Short-Run and Long-Run Dynamics of Resource. Commodity Prices

Transcription

1 Shor-Run and Long-Run Dynamics of Resource Commodiy Prices (Working Paper) Manle Lei Deparmen of Economics Universiy Of Guelph Guelph, ON N1G 2W1 Phone: , ex Glenn Fox Deparmen of Agriculural Economics & Business Universiy Of Guelph Guelph, ON N1G 2W1 Phone: , ex April 2004

2 Absrac This paper discusses he shor-run and long-run dynamics of price for sorable naural resource commodiies. I deals wih he inerrelaionships beween price, invenory, and price volailiy; he effecs of invenory and producers operaing flexibiliy on he dynamics of price in he shor-run; and he evoluion of he long-run equilibrium price. The paper describes a model ha explains he dynamics of commodiy prices based on demand for invenory and on hreshold prices ha would induce producers o expand or conrac operaions. The paper concludes ha in he shor-run, producers operaing flexibiliy reduces price volailiy when he spo price is higher han he hreshold price which would cause expansion in he scale of operaions. I furher concludes ha operaing flexibiliy will raise price volailiy when he spo price is lower han he hreshold price which would resul in a conracion of operaions. I concludes ha in he shor run, commodiy prices end o rever o a shor-run equilibrium price, whereas in he long run, hey end o rever o a long-run rend. The paper demonsraes he failure of currenly used parameric models in describing he sochasic process of commodiy prices, and suggess using non-parameric mehods. I also recommends including he ime rend in such a model. As an illusraion he paper presens an analysis of he price processes for lumber and coon. 2

3 1. Inroducion The sochasic behavior of commodiy prices plays an imporan role in models for evaluaing resource invesmens 1. Usually we represen he underlying price as a coninuous-ime diffusion process, saisfying a ime-homogeneous sochasic differenial equaion 2, dp = µ ( P) d + σ ( P) dz (1.1) where µ (P) is he drif funcion, σ (P) is he diffusion funcion, and Z is a sandard one- dimensional Brownian moion. Here we assume ha boh drif erm and diffusion erm are funcions only of he underlying price. Equaion (1.1) can be esimaed by eiher parameric or non-parameric mehods. However, in he real opions lieraure, mos sudies have been conduced using parameric mehods. In resource economics, he mos popular models used o describe he sochasic process of commodiy prices are he geomeric Brownian moion model and mean revering model. For example, Clarke and Reed (1989) derived an opimal harvesing rule for he single roaion problem when price is assumed o follow geomeric Brownian moion. Insley (2001) esimaed he opimal cuing ime assuming he lumber prices are mean revering. Recen sudies also inroduced some alernaive models 3, such as sochasic volailiy process 4 and Schwarz and Smih s (2000) wo-facor model. One advanage of parameric models is ha i may be possible o obain analyical soluions for opimal invesmen rules. However, no all parameric models have such 1 See Schwarz and Smih (2000). 2 See Sanon (1997). 3 Since mos jumps have been smoohed ou in monhly or quarerly daa, we do no use jump models in his paper. The discussion on he ime smoohing effec can be found in Ai-Sahalia (2003). 3

4 analyical soluions. In any case, our objecive is no o find an analyical soluion, bu a soluion ha is reliable. Thus, a quesion arises: are he prices of resource commodiies and oher goods bes modeled as geomeric Brownian moion, mean revering processes, or some alernaive process? A good parameric model describing he sochasic process of commodiy prices should reflec he dynamics of commodiy prices. To answer he above quesion, a deailed analysis of he dynamics of commodiy prices is necessary 5. Pindyck (2001) sudied he shor-run dynamics of resource commodiy spo markes, and explained he inerrelaionships among prices, raes of producion and invenory levels. In his paper we use Pindyck s framework o consider he effecs of producers abiliy o expand or conrac producion wihin a fairly shor ime frame. In some indusries, a monh or a quarer is a period long enough for producers o begin he adjusmen of heir operaing scales by hiring more labor or acquiring more capial equipmen or, alernaively, by emporarily shuing down producion. In his paper we consider he impac of his ype of operaing flexibiliy on he dynamics of resource commodiy prices. The main purposes of his paper are o analyze he shor-run and long-run 6 dynamics of resource commodiy prices for he evaluaion of resource invesmens, and o provide a heoreical foundaion for modeling and esimaing he sochasic process of resource commodiy prices. The ouline of his paper is as follows: Secion 2 describes he 4 See Deng (1999). 5 This paper is relaed o a foresry projec sudying opimal managemen under uncerainy. Thus, in his paper, we ry o use Canadian and US lumber prices as examples. To illusrae a non-horizonal long-run rend, we pick up he US coon price for convenience. We are concerned only wih sorable resource commodiies such as lumber, woodpulp, and coon. Since elecriciy is non-sorable, i is no included here. Readers ineresed in he sochasic models of elecriciy can see Deng (1999). 6 In his paper, shor-run refers o he period in which he producers face only one shock and have ime o adjus heir producion scale, and long-run refers o he period ha he producers face a series of shocks. 4

