VOLATILITY DYNAMICS OF NYMEX NATURAL GAS FUTURES PRICES

VOLATILITY DYNAMICS OF NYMEX NATURAL GAS FUTURES PRICES Hiroaki Suenaga Reearch Fellow School of Economic and Finance Curin Buine School Curin Univeriy of Technology Aaron Smih Aian Profeor Deparmen of Agriculural and Reource Economic Univeriy of California Davi and Member Giannini Foundaion Jeffrey William Daniel Baron DeLoach Profeor Deparmen of Agriculural and Reource Economic Univeriy of California Davi and Member Giannini Foundaion Correpondence o: Hiroaki Suenaga School of Economic and Finance Curin Buine School Curin Univeriy of Technology GPO Box U987 Perh WA 6845 Auralia. e mail: hiroaki.uenaga@cb.curin.edu.au. Phone: +6 8 966 4480. Fax: +6 8 966 306.

VOLATILITY DYNAMICS OF NYMEX NATURAL GAS FUTURES PRICES ABSTRACT We examine he volailiy dynamic of daily naural ga fuure raded on he NYMEX via he parially overlapping ime erie POTS model of Smih 005 Journal of Applied Economeric. We how ha aide from a ime o mauriy effec volailiy exhibi wo imporan feaure ha are cloely relaed o he eaonal cycle of US naural ga demand and orage. Fir volailiy i greaer in he winer han in he ummer. Second he perience of price hock and hence he correlaion among concurrenly raded conrac diplay ubanial eaonal and cro ecional variaion in a way conien wih he heory of orage. We demonrae ha by ignoring he eaonaliy in he volailiy dynamic of naural ga fuure price previou udie have uggeed ubopimal hedging raegie.

. INTRODUCTION Naural ga demand in he US peak beween December and March due o reidenial heaing during he winer. Alhough hi unremarkable fac implie higher average price in winer ineremporal arbirage miigae he price effec; he winer price equal he off peak pring and ummer price plu he co of carry. Thi paern i implied by he heory of orage which aer ha he equilibrium conellaion of po and fuure price repreen he price a which he marginal benefi of curren conumpion equal he expeced marginal benefi of oring a commodiy for fuure conumpion William and Wrigh 99. Thi relaionhip doe no hold beween wo conrac price if invenory i effecively zero a any poin beween heir mauriy dae. The ypical eaonaliy in he conellaion of naural ga price i illuraed in Figure which diplay all 7 NYMEX fuure price on April 003. Thoe conrac alhough reching ix year ino he fuure are aligned in Figure o he convenional April March year in naural ga. Thi alignmen reveal he rong increae in price during he fall and early winer preciely when ock are peaking. Unlike for mo commodiie orage faciliie for naural ga can reach collecive capaciy during hi period which lead o price relaionhip ha imply ignifican eaonaliy in he marginal co of orage. Srong eaonaliy in demand and orage alo implie highly nonlinear volailiy dynamic. Volailiy i naurally high in he winer conrac becaue he high marginal co of naural ga producion and he inelaic winer demand mean ha hock of even a mall magniude can caue a large price wing. A he ame ime he high naural ga invenory inended for all winer monh reduce volailiy of he early winer conrac becaue hock can be accommodaed albei parially by releaing or aborbing invenory. Such flexibiliy i ineviably

lower laer in he winer. The eaonal orage paern alo implie ha price of winer conrac ar flucuaing a early a he preceding off peak demand period during which informaion arrive abou he fuure availabiliy of ored ga. In conra price of pring and ummer conrac hould no exhibi ubanial movemen unil he end of he preceding peak demand eaon becaue lile invenory i carried over from winer o pring in a normal year. In hi paper we examine he volailiy dynamic of he NYMEX naural ga fuure price uing he parially overlapping ime erie POTS model of Smih 005. Unlike convenional model of commodiy price dynamic he POTS model rea he daily price on a given conrac a a ingle ime erie. A ime proceed ome conrac reach delivery and ceae o exi while oher are born and begin rading. In hi ene a e of conrac coniue a e of parially overlapping ime erie. Each of hee ime erie behave a a maringale proce which allow he model o be applied direcly o he fir difference of he commodiy fuure price and herefore o avoid mi pecifying he price level dynamic. To accoun for he volailiy dynamic he POTS model ue nonparameric funcion. Thi flexible pecificaion capure he ime o mauriy effec eaonal volailiy and oher nonlinear volailiy dynamic reuling from he peculiariie of naural ga a a commodiy. To illurae he pracical imporance of hee feaure of he POTS model we apply our analyi of he volailiy dynamic of naural ga fuure price o he andard opimal hedging raegy. We illurae ha eaonal and cro ecional variaion in he degree of perience of he price hock ogeher wih he eaonaliy in he arrival of informaion imply ha he December and ummer June hrough Augu conrac are more effecive han oher conrac in minimizing he variance of porfolio reurn. We conra hi reul wih he reul from previou udie which ignore eaonal and cro ecional variaion in volailiy of naural ga price and which recommend he ue of he adjacen delivery conrac for hedging.

