A Markov Regime Switching Approach for Hedging Energy Commodities

Transcription

1 A Markov Regime Swiching Approach for Hedging Energy Commodiies Amir Alizadeh, Nikos Nomikos & Panos Pouliasis Faculy of Finance Cass Business School London ECY 8TZ Unied Kingdom Slide

2 Hedging in Fuures Markes The objecive of hedging is o reduce he risk of price changes in he physical marke. To achieve his, he hedger deermines a hedge raio i.e. he raio of fuures conracs o buy or sell for each uni of he underlying asse. Hedge raios can be saic or dynamic:. The Saic Hedge Raio: Ederingon (979), shows ha he hedge raio ha minimises he risk of a hedged posiion is he slope coefficien,, in he following regression: ΔS = 0 + ΔF + u ; u iid(0, σ 2 ) 2. The Dynamic Hedge: Kroner and Sulan (993) argue ha since spo and fuures reurns are characerised by ime-varying disribuions, opimal hedge raios should be ime-varying.. Typically esimaed using GARCH models 2. performance, in erms of risk reducion, varies from marke o marke 3. end o ouperform consan hedge raios bu occasionally he benefis from using GARCH seem o be minimal o warran he addiional esimaion effor Slide 2

3 Regime Shifs and Hedging Regime Shifs Lamoureux and Lasrapes (990): associae high levels of volailiy persisence wih srucural breaks or regime shifs in he volailiy process. Sarno and Valene (2000): use regime swiching models o model he relaionship beween spo and fuures prices in he FTSE-00 and S&P-500 markes. Fong and See (2002): examine he condiional volailiy of crude oil fuures and repor significan regime shifs in he volailiy of oil fuures conracs which dominae he GARCH effecs. Regime Shifs and Hedging Alizadeh and Nomikos (2004): use Markov Regime Swiching models o esimae sae dependen hedge raios in he FTSE 00 and S&P500 sock index markes Lee and Yoder (2005): use mulivariae Markov Regime Swiching GARCH models o esimae he hedge raios hrough he second momens of fuures and spo prices in he corn and nickel commodiy markes. Slide 3

4 Conribuion We develop a procedure ha generaes hedge raios which are regime dependen and change as marke condiions change. We allow he regime probabiliies o form a ime-varying hedge raio. We also allow he volailiy o be dependen on he regime and esimae he hedge raios hrough he condiional second momens. We consider boh univariae and bivariae MRS models. We evaluae he hedging effeciveness of Markov models using boh in- and ou-of-sample ess. The ou-of-sample ess, are performed by forecasing boh he regime probabiliies and variance/covariance marix. The performance of he Markov regime swiching hedge raios is compared o ha of alernaive models GARCH, Error-Correcion and OLS models. Slide 4

5 Univariae MRS Models and Hedge Raios We exend he basic hedging equaion o a wo-sae MRS model in which we allow he model o swich beween wo differen processes, dicaed by he sae of he marke ΔS = + ΔF + ε 2 ~ iid(0, σ 0, s, s, s,s ε, s where, s =,2 indicaes he sae in which he marke is in., and,2 represen he minimum variance hedge raios, given he sae of he marke. The link beween he wo saes of he marke is provided hrough a firs order Markov process wih he following ransiion probabiliies. ; ε ) Pr( s Pr( s = s = 2 s = 2) = ) = = P P 2 2,, Pr( s Pr( s = 2 s = s = 2) = P = ) = P 22 = ( P = ( P 2 2 ) ) P 2 (P 2 ) gives he probabiliy ha sae (2) will be followed by sae 2 () P and P 22 give he probabiliies ha here will be no change in he sae of he marke in he following period. Slide 5

6 MRS GARCH Models and Hedge Raios An alernaive way o esimae he opimum hedge raio would be o use he condiional second momens of spo and fuures reurns which are measured by he family of ARCH models, inroduced by Engle (982). For he condiional mean equaions we employ a wo-sae MRS VAR model in which we allow he inercep erm o swich beween wo differen processes, dicaed by he sae of he marke: p ε S,, s ΔX = μ s + Γ iδx + ε,s ; ε,s = Ω ~ IN(0, H,s ) i= ε F,, s For he condiional second momens, we follow an augmened BEKK represenaion (Engle and Kroner, 995), wih sae-dependen coefficiens: H, s = C scs + A sε ε A s + Β sh Β s where, s =,2 indicaes he sae in which he marke is in. Transiion Probabiliies remain as defined before. Slide 6

