sum of dis ounted ash ows from stage i to stage TProof : In this ase, u�1(E �i (u(V Id;uk+1 ))) = V Id;uk+1 for all k = i; :::; T � 1, and, therefore, V Id;ui (G) =Gi + �iV Id;ui+1 =PTk=i e�r(�k��i)Gk, by a simple re ursion.�The onsequen e is that, in a omplete market setting (i.e., deterministi demand and no liquidity onstraints), Ji is at least equal to the arbitrage pri e of the portfolio.Property 1.3.4: In a situation of market ompleteness, Ji(xi) is equal to the arbitrage pri eof the portfolio Japi (xi) = Max(qk)k�i2A(xi)EQ�i (PTk=i e�r(�k��i)Gk), where Q is the (unique) risk-neutralmeasureProof : This property is derived from the following observations:- Ji(xi) � Max(qk)k�i2A(xi)V Id;Idi (G), as exhibited in property 1.3.2- Max(qk)k�i2A(xi)V Id;Idi (G) = Japi (xi), be ause the optimal value of the risk-neutral retailer's problemis equal to the portfolio's arbitrage pri e- Ji(xi) � Japi (xi), as shown in property 1.3.3.�Property 1.3.5: If markets are omplete and u stri tly on ave, then the risk of the optimalstrategy (q�k)k�i is null.Proof : The equality between Ji(xi) and Jrni (xi) implies an equality in equation (1.16) for ea hX = Vi+1, and, be ause the fon tion u is stri ly on ave, the equality is possible only if un ertaintyon all Vt is null.� 23

Consequently, we obtain the satisfa tory property that the optimization programme also providesa hedging strategy.To on lude this paragraph, we give a way of estimating the ask and bid pri es of a physi al assetor �nan ial ontra t in in omplete markets: as often done in the literature , we de�ne the ask (bid)pri e as the di�eren e of the values of Jt, with and without the bought (sold) asset. Under thisde�nition, the bid and ask pri es of an asset depend not only on the risk aversion of the managerbut also on her initial portfolio, a lassi al property in a situation of in ompleteness.1.3.6 A model for the evolution of the forward urve and demandWe assume a lassi al one-fa tor evolution model for the market forward urve F (i; j):F (i; j) = F (i� 1; j)Mi;jexp(e�ki(�j��i)Xi) 8j � i 8i � 2 (1.17)where (Xi)i�2 is a dis rete-time sto hasti pro ess omposed of independent variables with lawN(0; (�Xi )2), (ki) are positive parameters, and (Mi;j)j�i are positive onstants ensuring that F (i; j)i�jare martingale pro esses. In this model, only one type of sho k is allowed for the forward urve,namely translations, with an amplitude vanishing with time to delivery.Regarding the demand pro ess (di)i�2, we assume that it is driven by a dis rete-time sto hasti pro ess (Yi) (typi ally the temperature), omposed of independent variables with law N(0; (�Yi )2)positively orrelated with the pri e pro ess with orrelation oeÆ ients (�i):di = max(fi; �di + Yi) (1.18)where (fi) are positive oors ensuring that the demand pro ess is positive, and ( �di) are the averagedemands at ea h period.As a on lusion, to simulate the joint evolution of forward urve and demand at periods (pi), we24

only need to jointly simulate the random variables (Xi) and (Yi) for i = 1; :::; T and then useformulas (1.17) and (1.18).1.4 Numeri al resultsThree di�erent methods to numeri ally solve sto hasti ontrol problemsDue to the �nite horizon and the omplexity of the onstraints, the problem must be solved nu-meri ally. Three di�erent numeri al methods are possible:1. The �rst approa h onsists in using the dynami programming equation (1.15) to solve theproblem re ursively from stage i = T to stage i = 1. The advantage of this approa h is thatthe al ulation time is linear with respe t to the number of time steps. The drawba k isthat omplexity explodes exponentially with the dimension of the state xi and the numberof de ision variables. In the retailer's ase, the dimension of the state spa e at stage i� 1 is(T � i+ 1) (number of forward positions with delivery after period i� 1) + 1 (forward pri erisk) + 1 (demand risk) + 1 (storage level). We easily understand that this dimension an bevery large, and an be even larger if we onsider several storages and/or ontra ts. Longsta�and S hwartz (2001) proposed a numeri al method, ombining Monte-Carlo simulations anddynami programming; this method allows to handle large dimension in the sto hasti pro ess� but not in the de ision spa e, and therefore is not appropriate in our setting.2. The se ond method that an potentially be used here is the one used by Unger (2002) forhydro power risk management. It onsists in using Monte-Carlo simulations to adjust a priorifeedba k rules (i.e. rules relating de isions to realizations of �), determined by a �nite set ofparameters. This method has the advantage to potentially apture any type of dynami s for25

the pro ess � (jumps, sto hasti volatility...) in a very easy way, but is not appropriate to ompute onditional expe tations and impose to settle forms of feedba k rules a priori, whi his a diÆ ult task in our ase, due to the number of parameters intervening in de isions3. The third method whi h an be used is sto hasti programming, where a �nite number ofs enarios are represented on an event tree, and where de isions have to be taken at everynode of the tree. The main drawba k of this method is the exponential growth of al ulationtime with the number of time steps. In spite of this major drawba k, this method has threeadvantages that make it the most adapted to our problem: �rst, the al ulation time does notexplode with the dimension of de ision spa e; se ond, the onditional expe tations that arethe ore of our optimization obje tive an be assessed very easily on an event tree; thirdly,we do need to provide any form of feedba k laws but optimize dire tly our obje tive on theset of all possible de isions at ea h node.1.4.1 Expression of the retailer's problem on an event treeRe ursive evaluation of obje tive on the tree: Let Nt denote the set of time t nodes of thetree. Mathemati ally, a node n 2 Nt is an event representing a realization of the pro ess (�) fromtime 1 to time t. We all N the set of all nodes belonging to the tree. The root node n = 1represents the departing point.Note that Nt+1 is a re�nement of Nt and therefore, every node n of time t � 2 has a unique an estor(or father) node a(n) at time t� 1, de�ned as the unique member of Nt�1 whi h in ludes node n.Nodes n belonging to the set NT are alled leaves. A s enario orresponds to a path from the rootto a leaf. The su essors (or sons) to node n form the set S(n); NT = fn : S(n) = ;g.With the given transition probabilities �nm from node n to node m 2 S(n), we de�ne a probability26

�n for ea h node n by the re ursion �n := �a(n)�a(n)n and by �1 = 1. Clearly, Pn2Nt �n = 1 holdsfor ea h t = 1; :::; T .Vn = Gn 8n 2 NT (1.19)Vn = ��1 ��(Gn) + � Æ u�1(En(u(Vm))) (1.20)= ��18<:�(Gn) + � Æ u�1( Xm2S(n) �nmu(Vm))9=; 8n 2 Nt; t � T � 1 (1.21)Expression of s enario onstraints on the tree:Stati onstraints bounding de isions of stage i only have straightforward translations on orre-sponding de isions on the tree. However, s enario onstraints (1.6), (1.9) and (1.10) will be ex-pressed via the variables Ln, and Nn(p), representing respe tively the storage level and the forwardposition for delivery in period p at node n; these variables are de�ned by the following relations(remember that n = 1 is the root node of the tree):L1 = Linit + qinj1 � qdraw1 (1.22)Ln = La(n) + qinjn � qdrawn 8n 2 Nt; t � 2 (1.23)N1(p) = nb1(p)� ns1(p) (1.24)Nm(p) = Na(m)(p) + nbm(p)� nsm(p) 8t � 2; 8m 2 Nt; 8p � t (1.25)The orresponding onstraints are:0 � Ln � Lmax 8n 2 Nt; 1 � t � T (1.26)Ln � Lend 8n 2 NT (1.27)dm = Nm(t) + qdrawm � qinjm 8m 2 Nt; 1 � t � T (1.28)We de�ne the set Atree of admissible tree strategies as de isions(qm)m2N = (qdrawm ; qinjm ; nbm(p); nsm(p))m2N whi h respe t stati onstraints spe i�ed in 1.3.1 and27

dynami onditions (1.22) to (1.28).Optimization problem of the retailer Max(qn)n2N2AtreeV1 (1.29)We are left with a large-s ale ontinuous non linear optimization programme on variables (qn) inevery node of the tree. We solve this problem numeri ally using a large-s ale non linear solver.1.4.2 Building the event treeTo build the event tree, we use a two-dimensional latti e (see Webber (1997)), repli ating exa tlythe �rst two moments of the pro ess (X;Y ) at ea h time step.The four vertexes of the unit square �rst provide the equiprobable joint realizations of a ve tor~Z = ( ~X; ~Y ) of two un orrelated zero mean unit varian e random variables:-

6 ÆÆ

ÆÆ

(1; 1)(1;�1)

(�1; 1)(�1;�1)Figure 1.1: S enarios for two un orrelated random variablesThe extension to two orrelated variables is straightforward: onsidering a ve tor of two un or-related unit varian e variables ~Z = ( ~X; ~Y ), the ve tor of random variables Z = (X;Y ) = A ~Z with28

A = 0BB� �x 0��y p1� �2�y 1CCA have zero mean and ovarian e matrix � = 0BB� (�x)2 ��x�y��x�y (�y)2 1CCA.Therefore, we pro eed in the following way to build the event tree on the pri e/demand pro ess:- �rst, using the matrix M = 0BB� 1 1 �1 �11 �1 1 �1 1CCA, whose olumns represent the four joint re-alizations of a ve tor ( ~X; ~Y ) of two un orrelated zero mean, unit varian e variables, we form the2� 4 matrix N = AM , whose olumns are the realizations of the ve tor (X1; Y1), representing thepri e/demand nodes at time 1- then, we atta h to ea h node of period 1 the son nodes given by the matrix N = AM , and so on,until the last period- �nally, we apply formulas (1.17) and (1.18) to get the forward urve and the demand at ea hnode, the term Mi;j being determined by the martingale ondition at node n:Fn(i� 1; j) = En(Fm(i; j)) = Xm2S(n) 14Fm(i; j) (1.30)whi h gives: Mi;j = 1Pm2S(n) 14 exp(e�ki(�j��i)Xmi ) (1.31)It is important to point out here that the term M depends only on i and j and not of node nbe ause the variables (Xi; Yi) are independent of (Xi�1; Yi�1), hen e the sets fXmi ; m 2 S(n)g arethe same for every node n of date �i�1.We obtain 4T�1 di�erent s enarios from period 1 to period T .29

(a) Realizations of the forward urve (e/MWh) (b) Realizations of demand (TWh)

( ) Two-dimensional representation of the pri e and demand pro esses (X;Y ) at ea h timestep: the realizations of the pri e pro ess X an be read on the x-axisFigure 1.2: Event tree

30

1.4.3 The settingWe assume the following setting:- the retailer is trading an energy produ t, whose pri e is expressed in e/MWh- there are �ve periods of one quarter ea h: during the �rst quarter, the retailer fa es no demandand replenishes her storage fa ility using the spot market in order to meet the unknown lient'sdemand in the following year- the storage has an initial level at 20 TWh, a maximal inje tion/withdrawal per period of 10 TWh,a maximal (resp. minimal) storage level of 50 TWh (resp. 0), and a minimal end level of 20 TWh- the forward pri e dynami s are represented by the model des ribed in equation (1.17) with pa-rameters ki = 2 years�1 and volatility �Xi = 0:2 8i � 2; the initial forward urve is supposed tobe at at the level 20 e/MWh; in parti ular, the initial spot pri e equals 20 e/MWh- the maximal allowed traded volume in the market de reases with time-to-delivery: it equals 30TWh for ontra ts delivering in the present quarter ("spot" transa tion), 10 TWh for ontra tsdelivering in the next quarter, 5 TWh for ontra ts delivering in two quarters, and 0 TWh for ontra ts delivering in the following periods- the selling pri e on the sale ontra t is 21 e/MWh (hen e a margin of 5% with respe t to theaverage market forward pri e); regarding the demand hara teristi s, we suppose that d1 = 0, and8i � 2: �Yi = 10 TWh, �di = 20 TWh, fi = �di3 , and �i = 0:5. The realizations of (X;Y ) at ea htime step are represented on �gure (1.2( )): we note that there are four di�erent realizations forthe demand pro ess and two only for the pri e pro ess- we adopt CARA utility fun tions u(x) = �e��x and �(x) = �e��x to represent risk aversion andsubstitution preferen es, with varying risk aversion and substitution parameters � and �; interestrates are set to 0. 31

Figures (1.2(a)) and (2.24(b)) show the forward urve and demand s enarios. The mean-revertingnature of the spot pri e is visible.1.4.4 E�e t of optimal strategies on the �nal and minimal wealthsFigure (1.3(a)) shows the mean varian e trade-o� in the �nal wealth obtained when risk aversionvaries and the fun tion � remains equal to identity. When the risk is de�ned as the ConditionalValue at Risk11 on the �nal wealth WT 12:CV aRq(W ) = E(�WT j �WT > V aRq(W )) (1.32)the expe ted mean is an in reasing fun tion of risk, as shown in �gure (1.3(a)). For example, ade rease of the 0.5% (resp. 5%) CVaR on �nal wealth from 611 (resp. 505) to 371 (resp. 291)Me implies a de rease of the expe ted �nal wealth from 67 to 15 Me. Figure (1.3(b)) representsthe trade-o� between the risks of the �nal wealth and temporal minimal wealth13. Figure (1.3(b))shows that it is possible to ex hange bankrupt y risk for �nal wealth risk by de reasing the ratio ofparameter � to parameter �. For example, to ut the 0.5% (resp. 5%) CVaR on temporal minimalwealth from 1059 to 545 (resp. 473) Me, one has to a ept a rise of the 0.5% (resp. 5%) CVaRon �nal wealth from 365 (resp. 296) to 516 (resp. 458) Me. However, the ex hange of bankrupt yrisk for �nal wealth risk has limits: Figure (1.3(b)) shows in parti ular that it is not possible tobring down the 0.5% (resp. 5%) CVaR on temporal minimal wealth below a ertain threshold, orresponding to the pair (� = 0:1; � = 0:001) (resp. (� = 0:01; � = 0:0005)).Figures (1.4(a)) shows the umulative fun tion of the �nal wealth over the 256 tree s enarios used11V aRq(W ) is the well-known Value-at-Risk asso iated to quantile q12the wealth Wi at the end of period pi is de�ned as the umulative sum of ash ows from period p1 to period pi13Temporal minimal wealth is de�ned as mini2f1;2;3;4;5gWi; the temporal minimal wealth distribution is thusdire tly linked to bankrupt y risk 32

(a) Expe ted �nal wealth in terms of CVaR (in Me); ea h urve orresponds to a di�erentCVaR quantile and is onstru ted with � taking the values f0; 0:001; 0:005; 0:01; 0:02g

(b) CVaR of the temporal minimal wealth in terms of CVaR of the �nal wealth (in Me);ea h urve orresponds to a di�erent CVaR quantile and is onstru ted with (�; �) takingthe values (0:1; 0); (0:05; 0:0001); (0:02; 0:0001); (0:01; 0:0001); (0:1; 0:001); (0:01; 0:005);(0:01; 0:001); (0:001; 0:0001)Figure 1.3: Trade-o�s between expe ted wealth/�nal wealth risk and �nal wealth risk/bankrupt yrisk33

in the optimization pro edure under di�erent values of risk aversion. In �gure (1.4(a)), we observethat a risk aversion of 0:02 allows to signi� antly redu e the left tail up to 5% of the distributionobtained under a risk-neutral strategy. The ost of a higher risk aversion is that the main partof the �nal wealth distribution (to the right of the 10% quantile) is signi� antly moved upright.Figure (1.4(b)) shows the distribution of the minimal wealth over time: we see that a more on avefun tion � signi� antly redu es the likelihood of a very negative minimal temporal wealth, whi h isa onsequen e of the smoothing of ash ows in the time dimension. However, as shown by �gure(1.4(a)), if the ratio �� be omes too high (e.g.(� = 0:01; � = 0:0005)), the �nal wealth distributionexhibits a large left tail. If the portfolio manager seeks to strike a balan e between �nal wealthand bankrupt y risk management, she may hoose (� = 0:1; � = 0:001) or (� = 0:01; � = 0:0001).Figure (1.5) represents the intermediate wealths obtained at the di�erent nodes of the event treefor di�erent ouples of (�; �) and on�rms the above on lusions: hoosing (� = 0:01; � = 0:0005)allows one to ontrol the intermediate wealth risk but implies a great dispersion of the �nal wealth; onversely, hoosing (� = 0:02; � = 0) o�ers a very narrow range of �nal wealths but with a highbankrupt y risk at the end of the se ond period; the hoi e (� = 0:01; � = 0:0001) represents atrade-o� between and �nal and intermediate wealth risks.

