Co-integration in Crude Oil Components and
the Pricing of Crack Spread Options
Jin-Chuan Duan† & Annie Theriault‡
†Risk Management Institute and Dept of Finance, National U of Singapore
[email protected]
http://www.rmi.nus.edu.sg/DuanJC
and
‡Northwater Capital Management
February 2009
Co-integration in Crude Oil Components ...
www.rmi.nus.edu.sg
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
What is cointegration?
Two time series are cointegrated if they are individually unit
root processes (with or without drift) and some linear
combination of them becomes a stationary time series.
A discrete-time example
∆X1,t = (µ1 + δ1 Zt−1 ) + σ1 ε1,t
∆X2,t = (µ2 + δ2 Zt−1 ) + σ2 ε2,t
Zt = a + bt + X1,t + cX2,t
Naturally, if Zt can somehow be made into a stationary time
series, then X1,t and X2,t are cointegrated.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Compute
Zt = a + bt + X1,t−1 + µ1 + δ1 Zt−1 + σ1 ε1,t
+c (X2,t−1 + µ2 + δ2 Zt−1 + σ2 ε2,t )
= µ1 + cµ2 + a + bt + (1 + δ1 + cδ2 )Zt−1 + σ1 ε1,t + cσ2 ε2,t
The stationarity condition for the above AR(1) process is
obviously |1 + δ1 + cδ2 | < 1.
Note: The stationary condition has nothing to do with the
correlation between ε1,t and ε2,t . Thus, one can have the longand short-run relationships differ in sign.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Simulated data using µ1 = 0.1, µ2 = a = b = 0, δ1 = −0.5,
δ2 = 1, σ1 = σ2 = 1, ρ = −0.5, c = −1 and X1t = X2t = 100.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Time series plots of cointegrated vs. highly correlated data
series
Note: The right plot uses the following parameter values: µ1 = 0.1,
µ2 = a = b = 0, δ1 = δ2 = 0, σ1 = σ2 = 1, ρ = 0.9 and c = −1.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
The continuous-time counterpart
dX1,t = (µ1 + δ1 Zt )dt + σ1 dB1,t
dX2,t = (µ2 + δ2 Zt )dt + σ2 dB2,t
Zt = a + bt + X1,t + cX2,t
The stationary condition can be derived by applying Ito’s
lemma on Zt .
By the standard option pricing theory, cointegration has no
impact on option pricing except for perhaps a statistical
estimation effect.
But with the GARCH feature (or stochastic volatility),
option pricing will explicitly depend on cointegration.
The above results are from Duan and Pliska (2004, Journal
of Economic Dynamics and Control)
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
How does this paper fit in?
Adapt the Duan and Pliska (2004) cointegration-GARCH
option pricing model to work for crude oil crack spread
options.
Conduct an empirical study using Heating Oil/Crude and
Gasoline/Crude spread options traded at the NYMEX.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Empirical evidence for cointegration in the crude oil complex
Rolling series of alternating March and September delivery futures
prices for crude oil together with heating oil and gasoline (one-to-one
volume-adjusted)
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Test statistics
The time-series of alternating March and September delivery
contracts from June 1995 to June 2005.
Heating Oil/Crude
Correlation of
Prices
0.986
Correlation of
Returns
0.873
Phillips-Ouliaris
Statistics
-73.780
Gasoline/Crude
0.988
0.874
-79.666
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
A bivariate model for futures prices in the crude oil complex
A bivariate cointegration-GARCH system under measure P
Adapt the model of Duan and Pliska (2004) to futures prices.