Much of he previou reearch on naural ga no o menion oher energy fuure ha no accouned for complex volailiy dynamic reuling from he peculiariie of hee commodiie. One common approach in previou udie of energy fuure price ha been o pecify he po price a a funcion of underlying ae variable following ipulaed ochaic procee. The uggeed model i hen ued o derive he valuaion formula of fuure and oher derivaive conrac any difference from he oberved fuure price being inerpreed a repreening a rik premium. Example of hi approach are Schwarz 997 for he NYMEX crude oil and Lucia and Schwarz 00 for he Nordpool wholeale elecriciy marke. For naural ga Manoliu and Tompaidi 00 have applied one and wo facor model o NYMEX fuure price and repor ha he eimaed model exhibi rong eaonal variaion while deviaion from hi eaonal mean rever o zero. A econd common approach ha been o examine direcly he relaionhip beween he po and fuure price wihou ipulaing he ochaic procee of he oberved price erie. Two queion are commonly addreed in uch udie: i if he fuure price i unbiaed in forecaing ubequenly realized po price and ii if he po price and he price of concurrenly raded fuure conrac exhibi long run relaionhip and if o how quickly hey rever o he long run relaionhip afer deviaing from i. The fir queion i uually addreed by examining he aiical ignificance of he difference beween he po or nearby fuure and he fuure price oberved much earlier. Several udie have repored ha he NYMEX naural ga fuure price are downward biaed in forecaing he ubequenly realized po price and inerpreed hi bia a repreening a rik premium Wall 995; Modjahedi and Movaagh 005; Movaagh and Modjahedi 005. For he econd queion Roo and Lien 003 and Lien and Roo 999 ue variou coinegraion mehod o examine he long run relaionhip beween he po price and he price of concurrenly raded fuure conrac. 3

They conclude ha hee price hare a common ochaic rend and repond o deviaion from heir long run relaionhip in uch a way ha he difference beween he price converge o zero. Thi reul implie ha imulaneouly raded fuure price are in one o one relaionhip and herefore have zero bai in he long run. Improper or incomplee underanding of volailiy dynamic can lead uch udie o erroneou concluion abou he marke rik premium or more generally he efficiency of exiing fuure marke. William and Wrigh 99 how ha he equilibrium fuure price derived from a dynamic raional expecaion model of a eaonal orable commodiy are highly nonlinear and non mooh which i o ay ha hey canno be expreed in a reduced form. Clearly pecified eaonal price cycle can only approximae he rue price dynamic. Srong eaonaliy in orage implie ha he difference beween he po price and price of concurrenly raded fuure conrac or he bai repreening he co of carry varie acro conrac delivery monh. Previou udie on po fuure relaionhip have no incorporaed uch eaonal variaion in he bai. In conra he POTS model avoid uch mipecificaion by differencing ou he price level dynamic. Aide from hee modeling iue mo previou udie have uilized only a ube of price daa available from organized exchange where muliple conrac wih differen mauriy dae are concurrenly raded. One pracice i o conruc a monhly daa erie by acking he fuure price from ome paricular day of he monh. Thi pracice i paricularly common o e of bia in fuure price for mo energy fuure conrac are defined in monhly block. Anoher pracice i o conruc a daily erie by plicing ogeher he nearby fuure conrac. Eiher pracice no only dicard much informaion bu alo dior he emporal dynamic of he oberved price erie due o wiching from one delivery monh o anoher. 4

. Parially Overlapping Time Serie Model The parially overlapping ime erie POTS model of Smih 005 i a laen facor model of daily fuure price change. The laen facor repreen he fac ha conemporaneou fuure price are ied ogeher by ineremporal arbirage. In a ingle facor eing he model ake he following form ΔF = ε + λ u where ΔF i an n vecor of daily fuure price change wih n repreening he number of conrac raded on day. I comprie ΔFm = Fm Fm where he wo ubcrip repreen he mauriy and rading dae repecively. The wo ubcrip implicily define he number of day o mauriy a d = m. The calar ε i a laen facor and i n vecor of facor loading. The n vecor u denoe he idioyncraic error wih u ~ N0 I n and λ ignifie an n n diagonal marix ha deermine he variance of he idioyncraic error. The componen of he marice and λ are pecified a funcion of mauriy dae and ime o mauriy of conrac. Tha i m = d m λm = λd m For idenificaion we pecify E[ε ] = E[ε um] = 0 for all m and and E[ΔF I ] = 0 where I denoe he informaion e available a. Thee aumpion aer ha a erie of daily fuure price change follow a maringale difference equence which implie a zero rik premium. Following Smih 005 we pecify he condiional variance of ε by he GARCH proce 5

