THE STOCHASTIC SEASONAL BEHAVIOR OF THE NATURAL GAS PRICE

Transcription

1 E SOCASIC SEASONAL BEAVIOR OF E NAURAL GAS PRICE. Irouco Aré García ra Fracco Javr Poblacó García Prlmary Draf Ju-5 Acamc a pracor hav b rcly payg ao o a facal grg u: h valuao a hgg of commoy cog clam. h ochac bhavor of commoy prc play a cral rol h ara. Early u o h ochac bhavor of commoy prc aum ha po prc follow a gomrc Browa moo [Bra a Schwarz 985]. owvr, h or of mol ha om urabl propr l allowg prc o r a a coa ra a xhbg coa volaly fuur prc. Emprcal vc ugg ma-rvro po prc a crag-volaly fuur prc. o olv h fcc ffr o-facor mol ha b propo h lraur [Laugho a Jacob 993 a 995, Ro 995]. Nvrhl, a mplcao of all mol ha cor a gl ourc of ucray ha fuur prc for ffr maur houl b prfcly corrla, whch f xg vc. Loog for mor ralc rul, mul-facor mol hav b vlop [Schwarz 997, Schwarz a Smh, Corazar a Schwarz 3, Corazar a Narao 3]. o of h arcl ma h paramr of hr mol for ol. owvr, h umbr of papr arg h pcfc valuao a hgg problm of aural ga cog clam ll carc. h lac of coomcal raporao a h lm orably of aural ga ma upply uabl o chag vw of aoal varao of ma. hrfor, aural ga prc ar rogly aoal. hr ar u ha a o accou h aoal bhavor of om commoy prc [Luca a Schwarz ], bu o of hm cor aoaly a a ochac facor. O of h mo clar way o vualz h aoaly h aural ga prc hrough h forwar curv. I h x char clar ha h prc of aural ga ry ub xpc o b hghr urg wr moh a lowr urg ummr moh. Prc of Naural Ga ry ub 8 Spo Prc Forwar Curv 6 4 oc-89 ul-9 abr-9-9 oc-9 u-93 mar-94 c-94 p-95 u-96 mar-97 c-97 p-98 may-99 fb- ov- ago- may- fb-3 ov-3 ago-4 abr-5-6 oc-6 ul-7 abr-8-9 oc-9 ul- mar- $/Bu

2 I alo pobl o oc ha h po prc horcal r h hghr prc hav b rach wr whl h lowr prc appar ummr. I h papr w vlop a gral m-facor mol ha cor h aoaly a a ochac facor. W wll h apply hr, four a fv-facor mol o all aural ga ry ub fuur corac ra a NYE. h Kalma flr mhoology wll b u o ma h paramr of h mol. Ug h ma paramr, w aaly h mol goo of f o h rm rucur of fuur prc a volal. h gral m-facor mol aum ha h log-po prc um of m ochac facor o-aoal a m aoal. h o-aoal facor wll b h facor of h mol mo abov. h aoal facor wll b rgoomrc compo gra by ochac proc.. Naural Ga ry ub Prc Saoaly I h char abov m clar ha h prc of aural ga ry ub aoal wh a o yar pro. A vry mpl algorhm ca b vlop o ura mor clarly. S ha b ablh a h po prc a Y h cr movg avrag a yar of S. If {S },,3, h po prc m r wh mohly frqucy, h Y.5S 6 S 5 S 4 S S -5.5S -6 /. L τ S /Y, m for moh m h avrag of τ for moh m a h calg facor for moh m r m m /.... I clar ha r r r, a alo clar ha f r m grar ha o h prc moh m ar hghr ha h avrag prc a f r m l ha o h prc moh m ar l ha h avrag prc. A ca b h x char, h po prc a h forwar curv calg facor pr h am par: I h wr moh hy ar hghr ha o a h ummr moh hy ar l ha o. h clar vc of aoaly h prc of aural ga ry ub. Spo Prc Scalg Facor Ja Fb ar Apr ay Ju Jul Aug Sp Oc Nov Dc

