Misspecification in term structure models of commodity prices: Implications for hedging price risk

Transcription

1 19h Inernaional Congress on Modelling and Simulaion, Perh, Ausralia, December 2011 hp://mssanz.org.au/modsim2011 Misspecificaion in erm srucure models of commodiy prices: Implicaions for hedging price risk H. Suenaga School of Economics and Finance, Curin Universiy, GPO Box U1987, Perh, WA 6845, AUSTRALIA. Absrac Sochasic dynamics of commodiy prices and valuaion of derivaive conracs have long been sudied in he field of financial economics. In he lieraure a common approach is o specify a sochasic dynamics of he underlying asses and derive from he suggesed model valuaion formulas of various derivaive conracs whose payoff depends on he realizaion of he underlying asse value. Recenly, models wih addiional laen facors and more flexible sochasic process of each facor have been suggesed. Alhough hese complex models end o fi beer o he observed daa, i is ofen undersaed ha his modeling approach only approximaes he rue sochasic process of he underlying asse values. This approximaion bias can be subsanial in magniude for a sorable commodiy wih significan demand and/or supply seasonaliy, for which equilibrium pah of spo and fuures prices canno be expressed in reduced form, as shown by he heory of sorage. This sudy examines convenional erm-srucure models of commodiy prices. In paricular, his sudy quanifies he approximaion bias of hese convenional models hrough comparing hem wih an alernaive approach of modeling he variance of daily fuures reurns direcly by flexible non-parameric funcions so as o allow he model o replicae highly non-linear price dynamics of sorable commodiies. Empirical applicaions of he models reveal ha, for all four commodiies (gold, crude oil, naural gas, and corn) examined in his sudy, he volailiy of daily fuures prices is more complex han he paern as implied by he dynamics sipulaed in he convenional erm-srucure models. In paricular, all four commodiies exhibi a srong ime-o-mauriy effec as well as a significan seasonal paern in boh levels and composiions (among wo facors and idiosyncraic errors) of volailiy of daily fuures reurns. Convenional ermsrucure models draw incorrec porrais of volailiy dynamics as well as incorrec correlaion among concurrenly raded conracs wih differen mauriy daes, which lead hese models o sugges hedging sraegies ha are considerably less effecive han he sraegy based on he model of fuures reurns. Keywords: commodiy prices, erm-srucure model, volailiy, hedging 1767

2 1. INTRODUCTION Recen increases in he level and volailiy of primary commodiy prices have creaed remendous uncerainies for producers, consumers, and oher raders of hese commodiies. Sochasic dynamics of commodiy prices and valuaion of derivaive conracs have long been sudied in he field of financial economics. In he lieraure, a common approach is o specify he sochasic dynamics of he underlying asse, usually he spo price of he commodiy under invesigaion, and derive from he suggesed model valuaion formulas of various derivaive conracs whose payoff depends on he value of he underlying asse realized a conrac mauriy dae (see Hull, 2000). In a seminal sudy, Schwarz (1997) suggess a model in which he underlying spo price is specified as a linear combinaion of hree sochasic componens, represening long-erm facor, convenience yield facor, and ineres rae. Recen advancemens in his modelling approach have been aained hrough increasing he number of laen facors and/or sipulaing flexible sochasic process for each facor. These flexible models generally fi beer o he observed price daa, ye i is ofen undersaed ha hese models only approximae rue sochasic dynamics of commodiy prices. In paricular, as shown by he heory of sorage (Williams and Wrigh, 1991), for a sorable commodiy wih