3 E[ε I ] = h = ω + β h + α E[ε I ] Becaue he uncondiional variance of ε equal uniy we have ω = α β. The la erm in 3 i E[ε I ] = ε + P where ε = E[ε I ] and P = E[ε ε I ] which are obained hrough he Kalman filer Hamilon 994. The POTS model a defined in hrough 3 i economerically imilar o facor model of commodiy price dynamic uch a Schwarz 997 and Manoliu and Tompaidi 00 a applied o naural ga. Among he major difference i ha he POTS model view he daily price on a given conrac a a ingle ime erie and pecifie he dynamic of daily price change. In conra convenional facor model pecify he dynamic of daily price level and no price change. By modeling daily price change which are a maringale difference equence he POTS model avoid pecifying eaonal or any oher deerminiic variaion in he underlying po price level and hence i free of approximaion bia. Thi bia could be ofen large in commodiy price model epecially for commodiie ha exhibi uch a rong eaonal paern a naural ga Suenaga 005. The POTS model alo employ flexible funcional form in pecifying he facor loading m and he variance of he idioyncraic error λm. In conra convenional facor model e.g. Schwarz 997 pecify a ochaic proce of he po price dynamic wih a mall number of parameer hereby impoing a paricular rucure on he facor loading in he fuure pricing equaion. Even he mo complex of hee model including an increaed number of facor and/or pecifying dynamic of each facor by more complex ochaic proce ill pecify a imple variance erm for he idioyncraic error wih a mo cro ecional variaion acro 6

conrac delivery monh. In conra he flexible funcional form employed in he POTS model inend o capure ime o mauriy effec and eaonal volailiy a well a oher nonlinear volailiy dynamic of naural ga fuure price reuling from peculiariie of he commodiy. 3. DATA AND ESTIMATION RESULTS 3. Daa We eimae he model in and uing daily elemen price daa from he NYMEX naural ga fuure marke. The NYMEX ared rading naural ga fuure conrac in April 990. Conrac raded in hi marke are defined in monhly block wih each conrac providing for he delivery of 0000 million Briih Thermal uni BTu of naural ga a he Henry Hub locaed in Louiiana. The marke iniially raded conrac a far a year before he fir calendar day of he delivery monh bu ha gradually exended hi horizon o 6 year. Conrac were iniially raded unil even buine day prior o he fir calendar day of he delivery monh bu rading wa laer exended o 3 buine day prior o he delivery monh. We analyze he daily change in he logarihm of elemen price and our ample pan January 99 o December 3 003. Becaue dian delivery conrac are no alway acively raded we drop conrac of more han monh o mauriy. Excluding hee obervaion we have a ample of 4068 price among 75 conrac. 3. Eimaion Reul In applying he POTS model o he NYMEX naural ga fuure price daa we pecify he facor loading and he variance of idioyncraic error in by he following rigonomeric funcion 7

5 m m m = = + + m d a0 a d a j= λ m m = λ d = b m 0 + b m d + πjd + m πjd in + a j co dmax d m j max πjd + 5 m m b + j in b j co j= dmax dmax πjd where dmax = 365. To capure eaonaliy we eimae one uch funcion for each conrac monh i.e. for all m =. Specified a in he wo funcion become more flexible a he number of rigonomeric erm increae. Alhough hi exra flexibiliy allow he model o fi he oberved daa beer i alo make he coefficien eimae more eniive o exreme obervaion. By uing only five rigonomeric erm we allow ufficien flexibiliy o capure eaonaliy and ime o delivery effec bu we avoid exce eniiviy o oulier. We eimae he model by he mehod of Maximum Likelihood wih he iniial value obained by he ieraive approximae EM mehod of Smih 005. 3 Table ummarize he coefficien eimae of he GARCH parameer. In he able he coefficien eimae of α and β are 0.09 and 0.80. The um of he wo coefficien 0.90 i le han uniy indicaing ha he condiional volailiy of he common underlying facor i highly perien ye i aionary. Figure plo he uncondiional variance of daily price change of each of conrac which we compue a + λ for d ranging from 0 o 365 day o mauriy uing he eimaed m m d m md facor loading m and he variance of he idioyncraic error λm. The figure exhibi a lea four inereing feaure. Fir on any given dae he volailiy of conrac ha are cloer o mauriy exceed ha of more dian conrac. Thi feaure ofen called he Samuelon effec indicae ha hock o po price are expeced o diipae omewha over ime. Second volailiy of cloe o mauriy conrac i ubanially higher for winer conrac han for lae pring or ummer conrac. Third during he period from early May o lae Sepember 8

volailiy increae for all conrac mauring before he end of he following peak eaon. Finally during he early winer early November o mid January volailiy rie for all conrac alhough he January o April conrac diplay much larger increae han he oher conrac. The la hree feaure follow naurally from eaonaliy in he demand for naural ga. Nearby winer conrac exhibi high price volailiy becaue demand reache conemporaneou upply capaciy in winer. Even hough high winer invenory allow price hock of ome magniude o be aborbed he high winer volailiy depiced in Figure indicae ha uch price buffering i only limied. Summer conrac exhibi low nearby volailiy becaue low demand relaive o upply mean ha demand hock can be aborbed by alering he amoun injeced ino orage. The gradual volailiy increae in he lae pring and ummer May o lae Sepember reflec he arrival of imporan marke informaion. During hi period invenory coninuouly change a exce naural ga producion i ored for ue in he ubequen peak demand eaon. Ga ored in hi period will no be carried over o he following off peak demand eaon unle ga demand in he coming peak demand eaon i unuually low. Such informaion will no be revealed unil acual demand and/or upply condiion are realized. Hence price volailiy increae during he May o lae Sepember rading period bu only for he conrac deliverable o he coming peak demand eaon and earlier. During early winer new informaion provide rong ignal abou he imminen peak eaon a well a he likely amoun invenory carryover from he curren peak demand eaon o ubequen monh. Nonehele he amoun of ga carried over from he peak o he ubequen off peak eaon i very mall relaive o he amoun ored over he off peak eaon in a normal year. Thu nearby conrac exhibi high volailiy during hi early winer period 9