34

(a) Final wealth umulative fun tion (in Me); the ase � = 0 (resp. � = 0) orresponds to a fun tion u (resp. �) equal to identity

(b) Temporal minimal wealth (in Me) umulative fun tion in in omplete mar-kets; the ase � = 0 (resp. � = 0) orresponds to a fun tion u (resp. �) equalto identityFigure 1.4: Final and temporal minimal wealth umulative fun tions for di�erent risk aversion andsubstitution parameters35

(a) Wealth pro�le in the ase (0,0) (b) Wealth pro�le in the ase (0.02,0)( ) Wealth pro�le in the ase (0.01,0.0001) (d) Wealth pro�le in the ase (0.01,0.0005)Figure 1.5: Cumulative wealths (in Me) in the di�erent nodes of the event tree for di�erent pairs(�; �)1.4.5 Portfolio valueFigure (1.6(a)) represents the portfolio value de�ned in se tion 1.3.5 for di�erent risk aversionparameters. The portfolio value is a de reasing fun tion of the risk aversion parameter. The spreadbetween the risk-neutral and positive risk aversion values an be interpreted as a risk premium,whose value in reases logi ally with the risk aversion parameter.The value of the sale ontra t, obtained by setting the storage exibility to zero in the originalportfolio14, behaves similarly. The storage value, obtained by setting the lient's demand to zero in14Setting the storage exibility to zero may ause the problem to be infeasible in the ase of illiquid markets andnon-interruptible lients; estimating the sale ontra t value may thus require in some situations the introdu tion ofarti� ial interruption/emergen y supply osts to relax the possibly too restri tive volume onstraints; in our example,36

the retailer's portfolio, does not depend on the risk aversion parameter: this is due to the fa t that,under the liquidity assumptions made in se tion 1.4.3, the storage fa ility has a unique arbitragevalue (here 55.26 Me) whi h an be se ured by appropriate forward transa tions; in this ontext,the optimum J1 of the storage management problem redu es to the storage arbitrage value, asexplained in se tion 1.3.5. The synergy value, de�ned as the spread between the storage portfoliovalue de�ned in se tion 3.5 and the storage arbitrage value, is null for a risk-neutral retailer andin reases with the risk aversion parameter, whi h expresses the fa t that the synergy between sale ontra t and storage fa ility is in term of risk management rather than in term of expe ted return.Figure (1.6(b)) represents the synergy value in term of the risk aversion parameter under di�erentdemand volatilities. It is observed that the synergy value in reases with demand volatility, whi hmeans that the storage fa ility's value-added in the retailer's portfolio in reases with the volumeun ertainty. Figure (1.7) shows that the storage's value-added be omes null in a ontext of highforward market liquidity, even in the presen e of volume un ertainty: the synergy e�e t arisesonly under an illiquid forward market. In addition, the portfolio value varies from �89 to 37 Me,depending on the forward market liquidity, whi h points out the importan e of liquidity assumptionfor portfolio valuation.

the lients' demand ould be met in every s enario only with the illiquid market37

(a) De omposition of portfolio value for di�erent risk aversion parameters

(b) Synergy value in term of risk aversion parameter for di�erent demandvolatilitiesFigure 1.6: De omposition of J1(x1) = Max(qk)k�12Aliq(x1)V Id;u1 (G) (in Me) and synergy value fordi�erent risk aversion parameters and di�erent demand volatilities38

Figure 1.7: Portfolio and synergy values (in Me) for the di�erent settings of forward marketliquidity des ribed in table (1.1) (with � = 0:01 and demand volatility � = 10 TWh)Q0 Q1 Q2 Q3 Q4low liquidity setting 30 10 5 0 0medium liquidity setting 30 10 10 10 10high liquidity setting 30 30 30 30 30Table 1.1: Des ription of the three liquidity settings: Q0 represents the maximal volume of "spot"transa tions, Q1 the maximal volume for delivery in the next quarter, Q2 the maximal volume fordelivery in the next following quarter...1.5 Con lusionWe have developed in this paper a tra table model to introdu e time- onsisten y and inter-temporalwealth management in optimizing a ommodity portfolio. In this order, we ompared stati riskmeasures expressed on �nal wealth with utility-type dynami risk measures: only the latter leadto time- onsistent optimal strategies and disentangle the omponents of temporal substitution and39

risk a ross states of nature. These properties are illustrated on a numeri al example. The useof the model signi� antly redu es the left tail in the �nal wealth distribution, and leads to asatisfa tory trade-o� between �nal wealth risk and expe ted wealth when risk is represented byConditional Value at Risk. In addition, the model allows one to de�ne an optimal strategy betweende reasing the risk of the �nal wealth and redu ing the likelihood of a bankrupt y within thetime horizon. Lastly, our approa h allows one to assess the synergy value between the di�erentphysi al assets omposing a portfolio, with important appli ations in term of ommodity portfoliostru turing. Our urrent areas of investigation on ern the improvement of the omputing timeof the time- onsistent strategies and the omparison through simulations of strategies based onre ursive utilities with strategies based on stati risk measures. Regarding the �rst point, thenumeri al te hnique that is urrently used ex ludes for the moment a number of de ision stepshigher than 5, due to the non-linearity of the obje tive fun tion and the explosion of the number ofde ision variables with the number of time steps; approa hing the obje tive fun tion by a pie e-wiselinear on ave fun tion (whi h is theoreti ally possible thanks to the on avity property presentedin se tion 1.3.4) ould be a way of redu ing the problem to a linear programming one, whi h wouldpermit the in orporation of more than 10 de ision steps (the event tree would then ontain severalmillion nodes, implying a number of de ision variables whi h is ertainly ompatible with urrentlinear programming te hniques). Con erning the se ond area of resear h, the �rst step onsists inbuilding a simulator apable of dynami ally reprodu ing strategies based on di�erent risk measures(e.g, re ursive utilities, expe ted �nal wealth, expe ted �nal wealth/CVaR on the �nal wealth...)under pri e/demand s enarios whi h are independent of the ones used for the optimization. Then,the obje tive will be to assess the value-added of a temporally onsistent measure with respe tto a stati risk measure in term of de ision planning robustness, of expe ted return, and of inter-40

temporal risk management.1.6 Annex: proof of the onvexity resultWe use here the three following lemmas:Lemma 1.6.1 Let f : Rn � Rm ! R be a on ave fun tion. Let A 2 Rm�n and b 2 Rm . De�neP (b) = fx 2 Rn ; Ax � bg and g(b) = maxs:t:x2P (b)f(x; b) (1.33)Let Q = fb 2 Rm ; P (b) 6= ;g. Then Q is a onvex set and g : Q! R is a on ave fun tion.The proof an be found in Martinez-de-Albeniz and Sim hi-Levi (2005).Lemma 1.6.2 On a probability spa e (;F ;P), denote by L() the set of random variables withvalues in Rm . Let f : Rn � Rm ! R?� be a on ave fun tion with respe t to its �rst argumentand X be a random variable in L(), su h that 8� � 1; 8x 2 Rn ; E ((�f(x;X))�) < 1. Then,for CARA utilities u(x) = �e��x and �(x) = �e��x with 0 < � � �, the fun tion de�ned byg(x) = � Æ u�1 �E �u Æ ��1(f(x;X))��, is on ave.Proof : Straightforward al ulations lead to g(x) = �[E ((�f(x;X))�� )℄��So the problem is equivalent to showing that ~g(x) = [E ( ~f (x;X)�)℄ 1� is onvex for ~f onvex withrespe t to x and � � 1.To prove this property, we shall show that:8(x1; x2) 2 Rn ; 8� 2 [0; 1℄; ~g(�x1 + (1� �)x2) � �~g(x1) + (1� �)~g(x2) (1.34)First, by the onvexity of ~f :~g(�x1 + (1� �)x2) � [E ((� ~f (x1;X) + (1� �) ~f(x2;X))�)℄ 1� (1.35)41

Then, from Minkowski's inequality, whi h is valid for � � 1, we get:[E ((� ~f (x1) + (1� �) ~f(x2;X))�)℄ 1� � [E((� ~f (x1;X))�)℄ 1� + [E (((1 � �) ~f(x2;X))�)℄ 1�= �[E (( ~f (x1;X))�)℄ 1� + (1� �)[E (( ~f (x2;X))�)℄ 1�= �~g(x1) + (1� �)~g(x2) (1.36)The ombination of (1.35) and (1.36) leads to (1.34).�Lemma 1.6.3 On a probability spa e (;F ;P), denote by L() the set of random variables withvalues in Rm . Let f : Rn � Rm ! R be a on ave fun tion with respe t to its �rst argument andX be a random variable in L(), su h that 8� 2 R?+ ;8x 2 Rn ; E (exp(��f(x;X))) < 1. Then,for CARA utility u(x) = �e��x (� > 0), the fun tion de�ned by g(x) = u�1 (E [u(f(x;X)℄), is on ave.The proof an be found in Chen et al. (2004).Remark: Lemma 1.6.3 an be interpreted as the limit ase of lemma 1.6.2 when � onverges to zero.To prove the on avity of ��(Gi(qi)) + �� Æ u�1(E �i (u(Ji+1(xi+1))))15 with respe t to de isionsqi, we pro eed by ba kward re ursion on i:- Let us begin with the ase i = TWe denote by kT the number of de ision variables at time �T and de�ne:BliqT (dT ) = n(LT ; N(T � 1; :)); AliqT (LT ; N(T � 1; :); dT ) 6= ;o (1.37)15The proof that follows is valid for both ases:- �(x) = �e��x with � � �- � = Id 42

De�ne fun tion ~JT by:BliqT (dT ) 7! R(LT ; N(T � 1; :)) 7! ~JT (LT ; N(T � 1; :)) = MaxqT2AliqT (LT ;N(T�1;:);dT )�(GT (qT ))The set of onstraints an be put under the equivalent form:qT 2 AliqT (xT )() 8>><>>: 0 � LT + qinjT � qsoutT � LmaxN(T � 1; T ) + nb(T; T )� ns(T; T ) + qsoutT � qinjT = dTHere, we omitted all bounding onstraints on individual de isions qinjT ; qsoutT ; nb(T; T ); ns(T; T ).AliqT (LT ; N(T � 1; T ); dT ) an thus be put under the linear form:AqT � B(LT ; N(T � 1; T )) + CA, B, and C being appropriate matri es and ve tor. Hen e, using lemma 1.6.1, we know that ~JT is on ave with respe t to B(LT ; N(T � 1; T )) +C, and thus also with respe t to ve tor (LT ; N(T �1; T )) on BliqT (dT ).In addition, we see that:(LT ; N(T � 1; :)) 2 BliqT (dT )()

