Natural Gas Storage Valuation. A Thesis Presented to The Academic Faculty. Yun Li

Transcription

1 Naural Gas Sorage Valuao A Thess Preseed o The Academc Faculy by Yu L I Paral Fulfllme Of he Requremes for he Degree Maser of Scece he School of Idusral ad Sysem Egeerg Georga Isue of Techology December 2007

2 Naural Gas Sorage Valuao Approved by: Dr. Shje Deg, Advsor School of Idusral ad Sysem Egeerg Georga Isue of Techology Dr. Davd M. Goldsma School of Idusral ad Sysem Egeerg Georga Isue of Techology Dr. Sephe Demko School of Mahemacs Georga Isue of Techology Dae Approved: November 2, 2007

3 ACKNOWLEDGEMENTS I graefully ackowledge he advce ad suppor of my hess commee members. I am debed o Dr. Shje Deg for coually challegg me o grow professoally ad explore ew ad ufamlar felds, ad o Dr. Davd Goldsma ad Dr. Sephe Demko for her srucve suggesos. I also apprecae he help o my courses ad research from Dr. Deg ad Dr. Demko durg my say a Georga Isue of Techology. Mos of he work preseed hs hess s based o he research coduced a FPL Eergy where I served as a Seor Quaave Aalys uder Dr. Sephe Kerma s supervso durg he year of I am debed o Dr. Kerma for hs ecourageme, advce, ad suppor. I wsh o ackowledge he valuable asssace ad dscussos of my co-workers a FPL eergy. I apprecae Dr. Feg Ya, a Seor Quaave Aalys, for sharg hs mahemacal ad MATLAB sklls. I apprecae Mr. Mar Gajewsk, he Maager of Gas Tradg Desk, for sharg hs radg experece ad hs vews o aural gas marke. I also apprecae he dscussos wh Mr. Dael Kaufma ad Dr. Davd Emauel. I grealy apprecae Mr. Rya Clark ad Mr. Dael Kaufma s careful edg ad proofreadg of my hess.

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES.. v LIST OF FIGURES v SUMMARY... v CHAPTER : INTRODUCTION CHAPTER 2: THE BASICS OF THE NATURAL GAS MARKET IN THE US. 4 CHAPTER 3: SIMULATE FORWARD AND SPOT PRICES.. 7 CHAPTER 4: VALUATION METHODOLOGIES Problem Descrpo Moe Carlo wh Sochasc Dual Dyamc Programmg Moe Carlo wh Ordary Leas Square Prcpal Compoe Aalyss Revsed Implemeao Procedure. 23 CHAPTER 5: RESULTS AND DISCUSSIONS Hsorc Daa ad Parameer Esmao Resuls The Sorage Corac The Resuls by he Mehod of Irsc Rollg wh Spo ad Forward The Resuls by Spo Oly Mehods Valuao wh a Chagg Bd-ask Spread Furher Dscussos o he Mehodologes.. 48 CHAPTER 6: CONCLUSIONS 5 REFERENCES.. 54 v

5 LIST OF TABLES Table 5. Covarace Marx of he Spo ad Forward Prces 30 Table 5.2 Correlao Marx of he Spo ad Forward Prces 3 Table 5.3 Egevecors of he Covarace Marx of he Spo ad Forward Prces. 32 Table 5.4 Egevecors of he Covarace Marx of he Forward Prces Table 5.5 Egevalues of he Covarace Marx of he Spo ad Forward Prces. 33 Table 5.6 Egevalues of he Covarace Marx of he Forward Prces 33 Table 5.7 Table 5.8 Table 5.9 Table 5.0 Table 5. Table 5.2 Table 5.3 Table 5.4 The Value of he Sorage Corac Esmaed by he Mehod of Irsc Rollg wh Spo ad Forward o Jue 28, The Value of he Sorage Corac Esmaed by he Mehod of Irsc Rollg wh Spo ad Forward o Jue 29, The Value of he Sorage Corac Esmaed by he Mehod of Irsc Rollg wh Forward Oly The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Ordary Leas Square Regresso o Jue 28, The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Ordary Leas Square Regresso o Jue 29, The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Sochasc Dual Dyamc Programmg o Jue 28, The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Sochasc Dual Dyamc Programmg o Jue 29, The Valuao of he Sorage Corac wh Chagg Bd-ask Spread 48 v

6 LIST OF FIGURES Fgure 5. Hsorc Hery Hub Naural Gas Spo Prce.. 28 Fgure 5.2 NYMEX Naural Gas Prces o Jue 28 ad Jue 29, Fgure 5.3 Smulaed Spo Prces from July 07 o Jue Fgure 5.4 Sample Opmzao Resul o Day ad Day Fgure 5.5 The Hsogram of Sorage Value from Oe Ru of 000 Smulaos v

7 SUMMARY I hs hess, oe mehodology for aural gas sorage valuao s developed ad wo mehodologes are mproved. The all of he hree mehodologes are appled o a sorage corac. The frs mehodology s called rsc rollg wh spo ad forward, whch akes boh he spo ad forward prces o accou he valuao. Ths mehod s based o he radg sraegy by whch a rader locks he spo ad forward posos by solvg a opmzao problem based o he marke formao o he frs day. I he followg days, he rader ca oba added value by adjusg he posos based o ew marke formao. The sorage value s he sum of he frs day s value ad he added values he followg days. The problem ca be expressed by a Bellma equao ad solved recursvely. A crucal ssue he mplemeao s how o compue he expeced value he ex perod codoed o he formao curre perod. Oe way o compue he expeced value s Moe Carlo smulao wh ordary leas square regresso. However, f all of he sae varables, spo, ad forward prces are corporaed he regresso here are oo may erms, ad he regresso becomes ucorollable. To solve hs ssue, hree rsk facors are chose by performg prcple compoe v

