Jumps and regime switching in commodity prices

Jumps and Regime Switching in Commodity Prices Tanmay Satpathy (B.Tech., IIT Mumbai) A Thesis Submitted For The Degree of Doctor Of Philosophy Department Of Finance National University of Singapore 2011 i Acknowledgements I can say with certainty that the journey of PhD has been the most enriching and valuable experience for me till date. This has been a process of self discovery, particularly about my strengths, weaknesses, and attitudes. For me, however cliché it may sound, the most important takeaway is to question any phenomena or observation and ask myself what I know (if at all I know anything) and what should I know? Any amount of gratitude to my dissertation committee chair Prof. Jin-Chuan Duan would be insufficient. He has been immensely patient with me throughout the whole process, giving me just about the right amount of hint and guidance that would lead me to think about the problem rather than holding my hand and bringing me closer to the solution. He has always encouraged me to try to find the heart of the problem and not be lost in the trivial details surrounding the issue. A special mention goes towards his efforts in organizing a study group of fellow researchers meeting fortnightly to discuss important topics in quantitative finance and forcing us to drill deep into elementary concepts, understand the audience and improve our communication skills. I sincerely believe that this training cannot be obtained by reading journals or attending classes. For me, this has been particularly helpful for finding a quantitative job. I will always look up to him as a great scholar and mentor and hopefully be able to imbibe a few of his traits. ii I would also like to thank my dissertation committee members Dr Nan Li and Prof. Deng Yongheng for raising important questions, providing suggestions on various occasions and helping me to refine my problem statement. I have had a chance to interact with many other professors who deserve a special mention. Dr Anand Srinivasan and Dr Srinivasan Sankaraguruswamy have been the PhD coordinators during my stay and have been always willing to help in times when I have been lost and needed guidance. I have been extremely fortunate to interact freely with Prof. Joseph Cherian, Prof. Aaron Low and Ganesh Ramachandran who have shared their invaluable industry insights and often guided me with my job search. I would also like to thank the Finance Department and RMI who have provided me with the data and resources necessary for the research. In particular E’leen from RMI, Richa and Callie from Finance department and Hamidah and Cheow Loo from the Business School have provided invaluable assistance throughout my stay in the program. Apart from academic help, I have been fortunate enough to be invited by the Finance Department and RMI for lunch and dinner on special occasions and conferences. A PhD student invariably spends most of his time in his office and the association with fellow students developed during this time is a lifelong one. I consider myself very lucky to be surrounded by smart, friendly and helpful people, not only from the Finance Department but also from all other departments in the business school. I cannot justice to all by taking a few names here but I would iii still like to thank my seniors Shirish and Anna and my juniors Eugene and Michelle who have been with me through some of the most difficult times. I would also like to thank Sajid Memon, my colleague at Nomura who helped me with some technical sections of the thesis. Finally, I am indebted to my parents, for their eternal, selfless love and sacrifices they have made for me, and to my younger brother for his care and affection. Without their support, this journey could not have even started. This thesis is dedicated to them. iv Table of Contents 1. Introduction………………………………………………………………… 01 2. The Model…………………………………………………………………….05 2.1 Correlated Jump model………………………………………………08 2.2 Regime Switching model…………………………………………….13 2.3 Correlated Jump Regime Switching