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:Lend � LT � LmaxN(T � 1; T ) � dT � nmaxb (T; T )�QdrawTN(T � 1; T ) � dT + nmaxs (T; T ) +QinjTN(T � 1; T ) + LT � dT + nmaxs (T; T ) + LmaxN(T � 1; T ) + LT � dT � nmaxb (T; T ) + LendHen e, 8dT ; BliqT (dT ) has the form:8>>>>>><>>>>>>: LT 2 [0;Lmax℄N(T � 1; T ) 2 [minT (dT ); maxT (dT )℄; N(T � 1; T ) + LT 2 [�minT (dT ); �maxT (dT )℄43

where �minT is a onstant, and minT , maxT , �minT , �maxT are fun tions of the �nal demand.- Now, let us denote by ki the number of de ision variables at stage i and de�ne:Bliqi (di) = n(Li; N(i� 1; :)); Aliqi (Li; N(i� 1; :); di) 6= ;o (1.38)Suppose that 8 di+1; Bliqi+1(di+1) is of the form:8>>>>>>>>>><>>>>>>>>>>:Li+1 2 [0;Lmax℄N(i; p) 2 [mini+1 (di+1; p); maxi+1 (di+1; p)℄ 8p � i+ 1;N(i; i + 1) + Li+1 2 [�mini+1 (di+1); �maxi+1 (di+1)℄ (1.39)

where �mini+1 is a onstant, mini+1 and maxi+1 are fun tions of i + 1 demand and delivery period p,�mini+1 and �maxi+1 are fun tions of period pi+1 demand. Assume also that, for every realization of�i+1, the fun tion ~Ji+1(Li+1; N(i; :)) = �(Ji+1(Li+1; N(t; :); �i+1)) is on ave with respe t to ve tor(Li+1; N(i; :)). De�ne fun tion ~Ji by:Bliqi (di) 7! R(Li; N(i� 1; :)) 7! Maxqi2Aliqi (Li;N(i�1;:);di)�(Gi(qi)) + �� Æ u�1(E �i (u(Ji+1(xi+1))))(with xi+1 = (Li + qinji � qsouti ; N(i� 1; :) + nb(i; :) � ns(i; :); �i+1))Remark that ~Ji(Li; N(i � 1; :)) = �(Ji(Li; N(i� 1; :); �i)).Lemma 1.6.2 (resp 1.6.3) ensures that the fun tion � Æ u�1 �E �i hu Æ ��1( ~Ji+1(xi+1))i� (resp.u�1 �E �i hu( ~Ji+1(xi+1)i�) is on ave with respe t to variables Li+1, and N(i; p)p�i+1.As (Li+1; N(i; :)) = (Li + qinji � qsouti ; N(i � 1; :) + nb(i � 1; :) � ns(i � 1; :)), this fun tion isjointly on ave with respe t to de isions qi and state (Li; N(i� 1; :)) on Rki �Bliqi (di). Hen e, thedate �i obje tive fun tion �(Gi(qi)) + �� Æ u�1(E i(u(Ji+1(xi+1)))) = �(Gi(qi)) + �� Æ u�1(E �i (u Æ��1( ~Ji+1(xi+1))) is the sum of the on ave fun tion �(Gi(qi)) and of the on ave fun tion � Æ44

u�1 �E �i hu Æ ��1( ~Ji+1(xi+1))i�. It is thus jointly on ave with respe t to de ision variables qi andve tor (Li; N(i � 1; :)) on Rki � Bliqi (di). Let us now examine the set of onstraints. Denoting byHi(di) the (�nite) set of date �i+1 demands bran hing from date �i demand di, we have:qi 2 Aliqi (Li; N(i � 1; :); di)()8>><>>: (Li + qinji � qsouti ; (N(i � 1; p) + nb(i; p)� ns(i; p))p�i+1) 2 Bliqi+1(di+1) 8di+1 2 Hi(di)N(i� 1; i) + nb(i; i) � ns(i; i) + qsouti � qinji = di (1.40)Here again, we omitted bounding onstraints on de isions qi.First, the ompatibility between (1.40) and the bounding onstraints on (qi) implies that Bliqi (di)is of the form (1.39). The form (1.39) hen e passes from i+ 1 to i.In addition, as 8di+1; Bliqi+1(di+1) is of the form (1.39), the set of onstraints Aliqi (Li; N(i� 1; :); di) an be put under the linear form:Aqi � B(Li; N(i � 1; p)i�p�T ) + CA, B, and C being appropriate matri es and ve tor.Using lemma 1.6.1, we know that fun tion ~Ji is on ave with respe t to variables B(Li; N(i �1; p)i�p�T )+C, and thus also on ave with respe t to variables (Li; N(i� 1; p)p�i), whi h ends theproof. �1.7 Referen esArtzner P.,Delbaen F., Eber J.M., Heath D. (1999), Coherent Measures of Risk, Mathemati alFinan e 9: 203-228Artzner P., Delbaen F., Eber J.M., Heath D. , Ku H. (2002), Coherent Multiperiod Risk Measure-ment, Working Paper 45

Chen X., Sim M., Sim hi-Levi D., Sun P. (2004), Risk Aversion in Inventory Management, a eptedby Operations Resear hEi hhorn A., Romis h W. (2005), Polyhedral Risk Measures in Sto hasti Programming, SIAM J.Optim. 16 , pp. 69-95Epstein G., Zin S. (1989), Substitution, Risk Aversion, and the Temporal Behavior of Consumptionand Asset Returns: A theoreti al framework, E onometri a, Vol. 57, No. 4, pp. 937-969Frittelli M., Rosazza Gianin M.(2002), Putting order in risk measures, Journal of Banking andFinan e, Vol. 26 pp. 1473-1486Frittelli M., Rosazza Gianin M.(2004), Dynami onvex risk measures, New Risk Measures for the21th Century, G. Szego ed., John Wiley & Sons, pp. 227-248Lide Li, Paul R. Kleindorfer (2004), Multi-Period VaR-Constrained Portfolio Optimization withAppli ations to the Ele tri Power Se tor, Energy JournalLongsta� F., S hwartz E.S (2001), Valuing Ameri an Options by Simulation: A Simple LeastSquares Approa h, The Review of Finan ial Studies, Vol. 14, pp. 113-147Martinez-de-Albeniz V., D. Sim hi-Levi (2003), Mean-Varian e Trade-o�s in Supply Contra ts,Working paper, MITMartnez de Albniz, V. Sim hi-Levi, D. (2005), A portfolio approa h to pro urement ontra ts, Pro-du tion and Operations Management, Vol. 14, No 1, pages 90-114Unger G. (2002), Hedging Strategy and Ele tri ity Contra t Engineering, PhD Thesis, ETH Zri hWang, T. (2000), A Class of Dynami Risk Measures, under revisionWebber N., M Carthy L. (1997), An I osahedral Latti e Method for Three Fa tor Models, WorkingPaper46

Chapter 2A new dependen e model for ommodity forward urves;appli ation to the US natural gas andoil markets1The goal of this paper is to present and alibrate a model for the joint evolution of orrelated ommodity forward urves. The main originality of the model is that it aptures both the lo aland global dependen e stru tures of two forward urves, through an error- orre ting term in therisk-premia of the forward pri e returns. The model is applied here to the US oil and gas forwardmarkets, whi h have strong e onomi relations, from the demand and supply sides.1I thank Gaz de Fran e for providing me with the data, C�eline Jerusalem for helping me to treat them, GregoryBenmenzer, Olivier Bardou, and Jean-Ja ques Ohana for stimulating dis ussions47

2.1 Introdu tionSurprisingly, the modeling of the o-movements of ommodity forward urves has re eived very littleattention in the �nan ial literature. Yet, this is a subje t of onsiderable importan e for the pri ing,risk management, and optimization of portfolios omposed of multi- ommodity assets su h as gas-�red power plants, oil-indexed natural gas ontra ts, or oil re�neries. Indeed, the �nan ial value ofa multi- ommodity asset is a fun tion of the entire forward urves and the hedging strategies formulti- ommodity portfolios are based on futures ontra ts rather than spot transa tions. As a on-sequen e, a model des ribing the evolution of ommodity spot pri es only, provides a partial viewof the risks/value entailed in su h portfolios and of the possible a tions of the portfolio manager.A model des ribing the joint evolution of two ommodity forward urves should apture at thesame time their global and lo al dependen e stru tures. The lo al dependen e stru ture des ribesthe volatilities, the marginal densities and the orrelations of the daily forward urve moves. Aframework of analysis for this type of dependen e was des ribed in Clewlow and Stri kland (2000),who propose to extend the lassi al PCA on one ommodity forward urve to a PCA on the returnsof two ommodity forward urves, thus obtaining several types of o-movements of the two forward urves. By ontrast with the lo al dependen e stru ture, the global dependen e stru ture des ribesthe long-term relations existing between ommodity pri es2. Mu h attention has been devotedto the study of ointegration between series of di�erent spot/futures ommodity pri es3, with a2two frequent examples of long-term intera tions between ommodity markets are the possibility to use a given ommodity to produ t another one (natural gas to produ e power, rude oil to produ e heating oil...) or to use agiven ommodity as a substitute to another one (e.g. heating oil instead of natural gas for heating, oal instead ofnatural gas to produ e power)3see e.g. Alexander (1999) for a study of the ointegration between gas/oil spot and futures pri es on the NYMEX,Ates and Wang (2005) for an analysis of the relations between spot and �rst-near by natural gas pri es in the US,48

view to des ribing the intera tion between several parti ular points in the same forward urve orin di�erent forward urves (for example the relations between the front-month pri es of a pair of ommodities or the relations between the spot and front-month pri es of the same ommodity).There is extensive work also on the evolution of a single interest rate or ommodity forward urve,either for fore asting (see Diebold and Li (2003)) or VaR al ulation (see e.g. Brooks (2001)). Butno work, to our knowledge, has ever proposed a framework to simulate the evolution of two entire ommodity forward urves, des ribing the way the two urves "revert to ea h other". The retainedapproa h for this problem follows Pilipovi (1997), Manoliu and Tompaidis (2002), S hwartz andSmith (2000), and Geman and N'Guyen (2005), who de ompose the daily deformations of a forward urve into a short-term sho k, a�e ting only the �rst maturities, and a long-term sho k, onsist-ing of an overall translation of the forward urve. Regarding the lo al dependen e stru ture, themodel aptures, on the one hand, the ausal relations between the daily short-term and long-termsho ks of the two ommodities, and on the other hand, the time-dependent volatilities of the four omovements (see e.g. Geman and Nguyen (2005), Ri hter and Sorensen (2000), and DuÆe (2002),for eviden e of sto hasti ity of the volatility of ommodity pri es, and Blix (2003) for eviden e ofseasonality of natural gas impli it volatility), and their possibly non Gaussian dependen e stru ture(see e.g. Eydeland (2003) for eviden e of the non-normality of energy (log) pri es). The approa hto apture the long-term relations between two forward urves an be viewed as an extension of the on ept of ointegration to forward urves. The de omposition of the forward urve daily movesSiliverstovs et al. (2005) for an analysis of ointegration between Japanese, European, and North Ameri an gaspri es, Nguyen (2002) for the analysis of the ointegration between the futures pri es of metals on the London MetalEx hange, Pekka and Antti (2005) for the study of ointegration between spot and futures ele tri ity pri es on theNordPool 49

translates into a de omposition of the shape of the forward urve into a seasonal term, slope4 andlevel5. The long-term relationships between the two ommodity forward urve slopes and levelsare looked for and the deviations to these equilibriums be ome predi tive variables for the futurerelative evolution of the two urves. The model is applied here to the US natural gas, rude oiland heating oil markets during the period 99-2004. These three markets, in spite of their di�er-en es, are intertwined by e onomi relations, from the onsumption side and the produ tion side.Regarding the lo al dependen e stru ture, we �nd eviden e of ausal relations between naturalgas and oil sho ks, sto hasti volatility for the di�erent sho ks, seasonal volatility for natural gasand heating oil short-term sho ks only, and positive orrelations between the o-movements of oiland gas forward urves. Regarding the global dependen e stru ture, our analysis highlights theexisten e of a strong long-term relationship between the levels of natural gas and oil (with twobreak points o urring in the beginning of year 2000 and in the middle of year 2003), and of aweaker long-term relationship between natural gas and oil slopes. The analysis of the temporalstability of the model parameters reveals that the error- orre tion me hanisms have been strongersin e 2002, that the orrelations between the daily o-movements of oil and gas forward urveshave in reased signi� antly throughout the period 1999-2004 and that the orrelation between theshort-term sho ks of natural gas and heating oil peaked during the tight market winters 2000-2001,2002-2003, and 2003-2004.I view the ontribution of this paper as threefold: from an e onomi standpoint, the presentedforward urve model sheds light on the relations between the natural gas and oil markets in theUS, spe ifying at the same time the short-term and long-term relations between the three energies;4depending on the sign of the slope, the urve will be said to be in ontango or in ba kwardation5from an e onomi standpoint, the level is linked to the stru tural pri e of the ommodity, as observed in thequotation date, and the slope to the short-term situation of inventory, produ tion, and demand50

from a statisti al standpoint, the model proposed here opens a new avenue for the modeling ofthe joint evolution of several orrelated forward urves, giving a simple way to apture in a singlearbitrage-free model the long-term relations between the shapes of di�erent forward urves and thelo al statisti al relations between their daily o-movements. Lastly, from a �nan ial standpoint,the model developed here has important appli ations in terms of multi- ommodity asset pri ing,of ommodity portfolio risk management and of ommodity portfolio optimization; as far as assetpri ing is on erned, Duan and Pliska (2004) have shown that the ombination of ointegration andsto hasti volatility has an impa t on asset pri es: thus the model would lead to di�erent pri ingresults than standard lo al dependen e models without risk-premia. Regarding risk management,the model, be ause it aptures the long-term relations between two urves, allows one to realisti- ally simulate the evolution of two forward urves on long time horizons, whi h is important forthe estimation of portfolios' Earning-at-Risk on a long-term perspe tive. With respe t to portfo-lio optimization, our error- orre tion model allows the portfolio manager to fore ast the relativeevolutions of the two onsidered forward urves given their initial slopes and levels, a propertywhi h has numerous impli ations; hedge funds will be provided with dire tional strategies based onlong/short positions on the two urves while physi al portfolio managers will have a way to hoosethe best moments to lo k in the margin of their assets with futures ontra ts.The rest of this paper is organized as follows. In se tion 2, we des ribe the e onomi relationsbetween oil and natural gas markets in the US, from the demand side and the o�er side. In se tion3, we present the two-fa tor model and des ribe the global and lo al dependen e stru tures betweenoil and gas forward pri es in the US. In se tion 4, the model is pre isely alibrated and the temporalstability of the model parameters is studied. Se tion 5 ontains on luding omments.51

2.2 The e onomi relations between oil and natural gas in the USEven though there are strong di�eren es between the natural gas and oil markets in the US (thenatural gas market is a ompetitive and lo al market whereas the rude oil market is an oligopolisti and world market), the natural gas and oil pri es are intertwined by strong e onomi relations, bothfrom the demand and supply sides.2.2.1 Dependen e through the demandFigure (2.2( )) shows the repartition of natural gas and petroleum onsumption in the US by end-use. It appears that industry is the se tor where the overlapping of the two energies is the mostsigni� ant, the se ond one being the segment of residential/ ommer ial ustomers. As shown in�gure (2.1(b)), industry represents approximately 1/3 of the global US gas onsumption. Around30% of the natural gas onsumption of this se tor, where we �nd petroleum, hemi als, paper, andmetal industries, are represented by ustomers with dual-fuel apa ity (essentially old power plantsand boilers), who are able to swit h from natural gas to oil (generally distillate or residual fuel oil)depending on the market pri es of the two energies6. Figure (2.2(a)), taken from the Ameri anGas Foundation (2003), shows that the natural gas demand urve is omposed of in exible and exible parts. When natural gas pri es are higher than distillate pri es, all the industrial andpower generation ustomers with dual-fuel apability are o�-gas and the natural gas demand, dueto residential and ommer ial ustomers, shows very little pri e elasti ity. In this ontext, thedemand is mainly impa ted by the weather as residential and ommer ial ustomers use naturalgas for heating purposes essentially (indu ing the seasonal natural gas onsumption pro�le shown in6Note that the re ent environmental regulations, imposing air pollutant emission onstraints to industrials, tendto prevent them from using distillate fuel or oal as a substitute to natural gas52