8 aalyss. Dmeso of he regresso s grealy reduced by oly corporag he hree rsk facors. Boh he secod mehodology ad he hrd mehodology oly cosder he spo prce he valuao. The secod mehodology uses Moe Carlo smulao wh ordary leas square regresso, whch s based o he work of Booger ad Jog (2006). The hrd mehodology uses sochasc dual dyamc programmg, whch s based o he work of Brgedal (2003). However, boh mehodologes are mproved o corporae bd ad ask prces. Prce models are crucal for he valuao. Forward prces of each moh are assumed o follow geomerc Browa moos. Fuure spo prce s also assumed o follow a geomerc Browa moo bu for a specfc moh s expecao s se o he correspodg forward prce o he valuao dae. Sce he smulao of spo ad forward prces s separaed from he sorage opmzao, alerave spo ad forward models ca be used whe ecessary. The resuls show ha he value of he sorage corac esmaed by he frs mehodology s close o he marke value ad he value esmaed by he Facal Egeerg Assocaes (FEA) provded fuco. A much hgher value s obaed whe oly spo prce s cosdered, sce he hgh volaly of he spo curve makes freque poso chage profable. However he realy raders adjus her posos less frequely. v

9 CHAPTER INTRODUCTION Naural gas sorage valuao s a complcaed opc he feld of asse ad dervave valuao. Oe ca hk of aural gas sorage as a dyamc baske of caledar spreads, cludg o oly he spreads amog forwards, bu also he spreads bewee spo ad forwards. O he oe had, he operao of aural gas sorage s subjec o may cosras, whch make he valuao of he sorage more complcaed ha a pure facal srume. O he oher had, s dffcul o smulae he spo ad forward curves of aural gas, especally whe oe was o ake boh spo ad forwards o accou a he same me. Therefore, oly he spo prce or forward prces sead of boh are used mos of he leraure o aural gas sorage valuao. There s more leraure usg he spo prce oly, such as Booger ad Jog (2006), Che ad Forsyh (2006), Jog ad Wale (2004), Thompso, Davso, ad Rasmusse (2003), Brgedal (2003), ad Weso (2002). Alhough s challegg o develop a model ha ca capure he shor ad log-erm dyamcs (such as mea-reverg ad jumps) of aural gas spo prce, he spo prce mehod has some advaages. There are very lmed sae varables he sochasc corol problem, usually oly he spo prce ad sorage level, herefore s relavely easy o fd a decso rule (jec or whdraw) ad

10 esmae he value of a sorage. I comparso o forward prces, spo prces are more volale, herefore some auhors, cludg Booger ad Jog (2006), argue ha he value prced by spo prces s he rue value, whch s usually greaer ha he value prced by forward prces. Some auhors use forward prces aural gas sorage valuao, such as Eydelad ad Wolyec (2002), Gray ad Khadelwal (2004), ad Blaco, Soroow ad Sefszy (2002). A eresg po o oe s all of he auhors are workg or have experece he eergy radg dusry. All of hem apply he rsc rollg sraegy, where a rader locks he posos of aural gas forward coracs ad acheves he rsc value of he sorage gve he forward curve ad cosras o he frs day. Durg he followg days, he rader ca adjus he posos based o ew forward curve ad cosras o oba more values. Ths mehod seems more realsc, sce caledar spreads are frequely raded whe sorage s raded. Aoher advaage of forward prce mehod s ha he mohly forward prce curve s relavely easy o smulae. However, usually requres more compuer resources o coduc he opmzao every sage. For example, eve f we oly cosder 2 forward coracs, here are 66 spreads we eed o cosder he opmzao. If we cosder 24 coracs, he he umber of spreads creases o 276. The reasos ha mos auhors use spo oly or forwards oly are o oly because he sochasc corol problem becomes more complcaed whe boh are cosdered, bu also due o he fac ha s dffcul o make he spo prce ad forward prces 2

11 cosse. I oher words, s a challege o develop spo ad forward models ad sasfy he o-arbrage codo. Theorecally, oe ca develop spo models ad calbrae hem by he forward prces o make he spo prce cosse wh he forward prces. Che ad Forsyh (2006) calbraed 3 dffere aural gas spo models by he forward prces. However, he sably of he parameers s uclear sce he sgfca levels are o provded her paper. Eve for he bes model (he regme-swch GBM model), s ably of predcg forwards of he model eed furher verfcao, sce oly oe observao of he marke prces s compared wh he predced prces. Aoher problem calbrag spo models by forwards s ha usually dffere resuls for he parameers are obaed whe he spo model s calbraed by he spo prces. Sce a sorage rader ca rade boh forwards ad spo a he same me realy, we wll compare he valuao resuls by varous mehods. We combe he forwards ad spo a smple bu praccal way he sorage valuao hs paper. Ths hess s orgazed as follows: Chaper 2 provdes a roduco o he bascs of aural gas sorage marke he US. Chaper 3 descrbes how o coec he spo prce ad forward prces ad her smulao. Chaper 4 roduces he mehodologes for sorage valuao. Chaper 5 descrbes he daa ad parameer esmaos ad he resuls. Chaper 6 cocludes. 3

12 CHAPTER 2 THE BASICS OF THE NATURAL GAS MARKET IN THE US Naural gas marke he US has s ow characerscs. Naural gas prces are locao based prces. Hery Hub prce s he bechmark ad almos all of he prces are derved by addg adders o he Hery Hub prce. Hery Hub s called he backboe ad he adders are called he bases. The spo marke s raded every busess day ad seled o he ex busess day ad o-busess days f here are ay before he ex busess day. For sace, aural gas raded o July 26, 2007 (Thursday) s seled o July 27, 2007 (Frday), ad aural gas raded o July 27, 2007 s seled o July 28 (Saurday), July 29 (Suday), ad July 30, 2007 (Moday). The radg dae s also called he rasaco dae, ad he seleme dae(s) s (are) called he flow dae(s). The forward coracs are wdely raded o he New York Mercale Exchage (NYMEX) ad some oher o-le markes, such as ICE. NYMEX coracs are based o Hery Hub prce. The prces of 72 forward coracs are avalable o every busess day, bu here are oly abou 24 or less frequely raded coracs. The radg volume of aural gas forwards has a seasoal paer. Mos frequely raded coracs are promp moh (he eares moh from spo) ad a few followg mohs, Ocober, Jauary, March, ad Aprl. Aoher feaure of forward coracs s 4

