Modeling Commodity Spreads with Vector Autoregressions

Ted KuryThe Energy Authority ®

ICDSA 2007June 1, 2007

Rule #1 of Pricing Models

Pricing models can offer valuable insight into the behavior of simple or complex markets

Rule #2 of Pricing Models

“Markets tend to be rational in the long run, but markets can stay irrational longer than you can stay solvent”

J.M. Keynes (attributed)

The Problem

• Model the prices of two closely-related commodities– Natural gas at two different delivery points– Interest rates

• Prices usually tied together by some fundamental factor (e.g. transportation rates)

• Capture not only the evolution of prices, but the relationship between prices

Natural Gas Time SeriesN a tu ra l G a s P ric e H is to ry

-1 0 .0 0

-5 .0 0

0 .0 0

5 .0 0

1 0 .0 0

1 5 .0 0

2 0 .0 0

2 5 .0 0

01/0

3/94

05/1

3/94

09/2

3/94

02/0

9/95

06/2

1/95

10/3

1/95

03/1

4/96

07/2

5/96

12/0

5/96

04/2

1/97

08/2

9/97

01/1

4/98

05/2

7/98

10/0

6/98

02/2

2/99

07/0

2/99

11/1

1/99

03/2

7/00

08/0

8/00

12/1

9/00

05/0

4/01

09/1

4/01

01/3

0/02

06/1

2/02

10/2

4/02

03/1

1/03

07/2

1/03

12/0

1/03

04/1

6/04

09/0

1/04

01/1

8/05

05/3

1/05

10/2

5/05

03/1

0/06

07/2

4/06

12/0

4/06

$/M

MB

tu

H H G a s T Z 4 G a s S p re a d

Gas Price Characteristics

Since 1994 Since 2000

Henry HubTransco Zone 4 Basis Henry Hub

Transco Zone 4 Basis

1st %tile 1.44 1.46 -0.20 2.03 2.10 -0.125th %tile 1.59 1.60 -0.06 2.41 2.46 -0.0410th %tile 1.74 1.76 -0.03 2.82 2.86 -0.0125th %tile 2.17 2.19 0.00 3.94 3.98 0.0275th %tile 5.61 5.66 0.07 6.66 6.81 0.1290th %tile 7.15 7.27 0.18 7.79 8.05 0.2795th %tile 8.01 8.22 0.30 9.64 9.92 0.4099th %tile 12.82 13.28 0.72 13.68 14.28 0.95

Presentation Outline

• Modeling Considerations• Traditional Energy Models

– Modeling Prices– Modeling Spreads

• Vector Autoregression Framework

Presentation Outline

• Modeling Considerations• Traditional Energy Models

– Modeling Prices– Modeling Spreads

• Vector Autoregression Framework

Modeling Considerations

• Relative Prices– Spread Option

• Absolute and Relative Prices– Barrier Option– Cash Flow at Risk of Forward Purchase or Sale– Absolute Product (Natural Gas) Cost

Presentation Outline

• Modeling Considerations• Traditional Energy Models

– Modeling Prices– Modeling Spreads

• Vector Autoregression Framework

The Usual Suspects

• Closed-Form Spread Option Formulas• Geometric Brownian Motion Price Model• Single Factor Mean Reversion Price Model

Traditional Spread Models

• Model such as Margrabe (1978)• Derived from Black-Scholes, so it shares its

assumptions– Lognormal price returns– Independent and identically distributed shocks– No transaction costs

Margrabe Valuation for Spread Options

• Similar to Black and Black-Scholes

where:)()(),,( 221121 dNxdNxtxxw −=

( )t

txxdσ

σ 22

121

1/ln +

=

tdd σ−= 12

2122

21

2 2 σρσσσσ −+=

Geometric Brownian Motion Model (GBM)

• Use the log formulation since we’re modeling a trending series

where:ttt SS ε+= −1lnln

),0(~ 2σε Nt

Price Evolution Under GBM

• Let’s just assume that p is the log price• At time 1

• At time 2

• At time t, collecting the shock terms

101 ε+= pp

[ ] 2102 εε ++= pp

∑=

+=t

iit pp

10 ε

Price Variance Under GBM

• Variance of each individual shock term

• So the variance of t terms

2)( σε =tVar

2)( σtpVar t =

Modeling the Spread Between Two Prices

• Assume two prices, p and q, where

and:η+= pq

),0(~ 2ppt N σε

),0(~ 2qqt N σε

),( qpcorr εερ =

Spread between GBM Prices

• So the basis at time t

• or

• With variance

∑∑==

−+−+=−t

ipi

t

iqitt pppq

11000 εεη

∑∑==

−+=−t

ipi

t

iqitt pq

110 εεη

( ) ( )qpqptt tpqVar σρσσσ 222 −+=−

Sample GBM SimulationG B M S im u la tio n

-2 .0 0

0 .0 0

2 .0 0

4 .0 0

6 .0 0

8 .0 0

1 0 .0 0

05/01/0705/03/0705/05/0705/07/0705/09/0705/11/0705/13/0705/15/0705/17/0705/19/0705/21/0705/23/0705/25/0705/27/0705/29/0705/31/0706/02/0706/04/0706/06/0706/08/0706/10/0706/12/0706/14/0706/16/0706/18/0706/20/0706/22/0706/24/0706/26/0706/28/0706/30/07