model………………………….15 3. Model estimation…………………………………………………………… 18 3.1 State space formulation for the Correlated Jump model…………… 18 3.2 State space formulation for the Regime Switching model………… .21 3.3 State space formulation for the Correlated Jump Regime Switching model…………………………………………………………………23 3.4 Particle Filter…………………………………………………………25 3.5 Parameter Estimation using EM algorithm………………………… 28 4. Empirical Analysis………………………………………………………… .31 4.1 Data………………………………………………………………… 31 4.2 Implementation assumptions……………………………………… .32 4.3 Implementation results……………………………………………….33 5. Conclusion……………………………………………………………………40 References……………………………………………………………………….43 Appendix A…………………………………………………………………… .49 Appendix B…………………………………………………………………… .50 v Summary This article explores the role of jumps and changes in regime in commodity prices. Unlike equity and foreign exchange markets, most widely used models in pricing commodity derivatives use dynamic latent factor models. Even though commodity prices have been shown to have regime switching behavior, not many have used it to price derivatives. Similarly, theoretical models using jumps in latent factors have been suggested before, but very few empirical studies have been done in this regard owing to the complexity of estimating these latent factor models. Moreover, models combining these two phenomena haven’t been explored for commodities before. We extend the Schwartz and Smith (2000) idea of representing factors by permanent and transitory components to build a sequence of models to understand the individual and collective importance of these two features. Due to the non-Gaussian nature of the models, we adopt particle filtering to estimate the latent factors and the model parameters. Analyzing weekly crude oil futures data from 1990-2008, we find strong evidence of high and low volatility regimes which are very persistent. At the same time, we find strong statistical evidence for and economic significance of jumps in both the permanent and transitory factors with the latter being relatively more important. Also, jumps are much more likely to occur in the high volatility regime which is associated with periods of political and economic instability. On average, the final model shows a 19% improvement in fit over the model without jumps or regime switching. vi List of Tables Table1 - Sample Characteristics of changes in log futures prices……………….56 Table2 - MLE for SS and CJ model using Kalman and Particle Filter………… 57 Table3 - MLE for RS model using Jacobian and Particle Filter…………………59 Table4 - MLE for CJRS model using Jacobian and Particle Filter………………61 Table5 – Log likelihood ratio comparison across models……………………….63 Table6 - Analysis of measurement errors….…………………………………….64 vii List of Figures Figure - Changes in log futures prices for the NYMEX crude oil futures with 1and 18 month maturities (F1 and F5), 1990-2008……………………………….65 Figure - NYMEX crude oil futures term structures during the run-up to the 1991 Gulf War…………………………………………………………… .………….65 Figure - CJRS model implied smoothed probability of being in Regime (high vol regime), 1990-2008 ………………………………………………………….66 Figure - Actual and predicted futures prices on14/7/2008 ………………….…67 Figure - Actual and predicted futures prices on 21/7/2008…………………….67 viii List of symbols used Prices St Spot Price at time t Rt Regime at time t, takes values 0, 1. χt ξt Short term factor Equilibrium or long term factor F1,F2,F3,F4,F5 Futures contracts prices of 1, 5,9,13 & 18 month maturities Probability Measures Physical probability measure P Risk Neutral probability measure Q Drift parameters µξ Equilibrium drift rate (under P , annualized); same for R0 , R1 µξ* Equilibrium drift rate (under Q , annualized); same for R0 , R1 κ Coefficient of mean reversion for short term factor (annualized); same for R0 , R1 Market price of risk for short term factor (annualized); same for R0 , R1 λχ µξR Equilibrium drift rate in Rt (under P , annualized) t µξ*R Equilibrium drift rate in Rt (under Q , annualized) κR Coefficient of mean reversion for short term