A Vecor Auoregression Framework for he Modeling of Commodiy Sreads Ted Kury The Energy Auhoriy ICDSA 007 June, 007
Rule # of Pricing Models Pricing models can offer valuable insigh ino he behavior of simle or comlex markes
Rule # of Pricing Models Markes end o be raional in he long run, bu markes can say irraional longer han you can say solven J.M. Keynes (aribued)
The Problem Model he rices of wo closely-relaed commodiies Naural gas a wo differen delivery oins Ineres raes Prices usually ied ogeher by some fundamenal facor (e.g. ransoraion raes) Caure no only he evoluion of rices, bu he relaionshi beween rices
0/03/94 05/3/94 09/3/94 0/09/95 06//95 0/3/95 03/4/96 07/5/96 /05/96 04//97 08/9/97 0/4/98 05/7/98 0/06/98 0//99 07/0/99 //99 03/7/00 08/08/00 /9/00 05/04/0 09/4/0 0/30/0 06//0 0/4/0 03//03 07//03 /0/03 04/6/04 09/0/04 0/8/05 05/3/05 0/5/05 03/0/06 07/4/06 /04/06 $/MMBu Naural Gas Time Series Naural Gas Price Hisory 5.00 0.00 5.00 0.00 5.00 0.00-5.00-0.00 HHGas TZ4Gas Sread
Gas Price Characerisics Since 994 Since 000 Henry Hub Transco Zone 4 Basis Henry Hub Transco Zone 4 Basis s %ile.44.46-0.0.03.0-0. 5h %ile.59.60-0.06.4.46-0.04 0h %ile.74.76-0.03.8.86-0.0 5h %ile.7.9 0.00 3.94 3.98 0.0 75h %ile 5.6 5.66 0.07 6.66 6.8 0. 90h %ile 7.5 7.7 0.8 7.79 8.05 0.7 95h %ile 8.0 8. 0.30 9.64 9.9 0.40 99h %ile.8 3.8 0.7 3.68 4.8 0.95
Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework
Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework
Modeling Consideraions Relaive Prices Sread Oion Absolue and Relaive Prices Barrier Oion Cash Flow a Risk of Forward Purchase or Sale Absolue Produc (Naural Gas) Cos
Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework
The Usual Susecs Closed-Form Sread Oion Formulas Geomeric Brownian Moion Price Model Single Facor Mean Reversion Price Model
Tradiional Sread Models Model such as Margrabe (978) Derived from Black-Scholes, so i shares is assumions Lognormal rice reurns Indeenden and idenically disribued shocks No ransacion coss
Margrabe Valuaion for Sread Oions Similar o Black and Black-Scholes where: ) ( ) ( ),, ( d N x d N x x w x x x d / ln d d
Geomeric Brownian Moion Model (GBM) Use he log formulaion since we re modeling a rending series where: ln S ln S ~ N(0, )
Price Evoluion Under GBM Le s jus assume ha is he log rice A ime A ime 0 0 A ime, collecing he shock erms 0 i i
Price Variance Under GBM Variance of each individual shock erm Var( ) So he variance of erms Var( )
Modeling he Sread Beween Two Prices Assume wo rices, and q, where and: q q ~ ~ N(0, N(0, q ) ) corr(, q )
Sread beween GBM Prices So he basis a ime or q 0 0 0 i qi i i Wih variance q 0 qi i i Var q q i q
$/MMBu Samle GBM Simulaion GBM Simulaion 0.00 8.00 6.00 4.00.00 0.00 -.00 05/0/07 05/03/07 05/05/07 05/07/07 05/09/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 Henry Hub TZ4 Sread 05/3/07 06/0/07 06/04/07 06/06/07 06/08/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/30/07
Single Facor Mean Revering Model (SFMR) Framework of Pindyck (999) and Schwarz (997) where: ln ln S ( ln S S ) ~ N(0, is mean reversion rae is log of he long run equilibrium rice )
Price Evoluion Under SFMR Again, assuming ha is he log rice A ime, rearranging erms ( A ime ) ( )[ ( ) 0 ] A ime, collecing erms i 0 i 0 0 i i i
