Vol 4, No, 01 ISSN: 1309-8055 (Online STEEL PRICE MODELLING WITH LEVY PROCESS Emre Kahraman Türk Ekonomi Bankası (TEB A.Ş. Direcor / Risk Capial Markes Deparmen emre.kahraman@eb.com.r Gazanfer Unal Yediepe Universiy Financial Economics Coordinaor / Financial Economics Graduae and PhD Programs gunal@yediepe.edu.r Absrac The aim of his sudy is o model seel price reurns by Lévy process. The daily LME Seel Billes Spo Prices beween 04.01. 010 and 31.10.011 are analyzed and AR[1] ~ GARCH[1,1] discree model is found o be he bes candidae aking all indicaors ino accoun. Then he coninuous analogue of he discree model is derived from he discree model parameers. During he overall sudy, ime (pahwise, disribuional and specral analysis performed. Finally, i is shown ha he volailiy simulaed from boh discree and coninuous models shows similar volailiy paerns. The resuls of he sudy could be uilized o predic he behavior of fuure seel prices moves. In addiion, he finding could be a good reference specialis and researchers who are ineresed in seel marke. Key Words: Seel Modelling, ARMA, GARCH, COGARCH, Lévy Processes, Discree Time Models, Coninuous Time Models, Sochasic Modelling JEL Classificaion: C01, C51 1. INTRODUCTION The volailiy in he commodiy marke increases he imporance of modeling sudies in hose area and he models which have a success in forecasing he commodiy prices receive grea aenion in he las decade. Geman (005:4 analyzed commodiy marke and relaionships beween commodiies in his book. In addiion o general commodiy marke aracion, he seel prices are becoming more and more imporan nowadays and hey direcly or indirecly affec he economy in world. If he hisorical rends are analyzed, high volailiies, 101
Vol 4, No, 01 ISSN: 1309-8055 (Online upward/downward jumps and drifs could be easily observed. This means ha here is no equilibrium in he seel marke. Alhough he researches and analyses on he behavior of seel prices ogeher wih modeling sudies are increasing, i is also a realiy ha seel price modelling is scarce and i could be defined as quie new subjec in modeling. Those models are really imporan especially in hedging and risk managemen purposes as well as in rading. To be able undersand he characerisic of seel marke, he essenial crieria is o figure ou he sochasic models of seel prices. Since he confidence inervals of he models could change wih ime, he accuracy of he models can be improved wih variance of error models. I comes wih heeroskedasiciy concep in error erms. If he condiional heeroskedasiciy of seel prices could be capured wih sochasic volailiy models (GARCH, hen he accuracy of he model could be improved easily. I is also ime o menion he imporance of ime horizon selecion in he analysis. I plays a vial role in he appropriae model selecion procedure and should be aken ino accoun during all sudies. In financial economerics, mos of he volailiy models are in discree ime, namely GARCH models. Those discree ime models have been widely used in various modeling sudies o be able o capure he characerisics of financial daa. Nelson (1990:7 and Duan (1997:3 sudies are only some of hose. They ried o model he financial daa characerisics by GARCH diffusion approximaions. Because of he fac ha coninuous ime models allow closed form soluions, hey have advanages wih respec o discree ones and hose advanages are ried o be uilized wih several sudies carried ou wih coninuous models. Klüppelberg (004:5 worked on a new coninuous ime GARCH model, namely COGARCH by adaping he single noise process idea.. METHODOLOGY Firs of all, he saionariy of he daa has o be check before saring he discree modelling par. The rend analysis, Augmened Dickey-Fuller es, auocorrelaion and parial auocorrelaion funcions are uilized o be able o idenify wheher he daa is saionary or no. If he daa is no saionary, hen eiher difference or logarihmic difference is applied o make i saionary. The nex sep is o execue discree modeling via Hannan-Rissanen algorihm. In his approach, he resuls of he algorihm are checked wih AIC and BIC values o find he bes candidae model. The bes model should has he lowes 10