�gure (2.1(a))); this demand is slow in rea ting to higher gas pri es be ause it is mostly representedby non-interruptible lients without dual-fuel apa ity and be ause the utility rates for this typeof lients respond to market pri es with a lag. When gas pri es are ompetitive with residual fueloil and/or distillate fuel oil, natural gas demand is mu h more pri e elasti , as the industrial andpower generation ustomers with dual-fuel apability hoose the less expensive fuel to run theira tivities. Lastly, when natural gas pri es are below the point at whi h most dual-�red apa ityhas swit hed from oil to natural gas, the demand is again insensitive to pri e variations. Themiddle graph in �gure (2.2(b)) shows that, in a situation of "stable pri es", a demand sho k,resulting in a right translation of the demand urve, leads to a moderate natural gas pri e in rease.Natural gas ustomers with dual-fuel apa ity (8-10 % of the global natural gas onsumption),who have swit hed from natural gas to oil have ompensated for the in reased onsumption ofin exible natural gas onsumers. However, starting from the middle situation ("tighter naturalgas market"), when most of the fuel swit hable apa ity has already swit hed away from naturalgas, a further rise of the natural gas demand, whi h ould be provoked for instan e by a olderweather, an lead to a pri e spike. Of ourse storage, whi h has an e�e t on the supply urve inthe winter (see �gures (2.1(a)) and (2.2(a))) is a ru ial adjustment variable in this ontext. Theseobservations a ount for the variability of natural gas pri e volatility observed in the US market,a phenomenon whi h will be studied in details in the next se tions. Figure (2.3) shows that anoil pri e rise results in an upward translation of the demand urve, whi h in turn, leads to anin rease of the natural gas pri e. This �gure a ounts for the positive orrelations between theUS natural gas and oil pri es. Note that the e�e t illustrated in �gure (2.3) an happen both inthe very short term (for industrial onsumers already having exible swit hing apa ities), ausingpositive orrelations/ ausalities in the daily movements of natural gas and oil pri es, and in the53

medium-long-term (for industrial onsumers who progressively instal a fuel swit hing-te hnology toadapt to a ontext of a heap/expensive gas/oil pri es in the long run), ausing long-term relationsbetween the pri es of the two energies.In addition, we expe t a spe i� orrelation between natural gas and heating oil pri es as the demandof these two energies is in uen ed by the weather and the US heating oil market an dis onne t fromthe rude oil world market in the short-term due to ongestion/disruption problems in the re�ningsystem; �gure (2.2.1) shows that the apa ity surplus in the US re�ning industry is progressivelydisappearing, a phenomenon responsible for the frequent ongestions in the US re�ning system andinstable ra k spreads between rude oil and the re�ned produ ts observed in the re ent years.The EIA (2001) provides a detailed analysis of the orrelation between natural gas and heatingoil markets during winter peaks; in parti ular, in January 2000, where the market was tight fornatural gas and heating oil in the US, interruptible natural gas onsumers whose deliveries wereinterrupted due to severe weather onditions, pur hased and burned fuel oil instead of natural gasfor heating, whi h led to a surge in the heating oil pri es in the US (even though the orrespondingvolumes were very low).In addition, be ause industrials often lo k in their margins using the forward markets, we expe tpositive orrelations not only between oil and gas short-term pri es but between oil and gas forwardpri es as well; for example, if industrials with dual-fuel apa ity observe that gas forward pri es areabove oil substitutes for a given maturity, they will pur hase oil forward ontra ts instead of gasforward ontra ts for that maturity, leading to a orre tion of the spread between the two forwardpri es. This onvergen e between gas and oil forward pri es is reinfor ed by the urrent behaviourof hedge funds and �nan ial investors, who tend more and more to onsider the di�erent ommoditymarkets as a uni�ed asset lass (see Geman (2005)).54

(a) Monthly natural gas onsumption, produ tion, and net im-ports in the US (Sour e: EIA) (b) Repartition of natural gas onsumption in the USby end-use (Sour e: EIA)Figure 2.1: Natural gas onsumption, produ tion, and imports in the US; in the left graph, thegap between the seasonal onsumption and the at produ tion/import urves is �lled by storagemovements

55

(a) demand and supply urves for natural gas (Sour e:AGA) (b) volatility of natural gas pri es in di�erent market on-ditions (Sour e: AGA)

( ) Natural gas and oil onsumptions by end-use (Sour e: EIA)Figure 2.2: E onomi relationships between oil and gas: demand side56

Figure 2.3: Impa t of an oil pri e rise on natural gas pri e2.2.2 Dependen e through the supplyThe dependen e between oil and natural gas pri es in the US is also originating in the supply side.The Gulf of Mexi o on entrates indeed major gas and oil �elds, gas pro essing plants, and oilre�neries (see �gure (2.5(a))). When the hurri anes Katrina, Rita, and Wilma stru k this region,they a�e ted at the same time the produ tion of natural gas, rude oil, and re�ned produ ts in theUS, thus ausing a rise in pri es of all these energies (see �gures (2.5(b)), (2.5( )), and (2.5(d))).The supply of re�ned produ ts in the US was impa ted be ause the Gulf of Mexi o on entratesa large part of the US re�ning apa ity and be ause the import of re�ned produ ts from abroadrepresents only 3% of the total US demand. The same applied to the US natural gas market, whi hrelies at more than 80% on the US produ tion (18% of whi h omes from the Gulf of Mexi o) tomeet the domesti demand, and where gas pro essing plants play a ru ial role in the treatment ofextra ted natural gas. The rude oil pri es were also a�e ted, due to the tightness of the rude oilworld market at the time of Katrina's landfall.57

Figure 2.4: Re�ning apa ity in the US (Sour e: EIA)

58

(a) Natural gas �elds and pro essing plants in the US (Sour e:EIA) (b) Natural gas and oil produ tion interrup-tions after the hurri anes (Sour e: EIA)

( ) Impa ts of the hurri anes on natural gas and rude oil pri es (Sour e: EIA) (d) Impa ts of the hurri anes on gasoline andheating oil pri es (Sour e: EIA)Figure 2.5: E onomi relations between oil and gas: supply side59

2.3 Empiri al observation of the dependen e between oil and gasforward urves in the US2.3.1 Data des riptionThe data used here are the NYMEX daily futures pri es of natural gas, rude oil, and heatingoil, from January 1999 to the end of O tober 2004. For the three energies, the pri es are the 1stmonth, 2nd month,...,15th month futures pri es. Con erning natural gas, the pri e is based ondelivery at the Henry Hub in Louisiana, the rossing of 16 intra- and interstate natural gas pipelinesystems that draw supplies from the region's proli� gas deposits. The pipelines serve marketsthroughout the U.S. East Coast, the Gulf Coast, the Midwest, and up to the Canadian border.The futures pri es are expressed in dollars per Million British Thermal Units (MMBtu). For rudeoil, the NYMEX futures ontra ts's delivery point is Cushing, Oklahoma, whi h is also a essibleto the international spot markets via pipelines and the pri es are expressed in dollars per barrel.The NYMEX rude oil futures ontra t is the world's largest-volume futures ontra t trading on aphysi al ommodity. Be ause of its ex ellent liquidity and pri e transparen y, the ontra t is usedas an international pri ing ben hmark. Lastly, the Heating Oil futures ontra ts are based upondelivery in New York harbor, the prin ipal ash market trading enter, and the pri es are expressedin dollars per Gallon (1 Gal = 42 Barrel).

60

(a) Crude Oil futures pri es in dollars/Barrel (b) Heating Oil futures pri es in dollars/Gal

( ) Natural Gas futures pri es in dollars/MMBtuFigure 2.6: Pri e traje tories from January 99 to O tober 2004

61

Figure (2.6) represents the traje tories of 1st month and 15th month futures pri es for the threeenergies:� the traje tories of Crude Oil and Heating Oil 15th month futures pri es are quasi identi al� there are di�eren es in the short term between rude and heating oil� the trends of natural gas and oil 15th month futures display a parallel dire tion� even though the 1st month natural gas futures pri e exhibits mu h larger moves than oilwithin the period, oil and gas approximately share the same ba kwardation and ontangoperiods7� the period 1999-2004 an be separated in several subperiods:{ from January 1999 to end of 2001, the three traje tories display a "bump": they �rstfollow an upward trend until the end of 2000, and then a de ay until the end of 2001{ in the years 2002-2003, gas pri es start rising while oil pri es remain stable{ from the beginning of 2004 to now, the three energies display a very lear surge, withan exponential speed for oil and a linear speed for natural gas2.3.2 De omposition of daily forward urve moves into short term and long-term sho ksJusti� ation and interpretation of the de ompositionIn �gure (2.7), it appears that forward urve moves de ompose into a long-term sho k, whi hprovokes a global upward or downward translation of the forward urve, and a short term sho k,7There are a few notable ex eptions to this rule su h as the summer 2004, when the gas forward urve was in ontango and the heating oil and rude oil urves were ba kwardated62

whi h only impa ts the short term futures pri es, with an amplitude that de ays with time-to-maturity. In e onomi terms, the interpretation of the de omposition is the following:� the short term sho k refers to events that are expe ted to a�e t the market for a limited periodof time (temperature hange, transitory supply shortage or transportation ongestion...)8� the long-term sho k relates to events or news that potentially impa t the long-term energypri e (news about the likelihood of a war or politi al instability in an oil produ ing ountry,dis losure of lower than expe ted reserves...)

8One ould wonder why events of weekly time s ale su h as a temperature drop or a bottlene k in the transportationsystem should a�e t the pri es of the ontra ts delivering in the following months; this link between spot and forwardmarkets is explained by the storability of the three onsidered energies. Indeed, tensions in the day-ahead marketprompt utilities and distribution ompanies to pump on their reserves in order to take advantage of high spot pri esor be able to deliver their �rm lients; this in turn reates a situation of s ar ity in the medium term, whi h, asexplained by the theory of storage, has a dire t impa t on the slope of the monthly forward urve63

(a) Natural Gas futures pri es (in $/MMBtu) as a fun tion of time to maturity (in months)from January 4th to January 19th, 1999

(b) Natural gas futures pri es returns as a fun tion of time to maturity (in months) fromJanuary 5th to January 19th, 1999Figure 2.7: De omposition of returns into a short and long term sho ks64

Mathemati al formulation of the de ompositionWe denote Fe(t; T ) the futures pri e at time t of energy e for delivery at month T . For the sake ofsimpli ity, we assume that delivery o urs the last trading day of the futures ontra t. We assumethe following arbitrage-free daily evolution model for the forward urve of energy e:�Fe(t; T )Fe(t; T ) = e�ke(T�t)�Xet +�Y et (2.1)�Xet = �X;te + �X;te �e;Xt�Y et = �Y;te + �Y;te �e;Ytwhere:� (�X;te ) and (�Y;te ) are (Ft)-adapted pro esses representing the drifts� (�X;te ) and (�Y;te ) are (Ft)-adapted pro esses representing the volatilities� (�e;Xt ) and (�e;Yt ) are orrelated pro esses formed of i.i.d variables� 1ke represents the hara teristi time of the short term sho kCal ulation of the short term and long-term sho ksAssuming that the short term sho k does not a�e t the 14th month return9, the short term andlong-term sho ks an be readily derived from the observed short term and long-term returns ofenergy e: �Y et = ��Fe(t; T14)Fe(t; T14) �obs (2.2)�Xet = eke(T1�t)��Fe(t; T1)Fe(t; T1) � �Fe(t; T14)Fe(t; T14) �obs9This is equivalent to the assumption 3� 1ke < 14 months65

where the variable Ti denotes the last trading day of the i-th month futures ontra t observed atdate t.Estimation of the short term hara teristi time for the three energiesTo estimate ke for the three energies, we minimize the root mean squared errors (RMSE) i.e., theroot of the mean squared di�eren es between the observed returns and the model implied returns��Fe(t; Ti)Fe(t; Ti) �model = ��Fe(t; T14)Fe(t; T14) �obs + e�ke(Ti�T1)��Fe(t; T1)Fe(t; T1) � �Fe(t; T14)Fe(t; T14) �obs (2.3)Therefore, we solve, for ea h energy e, the following programme:Mke inRMSE =vuut 1N � 14 NXt=1 14Xi=1 ���Fe(t; Ti)Fe(t; Ti) �obs ���Fe(t; Ti)Fe(t; Ti) �model�2 (2.4)where N is the number of observations. rude oil heating oil natural gaske 2:52 3:34 3:331ke (in months) 4:77 3:59 3:60Table 2.1: ke for the three energies rude oil heating oil natural gaske 2:52 3:10 2:971ke (in months) 4:77 3:87 4:04Table 2.2: ke for the three energies when ex eptional deformations are removed (the date t isremoved if RMSEt =r 114P14i=1 h��Fe(t;Ti)Fe(t;Ti) �obs � ��Fe(t;Ti)Fe(t;Ti) �modeli2 > RMSR)66

Table (2.1) reports the short term hara teristi times of the three energies. As a �rst obser-vation, these hara teristi times are ompatible with the assumption 3 � 1ke < 14 months, whi hhelped us al ulate the short term and long-term sho ks. In addition, we observe that the shortterm hara teristi times of natural gas and heating oil are similar and signi� antly smaller thanthe one of rude oil. The e onomi interpretation is that the short term sho ks in the heating oiland natural gas lo al markets are linked to very short-lived events (e.g., sudden drop of temperaturein the US, bottlene k in the US re�ning system et ...) whereas the short term sho ks in the global rude oil market orrespond to events with a longer time s ale (e.g. dis losure of a lower thanexpe ted world inventory).