factor in Rt (annualized) Market price of risk for short term factor in Rt (annualized) t t λχR t Jump Parameters Poisson process under P , Q (same for R0 , R1 ) Pt , Pt* Poisson process under P , Q with regime dependent intensity Pt Rt , Pt* Rt J t = [ J χ t J ξ t ]' Normally distributed jump sizes under P (same for R0 , R1 ) J t* = [ J χ* t J ξ*t ]' Normally distributed jump sizes under Q (same for R0 , R1 ) J tRt = [ J χRtt J ξRtt ]' Normally distributed jump size in Rt under P J t* Rt = [ J χ* tRt J ξ*tRt ]' Normally distributed jump size in Rt under Q µχ J Mean jump size(under P ) of short term factor (same for R0 , R1 ) µ *χ Mean jump size(under Q ) of short term factor (same for R0 , R1 ) J µ χRJ Mean jump size in Rt (under P ) of short term factor, t µ *χ R t J Mean jump size (under Q ) of short term factor in Rt ix wt(i ) = i) p ( xt(i ) , xt(−∆ ) t yt , yt −∆t , (i ) i q ( xt yt ) pˆ ( xt −∆t yt −∆t , ) i) p ( yt , xt(i ) , xt(−∆ ) t yt −∆t , ∝ (i ) i q ( xt yt ) pˆ ( xt −∆t yt −∆t , ) ≈ i) ˆ i ) p ( yt xt(i ) ) p ( xt(i ) | xt(−∆ t ) p ( xt −∆t yt −∆t , q ( xt(i ) yt ) pˆ ( xti−∆t yt −∆t , )  ∂xt(i )  i) p ( yt xt(i ) ) p ( xt(i ) | xt(−∆ ) det  (i )  t  ∂ε t  = φ (ε t(i ) ) The density p ( yt xt(i ) ) is given by the measurement equation in (10) and i) p ( xt(i ) | xt(−∆ t ) is given by equation (9). M Step 3: Compute the probability: wt(i ) = wt(i ) / ∑ wt(i ) . Re-sample particles using i =1 (i ) t w to generate a sample of M equally weighted particles. Go back to Step to advance to the next time point. A complete-data log-likelihood function The complete data can be chosen as (VTR , YT ) where VTR = (v∆Rt , v2R∆t , , vnR∆t ) . Let xt* = ( BtR )−1 ( ytR − AtR − vtR ) . The log-likelihood function is ln L(VTR , YT Θ) n = ∑ ln L(viR∆t , yi∆t V(iR−1) ∆ t , Y( i −1) ∆t , Θ) i =1 n n i =1 i =1 = ∑ ln p( yi∆t | viR∆t ,V( iR−1) ∆ t , Y(i −1) ∆t , Θ) + ∑ ln p (viR∆t | Θ) n n R R R R R R = ∑ ln p( yiNR ∆t | yi∆t , vi∆t , V( i −1) ∆ t , Y( i −1) ∆t , Θ) + ∑ ln p ( yi∆t | vi∆t , V( i −1) ∆ t , Y( i −1) ∆t , Θ) i =1 i =1 n + ∑ ln p (viR∆t | Θ) i =1 n n n i =1 i =1 i =1 n n * R R * R = ∑ ln p( yiNR ∆t | xi∆t = xi∆t , Θ) + ∑ ln p ( yi∆t | vi∆t , x( i −1) ∆t = x( i −1) ∆t , Θ) + ∑ ln p ( vi∆t | Θ) * * *  R −1  = ∑ ln p( yiNR ∆t | xi∆t = xi∆t , Θ) + ∑ ln p (xi∆t = xi∆t | x( i −1) ∆t = x( i −1) ∆t , Θ) det  ( Bi∆t )  i =1 i =1 n + ∑ ln p (viR∆t | Θ) i =1 52 Note that p ( yiNR ∆t | xi∆t , Θ) is determined by a subset of the measurement equation in (26) and p ( xi∆t | x(i −1) ∆t , Θ) is given by equation (25). Applying fixed-lag smoothing means that E ( ln L(VTR , YT Θ) | YT , Θ( j ) ) n * ( j) ≅ ∑ E ( ln p ( yiNR ) ∆t | xi∆t = xi∆t , Θ) | Ymin(( i + l ) ∆t ,T ) , Θ i =1 n ( + ∑ E ln p ( xi∆t = xi*∆t | x( i −1) ∆t = x(*i −1) ∆t , Θ) det ( BiR∆t ) −1  | Ymin(( i +l ) ∆t ,T ) , Θ( j ) i =1 ) n + ∑ E ( ln p (viR∆t | Θ) | Ymin((i + l ) ∆t ,T ) , Θ( j ) ) i =1 The conditional expectation of any item is immediately computed by summing over particles when the particle filter just reaches past l periods. B.2. Filtering and likelihood for RS and CJRS models In addition to the variables present in the CJ model, we have rt = 0,1: Values which the regime takes at time t. The measurement equations in (29) and (33) can be written as two sub-systems: ytR = AtR (rt ) + BtR xt + vtR NR t y (B3) = A (rt ) + B x + v NR t NR t t NR t (B4) A localized particle filter The steps for the derivation of the weights are very similar to the CJ model except for the fact that the state now includes the regime in addition to the latent factors. i) (i ) (i ) (i ) M ) . Follow the steps below: Now we consider ( xt(−∆ t , rt −∆t , xt , rt ) (i = 1, 2, i) (i ) Step 1: Start from an equally weighted sample, ( xt(−∆ t , rt −∆t ) (i = 1, 2, M ) which represents the filtering density at time t-1; that is, p( xt −∆t , rt −∆t yt −∆t , ) . i) Corresponding to each xt(−∆ t , sample a two-dimensional independent standard normal random variables, ε t(i ) . Thus, vtR (i ) = H ε t( i ) is the vector of the measurement errors for the second and fourth futures contracts. (Using