srong seasonaliy in demand and/or supply, he equilibrium pah of spo and fuures prices exhibi highly non-linear dynamics ha canno be expressed in reduced form. Sochasic processes of he underlying spo price sipulaed in models of commodiy price dynamics, even recenly developed complex models, canno induce a fuures price formula ha replicaes key feaures of commodiy fuures prices implied by he heory of sorage. Thus, no maer how flexible a model is specified, i is inrinsically subjec o approximaion bias. This bias may resul in inaccurae pricing of derivaive conracs, lead o hedging sraegies ha are subopimal, and erroneously conclude ha he exising marke is inefficien. This sudy examines he convenional erm-srucure models of commodiy prices hrough comparing hem wih a model of daily fuures reurns. In his alernaive model, I follow he same approach as Smih (2005) and Suenaga and Smih (2011) and specify facor loadings direcly by flexible, non-parameric funcions, raher han deermining hem by a small number of parameers characerizing he emporal dynamics of he underlying sochasic facors. These flexible funcions allow he model o replicae highly non-linear price dynamics of commodiies wih significan sorage coss and srong seasonaliy in demand and/or supply. I quanify he approximaion bias in convenional erm-srucure models hrough comparing hem wih he model of daily fuures reurns. Empirical applicaions o four commodiies (gold, crude oil, naural gas, and corn) illusrae ha he convenional erm-srucure models are subjec o misspecificaion bias of considerable size, which can lead o hedging sraegies ha are less effecive han he sraegy based on he model of fuures reurns. 2. COMPARISON OF TERM-STRUCTURE MODELS AND MODELS OF FUTURES RETURNS 2.1 Term Srucure Models of Commodiy Prices As an example of erm-srucure model of commodiy prices, Schwarz and Smih (2000) consider a wofacor model in which log of he spo price, S, is a linear combinaion of he long-run equilibrium price, ξ and he shor-erm deviaion from he long-run equilibrium price, χ : 1 (1) ln S = ξ + χ dξ = μξd+ σξdzξ dχ = κχd+ σχdz dz dz = ρ d ξ χ χξ χ In (1), he long-erm (LT) and shor-erm (ST) facors follow a Geomeric Brownian Moion (GBM) and mean-revering (MR) process, respecively, and wo incremens o he sandard Brownian moion are allowed o correlae hrough he fourh equaion. MR process allows he log of spo price o rever o long-run mean for κ > 0. As argued by Schwarz (1997), he process is appropriae in modeling commodiy prices, for which demand and supply response force prices, if unusually high or low, o rever o he long-run equilibrium level. 1 Schwarz and Smih (2000) show ha he model defined in (1) is equivalen o he wo-facor model of Schwarz (1997) in which wo laen facors represen he spo price and convenience yield. 1768

3 Two-facor models similar o (1) have been applied o various commodiies, such as meals and energy commodiies. 2 A noable modificaion on model (1) as considered in previous sudies is o allow mean price o exhibi seasonal variaion in he righ-hand side of he firs equaion in (1) when applying he model o a commodiy wih subsanial seasonaliy in demand or supply. To derive he pricing formula of a financial conrac, he sochasic process of he underlying spo price in (1) is ransformed ino a process ha is maringale under he risk-neural probabiliy measure: (1 ) dχ = ( κχ λ ) d+ σ dz χ ξ χξ χ χ χ dξ = ( μ λ ) d+ σ dz dz dz ξ ξ ξ ξ = ρ d where dz χ and dz ξ are incremens of he sandard Brownian moion under risk neural probabiliy measure and he wo coefficiens λ χ and λ ξ are ofen inerpreed as represening he marke price of risk associaed wih each sochasic facor. The price in period of he fuures conrac mauring in period T is obained as he period condiional expecaion of period T spo price. I is obained as, for he spo price following he sochasic process (1), (2) κτ 2κτ 2 2 κτ (1 e ) λ (1 e ) σ σ τ (1 e ) ρ σσ κτ χ χ ξ χξ χ ξ ln FT (, ) = ξ + e χ λ τ + μ τ ξ ξ κ κ κ 4 2 The model parameers, Ω= { κσ,, σ, λ, λ, ρ }, are usually esimaed wih he fuures price daa. In many χ ξ χ ξ χξ organized exchanges, muliple conracs wih differen mauriy daes are raded per each commodiy. To fi equaion (2) ino muliple prices observed on day wih differen ime-o-mauriy (T), an error erm, ofen called measuremen error, is added o he righ-hand side of (2), which makes he values of he sochasic facors χ and ξ no idenifiable. The model is pu ino a sae-space form and he parameers are esimaed usually by a filering mehod, (3) ln F ( T, ) = ξ + e χ λ τ + μ τ u κ 4κ 2 κ χ = e χ + ω κ 1 1, ξ = μ + ξ + ω ξ 1 2, κτ 2κτ 2 2 κτ (1 e ) λχ (1 e ) σχ σξτ (1 e ) ρ κτ χξσχσ ξ ξ ξ T, where E[u,T ] = 0 and V[u,T ] = σ T 2 and T. ω = {ω 1,, ω 2, } is iid bivariae normal wih E[ω ] = 0 and E[ω ω ] = H where H is he symmeric marix wih σ x 2 and σ ξ 2 on he main diagonal and ρσ x σ ξ, off diagonal. 2.2 Models of Price Reurn (POTS model) Parially overlapping ime-series (POTS) model, inroduced by Smih (2005), is a laen facor model of daily fuures price changes. In his model, daily reurn of a fuures conrac is decomposed ino he common laen facors and an idiosyncraic erm. The model, as applied o he NYMEX energy fuures conracs in Suenaga and Smih (2011), is expressed in he following form, (4) ΔF,m = θ 1 (m,d) ε 1, + θ 2 (m,d) ε 2, + θ 3 (m,d) u,m where ΔF,m = F,m F 1,m is he daily price change on rade dae of fuures conrac ha maures a m, ε 1, and ε 2, are he laen facors ha affec all he conracs raded on, u,m is he idiosyncraic error ha is specific o he conrac mauring a m, θ 1 (m,d) and θ 2 (m,d) are he facor loadings deermining he exen ha he underlying shocks, ε 1, and ε 2,, are refleced ino he change in he price of fuure conrac mauring in m, and θ 3 (m,d) deermines he sandard deviaion of he shock specific o he conrac mauring in m. The hree erms, θ i (m,d), i = 1, 3, are specified as funcions of he conrac delivery monh (m) and he ime o delivery of he conrac (d = m ), 2 See Lauier (2005) for a comprehensive review on applicaions of erm-srucure models of commodiy prices o various commodiies. 1769

4 2πkd 2πkd K (5) θ ( md, ) = exp sin cos i a + a d+ i, m,0 i, m,1 a + a i, m,2 k i, m,2k+ 1 k= 1 d d max max where d max is he maximum days o mauriy for which he model is esimaed. For idenificaion, he consrain is imposed as a K 2,,0 = 1 2,,2 1 2,,1 max 10 m a a d so ha θ ( md, ) 0 for all m. The k= m k+ m 2 max condiion is equivalen o he one used in Schwarz and Smih (2000), which allows he wo facors o be inerpreed as represening he LT and ST facor, respecively. 2.3 Model Comparison A major difference beween he convenional erm-srucure models of commodiy prices and he POTS model is ha he former specifies he dynamics of price level whereas he laer specifies he dynamics of price reurn. By modeling price reurn raher han level, he POTS model does no specify seasonal and any oher deerminisic variaion of he commodiy price ha resul from demand/supply seasonaliy and oher peculiariies of he underlying commodiy. Thus, he model is free from bias in approximaing such deerminisic price variaion. To compare he wo models in furher deails, ake he firs difference of he fuures price formula in (2), σ ρ (1 ρ ) σ Δ = Δ κ 4κ 2 s 2 2s 2 2 ρ (1 ρ) χ ξ s m, λ χ ρ χξ σ χ σ ξ λ ξ ρ ω 1, ω 2, u, T (6) ln F ( ) where s = m is he ime o mauriy of he conrac, ρ = e κ, and he physical dynamics of he sae variables in (3) are used o simplify he expression. In (6), Δu,T ~ iid N(0, 2σ 2 T ) since Δu,T = u,t u 1,T. Comparison of (6) and (4) reveals hree shorcomings of he convenional approaches in modelling erm srucure of commodiy prices. Firs, he facor loadings are deermined by a small number of parameers defining he dynamics