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( ) RMSE/RMSR (in %) in terms of ke for natural gasFigure 2.8: RMSE/RMSR (in %) in terms of ke68

Figure (2.8) shows the behavior of the relative error fun tion (i.e., the square root of the meansquared errors (RMSE) divided by the square root of the mean squared returns (RMSR)) in termsof ke. We see that the performan e of the model in explaining the varian e of the observed returnsis signi� antly lower for natural gas (with a ratio varian e of the residualstotal varian e = �RMSERMSR�2 of 7%) than foroil (with a ratio varian e of the residualstotal varian e of 3%). A �rst explanation is that the relative importan eof "twist" moves (whi h are not a ounted for in the two fa tor model) in the global forward urve volatility is more pronoun ed for natural gas than for oil. This is on�rmed by a Prin ipalComponent Analysis on the 14 series of forward urve returns, whose results are displayed in table(2.3): rude oil heating oil natural gas1st fa tor 95:95% 95:13% 92:54%2nd fa tor 2:94% 3:77% 4:95%3rd fa tor 0:41% 0:81% 1:34%Table 2.3: Proportion of overall varian e explained by the 1st (translation), 2nd (rotation), and3rd fa tors (twists) for the three energiesA se ond explanation ould be that kgaz is more variable than k rude and koil. Figure (2.9)represents the evolution of the optimal ke 10 for the three energies with a 6 months time step fromsummer 99 to summer 2004. We onsider that summer months are April, May,..., September andwinter months are O tober, November,..., Mar h. We had to remove the dates when the meansquared error of the two-fa tor model was greater than the mean squared returns over the wholeperiod, be ause otherwise, the optimal ke took abnormally high values at some point during the10the optimal ke on period p is the one minimizing the RMSE of the two-fa tor model on period p69

horizon. Thus, the plotted traje tories orrespond to those ke that �t the "normal" movementsof the forward urve. We see in table (2.2) that the removal of the ex eptional moves leads tosigni� antly smaller estimations of ke for heating oil and natural gas. The parameter ke is seasonal(with higher values during the winter season than during the summer season) and volatile fornatural gas and heating oil and more stable for rude oil. The higher winter values of ke for heatingoil and natural gas are explained by the fa t that the winter demand is more weather-sensitive hen emore volatile than the summer demand and that inventory withdrawals are needed to ful�ll thedemand for heating during the winter season, hen e the short term winter sho ks are more brutaland short-lived than the short-term summer sho ks. The winter 2001-2002 ( orresponding to the6th point in �gure (2.9)) was ex eptionally mild and experien ed higher than normal inventorylevels, ausing an abnormally low ke for heating oil and natural gas. This is on�rmed by theobservation of the natural gas and heating oil futures pri es during the winter 2001-2002, when thetwo urves are in ontango and the 1st month futures pri es of heating oil and natural gas displaya relative stability. On the whole, it is the heating oil's parameter that is the most variable duringthe period, with ample seasonal variations and higher average values in the period 99-2001 andlower values in the period 2002-2004. As a onsequen e, the lower performan e of the two-fa tormodel for natural gas ompared to heating oil is not due to more variable and seasonal ke but tothe higher relative importan e of twist moves in the forward urve deformations. In what follows,we will use the onstant values of ke given by table (2.1), minimizing the total RMSE over thewhole period.70

Figure 2.9: Evolution of ke with a 6 months time step from summer 1999 to summer 20042.3.3 Slope and level: two state variables for the shape of the forward urveThe evolution model (2.1) implies a forward urve shape model. Indeed, if we negle t the se ond-order terms: �lnFe(t; T ) � �Fe(t; T )Fe(t; T ) = e�ke(T�t)�Xet +�Y etwe obtain the following expression for the shape of the forward urve at date t:lnFe(t; T ) = lnFe(0; T ) + tXs=0 e�ke(T�s)�Xes + tXs=0�Y es (2.5)Let us assume that the shape of the initial forward urve is of the type:lnFe(0; T ) = Q(T ) + e�keT �Xe0 + Y e0 (2.6)where T takes integer values representing months and Q is a fun tion of period one year and zeromean. Then, equation (2.5) leads to:lnFe(t; T ) = Q(T ) + e�ke(T�t) �Xet + Y et (2.7)71

with: �Xet = �X0e�ket + tXs=1 e�ke(t�s)�Xes (2.8)Y et = Y0 + tXs=1�Y es (2.9)Equation (2.7) shows that, under model (2.1), the shape of the forward urve at any date t is thesuperposition of a seasonal fun tion Q(T ), a slope �Xt, and a level Yt. The slope and level anbe derived from the daily sho ks (�Xet ;�Y et ) via (2.8)-(2.9). The slope follows a mean-revertingpro ess driven by the short term sho ks and the level a random walk driven by the long-termsho ks: �Xet = �Xet��te�ke�t +�Xet (2.10)Y et = Y et��t +�Y et (2.11)E onomi interpretation of the season, slope and levelThe forward urve model (2.7) has very lassi al e onomi interpretations: the seasonality of theforward urve is explained by a stru tural imbalan e between winter and summer onsumptionsand by the small number of market parti ipants having a ess to storage reservoirs; the level isrelated to the long-term pri e of the ommodity and the slope to the bene�t ( lassi ally referredto as the " onvenien e yield") of holding the physi al ommodity vs holding a ontra t for futuredelivery. Fama and Fren h (1988) in parti ular use the slope of the forward urve as a proxy forthe inventory level. 72

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(b) Natural GasFigure 2.10: Seasonal fun tions Q for heating oil and natural gasEstimation of the seasonality for natural gas and heating oilTo estimate the seasonality of the natural gas and petroleum forward urves, I ompute, for ea h ommodity, the average log futures pri es of the ontra ts with a time-to-maturity below one yearfor delivery in January, February,..., De ember from the beginning of January 99 to the end ofDe ember 2003. This way, I obtain, for ea h energy e, the average log forward urve fe(T );T = 1; :::; 12. To obtain the zero mean fun tion Q, I subtra t the mean of fe to fe. As expe ted, I�nd no seasonality for the rude oil forward urve. The obtained seasonal fun tions for heating oiland natural gas are displayed on �gure (2.10). The winter peaks are mu h more pronoun ed fornatural gas than for heating oil.Initial slope and levelIn order to al ulate the slope and level at all dates t, we need the daily short and long-term sho kson the one hand and the initial slope and level on the other hand. On e the parameter ke is known,73

the short term and long-term sho ks are obtained from equation (2.2). Regarding the initial slopeand level, they are obtained by "inversion" of formula (2.6) using the 1st month and 13th monthlog futures pri es observed on January 4th 1999:�Xe0 = eke(T1�0)ln(Fe(0; T1)=Fe(0; T13))Y e0 = lnFe(0; T13)�Q(2)From the estimated slope and level, I an re onstru t an estimated log forward urve using formula(2.6). Figure (2.11) displays the observed and estimated log forward urves on January 4, 1999. For rude oil, the two-fa tor model fails to reprodu e orre tly the shape of the initial forward urve.

74

1 3 5 7 9 11 13 152.51

2.53

2.55

2.57

2.59

2.61

2.63

2.65

2.67

(a) Crude Oil 1 3 5 7 9 11 13 15-1.06

-1.04

-1.02

-1.00

-0.98

-0.96

-0.94

-0.92

-0.90

-0.88

-0.86

(b) Heating Oil

1 3 5 7 9 11 13 150.67

0.71

0.75

0.79

0.83

0.87

0.91

( ) Natural GasFigure 2.11: Comparison of observed (bla k) and estimated (red) log forward urves75

Comparison of estimated slope and level with dire t estimatorsFrom the initial slope and level and the daily short and long-term sho ks, I an ompute the slopeand level at any date using formula (2.10)-(2.11). I ompare this (indire t) estimation with a dire testimation by "inversion" of formula (2.7) using the observed 1st month and 13th month log futurespri es at date t: �Xet = eke(T1�t)ln(Fe(t; T1)=Fe(t; T13))Y et = lnFe(t; T13)�Q(t+ 1 month)The traje tories of dire t and indire t estimators of slope and level for the three energies aredisplayed in �gures (2.12) and (2.14). There are remarkable similarities between the slopes of thethree energies, whi h are most of the time of the same sign. The di�eren es between dire t andindire t slope estimators are due to the existen e of twists in the forward urve: these twists, whi hlead to an inexa t estimation of the slope, have only a temporary e�e t on the dire t estimator, asthe e�e t disappears when the twist is no longer visible, and a longer term e�e t on the indire testimator, due to its re ursive form (2.10). This is illustrated in �gure (2.13): from August 20,1999 to August 21, 1999, the dire t estimator jumps upward after the last trading day of theSeptember 1999 ontra t due to the bump observed in the August forward urve; however, theindire t estimator is ontinuous as it is only impa ted by the very small variation of the 1st monthfutures pri e, i.e., the O tober ontra t pri e. This is the ause of the �rst break point betweenthe rude oil dire t and indire t slope estimators observed in �gure (2.12(a)). Regarding the levels,there are big dis repan ies between the dire t and indire t estimators, always in the same dire tionfor the three energies. This is be ause the two-fa tor model implies that the forward urve isalmost at above a maturity of 3ke � 12 months, whereas the observed forward urves are often76

ba kwardated at this time s ale; as a onsequen e, at ea h rolling date when ba kwardation isobserved in long-term futures pri es, the dire t estimator of the level jumps down whereas theindire t one remains steady, and the errors a umulate over time. The e�e ts of this bias howeverseem to be very omparable for the three energies, whi h allows one to use indi�erently the dire tand the indire t estimators for the analysis of the long-term relations between gas and oil levels.From now on, the global dependen e stru ture will be analyzed using the dire t estimator for theslopes and the indire t estimator for the levels. The reason for this hoi e is the following: for theslopes, we saw that the dire t estimator, ontrary to the indire t one, is only temporarily a�e tedby the existen e of twists in the forward urve; for the levels, the indire t estimator is preferredbe ause it re e ts the gains of an investor entering a 15-th month futures ontra t and rolling overhis position at ea h last trading day11.

11the gain of the onsidered investor would be Ptk=1�F (k; T14) and the indire t level estimator is Yt = Y0 +Ptk=1 �F (k;T14)F (k;T14) 77

−0.

10.

00.

10.

20.

30.

4

1999 2000 2001 2002 2003 2004 2005(a) Crude Oil−

0.2

0.0

0.2

0.4

1999 2000 2001 2002 2003 2004 2005(b) Heating Oil−

0.4

−0.

20.

00.

20.

40.

60.

8

1999 2000 2001 2002 2003 2004 2005( ) Natural GasFigure 2.12: Comparison of dire t estimation (bla k) and indire t estimation (red) of the slopes78

1 3 5 7 9 11 13 1518.6

19.0

19.4

19.8

20.2

20.6

21.0

21.4

21.8

22.2

Figure 2.13: Crude oil forward urves on August 20 (bla k) and August 21 (red) 1999 (August 20being the last trading day of the futures ontra t delivering in September 1999

79

2.5

3.0

3.5

4.0

4.5

1999 2000 2001 2002 2003 2004 2005(a) Crude Oil −1.

0−

0.5

0.0

0.5

1999 2000 2001 2002 2003 2004 2005(b) Heating Oil1.

01.

52.

02.

5

1999 2000 2001 2002 2003 2004 2005( ) Natural GasFigure 2.14: Comparison of dire t estimation (bla k) and indire t estimation (red) of the levels80

2.3.4 De�nition of lo al and global dependan e stru turesFrom now on, we will refer to the relations between slopes and levels as the global dependen estru ture and to the orrelation between o-movements as the lo al dependen e stru ture. Theglobal dependen e stru ture des ribes the long-term relations between slopes and levels of thethree energies and the error orre tion me hanism that insures the reversion of slopes and levelsto the long-term equilibrium. It will translate into an error orre tion term in the drifts �et . Thelo al dependen e stru ture des ribes the relations between the o-movements of the three energiesas well as the dependen e of date t returns on date t ��t returns, whi h will add a term in thedrifts �et .2.3.5 Analysis of lo al dependen e stru tureFigures (2.15) to (2.19) present the lo al dependen e stru ture between oil and gas futures pri es.Dependen e on past sho ksRegarding the ross-energy dependen e on past sho ks, we �nd that the ausality generally runsfrom oil to natural gas and is negative12, ex ept for the natural gas and heating oil short-termmoves, where the ausality runs from gas to heating oil and is positive13. Regarding the inter-temporal dependen e on past sho ks, the ausality runs both ways between the short-term and thelong-term, but is positive in the dire tion long-term ,! short-term and negative the other way14.Lastly, regarding the auto- orrelation of sho ks, we �nd that in the short-term, oil markets tend to12The e onomi intuition being that the natural gas pri e �rst over-rea ts to the oil pri e and then is subje t to a orre tion the next trading day13This �nding is in line with the analysis of natural gas and heating oil markets led in EIA (2001)14The e onomi interpretation being that the long-term �rst over-rea ts to the short-term moves and is subje t toa orre tion the next trading day 81

amplify the previous move, whereas in the long-term, they are more likely to orre t it.Dependen e between o-movementsRegarding the ross-energy o-movements, we obtain the expe ted positive orrelations betweenthe natural gas and oil markets. Regarding the inter-temporal o-movements, we an see that the orrelations between the short-term and long-term sho ks are positive for the three energies (the orrelations being of 40% for rude oil and natural gas and of 30 % for heating oil)

82

−10 −5 0 5 10

−0.

10−

0.05

0.00

0.05

0.10

0.15

Lag

AC

F

st_shocks_gas & st_shocks_crude

(a) gas short-term sho ks/ rude short-termsho ks−10 −5 0 5 10

−0.

050.

000.

050.

100.

150.

200.

25

Lag

AC

F

st_shocks_gas & lt_shocks_crude

(b) gas short-term sho ks/ rude long-termsho ks

−10 −5 0 5 10

−0.

050.

000.

050.

100.

150.

20

Lag

AC

F

lt_shocks_gas & st_shocks_crude

( ) gas long-term sho ks/ rude short-termsho ks−10 −5 0 5 10

−0.

050.

000.

050.

100.

150.

200.

25

Lag

AC

F

lt_shocks_gas & lt_shocks_crude

(d) gas long-term sho ks/ rude long-termsho ksFigure 2.15: Cross orrelation fun tions between natural gas and rude oil daily sho ks with lagsof one to ten days; the ross orrelation fun tions with lag i (resp. �i) represent the orrelationbetween rude oil at time t and gas at time t+ i (resp. t� i)83

−10 −5 0 5 10

−0.

050.

050.

100.

150.

200.

250.

30

Lag

AC

F

st_shocks_gas & st_shocks_heat

(a) gas short-term sho ks/heating oil short-term sho ks−10 −5 0 5 10

−0.

050.

000.

050.

100.

150.

200.

25

Lag

AC

F

st_shocks_gas & lt_shocks_heat

(b) gas short-term sho ks/heating oil long-term sho ks

−10 −5 0 5 10

−0.

100.

000.

050.

100.

150.

20

Lag

AC

F

lt_shocks_gas & st_shocks_heat

( ) gas long-term sho ks/heating oil short-term sho ks−10 −5 0 5 10

0.0

0.1

0.2

0.3

Lag

AC

F

lt_shocks_gas & lt_shocks_heat

(d) gas long-term sho ks/heating oil long-term sho ksFigure 2.16: Cross orrelation fun tions between natural gas and heating oil daily sho ks withlags of 1 to 10 days; the ross orrelation fun tions with lag i (resp. �i) represent the orrelationbetween heating oil at time t and gas at time t+ i (resp. t� i)84

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

FSeries st_shocks_gas

(a) auto orrelation of gas short-termsho ks0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series lt_shocks_gas

(b) auto orrelation of gas long-term sho ks

−10 −5 0 5 10

−0.

10.

00.

10.

20.

30.

4

Lag

AC

F

st_shocks_gas & lt_shocks_gas

( ) gas short-term sho ks/gas long-termsho ksFigure 2.17: Auto and ross- orrelation fun tions gas short-term sho ks/gas long-term sho ks withlags of 1 to 10 days; the ross orrelation fun tion with lag i (resp. �i) represents the orrelationbetween the long-term sho k at time t and the short-term sho k at time t+ i (resp. t� i)85

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series st_shocks_crude

(a) auto orrelation of rude oil short-termsho ks0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series lt_shocks_crude

(b) auto orrelation of rude oil long-termsho ks

−10 −5 0 5 10

−0.

10.

00.

10.

20.

30.