the normality sampler does not mean that the measurement errors are normally distributed. The importance weight will automatically adjust for the mismatch in distributions.). Corresponding to each rt(−∆i ) t , sample rt(i ) using the transition probability matrix Step 2: Use equation (B3) to invert to obtain the implied latent (i ) factors: xt . Denote the density of this localized sampler by 53 i) (i ) p (rt(i ) | rt(−∆ t )φ (ε t ) where φ ( x) is  ∂xt(i )  det  ( i )   ∂ε t  the bivariate independent standard normal density function and ∂xt(i ) = −( BtR )−1 H . Compute the importance weight for (i ) ∂ε t (i ) t q( x , rt (i ) (i ) t yt ) . Clearly, q ( x , rt (i ) yt ) = i) (i ) (i ) (i ) ( xt(−∆ t , rt −∆t , xt , rt ) which is also the same for the marginal, i.e., ( xt(i ) , rt (i ) ) . The weight equals w t( i ) = p ( x t( i ) , rt ( i ) , x t(−i )∆ t , rt (−i∆) t y t , y t − ∆ t , ) q ( x t( i ) , rt ( i ) y t ) pˆ ( x ti − ∆ t , rt (−i ∆) t y t − ∆ t , ) ∝ p ( y t , x t( i ) , rt ( i ) , x t(−i )∆ t , rt (−i∆) t y t − ∆ t , ) q ( x t( i ) , rt ( i ) y t ) pˆ ( x ti − ∆ t , rt (−i∆) t y t − ∆ t , ) ≈ ≈ p ( y t x t( i ) , rt ( i ) ) p ( x t( i ) , rt ( i ) | x t(−i )∆ t , rt (−i∆) t ) pˆ ( x ti − ∆ t , rt (−i∆) t y t − ∆ t , (i ) t q ( x , rt (i ) t (i ) p ( y t x , rt ) p ( x (i ) (i) t (i ) t y t ) pˆ ( x (i) | rt , x q ( x , rt (i ) i t − ∆t (i) t − ∆t (i ) t − ∆t ,r ) p ( rt y t ) pˆ ( x (i) i t − ∆t yt − ∆t , (i) t − ∆t |r (i ) t − ∆t ,r ) ) ) pˆ ( x ti − ∆ t , rt (−i∆) t y t − ∆ t , yt − ∆t , ) )  ∂x (i )  p ( y t x t( i ) , rt ( i ) ) p ( x t( i ) | rt ( i ) , x t(−i )∆ t ) det  t( i )   ∂ε t  = (i ) φ (ε t ) The density p ( yt xt( i ) , rt(i ) ) is given by the measurement equations (29) i) and (33) and p ( xt(i ) | rt(i ) , xt(−∆ t ) is given by equations (28) and (31) for RS and CJRS models respectively. M Step 3: Compute the probability: wt(i ) = wt(i ) / ∑ wt(i ) . Re-sample particles using i =1 (i ) t w to generate a sample of M equally weighted particles. Go back to Step to advance to the next time point. A complete-data log-likelihood function The complete data can be chosen as (VTR , RT , YT ) where VTR = (v∆Rt , v2R∆t , , vnR∆t ) and RT = (r∆t , r2 ∆t , ., rn∆t ) . Let xt* = ( BtR ) −1 ( ytR − AtR (rt ) − vtR ) . The log-likelihood function is 54 ln L(VTR , RT , YT Θ) n = ∑ ln L(viR∆t , yi∆t , ri∆t V( iR−1) ∆ t , Y(i −1) ∆t , R( i −1) ∆t Θ) i =1 n n = ∑ ln p( yi∆t | viR∆t ,V( iR−1) ∆ t , Y(i −1) ∆t , ri∆t , R(i −1) ∆t , Θ) + ∑ ln p (viR∆t | Θ) i =1 i =1 n + ∑ ln p(ri∆t | r( i −1) ∆t , Θ) i =1 n R R R = ∑ ln p( yiNR ∆t | yi∆t , vi∆t , ri∆t , V( i −1) ∆ t , Y( i −1) ∆t , R( i −1) ∆t , Θ) i =1 n + ∑ ln p( yiR∆t | viR∆t , ri∆t , V( iR−1) ∆ t , R( i −1) ∆t , Y(i −1) ∆t , Θ) i =1 n n i =1 i =1 + ∑ ln p(viR∆t | Θ) + ∑ ln p (ri∆t | r( i −1) ∆t , Θ) n n * * R R = ∑ ln p( yiNR ∆t | xi∆t = xi∆t , ri∆t , Θ) + ∑ ln p ( yi∆t | vi∆t , ri∆t , x( i −1) ∆t = x( i −1) ∆t , Θ) i =1 i =1 n n i =1 i =1 + ∑ ln p(viR∆t | Θ) + ∑ ln p (ri∆t | r( i −1) ∆t , Θ) n * = ∑ ln p( yiNR ∆t | xi∆t = xi∆t , ri∆t , Θ) i =1 n + ∑ ln p(xi∆t = xi*∆t | x( i −1) ∆t = x(*i −1) ∆t , ri∆t , Θ) det ( BiR∆t ) −1  i =1 n n + ∑ ln p(viR∆t | Θ) + ∑ ln p (ri∆t | r( i −1) ∆t , Θ) i =1 i =1 NR i∆t Note that p ( y | xi∆t = xi*∆t , ri∆t , Θ) is determined by a subset of the measurement equations in (27) and (31) and p ( xi∆t | x( i −1) ∆t , ri∆t , Θ) is given by equations (26) and (30). Applying fixed-lag smoothing means that E ( ln L(VTR , YT , RT Θ) | YT , Θ( j ) ) n * ( j) ≅ ∑ E ( ln p ( yiNR ) ∆t | xi∆t = xi∆t , ri∆t , Θ) | Ymin(( i + l ) ∆t ,T ) , Θ i =1 n ( + ∑ E ln p ( xi∆t = xi*∆t | x( i −1) ∆t = x(*i −1) ∆t , ri∆t , Θ) det ( BiR∆t ) −1  | Ymin(( i +l ) ∆t ,T ) , Θ( j ) i =1 n n i =1 i =1 ) + ∑ E ( ln p (viR∆t | Θ) | Ymin(( i +l ) ∆t ,T ) , Θ( j ) ) + ∑ E ( ln p(ri∆t | r( i −1) ∆t , Θ) | Ymin(( i +l ) ∆t ,T ) , Θ( j ) ) The conditional expectation of any item is immediately computed by summing over particles when the particle filter just reaches past l periods. 