Co-integration in Crude Oil Components... the Pricing of Crack Spread Options

Transcription

1 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] and Northwater Capital Management February 2009 Co-integration in Crude Oil Components...

2 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 X 1,t = (µ 1 + δ 1 Z t 1 ) + σ 1 ε 1,t X 2,t = (µ 2 + δ 2 Z t 1 ) + σ 2 ε 2,t Z t = a + bt + X 1,t + cx 2,t Naturally, if Z t can somehow be made into a stationary time series, then X 1,t and X 2,t are cointegrated.

3 Compute Z t = a + bt + X 1,t 1 + µ 1 + δ 1 Z t 1 + σ 1 ε 1,t +c (X 2,t 1 + µ 2 + δ 2 Z t 1 + σ 2 ε 2,t ) = µ 1 + cµ 2 + a + bt + (1 + δ 1 + cδ 2 )Z t 1 + σ 1 ε 1,t + cσ 2 ε 2,t The stationarity condition for the above AR(1) process is obviously 1 + δ 1 + cδ 2 < 1. Note: The stationary condition has nothing to do with the correlation between ε 1,t and ε 2,t. Thus, one can have the longand short-run relationships differ in sign.

4 Simulated data using µ 1 = 0.1, µ 2 = a = b = 0, δ 1 = 0.5, δ 2 = 1, σ 1 = σ 2 = 1, ρ = 0.5, c = 1 and X 1t = X 2t = 100.

5 Time series plots of cointegrated vs. highly correlated data series Note: The right plot uses the following parameter values: µ 1 = 0.1, µ 2 = a = b = 0, δ 1 = δ 2 = 0, σ 1 = σ 2 = 1, ρ = 0.9 and c = 1.

6 The continuous-time counterpart dx 1,t = (µ 1 + δ 1 Z t )dt + σ 1 db 1,t dx 2,t = (µ 2 + δ 2 Z t )dt + σ 2 db 2,t Z t = a + bt + X 1,t + cx 2,t The stationary condition can be derived by applying Ito s lemma on Z t. 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)

7 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.

8 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)

9 Test statistics The time-series of alternating March and September delivery contracts from June 1995 to June Correlation of Correlation of Phillips-Ouliaris Prices Returns Statistics Heating Oil/Crude Gasoline/Crude

10 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. ( ) F1,t,T1 ln = λ 1 h1,t 1 F 1,t 1,T1 2 h 1,t + δ 1 Z t 1 + h 1,t ɛ 1,t, ( ) F2,t,T2 ln = λ 2 h2,t 1 2 h 2,t + δ 2 Z t 1 + h 2,t ɛ 2,t, F 2,t 1,T2 h 1,t = β 1,1 h 1,t 1 + β 1,2 h 1,t 1 (ɛ 1,t 1 θ 1 ) 2 +β 1,0 (T 1 t) γ1 + η 1 SW 1,t, h 2,t = β 2,1 h 2,t 1 + β 2,2 h 2,t 1 (ɛ 2,t 1 θ 1 ) 2 +β 2,0 (T 2 t) γ2 + η 2 SW 2,t, Z t = a + bt + c ln (F 2,t,T2 ) + ln (F 1,t,T1 ), where ɛ 1,t and ɛ 2,t follow the bivarate standard normal distribution with correlation ρ.

11 A bivariate model for futures prices in the crude oil complex Corresponding cointegration-garch system under measure Q ( F1,t,T1 ln ln F 1,t 1,T1 ( F2,t,T2 F 2,t 1,T2 ) ) = 1 2 h 1,t + h 1,t ξ 1,t, = 1 2 h 2,t + h 2,t ξ 2,t, ( Z t 2 h 1,t = β 1,1 h 1,t 1 + β 1,2 h 1,t 1 ξ 1,t 1 λ 1 θ 1 δ 1 h 1,t 1 +β 1,0 (T 1 t) γ1 + η 1 SW 1,t, ( ) 2 Z t 2 h 2,t = β 2,1 h 2,t 1 + β 2,2 h 2,t 1 ξ 2,t 1 λ 2 θ 2 δ 2 h 2,t 1 +β 2,0 (T 2 t) γ2 + η 2 SW 2,t, Z t = a + bt + c ln (F 2,t,T2 ) + ln (F 1,t,T1 ), where ξ 1,t and ξ 2,t follow the bivarate standard normal distribution with correlation ρ. ) 2