5 commodiy marke and illusraes invenory and ne demand, he propery of price volailiy, and producers operaing flexibiliy. Secion 3 and secion 4 illusrae he shorrun and long run dynamics of commodiy prices, respecively. Secion 5 discusses he performance of parameric models in describing he sochasic process of commodiy prices and implicaions for he modeling of price processes, using lumber and coon as examples. Secion 6 presens conclusions. 2. Commodiy Markes and Producers Operaing Flexibiliy In his secion, we firs review Pindyck s model of he shor run dynamics of commodiy spo markes. Then we will describe producers operaing flexibiliy in erms of real opions Commodiy Markes Pindyck (2001) explained he funcion of invenories as follows. In a compeiive commodiy marke, boh producers and indusrial consumers hold invenories in order o miigae he impacs of sochasic flucuaions in producion and consumpion. For producers, invenories can reduce he coss of adjusing producion over ime, faciliae producion and delivery scheduling and help o avoid sockous. Changing he rae of producion may involve adjusmen coss such as he coss of hiring and raining new or emporary workers, leasing addiional capial, ec. The selling off of invenory during 5

6 high-demand periods and replenishing invenories during low-demand periods can reduce hese adjusmen coss. Pindyck (2001) illusraed how boh he price level and volailiy affec he demand for invenory. An increase in price volailiy implies an increase in he demand for invenory. Oher hings equal, marke paricipans will wan o hold greaer invenories when prices are more volaile in order o buffer he effecs of flucuaions in producion or consumpion. Denoe N D / σ > 0. D N as he demand for invenory, σ as he price volailiy, and hen The demand for invenory also depends on he spo price of he commodiy. Oher hings equal, one should be willing o pay more o sore a higher-priced good 7 han a lower-priced one, i.e., N D / P > 0. Thus we can wrie he demand of invenory as N D D = N P; σ, z ) (2.1) ( N where P is he spo price, and z N is a vecor of variables ha can affec he demand of invenory oher han spo price and price volailiy. When invenory holdings can flucuae, producion in any period need no equal consumpion. As a resul, he marke-clearing price in he spo marke is deermined no only by curren producion and consumpion, bu also by changes in invenory holdings. Denoing Ne Demand as he difference beween producion and consumpion, Pindyck (2001) characerized he cash marke 8 as a relaionship beween he spo price and Ne 7 Of course, one of he hings being held equal would have o be he cos of he good in quesion. Anoher would be he cos of sorage. So ha, by a high priced good we mean a good for which he expeced profis from sorage are high. 8 The cash marke refers o purchases and sales of a commodiy for immediae delivery. 6

7 Demand. Wriing he demand funcion for curren consumpion and he supply funcion for curren producion for he cash marke as: D = D P ; z, ε ) (2.2) ( D D and S = S P ; z, ε ) (2.3) ( S S where P is he spo price, z is a vecor of demand-shifing variables, z is a vecor of supply-shifing variables, andε D D andε S are random shocks a ime. Leing N denoe he invenory level a ime. The change in invenories a ime is given by he accouning ideniy: S dn = S P ; z, ε ) D( P ; z, ε ) (2.4) ( S S 2 D D Consider SPz ( ; S, ε S 2 ) DPz ( ; D, ε D ) as ne supply of invenories and dn as he change in invenory or ne demand for producion in excess of consumpion. Thus his equaion jus says ha a seady sae, i.e., dn = 0, curren consumpion equals curren producion. Rewrie equaion (2.4) in erms of he following inverse ne demand funcion: P = P dn ; z, z, ε, ε ) (2.5) ( D S D S Marke clearing in he cash marke herefore implies a relaionship beween he spo price and he change in invenories. Because S / P > 0and D / P < 0, he inverse ne demand funcion is upward sloping in dn, i.e., a higher price corresponds o a larger S and smaller D, and hus a larger dn. 7