of he spo price series in he convenional erm-srucure models. Specifically, he facor loadings are exponenially decreasing in ime-o-mauriy for he ST facor, whereas hey are uniy for he LT facor. In conras, facor loadings for boh he LT and ST facors are specified by flexible, nonparameric funcions in (4), which allows he exen ha curren marke shocks are refleced ino fuures prices o exhibi very complex paerns and vary across conracs wih differen mauriy daes. Second, while he variance of measuremen error u,m is allowed o vary across he conracs wih differen mauriy dae, i is assumed o be consan over he ime-o-delivery, s, in convenional erm-srucure models. In conras, he variance of idiosyncraic error in (4) is specified as a funcion of s and his funcion is allowed o vary across conrac delivery monhs. Third, he innovaions o he sae variables, ω i,m (i = 1, 2), are assumed homoskedasic in he erm srucure models whereas heir condiional variance is specified o follow a GARCH process in he POTS model. These resricions imposed on he sochasic dynamics of he underlying facors, facor loadings, and variance of measuremen errors can ogeher resul in biased esimaes of no only underlying facors and measuremen errors bu also he parameers deermining seasonal mean price and he marke price of risk parameers in convenional erm-srucure models. These biases are expeced o be of considerable size for commodiies wih significan sorage coss and srong seasonaliy in demand/supply, which are o be examined empirically in he subsequen secion. 3. DATA AND ESTIMATION 3.1 Daa and esimaion mehods In his sudy, I esimae hree models. The firs model is he wo-facor model of Schwarz and Smih (2002). Here, I esimae he subse of he model parameers ha appear on he model s firs difference form (6). Firs differencing eliminaes he deerminisic variaion in mean price level (as specified by f(t)), hus, i emphasizes he bias resuling from he resricive specificaions of facor loadings, emporal dynamics of facor variances, and variance of measuremen errors. Firs differencing also makes he model direcly comparable o he second model I esimae; he POTS model, as defined in (4). The hird model is a composie model in which facor loadings are specified as in Schwarz-Smih 2-facor model (hence imposing resricive specificaion on he sochasic dynamics of he laen facors), whereas he variance of measuremen errors are specified by a non-parameric funcion as in he POTS model. Comparison of his model wih he POTS model allows us o disinguish he bias in he esimaed facor loadings and he bias in he esimaed dynamics of he laen facors ino wo sources; he biases resuling from he misspecificaion of heir own and hose resuling from misspecifying he variance of measuremen errors. 1770

5 I esimae he hree models for fuures prices of he following four commodiies wih differen peculiariies: Crude oil: consumpion goods wih very weak seasonaliy in demand and supply, Naural gas: consumpion goods wih srong seasonaliy in demand, Corn: consumpion goods wih srong seasonaliy in supply, and Gold: invesmen goods wih virually no seasonaliy eiher in demand or supply. Daa examined in his sudy are daily selemen prices of fuures conracs raded a he NYMEX (crude oil, naural gas, and gold) and CBOT (corn) for he period beween 1984/1/1 and 2007/12/31 for corn and gold, 1984/4/1 and 2007/12/31 for crude oil, and 1991/4/1 and 2007/12/31 for naural gas. Because long-daed conracs do no rade acively, price of conracs for more han welve monhs o mauriy are excluded from he analysis. Excluding hese observaions leaves 70,800 prices among 307 conracs for crude oil, 52,780 prices among 223 conracs for naural gas, 43,831 prices among 168 conracs for gold, and 48,762 prices among 142 conracs for corn. 3.2 Esimaion Resuls Table 1 summarizes he resuls from he specificaion es. In shor, he wo facor