p
p
F1,t,T1
1
ln
= λ1 h1,t − h1,t + δ1 Zt−1 + h1,t 1,t ,
F1,t−1,T1
2
p
p
F2,t,T2
1
= λ2 h2,t − h2,t + δ2 Zt−1 + h2,t 2,t ,
ln
F2,t−1,T2
2
h1,t
=
β1,1 h1,t−1 + β1,2 h1,t−1 (1,t−1 − θ1 )
2
+β1,0 (T1 − t)γ1 + η1 SW1,t ,
h2,t
=
β2,1 h2,t−1 + β2,2 h2,t−1 (2,t−1 − θ1 )
2
+β2,0 (T2 − t)γ2 + η2 SW2,t ,
Zt
=
a + bt + c ln (F2,t,T2 ) + ln (F1,t,T1 ) ,
where 1,t and 2,t follow the bivarate standard normal distribution
with correlation ρ.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
A bivariate model for futures prices in the crude oil complex
Corresponding cointegration-GARCH system under measure Q
F1,t,T1
ln
=
F1,t−1,T1
F2,t,T2
ln
=
F2,t−1,T2
h1,t
h2,t
Zt
p
1
− h1,t + h1,t ξ1,t ,
2
p
1
− h2,t + h2,t ξ2,t ,
2
2
Zt−2
β1,1 h1,t−1 + β1,2 h1,t−1 ξ1,t−1 − λ1 − θ1 − δ1
h1,t−1
+β1,0 (T1 − t)γ1 + η1 SW1,t ,
2
Zt−2
= β2,1 h2,t−1 + β2,2 h2,t−1 ξ2,t−1 − λ2 − θ2 − δ2
h2,t−1
γ2
+β2,0 (T2 − t) + η2 SW2,t ,
=
=
a + bt + c ln (F2,t,T2 ) + ln (F1,t,T1 ) ,
where ξ1,t and ξ2,t follow the bivarate standard normal distribution
with correlation ρ.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Pricing crack spread options
Numerical pricing procedure
Since crude oil crack spread options are American-style, their
prices can be computed by solving
Pt (F1,t,T1 , F2,t,T2 , h1,t+1 , h2,t+1 )
=
sup E Q e−rτ max [ω (42 ∗ F2,τ,T2 − F1,τ,T1 − K) , 0] |Ft ,
t≤τ ≤T op
where F1,τ,T1 is the futures price per barrel of crude oil at any
stopping time, τ , F2,τ,T2 is the futures price per gallon of
heating oil (or gasoline) at any stopping time, τ .
Apply a primal simulation technique. Specifically, we adapt the
Stentoft (2005, Journal of Empirical Finance) approach for the
univariate GARCH model to the bivariate GARCH system.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Pricing crack spread options
Comparing five models
1
2
3
4
5
Constant volatility (CV) model (i.e., the standard bivariate
lognormal model)
Standard bivariate NGARCH(1,1) (G11) model
Bivariate NGARCH(1,1) with cointegration (G11-C) model
Bivariate NGARCH(1,1) with two maturity effects model
(GM)
Bivariate NGARCH(1,1) with two maturity effects and
cointegration model (GM-C)
Note: The first maturity effect is the so-called Samuelson effect and
the second is related to switching over to new futures. The switching
dummy is set to 1 if the futures’ remaining maturity is between 24
and 32 trading days for crude oil, between 25 and 39 trading days for
heating oil, and between 25 to 38 days for gasoline.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
ML parameter estimation
The time-series of alternating March and September delivery
contracts from June 1995 to June 2005.
Heating Oil/Crude
CV
G11
G11-C
GM
Mean Parameters
λ1 0.0288 0.0338 0.0380 0.0250
(1.36) (1.55)
(1.76)
(1.18)
λ2 0.0296 0.0361 0.0396 0.0474
(1.43) (1.68)
(1.83)
(2.23)
δ1
-0.0084
(-1.03)
δ2
0.0238
(2.79)
Co-integration in Crude Oil Components ...
GM-C
0.0095
(7.60)
0.0134
(8.94)
-0.0034
(-0.39)
0.0031
(0.35)
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
ML parameter estimation
Heating Oil/Crude
CV
G11
G11-C
GM
Volatility Parameters
β1,0 4.03E-04 3.73E-05 5.10E-05 3.67E-05
(70.72)
(6.17)
(4.34)
(8.03)
β1,1
0.8694
0.7995
0.9459
(50.43)
(17.51)
(158.57)
β1,2
0.0329
0.0054
0.0154
(8.06)
(2.78)
(7.01)
θ1
-0.4212
-3.5608
0.3278
(-4.82)
(-2.74)
(1.93)
γ1
-0.2775
(-9.72)
η1
1.75E-05
(4.09)
Co-integration in Crude Oil Components ...