8 Pindyck (2001) poined ou ha price volailiy has negaive relaionship wih invenory level, i.e., an increase in invenory level can reduce price volailiy 9. Suppose a emporary shock causes he price level o rise. Such shock, eiher a emporary increase in demand or a emporary decrease in supply, will also usually cause a drawdown in invenories and hence a decrease in he invenory levels 10. So a price increase will be expeced o be accompanied by a decrease in invenory level. Thus, price volailiy will be posiively relaed o price Producers Operaing Opions In addiion o alering invenory levels, producers generally have some flexibiliy o change he scale of operaions by hiring or firing workers or acquiring or mohballing capial equipmen. If marke condiions are more favorable han expeced, producers can expand he scale of producion, or he conversse if condiions are less favorable han expeced, producers can reduce he scale of operaions. In exreme cases, producion may be shu down and possibly resared laer. To simplify his illusraion, we assume ha a any operaing scale, he individual producer s producion is no elasic. Iniially, he marke price is P*, and he base (or exising) scale 11 of he producer is Q*, as shown in Figure Demand for invenory (discussed in previous subsecion) and invenory level are differen. Demand for invenory is he desired level of invenory, while invenory level is he acual level of invenory. Since marke paricipans will wan o hold greaer invenories when prices are more volaile in order o buffer he effecs of flucuaions in producion or consumpion, demand for invenory is posiively relaed o price volailiy. However, invenory level is negaively relaed o price volailiy. 10 Producers have o judge wheher he price increase is caused by a shock or by regular flucuaion. Thus, producers will no adjus heir operaion immediaely, and producion will increase laer han price rises. 11 Base scale refers o he desired scale of operaions. 8

9 If he marke condiions urn ou o be more favorable han expeced, i.e., he commodiy price increases significanly (above Pe in Figure 2-1), hen producers can expand he scale of producion by incurring an invesmen, Ie. This managerial flexibiliy is similar o a call opion o add addiional capaciy by paying Ie as he exercise price 12. If we denoe V as he presen value of he compleed projec s expeced operaing cash flow, x as he percenage increase of he producion scale, hen he invesmen opporuniy wih he opion o expand can be viewed as he base-scale projec plus a call opion on fuure invesmen, i.e., V + max( xv Ie,0). If marke condiions are weaker han originally expeced, i.e., he commodiy price drops below he average oal cos of producion (below Pc in Figure 2-1), hen he producer can reduce he scale of operaions o save par of fuure invesmen oulays needed o mainain he curren scale of operaions, I. This flexibiliy o miigae loss is analogous o a pu opion on par of he base-scale projec, wih exercise price equal o he poenial cos savings, Ic. If we denoe c as he percenage decrease of he producion scale, hen he invesmen opporuniy wih he opion o conrac can be viewed as he base-scale projec plus a pu opion on fuure invesmen, i.e., V I + max( Ic cv,0). Producers do no have o operae in each and every period. In case he price drops such ha cash revenues are no sufficien o cover variable operaing coss (below Ps in Figure 2-1), i migh be beer no o operae emporarily. Producers can resar operaions laer, once prices rise sufficienly. If we denoe R as he annual cash revenues, Iv as he variable coss of operaing, hen operaion in each year is similar o a call opion o 12 A deailed discussion can be found in Trigeorgis (1996). 9