model of Schwarz and Smih (2000) sipulaing resricive srucures on he facor loadings is empirically no suppored. Comparing beween he POTS and composie model, he POTS model is generally preferred o he composie model for all commodiies. Only when he SIC is used, he composie model is preferred o he POTS model for crude oil which exhibis lile or no seasonaliy in demand and supply. Wha is surprising is ha he POTS model is preferred o he composie model for gold for which he sorage cos is no significan and no demand or supply seasonaliy is expeced. Figures 1 hrough 3 illusrae he facor loadings, variance of idiosyncraic error, and oal variance of naural gas prices as implied by he esimaes of he hree models. The figures show hese resuls only for naural gas, for which he difference across he hree models in he esimaed facor loadings and variance of idiosyncraic errors is mos noable. 3 In panel (a) of figure 1, he facor loadings in he esimaed POTS model indicae wo noable feaures. Firs, he esimaed facor loadings increase as he conrac approaches o he mauriy dae for all welve conracs. Second, he esimaed facor loadings in he las few monhs of rading are higher for he conracs mauring in winer han hose in summer. The esimaed facor loadings for he ST facor in panel (b) exhibi he same feaures bu in greaer magniude han hose observed for he LT facor. In addiion, for all welve conracs, he esimaed facor loadings for he ST facor sar increasing rapidly in May, before which hey are virually zero for all welve conracs. This is because he naural gas price ends o be high in winer peak demand season and here is no invenory carried over from he end of winer o he spring when he price is he lowes. In oher words, he iner-emporal price linkage breaks a he end of winer peak demand season. In panel (c), he variance of idiosyncraic error in he esimaed POTS model exhibis srong seasonaliy as well as he ime-o-mauriy effec. Unlike he wo common facors, he idiosyncraic errors are no conemporaneously correlaed across concurrenly raded conracs. This indicaes ha high volailiy in las one monh of rading, paricularly for winer conracs, represens he markes shocks ha are specific o each conac and are of very shor-erm naure. Figure 2 indicaes he facor loadings, variance of idiosyncraic error, and oal variance of naural gas prices as implied by he esimaed Schwarz-Smih wo-facor model. The model imposes he LT and ST facors o follow Brownian Moion and MR process, respecively, which resuls in he facor loadings of he LT facor o be idenical for all welve conracs and consan over one year of rading horizon whereas hose of he ST facor o decrease exponenially wih he ime-o-mauriy of he conrac a he idenical rae for all welve conracs. The model also imposes he variance of he measuremen error o be consan over he rading horizon while i allows he variance o vary by conrac delivery daes. The figure indicaes ha, for naural gas, he idiosyncraic error dominaes in magniude he wo laen facors in deermining he variance of he daily fuures price changes. Even hough he esimaed facor loadings depic clear mean-revering behaviour in panel (b), heir magniude even on he las day of rading is marginal relaive o he variance of idiosyncraic error. Togeher wih he low facor loadings of he LT facor (resuling from he low variance of he LT facor esimaed in he model), he model implies ha he daily price changes exhibi virually no correlaion across concurrenly raded conracs wih differen mauriy daes, which will imply virually no opporuniy for hedging price risks by aking spread posiions. Figure 3 indicaes he resuls for he composie model. Unlike for he Schwarz-Smih wo-facor model, he esimaed variance of idiosyncraic error exhibis srong seasonaliy and ime-o-mauriy effecs. The 3 The resuls obained for he oher hree commodiies are no presened here bu are available from he auhor upon reques. 1771