GM-C
3.09E-06
(4.96)
0.9753
(357.52)
0.0095
(7.60)
0.7232
(5.03)
-0.1747
(-2.91)
4.70E-05
(18.23)
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
ML parameter estimation
Heating Oil/Crude
CV
G11
G11-C
GM
Volatility Parameters
β2,0 4.21E-04 4.57E-05 4.73E-05 8.19E-06
(104.03)
(9.58)
(8.92)
(3.69)
β2,1
0.8156
0.8121
0.9600
(55.60)
(44.82)
(259.65)
β2,2
0.0668
0.0258
0.0140
(11.37)
(6.87)
(8.43)
θ2
-0.4094
-1.3624
0.1383
(-6.99)
(-7.52)
(1.57)
γ2
-0.0704
(-1.21)
η2
3.96E-05
(12.97)
Co-integration in Crude Oil Components ...
GM-C
1.02E-05
(4.08)
0.9677
(335.63)
0.0134
(8.94)
0.2674
(3.05)
-0.2628
(-5.35)
3.55E-05
(20.75)
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
ML parameter estimation
Heating Oil/Crude
CV
G11
G11-C
GM
Correlation and Co-integration Parameters
ρ
0.8790
0.8747
0.8790
0.8767
(489.30) (332.99)
(352.32)
(336.00)
a
-3.5421808
(-1088)
b
-2.70E-05
(-18.94)
c
-0.9558
(-271.29)
Log-Lik
Stat1
Stat2
CS
18975
GM-C
0.8800
(347.86)
-3.542181
(-1088)
-2.70E-05
(-18.94)
-0.9558
(-271.29)
18989
19034
19125
19128
0.908
0.894
0.873
0.886
0.969
0.963
0.974
0.990
0.982
0.994
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Option data & pricing errors
Crack spread option data
The option data are the NYMEX daily settlement prices
between January 2004 and December 2005.
18 consecutive expiration months for Heating Oil/Crude
spread options and 12 consecutive expiration months for
Gasoline/Crude spread options.
Moneyness range: 0.9 to 1.1. The sample has 3002 Heating
Oil/Crude spread option prices and 3430 Gasoline/Crude
spread option prices.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Option data & pricing errors
Pricing errors using the entire sample
MPE MdPE MAPE MdAPE RRMSE
A. Heating Oil/Crude Futures Spread Options
CV
1.69
-7.35
29.99
21.72
45.14
G11
6.12
-4.36
30.39
20.27
47.90
G11-C
5.61
-4.88
30.35
20.02
48.02
GM
4.08
-1.79
24.57
17.92
39.70
GM-C
0.58
-5.87
25.24
19.78
39.10
B. Gasoline/Crude Futures Spread Options
CV
-1.31
-5.85
26.45
24.87
30.67
G11
6.51
6.12
28.24
26.39
32.93
G11-C
9.91
7.20
27.25
24.58
32.95
GM
-1.43
-2.55
21.26
19.26
25.46
GM-C
4.12
2.83
22.12
18.28
28.14
The performance improvement of the GM an GM-C model mainly
concentrate in longer-term options (61 trading days or longer) across
all moneyness.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Option data & pricing errors
Absolute percentage pricing error regression
AP E
2
= a1 + a2 (T − t) + a3 (T − t)2 + a4 Mt + a5 Mt2 + a6 σ1,t + a7 σ1,t
+a8 ρt + a9 (T − t)I P + a10 (T − t)2 I P + a11 Mt I P + a12 Mt2 I P
+a13 σ1,t I P + a14 ρt I P + νt ,
where (T − t) is the time to maturity of the options contract in years,
Mt is the moneyness of the option, σ1,t is the 20-day standard
deviation of crude oil futures, ρt is the 20-day correlation between
crude oil futures and heating oil (or gasoline) futures, I P is an
indicator equal to 1 for put options and 0 otherwise and νt is an error
term.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Option data & pricing errors
A. Heating Oil/Crude Futures Spread Options
CV
G11
G11-C
a1
0.97
0.54
0.80
0.41
0.85
0.44
a2
-0.39
-2.54
-0.19
-1.11
-0.25
-1.46
a3
1.24
5.24
1.08
4.17
1.13
4.38
a4
-1.79
-0.50
-2.47
-0.63
-2.64
-0.68
a5
0.53
0.29
0.86
0.44
0.95
0.49
a6
-3.82
-7.25
-2.89
-5.02
-3.49
-6.08
a7
5.30
7.02
3.44
4.18
4.41
5.36
a8
1.36
8.81
1.78
10.56
1.92
11.41
a9
-0.42
-2.35
-0.55
-2.80
-0.59
-3.00
a10
0.43
1.67
0.60
2.16
0.65
2.35
a11
1.64
3.58
1.85
3.70
1.96
3.93
a12
-0.12
-0.43
-0.20
-0.66
-0.27
-0.88
a13
0.67
4.17
0.69
3.94
0.74
4.19
a14
-1.94
-7.83
-2.08
-7.69
-2.14
-7.90
2
R
0.301
0.308
0.319
Co-integration in Crude Oil Components ...