10 acquire ha year s cash revenues by paying he variable coss of operaing as he exercise price, i.e., max( R Iv,0). 3. The Shor-Run Dynamics of Commodiy Prices Since a resource invesmen ypically lass for several decades, we need o consider he evoluion of commodiy prices over a long period. In such a long period, mos ofen only monhly or quarerly daa are available. For producers, a monh or a quarer may be a period long enough o begin he adjusmen of heir operaing scales. Thus, his ype of operaing flexibiliy should be considered in a shor-run dynamic analysis. For his reason, his paper will use shor-run o denoe he period in which he producers face only one shock and have ime o adjus heir producion. In order o analyze he shor-run dynamics of commodiy prices in more deail, we will consider how invenory, price and price volailiy change in response o small or large exogenous shocks 13. We define a shock as posiive if i raises price and negaive if i decreases price. Boh posiive and negaive shocks can originae eiher on he supply or he demand side of he marke. For example, counry M s high economic growh rae increases he demand for lumber. A fores fire in counry N reduces he lumber supply. Boh evens will raise he lumber price, so hey are posiive shocks. On he conrary, if a new maerial is invened o replace lumber in furniure, or a new bioechnology can be 13 Pindyck (2001) did no consider he issue of he magniude of shocks. We differeniae beween small and large shocks. Small shocks do no change he price enough o cause producers o change operaing scales, while large shocks do. 10

11 applied o significanly increase he growh rae of rees, hen he lumber price will be pushed down. These evens are called negaive shocks in his paper. In his secion, we analyze he shor-run evoluion of commodiy price when a posiive or negaive shock occurs, as well as he behavior of price volailiy Posiive Shocks 14 A Small Posiive Shock Suppose a small posiive shock occurs. According o Pindyck (2001), he dynamics of price and invenory change can be expressed by he inverse ne demand funcion, P(dN), as shown in Figure 3-1. Iniially, he ne demand funcion is P ( dn 1 ), and ne demand is zero. Denoe he curren spo price as P*. A small posiive shock will shif he inverse ne demand funcion P ( dn ) upward o P ( dn) 1 2. Before producers can adjus producion, invenories will decrease (dn1 < 0), and his will limi he size of he price increase (from P* o P1 in Figure 3-1). Poin B is no a seady sae, (because invenory mus be posiive, i canno keep decreasing.) The new seady sae equilibrium can only occur when he ne demand curve crosses he verical axis and dn = 0), so price will move along he P ( dn 2 ) curve upward o P**, he new equilibrium price. Since he price does no rise over he hreshold price o expand, Pe, he producer will no change he scale of he operaion, bu will adjus is producion a he base scale level. As 14 The definiion of shock in his paper is differen from Pindyck (2001). Pindyck analyzed he overall effecs of a emporary even on he dynamics of commodiy price. We find ha i is more convenien o use wo separae shocks o describe an even: one shock a he occurrence of he even, and one shock when he effec of he even disappears. I is useful when he effec of a emporary even migh las several years. 11

12 producion increases, he invenory level increases. Since he invenories will be reaccumulaed 15, producion will have o exceed consumpion (i.e., dn > 0). Thus in Figure 3-1, he spo price rises o P2 > P** (from B o C in Figure 3-1). This accumulaion of invenories will coninue unil he invenory level reaches N D (P **), he new balanced invenory level, i.e., dn D ( P **) = 0. When here is no furher accumulaion of invenories, he spo price will fall o P** (from C o P** in Figure 3-1). Figure 3-3 shows he shor-run behavior of price volailiy, which is indicaed by meanreversion model 16. Iniially, a spo price P*, he price volailiy is E. If producers have no abiliy o expand or conrac producion, he diffusion curve will be he ABECD line in Figure 3-3. A small posiive shock raises he price, and causes he price volailiy goes upward along he EC line. A Large Posiive Shock When he shock is large, i.e., he shock pushes he price o rise above he hreshold o expand, he consequence of such a shock is no like ha of small shock. Firs, he shock will immediaely push up he spo price. The inverse ne demand funcion P(dN) will shif upwards o 2 ( dn) in Figure 3-2. If we neglec he producer s opion o expand, P LP 15 To miigae he posiive shock, invenory has been reduced o he required level. 16 On modeling he commodiy price, Pindyck (2001) concluded ha, over he long run, price behavior seems consisen wih a model of slow mean reversion, i.e., dp = B ( P P ) d + σ P dz, where B is he revering rae, P is he mean of price, and σ is a consan. This model indicaes ha price volailiy curve is linear wih posiive sloping. 12