13 ha hey expre o he hrd from he las busess day of he prevous moh. For example, 2007 Augus corac expres o July 27, 2007 ad 2007 Sepember corac expres o Augus 29, Usually, a rader closes he posos before hey expre, bu a sorage rader akes he physcal aural gas. Naural gas forward coracs are seled a specal way ha a buyer receves he physcal gas a a specfc locao a equal amou o each day of he corac moh. For example, 2007 Augus corac expres o July 27, 2007, bu a buyer does o receve he gas ay day July, sead he buyer receves gas every day Augus, If a rader buys a corac, he he rader receves aural gas /3 corac every day Augus. If he rader buys 2 coracs, he he or she receves gas 2/3 corac every day. A fraco of a corac, such as a quarer, half, or hree quarers of a corac ca be raded. A aural gas corac has a eergy value of 0,000 MMBu. There are varous kds of sorages, such as depleed ol reservors, aqufers, sal cavers, ad LNG sorages. These sorages have dffere physcal characerscs. For more dealed formao, oe ca refer o he Eergy Iformao Agecy (EIA) webse of he US goverme: (November 2, 2007). A aural gas sorage corac ca have a erm from a few mohs o a few years. The corac also covers some physcal cosras ad operaoal coss, such as al workg gas capacy, maxmum workg gas capacy, maxmum daly jeco ad whdrawal raes, u jeco ad whdrawal coss. Usually, a sorage corac s also based o he aural gas prce a a specfc locao, such as Hery Hub, Houso 5

14 Shp Chael ec. Fally, lke all oher facal coracs, oe of he mos mpora parameers s he premum of he corac. Ths s he arge of our exercse. Here s a ypcal aural gas sorage corac: Term: 2//2008-6/30/202 Bass: Hery Hub Premum: $X/MMBu-moh Maxmum workg gas capacy:,000,000 MMBu ~ bllo cubc fee Ial workg gas: 0 MMBu Maxmum jeco rae: 35,000 MMBu/day Maxmum whdrawal rae: 75,000 MMBu/day Operag coss:.5% of he fuel cos o jeco As he corac dcaed, akes almos 2 weeks o whdraw ad almos a moh o jec he full amou of he sorage. So eve hough oe ca lock he posos based o a forward curve oe perod, usually ca be mplemeed oe perod, hs s dffere from a pure facal srume. Lke oher facal srumes, a rader has o pay he ask prce whe he or she was o jec gas whle he rader receves he bd prce whe he or she whdraws he gas. A sorage rader ca rade aural gas oly o busess days, bu physcal jeco ad whdrawal ca happe o every day, cludg he weekeds ad holdays. 6

15 CHAPTER 3 SIMULATE FORWARD AND SPOT PRICES If we smulae forward prces of varous mohs by a sgle model, he may facors, such as mea-reverg ad jumps, should be ake o accou. For example, he magudes ad volales of March ad Aprl coracs ca be very dffere. I hs hess, we smulae each forward corac by a specfc model. I oher words, we rea he forward coracs for dffere mohs as dffere commodes. For example, we use oe model o smulae he March corac ad use aoher model for Aprl s corac. Sce each moh has s ow forward curve, mea-reverg ad jumps ca be gored. I oher words, we expec here s a sgfca drop from March prces o Aprl prces, bu we do expec a spke or drop wh March prces or Aprl prces uder he ormal marke codos. Thus, relavely smple models ca be creaed for each forward corac. Ths s oe reaso ha we use dffere models for dffere coracs. Aoher reaso s ha he prce for each forward corac s avalable he marke. I hs exercse, we assume ha all of he forward coracs follow a geomerc Browa moo process: df = F σ dw, T, T, (3.) 7

16 Where =,, ( for Jauary,, 2 for December, 3 for ex Jauary corac, ), T s he exprao dae of he h corac, F, T s he prce of he h moh forward corac a me, σ s he volaly of he h corac, ad W, s he Browa moo assocaed wh he h corac. By Io s lemma, we have d l F, T = df, T F, T 2( F, 2 = σ dw, σ d 2 Therefore, we have l F F l F T ) = σ + σ W 2 2 ( df 2, T 0, T, = F 2 exp( σ + σ W, ) 2, T 0, T Where, T, or ) 2 F 0,T s he observed forward prce of he h corac o he valuao dae. Sce ~ N(0, ), we ca rewre he above equao as W, 2 F, T = F0, T exp( σ +σ ε ) ε ~ N(0,) 2 (3.2) 8

17 As we meoed earler, heorecally oe should be able o develop a spo prce model so ha s expeced value equals he forward prce for ay erm corac (from promp moh o he 72 d moh) uder a rsk-eural probably measure. However, s o praccable realy, sce he aural gas marke s very dffere from a pure facal marke. Frs, he seleme of aural gas marke s very specal,.e., a buyer does o receve he gas whe a corac expres ad does o receve he gas o he same day. Secod, he demad of he aural gas marke dcaes a seasoal paer ha s very dffcul o be capured by a mahemacal model. Fally, he marke becomes less lqud as he exprao of a corac creases. I oher words, he marke s o complee ad we may o fd a rsk-eural probably measure for spo ad all of he forward coracs he aural gas marke, especally for he logerm forward coracs. Therefore, we oly coec spo prce o he forward prces he same moh hs hess. We assume ha spo prce also follows a pece-wse geomerc Browa moo. Specfcally, sce he spo prce s avalable he marke for he valuao dae, durg he same moh, we assume ha he spo prce follows a geomerc Browa moo where expeced value equals he spo prce from he marke. Durg he followg mohs, we se he expeced value same as he forward prces ha are avalable from he marke o he valuao dae. Followg he assumpos, we have 2 S0 exp( σ s + σ s 2 S = 2 F0, exp( σ s + σ s 2 ε ) ε ) for he valuao moh ( a) for he h moh corac ( b) ε ~ N(0,) (3.3) 9

18 Where S s he spo prce a, S0 s he spo prce o he valuao dae, F o, s he forward prce for he h moh corac o he valuao dae, ad σ s s he volaly of spo prce. Equao (3.2) ad (3.3) look very smlar, bu here are dffereces. I equao (3.2), he forward volaly σ chages from moh o moh whle he spo volaly σ s equao (3.3) s cosa over he erm of a sorage corac. Ths s cosse wh wha we observed from he aural gas marke. Namely, spo prces always have hgh volales, bu he volaly of a forward corac decreases as he maury creases. Noe ha eve he same corac moh has dffere volales f s dffere years. We ca expec ha he volaly for Jauary 2008 s greaer ha he volaly of Jauary 2009 whe we are o a dae before Jauary If he erm of a sorage corac s very log ad oly some of he forward prces are avalable from he marke o he valuao dae, he he forward prces for he remag mohs are derved from he forward prces of he same mohs he prevous year assumg he growh rae s he same as he rsk-free eres rae. For example, a sorage corac has a erm from Sepember s, 2007 o Augus 3 s, 202 ad he valuao dae s Augus s, O he valuao dae, assume ha all of he prces we ca oba from he marke are spo o Augus s, 2007 ad he forward prces from Sepember 2007 o Augus 200, he he forward prce of Sepember 200 equals he produc of he forward prce of Sepember 2009 ad ( + rsk-free 0