3 . Forwar Curv Scalg Facor Ja Fb ar Apr ay Ju Jul Aug Sp Oc Nov Dc A mor ophca aaly ca b vlop o compl h rul. h pcrum of a aoary proc f a h Fourr raformao of h aboluly ummabl auocovarac fuco of h proc. I ca b prov W ha h aoaly appar h pcrum a pa vral frquc a f h aoal proc a rmc proc h pa appar h pcrum a cr way whra f h aoal proc a ochac o h pa appar h pcrum a couou way. h mpl ha, aa aaly, a harp p h ampl pcrum may ca a pobl rmc cyclcal compo, whl broa pa of mply a ormc aoal compo. I ca b prov W ha a gral aoary ARAp,q mol: ψ p LS θ q L, whr wh o wh varac, h pcrum gv by: θ q f w π ψ p w w I o crazy o aum ha h po prc of aural ga ry ub follow a AR wh yarly ochac aoaly: -φl-φl S, whr L h lag fuco a wh o wh varac. I h ca h pcrum : f w π Φ Φ cow co w hu, for Φ>, ohr ha a pa a w, h pcrum alo xhb pa a aoal harmoc frquc π/ for,, 3, 4, 5 a 6 a rough a frquc π-/ for,, 3, 4, 5 a 6. I h x char h pcrum of h mol for φ.9 a Φ. horcal compar wh h acual pcrum of h po prc of h aural ga ry ub. Spcrum of h Spo Prc of h Naural Ga ry ub horcal Acual Frcucy 3

4 I m clar h char ha h prc of aural ga ry ub ha a yarly ochac aoal compo. 3. Gral ol A mo abov, h gral m-facor mol aum ha h log-po prc um of m ochac facor o-aoal a m aoal. m No-aoal facor a ar gog o b h am a h o ha Corazar a Narao u hr papr Corazar-Narao 3. h ochac ffral quao SDE of h facor ar: µ,, 3,..., - 3 κ Each aoal facor gog o b moll hrough a rgoomrc compo. h rgoomrc SDE a complx o: a a R a bg a a complx facor a, R a complx umbr R R R a W a a complx Browa moo W a W W. I clar ha whou lac of graly pobl o aum ha W a W ar p. L R b xpr polar R θ whr h moul a θ h pha. I h appx A h proof ha θ guhabl. hrfor h SDE : a a a a quallg ral compo wh ral compo a magary compo wh magary compo h prvou quao follow wo SDE for ach aoal facor: 4 whr W a W ar p.,, 3,..., m 5 o valu fuur a opo corac h r-ural vro of h mol ha o b vlop. h SDE for h facor ur h quval margal maur ca b xpr a: µ κ,, 3,..., - 3 4

5 5 4,, 3,..., m 5 whr,, y ar h r-prmum of ach facor a W, W, W a W ar h Browa moo ur h quval margal maur of ach facor. h Browa moo ar corrla xcp W a W whch ar ucorrla,, 3,..., m. From quao??, a m h cooal rbuo of a m ur h r-ural maur ormal. h, from h propr of h log-ormal rbuo, ow a m, h fuur prc of h commoy wh maury a ha o b: / xp, Var E F whr E a Var ar h ma a varac of cooal o h formao avalabl ur h quval margal maur. Followg h mhoology xpo appx B ay o g h xpro for E a Var. h xpro for a gral mol wh m facor ar oo log o wr h arcl. I h followg co ar h xpro for,,3 a m. I h gral mol ca, f ach aoal facor ucorrla wh h ohr facor h: [ ] m m E co co ' µ > m Var 3 hrfor: [ ] > / co co, l m m m F µ 4

6 4. Parcular ol A a bfor, hr ar gog o b hr parcular mol. Du o h aaly vlop co, h parcular mol ar gog o hav oly o aoal facor a xpc ha h ma pha φ b o yar. I all h ca h o-ochac facor ar gog o b h am facor u h orgal mol, Schwarz 997 for h hr-facor mol, Schwarz-Smh for h four-facor mol a Schwarz-Corazar 3 for h fv-facor mol u o h aoal facor h o-facor mol bcom a hr-facor mol, h wo-facor mol bcom a four-facor mol a h hr-facor mol bcom a fv-facor mol. h hr-facor mol I h mol h log-po prc h um of wo ochac facor: a horrm compo a a aoal compo, a o rmc facor: a log-rm compo. 6 h hr ochac facor gog o b h ohr aoal facor whch complm o. h SDE of h facor ar: µ 8 κ 7 9 h o facor mol Schwarz 997 ha oly o SDE, whl hr h SDE parag o wo quao: a ochac ffral quao - h quao 7???- a a rmc quao - h quao 8??? - bu ohg chag, h ohr way o wr h am SDE. h r-ural SDE ar: µ κ 3 4 Ug h mhoology crb h appx B, ow a m, h fuur prc of h commoy wh maury a ha o b: 6

7 l [ F, ] co µ.5 co /.5 { co } co h four-facor mol / / I h mol h log-po prc h um of hr ochac facor: a log-rm compo, a hor-rm compo a a aoal compo. 5 h fourh ochac facor gog o b h ohr aoal facor whch complm o. h SDE of h facor ar: µ 7 6 κ 8 9 h quao 6??? a 7??? ar cal o quao a Schwarz-Smh. h r-ural SDE ar: µ ' κ 3 whr µ µ - h r-ural rf. I h ca, ow a m, h fuur prc of h commoy wh maury ha o b: 7