13 we would hink ha he inverse ne demand funcion say a 2 ( dn), and conclude ha, similar o he small posiive shock case, he marke will reurn o equilibrium a price. However, since a dn = 0, P > P, he producer will choose o expand he scale of 2 e operaion. Before spo price reaches Pe, he producer will no exercise he opion o expand, and he invenories will decrease (dn1 < 0). When he spo price reaches Pe, he producer will expand he scale of operaion, and he inverse ne demand funcion will shif downwards o P LP 3 ( dn). Thus, in Figure 3-2, insead of P* A B P2, he dynamics of commodiy price will be P* C D P **. We observe in he Figure 3-2 ha when we ake accoun of a producer s abiliy o expand producion capaciy, he price increase caused by he large posiive shock is mued and invenory changes are less han when we ignore his flexibiliy When a firm faces a posiive shock, is producion increases by a greaer amoun wih operaing opions han wihou, herefore is invenory level wih operaing opions will be higher han he invenory level wihou. Thus, he abiliy o expand operaing scale will reduce price volailiy level, and bend he volailiy curve downward from CD o CD1 or CD2 shown in Figure 3-3. P LP P Negaive Shocks The consequence of a negaive shock is similar o ha of a small posiive shock, bu in he opposie direcion. The dynamics of price and change of invenory for a small negaive shock is shown in Figure 3-4 as P* B C P**. As shown in Figure 3-3, a 13

14 small negaive shock decreases he price, and causes he price volailiy o go downward along he EB line. The dynamics of price and change of invenory in a large negaive shock is shown in Figure 3-5. Wihou considering he conrac opion, he dynamics is P* J K P 2. However, if we consider he conrac opion, he dynamics will be P* L M P ***. The conracion of operaing scale will raise he price volailiy level and bend he diffusion curve upward from AB o B or B as shown in Figure 3-3. If he negaive shock is so large ha he spo price may be less han some producers hreshold price o emporarily shu down a some poin before he marke reaches a new balance, hen hese producers will choose o suspend producion for a while, and resume operaions laer when he spo price rises again. Since he effec of he operaing opion o emporarily shu down on he commodiy marke is similar o he effec of he operaing opion o conrac, we will no alk abou his process in deail. A 1 A Empirical Examples The locus of price volailiy depends on he properies of he paricular commodiy marke. Of course, producers managerial flexibiliy will depend on he characerisics of he indusry so we would expec o see ha differen commodiies have differenly shaped diffusion curves. 14

15 Figure 3-6 and 3-7 presen ime series daa of real Canadian sofwood lumber prices and real US lumber prices, respecively. The Canadian price daa is a Monhly Price Index for Canadian sofwood lumber from January 1956 o December I is a nominal price index defined so ha P 1997 =100 and deflaed by he monhly Canadian Consumer Price Index 17. The US lumber price daa, from January 1947 o December 2003, was obained by deflaing he monhly nominal price index of US lumber by he monhly US Consumer Price Index. Figure 3-8 and figure 3-9 compare esimaes of drif and diffusion erms for he real prices of Canadian and US lumber esimaed by a non-parameric mehod and using a mean revering model 18. For non-parameric esimaion, we employ a local linear mehod, and choose bandwidh h = 6* n 2 / 9 * sd( P), where n is he number of observaions. The figures indicae ha boh price volailiy curves are roughly in line wih he approximae S-shape shown in Figure They also indicae ha esimaion by he mean-revering model migh give us biased resuls on he diffusion esimaion. 17 I seems ha choosing differen deflaors does no have significan effec on he resuls. If we use indusrial price index o be he deflaor, we can ge similar curves. 18 Mean Revering Model: dp = η ( P P ) d + σ P dz 19 Some researches poined ou ha non-parameric esimaion of price volailiy migh cause spurious nonlineariy. (See Fan and Zhang, 2001). To es he hypohesis ha he non-lineariy of he volailiy is spurious, we esimae he price process and use Mone Carlo mehod o generae 500 simulaions wih same size. The quaniles of esimaions seem o indicae ha he S shape of volailiy curve canno be explained only by non-lineariy. The resuls of he es can be seen in Appendix I. 15