GM
2.40
1.30
-0.09
-0.57
0.31
1.28
-5.61
-1.51
2.34
1.26
-3.72
-6.81
5.14
6.57
1.93
12.05
-0.32
-1.70
0.24
0.89
1.58
3.34
-0.11
-0.37
0.33
1.98
-1.77
-6.88
0.121
GM-C
2.35
1.32
-0.23
-1.50
0.50
2.09
-5.75
-1.61
2.39
1.34
-1.91
-3.62
2.67
3.55
1.77
11.48
-0.24
-1.33
-0.02
-0.07
1.89
4.14
-0.26
-0.92
0.24
1.50
-1.92
-7.74
0.111
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
Option data & pricing errors
B. Gasoline/Crude Futures Spread Options
CV
G11
G11-C
a1
3.70
5.13
2.83
3.48
2.04
2.33
a2
-0.99
-13.85
-0.19
-2.32
-0.32
-3.70
a3
2.50
20.70
1.55
11.39
1.87
12.73
a4
-3.55
-2.45
-2.26
-1.39
-2.35
-1.34
a5
1.41
1.95
0.74
0.90
0.81
0.91
a6
-0.77
-3.63
-0.27
-1.11
-0.67
-2.58
a7
1.69
5.12
1.14
3.05
1.50
3.72
a8
-1.27
-10.62
-1.19
-8.83
-0.24
-1.61
a9
0.98
10.62
0.36
3.49
0.57
5.04
a10
-2.04
-14.12
-1.25
-7.67
-1.55
-8.79
a11
-0.61
-2.01
-2.82
-8.21
-0.70
-1.88
a12
1.00
5.94
2.09
11.07
1.00
4.90
a13
-0.29
-3.99
-0.66
-8.23
-0.57
-6.52
a14
-0.41
-2.49
0.99
5.32
-0.16
-0.81
R2
0.363
0.322
0.338
Co-integration in Crude Oil Components ...
GM
2.92
3.85
-0.69
-9.16
1.37
10.80
-4.09
-2.69
1.78
2.33
-1.00
-4.45
1.93
5.55
-0.24
-1.93
0.40
4.16
-1.18
-7.80
-1.28
-3.99
1.19
6.75
-0.11
-1.51
0.10
0.55
0.135
GM-C
2.10
2.15
-0.53
-5.50
1.18
7.26
-3.43
-1.75
1.48
1.50
-0.41
-1.43
0.88
1.96
0.16
0.99
0.31
2.47
-1.08
-5.54
-0.68
-1.65
0.88
3.86
-0.01
-0.08
-0.23
-1.00
0.070
JC Duan & A Theriault, 02/2009
Background...
Evidence for cointegration
Model & methodology
Data & results
Conclusion
MLE analysis of futures prices reveals the importance of
incorporating GARCH, cointegration and maturity effects
(the Samuelson effect and contract-switching effect) for
paired components in the crude oil complex.
Crude oil crack spread options can be priced better with
the cointegration-GARCH model with built-in maturity
effects. Performance improvement mainly concentrates on
longer-term spread options (61 trading days to maturity or
longer).
Pricing errors show the least systematic bias when
cointegration, GARCH and maturity effects are
incorporated into the model.
Co-integration in Crude Oil Components ...
JC Duan & A Theriault, 02/2009