Vol 4, No, 01 ISSN: 1309-8055 (Online AIC and BIC values. The resuled model is conrolled for boh saionariy and auocorrelaions lef in he residuals. The ARCH effecs in he residuals will be eliminaed wih ARCH/GARCH modeling by selecing he bes GARCH model. Finally, he non-negaiviy consrains of GARCH process and he covariance saionary condiion are conrolled before moving o he coninuous modeling. In he coninuous side, discree model parameers are used o find he parameers for COGARCH. The simulaions are carried ou for discree and coninuous models and resuls are compared wih each ohers. Overall, he sudy is covering no only ime analysis, bu also disribuional and specral analysis on he daa. Nelson (1990:7 worked on GARCH diffusion approximaion wih a differen way. In he model, here are wo differen and independen Brownian moions which drive he diffusion, alhough he process is driven by a single noise sequence. G and volailiy process can be represened as: dg db 0 [1] (1 ( ( d db 0 [] where β > 0, η 0, and φ 0 are consans. Klüppelberg (004:5 and Ross (008:8 / 009:9 ogeher wih all ohers model COGARCH wih a direc analogue of GARCH. The model is based on Lévy process and he model consrucion is done by aking limis of an explici represenaion of he discree ime GARCH process. The COGARCH process ( G 0 is defined in erms of is sochasic differenial dg, such ha, dg dl 0 [3] (, d d ( d L L d > 0 [4] where β > 0, η 0, and φ 0 are consans. ( d, L L is a quadraic variaion process of L (Lévy process which is defined as; 103
Vol 4, No, 01 ISSN: 1309-8055 (Online N, Vi where L L 1 0s i1 ( d L L ( Ls L for 0 [5] The process G jumps a he same ime as L (Lévy process does, and has jump sizes; G 0 [6] L Deriving a recursive and deerminisic approximaion for he volailiies a he jump imes, Klüppelberg (004:5 shows; s ds s ( L [7] i i1 0 0s Since s is laen Ls is usually no observable, hence using Euler approximaion for he inegral we ge; 0 s ds 1 [8] 0s ( L ( G G [9] s 1 Therefore, for he volailiy esimaion, we end up wih; i ( 1 i1 ( G G 1 [10] The bivariae process (, G 0 is Markovian. If ( 0 is he saionary version of he process wih 0, hen ( G 0 is a process wih saionary incremens. 3. DATA ANALYSIS 3.1. Analysis in Time Domain In his sudy, LME Seel Billes Spo Prices (in US dollars have been analyzed. The analysis has been performed by using he daily close daa over he period from January 4, 010 o Ocober 31, 011. The daa conains 46 observaions and he graphical represenaion is given on Figure-1. 104
Vol 4, No, 01 ISSN: 1309-8055 (Online Figure-1: LME Seel Billes Spo Prices / Time Series Plo 700 LMESeel Spo Bille Prices (USD 650 600 550 500 450 400 350 04.01.10 04.03.10 04.05.10 04.07.10 04.09.10 04.11.10 04.01.11 Time ( 04.03.11 Source: Bloomberg / LMFMDY Commodiy 04.05.11 04.07.11 04.09.11 I is visually clear from he graph ha he price series have a rend and are mean non-saionary. The daa is checked for a rend and he resul shows ha here is a rend which means i is no saionary. [-9673.+0.51505 ] I is also checked by Augmened Dickey-Fuller es. Tes resuls confirm ha he series is no saionary. (Table-1 Table 1: Uni Roo Tes for LME Seel Billes Spo Prices -Saisic Prob.* Augmened Dickey-Fuller es saisic -.183019 0.19 Tes criical values: 1% level -3.444311 5% level -.86759 10% level -.570055 Source: Own Sudy Finally, auocorrelaion and parial correlaion funcion graphs have been given on Figure-. While auocorrelaion values decrease slowly, parial auocorrelaion values sharply converge o almos zero levels. The decrease in auocorrelaion could be hough as he fac ha random shocks o he sysem dissipae wih ime. I could be concluded ha LME Seel Billes Spo Prices have a rend and so are mean non-saionary. Finally, considering he slow decrease in auocorrelaion values, i could be concluded ha here is a long memory srucure in he daa. Figure-: Auocorrelaion & Parial Auocorrelaion Funcions of LME Spo Prices 1.0 1.0 Auo Correlaion Funcion(ACF 1.00 0.80 0.60 0.40 0.0 Parial Auo Correlaion Funcion (PACF 1.00 0.80 0.60 0.40 0.0 0.00 0.00 0 5 10 15 0 5 30 LAG Source: Own Sudy -0.0 0 5 10 15 0 5 30 LAG 105