8 l [ F, ] co µ '.5.5 co /.5 / { co } co / co / h fv-facor mol I h mol h log-po prc h um of four ochac facor: a logrm compo, wo hor-rm compo a a a aoal compo. 5 h ffh ochac facor gog o b h ohr aoal facor whch complm o. h SDE of h facor ar: µ 7 6 κ κ I appx C, ca b ha h o-aoal par of h mol quval o h hr facor mol propo Corazar-Schwarz 3. h r-ural SDE ar: µ ' κ κ 3 whr µ µ - h r-ural rf. I h ca, ow a m, h fuur prc of h commoy wh maury ha o b: 8

9 9 [ ] co co /.5 co co / /.5 co / / / co / /.5.5 ' co, l µ F 5. Emao mhoology h Kalma Flr A prvou u, h ma ffculy mag h mol paramr ha h facor or a varabl ar o rcly obrvabl a mu b ma from po a/or fuur prc. Iuvly, h o-aoal facor log rm a or-rm facor ar gog o b ma ba o h rlaohp bw log-maury fuur a hor-maury fuur or po prc a h aoal facor ar gog o b ma ba o h rlaohp bw fuur who maury occur ffr moh. h formal way o o h hrough h Kalma flrg mhoology. h mhoology abl h calculao of h llhoo of a aa r gv a parcular of mol paramr a a pror rbuo of h varabl whch prm h mao of h paramr ug maxmum llhoo chqu. h Kalma flr mhoology a rcurv mhoology ha ma h uobrvabl m r, h a varabl, ba o obrvabl m r Y whch p o h a varabl. h rlaohp bw h obrvabl m r a h a varabl crb hrough h maurm quao: Y η,, N whr Y a vcor, a vcor, a m marx, a m vcor m h umbr of a varabl or facor ug h mol a η a vcor of rally ucorrla Gaua urbac wh ma cro a covarac marx. h voluo of h a varabl crb hrough h rao quao: c ψ,, N

10 whr c a m vcor, a mm marx a ψ a m vcor of rally ucorrla Gaua urbac wh ma cro a covarac marx Q. I h raoal vro of h mhoology cary ha hr o b ay mg po h aa a h lgh of vcor Y b p of. owvr, a mprov vro of h mhoology ha b vlop Corazar- Narao 3 o hal wh compl aa a vcor Y who lgh p o. I Corazar-Schwarz 3 a alrav o h Kalma flr mhoology ha b vlop o ma h mol paramr a h a varabl. h chqu a mpl o whch oly a prah o b mplm. Dcrzao o apply h Kalma flr chqu, or h Corazar-Schwarz 3 chqu, o ma h paramr of h mol xpo abov, cary o cr hm. h mhoology whch gog o b follow o o gog o b ba o h mhoology u o g h fuur prc xpro vlop appx C. I h appx prov ha f h vcor of facor of a parcular mol follow h x SDE quao: b A Ω h : Ω b a alo prov ha ormal rbu wh: E Var b Ω Ω hu, f h ffr bw h curr pro a h al pro o pro m, h prvou quao ca b wr: c ψ,, N whr c b, a ψ a m vcor of rally ucorrla Gaua urbac wh ma cro a covarac marx Q Ω Ω. h quao gog o b h rao quao of ach mol.

11 h maurm quao u h xpro of h log-fuur prc rm of h facor. o avo h alg wh a gra amou of paramr, h covarac marx gog o b aum agoal a h agoal gog o b compou oly for a fw ffr lm. Followg wha wa vlop h co h cr vro of h hr mol ar: hr-facor mol rao quao: c ψ,, N whr, c µ, co co a Var co co co co / ψ aurm quao: Y η,, N whr F F Y l l, A A, co co /4 / co /.5 µ A co co Four-facor mol rao quao:

12 c ψ,, N whr, c µ, co co a Var co co / co co co / co / / ψ aurm quao: Y η,, N whr F F Y l l, A A, µ co co co / /.5 / co /.5.5 ' A co co Fv-facor mol rao quao: c ψ,, N

13 3 whr, c µ, co co a co co co co / / / ψ Var aurm quao: Y η,, N whr F F Y l l, A A, µ co co co / /.5 co co / / /.5 / co / /.5.5 ' A co co