$/M

MB

tu

H e n ry H u b T Z 4 S p re a d

Single Factor Mean Reverting Model (SFMR)

• Framework of Pindyck (1999) and Schwartz (1997)

where:

α is mean reversion rateμ is log of the long run equilibrium price

tttt SSS εμα +−+= −− )ln(lnln 11

),0(~ 2σε Nt

Price Evolution Under SFMR

• Again, assuming that p is the log price• At time 1, rearranging terms

• At time 2

• At time t, collecting terms

101 )1( εααμ +−+= pp

2102 ])1()[1( εεααμααμ ++−+−+= pp

( ) ( ) ( )∑∑=

−−

=

−+−+−=t

i

iti

tt

i

it pp

10

1

0111 αεαααμ

Price Variance Under SFMR

• Variance of each individual shock term

• So the variance of t terms

• Which reduces to

2)( σε =tVar

( )∑=

−−=t

i

ittpVar

1

)(22 1)( ασ

( )( )

22

2

1111)( σ

αα−−−−

=t

tpVar

Variance ComparisonsV a ria n c e G ro w th R a te s

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

t 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

T im e

Varia

nce

N o M e a n R e ve rs io n S lo w e r M e a n R e ve rs io n F a s te r M e a n R e ve rs io n

Price Comparisons

0 .00

2 .00

4 .00

6 .00

8 .00

10 .00

12 .00

14 .00

16 .00

08/0

1/06

08/0

8/06

08/1

5/06

08/2

2/06

08/2

9/06

09/0

5/06

09/1

2/06

09/1

9/06

09/2

6/06

10/0

3/06

10/1

0/06

10/1

7/06

10/2

4/06

10/3

1/06

11/0

7/06

11/1

4/06

11/2

1/06

11/2

8/06

12/0

5/06

12/1

2/06

12/1

9/06

12/2

6/06

01/0

2/07

01/0

9/07

01/1

6/07

01/2

3/07

01/3

0/07

G B M - 95 th % tile G B M - 5 th % tile S F M R - 95 th % tile S F M R - 5 th % tile

Spread between SFMR Prices

• So the basis at time t

• With variance

( ) ( )( )

( )( )

( )( )

( )( ) q

q

tq

pp

tp

qq

tq

pp

tp

tt pqVar σαα

σαα

ρσαα

σαα

2

2

2

22

2

22

2

2

1111

1111

21111

1111

−−

−−

−−

−−−

−−

−−+

−−

−−=−

( ) ( )∑∑−

=

−−−

=

−−−+1

0

1

011

t

i

itppi

itq

t

iqi αεαε

[ ] [ ]ηαηαpμααpq tq

tqp

tq

tptt )1(1)1()()1()1( 00 −−+−+−−−−=−

Sample SFMR SimulationS F M R S im u la tio n

-2 .0 0

0 .0 0

2 .0 0

4 .0 0

6 .0 0

8 .0 0

1 0 .0 0

1 2 .0 0

1 4 .0 0

1 6 .0 0

05/01/0705/03/0705/05/0705/07/0705/09/0705/11/0705/13/0705/15/0705/17/0705/19/0705/21/0705/23/0705/25/0705/27/0705/29/0705/31/0706/02/0706/04/0706/06/0706/08/0706/10/0706/12/0706/14/0706/16/0706/18/0706/20/0706/22/0706/24/0706/26/0706/28/0706/30/07

$/M

MB

tu

H e n ry H u b T Z 4 S p re a d

Behavior of SFMR Expected BasisB a s is E vo lu tio n U n d e r D iffe re n t M e a n R e ve rs io n R a te s

0 .0 0

1 .0 0

2 .0 0

3 .0 0

4 .0 0

5 .0 0

6 .0 0

7 .0 0

1 20 39 58 77 96115

134153

172191

210229

248267

286305

324343

362381

400419

438457

476495

514533

552571

590

T im e

Pric

e

0 .9 8

1 .0 0

1 .0 2

1 .0 4

1 .0 6

1 .0 8

1 .1 0

1 .1 2

Bas

is (P

ropo

rtio

nal)