4

Lag

AC

F

st_shocks_crude & lt_shocks_crude

( ) rude short-term sho ks/ rude long-term sho ksFigure 2.18: Auto and ross- orrelation fun tions rude oil short-term sho ks/ rude oil long-termsho ks with lags of 1 to 10 days; the ross orrelation fun tion with lag i (resp. �i) represents the orrelation between the long-term sho k at time t and the short-term sho k at time t + i (resp.t� i)86

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series st_shocks_heat

(a) auto orrelation of rude oil short-termsho ks0 2 4 6 8 10

−0.

20.

00.

20.

40.

60.

81.

0

Lag

AC

F

Series lt_shocks_heat

(b) auto orrelation of heating oil long-termsho ks

−10 −5 0 5 10

−0.

10.

00.

10.

20.

3

Lag

AC

F

st_shocks_heat & lt_shocks_heat

( ) heating oil short-term sho ks/heatingoil long-term sho ksFigure 2.19: Auto and ross- orrelation fun tions heating oil short-term sho ks/heating oil long-term sho ks with lags of 1 to 10 days; the ross orrelation fun tion with lag i (resp. �i) representsthe orrelation between the long-term sho k at time t and the short-term sho k at time t+ i (resp.t� i)87

VolatilitiesFor the three energies, the volatilities of the short-term and long-term sho ks are estimated bythe standard deviation of the sho ks within a 50-days sliding window. The obtained traje toriesare displayed on �gure (2.20): all sho ks exhibit volatility lusters, jumps, and the natural gasand heating oil short-term volatilities follow a seasonal pattern, with high values in winter (60% in normal winters for natural gas, 25 % in normal winters for heating oil) and lower valuesin summer (20 % for natural gas, 10% for heating oil). The phenomenon of sto hasti volatility,observed in most ommodity markets, is linked to the temporal variations of some key indi ators ofthe supply exibility, su h as the deviation to "normal" storage level, and the proportion of spareprodu tion/re�ning apa ity. Note also that the short-term volatility peaks orrespond to periodsof high positive forward urve slopes, an observation whi h is onsistent with the theory of storage(Kaldor (1939)), and whi h was also observed by Ates and Wang (2005) in the US gas market.The seasonal pattern of natural gas and heating oil short-term volatilities an be explained by thefa t that the demand is more sensitive to the temperature during the heating season and that thedemand and produ tion sho ks have more impa t on the pri es during the winter, when storage ispart of the supply urve and the market is tight, than during the summer, when storage is part ofthe demand urve and the market is loose. The seasonal behavior of gas impli it volatilities wasalready observed by Blix (2003) in the US gas market.

88

2040

6080

100

natu

ral g

as s

hort

term

vol

atili

ty in

%

1999 2000 2001 2002 2003 2004(a) natural gas short-term volatility 1020

3040

natu

ral g

as lo

ng te

rm v

olat

ility

in %

1999 2000 2001 2002 2003 2004(b) natural gas long-term volatility10

1520

2530

3540

crud

e oi

l lon

g te

rm v

olat

ility

in %

1999 2000 2001 2002 2003 2004( ) rude oil short-term volatility 1520

2530

35

crud

e oi

l lon

g te

rm v

olat

ility

in %

1999 2000 2001 2002 2003 2004(d) rude oil long-term volatility

2030

4050

60

heat

ing

oil s

hort

term

vol

atili

ty in

%

1999 2000 2001 2002 2003 2004(e) heating oil short-term volatility 1520

2530

35

heat

ing

oil l

ong

term

vol

atili

ty in

%

1999 2000 2001 2002 2003 2004(f) heating oil long-term volatilityFigure 2.20: short-term and long-term volatilities (in %) of the three energies estimated with a50-days sliding window89

2.3.6 Analysis of global dependen e stru tureStationarity properties of the slopes and levelsTable (2.4) reports the results of the Phillips-Perron unit root tests on the slopes and levels of thethree energies. Not surprisingly, the slopes are found to be mean-reverting while the levels displaya random walk behavior.

90

Crude oil slopeDi key-Fuller Lag Parameter p-value�3:914 7 0:01319Heating oil slopeDi key-Fuller Lag Parameter p-value�3:5531 7 0:03911Natural gas slopeDi key-Fuller Lag Parameter p-value�3:4437 7 0:04771Crude oil levelDi key-Fuller Lag Parameter p-value�1:2474 7 0:897Heating oil levelDi key-Fuller Lag Parameter p-value�1:2974 7 0:8757Natural gas levelDi key-Fuller Lag Parameter p-value�1:7799 7 0:6715Table 2.4: Philipps-Perron unit root tests on the slopes and levels of the three energies; the test-statisti s, trun ation lag parameters, and p-values of the tests are reported91

Long-term relation between forward urve slopesFigure (2.21) displays the relation between natural gas and rude oil/heating oil slopes. We seethat, when the natural gas forward urve is in ba kwardation (positive slope), the oil forward urve is also in ba kwardation15. However, a ba kwardated oil urve does not ne essarily imply aba kwardated natural gas forward urve. In parti ular, year 2002 experien ed a ba kwardated oil urve and a natural gas forward urve in ontango. The results of the linear regression of naturalgas slope on rude oil and heating oil slopes are reported on table (2.5). Note that the regression oeÆ ients are not signi� antly di�erent from 1, the regression R2 being around 30% for rude and40% for heating oil. Estimate Std. Error t value Pr(> jtj)b rude �0:137 0:00710 �19:23 < 2:10�16a rude 1:027 0:0414 24:79 < 2:10�16R2 = 29:76%bheat �0:0979 0:00489 �20:01 < 2:10�16aheat 0:987 0:0298 33:10 < 2:10�16R2 = 43:04%Table 2.5: Linear regression of natural gas slope on rude oil and heating oil slopes: a denotes theregression oeÆ ient and b the inter ept; the estimated oeÆ ients, standard deviations, t-statisti s,and two-sides p-values are reported15note however that there are outliers in the linear relation: for instan e, during the winters 2000-2001 and 2002-2003, the natural gas slope was very high while the oil slope was mildly positive92

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.5

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(a) natural gas slope in terms of rude oil slope -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.5

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(b) natural gas slope in terms of heating oil slopeFigure 2.21: Natural gas slope in terms of rude oil (left) and heating oil (right) slopes; the linear�t is displayed in green

2.4 2.8 3.2 3.6 4.0 4.4 4.80.3

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(b) natural gas level in terms of heating oil levelFigure 2.22: Natural gas level in terms of rude oil (left) and heating oil (right) levels; the linear �tis displayed in green; the best three-lines �t is displayed in red; in dark blue: year 99; in mediumblue: year 2000; in lear blue: year 2001; in pink: year 2002; in yellow: year 2003; in bla k: year2004 93

Long-term relation between forward urve levelsFigure (2.22) displays the relation between natural gas and rude oil/heating oil levels: in both ases, a pie ewise-linear relation appears, with two break points o urring at the beginning of year2000 (where gas long-term futures pri es start rising at an in reased pa e, the pa e of the oil pri erise remaining stable), and in the middle of year 2003 (where oil long-term futures pri es startrising sharply, while the gas pri es rise remains stable).Tables (2.6) and (2.7) report the results of the lassi al linear regression and the pie ewise-linearregression of gas level on rude oil and heating oil levels. First, we see that the R2 is mu h higherthan for the regression on the slopes: the long-term equilibrium between the levels is mu h strongerthan the long-term relation between the slopes. Se ond, the pie ewise-linear regression oeÆ ientsare signi� ant, whi h on�rms the validity of the pie ewise linear model, and of the same negativesign, ausing the gas long-term pri e to be less sensitive to the variations of oil long-term pri eabove the up-threshold �Y and below the down-threshold Y. Lastly, table (2.8) shows that the unit-root hypothesis an be reje ted for the residuals of the pie ewise linear relation between gas andoil levels but not for the residuals of the linear relation between gas and oil levels. As a on lusion,only the pie ewise linear relation allows one to obtain the desired stationary residuals.

94

Estimate Std. Error t value Pr(> jtj)b rude �2:277 0:0314 �72:44 < 2:10�16a rude 1:095 0:0089 123:04 < 2:10�16R2 = 91:26%bheat 1:774 0:00355 499:7 < 2:10�16aheat 1:174 0:0086 136:0 < 2:10�16R2 = 92:73%Table 2.6: Linear regression of natural gas level on rude oil (up) and heating oil (down) levels: adenotes the regression oeÆ ient and b the inter ept; estimated oeÆ ients, standard deviations,t-statisti s, and two-sided p-values are reported

95

Estimate Std. Error t value Pr(> jtj)b rude �4:323 0:0648 �66:69 < 2:10�16a rude 1:679 0:0185 90:89 < 2:10�16a� rude �1:323 0:0425 �31:16 < 2:10�16a+ rude �1:004 0:032 �31:22 < 2:10�16Y rude = 3:15 �Y rude = 3:73 R2 = 95:12%bheat 1:885 0:00412 457:71 < 2:10�16aheat 1:736 0:0170 102:05 < 2:10�16a�heat �1:287 0:0432 �29:80 < 2:10�16a+heat �1:045 0:0297 �35:22 < 2:10�16Yheat = �0:53 �Yheat = 0:031 R2 = 96:18%Table 2.7: Pie ewise-linear regression of natural gas level on rude oil and heating oil levels; theregression variables are Ye, (Ye � Ye)� = Min(0;Ye � Ye), and (Ye � �Ye)+ = Max(0;Ye � �Ye),with e= rude oil (up) or e=heating oil (down); a denotes the di�erent regression oeÆ ients andb the inter epts; the thresholds Ye and �Ye are determined by the minimization over the ouples(Ye; �Ye) of the sum of squared residuals of the regression of Ygas on the variables Ye, (Ye � Ye)�,and (Ye� �Ye)+; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported96

Residuals of the rude-gas pie ewise linear relationDi key-Fuller Lag Parameter p-value�3:507 7 0:0417Residuals of the rude-gas linear relationDi key-Fuller Lag Parameter p-value�2:428 7 0:397Residuals of the heating oil-gas pie ewise linear relationDi key-Fuller Lag Parameter p-value�4:251 7 0:01Residuals of the heating oil-gas linear relationDi key-Fuller Lag Parameter p-value�2:514 7 0:361Table 2.8: Philipps-Perron unit root tests on the residuals of the pie ewise linear and linear relationsbetween gas and oil levels; the test-statisti s, trun ation lag parameters, and p-values of the testare reported2.4 A new dependen e model for ommodity forward urves2.4.1 Formulation of the modelWe want to introdu e an error- orre tion me hanism on the levels and on the slopes between theenergies e and e0. Therefore, we postulate that the drifts are the sums of a onstant part, a term97

expressing dependen e on past returns, and an error- orre tion term:0BBBBBBBBBB��Xet�Xe0t�Y et�Y e0t

1CCCCCCCCCCA = 0BBBBBBBBBB��X;e�X;e0�Y;e�Y;e0

1CCCCCCCCCCA+ �0BBBBBBBBBB��Xet�1�Xe0t�1�Y et�1�Y e0t�1

1CCCCCCCCCCA+0BBBBBBBBBB���X;eRXt�lX�X;e0RXt�lX��Y;eRYt�lY�Y;e0RYt�lY

1CCCCCCCCCCA+0BBBBBBBBBB��X;et�X;e0t�Y;et�Y;e0t

1CCCCCCCCCCA (2.12)RXt = �Xet � f e;e0X ( �Xe0t )RYt = Y et � f e;e0Y (Y e0t )In the model (2.12):� e stands for natural gas and e0 stands alternatively for rude oil and heating oil� � = (�X;e; �X;e0 ; �Y;e; �Y;e0) is the 1� 4 ve tor omposed of the onstant part of the drifts� � is a 4� 4 matrix� � = (�X;e; �X;e0 ; �Y;e; �Y;e0) is the 1� 4 ve tor omposed of the error- orre tion speeds� �Xet and Y et denote the slope and level of the forward urve of the energy e� lX and lY refer to the lags between an observed deviation to the long-term equilibrium andthe orre tion of this deviation� x! f e;e0X (x) is the relation between the slopes of energy e and e0 (in the ase of gas and oil,fX is a linear fun tion)� x ! f e;e0Y (x) is the relation between the levels of energy e and e0 (in the ase of gas and oil,fY is pie ewise linear fun tion) 98

� (RXt ) is the pro ess omposed of the deviations to the long-term relation between the slopes� (RYt ) is the pro ess omposed of the deviations to the long-term relation between the levels� the pro esses (�X;et = �X;et �X;et ; �X;e0t = �X;e0t �X;e0t ; �Y;et = �Y;et �Y;et ; �Y;e0t = �Y;e0t �Y;e0t ) followindependent GARCH pro esses; we in lude a seasonal omponent in the GARCH pro essfollowed by natural gas and heating oil short-term sho ks� the residual sho ks (�X;et ; �X;e0t ; �Y;et ; �Y;e0t ) are assumed to be i.i.dWe use the 4 � 1 ve tor pro ess �Zt = (�Xet ;�Xet 0;�Y et ;�Y et 0)0. A few omments are requiredhere. First, keeping in mind equations (2.10) and (2.11), assuming linear fun tions fX and fY ,and making abstra tion of the dependen e between �X and �Y indu ed by the term ��Zt�1,the model (2.12) implies a ve tor autoregressive model (VAR) for the slopes and a ve tor error- orre tion model (VECM) for the levels, whi h makes sense from an e onomi standpoint. Se ond,we believe that the model (2.12) is suÆ iently general to a ount for the evolution of any pair ofrelated ommodity forward urves, with appropriate long-term relations f e;e0X and f e;e0Y . However, aswe hoose to model the pro esses (�X;et �X;et ; �X;e0t �X;e0t ; �Y;et �Y;et ; �Y;e0t �Y;e0t ) as independent GARCHpro esses, we ex lude from our s ope the relations slope/volatility (whi h are studied by Atesand Wang (2005) in the US gas market) and the e�e t of volatility transmission between thetwo ommodity pri es, an e�e t whi h was highlighted before in the litterature on erning gasand oil markets (see Pindy k (2004) and Ewing et al. (2003)). Lastly, we assume a onstantdependen e stru ture between the residuals (�X;et ; �X;e0t ; �Y;et ; �Y;e0t ), thus negle ting the possible orrelation lustering (see Eydeland and Wolynie (2003)) and the potential relations between orrelation and volatilities (see e.g. Goorbergh et al. (2005)).99

2.4.2 Calibration of the modelTo alibrate the model, we pro eed in three steps: �rst, we �nd the lags lX and lY and we estimate� and � by a linear regression of �Zt on �Zt�1, RXt�lX , and RYt�lY ; se ond, we apply independentGARCH models to the residuals of this linear regression; third, we study the dependen e stru turebetween the standardized residuals of the independent GARCHmodels. The hoi e of this imperfe tpro edure, whi h is also done by Ng and Pirrong (1994) and Ates and Wang (2005), was motivatedby the high number of parameters to be estimated.Estimation of lX , lY , � and �Figures (2.23) and (2.24) represent the ross- orrelations between the short-term (resp. long-term)sho ks and the residuals of the long-term relations on the slopes (resp. levels). For the pair rude oil-natural gas, we observe that rude oil and natural gas short-term sho ks both orre tthe deviations to the long-term relation on the slopes with a lag of one day and that only naturalgas long-term sho ks orre t the deviations to the long-term relation on the levels (with no lag).We therefore hoose lX = 1, lY = 0 for the pair rude oil-natural gas and on lude that rude oilplays the leading role in the long-term pri e dis overy. For the pair heating oil-natural gas, weobserve that only the natural gas short-term sho ks orre t the deviations to the long-term relationon the slopes (with no lag), and that both heating oil and natural gas long-term sho ks orre tthe deviations to the long-term relation on the levels (with no lags). We therefore hoose lX = 0,lY = 0 for the pair heating oil-natural gas and on lude that heating oil plays the leading role inthe short-term pri e dis overy. Tables (2.9) to (2.16) report the results of the 8 linear regressions:100

we �nd many signi� ant dependen e relations on past returns16 and a on�rmation of the error- orre tion me hanism des ribed above; note also the positive trends on the long-term moves forthe three energies.