55 Table 1: Sample characteristics of changes in log futures prices (1990-2008) Table reports the mean, standard deviation, skewness and kurtosis of changes in the log prices of the NYMEX crude oil futures contracts with maturities 1, 5, 9, 13 and 18 months (F1, F2, F3, F4 and F5). The data are weekly from 1990 to 2008. Characteristic Mean Standard Dev Skewness Kurtosis F1 0.0009 0.0530 -0.5883 9.5040 Test Statistic Critical value alpha .0649 .0336 .01 F2 F3 0.0010 0.0011 0.0426 0.0380 -0.7666 -0.8287 7.7240 7.9962 Lilliefors Test results .0693 .0660 .0335 .0335 .01 .01 F4 0.0011 0.0349 -0.8364 8.3353 F5 0.0012 0.0326 -0.7001 7.5900 .0620 .0335 .01 .0622 .0334 .01 The Lilliefors test rejects normality of changes in log prices for all maturities at 99% confidence level. 56 Table 2: ML estimates for SS and CJ model Table reports the maximum likelihood estimates for the SS and CJ model using weekly data of NYMEX crude oil futures from 1990 to 2008. Maturities of 1, 5,9,13 and 18 months are used for estimation. The SS model is estimated using the Kalman filter and prediction error decomposition and the CJ model is estimated using the localized particle filter and EM algorithm. Numbers in parenthesis indicate standard errors. *, and ** indicate statistical significance of parameters at 10% and 5% levels respectively. Parameter under P measure µξ κ σχ σξ ρ ω µχ J σ χJ µξ J σξJ ρJ Equilibrium drift rate (annualized) Short term mean reversion coefficient (annualized) Short term factor volatility (annualized) Long term factor volatility (annualized) Correlation coefficient (annualized) Common jump intensity (annualized) Mean jump size of short term factor Std deviation of jump size for short term factor Mean jump size of long term factor Std deviation of jump size for long term factor Correlation between jump sizes Risk Premium Parameters (annualized) Market price of risk for µξ - µξ* long term factor Market price of risk for λχ short term factor * δ = ω / ω Jump intensity premium µ χ J - µ χ* J Short term jump size premium SS 0.0731* (0.0432) 0.9591** (0.0116) 0.2894** (0.0073) 0.1856** (0.0035) 0.0076 (0.0301) Correlated Jump 0.1243** (0.0423) 1.0296** (0.0011) 0.2101** (0.0080) 0.1439** (0.0047) 0.0408* (0.0468) 6.0754** (1.0102) -0.0102 (0.0088) 0.0760** (0.0059) -0.0039 (0.0060) 0.0492** (0.0050) -0.1324 (0.1307) 0.0857** (0.0432) 0.0895 (0.0661) 0.1376** (0.0571) -0.4573** (0.0890) 0.6128** (0.1688) 0.1788** (0.0278) 57 µξ J - µξ*J Long term jump size premium Measurement Error Parameters e1 Std deviation of ME (F1) e2 Std deviation of ME (F2) e3 Std deviation of ME (F3) e4 Std deviation of ME (F4) e5 Std deviation of ME (F5) Number of observations Log Likelihood -0.0067 (0.0251) 0.0446** (0.0007) 0.0087** (0.0003) 0.0028** (0.0002) 0.0000 (0.0341) 0.0071** (0.0002) 986x5 13945 0.0380** (0.0005) 0.0041** (0.0003) 0.0042** (0.0001) 0.0002** (0.0001) 0.0085** (0.0002) 986x5 14077 58 Table 3: ML estimates RS model using Transformed-data and Filtering methods Table reports the maximum likelihood estimates for RS model using the weekly data of NYMEX crude oil futures from 1990 to 2008. Columns & present results using the Jacobian method for (F1, F5) and (F2, F4) contracts. Column shows the estimates using the localized particle filter with the EM algorithm. Numbers in parenthesis indicate standard errors. *, and ** indicate statistical significance of parameters at 10% and 5% levels respectively. Parameters independent of regime (annualized) Equilibrium drift rate µξ µξ - µξ* Market price of risk for long term factor κ λχ Short term mean reversion coefficient F1,F5 F2,F4 All 0.0769** (0.0335) 0.0805** (0.0380) 0.0916** (0.0007) 0.1086** (0.0372) 0.1045** (0.0405) 0.1021** (0.0005) 1.3544** (0.1476) 0.1055 (0.0759) 1.1601** (0.1246) 0.0780 (0.0629) 1.0902** (0.0002) 0.1470** (0.0012) 0.4546** (0.0204) 0.2933** (0.0136) -0.0714 (0.0760) 0.5342** (0.0099) 0.2916** (0.0109) -0.0297 (0.0456) 0.1940** (0.0093) 0.1234** (0.0045) 0.1174 (0.0770) 0.1881** (0.0060) 0.1244** (0.0036) 0.0975** (0.0454) 0.8296** (0.0296) 0.8476** (0.0476) 0.9277** (0.0140) 0.9354** (0.0219) 0.6218** (0.2621) 0.6212** (0.0027) 0.8265** (0.1272) 