16 4. The Long-Run Dynamics of Commodiy Price Marke uncerainy can be regarded as he resul of an endless series of posiive and negaive shocks, caused by shifs in he supply funcion and/or demand funcion. The effecs of hese shocks can be caegorized as emporary effecs and permanen effecs, based on heir duraion. A emporary exogenous effec can be regarded as consising of wo shocks of opposie direcion bu wih he same srengh. As he exogenous effec disappears, producion, consumpion and invenory levels, as well as commodiy price and price volailiy will reurn o heir iniial levels. Thus, alhough emporary exogenous effecs cause shor erm variaions in price, hey will no change he value of he long-run equilibrium price. A permanen exogenous effec is caused by a single shock which changes hese marke variables permanenly. The new long run equilibrium price will depend on consumer and producer responses o he shock. For example, if counry A s imber cuing regulaion have a posiive permanen effec on lumber price, boh producers and consumers will gradually respond. Producers will ry o employ new echnology o increase he oupu, and consumers will ry o find cheaper subsiues for lumber. If here is no long-run upward or downward rend in he long run equilibrium price, his implies ha he permanen exogenous effecs are symmerically disribued and neuralize each oher. Thus, in he long run, he evoluion of long-run equilibrium price will exhibi mean reversion. 16

17 However, a long-run upward or downward rend does exis in some resource prices 20. The long run price rend will be deermined by he properies of he underlying resource, such as he possibiliy of subsiuing and being subsiued by oher maerials, he possibiliy of increasing he producion rae, and he possibiliy of exincion. Figure 3-6 and Figure 4-1 show ime series daa of he real prices of Canadian sofwood lumber and US coon, which appear o indicae no rend in he lumber price and a downward rend in he coon price. I is imporan o consider he exisence of a ime rend when modelling resource commodiy prices. Noe ha he responses of producers and consumers o he change of long-run equilibrium price migh las several years, even decades. When we es he upward or downward rend of price, we mus be careful o examine a long enough ime span of daa. For example, Figure 4-2 shows he lumber price index from January 1973 o December 1992 and here seems o be a downward rend in he lumber price. However, if we invesigae a longer ime series from January 1956 o December 2003 as shown in Figure 3-6, he downward rend seems o disappear. When we evaluae foresry invesmen, if he period of he available daa is less han he opimal harvesing age, we should be cauious o include a ime rend in modeling he sochasic process of lumber price. 20 The forecasing of long-run rend is beyond his paper. Relaed discussion can be found in Pindyck (1999). 17

18 5. Modeling and Esimaing Commodiy Price Process A good parameric model describing he sochasic process of commodiy prices should capure he characerisics of he evoluion of commodiy prices. In his secion, we will examine wheher hose currenly used parameric models can reflec he dynamics of commodiy prices Parameric Models The mos popular parameric models describing he sochasic process of commodiy prices in he real opions lieraure include geomeric Brownian moion (GBM) and some ype of mean revering (MR) process. GBM MR dp dp = B P d + σ P dz (5.1) = B ( P P) d + σ P dz (5.2) where B, σ are assumed consan, and P is he mean of price. Oher parameric examples used in he finance lieraure include models from Vasicek (VAS) (1977), Cox, Ingersoll and Ross (CIRSR) (1985), Cox, Ingersoll and Ross (CIRVR) (1980), and Chan, Karolyi, Longsaff and Sanders (CKLS) (1992). VAS dp = ( A + B P ) d + σ dz (5.3) CIR SR dp = ( A + B P ) d + σ P dz (5.4) 3 / 2 CIR VR dp = σ P dz (5.5) γ CKLS dp = ( A + B P ) d + σ P dz (5.6) In model (5.3) (5.6), A, B,σ, γ are consan. Comparing model (5.1) (5.6), we can find ha model (5.1) (5.5) are generalized by model (5.6). 18