7 Markov Regime Swiching Hedge Raios Univariae MRS: his yields wo hedge raios which can be considered as he upper and lower bounds of he opimum hedge raio The opimum hedge raio a any poin in ime is calculaed as he weighed average of he wo hedge raios (, and,2 ), weighed according o heir respecive regime probabiliies: * = P,, + ( P ) Bivariae MRS-BEKK: This yields wo variance/covariance marices, given he sae of he marke. From hese wo var/covar marices we form he collapsed var/covar marix by inegraing he unobserved sae variable s, following he procedure of Gray (996) and Lee & Yoder (2005). The hedge raio is hen a funcion of he aggregae var/covar marix:,,2 * = Cov ( Δ S Var ( Δ, Δ F F ) ) Slide 7

8 Descripion of he Daa Weekly NYMEX WTI, Unleaded Gasoline and Heaing Oil fuures and spo prices for he period 23 January 99 o 28 July Daa are colleced from CRB-InfoTech CD. Wednesday prices; when a holiday occurs on Wednesday, Tuesday s observaion is used. In sample analysis: 23 January July 2003 (654 observaions) Ou-of-sample analysis: 6 Augus July 2004 (52 observaions) Rolling over he conrac Volume and Open ineres are used as indicaors of marke aciviy and liquidiy. Rolling over o he fron monh conrac occurs he business day following he day ha boh rading volume and open ineres exceed ha of he neares o expiry conrac. Slide 8

9 TABLE I. Esimaes of Univariae MRS Hedge Raios for NYMEX Energy Commodiies Wes Texas Inermediae Unleaded Gasoline Heaing Oil # 2 Mean Equaion 0, (0.45) (0.08) (0.36), 0.87 (0.0) * (0.048) * 0.90 (0.002) * 0, (0.00) (0.057) (0.00) *,2.004 (0.005) *.3 (0.023) *.036 (0.02) * Variance Equaion σ (0.366) * (0.252) * (.63) * σ (0.08) *.005 (0.02) * (0.00) * Transiion Probabiliies P (0.045) * (0.08) * 0.7 (0.00) * P (0.056) * (0.07) * (0.00) * LR es 2 H 0 ~ x () 2 H 0 ~ x () 2 H 0 ~ x (), =, * 4.70 * 28.6 * σ = σ * 27.7 * 4.8 * OLS HR Log-L SBIC Slide 9

10 TABLE II. Esimaes of MRS-BEKK Hedge Raios for NYMEX Energy Commodiies Wes Texas Inermediae Unleaded Gasoline Heaing Oil # 2 Mean Equaion α S,s= (0.423) (0.43) (0.843) * α S,s= (0.64) 0.76 (0.203) (0.39) *** b S, (0.090) * (0.077) * (0.064) * S, (0.00) * 0.25 (0.098) ** (0.086) * α F,s= (0.396) (0.452) (0.654) * α F,s= (0.66) 0.55 (0.96) (0.42) *** b F, (0.097) (0.062) (0.076) F, 0.04 (0.2) (0.296) (0.090) Variance Equaion c,s= 2.06 (0.358) * 2.3 (0.374) *.869 (0.920) ** c 2,s=.324 (0.366) *.653 (0.243) * (.45) *** c 22,s= -.82 (0.229) *.250 (0.28) * 0.92 (.42) α,s= (0.024) ** (0.034) 0.66 (0.032) * α 22,s= 0.02 (0.06) 0.09 (0.054) *** (0.035) β,s= (0.265) *** (0.227) * (0.288) * β 22,s= (0.73) * (0.32) * 0.50 (0.537) c,s=2.683 (0.87) *.477 (0.9) *.247 (0.42) * c 2,s=2.655 (0.96) *.543 (0.26) *.287 (0.87) * c 22,s= (0.045) * 3x0-6 (0.026) (0.04) * α,s=2 0.2 (0.052) ** (0.02) 0.09 (0.09) * α 22,s=2 0.3 (0.053) ** (0.09) (0.09) * β,s= (0.57) ** 0.65 (0.06) * (0.062) * β 22,s=2 0.4 (0.6) ** (0.098) * (0.089) * Transiion Probabiliies P (0.229) (0.069) * (0.082) * P (0.098) (0.058) * (0.022) * Log-L SBIC Slide 0

11 Regime Probabiliies, Basis & Time-Varying Hedge Raios for WTI Crude Oil REGIME_PROB MRS_BEKK_HR VECM_GARCH_HR CONSTANT_OLS_HR Slide