Vol 4, No, 01 ISSN: 1309-8055 (Online The nex sep is o make he daa saionary. To achieve mean saionary, difference of he series could be used. Bu, if he series show non-linear rend, he differencing creaes non-saionary variance. So, o achieve boh mean and variance saionary, firs he logarihm should be aken and hen difference of he series, which is he logarihmic reurn. The resuled series is given on Figure-3. Figure-3: Log Reurn of LME Seel Billes Spo Prices 0.08 Log Reurn of LME Seel Spo Bille Prices 0.06 0.04 0.0 0-0.0-0.04-0.06-0.08-0.1-0.1 04.01.10 04.03.10 04.05.10 04.07.10 Source: Own Sudy 04.09.10 04.11.10 04.01.11 Time ( 04.03.11 04.05.11 04.07.11 04.09.11 The logarihmic reurn of seel price series seems o have no rend. The nex sep is o check ha wheher i is a non-saionary or no. Augmened Dickey-Fuller es resuls confirm ha he series is no saionary. (Table- Table : Uni Roo Tes for Log Reurns -Saisic Prob.* Augmened Dickey-Fuller es saisic -3.75465 0.0000 Tes criical values: 1% level -3.44434 5% level -.867603 10% level -.570063 Source: Own Sudy Finally, he auocorrelaion and parial auocorrelaion resuls are obained. (Figure-4 I seems ha here is relaionship in an order of one. Figure-4: Auocorrelaion & Parial Auocorrelaion Funcions of Log Reurns Auo Correlaion Funcion(ACF 1.0 1.00 0.80 0.60 0.40 0.0 0.00 Parial Auo Correlaion Funcion (PACF 0.15 0.10 0.05 0.00-0.05-0.10-0.0 0 5 10 15 0 5 30-0.15 0 5 10 15 0 5 30 Source: Own Sudy LAG LAG 106
Vol 4, No, 01 ISSN: 1309-8055 (Online The nex sep is o apply Hannan-Rissanen algorihm o decide he mean equaion. The idea behind he procedure is firs o fi an AR model o he daa in order o obain he esimaes of he noise or innovaion. When his esimaed noise is used in place of he rue noise, i enables us o esimae ARMA parameers using he less expensive mehod of leas squares regression. The orders are deermined wihin he procedure iself using an informaion crierion. I gives AR[1] as he bes fi. The resuls for he firs eigh models are given a Table-3. Table 3: Hannan-Rissanen Resuls and AIC&BIC Values Ranking Model Model Parameers AIC BIC 1 AR [1] [{-0.10947},0.000368701] -7.89-7.871 MA [1] [{-0.118533},0.000368849] -7.8918-7.8717 3 AR [] [{-0.109846,-0.0564174},0.000366953] -7.8836-7.8435 4 ARMA [1,1] [{0.89847},{-0.4085},0.00036783] -7.881-7.8411 5 MA [] [{-0.117819,-0.03367},0.00036903] -7.8780-7.8378 6 MA [4] [{-0.113389,-0.0375167,0.086335,0.113773},0.00035977] -7.8768-7.7966 7 AR [4] [{-0.11514,-0.0447406,0.0698673,0.15534},0.00036138] -7.87-7.790 8 AR [3] [{-0.108007,-0.0507689,0.0568196},0.00036638] -7.8718-7.8116 Source: Own Sudy Moreover, Saionary-Q es resul for AR[1] model saes ha i is saionary. Finally, he Pormaneau es is used o see wheher here is any auocorrelaion is lef. AR[1] Pormaneau saisics for he firs 35 auocorrelaions is 30.53, while 95% confidence level Chi-Square Disribuion resul is 48.60. In oher word, here is no enough evidence o sae ha here is auocorrelaion lef in he residuals. 3.. Analysis in Frequency Domain The aim in his secion is o analyze he daa in frequency domain. I is also called he ime series analysis in Fourier space. I enables us o work wih he same daa in differen represenaion and all should give he same resul. When he specrum analysis has been carried ou, i could be observed ha while he pahs are same for boh he smoohed specrum of he daa and he specrum of AR[1] esimaes, here are some noise due he GARCH effec. (Figure-5 Figure-5: Smoohed Specrum of Daa vs Hanning Window Specrum of AR[1] c1 0.00008 0.00007 0.00006 0.00005 0.0 0.5 1.0 1.5.0.5 3.0 Source: Own Sudy 107