Price Variance Under SFMR Variance of each individual shock erm Var( ) So he variance of erms Var( Which reduces o Var( ) i ) ( i)
Variance Variance Comarisons Variance Growh Raes 90 80 70 60 50 40 30 0 0 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 Time No Mean Reversion Slower Mean Reversion Faser Mean Reversion
08/0/06 08/08/06 08/5/06 08//06 08/9/06 09/05/06 09//06 09/9/06 09/6/06 0/03/06 0/0/06 0/7/06 0/4/06 0/3/06 /07/06 /4/06 //06 /8/06 /05/06 //06 /9/06 /6/06 0/0/07 0/09/07 0/6/07 0/3/07 0/30/07 Price Comarisons 6.00 4.00.00 0.00 8.00 6.00 4.00.00 0.00 GBM - 95h %ile GBM - 5h %ile SFMR - 95h %ile SFMR - 5h %ile
Sread beween SFMR Prices So he basis a ime Wih variance q q q q q q q Var 0 0 i i i i q i qi η α η α μ α α q q q q ) ( ) ( ) ( ) ( ) ( 0 0
$/MMBu Samle SFMR Simulaion SFMR Simulaion 6.00 4.00.00 0.00 8.00 6.00 4.00.00 0.00 -.00 05/0/07 05/03/07 05/05/07 05/07/07 05/09/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 Henry Hub TZ4 Sread 05/3/07 06/0/07 06/04/07 06/06/07 06/08/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/30/07
Price Basis (Proorional) Behavior of SFMR Execed Basis Basis Evoluion Under Differen Mean Reversion Raes 7.00. 6.00.0 5.00.08 4.00.06 3.00.00.00 0.00 0 39 58 77 96 5 34 53 7 9 0 9 48 67 86 305 Time 34 343 36 38 400 49 438 457 476 495 54 533 55 57 590 Price wih Lower Mean Reversion Rae Price wih Higher Mean Reversion Rae Basis wih Bias Basis wihou Bias.04.0.00 0.98
Sandard Normal Shock Behavior of SFMR Shocks Decay of SFMR Shocks 0.000 0.0000-0.000-0.4000-0.6000-0.8000 -.0000 -.000 -.4000 -.6000 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 8 84 87 90 93 96 99 Time Slower Mean Reversion Rae Faser Mean Reversion Rae Difference
Sandard Normal Shock Behavior of SFMR Shocks Decay of SFMR Shocks.0000.8000.6000.4000.000.0000 0.8000 0.6000 0.4000 0.000 0.0000 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 8 84 87 90 93 96 99 Time Slower Mean Reversion Rae Faser Mean Reversion Rae Difference
Sandard Normal Shock Behavior of SFMR Shocks Decay of SFMR Shocks 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.000 0.000 0.0000 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 8 84 87 90 93 96 99 Time Slower Mean Reversion Rae Faser Mean Reversion Rae Difference
Margrabe Sread Model Tradiional Energy Models Shares assumions, boh good and bad, wih Black-Scholes Geomeric Brownian Moion Fixed execed basis equal o oday s basis Infinie variance of sread Single Facor Mean Revering Variable execed basis Finie variance of sread, bu differen mean reversion raes can lead o much differen decay raes Difference in shocks can increase and may diverge from wha is seen in realiy
Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework
Vecor Auoregressions - The Beer Mousera Vecor auoregression framework allows greaer flexibiliy Esablished mehodology Robus diagnosic esing Mulile mehodologies o handle shocks Fuure ah of rices deends on Hisorical ah of all modeled rices; and Fuure ah of oher rices
Vecor Auoregression Model (VAR) Models rices of goods ha are close subsiues q q q q where: q ~ ~ N(0, N(0, q ) ) corr(, q )
Price Evoluion under VAR Marix reresenaion q Change noaion o Price a in erms of P 0 P Α ΒP q Ε q P Β P 0 i 0 Β i ( Α Ε i )
The Sabiliy of he Weighing Marix Given he weighing marix Β 0.76 0.7 Subsequen owers are 0.53 0.8 Β 0.57 0.66 0.39 0.79 3 Β 0.48 0.36 0.467 0.659 5 Β 0.399 0.356 0.55 0.598 0 Β 0.346 0.355 0.54 0.544