0.8455** (0.0030) Market price of risk for short term factor Parameters under Regime (annualized) Short term factor volatility 0.5154** σ χ0 (0.0233) Long term factor volatility 0.3096** σξ (0.0169) Correlation coefficient 0.0448 ρ (0.0801) Parameters under Regime 1(annualized) Short term factor volatility 0.2215** σ 1χ (0.0093) Long term factor volatility 0.1175** σξ (0.0039) Correlation coefficient 0.1172 ρ (0.0745) Transition Probabilities (per week) Probability(under P ) of moving p00 0.8088** from R0 to R0 (0.0340) Probability (under P ) of moving p11 0.9295** from R1 to R1 (0.0136) * Probability(under Q ) of moving p00 0.6378** from R0 to R0 (0.3171) * Probability (under Q ) of moving p11 0.8065* from R1 to R1 (0.4311) 59 Measurement error parameters e1 Std deviation of ME (F1) e2 Std deviation of ME (F2) e3 Std deviation of ME (F3) e4 Std deviation of ME (F4) e5 Std deviation of ME (F5) Number of observations Log Likelihood 986x2 4275 986x2 5143 0.0377** (0.0005) 0.0032** (0.0002) 0.0042** (0.0001) 0.0004** (0.0000) 0.0086** (0.0002) 986x5 14122 60 Table 4: Results for CJRS model using Transformed-data and Filtering methods Table reports the maximum likelihood estimates for CJRS model using the weekly data of NYMEX crude oil futures from 1990 to 2008. Columns & present results using the Jacobian method for (F1, F5) and (F2, F4) contracts. Column shows the estimates using the Localized Particle Filter with the EM algorithm. Numbers in parenthesis indicate standard errors. *, and ** indicate statistical significance of parameters at 10% and 5% levels respectively. Parameters independent of regime Equilibrium drift rate µξ (annualized) * Market price of risk for long µξ - µξ term factor (annualized) Short term mean reversion κ coefficient (annualized) Market price of risk for short λχ term factor (annualized) Mean jump size of short term µχ J factor Std deviation of jump size for σ χJ short term factor Mean jump size of long term µξ J factor σξJ µ χ J - µ χ* J Std deviation of jump size for long term factor Correlation between jump sizes Short term jump size premium µξ J - µξ*J Long term jump size premium ρJ Parameters under Regime Short term factor volatility σ χ0 (annualized) Long term factor volatility σξ (annualized) Correlation coefficient ρ (annualized) ω0 δ = ω *0 / ω Common jump intensity (annualized) Jump intensity risk premium F1,F5 0.0207 (0.0363) 0.0055 (4.6692) 1.0527** (0.1389) 0.0241 (3.8802) -0.0093 (0.0142) 0.1002** (0.0166) F2,F4 0.0630* (0.0361) 0.0021 (15.3722) 1.0471** (0.1271) 0.1751 (13.7670) -0.0041 (0.0164) 0.0704** (0.0109) -0.0014 (0.0098) 0.0422** (0.0113) 0.1072 (0.2230) -0.0008 (0.1502) -0.0001 (0.1794) -0.0056 (0.0096) 0.0443** (0.0104) 0.1206 (0.3178) -0.0047 (0.7875) -0.0008 (0.8828) 0.3428** (0.0287) 0.2287** (0.0146) 0.3633** (0.0365) 0.2381** (0.0202) -0.2156** (0.1005) 8.2029** (3.4796) 0.8387 -0.2512** (0.1116) 8.4873* (5.8141) 0.2094 All 0.0454* (0.0265) 0.0811** (0.0226) 1.0507** (0.0001) 0.2608** (0.0163) -0.0008 (0.0047) 0.0852** (0.0130) 0.0054** (0.0010) 0.0577** (0.0138) 0.2042 (0.3772) -0.0008 (0.005) -.0051** (0.0008) 0.3888** (0.0238) 0.2467** (0.0145) 0.2163** (0.0810) 6.1796** (0.0287) 0.5129 61 Parameters under Regime Short term factor volatility σ 1χ (annualized) Long term factor volatility σξ (annualized) Correlation coefficient ρ (annualized) Common jump intensity ω (annualized) *1 δ = ω / ω Jump intensity risk premium Transition Probabilities (per week) Probability(under P ) of p00 moving from R0 to R0 Probability (under P ) of p11 moving from R1 to R1 * Probability(under Q ) of p00 moving from R0 to R0 Probability (under Q ) of p11* moving from R1 to R1 Measurement error parameters e1 Std deviation of ME (F1) e2 Std deviation of ME (F2) e3 Std deviation of ME (F3) e4 Std deviation of ME (F4) e5 Std deviation of ME (F5) Number of observations Log Likelihood (216.376) (352.64) (1.2676) 0.2267** (0.0142) 0.1085** (0.0047) -0.0102 (0.1246) 0.6546 (0.7967) 0.1925** (0.0096) 0.1156** (0.0047) 0.0682 (0.1039) 0.6989 (1.2313) 0.1864** (0.0078) 0.1231** (0.0044) 