bu volailiy alo increae albei only marginally for conrac ha maure in he ummer. By mid January mo of he informaion abou he curren peak eaon ha arrived and price volailiy begin o decline. Thee eaonal paern in price variance correpond o he eaonal paern in ga orage. Temporal arbirage induce a fair amoun of naural ga o be carried over from he off peak demand eaon April hrough June o he peak demand eaon December o March o ha he price difference beween hee eaon i on average equal o he co of carry. Figure 3 illurae hi orage paern. In he figure he US naionwide working ga orage i he lowe in March afer which i gradually increae hroughou pring and ummer unil i reache an annual high in Ocober or November. A rapid decreae of invenory from November hrough he following February indicae ha a large amoun of ored ga i wihdrawn o mee wih a high demand for heaing energy. Table ummarize he proporion of he variance explained by he common facor ε. I indicae ha he model explain on average abou 83% of daily price variaion. In general pring and ummer conrac are more cloely relaed o he common facor han winer conrac. Figure 4 illurae for each of he mauriie how he proporion of oal price variance explained by he common facor change over he ime ahead of expiraion. We meaure hi proporion by / + λ. For all conrac hi proporion drop m m d m md m md ubanially below 00 percen in he la few monh before mauriy. Thi paern indicae ha a delivery approache he fuure price increaingly reflec local condiion a he Henry Hub and le condiion in naural ga marke in he re of Norh America. Such a paern i no urpriing for naural ga. Becaue he fuure conrac for naural ga repreen a monhlong flow hrough he Henry Hub raher han he more radiional warehoue receip for he commodiy in ore local congeion in pipeline can diconnec nearby fuure from marke 0

elewhere. For he ame reaon of nework congeion hoe local marke elewhere omeime have po or nearby forward price eiher a eep dicoun or eep premium o he average hroughou he nework. Figure 4 alo how ha for conrac ha deliver beween Ocober and March he hare explained by he common facor drop ubanially around he middle o he end of Augu and keep decreaing aferward. Tha i much of he po Augu price variaion in hee conrac emanae from informaion of a hor erm naure ha doe no affec he amoun of carryover o he following off peak eaon. Coupled wih he high volailiy of hee conac in hi period ee Figure hi obervaion implie ha he high invenory accumulaed by lae ummer allow only limied buffering of curren marke hock. Thi lack of abiliy o aborb hock likely arie from he high marginal co of invenory adjumen which increae wih he invenory level becaue injecing ga back ino underground orage require he ga o be a ever higher preure. In hi ame po Augu period conrac ha deliver afer he end of he following peak eaon exhibi low oal volailiy ee Figure bu a high proporion of heir variaion i capured by he common facor ee Figure 4. Thu he common facor repreen informaion abou marke condiion in he following off peak eaon bu relaively lile uch informaion arrive in hi period. The fac ha he Ocober o March conrac are weakly relaed o he facor in hi period i conien wih he mall invenory carryover from he peak eaon o he following off peak eaon. Thi lack of carryover reduce he poenial for arbirage acro hee wo period and herefore i weaken he link beween he price of conrac ha deliver before he end of he peak eaon and hoe ha deliver afer he end of he peak eaon. In um he eimaed POTS model reveal ha he volailiy of naural ga fuure price exhibi boh he Samuelon effec and rong eaonaliy. Volailiy i higher for winer conrac

han for oher conrac. Volailiy alo increae from early May o Sepember for all conrac mauring by he end of he following peak demand eaon and from early November hrough mid January for he January o April conrac. The correlaion beween he daily price change of concurrenly raded conrac end o be highe during hee wo period. Overall our model illuminae he complex dynamic of naural ga price volailiy which previou udie were unable o dicern becaue heir economeric model preumed a volailiy pecificaion ha i oo imple. 4 OPTIMAL HEDGING STRATEGY In hi ecion we exend our analyi of he volailiy dynamic of naural ga fuure price o inveigae he implicaion for hedging. We conider a imple hedging raegy in which a hedger ha a po poiion Q a ime and imulaneouly ake a hor poiion in X fuure conrac for delivery a τ >. A + k < τ he hedger clear i poiion by elling Q uni in he po marke and buying X fuure conrac for delivery a τ. The hedger change in wealh from o + k ignoring he inere rae i 4 W = S S Q F F X = ΔS F Q + k + k + k τ τ + k η Δ + k τ where Si i he po price a i η = X/Q i he hedge raio. The variance of W + k i F i τ i he period i price of he fuure conrac for delivery a τ and 5 V[ W ] = V[ ΔS ] + V[ ΔF ] η cov[ ΔS F ] Q + k + k η + k τ + k Δ + k τ which i minimized by * 6 η τ = cov[ ΔS + k ΔF + k ]/ V[ ΔF + k ] τ τ