6 esimaed facor loadings exhibi he same paern as for he Schwarz-Smih wo facor model, ye hey are much greaer in magniude for he composie model han for he Schwarz-Smih model. The esimaed oal variance indicaes srong seasonaliy and ime-o-mauriy effecs in paerns similar o hose revealed by POTS model in figure 1. The resuls signify he imporance of allowing flexible funcional forms in specifying he variance of idiosyncraic error, even when a researcher esimaes parameers in convenional erm- srucure models. 4. CONCLUSION In his sudy, I compare a convenional wo-facor erm-srucure model of commodiy fuures prices wih an alernaive approach of modeling he variance of daily fuures reurns direcly by flexible, non-parameric funcions. Empirical esimaion of he models o four commodiies; gold, crude oil, naural gas, and corn, reveal ha he volailiy of daily fuures prices exhibis srong ime-o-mauriy effecs as well as significan seasonal paern in is levels and composiions (among wo facors and idiosyncraic errors). These complex paerns in he volailiy of commodiy fuures prices canno be replicaed by he convenional wo-facor Gaussian model, due o he resricive dynamics sipulaed ono he underlying sochasic facors. The composie model performs reasonably well in capuring srong seasonaliy in he volailiy of commodiy prices. The resul signifies he imporance of specifying he variance of idiosyncraic error by flexible funcional forms; allowing capure of boh he ime-o-mauriy effec and he seasonaliy in paerns differen among conracs wih differen mauriy dae. Misspecifying he variance of idiosyncraic error can resul in very differen parameer esimaes in he convenional erm-srucure models, which cauions recen rend of allowing more flexible sochasic processes of laen facors while sipulaing resricive srucures on he variance of idiosyncraic errors. Incorrec porrai of volailiy dynamics as well as correlaion among concurrenly raded conracs wih differen mauriy daes can lead he convenional erm-srucure models o sugges hedging sraegies ha are less effecive han he sraegy based on he model of fuures reurns. REFERENCES Hull, J.C. (2002). Opions, Fuures, and Oher Derivaives. 5 h ed. Prenice Hall. Lauier, D. (2005). Term srucure models of commodiy prices: A review. Journal of Alernaive Invesmens, 8(1), Schwarz, E.S. (1997). The Sochasic behavior of commodiy prices: Implicaions for valuaion and hedging. Journal of Finance, 52, Schwarz, E.S., and J.E. Smih. (2000). Shor-erm Variaions and Long-erm Dynamics in Commodiy Prices. Managemen Science, 46, Smih, A. (2005). Parially overlapping ime series: A new model for volailiy dynamics for commodiy fuures. Journal of Applied Economerics, 20, Suenaga, H., and A. Smih. (2011). Volailiy Dynamics and Seasonaliy in Energy Prices: Implicaions for Crack-Spread Price Risk. Energy Journal, 32(3), Williams, J.C., and B.D. Wrigh. (1991). Sorage and Commodiy Markes. Cambridge Universiy Press, New York. ACKNOWLEDGEMENT This research was conduced wih he financial suppor from he ARC Discovery Gran (DP ). APPENDIX TABLE AND FIGURES Table 1. Model Selecion Tes CL NG GC C AIC POTS Schwars-Smih Composie SIC POTS Schwars-Smih Composie

7 (a) Facor Loading 1 1 (b) Facor Loading 2 1 (c) Sandard Deviaion of Idiosyncraic Error 1 (d) Toal Variance Figure 1. Esimaed facor loadings, variance of idiosyncraic error and oal variance for POTS model (a) Facor Loading (b) Facor Loading (c) Sandard Deviaion of Idiosyncraic Error (d) Toal Variance Figure 2. Esimaed facor loadings, variance of idiosyncraic error and oal variance for convenional ermsrucure model (Schwarz-Smih, wo-facor model) (a) Facor Loading 1 1 (b) Facor Loading 2 1 (c) Sandard Deviaion of Idiosyncraic Error (d) Toal Variance Figure 3. Esimaed facor loadings, variance of idiosyncraic error and oal variance for composie model 1773