P rice w ith Lo w e r M e a n R e ve rs io n R a te P rice w ith H ig h e r M e a n R e ve rs io n R a te B a s is w ith B ias B as is w ithou t B ias

Behavior of SFMR ShocksD e c a y o f S F M R S h o c k s

-1 .6 0 0 0

-1 .4 0 0 0

-1 .2 0 0 0

-1 .0 0 0 0

-0 .8 0 0 0

-0 .6 0 0 0

-0 .4 0 0 0

-0 .2 0 0 0

0 .0 0 0 0

0 .2 0 0 0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

T im e

Stan

dard

Nor

mal

Sho

ck

S lo w e r M e a n R e ve rs io n R a te F a s te r M e a n R e ve rs io n R a te D iffe re n ce

Behavior of SFMR ShocksD e c a y o f S F M R S h o c k s

0 .0 0 0 0

0 .2 0 0 0

0 .4 0 0 0

0 .6 0 0 0

0 .8 0 0 0

1 .0 0 0 0

1 .2 0 0 0

1 .4 0 0 0

1 .6 0 0 0

1 .8 0 0 0

2 .0 0 0 0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

T im e

Stan

dard

Nor

mal

Sho

ck

S lo w e r M e a n R e ve rs io n R a te F a s te r M e a n R e ve rs io n R a te D iffe re n ce

Behavior of SFMR ShocksD e c a y o f S F M R S h o c k s

0 .0 0 0 0

0 .1 0 0 0

0 .2 0 0 0

0 .3 0 0 0

0 .4 0 0 0

0 .5 0 0 0

0 .6 0 0 0

0 .7 0 0 0

0 .8 0 0 0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

T im e

Stan

dard

Nor

mal

Sho

ck

S lo w e r M e a n R e ve rs io n R a te F a s te r M e a n R e ve rs io n R a te D iffe re n ce

Traditional Energy Models

• Margrabe Spread Model– Shares assumptions, both good and bad, with Black-Scholes

• Geometric Brownian Motion– Fixed expected basis equal to today’s basis– Infinite variance of spread

• Single Factor Mean Reverting– Variable expected basis– Finite variance of spread, but different mean reversion rates can lead to

much different decay rates– Difference in shocks can increase and may diverge from what is seen in

reality

Presentation Outline

• Modeling Considerations• Traditional Energy Models

– Modeling Prices– Modeling Spreads

• Vector Autoregression Framework

Vector Autoregressions - The Better Mousetrap

• Vector autoregression framework allows greater flexibility– Established methodology– Robust diagnostic testing– Multiple methodologies to handle shocks

• Future path of prices depends on– Historical path of all modeled prices; and– Future path of other prices

Vector Autoregression Model (VAR)

• Models prices of goods that are close substitutes

where:

qtttt

ptttt

qpq

qpp

εββα

εββα

+++=

+++=

−−

−−

1221212

1121111

),0(~ 2ppt N σε

),0(~ 2qqt N σε

),( qpcorr εερ =

Price Evolution under VAR

• Matrix representation

• Change notation to

• Price at t in terms of P0

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

−

−

qt

pt

t

t

t

t

qp

qp

εε

ββββ

αα

1

1

2221

1211

2

1

t1tt ΕΒPΑP ++= −

)(1

0it0t ΕΑΒPΒP −

−

=

++= ∑t

i

it

The Stability of the Weighting Matrix

• Given the weighting matrix

• Subsequent powers are

⎥⎦

⎤⎢⎣

⎡=

822.0171.00.2530.7261Β

⎥⎦

⎤⎢⎣

⎡=

598.0356.00.5250.3995Β

⎥⎦

⎤⎢⎣

⎡=

659.0316.00.4670.4823Β⎥

⎦

⎤⎢⎣

⎡=

719.0266.00.3920.5712Β

⎥⎦

⎤⎢⎣

⎡=

544.0355.00.5240.34610Β

Sample VAR SimulationV A R S im u la tio n

-2 .0 0

0 .0 0

2 .0 0

4 .0 0

6 .0 0

8 .0 0

1 0 .0 0

1 2 .0 0

05/01/0705/03/0705/05/0705/07/0705/09/0705/11/0705/13/0705/15/0705/17/0705/19/0705/21/0705/23/0705/25/0705/27/0705/29/0705/31/0706/02/0706/04/0706/06/0706/08/0706/10/0706/12/0706/14/0706/16/0706/18/0706/20/0706/22/0706/24/0706/26/0706/28/0706/30/07