Subiuing 6 ino 5 yield he minimized variance * 7 V[ W ] = V[ ΔS ] ρ Q + k η τ + k + k τ where ρ k τ + i he correlaion beween po and fuure conrac for delivery τ in heir price change over period o + k. Alernaively for a hedger minimizing he variance of he porfolio reurn r = rs η r rf he opimal hedge raio and he aociaed minimum variance are r* 8 ητ = cov[ Δ lns + k Δ lnf + k ]/ V[ Δ lnf + k ] τ τ + k η τ + k + k τ r* r 9 V[ r ] = V[ Δ lns ] ρ Q where r ρ k τ + repreen he correlaion beween he log po price and log price of fuure conrac for delivery a τ over he period o + k. Three remark hould be made abou hee hedging raegie. Fir many organized exchange concurrenly rade muliple conrac wih differen mauriy dae. Thu alhough he fuure conrac o be included ino he porfolio τ i exogenou o he above raegy i hould be a deciion variable for he hedger. The expreion 6 and 8 indicae ha given he ime of enry and hedging horizon k he hedger hould include ino i porfolio he fuure conrac for which he price change ha he highe correlaion wih he po price change. Becaue only a finie number of conrac are raded on any given day one need o calculae he opimal hedge raio only for he fuure conrac wih he highe correlaion o he po price. Second he ime of enry and hedging horizon k hould be alo endogenou o he hedger deciion. The choice of and k i imporan paricularly for commodiy wih rong eaonaliy in mean price. Given a priori knowledge of uch a eaonal paern a hedger hould 3

no hold a po poiion from he peak o he off peak demand eaon for uch pracice would yield a lea on average a negaive reurn. The hedge raio in 6 and 8 are opimal only given ha he deciion o carry over from o + k i predeermined. In pracice a poiion hould be held only if he expeced wealh E[W+k] exceed he value of rik aociaed wih he minimum variance of porfolio reurn. Finally he expreion 6 hrough 9 indicae ha boh he opimal hedge raio and he fuure conrac included ino he porfolio depend on wo aribue: i he covariance of po and fuure price and ii he variance of fuure price change. Specificaion on he volailiy dynamic of po and fuure price play key role in deermining empirical eimae of hee aiic and hence he opimal hedging raegy. The POTS model by allowing eaonal and cro ecional variaion in he facor loading and he variance of idioyncraic error yield an opimal hedge raio ha varie by τ and k. In conra convenional model of commodiy price dynamic deermine facor loading by he ime o mauriy of he conrac and a mall number of parameer defining he ochaic procee of he underlying facor. Due o hi rericive pecificaion he opimal hedge raio implied by hee model doe no vary by conrac delivery dae. In paricular a imple one facor mean reverion model conidered by Schwarz 997 and he wo facor model by Manoliu and Tompaid 00 boh imply ha he opimal porfolio alway include he nearby conrac. In oher word he pecificaion choice for he ochaic procee of he underlying facor deermine he opimal hedging raegy. The andard regreion model conidered for he analyi of he po fuure price relaionhip are imilarly incapable of implying cro ecional and eaonal variaion in he opimal hedge raio due o a imple variance rucure aumed for he diurbance erm. 4

4. Opimal Hedging Sraegy Implied by he POTS Model We evaluae he opimal hedge raio baed on he uncondiional variance of daily price change implied by he eimaed POTS model. 4 In doing o we ue he nearby fuure price a a proxy for he po price. Baed on our dicuion in Secion 3. a delivery approache he fuure price increaingly reflec condiion a he Henry Hub. In hi ene he nearby conrac approximae a po price again which marke paricipan may wan o hedge. We conider he cae where and k are predeermined and he hedger deciion i o chooe he fuure conrac o be included ino he porfolio τ and he hedge raio. We find he opimal oluion for he wo cae wih differen holding period: i a hedger who carrie a hor poiion only for a ingle day k = and ii a hedger who carrie a hor poiion for one monh aring a he fir day of each calendar monh and ending a he la day of he ame monh. We conider hee wo cae o deermine wheher very hor holding period are le eniive o he modeling of price dynamic. From he POTS model he uncondiional variance of daily fuure price change i given a 0 E[ΔFΔF ] = + λ λ Wih he aumpion ud iid ~ N0 he diagonal and off diagonal elemen are E[ Δ ] = d m + λd m F m E m [ Δ F ΔF ] = d m δ for m > and m. where d = m and δ =. Uing he expreion he correlaion beween daily price change of wo fuure conrac for delivery and i 5