16On the whole, the results on�rm the results of se tion 2.3.5, ex ept that natural gas moves are now found to bepositively auto- orrelated 101

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(d) residuals of the long-term relation in thelevels/ rude oil long-term sho ksFigure 2.23: Cross orrelation fun tions between the sho ks and the residuals of the rude oil-natural gas long-term relations (in the slopes and levels) with lags of 1 to 30 days; the ross orrelation fun tions with lag i (resp. �i) represent the orrelation between the residuals at timet and the sho ks at time t+ i (resp. t� i)102

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(d) residuals of the long-term relation in thelevels/heating oil long-term sho ksFigure 2.24: Cross orrelation fun tions between the sho ks and the residuals of the heating oil-natural gas long-term relations in the slopes and levels with lags of 1 to 30 days; the ross orrelationfun tions with lag i (resp. �i) represent the orrelation between the residuals at time t and thesho ks at time t+ i (resp. t� i)103

Estimate Std. Error t value Pr(> jtj)�X;gas 0:000161 0:000812 0:199 0:843�1;1 0:104 0:0297 3:513 0:000457 ***�1;2 �0:169 0:0585 �2:896 0:00384 **�1;3 �0:167 0:0601 �2:776 0:00558 **�1;4 �0:103 0:0647 �1:599 0:110��X;gas �0:0110 0:00509 �2:165 0:0305 *Table 2.9: Linear regression of the natural gas short-term sho ks on �Zt�1 and RXt�lX : pair rudeoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported

104

Estimate Std. Error t value Pr(> jtj)�X; rude 0:000565 0:000397 1:424 0:155�2;1 0:0267 0:0145 1:839 0:0661 Æ�2;2 0:0917 0:0286 3:212 0:00135 **�2;3 �0:0886 0:0294 �3:019 0:00258 **�2;4 0:144 0:0316 4:570 5:3:10�6 ***�X; rude 0:00543 0:00249 2:187 0:0289 *Table 2.10: Linear regression of the rude oil short-term sho ks on �Zt�1 and RXt�lX : pair rudeoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported Estimate Std. Error t value Pr(> jtj)�Y;gas 0:00117 0:000408 2:867 0:004207 **�3;1 �0:0499 0:0149 �3:348 0:0000835 ***�3;2 �0:0588 0:0293 �2:004 0:00453 *�3;3 0:0547 0:0303 1:809 0:0706 Æ�3;4 �0:0229 0:0326 �0:704 0:482��Y;gas �0:0119 0:00408 �2:923 0:00352 **Table 2.11: Linear regression of the natural gas long-term sho ks on �Zt�1 and RYt�lY : pair rudeoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported 105

Estimate Std. Error t value Pr(> jtj)�Y; rude 0:00148 0:000367 4:003 6:56:10�5 ***�4;1 0:00787 0:0135 0:584 0:559�4;2 �0:0745 0:0265 �2:810 0:00503 **�4;3 �0:00647 0:0273 �0:237 0:813�4;4 �0:107 0:0294 �3:640 0:000282 ***�Y; rude 0:00582 0:00369 1:580 0:114Table 2.12: Linear regression of the rude oil long-term sho ks on �Zt�1 and RYt�lY : pair rudeoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported

106

Estimate Std. Error t value Pr(> jtj)�X;gas 0:000188 0:000812 0:231 0:817�1;1 0:126 0:0303 4:168 3:26:10�5 ***�1;2 �0:111 0:0498 �2:238 0:0254 *�1;3 �0:172 0:0601 �2:867 0:00420 **�1;4 �0:142 0:0563 �2:524 0:0117 *��X;gas �0:0152 0:00569 �2:661 0:00787 **Table 2.13: Linear regression of the natural gas short-term sho ks on �Zt�1 and RXt�lX : pairheating oil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sidedp-values are reported Estimate Std. Error t value Pr(> jtj)�X;heat 0:000730 0:000456 1:602 0:109�2;1 0:0538 0:0170 3:161 0:00160 **�2;2 0:0523 0:0279 1:873 0:0613 Æ�2;3 �0:0456 0:0337 �1:354 0:176�2;4 0:153 0:0316 4:845 1:40:10�6 ***�X;heat 0:000901 0:00319 0:282 0:778Table 2.14: Linear regression of the heating oil short-term sho ks on �Zt�1 and RXt�lX : pair heatingoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported 107

Estimate Std. Error t value Pr(> jtj)�Y;gas 0:00122 0:000407 2:991 0:00283 **�3;1 �0:0402 0:0151 �2:652 0:00808 **�3;2 �0:0778 0:0249 �3:125 0:00181 **�3;3 0:0594 0:0302 1:967 0:0494 *�3;4 �0:0432 0:0283 �1:524 0:128��Y;gas �0:0131 0:00461 �2:840 0:00458 **Table 2.15: Linear regression of the natural gas long-term sho ks on �Zt�1 and RYt�lY : pair heatingoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reported

108

Estimate Std. Error t value Pr(> jtj)�Y;heat 0:00147 0:000402 3:649 0:000272 ***�4;1 0:0160 0:0150 1:071 0:284�4;2 �0:0692 0:0246 �2:813 0:00497 **�4;3 �0:0198 0:0298 �0:663 0:507�4;4 �0:170 0:0280 �6:076 1:58:10�9 ***�Y;heat 0:0108 0:00455 2:370 0:0179 *Table 2.16: Linear regression of the heating oil long-term sho ks on �Zt�1 and RYt�lY : pair heatingoil-natural gas; the estimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-valuesare reportedGARCH models for the volatilitiesIn this se tion, we model the volatility pro esses of the residuals of the eight previous regres-sions (i.e. the pro esses (�t) in model (2.12))17. Ljung-Box tests on the pro esses (�2t ) show theheteros edasti ity of the di�erent residuals:

17Unsurprisingly, we obtain very similar volatility models for the natural gas residuals of the pair rude oil-naturalgas and for the the natural gas residuals of the pair heating oil-natural gas; in what follows, we report only the former109

data X2 df p-value(�X;gast )2 25:03 1 5:65:10�7data X2 df p-value(�Y;gast )2 20:24 1 6:827:10�6data X2 df p-value(�X; rudet )2 9:668 1 0:00187data X2 df p-value(�Y; rudet )2 36:178 1 1:801:10�9data X2 df p-value(�X;heatt )2 324:488 1 < 2:2:10�16data X2 df p-value(�Y;heatt )2 25:717 1 3:954:10�7Table 2.17: Box-Ljung tests on the pro esses (�X;gast )2,(�Y;gast )2,(�X; rudet )2, (�Y; rudet )2, (�X;heatt )2,and (�Y;heatt )2; the test-statisti s, degrees of freedom of the approximate hi-square distribution ofthe test statisti s, and p-values of the tests are reportedMoreover, �gure (2.20) exhibits a signi� ant seasonal omponent in the natural gas and heatingoil short-term volatilities. The following seasonal GARCH model, proposed by Diebold (2003) forthe modeling of temperature series, a ounts for this phenomenon:�t = �t�t (2.13)�2t+1 = ��2t + ��2t + (1� �� �)(a0 + a1 os(2�t=252) + b1sin(2�t=252)) (2.14)(�t) i:i:d (2.15)110

Note that the volatility of volatility is itself seasonal sin e the volatility sho ks �2t = �2t �2t havedi�erent average winter and summer values. This hara teristi is ompatible with the observationof gas and heating oil short-term volatilities, whi h mostly luster during the winters (see �gure(2.20)). This model was alibrated by Quasi-Maximum Likelihood on natural gas and heating oilshort-term residuals (�X;gast ) and (�X;heatt ). The log-likelihood of the model in the ase of Gaussianresiduals (�t) is: LL = � NXt=1 � �2t2�2t + log(�t)�� T2 log(2�)Estimate Std. Error t value Pr(> jtj)�gas 0:795 0:0376 21:171 < 2:10�16 ***�gas 0:127 0:0282 4:503 6:691:10�6 ***a0;gas 0:00100 0:000131 7:660 1:865:10�14 ***a1;gas 0:000368 0:000102 3:621 2:930:10�4 ***b1;gas �0:000348 0:0000977 �3:557 3:750:10�4 ***Table 2.18: Quasi-Maximum-Likelihood estimation of a seasonal GARCH model on (�X;gast ); theestimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-values are reportedTable (2.20) reports the results of the Jarque-Bera (resp. Ljung-Box) tests on the residuals(resp. squared residuals) of the two seasonal GARCH models. The Jarque-Bera tests allow usto reje t the hypothesis of Gaussian residuals. The Ljung-Box test indi ates that the squaredresiduals on heating oil are still auto orrelated, whi h shows that the seasonal GARCH model isnot appropriate to �lter out the heteros edasti ity of the heating oil returns. By ontrast, we annot reje t the hypothesis of independen e for the squared residuals of natural gas, whi h is anindi ation of the validity of the model for natural gas. We thus adopt an ARCH model rather than111

Estimate Std. Error t value Pr(> jtj)�heat 0:880 2:148:10�2 40:955 < 2:10�16 ***�heat 9:337:10�2 1:771:10�2 5:271 1:357:10�7 ***a0;heat 2:958:10�4 5:206:10�5 5:681 1:336:10�8 ***a1;heat 1:488:10�4 5:175:10�5 2:876 4:034:10�3 ***b1;heat �6:768:10�5 5:236:10�5 �1:292 1:962:10�1Table 2.19: Quasi-Maximum-Likelihood estimation of a seasonal GARCH model on (�X;heatt ); theestimated oeÆ ients, standard deviations, t-statisti s, and two-sided p-values are reporteda GARCH model for heating oil:�t = �t�t (2.16)�2t+1 = ��2t + (1� �)(a0 + a1 os(2�t=252) + b1sin(2�t=252)) (2.17)(�t) i:i:d (2.18)The estimated parameters are reported on table (2.21) and the tests on the residuals are reportedon table (2.22): we observe now that we annot reje t anymore the hypothesis of independen e forthe squared residuals. Figures (2.25) and (2.26) plot the traje tories of ����X;gast ��� and ����X;heatt ���,together with the volatilities �t predi ted by the seasonal GARCH models and the long-termseasonal varian e fun tions a0 + a1 os(2�t=252) + b1sin(2�t=252).112

Jarque-Beradata X2 df p-value�X;gast 1853:361 2 2:2:10�16Ljung-Boxdata X2 df p-value(�X;gast )2 0:0189 1 0:8907Jarque-Beradata X2 df p-value�X;heatt 61:5038 2 4:4:10�14Ljung-Boxdata X2 df p-value(�X;heatt )2 4:5605 1 0:0327Table 2.20: Jarque-Bera and Box-Ljung tests on the residuals of the seasonal GARCH models fornatural gas and heating oil; the test statisti s, degrees of freedom, and p-values of the tests arereported113

Estimate Std. Error t value Pr(> jtj)�heat 0:262 2:619:10�2 4:697 2:636:10�6 ***a0;heat 2:734:10�4 1:235:10�5 22:172 < 2:10�16 ***a1;heat 1:460:10�4 1:597:10�5 9:143 < 2:10�16 ***b1;heat 3:890:10�5 1:391:10�5 2:795 5:191:10�3 **Table 2.21: Quasi-Maximum-Likelihood estimation of a seasonal ARCH model on (�X;heatt ); theestimated oeÆ ients, standard deviations, t-statisti s, and tow-sided p-values are reportedJarque-Beradata X2 df p-value�X;heatt 147:947 2 < 2:2:10�16Ljung-Boxdata X2 df p-value(�X;heatt )2 0:992 1 0:319Table 2.22: Jarque-Bera and Box-Ljung tests on the residuals of the seasonal ARCH model forheating oil; the test statisti s, degrees of freedom, and p-values of the tests are reported

114

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Figure 2.25: Traje tories of ����X;gast ��� (bla k), volatility �t predi ted by a seasonal GARCH model(red), and square root of the long-term varian e a0 + a1 os(2�t=252) + b1sin(2�t=252) (green)

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Figure 2.26: Traje tories of ����X;heatt ��� (bla k), volatility �t predi ted by a seasonal ARCH model(red), and square root of the long-term varian e a0 + a1 os(2�t=252) + b1sin(2�t=252) (green)

116

We model the other series by lassi al GARCH models. The results are reported on tables(2.23)-(2.31). When the Ljung-Box test leads to reje t the independen e hypothesis on the squaredresiduals of the GARCH(1,1) model, we use instead the ARCH(1) model. This happens here forthe rude oil long-term sho ks.Estimate Std. Error t value Pr(> jtj)a0 1:214:10�6 4:203:10�7 2:888 0:00388 **a1 6:712:10�2 7:712:10�3 8:704 < 2:10�16 ***b1 9:305:10�1 7:113:10�3 130:817 < 2:10�16 ***Table 2.23: Quasi-Maximum-Likelihood estimation of a GARCH model �2t+1 = a0 + a1�2t + b1�2ton natural gas long-term sho ks; the estimated oeÆ ients, standard deviations, t-statisti s, andtow-sided p-values are reportedJarque-Beradata X2 df p-value�Y;gast 502:0754 2 < 2:10�16Ljung-Boxdata X2 df p-value(�Y;gast )2 0:1007 1 0:751Table 2.24: Jarque-Bera and Ljung-Box tests on the residuals of the GARCH model for naturalgas long-term sho ks; the test statisti s, degrees of freedom, and p-values of the tests are reported117