19 eres rae), ad he forward prce of Ocober 200 equals he produc of he forward prce of Ocober 2009 ad ( + rsk-free eres rae). Afer we derve all of he forward prces, we ca smulae forward ad spo prces by equao (3.2) ad (3.3) respecvely. Correlao s also very mpora for he smulao, sce we ca expec ha he forward prces are hghly correlaed ad also correlaed o he spo prce. Specfcally, whe we smulae he prces of dffere forward coracs ad he spo prce, we eed o draw correlaed samples for dw,. Boh volales ad correlaos ca be calbraed by hsorc forward prces.

20 CHAPTER 4 VALUATION METHODOLOGIES 4. Problem Descrpo The valuao of aural gas sorage s a sochasc corol problem. The value of a sorage corac s he maxmum value of he sum of he dscoued cash flows durg he erm of he corac, whch ca be expressed by he followg equao: V T 0 = max( β E[ π = s.. v + 0 v = v + Δv v Δv Δv Δv ( v, P ; Δv )] (4.) Where V0 s he value of he sorage a me =0, β s he dscou facor a, v s he sorage level a, P s he aural gas prces a, whch ca be spo prce, or he vecor of forward prces, or boh, Δ v s he jeco or whdrawal amou durg he perod, π s he prof or loss a, 2

21 E s he expecao operaor, v s he maxmum workg gas capacy, Δ v s he maxmum whdrawal rae, ad Δv s he maxmum jeco rae. Usually, hs problem ca be expressed by he Bellma equao: V + ( v, P ; Δv ) = max( π ( v, P ; Δv ) + βe[ V + ( v+, P + ; Δv ) F ] (4.2) Where V s he sorage value a, V + s he sorage value a +, also called he couao value of he sorage a, ad F s he formao fler a. Sce here are oo may sae varables he problem, cludg he sorage level, spo prce, ad forward prces, s very dffcul o solve he problem by ree models or by solvg a fe dfferece equao. Acually, he pece wse coeco bewee spo ad forwards preves us from usg he mehodology of solvg a fe dfferece equao. Therefore, we choose he Moe Carlo mehodology o solve he problem. Nex, we roduce he wo Moe Carlo mehodologes ha ca be appled he aural gas sorage valuao problem. 4.2 Moe Carlo wh Sochasc Dual Dyamc Programmg 3

22 Before roducg he sochasc dual dyamc programmg, we gve a revew of he radoal sochasc dyamc programmg by showg how ca be appled o he sorage valuao problem. The procedure s as follows: Ialze he couao value a T: V T + 0 for all of he scearos For = T,, m For each sorage level v = ( v, m,..., M ) k For each prce scearo P = ( P, k,..., K), solve he oesage problem = = Nex k m k m k V ( v, P; Δv ) = max( π ( v, P; Δv ) + βe[ V ( v s.. v + = v 0 v v m +Δv Δv Δv Δv k + +, P + ; Δv + ) F ] V ( v ) s he average of m V k or he probably weghed V k Nex Creae a complee V v ) curve for he prevous sage by erpolag ( over he dffere sorage levels Nex The radoal mehodology s resource cosumg sce requres he opmzao o each sorage level ad erpolao. To mprove he radoal mehodology, Perera, Campodoco, ad Kelma (999) developed sochasc dual dyamc 4

23 programmg ad appled o he hydrohermal schedulg problem. I hs mehodology, s assumed ha he couao value s lear for a specfc sorage level: V + = ϕ + v+ + δ + =,..., N Therefore, f a sorage capacy s broke o N pars, he he valuao s subjec o N lear cosras each sage. The slope coeffces ( ϕ + =,..., N ) are he smplex mulplers ha are assocaed wh he cosras of sorage level chage. I oher words, he slope of a sorage level s he shadow prce of he sorage level,.e, he creased value of he sorage gve here s oe more u gas he sorage. The mehod ca be appled o he sorage valuao problem by he followg procedure: Ialze he couao value a T: V T + 0 (or se ϕ, δ 0 ) T + = 0 T + = Se he umber of segmes N = he umber of sorage levels M For = T,, m For each sorage level v = ( v, m,..., M ) k For each prce scearo P = ( P, k,..., K), solve he oesage problem = = V k ( v s.. v + 0 v Δv Δv Δv V m k k m k k, P ; Δv ) = max( π ( v, P ; Δv ) + βe[ V + = v m + + Δv v ϕ v + + k + δ + smplex mulpler λ k + ( v +, P + ; Δv + ) F ] =,..., N (4.3) 5

24 Nex ϕ k k = p λ δ = p k V k ϕ v Nex Nex Brgedal (2003) appled hs mehodology o he valuao of aural gas sorage, bu oly he spo prce s cosdered ad dd o ake o accou he dfferece of bd ad ask prces. The smplex mulpler ca be approxmaed by he creme he value of he sorage whe here s a small crease he sorage level. 4.3 Moe Carlo wh Ordary Leas Square Aoher mehodology s he ordary leas square Moe Carlo smulao, whch s developed by Logsaff ad Schwarz (200). Oe of he key ssues usg Moe Carlo mehodology for dervave ad asse valuao s how o compue he expeced couao value a me ha s codoed o he formao a me. Logsaff ad Schwarz (200) used a lear combao of bass fucos o approxmae he couao value a. The bases are usually he sae varables ad kow a me. There are a lo of choces for he bass fucos, whch ca be Laguerre polyomals, Herme polyomals ad oher polyomals. Acually, he smple powers of he sae varables also work well. For example, whe valug a Amerca spread opo where S ad 2 S are he prces of he wo uderlyg a, he a se of bass fucos ca 6