6 δ λ δ δ λ δ δ δ ρ + + = = π π where δi = i and i π = + i i i i i i δ λ δ δ i he quare roo of he hare of he oal variance of conrac i explained by he common facor. Equaion indicae ha he variance of he porfolio reurn i minimized when he porfolio include he fuure conrac for which he large hare of price change i accouned for by he common facor. For a day long holding period he opimal hedge raio and he minimized variance are 3 * η = δ λ δ δ δ + 4 V[ * W η + ] = Q ρ δ λ δ + For he hedging raegy wih he monh long holding period over o we need he expreion for he uncondiional variance of daily fuure price change which wih he maringale propery aumed in he model i imply he um of he daily price change over hi period 5 = + = Δ Δ ʹ ʹ ] ʹ [ E λ λ F F wih i elemen 6 = + = Δ ] E[ m m m m m F λ = = Δ Δ ] [ E m m m F F for m > m. The correlaion beween he wo conrac over he horizon o i

7 7 ρ = + + = = = λ λ The opimal hedge raio and he aociaed minimum variance are 8 * η = = = + λ 9 ] V[ * W η = Q ρ λ + = Equaion 3 and 8 indicae ha he opimal hedge raio i a funcion of he ime of enry he hedging horizon k and he delivery period of he fuure conrac in he porfolio. I paricular he opimal hedge raio increae wih he variance of he nearby fuure price ha i aribuable o he common facor and/or he proporion of he fuure price variance explained by he common facor. In oher word a hedger hould ake a large hor poiion when he nearby fuure price i very volaile and i i rongly relaed o he fuure price of ubequen delivery. Even when he nearby fuure price i very volaile a hedger poiion i mall if i price movemen i no cloely relaed o he oher concurrenly raded conrac. 4. Opimal Hedging Sraegy for Naural Ga Figure 5 hrough 7 illurae he fuure conrac included ino he opimal porfolio he opimal hedging raio and he minimum variance aained by he opimal porfolio for each of he wo hedging horizon. Fir for a daily hedging raegy he opimal porfolio frequenly include four conrac: he December conrac for he period beween mid May and mid Augu and

eiher he June July or Augu conrac for he period beween mid Sepember and mid April in he following year. Thee conrac are ofen ued becaue hey exhibi he highe hare of heir price volailiy explained by he common facor in relevan period. Oher conrac are rarely included ino he porfolio. Becaue hor daed conrac exhibi ubanial idioyncraic volailiy he opimal porfolio never include hee conrac. The hree monh ahead conrac i he hore horizon and i ued only in he fir half of April and he fir half of Augu where he variance minimizing conrac wiche gradually from he Augu o winer and from he December o ummer conrac repecively. The predominance of he June July Augu and December conrac i conien wih our previou dicuion relaing eaonaliy of price volailiy o ha of naural ga demand and orage. From he beginning of he year he opimal porfolio include he July conrac and hen i wiche o he Augu conrac. Conrac for earlier mauriy are no ued due o heir high idioyncraic volailiy. Thi high idioyncraic volailiy repreen he Samuelon effec for he March conrac wherea i i due o low ga orage for he April o June conrac price movemen of hee conrac in repone o demand upply and oher marke hock are no linked o one anoher hrough available orage. In he middle of April he opimal conrac for hedging purpoe wiche from he Augu o dian conrac a he Augu conrac become ubjec o he mauriy effec. Thi raniion i raher quick wih he Augu conrac replaced by he December conrac by mid May. The Sepember hrough November conrac are ued only for a hor duraion becaue of heir high idioyncraic variance. The nex raniion ake place in he middle of Augu when he opimal porfolio wiche o he March conrac a idioyncraic variance of he December conrac increae rapidly. The January and February conrac are no ued in hi raniion becaue a large hare of heir price variaion i conrac pecific. The March and April conrac are opimal 8

only for a hor period becaue price movemen of hee wo conrac are conrac pecific due o he low invenory applicable during hoe wo monh. Figure 6a plo he opimal hedge raio a a funcion of he dae wihin he ga year. A hown in 3 he opimal hedge raio i imply he raio of he covariance beween he nearby and he fuure conrac o he variance of he fuure conrac included in he porfolio. In he figure he opimal hedge raio i alway above one indicaing ha he covariance alway exceed he variance of he fuure conrac. Thi i becaue he fuure conrac included ino he opimal porfolio are a lea hree monh away from delivery and hence diplay lile variaion a dominan hare of which reflec he common facor. Wheher a hedge raio above uniy make ene i a queion for he whole heory of opimal hedging. The opimal hedge raio i paricularly high in December and January during which he price variance of he nearby conrac i largely aribuable o he common facor yielding large covariance. Even hough he nearby fuure conrac exhibi high price volailiy from Sepember o November he opimal hedge raio i only moderae in hee monh becaue much of he price variaion i conrac pecific and hence ha low covariance. Figure 6b plo he opimal hedge raio for he porfolio including he econd poiion conrac which i he opimal oluion according o he one and wo facor model conidered by Manoliu and Tompade 00. 5 Two obervaion become clear wih hi comparion of he wo hedging raegie. Fir he opimal hedge raio i ubanially lower for he porfolio including he econd poiion han for he opimal porfolio implied by he POTS model. Thi i imply becaue he hare of he price variance explained by he common facor i maller for he econd poiion han for more dian conrac. Second he opimal hedge raio decreae a he conrac approache mauriy. Thi i again becaue he hare of he price variance explained by he common facor decreae due o he Samuelon effec. Thee reul are peculiar o he mean 9