0.1389** (0.0553) 0.6793** (0.2354) 91.5305 90.0157 (4729) 37.4116 (2561) ** (0.0020) 0.9247** (0.0202) 0.8749** (0.0297) 0.9137** (0.0403) 0.9432** (0.0155) 0.9251** (0.0200) 0.9404** (0.0300) 0.7407** (0.4161) 0.3021 (0.2299) 0.6830** (0.0018) 0.5619 (0.3720) 0.6310** (0.3251) 0.5536** (0.0107) 986x2 5143 0.0376** (0.0005) 0.0033** (0.0003) 0.0044** (0.0001) 0.0003** (0.0000) 0.0087** (0.0002) 986x5 14153 986x2 4275 62 Table 5: Likelihood ratio comparison across models Table compares the log likelihood ratio statistic across various models which are presented in Tables 2, and 4. DOF indicates the additional degrees of freedom of the first model compared to the second in column1. We assume that the appropriate degrees of freedom are a straight count of the number of additional parameters (i.e., we include the nuisance parameters as it is without making any adjustment). Doing so in essence makes the LR test more conservative, i.e., harder to reject the null hypothesis. Model comparison X Log likelihood Ratio CJ vs SS RS vs SS CJRS vs RS CJRS vs CJ 264 354 62 152 DOF(including nuisance parameters) 11 Chi-square dist Cutoff (5%) 16.9 14.07 19.7 16.9 63 Table 6: Analysis of measurement errors Table shows the results after analyzing the residuals from the measurement equation of the SS, CJ, RS and CJRS models. The pricing error (PE) is the dollar difference between the actual and predicted futures price using the maximum likelihood estimates for the sample period 1990-2008. For the CJ, RS and CJRS models the predicted futures price is calculated as an average over all particles, and the futures price for the ith particle is calculated using the pricing formula in Equations 7, 14 and 22 respectively. For the SS(2000) model, n (number of particles used in CJRS model) two dimensional latent variables are sampled from the Normal distribution using the Kalman filtered mean and variance, and the estimated futures price is the average of the implied futures price from these points. Months to expiry 13 18 Avg SS Mean Mean PE abs PE -0.376 -0.039 0.008 0.000 -0.047 -.091 1.059 0.181 0.040 0.000 0.166 0.289 CJ Mean PE -0.256 -0.013 0.013 -0.001 -0.068 -0.065 RS CJRS Mean Mean Mean Mean Mean abs PE abs PE PE abs PE PE 0.853 -0.359 0.880 -0.199 0.819 0.053 -0.017 0.039 -0.019 0.039 0.087 0.022 0.094 0.023 0.095 0.001 -0.000 0.001 -0.000 0.000 0.207 -0.089 0.217 -0.076 0.214 0.240 -0.088 0.246 -0.054 0.233 64 Figure 1: Changes in log futures prices (with standard deviation confidence bands) for the NYMEX crude oil futures with 1- and 18-month maturities (F1 and F5). Figure 2: NYMEX crude oil futures term structures during the run-up to the 1991 Gulf War. Futures Prices run upto Gulf war 40 18/6/1990 23/7/1990 6/8/1990 10/9/1990 8/10/1990 Futures Prices 35 30 25 20 15 10 12 Maturities (months) 14 16 18 65 Figure (I and II) CJRS model implied smoothed probability of being in Regime (high vol regime), 1990-2008 0.9 0.8 Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1990 1993 1995 1998 2005 2008 Date 0.9 0.8 Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1999 2002 Date 66 Figure 4: Actual and predicted futures prices on 14/7/2008. 148.5 148 Futures Prices 147.5 NoJump Jump+RegimeSwitching Actual 147 146.5 146 145.5 145 10 Maturity (months) 12 14 16 18 Figure 5: Actual and predicted futures prices on 21/7/2008. 134 133.5 Futures Prices 133 132.5 NoJump Jump+RegimeSwitching Actual 132 131.5 131 10 Maturity (months) 12 14 16 18 67 [...]