25 be S, 2 S, ( S ) 2, ( 2 ) 2 2 S, ad ( ) S, so he couao value a, V +, ca be S expressed by he followg lear regresso model: V = γ + γ S + γ S + γ ( S ) + γ ( S ) + γ ( S S ) + ε ε ~ N (0,) The mplemeao procedure s: frs, smulae N pahs of uderlg prces of ad 2 S by Moe Carlo smulao. Secod, coduc he valuao by backwardao duco. I each sage, N values of V + ca be compued by he smulaed prces, ad a regresso s coduced based o he above regresso model. The replace he codoal expeced value by he regresso model ad make he decso of early exercse. Fally, ge he value of he spread opo by akg he average of dscoued cash flow a me = 0. S The ordary leas square Moe Carlo mehodology ca be appled o aural gas sorage valuao by he followg procedure:. Smulae N depede prce pahs P,..., P,,..., N T = 2. Ialze he couao value a T: V T Coduc backward duco: For = T,,, m For each sorage level v = ( v, m,..., M ) = 7

26 Carry ou a ordary leas square regresso ad compue he codoal expeced couao value by he regresso resuls, Nex For each smulao =,, N = m For each sorage level v = ( v, m,..., M ), solve he oesage problem ad fd a decso rule, m m V ( v, P ; Δv ) = max( π ( v, P ; Δv ) + βev [ s.. v + = v 0 v m + +Δv v, P ; Δv ) F ] Δv Δv Δv (4.4) Nex Nex + ( v Nex For =,, N Compue he prese value of he sorage by summg he dscoued fuure cash flows followg he decso rule Nex 4. Sorage value s he average of he prese values uder pahs Booger ad Jog (2006) appled a smlar mehodology o aural gas sorage valuao, bu oly spo prce s cosdered, so a smple regresso model works well. They also gored he dfferece of bd ad ask prces. A aural dea s o corporae he sorage level o avod he erpolao over varous sorage levels, bu hey foud 8

27 he resuls are o sable whe sorage level s he regresso model. I hs hess, we ake o accou boh spo prce ad forward prces, so eve f we exclude he sorage level he regresso, here wll be oo may erms f he bass fucos cover he spo prce ad all of he forward prces. Sce here may be may prce curves o be cosdered he valuao, f we corporae all of he prces he regresso model, he model may become usable. Eve f we oly ake he prces, her squares, ad cross producs, here wll be 9 ( ercep + 2 prces + 2 squares of prces + 66 cross producs) erms he regresso model f here are 2 prce curves. Therefore we eed o reduce he dmeso of he regresso. Oe way o acheve hs goal s prcpal compoe aalyss, whch s descrbed followg seco. 4.4 Prcpal Compoe Aalyss Prcpal compoe aalyss (PCA) s a way o fd he paers compleed daa ad reduce he dmeso of he daa. PCA s wdely used may felds, such as mage aalyss, smulao, ec. For our purpose, PCA o oly ca be used o reduced he me of Moe Carlo smulao for he spo ad forward prces, bu more mporaly, we creae bass fucos based o he prcpal compoes obaed from PCA hus reduce he dmeso of he regresso. Here s a bref revew of PCA. 9

28 Suppose Q s he covarace marx of he log-reurs of aural gas spo ad 2 forward prces, he Q s square, symmerc, ad posve sem-defe. Le λ ( =,, ) be he egevalues of Q wh λ > λ... λ ad U ( =,, ) be he assocaed egevecors, he we have QU = UΛ 2 > Where λ λ Λ = λ3 The egevecors are orhogoal, herefore, he raspose of he egevecor marx s he same of s verse marx ad we have Q = UΛU T Le X = U Λz, where λ λ Λ =, ad λ3 z z2 z = s a vecor wh depede ad sadard... z3 ormally dsrbued compoes The we ca show ha he covarace marx of X s Q by he followg dervao: Q X = [ X E[ X ]][ X E[ X ]] = XX = U Λz( U = UΛU = Q T T ( zz Λz) T ( E[ X ] = 0) T = I, Λ T T = Λ) 20

29 So we ca rewre X as X = U λ z + U λ z U λ z The represeao s called he prcpal compoe expaso of X. The radom varables y = λ z ( =,, ) are called he prcpal compoes of he radom varable X. Sce he egevaluesλ are raked decreasg order, we ca approxmae X by he frs j erms he prcpal compoe expaso: X U λ z + U λ z U j λ z j j (4.5) Ths approxmao ca be used Moe Carlo smulao. By hs approxmao, we oly eed j samples o smulae all of he varables X, ad more mporaly, hese j samples are depede sadard ormally dsrbued, whch ca be easly mplemeed may sof packages. Usually, s accurae eough o pck up he 3 bgges egevalues ad he correspodg egevecors for facal ad eergy markes. These 3 rsk facors ca be explaed as parallel shf, slope, ad curvaure of he prce curves respecvely. These explaaos ca be foud from he work of Corazar ad Schwarz (994), Schwarz (997), Blaco, Soroow, ad Sefszy (2002), ad Lauer (2003). As we dscussed earler, a more mpora applcao of PCA s o reduce he dmeso of he regresso. Equao (4.5) dcaes ha every sae varable (prce) 2

30 s correlaed o he prcpal compoes, hus we ca use he prcpal compoes o replace sae varables he regresso. I hs case we choose 3 rsk facors, a possble regresso model ca be E[ V + F ] = α + α exp( y 0 + α exp(2y 5 2 ) + α exp( y ) + α exp(2y ) + α exp( y ) + α exp( y ) + α exp(2y + y y 3 ) ) (4.6) The reaso ha we use he expoeal fucos of he prcpal compoes he regresso sead of he prcpal compoes hemselves s ha usually prces sead of her log-reurs are used he regresso. Ths s dffere from Chalamadars (2007), who also used PCA o reduce he dmeso he regresso o compue he expeced value for mulcallable rage accruals. Wha we have derved s based o he covarace marx for a u me, ca be a day or a year or some oher me horzo. Sce we eed o smulae he whole prce pah, we eed o derve correspodg equaos of (4.5) ad (4.6) for ay me. We kow he covarace marx a s Q, so afer a smple dervao, we ca rewre he equaos (4.5) a ay me as: X U λ z + U λ z U λ z j j j (4.7) We do eed o rewre he equao (4.4), bu ow he prcpal compoes a are y λ, = z ( =,, j) 22