reverion proce aumed for he underlying facor in he model of Manoliu and Tompade 00. Thi paricular ochaic proce aure ha he facor loading increae monoonically a he conrac approache i mauriy dae wherea he variance of he idioyncraic error by pecificaion i conan over he enire horizon. Conequenly he price correlaion mu alway be he highe for he wo conrac wih he minimum diance in heir mauriy dae. Figure 7a illurae he variance of he opimal porfolio relaive o he variance of he unhedged porfolio i.e. he oal variance of he nearby fuure price. The figure how ha he opimal hedging raegy implied by he eimaed POTS model reduce price rik ubanially. The variance i reduced o le han 40% of he variance of he nearby fuure excep ha i i above 40% in Ocober November and February. The magniude of variance reducion i mall in hee hree monh a he nearby fuure price he November December and March conrac are inherenly very volaile and heir price volailiy i conrac pecific. Quie noiceably he variance of he opimal porfolio i below 35% of he variance of he nearby fuure price for December and January during which he nearby fuure January and February conrac price are very volaile. For each of he monh he minimum variance aained by he opimal porfolio i higher oward he end of each monh imply becaue he Samuelon effec raie he conrac pecific volailiy of he nearby fuure. Figure 7b compare he opimal hedging raegy implied by he POTS model wih he raighforward raegy of uilizing he econd poiion conrac. The figure indicae ha he raegy baed on he POTS model aain he porfolio variance ha i abou 5 o 45 percen below he variance of he porfolio uilizing he econd poiion conrac. Thi i becaue he fuure conrac included in he former raegy i le ubjec o conrac pecific variaion han in he econd poiion conrac. 0

The opimal hedging raegie for a monhly horizon yield eenially he ame reul a hoe for a daily horizon. Three ummer conrac June July and Augu and he December conrac are he mo commonly included ino he opimal porfolio wherea he re of he fuure conrac are le common wih he Ocober and November conrac ued in April and May repecively and he January and April conrac in Augu and Sepember. The opimal hedging raio and he minimum variance aained by he opimal conac are alo imilar o hoe for daily hedging raegy. They are almo idenical o he monhly average of heir correponding value for daily hedging horizon price rik i reduced o half he ize of price variance of he nearby conrac and he hedge raio range beween. and lighly above.0 due o a mall variance of he fuure conrac included in he porfolio. 5. CONCLUSION We examine he volailiy dynamic of NYMEX naural ga fuure price uing he parially overlapping ime erie POTS model of Smih 005. The eimaed POTS model reveal ha he NYMEX naural ga fuure price exhibi ime o mauriy effec and rong eaonal variaion in heir price volailiy volailiy rapidly increae in he la hree monh of rading period and i higher for winer conrac han for pring and ummer conrac. In addiion our analyi reveal ha he perience of price hock and hence he correlaion beween daily price change in concurrenly raded conrac exhibi ubanial eaonal and cro ecional variaion. Specifically price volailiy i relaively high in wo rading period: early November o mid January for he January o April conrac and early May o Sepember for all conrac mauring before he following March. Such volailiy dynamic are cloely relaed o he eaonal paern of he US naural ga orage in a way conien wih he heory of orage.

The depiced porrai of naural ga price volailiy dynamic implie ha a rader in need of hedging price rik hould cro hedge wih a fuure conrac of a lea hree monh o mauriy o avoid high conrac pecific volailiy in nearby conrac. In addiion hey hould include in heir porfolio he December conrac o hedge again po price rik during pring and ummer monh and eiher of he June July and Augu conrac in winer monh. The opimal hedge raio i high ranging from. o lighly above.0 becaue he price of he fuure conrac exhibi much maller movemen han he nearby conrac while hey hare much informaion regarding underlying marke condiion. Thee reul ugge ha he previou udie of he po fuure price relaionhip and he dynamic of naural ga fuure price are ubjec o mipecificaion bia in he variance rucure of he diurbance erm in heir regreion model. In paricular model of commodiy dynamic hould allow more flexible pecificaion in paricular eaonal and cro ecional variaion in he facor loading and he variance of he idioyncraic error. Alo he analyi of po fuure price relaionhip hould allow eaonal and cro ecional variaion in he variance of he diurbance erm. The aumpion of a conan more pecifically zero bai i clearly inappropriae for he co of carry i no conan for a orable commodiy wih rong eaonaliy in demand and/or upply. The opimal hedging raegie implied by hee mipecified model are noiceably ineffecive wih he variance of he porfolio reurn 8 o 80% higher han he minimum variance aained by he hedging raegy uggeed by he eimaed POTS model.