$/MMBu Samle VAR Simulaion VAR Simulaion.00 0.00 8.00 6.00 4.00.00 0.00 -.00 05/0/07 05/03/07 05/05/07 05/07/07 05/09/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 Henry Hub TZ4 Sread 05/3/07 06/0/07 06/04/07 06/06/07 06/08/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/30/07
$/MMBu Unsable VAR Simulaion VAR Simulaion 7,000 6,000 5,000 4,000 3,000,000,000 0 -,000 -,000-3,000-4,000 05/0/07 05/03/07 05/05/07 05/07/07 05/09/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 05//07 05/3/07 05/5/07 05/7/07 05/9/07 Henry Hub TZ4 Sread 05/3/07 06/0/07 06/04/07 06/06/07 06/08/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/0/07 06//07 06/4/07 06/6/07 06/8/07 06/30/07
Sandard Normal Shock Behavior of VAR Shocks Decay of VAR Shocks 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.000 0.000 0.0000-0.000 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 8 84 87 90 93 96 99 Time Firs Shock Term Second Shock Term Difference
Sandard Normal Shock Behavior of VAR Shocks Decay of VAR Shocks.5000.0000 0.5000 0.0000-0.5000 -.0000 -.5000 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 8 84 87 90 93 96 99 Time Firs Shock Term Second Shock Term Difference
Sandard Normal Shock Behavior of VAR Shocks Decay of VAR Shocks.8000.6000.4000.000.0000 0.8000 0.6000 0.4000 0.000 0.0000-0.000 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 54 57 60 63 66 69 7 75 78 8 84 87 90 93 96 99 Time Firs Shock Term Second Shock Term Difference
Srenghs of he VAR Consruc ahs for rices wihou associaed forward curves Several well esablished ess o deermine oimal number of lags Two mehods o correlae rice shocks Acual correlaion and normally disribued shocks Resamle acual hisorical shocks o ick u correlaion and any non-normal disribuions
Weaknesses of he VAR More rigorous rocess o deermine arameers More diagnosic esing of model Proer funcional form Sable sysem of equaions Number of arameers grows quickly (N L) and can erode your degrees of freedom, so a larger daa se may be required
Pieline Managemen Risk Assessmen Resource managemen roblem Value of ransoraion caaciy Risk deends on he rice of gas a 3 hubs Mos conservaive es shows he need for 00 lags
$/MMBu Pieline Model Simulaion Samle Model Ieraion 4.00.00 0.00 8.00 6.00 4.00.00 0.00 -.00 04/0/07 04/5/07 04/9/07 05/3/07 05/7/07 06/0/07 06/4/07 07/08/07 07//07 08/05/07 08/9/07 09/0/07 09/6/07 09/30/07 0/4/07 0/8/07 //07 /5/07 /09/07 /3/07 0/06/08 0/0/08 0/03/08 0/7/08 03/0/08 03/6/08 03/30/08 Transco Zone Transco Zone 3 Transco Zone 6 Z3-Z Sread Z6-Z3 Sread
Summary Tradiional models may no work well o model absolue rice levels and commodiy sreads Infinie variance Unsable mean sreads Differen raes of mean reversion can cause divergence over ime Vecor auoregression offers a flexible framework Beer caures rice ineracions Derive fuure ah for rices wihou forward curve Handle non-normal and heeroscedasic shocks
Quesions? Conac Info: Ted Kury kury@eainc.org (904) 360-444
References Margrabe, W., 978, The Value of an Oion o Exchange One Asse for Anoher, Journal of Finance 33:. 77-86 Pindyck, R., 999, The Long Run Evoluion of Energy Prices, The Energy Journal 0:,. -7 Schwarz, E., 997, The Sochasic Behavior of Commodiy Prices: Imlicaions for Valuaion and Hedging, Journal of Finance 5:3,. 93-973