31 Revsed Implemeao Procedure All of he procedures meoed above are oly applcable f he spo prce s he sole pu. Whe boh he spo ad forwards are cosdered, we eed o rewre he problem a each sage. Oe valuao mehod s o ake he sorage as a Amerca opo, so he value of he sorage a s he max of he curre value ad dscoued codoal expeced couao value. The oe sep problem s expressed by he followg equao: T T I j a T T I wh b T I T T T T T T T I v f v c P v f v c P v c P forward mohs for s cosra I v d v v d moh spo for cosra v d v v d I v v v s P P V 0,,..., 0 ) ( 0 ) ( ) (,..., 0,,...,.. ) ( max ) (,, 0,,,, 0,,, 0,,, = > Δ Δ + Δ Δ = Δ + = = Δ Δ Δ Δ Δ Δ = Δ = = = = π π (4.8) ] ) ( [ ) ( ], ) ( [ ) ( F P E V P he V F P E V P V If = β β Where I s he umber of forward mohs avalable a. If I =0, he oly he spo marke s avalable, T 0 d s he umber of remag days curre moh, whch chages over me, T d s he umber of days he h forward moh,

32 , s he h moh bd prce uder scearo a. If =0, he s he spo prce, b P, oherwse, s he h moh forward prce,, s he h moh ask prce uder scearo a. If =0, he s he spo prce, a P, oherwse, s he h moh forward prce, c wh s he whdrawal cos, ad cj s he jeco cos. Oe way o solve problem (4.8) s o rewre as V ( P ) = max π ( P ) s.. () 0 Δv (2) 0 Δv (3) 0 Δv (4) 0 Δv (5) (6) k= 0 k= 0 Δv Δv,,0 a,, b,, a,, k b,, k v d T0 d d T T k= 0 k= 0 Δv Δv Δv Δv Δv b,, k a,, k = 0,,..., I, jeco ad whdrawl for spo ad forward mohs =,..., I v =,..., I 0 =,..., I =,..., I I +,..., 2( I + ) cosra cosras cosras cosras cosras cosras for jecohespo for jeco forwardmohs for whdrawal forwardmohs for jeco for whdrawl for moh forwardmohs forwardmohs π = π + π wherep Δv + = = P.* Δv I = 0 = [ Δv, ( P = [ P a, a, a, Δv + c b, P b, ) Δv ], ad.* s he er produc j ], a, T, + I = 0, ( P b, c wh ) Δv b, T, = 0,,..., I (4.9) 24

33 I equao (4.9), jecos ad whdrawals are defed by separae varables ad whdrawals are redefed as posve varables. Ijecos are assocaed wh he ask prces ad whdrawals are assocaed wh he bd prces. Alhough he umber of varables o be solved s doubled, s easer o be mplemeed. Cosras defed by (5) are explaed as: he amou ha ca be jeced he h moh should be less ha or equal o he oal capacy mus he oal jecos mohs from 0 o - plus he oal whdrawals mohs from 0 o -. Cosras defed by (6) are explaed as: he amou ca be whdraw he h moh should be less ha or equal o he oal jecos mus he oal whdrawals mohs from 0 o -. Noe ha he cosras chage over ad he umber of cosras chages from moh o moh. So s ecessary o deerme he umber of cosras o be used he opmzao every me sep. Sce he curre value s he rsc value of he sorage gve he curre marke formao, hs mehod s very smlar o he so-called rsc rollg valuao or he forward dyamc opmzao mehod gve by Eydelad ad Wolyec (2002). However, spo prce s ake o accou our valuao. We ame hs mehod as rsc rollg wh spo ad forward. Ths mehod ca be mplemeed by he Moe Carlo wh ordary leas square mehod. The procedure s as follows: 25

34 . Smulae N depede prce pahs P,..., PT, =,..., N usg equaos (3.2), (3.3) ad (4.7) 2. Ialze he couao value a T: V T Coduc backward duco: For = T,,, Carry ou a ordary leas square regresso ad compue he codoal expeced couao value by equao (4.6), opmzao, Nex Deerme he umber of cosras o be cluded he For =,, N Nex Solve he oe-sage problem descrbed by equao (4.9) If V ( P ) β E[ V + ( P + ) F ], he V ( P ) = βe[ V + ( P + ) F ] V 0, =,, N s he sorage value uder he h pah 4. Sorage value s he average of he prese values uder pahs To ake o accou he bd ad ask prce, boh (4.3) ad (4.4) eed o be rewre a smlar way as (4.9). 26

35 CHAPTER 5 RESULTS AND DISCUSSIONS 5. Hsorc Daa ad Parameer Esmao The prce models are calbraed based o hsorc daa. Those hsorc daa are he spo ad forward prces from December s, 2004 o Jue 29 h, For each busess day, he prces of 24 forward moh coracs are colleced. Fgure 5. shows he hsorc spo prce a Hery Hub. The spo prce has a seasoal paer ad s very volale. There were wo spkes. Oe was Sepember 2005, whch was due o he hurrcae Kara. The oher was December 2005, whch was due o he hgher demad ad lower supply he wer. Fgure 5.2 shows he NYMEX aural gas spo ad forward prces o Jue 28 ad Jue 29, The seasoal paer of he forward prce curve s more obvous compared o he spo prce curve. 27

36 $/MMBu //2004 2//2005 4//2005 6//2005 8//2005 0//2005 2//2005 2//2006 4//2006 6//2006 8//2006 0//2006 2//2006 2//2007 4//2007 6//2007 Daes Fgure 5.: Hsorc Hery Hub Naural Gas Spo Prce 6/28/2007 6/29/ $/MMBu Spo Moh Fgure 5.2: NYMEX Naural Gas Prces o Jue 28 ad Jue 29,

37 Tables 5. o 5.6 show he covarace, correlao, egevecors, ad egevalues. For egevecors ad egevalues, oe s for he spo ad forwards ad he oher s for he forwards oly. Noe ha represes Jauary corac sead of he promp moh corac. Whe we ake boh spo ad forwards o accou, here are 3 egevalues ad he rao of he bgges hree facors o he oal s 79%. Whe oly forwards are cosdered, here are 2 egevalues ad he rao of he bgges hree facors o he oal s 80%. Fgure 5.3 s he 0 smulaos of he spo prce based o he covarace marx ad he spo ad forward prces o Jue 29, The red curve s he average of he 0 smulaos. 29