12 Pricing crack spread options Numerical pricing procedure Since crude oil crack spread options are American-style, their prices can be computed by solving P t (F 1,t,T1, F 2,t,T2, h 1,t+1, h 2,t+1 ) = sup t τ T op E Q { e rτ max [ω (42 F 2,τ,T2 F 1,τ,T1 K), 0] F t }, where F 1,τ,T1 is the futures price per barrel of crude oil at any stopping time, τ, F 2,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.

13 Pricing crack spread options Comparing five models 1 Constant volatility (CV) model (i.e., the standard bivariate lognormal model) 2 Standard bivariate NGARCH(1,1) (G11) model 3 Bivariate NGARCH(1,1) with cointegration (G11-C) model 4 Bivariate NGARCH(1,1) with two maturity effects model (GM) 5 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.

14 ML parameter estimation The time-series of alternating March and September delivery contracts from June 1995 to June Heating Oil/Crude CV G11 G11-C GM GM-C Mean Parameters λ (1.36) (1.55) (1.76) (1.18) (7.60) λ (1.43) (1.68) (1.83) (2.23) (8.94) δ (-1.03) (-0.39) δ (2.79) (0.35)

15 ML parameter estimation Heating Oil/Crude CV G11 G11-C GM GM-C Volatility Parameters β 1,0 4.03E E E E E-06 (70.72) (6.17) (4.34) (8.03) (4.96) β 1, (50.43) (17.51) (158.57) (357.52) β 1, (8.06) (2.78) (7.01) (7.60) θ (-4.82) (-2.74) (1.93) (5.03) γ (-9.72) (-2.91) η E E-05 (4.09) (18.23)

16 ML parameter estimation Heating Oil/Crude CV G11 G11-C GM GM-C Volatility Parameters β 2,0 4.21E E E E E-05 (104.03) (9.58) (8.92) (3.69) (4.08) β 2, (55.60) (44.82) (259.65) (335.63) β 2, (11.37) (6.87) (8.43) (8.94) θ (-6.99) (-7.52) (1.57) (3.05) γ (-1.21) (-5.35) η E E-05 (12.97) (20.75)

17 ML parameter estimation Heating Oil/Crude CV G11 G11-C GM GM-C Correlation and Co-integration Parameters ρ (489.30) (332.99) (352.32) (336.00) (347.86) a (-1088) (-1088) b -2.70E E-05 (-18.94) (-18.94) c ( ) ( ) Log-Lik Stat Stat CS

18 Option data & pricing errors Crack spread option data The option data are the NYMEX daily settlement prices between January 2004 and December 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.

19 Option data & pricing errors Pricing errors using the entire sample MPE MdPE MAPE MdAPE RRMSE A. Heating Oil/Crude Futures Spread Options CV G G11-C GM GM-C B. Gasoline/Crude Futures Spread Options CV G G11-C GM GM-C The performance improvement of the GM an GM-C model mainly concentrate in longer-term options (61 trading days or longer) across all moneyness.

20 Option data & pricing errors Absolute percentage pricing error regression AP E = a 1 + a 2 (T t) + a 3 (T t) 2 + a 4 M t + a 5 Mt 2 + a 6 σ 1,t + a 7 σ1,t 2 +a 8 ρ t + a 9 (T t)i P + a 10 (T t) 2 I P + a 11 M t I P + a 12 Mt 2 I P +a 13 σ 1,t I P + a 14 ρ t I P + ν t, where (T t) is the time to maturity of the options contract in years, M t is the moneyness of the option, σ 1,t is the 20-day standard deviation of crude oil futures, ρ t 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 ν t is an error term.

21 Option data & pricing errors A. Heating Oil/Crude Futures Spread Options CV G11 G11-C GM GM-C a a a a a a a a a a a a a a R

22 Option data & pricing errors B. Gasoline/Crude Futures Spread Options CV G11 G11-C GM GM-C a a a a a a a a a a a a a a R

23 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.