Vol 4, No, 01 ISSN: 1309-8055 (Online 3.3. GARCH Modelling The daa has o be checked for GARCH modeling. In oher word, i is conrolled o be able o see ha is here any GARCH effec of no. ARCH LM es resul shows ha here is GARCH effec in he series which is o be modeled. (LM Saisic: 6.884 & 95% confidence level Chi-Square Disribuion resul: 3.841 GARCH effec is analyzed and he resuls for differen models indicaed ha GARCH[1,1] is he bes fi via lowes AIC value. The parameers for consan, ARCH[1] and GARCH[1] coefficiens are 0.00000797061185, 0.07484309 and 0.9049436507 respecively. The esimaed GARCH obeys he non-negaiviy consrains of GARCH process, since all coefficiens are posiive. The model also saisfies he covariance saionary condiion ha sum of coefficiens is less han 1. X 0.000368701 0.10947 X 1 0.00000797061185 0.07484309 1 0.9049436507 1 3.4. Disribuion Analysis on Errors The aim of his secion is o find he bes disribuion which fi o GARCH model error series. Normal, Johnson SU, Weibull, Gumbel and Cauchy disribuions are being esed for disribuion fiing. Hisograms, probabiliy plos, Q-Q plos and cumulaive disribuion funcions are checked ogeher wih differen saisical ess including Anderson-Darling, Cramér-von Mises, Kolmogorov-Smirnov and Kulper. As a resul, Johnson SU is found o be he bes fi. (Figure-6 Figure-6: Hisorgram and Q-Q Plo for Johnson SU Disribuion and Errors 0.04 0.0 0.00 0.0 0.04 0.06 Source: Own Sudy 0.08 0.06 0.04 0.0 0.00 0.0 0.04 0.06 3.5. Coninuous Modelling When he discree ime GARCH[1,1] model had been esimaed, he coninuous ime COGARCH[1,1] model can be found from he discree model parameers. 108
Vol 4, No, 01 ISSN: 1309-8055 (Online The parameers of coninuous COGARCH [1,1] model in erms of discree ime GARCH[1,1] model can be wrien as: ln( / Therefore, he parameers of COGARCH [1,1] model can be calculaed as: 0.00000797061185 ln( 0.9049436507 0.099886016406098 / 0.07484309/0.9049436507 0.086816598741 Afer he parameer esimaion, he simulaion has been carried ou by using he numerical soluions for G and. The Lévy process driven by Johnson-SU process, which is found o be he bes fi, is uilized. Figure-7 shows he discree GARCH model and coninuous GARCH (COGARCH model. I could be easily realized ha boh models are showing similar paern in ime. Figure-7: Volailiy Graphs of COGARCH & GARCH & Log Reurns 0.0009 x a- Volailiy of COGARCH 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 x 100 00 300 400 b- Volailiy of GARCH 0.001 0.0010 0.0008 0.0006 0.0004 Source: Own Sudy 100 00 300 400 4. CONCLUSION In his sudy, log reurns of daily LME Seel Billes Spo Prices beween 04.01. 010 and 31.10.011 have been modeled wih AR[1] ~ GARCH[1,1] discree model which is he bes candidae. The discree model parameers are used o consruc he coninuous COGARCH[1,1] analogue. Then, he simulaed volailiy 109
Vol 4, No, 01 ISSN: 1309-8055 (Online resuls of boh discree and coninuous models are compared wih each oher. I was shown ha boh models follow he similar paerns especially in he jumps. BIBLIOGRAPHY Barndorff-Nielsen, Ole E., Neil Shepherd (001, Non-Gaussian Ornsein- Uhlenbeck Based Models and some of heir Use in Financial Economics (wih discussion, Journal of Royal Saisics Sociey Series B, Vol. 63, pp.167-41. Chrisian Kleiber and Samuel Koz (003, Saisical Size Disribuions in Economics and Acuarial Sciences, Wiley Series in Probabiliy and Saisics. Duan, Jin Chuan (1997, Augmened GARCH (p,q Process and Is Diffusion Limi, Journal of Economerics, Vol. 79, pp.97-17. Geman Hélyee (005, Commodiies and Commodiy Derivaives: Modelling and Pricing for Agriculurals, Meals and Energy, Wiley Finance. Klüppelberg Claudia, Alexander Lindner and Ross Maller (004, A Coninuous Time GARCH Process Driven by a Lévy Process: Saionary and Second Order Behaviour, Journal of Applied Probabiliy, Vol. 41, pp.601-6. Klüppelberg Claudia, Alexander Lindner and Ross Maller (006, Coninuous Time Volailiy Modelling: COGARCH versus Ornsein-Uhlenbeck Models,(in: Yuri Kabanov, Rober Lipser and Jordan Soyanov-Eds, From Sochasic Calculus o Mahemaical Finance, Springer:Berlin, pp.393-419. Nelson, Daniels B. (1990, ARCH Models as Diffusion Approximaion, Journal of Economerics, Vol. 45, pp.7-38. Ross A. M., Gerno M. and Alex S. (008, GARCH Modelling in Coninuous Time for Irregular Spaced Time Series Daa, Bernoulli, Vol. 14, pp.519-54. Ross A. Maller, Gerno Müller and Alex Szimayer (009, Ornsein-Uhlenbeck Processes and Exensions, Handbook of Financial Time Series. Tim Bollerslev (1986, Generalized Auoregressive Condiionally Heeroscedasiciy, Journal of Economerics, Vol. 31, pp.307-37. 110