$/M

MB

tu

H e n ry H u b T Z 4 S p re a d

Unstable VAR SimulationV A R S im u la tio n

-4 ,0 0 0

-3 ,0 0 0

-2 ,0 0 0

-1 ,0 0 0

0

1 ,0 0 0

2 ,0 0 0

3 ,0 0 0

4 ,0 0 0

5 ,0 0 0

6 ,0 0 0

7 ,0 0 0

05/01/0705/03/0705/05/0705/07/0705/09/0705/11/0705/13/0705/15/0705/17/0705/19/0705/21/0705/23/0705/25/0705/27/0705/29/0705/31/0706/02/0706/04/0706/06/0706/08/0706/10/0706/12/0706/14/0706/16/0706/18/0706/20/0706/22/0706/24/0706/26/0706/28/0706/30/07

$/M

MB

tu

H e n ry H u b T Z 4 S p re a d

Behavior of VAR ShocksD e c a y o f V A R S h o c k s

-0 .1 0 0 0

0 .0 0 0 0

0 .1 0 0 0

0 .2 0 0 0

0 .3 0 0 0

0 .4 0 0 0

0 .5 0 0 0

0 .6 0 0 0

0 .7 0 0 0

0 .8 0 0 0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

T im e

Stan

dard

Nor

mal

Sho

ck

F irs t S h o ck T e rm S e c o n d S h o c k T e rm D iffe re n ce

Behavior of VAR ShocksD e c a y o f V A R S h o c k s

-1 .5 0 0 0

-1 .0 0 0 0

-0 .5 0 0 0

0 .0 0 0 0

0 .5 0 0 0

1 .0 0 0 0

1 .5 0 0 0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

T im e

Stan

dard

Nor

mal

Sho

ck

F irs t S h o ck T e rm S e c o n d S h o c k T e rm D iffe re n ce

Behavior of VAR ShocksD e c a y o f V A R S h o c k s

-0 .2 0 0 0

0 .0 0 0 0

0 .2 0 0 0

0 .4 0 0 0

0 .6 0 0 0

0 .8 0 0 0

1 .0 0 0 0

1 .2 0 0 0

1 .4 0 0 0

1 .6 0 0 0

1 .8 0 0 0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

T im e

Stan

dard

Nor

mal

Sho

ck

F irs t S h o ck T e rm S e c o n d S h o c k T e rm D iffe re n ce

Strengths of the VAR

• Construct paths for prices without associated forward curves

• Several well established tests to determine optimal number of lags

• Two methods to correlate price shocks– Actual correlation and normally distributed shocks– Resample actual historical shocks to pick up correlation

and any non-normal distributions

Weaknesses of the VAR

• More rigorous process to determine parameters• More diagnostic testing of model

– Proper functional form– Stable system of equations

• Number of parameters grows quickly (N2L) and can erode your degrees of freedom, so a larger data set may be required

Pipeline Management Risk Assessment

• Resource management problem• Value of transportation capacity• Risk depends on the price of gas at 3 hubs• Most conservative test shows the need for 100 lags

Pipeline Model SimulationS a m p le M o d e l Ite ra tio n

-2 .0 0

0 .0 0

2 .0 0

4 .0 0

6 .0 0

8 .0 0

1 0 .0 0

1 2 .0 0

1 4 .0 0

04/01/0704/15/0704/29/0705/13/0705/27/0706/10/0706/24/0707/08/0707/22/0708/05/0708/19/0709/02/0709/16/0709/30/0710/14/0710/28/0711/11/0711/25/0712/09/0712/23/0701/06/0801/20/0802/03/0802/17/0803/02/0803/16/0803/30/08

$/M

MB

tu

T ra n s c o Z o n e 1 T ra n s c o Z o n e 3 T ra n s c o Z o n e 6 Z 3 -Z 1 S p re a d Z 6 -Z 3 S p re a d

Summary

• Traditional models may not work well to model absolute price levels and commodity spreads– Infinite variance– Unstable mean spreads– Different rates of mean reversion can cause divergence

over time

• Vector autoregression offers a flexible framework– Better captures price interactions– Derive future path for prices without forward curve– Handle non-normal and heteroscedastic shocks

Questions?

• Contact Info:Ted Kurytkury@teainc.org(904) 360-1444

References• Margrabe, W., 1978, “The Value of an Option to Exchange One Asset for

Another”, Journal of Finance 33:1 p. 177-186• Pindyck, R., 1999, “The Long Run Evolution of Energy Prices”, The

Energy Journal 20:2, p. 1-27• Schwartz, E., 1997, “The Stochastic Behavior of Commodity Prices:

Implications for Valuation and Hedging”, Journal of Finance 52:3, p. 923-973