38 Table 5.: Covarace Marx of he Spo ad Forward Prces Spo Ja Feb Mar Apr May Ju Jul Aug Sep Oc Nov Dec

39 Table 5.2: Correlao Marx of he Spo ad Forward Prces Spo Ja Feb Mar Apr May Ju Jul Aug Sep Oc Nov Dec

40 Table 5.3: Egevecors of he Covarace Marx of he Spo ad Forward Prces Table 5.4: Egevecors of he Covarace Marx of he Forward Prces

41 Table 5.5: Egevalues of he Covarace Marx of he Spo ad Forward Prces SUM Table 5.6 Egevalues of he Covarace Marx of he Forward Prces SUM NG Prce ($/MMBu) Avg Daes Fgure 5.3: Smulaed Spo Prces from July 07 o Jue Resuls 5.2. The Sorage Corac The resuls are from he valuao of a aural gas sorage corac: 33

42 Term: from July, 07 o Jue 30, 08 Locao: Hery Hub Workg gas capacy:,000,000 MMBu Sar volume: 0 Ed volume: 0 Maxmum daly jeco rae: 35,000 MMBu Maxmum daly whdrawal rae: 75,000 MMBu Whdrawal cos: 0 Ijeco cos:.5% of he fuel prce We also assume ha he ask prce equals md prce plus oe ce ad he bd prce equals md prce mus oe ce. The valuao s mplemeed wh MATLAB o a deskop persoal compuer wh 2.8 GHz CPU ad 2.5 GB RAM The Resuls by he Mehod of Irsc Rollg wh Spo ad Forward Table 5.7 shows he value of he sorage corac esmaed by he mehod of rsc rollg wh spo ad forward based o he marke formao o Jue 28, The smulaed value s abou $.7 mllo, whch s close o he value esmaed by FEA (Facal Egeerg Assocao) provded model MCSTORAGEOPT. As he umber of smulaos creases, he CPU me learly creases. For sace, 34

43 oe ru wh 000 smulaos akes abou 38 mues. Noe he erm of he corac s oe year. If creases, he he CPU me also wll crease. Foruaely, he value of he sorage coverges relavely well. So for valuao purpose 00 smulaos seem eough. MCSTORAGEOPT uses Moe Carlo smulao ad uses oe rsk facor model for prces. Is pus clude spo ad forward prce o valuao dae, volaly of forward mohs, ec. However, eher he covarace marx or he correlao marx s requred, so s uclear wheher he model akes correlao o accou. I addo, oher key ssues such as he mehod for compug codoal expecao are o dsclosed. 35

44 Table 5.7: The Value of he Sorage Corac Esmaed by he Mehod of Irsc Rollg wh Spo ad Forward o Jue 28, 2007 Value of Value of he CPU Tme CPU me Number of he s Ru 2d Ru of he s of he 2d FEA smulaos ($ 000) ($ 000) Ru (Sec) Ru (Sec) ($ 000) Fgure 5.4 shows a sample opmzao resul o day ad day 20, respecvely. Based o he smulaed prces o day, he opmzao model suggess oe urover durg he corac erm: jec MMBu July 07 ad MMBu Sepember 07 ad hem whdrawal he full amou Jauary 08. O day 20, he opmzao model suggesos wo urovers. 36

45 Fgure 5.5 shows he hsogram of sorage value from oe ru wh 000 smulaos. I shows he smulaed sorage values are ormally dsrbued Ijeco/Whdrawal (MMBu) Day Day Moh Fgure 5.4: Sample Opmzao Resul o Day ad Day 20 37

46 Frequecy sorage value ($) Fgure 5.5: The Hsogram of Sorage Value from Oe Ru of 000 Smulaos To sudy he mpac of marke formao, he value of he sorage based o he formao of Jue 29, 2007 s also esmaed, whch s lsed able 5.8. The mpac of marke formao s sgfca, he value from Jue 29 prce s much hgher ha he value from oe day before. Ths s due o a very low spo prce o Jue 29 compared o he forward prces (see fgure 5.2), whch creaes bgger spreads bewee he spo ad he forwards. 38

47 Table 5.8: The Value of he Sorage Corac Esmaed by he Mehod of Irsc Rollg wh Spo ad Forward o Jue 29, 2007 Value of Value of he CPU Tme CPU me Number of he s Ru 2d Ru of he s of he 2d FEA smulaos ($ 000) ($ 000) Ru (Sec) Ru (Sec) ($ 000) Table 5.9 shows he value of he sorage esmaed by forwards oly. The resuls show ha he esmaed sorage value decreases sgfcaly whe spo s o ake o accou. Sce he spo s moved away, he seleced egevalues ad egevecors wll be dffere, bu he same regresso model s appled. 39

48 Table 5.9: The Value of he Sorage Corac Esmaed by he Mehod of Irsc Rollg wh Forward Oly Value from Value from pus of pus of CPU Tme CPU me Number of 6/28/07 6/29/07 (6/28/07) (6/29/07) smulaos ($ 000) ($ 000) (Sec) (Sec) Prcpal compoe aalyss s helpful for oba a praccable regresso model, bu may reduce he value due o he reduced volaly, especally hs exercse, where he rao of he hree bgges egevalues o he oal egevalues s oly abou 80%. 40

49 5.2.3 The Resuls by Spo Oly Mehods Tables 5.0 ad 5. ls he valuao resuls by he mehod of Moe Carlo wh ordary leas square regresso o Jue 28 ad Jue 29, 2007 respecvely. Ths mehod shows ha he value of he sorage corac s abou $7 mllo, whch s much hgher ha he value esmaed by he prevous mehodology. Ths s mosly due o he hgher volaly of he spo prce, whch makes freque poso adjusme profable. However, sorage raders do adjus her poso very frequely. A ypcal frequecy of poso adjusme s aroud oce a week. Noe ha he spo curve s also ake o accou he decso process he mehod of rsc rollg wh spo ad forward, bu s smply reaed as a forward curve, so he value from he spo volaly ca o be fully capured he valuao. The volales are calbraed from hsorc daa, bu ca be replaced by mpled volales. Tables 5.2 ad 5.3 ls he valuao resuls by he mehod of Moe Carlo wh sochasc dual dyamc programmg o Jue 28 ad Jue 29, 2007 respecvely. Ths mehod also gves a sgfcaly hgher sorage value of abou $ mllo. The emprcal valuao shows ha he value of he sorage coverges very well by he sochasc dual mehodology. 00 smulaos are eough for a accurae valuao, whch akes oly abou 4 secods. I requres more smulaos (abou 4