14 6. Emprcal Rul h aa u o ma h paramr of h mol co of wly obrvao of prc for NYE aural ga fuur corac wh maury bw o a 7 moh. Fuur from 3// o 7//5 hav b u wh a oal of 64 obrvao. A mplf vro of h hr facor mol whr h corrlao a h r-prmum ar qual o zro a h varac-covarac marx of η agoal ha x paramr o b ma µ,, φ,, a η. Ug h Kalma flr mhoology wh fuur corac whch maury ar, 3, 5, 7, 9,, 4, 6, 8,,, 5, 7, 9, 3, 33, 35, 37, 39, 4, 43, 45, 47, 49, 5, 54, 56, 58, 6, 6, 64, 67, 69, 7 moh ha b fou ha: µ 7.7%,.3475, φ 6.368,.98 a.337 bg h log-llhoo qual o I clar ha h aoal pro o yar φ 6.368, whch co wh h rul of co. h x char compar h ral fuur corac volaly rm rucur of volaly wh ha whch h mol mpl for. Volaly % 9% 8% 7% 6% 5% 4% 3% % % % Fuur Corac Volaly aury oh Ral hr Facor ol h four facor mol aumg oly ha h varac-covarac marx of η agoal ha four paramr o b ma µ,, φ,,,,,,, µ,,, a η. Procg l h hr facor mol, ha b fou ha: µ 4.7%,.5738, φ 6.3,.966,.888,.9, , -.358, -.864, µ -.3%%,.35, -.,.3 a η.334 bg h log-llhoo qual o I h ca alo clar ha h aoal pro o yar φ 6.3, whch co wh h rul of co. h x char compar h ral fuur corac volaly rm rucur of volaly wh ha whch h mol mpl for. 4

15 7% Fuur Corac Volaly Volaly 6% 5% 4% 3% % Ral Four Facor ol % % aury oh h rao why h horcal volaly o o f h ral o rla o h fac ha h volaly of h fuur who maury o moh much mor volal ha h ohr a aumg ha h prco rror of all of hm ha h am varac. Elmag h fuur who maury o moh a mplf vro of h four facor mol h rul bcom: 6% Fuur Corac Volaly Volaly 5% 4% 3% % Ral Four Facor ol % % aury oh APPENDICES Appx A: Saoal facor A a abov, h ochac ffral quao of ach aoal facor : a a R a W a L R b xpr polar R θ whr h moul a θ h pha. Dfg θ W a a θ a, clar ha Wa θ a or quval:. Equallg ral compo wh ral compo a magary compo wh magary compo h la quao h: 5

16 W coθ θ W W θ coθ W From h prvou quao follow ha: W W Var Var I W W bg I h x y marx. hrfor, h pha θ guhabl. Appx B: Fuur prc mao mhoology I h gral mol h log-po prc ha b f a h um of m ochac facor o-aoal a m aoal. m h SDE for h facor ur h quval margal maur ca b xpr a: µ ' κ,, 3,..., - 3,, 3,..., m 5 whr W, W, W a W ar h Browa moo ur h quval margal maur. From h propr of h log-ormal rbuo, ow a m, h fuur prc of h commoy wh maury ha o b: { E / Var }, xp F o calcula E a Var h facor SDE hav o b olv. L b h vcor of all facor: 4 6

17 m m I a gral ca h SDE of ca b xpr a: b A Ω whr z a vcor of p Browa moo a hrfor Var ΩΩ Ω h rapo marx of Ω. I h gral mol of h papr b a A ar p of bu h mhoology gog o b vlop for a mor gral ca. ha: L: A P D P. h marx agoalzabl. hrfor ca b wr, whr D a agoal marx a D P P h xpoal of a agoal marx aohr agoal marx who agoal lm ar h xpoal of h lm of h orgal marx. Wh ha h oluo of h SDE : Ω b o mora, h gral rul for h rvao of h prouc of ochac compo ca b appl: Ω b b Ω b If A a commu, h rvav of wh rpc o : Ω A 7

18 I h ca ha A p of l h gral mol of h papr, A a commu. hu, h gral mol of h papr A. h ohr ffral h rva of a gral, hrfor: b Ω b Ω h fr ffral oly ha lm of yp, hrfor h prouc of h fr ffral m h co ffral zro. hu: A b Ω [ b Ω ] A b Ω hrfor, b Ω A lmary rul of h ochac calculu a ha J rmc fuco, ormally rbu wh:, whr J a hu, ormally rbu wh: E J Var J J J Io omry E Var b Ω Ω h: A l m, whr l L L, le lvar l E Var Appx C: Fv-facor mol rformulao I h fv-facor mol h log-po prc : 8

19 9 h facor SDE ca b xpr a: µ Dfg y κ a v clar ha: y y y κ κ κ κ κ κ κ κ κ v v L µ v v a µ, h ca: v v v v µ 3 µ 4 I ay o ha: v y 5 hrfor: y v v y v y µ µ 6 whr hu, h mol ca b xpr rm of h w varabl. h w facor ar:, y, v, a. h SDE of h w facor ar: y v y y v v µ 3

20 8 9 A ow obvou ha h mol a gralzao of h hr-facor mol vlop Corazar-Schwarz 3.