12 Esimaed Volailiies MRS BEKK WTI Crude Oil 7.3 Spo Price Volailiy /0/99 23/0/992 23/0/993 23/0/994 23/0/995 23/0/996 23/0/997 23/0/998 23/0/999 23/0/ /0/200 23/0/ /0/2003 high_var low_var var 7.3 Fuures Price Volailiy /0/99 23/0/992 23/0/993 23/0/994 23/0/995 23/0/996 23/0/997 23/0/998 23/0/999 23/0/ /0/200 23/0/ /0/2003 high_var low_var var Fuures - Spo Correlaion 23/ 0/ 99 23/ 0/ / 0/ / 0/ / 0/ / 0/ / 0/ / 0/ / 0/ / 0/ / 0/ / 0/ / 0/ 200 RHO high var sae RHO low var sae RHO Slide 2

13 In-sample Hedging Performance We evaluae he performance of he univariae and mulivariae MRS hedges, by consrucing porfolios using he compued hedge raios each week, and calculaing he variance of he reurns of hese porfolios over he sample Tha is we evaluae: Var(ΔS - *ΔF) where * are he compued hedge raios. Differen Hedge Raios are considered OLS Hedge Raio VECM Hedge Raio GARCH Hedge Raio Naïve Hedge Raio (i.e. seing * = ). Slide 3

14 TABLE III. In-Sample Hedging Effeciveness of MRS Agains he Alernaive Hedge Raio Models WTI li gh s wee Cr ude Oi l Unl eaded Gasoli ne Heai ng oil Vari ance Variance Improvemen of MRSBEKK Weekly Uiliy Variance Vari ance Improvemen of MRSBEKK Weekly U i l i y The i n-sa mpl e peri od is fromjanuary 23, 993 (654 observaions). An aserisk (*) indicaes he model ha provides he greaes variance reducion and/or increase in uiliy. Va r i ance Vari ance Improvemen of MRSBEKK Weekly U i l i y Unhedged % % % Naï ve % % % Cons an % % % VECM % % % GARCH % % % MRS % * % * % MRS- BEKK * * * * Slide 4

15 Ou-of-sample Hedging Performance Ou-of-Sample ess are carried ou over he period 6 Augus 2003 o 28 July Univariae MRS hedge raios a ime + are obained using a 2-sep procedure. Firs, we obain forecass of he regime probabiliies as follows pˆ pˆ 2 ( ) ( ) pˆ( s+ = ) pˆ( s+ = 2) = pˆ( s = ) pˆ( s = 2) pˆ 2 pˆ 22 Second, we obain he forecased hedge raios as: ( ) *, + = pˆ( s+ = ) pˆ( s+ = 2),2 Bivariae MRS hedge raios a ime + are more complex and require: Forecass of boh he sae dependen covar marix and he mean equaion, in order o inegrae he sae variable s a each sep of he recursive esimaion. Again, he collapsing procedure of he var/covar marix is based on Gray (996) and Lee & Yoder (2005). Then he -sep ahead forecas of he hedge raio is : * + Ω Ε = Ε [ hsf, + Ω ] [ h Ω ] FF, + Slide 5

16 TABLE IV. Ou-of-Sample Hedging Effeciveness of MRS Agains he Alernaive Hedge Raio Models WTI li gh s wee Cr ude Oil Unl eaded Gasoli ne Heai ng Oil #2 Vari ance Vari ance Impr ove men of MRS BEKK Weekly Uiliy Variance Vari ance Impr ove men of MRSBE KK We e kl y Uiliy Varianc The i n-sa mpl e peri od is fro mjanuary 23, 993 (654 observaions). An aserisk (*) indicaes he model ha provi des he greaes variance reduci on and/ or increase in uiliy. Vari ance Impr ove men of MRS BEKK Unhedged % % % Naï ve % % % Cons an % % % VECM % % % GARCH. 695* % * % % MRS % % % MRS- BEKK * * * * Weekly U i l i y Slide 6

17 Conclusions The effeciveness of univariae and mulivariae MRS hedge raios is invesigaed in he NYMEX WTI, Unleaded Gasoline & Heaing Oil markes. Hedge raios are significanly differen across differen saes of he marke; in fac, hedge raios appear o be significanly higher when he volailiy in he marke is low. In & ou-of-sample ess indicae ha MRS-BEKK hedge raios ouperform he GARCH, VECM and he OLS hedges in 2 ou of 3 cases in boh he in-sample and ou-ofsample cases. Overall he resuls indicae ha by using MRS models, marke agens may be able o increase he performance of heir hedges, measured in erms of variance reducion and uiliy enhancemen. Slide 7

18 A Markov Regime Swiching Approach for Hedging Energy Commodiies Amir Alizadeh, Nikos Nomikos & Panos Pouliasis Faculy of Finance Cass Business School London ECY 8TZ Unied Kingdom Slide 8