50 2000) o oba a accurae value for he leas square mehod. Booger ad Jog (2006) observed ha as few as 50 smulaos are requred o oba a precse value of sorage ad explaed s possbly due o he mea-reverso model. CPU me does o crease sgfcaly for he sochasc dual dyamc mehod as he umber of smulao creases. Ths s due o he fac ha here are oly 0 prce scearos all of he rus. Whe he umber of smulao s 0, he he frs scearo uses he lowes prce ad he secod scearo uses he secod lowes prce, ec. Whe he umber of smulao s 20, he frs scearo uses he average of lowes wo prces ad he secod scearo uses he average of he secod lowes wo prces, ec. 42

51 Table 5.0: The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Ordary Leas Square Regresso o Jue 28, 2007 Value of Value of he CPU Tme CPU me Number of he s Ru 2d Ru of he s of he 2d smulaos ($ 000) ($ 000) Ru (Sec) Ru (Sec)

52 Table 5.: The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Ordary Leas Square Regresso o Jue 29, 2007 Value of Value of he CPU Tme CPU me Number of he s Ru 2d Ru of he s of he 2d smulaos ($ 000) ($ 000) Ru (Sec) Ru (Sec)

53 Table 5.2: The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Sochasc Dual Dyamc Programmg o Jue 28, 2007 Value of Value of he CPU Tme CPU me Number of he s Ru 2d Ru of he s of he 2d smulaos ($ 000) ($ 000) Ru (Sec) Ru (Sec)

54 Table 5.3: The Value of he Sorage Corac Esmaed by he Mehod of Moe Carlo wh Sochasc Dual Dyamc Programmg o Jue 29, 2007 Value of Value of he CPU Tme CPU me Number of he s Ru 2d Ru of he s of he 2d smulaos ($ 000) ($ 000) Ru (Sec) Ru (Sec) The value of he sorage s also mpaced by he operaoal flexbles. Sorage creases value wh he level of flexbly. Icreased flexbles gve more value o he sorage whe spo oly mehods are used for valuao. 46

55 5.3 Valuao wh a Chagg Bd-ask Spread I above smulao, bd-ask spread s se as cosa, bu chages over me realy. The spo ad promp forward are very lqud, hus he assocaed spread ca be less ha ce. As he maury of he forwards creases, he spread creases as well. Also, he spread s deermed by he locao. Hery Hub marke s very lqud ad he oher markes are less lqud. Cosderg he specfc sorage corac, he followg bd-ask spread s appled: Bd-ask spread = ( )% of fuel prce (=0,,, ), If he prce of aural gas s $8/MMBu, he he above equao mples a spread of 0.8 ce for he spo ad a spread of 6.08 ces for he corac expres oe year. Table 5.4 shows he resuls wh chagg bd-ask spread. Those values are close o he value esmaed by a cosa bd-ask spread of 2 ces. 47

56 Table 5.4: The Valuao of he Sorage Corac wh Chagg Bd-ask Spread Irsc rollg Number of wh spo ad Leas-square MC, SDDP, spo oly smulaos forwards ($000) spo oly ($000) ($000) Furher Dscussos o he Mehodologes The rsc rollg may oly capure very lmed me value compared wh he rsc value obaed o he frs day. For sace, we have a corac ha has a erm from Sepember, 2007 o December 3, Whe we are Sepember, we ca adjus he posos ad capure he me value, bu o ad afer Sepember 26, he 48

57 Ocober corac expres, so he choces for jeco are lmed o he spo ad November corac, whch s usually more expesve ha Ocober corac. Thus, afer Sepember 26, he sorage value eds o decrease. Tha meas he mehod may o capure he value he prce volales afer movg o he ex moh. The problem s ha whe we compare he sorage values wo perods, hey always have a ew sar (empy sorage) bu have dffere choces (more or less forward coracs are avalable). Ths mehod ca be mproved by excludg he realzed acves whe comparg he values wo perods. For example, f he sorage value o Sepember 25 s 00 ad he sorage value o Sepember 26 s 90, he follow he orgally mehod we se he value o Sepember 25 as 00. Bu wha we should do s as follows: Frs, check he resuls of he opmzao o Sepember 25, f here are o acves assocaed wh he spo of Sepember 25 ad Ocober forward, he here s o eed for furher aalyss ad we sll se he value o Sepember 25 as 00. Secod, f here are acves o he spo of Sepember 25 ad Ocober forward, he we separae he value of he sorage o wo pars. Oe s he jeco o he spo of Sepember 25 ad Ocober forward, sayg -000 (sce s he cos), ad he oher s whdrawal o December forward, sayg

58 Thrd, redog he opmzao o Sepember 26 gve he marke formao o Sepember 26 ad addoal cosras ha are he resuls from he opmzao o Sepember 25 assocaed wh he spo o Sepember 25 ad Ocober corac. Fally, f he ew value of he sorage o Sepember 26 s less ha 00, he se he value o Sepember 25 as 00. If he ew value of he sorage o Sepember 26 s greaer ha 00, sayg 05, he se he value o Sepember 25 as 05. By hs way, he valuao ca capure more me value. However, hs mehod may ake more me. I hs exercse, we use very smple prce models. Sce he smulao of he spo ad forward prces are separaed from he opmzao models, alerave prce models ca be used he valuao. 50

59 CHAPTER 6 CONCLUSIONS Prcg aural gas sorage s a very challegg opc. I s ecessary o develop boh approprae prce models ad opmzao models. For he prce models, we assume ha each forward corac follows a geomerc Browa moo wh zero drf. The volaly of each forward curve s calbraed from he hsorc daa. The spo prce curve also follows a geomerc Browa moo, bu s expecao chages from moh o moh. Specfcally, s expecao a cera moh equals he correspodg forward prce o he valuao dae. Sce prce models do mpac he opmzao models ad her mplemeao, alerave models ca be seleced he valuao. The volales of he spo ad forward curves are calbraed based o he hsorc daa, bu hey ca be replaced by mpled volales ha ca be derved from he raded opos. We developed hree mehodologes for sorage valuao. The frs mehodology s called rsc rollg wh spo ad forward. I akes boh he spo ad forward prces o accou. Ths mehod apples he so-called Irsc rollg radg sraegy where a rader locks hs/her posos based o he marke formao o he 5