A Vector Autoregression Framework for the Modeling of Commodity Spreads

Transcription

1 A Vecor Auoregression Framework for he Modeling of Commodiy Sreads Ted Kury The Energy Auhoriy ICDSA 007 June, 007

2 Rule # of Pricing Models Pricing models can offer valuable insigh ino he behavior of simle or comlex markes

3 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)

4 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

5 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 HHGas TZ4Gas Sread

6 Gas Price Characerisics Since 994 Since 000 Henry Hub Transco Zone 4 Basis Henry Hub Transco Zone 4 Basis s %ile h %ile h %ile h %ile h %ile h %ile h %ile h %ile

7 Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework

8 Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework

9 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

10 Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework

11 The Usual Susecs Closed-Form Sread Oion Formulas Geomeric Brownian Moion Price Model Single Facor Mean Reversion Price Model

12 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

13 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

14 Geomeric Brownian Moion Model (GBM) Use he log formulaion since we re modeling a rending series where: ln S ln S ~ N(0, )

15 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

16 Price Variance Under GBM Variance of each individual shock erm Var( ) So he variance of erms Var( )

17 Modeling he Sread Beween Two Prices Assume wo rices, and q, where and: q q ~ ~ N(0, N(0, q ) ) corr(, q )

18 Sread beween GBM Prices So he basis a ime or q i qi i i Wih variance q 0 qi i i Var q q i q

19 $/MMBu Samle GBM Simulaion GBM Simulaion /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

20 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 )

21 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

22 Price Variance Under SFMR Variance of each individual shock erm Var( ) So he variance of erms Var( Which reduces o Var( ) i ) ( i)

23 Variance Variance Comarisons Variance Growh Raes Time No Mean Reversion Slower Mean Reversion Faser Mean Reversion

24 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 GBM - 95h %ile GBM - 5h %ile SFMR - 95h %ile SFMR - 5h %ile

25 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

26 $/MMBu Samle SFMR Simulaion SFMR Simulaion /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

27 Price Basis (Proorional) Behavior of SFMR Execed Basis Basis Evoluion Under Differen Mean Reversion Raes Time Price wih Lower Mean Reversion Rae Price wih Higher Mean Reversion Rae Basis wih Bias Basis wihou Bias

28 Sandard Normal Shock Behavior of SFMR Shocks Decay of SFMR Shocks Time Slower Mean Reversion Rae Faser Mean Reversion Rae Difference

29 Sandard Normal Shock Behavior of SFMR Shocks Decay of SFMR Shocks Time Slower Mean Reversion Rae Faser Mean Reversion Rae Difference

30 Sandard Normal Shock Behavior of SFMR Shocks Decay of SFMR Shocks Time Slower Mean Reversion Rae Faser Mean Reversion Rae Difference

31 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

32 Presenaion Ouline Modeling Consideraions Tradiional Energy Models Modeling Prices Modeling Sreads Vecor Auoregression Framework

33 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

34 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 )

35 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 )

36 The Sabiliy of he Weighing Marix Given he weighing marix Β Subsequen owers are Β Β Β Β

37 $/MMBu Samle VAR Simulaion VAR Simulaion /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

38 $/MMBu Unsable VAR Simulaion VAR Simulaion 7,000 6,000 5,000 4,000 3,000,000, ,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

39 Sandard Normal Shock Behavior of VAR Shocks Decay of VAR Shocks Time Firs Shock Term Second Shock Term Difference

40 Sandard Normal Shock Behavior of VAR Shocks Decay of VAR Shocks Time Firs Shock Term Second Shock Term Difference

41 Sandard Normal Shock Behavior of VAR Shocks Decay of VAR Shocks Time Firs Shock Term Second Shock Term Difference

42 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

43 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

44 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

45 $/MMBu Pieline Model Simulaion Samle Model Ieraion /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

46 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

47 Quesions? Conac Info: Ted Kury (904)

48 References Margrabe, W., 978, The Value of an Oion o Exchange One Asse for Anoher, Journal of Finance 33: 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,