ENDNOTE Suenaga 005 illurae ha hi difference in he level of complexiy of he ipulaed variance rucure of he laen facor and idioyncraic error ha a ubanial impac on he oher model parameer epecially he rik premium. We alo eimaed he model allowing for a nonzero mean in he log price change. The eimae of hi mean parameer i very mall in value and our main reul are unaffeced. 3 Thi approximae EM mehod involve ieraion of he following hree ep: obain he prediced value of he laen facor and GARCH condiional variance hrough Kalman Filer maximize he expeced complee daa likelihood wih repec o he pline parameer condiional on he prediced value of laen facor and GARCH condiional variance from he fir ep and 3 eimae he GARCH parameer holding he pline parameer a he value from he ep. 4 One can alo evaluae he opimal hedge raio uing he model implied condiional variance. Unlike he uncondiional variance ued in he main ex he condiional variance depend on he hiorical movemen of he naural ga fuure price. We only preen evaluaion baed on he uncondiional variance becaue our objecive here i o draw a general implicaion abou he need o model eaonaliy. 5 See Table 4 and 5 on page 38 and 39 of Manoliu and Tompaid 00. Number in Figure 6b were calculaed by uing he volailiy eimae of he POTS model in 3. 3

BIBLIOGRAPHY Hamilon J. D. 994. Sae Space Model in Engle R. F. and McFadden D. L. ed. Handbook of Economeric Vol. 4. Elevier Amerdam. Lien D. and Roo T. H. 999. Convergence o he long run equilibrium: The cae of naural ga marke Energy Economic : 95 0. Lucia J. J. and Schwarz E. S. 00. Elecriciy price and power derivaive: Evidence from he Nordic Power Exchange Review of Derivaive Reearch 5: 5 50. Manoliu M. and Tompaidi S. 00. Energy fuure price: Term rucure model wih Kalman filer eimaion Applied Mahemaical Finance 9: 43. Movaagh N. and Modjahedi B. 005. Bia in backwardaion in naural ga fuure price Journal of Fuure Marke 5: 8 308. Modjahedi B. and Movaagh N. 005. Naural ga fuure: Bia predicive performance and he heory of orage Energy Economic 7: 67 637. Roo T. H. and Lien D. 003. Can modeling he naural ga fuure marke a a hrehold coinegraed yem improve hedging and forecaing performance? Inernaional Review of Financial Analyi : 7 3. Schwarz E. S. 997. The ochaic behavior of commodiy price: Implicaion for valuaion and hedging Journal of Finance 5: 93 973. 4

Smih A. 005. Parially overlapping ime erie model: A new model for volailiy dynamic in commodiy fuure Journal of Applied Economeric 0:405 4. Suenaga H. 005. Analyi of po forward price relaionhip in he rerucured elecriciy marke Ph.D. Dieraion Univeriy of California Davi California. Wall W. D. 995. An economeric analyi of he marke for naural ga fuure Energy Journal 6: 7 83. William J. C. and Wrigh B. D. 99. Sorage and Commodiy Marke. Cambridge Univeriy Pre New York. 5

Table. Maximum Likelihood Eimae of GARCH Parameer Coefficien SE raio GARCH Parameer Wih repec o 0 Wih repec o α 0.09 0.0963 0.9464 α + β 0.9006 0.86 7.596 0.8383 Log Likelihood BIC.8E+08 359.38 6

Table. Proporion of he Variance Explained by a Common Facor Overall 0.857 By conrac 0.788 0.835 3 0.78 4 0.8580 5 0.8643 6 0.8605 7 0.867 8 0.8834 9 0.860 0 0.84 0.8088 0.7347 7

Figure. NYMEX naural ga elemen price a of April 003 5.5 003 004 5.0 Price $/MBT 4.5 004 005 005 006 009 006 007 4.0 008 009 007 008 3.5 4 5 6 7 8 9 0 3 Delivery Monh 8

Figure. Variance of daily log price change implied by he eimaed POTS model 0.0035 0.0030 0.005 0.000 3 0.005 0.000 0.0005 4 5 6 7 8 9 0 0.0000 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 3 Trading dae monh 9

Figure 3. US naural ga underground orage Monhly average of working ga for 976 005 3500 Monhly Average Working Ga Sorage Billion c 3000 500 000 500 000 500 0 3 4 5 6 7 8 9 0 Monh 30

Figure 4. Share of price variaion explained by he common facor 00% 90% 80% 70% 60% 50% 4 5 6 7 8 9 0 40% 30% 3 0% 0% 0% 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 3 Trading dae monh 3

Figure 5. Delivery monh of he fuure conrac included in he opimal porfolio Monhly Daily 0 8 6 4 0 4 5 6 7 8 9 0 3 Time of Enry Monh 3

Figure 6. Opimal hedge raio a Porfolio uggeed by he eimaed POTS model Monhly Daily.5.0.5.0 0.5 0.0 4 5 6 7 8 9 0 3 Time of Enry Monh b Porfolio including he econd poiion conrac Monhly Daily.5.0.5.0 0.5 0.0 4 5 6 7 8 9 0 3 Time of Enry Monh 33

Figure 7. Minimum variance aained by he opimal porfolio a Relaive o he variance of unhedged porfolio Monhly Daily 00% 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% 4 5 6 7 8 9 0 3 Time of Enry Monh b Relaive o he variance of porfolio including he econd poiion conrac Monhly Daily 00% 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% 4 5 6 7 8 9 0 3 Time of Enry Monh 34