19 Can hese models correcly reflec he dynamics of he price process? The diffusion erm in all hese parameric models assumes a consan funcional form, σ ( ) = σ P. γ Obviously, his funcional form canno describe he S shape of he volailiy curve. Even hough he dynamics of price shows mean reversion, since producers have operaing opions o expand or conrac, he mean revering rae a differen price level migh be differen. Thus, hese models migh be oversimplified Some sudies have considered alernaive models. One is he sochasic volailiy model. For ease of illusraion, consider a simple sochasic volailiy model: geomeric Brownian moion where σ in he diffusion erm follows Brownian moion. 21. dp = B P d + σ P dz (5.7) σ = κ d + υ (5.8) dw where B, κ and υ are consan and W is a sandard one- dimensional Brownian moion. Anoher alernaive is Schwarz and Smih (2000) s wo-facor model. Schwarz and Smih (2000) developed a wo-facor model of commodiy prices ha allows meanreversion in shor-erm prices and uncerainy in he long-erm equilibrium level o which prices rever. In heir parameric model, he long-run equilibrium price is assumed o evolve according o geomeric Brownian moion wih a drif erm reflecing expecaions of he exhausion of exising supply, improving producion echnology and fuure discoveries of he commodiy, inflaion, as well as poliical and regulaory effecs. The shor-erm deviaions, defined as he difference beween spo and equilibrium prices, are expeced o rever oward zero following an Ornsein-Uhlenbeck process. These 21 Empirical examples will be provided laer in his secion. 19

20 deviaions may reflec shor-erm changes in demand resuling from variaions such as inermien supply disrupions, and are empered by he abiliy of marke paricipans o adjus invenory levels in response o changing marke condiions. However, as is demonsraed in Appendix II, boh he sochasic volailiy model and Schwarz and Smih s wo-facor model indicae he posiive relaionship beween price volailiy and price. Thus, hey canno explain he S shape of he volailiy curve Impac of Misspecificaion on Invesmen Decisions All parameric esimaors have some model assumpions which, if incorrec, can lead o model misspecificaion. Once model misspecificaion exiss, bias can resul in parameer esimaes. Misspecificaion migh cause non-opimal in invesmen decisions. For example, suppose we wan o deermine he enry and exi hreshold prices of an indusry. Figure shows he enry and exi hreshold prices curves as funcions of he price volailiy of he underlying commodiy. The enry hreshold price has posiive relaionship wih he price volailiy, while he exi hreshold price has a negaive relaionship wih he price volailiy. (The inuiion here is ha, he higher he volailiy, he greaer he opion value o ener or o exi, and he laer he producer will exercise he opion. Thus, a higher volailiy will be accompanied by a higher enry hreshold price and a lower exi hreshold price.) Assume he real volailiy curve is ABCD in Figure 5-2, which is derived from Secion 3. However, we impose an assumpion ha he diffusion erm has a funcional form as 22 See Dixi & Pindyck (1994), p

21 σ ( ) = σ P, where σ is consan. The esimaed diffusion curve, based on he meanrevering model (equaion (2.6)), is OE in Figure 5-2. Suppose he enry hreshold price is P, and he exi hreshold price is P. In he H L neighborhood of P H, he esimaed price volailiy is lower han he real value, so he * esimaed enry hreshold price, P, will be lower han P ; In he neighborhood of P, H he esimaed price volailiy is higher han he real value, so he esimaed exi hreshold * price, P, will also be lower han P. We can see ha, in his special case, L misspecificaion migh cause invesors o ener oo early or o exi oo lae. L H L 5.3. Applicaion of Non-parameric Mehod In general, we do no know he exac funcional form of he drif erm and diffusion erm. We es hypoheses on he funcional form of he commodiy price process. However, based on he analysis on he dynamics of commodiy prices in secion 2, i seems ha he funcional form migh be very complicaed, so i is logical o check wheher or no a specific funcional form is necessary. Consider he value of a coningen claim, F( P, ), which is a funcion of an underlying commodiy price, P, and ime,, only. To evaluae he coningen claim, we need o solve he parial differenial equaion saisfied by he value funcion: F F F σ ( P) + µ ( P) + ρ F = 0 (5.9) 2 2 P P where σ (P) is he diffusion funcion, µ (P) is he drif funcion, and ρ is consan. To solve he parial differenial equaion (5.9) numerically, we only need o know he 21