... stochastic interest rates, and allowed jumps in the spot price According to their model, allowing for jumps in the spot price does not change concurrent futures prices as compared to a model without jumps However, jumps will impact option prices In our model, jumps do change futures price as indicated by the term, A(T , t ) Unlike jumps in the spot price, jumps in the short-term factor like in our model... done by choosing suitable starting initial values and convergence is obtained far away from these points 3.4 Particle Filter The key to handling a state-space model is to solving the filtering problem The models in Schwartz (1997) and Schwartz and Smith (2000) models are linear Gaussian state-space models Such models can be dealt with using the Kalman filter to handle the latent factors and using prediction... of the factors in these models can be handled by employing the standard Kalman filtering technique because the latent factors are assumed to be Gaussian processes and the transformations are in essence linear The latent variable approach for modeling commodities was initiated by Schwartz (1997) and has become a standard feature in empirical analysis of commodity models Either adding jumps to the latent... still allows two jumps to act independently Finally, we combine the Regime Switching and the Correlated Jump model and additionally allow the jump parameters to vary with the regime We refer to this model as the Correlated Jump Regime Switching (CJRS) model Similar types of models are popular for pricing electricity derivatives6 Electricity prices are often characterized by spiky behavior and the appearance... end This is indicative of the existence of a meanreverting component in the commodity price, a feature that has only been recently introduced into models with jumps, such as Crossby (2008), Dempster et al (2009) and Aiube et al (2008) Our model incorporates this feature through having differential jump amplitudes and the mean-reverting property of the short-term factor 12 2.2 Regime Switching model The... ideal for handling non-Gaussian state-space models 3 Christofferson et al (2010), Duan & Fulop (2009), and Johannes et al (2009) to name a few 3 We study weekly crude oil futures prices, corresponding to five maturities ranging from 1 to 18 months, from 1990 to 2008 Our research question is on the importance of jumps and changes in regime in modeling the behavior of crude oil futures prices We find evidence... high and low volatility regimes which are very persistent (expected duration of about 3 and 4 months respectively) with the high volatility regimes being associated with politically and economically unstable periods Bidirectional jumps occur in the both the factors and are 9 times more likely to occur in high volatility regime compared to the low volatility regime A one standard deviation jump in the... of the short and long term factors to be regime dependent We follow Hamilton (1988) and specify the unobserved regime to follow a first-order two-state Markov Process with a time invariant transition probability matrix This unobserved state variable is in addition to the existing latent short and long term factors This model is referred to as the Regime Switching (RS) model Regime switching models have... ξR t d Z ξ t (9 ) (1 0 ) The regime variable Rt is assumed to follow a continuous-time Markov Chain with a constant infinitesimal generator, G, and it is assumed to be independent of the Brownian motions The infinitesimal generator produces transition probabilities over any time interval of interest We will later use more intuitive transition probability (from regime i to regime j) over time ∆t , denoted... expectation term in the above formula can be computed with Monte Carlo simulations Simulating sample paths for the regime between current time and time to maturity and taking average over all such sample paths leads us to the value of log futures prices It can be immediately seen that in the case of no regime switching, the formula collapses to that of the SS model 2.3 Correlated Jump Regime Switching model . This article explores the role of jumps and changes in regime in commodity prices. Unlike equity and foreign exchange markets, most widely used models in pricing commodity derivatives use dynamic. empirical study for commodity derivatives using pricing models with regime switching is absent and we try to fill that gap. Secondly, we incorporate jumps into the two factors. The jumps are caused. Allowing volatility to be regime dependent and prices to jump have important implications for both derivative pricing and hedging practice. In the commodities sphere, most pricing models involve