Estimate Std. Error t value Pr(> jtj)a0 8:655:10�6 2:611:10�6 3:315 0:000916 ***a1 7:928:10�2 1:248:10�2 6:355 2:09�10 ***b1 8:827:10�1 2:028:10�2 43:530 < 2:10�16 **Table 2.25: Quasi-Maximum-Likelihood estimation of a GARCH model �2t+1 = a0 + a1�2t + b1�2ton rude oil short-term sho ks; the estimated oeÆ ients, standard deviations, t-statisti s, andtow-sided p-values are reportedJarque-Beradata X2 df p-value�X; rudet 62:5259 2 2:242:10�14Ljung-Boxdata X2 df p-value(�X; rudet )2 0:6365 1 0:425Table 2.26: Jarque-Bera and Ljung-Box tests on the residuals of the GARCH model for rude oilshort-term sho ks; the test statisti s, degrees of freedom, and p-values of the tests are reported

118

Estimate Std. Error t value Pr(> jtj)a0 1:677:10�5 5:692:10�6 2:946 0:00322 **a1 7:387:10�2 1:329:10�2 5:557 2:74�8 ***b1 8:543:10�1 3:406:10�2 25:083 < 2:10�16 ***Table 2.27: Quasi-Maximum-Likelihood estimation of a GARCH model �2t+1 = a0 + a1�2t + b1�2ton heating oil long-term sho ks; the estimated oeÆ ients, standard deviations, t-statisti s, andtow-sided p-values are reportedJarque-Beradata X2 df p-value�Y;heatt 75:6266 2 < 2:2:10�16Ljung-Boxdata X2 df p-value(�Y;heatt )2 0:196 1 0:658Table 2.28: Jarque-Bera and Ljung-Box tests on the residuals of the GARCH model for heating oillong-term sho ks; the test statisti s, degrees of freedom, and p-values of the tests are reportedDependen e stru ture of the standardized o-movementsWe model the dependen e stru ture of the residuals (�) using the opula representation:P(�X;et � z1; �X;e0t � z2; �Y;et � z3; �Y;e0t � z4) = C(FX;e(z1); FX;e0(z2); F Y;e(z3); F Y;e0(z4)) (2.19)where the opula fun tion C is de�ned in [0; 1℄4 with values in [0; 1℄, and (FX;e; FX;e0 ; F Y;e; F Y;e0)denote the marginal distributions of the residuals (�X;e; �X;e0 ; �Y;e; �Y;e0). We will use here the119

Ljung-Boxdata X2 df p-value(�Y; rudet )2 4:5107 1 0:0337Table 2.29: Ljung-Box test on the residuals of the GARCH(1,1) model for rude oil long-termsho ks; the test statisti s, degrees of freedom, and p-values of the tests are reportedEstimate Std. Error t value Pr(> jtj)a0 1:623:10�4 6:295:10�6 25:782 < 2:10�16 ***a1 1:590:10�1 2:593:10�2 6:132 8:67:10�10 ***Table 2.30: Quasi-Maximum-Likelihood estimation of an ARCH(1) model �2t+1 = a0 + a1�2t on rude oil long-term sho ks; the estimated oeÆ ients, standard deviations, t-statisti s, and tow-sided p-values are reportedGaussian opula de�ned by:C(u1; u2; u3; u4) = �4�(��1(u1);��1(u2);��1(u3);��1(u4)) (2.20)where �4� is a 4-variate normal distribution of orrelation matrix �, and ��1 is the inverse of theunivariate standard normal distribution. The log-likelihood of the opula model (2.19)-(2.20) is:LL(�; FX;e; ::; F Y;e0) = TXt=1 ln h (FX;e(zt1); :::; F Y;e0(zt4))i+ NXt=1 �ln �fX;e(zt1)�+ :::+ ln hfY;e0(zt4)i�(2.21)where (fX;e; fX;e0; fY;e; fY;e0) are the univariate densities, is the density of the 4-variate normal opula (2.20): (u1; u2; u3; u4) = �4C�u1:::�u4120

Jarque-Beradata X2 df p-value�X;heatt 62:5259 2 2:242:10�14Ljung-Boxdata X2 df p-value(�Y; rudet )2 0:6365 1 0:425Table 2.31: Jarque-Bera and Ljung-Box tests on the residuals of the ARCH(1) model for rude oillong-term sho ks; the test statisti s, degrees of freedom, and p-values of the tests are reportedand (zti)1�t�T are the observations of the i-th variable. As explained in Joe and Xu (1996), the alibration of the model (2.19) is done in two steps:- we �rst make a non parametri estimation of the marginal densities (fX;e; fX;e0 ; fY;e; fY;e0) of thedi�erent sho ks, using the Gaussian kernel estimator (Silverman (1986)):f(z) = 1Nh NXi=1 K(z � zih )where K(z) = 1p2�exp(� z22 ), (zi) are the observations, and h is an appropriate bandwidth; the ob-tained densities are plotted and ompared to the standard normal density in �gures (2.27)-(2.28);the univariate distributions (FX;e; FX;e0 ; F Y;e; F Y;e0) are obtained by numeri al integration of thedensities.- on e the marginal distributions are determined, the orrelation matrix � is estimated by maximiza-tion of the �rst part of the log-likelihood (2.21), whi h ontains the information on the dependen estru ture.The estimates of matrix orrelations � for the pairs gas/ rude oil and gas/heating oil are reportedon tables (2.32)-(2.33). We note that all orrelations are signi� antly positive and that the depen-121

den e stru ture is very similar for the two pairs, apart form the orrelation between the short-termsho ks, whi h is higher for the pair gas-heating oil than for the pair gas- rude oil. This higher orrelation was expe ted be ause weather is a ommon determinant of natural gas and heating oildemand urves.

122

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Figure 2.27: densities of natural gas short-term sho ks (bla k), natural gas long-term sho ks (red), rude oil short-term sho ks (green), rude oil long-term sho ks (blue), and normal density ( learblue)123

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Figure 2.28: densities of natural gas short-term sho ks (bla k), natural gas long-term sho ks (red),heating oil short-term sho ks (green), heating oil long-term sho ks (blue), and normal density ( learblue)124

short-term gas short-term rude long-term gas long-term rudeshort-term gas 1 0:258(0:024) 0:583(0:015) 0:278(0:024)short-term rude 0:258(0:024) 1 0:280(0:024) 0:480(0:019)long-term gas 0:583(0:015) 0:280(0:024)) 1 0:329(0:023)long-term rude 0:278(0:024) 0:480(0:019) 0:329(0:023) 1Table 2.32: Maximum-likelihood estimation of the orrelation matrix of the Gaussian opula forthe pair gas- rude oil (standard errors in parenthesis)short-term gas short-term heat long-term gas long-term heatshort-term gas 1 0:337(0:022) 0:584(0:015) 0:299(0:023)short-term heat 0:337(0:022) 1 0:309(0:023) 0:428(0:020)long-term gas 0:584(0:015) 0:309(0:023) 1 0:337(0:022)long-term heat 0:299(0:023) 0:428(0:020) 0:337(0:022) 1Table 2.33: Maximum-likelihood estimation of the orrelation matrix of the Gaussian opula forthe pair gas-heating oil (standard errors in parenthesis)2.4.3 Stability of the orre tion me hanismsThe obje tive here is to study the temporal stability of the di�erent orre tion me hanisms whi hwere found in se tion 3.4.2. As in Alexander (1999), �gure (2.29) shows the temporal evolution(with a two-year sliding window) of the t-statisti s on �X;gas and �X; rude for the pair gas- rude.We observe �rst that the oeÆ ient �X;gas has a positive trend during the period: as a onsequen e,from 2000 to 2002, the orre tion is done through rude oil rather than through natural gas, and the125

situation is inverted in the period 2002-2003. Se ond, we observe that the t-statisti s on �X;gas and�X; rude often move in opposite dire tions, whi h implies that the strength of the global orre tion,represented by the bla k traje tory, tends to remain lo ally stable. Yet, after a transitory periodin 2001-2002, the bla k urve takes higher values in the period 2002-2003 than in the period 2000-2002, indi ating that the long-term equilibrium was stronger in the latter period than in the former.Figures (2.30) to (2.32) present similar results. Most on lusions remain the same, in parti ularthe fa t that all orre tion me hanisms have been stronger sin e 2002.

126

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Figure 2.29: Stability of the orre tion me hanism on gas and rude oil slopes: traje tories (witha two-year sliding window) of the t-statisti s of �X;gas (gas short-term sho ks to deviations RXt )(blue), of �X; rude ( rude short-term sho ks to deviations RXt ) (red), mean of the two previoust-statisti s (bla k), 95% signi� an e level (green)127

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Figure 2.30: Stability of the orre tion me hanism on gas and rude oil levels: traje tories (witha two-year sliding window) of the t-statisti s of �Y;gas (gas short-term sho ks to deviations RYt )(blue), of �Y; rude ( rude short-term sho ks to deviations RYt ) (red), mean of the two previoust-statisti s (bla k), 95% signi� an e level (green)128

−2

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Figure 2.31: Stability of the orre tion me hanism on gas and heating oil slopes: traje tories (witha two-year sliding window) of the t-statisti s of �X;gas (gas long-term sho ks to the deviationsRXt ) (blue), of �X;heat (heating oil long-term sho ks to the deviations RXt ) (red), mean of the twoprevious t-statisti s (bla k), 95% signi� an e level (green)129

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Figure 2.32: Stability of the orre tion me hanism on gas and heating oil levels: traje tories (witha two-year sliding window) of the t-statisti s of �Y;gas (gas long-term sho ks to the deviationsRYt ) (blue), of �Y;heat (heating oil long-term sho ks to the deviations RYt ) (red), mean of the twoprevious t-statisti s (bla k), 95% signi� an e level (green)130

2.4.4 Evolution of the orrelationsThe obje tive here is to study the stability of the dependen e stru ture whi h was found betweenthe forward urves o-movements. Figure (2.33) represents the temporal evolution of Kendall's orrelation18 (with a one-year sliding window) between di�erent pairs of sho ks. All orrelationsdisplay an upward trend on the period. A possible explanation for this observation is the fa t that,in ommodity markets, in ontrast to equity markets, orrelation is generally bigger when pri esare rising. This ould also a ount for the fa t the orre tion me hanisms between oil and gas havebeen stronger sin e 2003, when gas and oil pri es started rising. Lastly, �gure (2.34) representsthe temporal evolution of Kendall's orrelation (with a 50-days sliding window) between naturalgas and heating oil short-term sho ks. Not surprisingly, stronger orrelations prevailed during thetight market winters 2000-2001, 2002-2003, and 2003-2004, when natural gas and heating oil urvesboth were in ba kwardation.

18The Kendall's orrelation � is linked to the orrelation matrix of the Gaussian opula estimated above throughthe relation � = 2� ar sin(�) (see Lindskog et al. (2001)) and is known to be more robust than the usual Pearsonlinear orrelation 131

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2000 2001 2002 2003 2004(d) long-term gas/long-term heating oilFigure 2.33: Kendall's orrelation (with a one-year sliding window)132

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Figure 2.34: Kendall's orrelation between gas and heating oil short-term sho ks (with a 50-dayssliding window)

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2.5 Con lusionThis paper has presented a new dependen e model for ommodity forward urves. Like popularmodels on single ommodity forward urves, it de omposes the forward urve moves into a short-term and a long-term sho ks, with sto hasti and possibly seasonal volatilities. The orrelationbetween the sho ks of the two urves is aptured through a non-Gaussian dependen e stru ture.The originality of the model is that, in addition to this lo al dependen e stru ture, it a ounts forthe long-term relations between the ommodity forward pri es through an error- orre tion termin the risk-premia of the forward pri e returns. The long-term relations are based on the statevariables des ribing the shape of a forward urve under the two-fa tor model, namely the slopeand level. Our urrent resear h on erns the extension of the model to three sto hasti fa tors forea h urve (as in Diebold and Li (2003)), the modeling of sto hasti dependen e stru ture, and theimpli ations of the model for multi- ommodity asset pri ing and portfolio optimization.2.6 Referen esAlexander C. (1999), Correlation and Cointegration in Energy Markets in Managing Energy Pri eRisk (2. nd. Edition) V. Kaminsky (ed.).Ates A., Wang G. (2005), Storage, Weather and Dynami s of Natural Gas Pri es in Futures andSpot Markets, Working PaperAmeri an Gas Foundation (2003), Natural Gas and Energy Pri e Volatility, Energy and Environ-mental Analysis, In .Blix M. (2003), The volatility Stru ture of Gas Futures Contra ts, Working Paper, Sto kholmS hool of E onomi s 134

Brooks R. (2001), Value at Risk Applied to Natural Gas Forward Contra ts, WP01-08-01, Depart-mentof E onomi s, Finan e and Legal Studies, University of AlabamaCampbell D., Diebold F. (2003), Weather Fore asting for Weather Derivatives, NBER WorkingPapers 10141, National Bureau of E onomi Resear h, In .PIER Working Paper No. 01-031.Clewlow L., Stri kland C. (2000), Energy Derivatives: Pri ing and Risk Management, La ima Pub-li ations (2000)Diebold F., Li C. (2003) Fore asting the Term Stru ture of Government Bond Yields, NBER Work-ing Papers 10048, National Bureau of E onomi Resear h, In Duan J., Pliska S. (2004), Option Valuation with Co-integrated Asset Pri es, Journal of E onomi Dynami s and Control, 28(4), 727-54DuÆe D. (1999), Volatility in Energy Pri es in Managing Energy Pri e Risk (2. nd. Edition) V.Kaminsky (ed.).Energy Information Administration (2001), Impa t of Interruptible Natural Gas Servi e on North-east Heating Oil DemandEwing B., Malik F., Oz�dan O. (2002), Volatility Transmission in the Oil and Natural Gas Markets,Energy E onomi s 24 (2002) 525-538Eydeland A., Wolynie K. (2003), Energy and Power Risk Management: New Developments inModeling, Pri ing, and Hedging, WileyEuropeFama E., Fren h K. (1988), Business Cy les and the Behavior of Metals Pri es, Journal of Finan e,Vol. 43, No. 5, pp. 1075-1093Geman H. (2005), Commodities and Commodity Derivatives : Modelling and Pri ing for Agri ul-turals, Metals and Energy, Wiley Finan eGeman H., Nguyen V. (2005), Soybean Inventory and Forward Curves Dynami s, Management135

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S hwartz E., Smith J. (2000), Short-Term Variations and Long-Term Dynami s in CommodityPri es, Management S ien eSiliverstovs B., Neumann A., LH�egaret G., Hirs hhausen C. (2005), International Market Inte-gration for Natural Gas? A Cointegration Analysis of Gas Pri es in Europe, North Ameri a andJapan, Energy E onomi s, Vol. 27, No. 4, 603-615.Silverman B. W. (1986) Density Estimation for Statisti s and Data Analysis, London: ChapmanHall.

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