22 condiional value of he drif funcion and diffusion funcion on price. However, we can use a non parameric mehod o obain he condiional value of he drif and diffusion. Thus, a funcional form for drif or diffusion is no necessary. One advanage of non-parameric esimaors is ha hey relax model assumpions, so he possible modeling biases are reduced. Because nonparameric esimaors require lile prior informaion relaing o he funcional form of he condiional expecaions, i is very convenien o employ hese non-parameric esimaors. Since he rend reflecs he long-run characerisic of he evoluion of commodiy price, which is no relaed o he shor-run flucuaion represened by he drif, we can separae he rend and he drif. Denoing he long-run equilibrium price change, or he rend, as * * L(), hen he price a ime can be decomposed as P = L ) + P, where is he derended price a ime. The change of price can be wrien as: ( P dp = L( ) d + µ ( P L( )) d + σ ( P ) dz = L( ) d + µ ( P ) d + σ ( P ) dz * (5.9) The de-rended drif can be idenified by: 1 * * * 1 lim E[ dp P = P ] = lim E[ dp d 0 d d 0 d 1 * * * * = lim E[ dp P = P ] = µ ( P ) d 0 d L( ) d P L( ) = P * ] (5.10) The diffusion σ (P) can be idenified by: 1 lim E[( dp ) d 0 d 2 P 2 = P] = σ ( P) (5.11) 22

23 Equaion (5.10) and (5.11) can be esimaed by Kernel mehods 23. The Kernel esimaors of he j h ( j = 1,2) condiional momens are: 1 Eˆ[( dp ) d j P i= 1 = P] = T T Pi P ( P K( ) * h i= 1 j i+ 1 Pi P K( ) h j P) d i j (5.12) h where is he opimal bandwidh in he j esimaor. h j Thus, he procedure for esimaing he diffusion process includes wo seps. Firs, esimae he rend L( ) d and second, use Kernel esimaors (5.12) on he no de-rended ime series daa o obain variance σ 2 ( P) and on he de-rended ime series daa o ge he de-rended drif µ ( P * ). The expecaion of price incremen can be obained by: E( dp P = P) = L( ) d + µ ( P * P * = P L( ) ) d (5.13) Figure 5-3 shows he esimaion of drif of he real price of US Coon wih and wihou de-rending by a non-parameric mehod using a mean revering model (5.2) and Model I, below. Model I dp = ( A + B P ) d + σ P dz (5.14) We can see ha he drif wihou de-rending does no show he mean revering propery - he esimaion of he drif erm using he mean revering model has posiive slope. The dashed line is he esimaion by using Model I, which relaxed he mean revering assumpion. However, afer de-rending, he drif erm shows mean revering propery. In his case, he esimaions, using Model I and using he mean-revering model, coincide. 23 See Fan and Zhang (2001). 23

24 Figure 5-4 compares he esimaion of he diffusion erm by a non-parameric mehod and using he mean-revering model. Again, i seems ha price volailiy curve is roughly in line wih he approximae S-shape shown in Figure 2-6, and ha esimaion by he meanrevering model migh give us biased resuls on he diffusion esimaion. In Appendix I we discuss wheher he non-parameric resuls presened in Figure 5-4 may be spurious and an arifac of non-parameric esimaion. We conclude ha he S-shaped volailiy curve canno be solely explained by spurious non-lineariy. 6. Conclusions This paper discusses several aspecs of he shor-run and long-run dynamics of commodiy prices and he inerrelaionships beween price, invenory and price volailiy. The paper illusraes he effecs of invenory and he producers operaing flexibiliy on he dynamics of price in he shor-run, as well as he evoluion of he long-run equilibrium price. The paper also discusses how o model and esimae he sochasic process of commodiy prices. Some conclusions can be made from he above discussion: In he shor-run, producers operaing flexibiliy pus downward pressure on volailiy a high price levels, and upward pressure on volailiy a low price levels, causing a plo of volailiy versus price o be somewha S-shaped. The paper illusraes he failure of currenly used parameric models in describing he sochasic process of commodiy prices, and suggess using non-parameric 24

25 mehods. Currenly used parameric models do no allow for he S-shape which we argue is reasonable given our model of he dynamics of commodiy prices. We demonsrae he esimaion of he sochasic process followed by coon and lumber prices. We find ha in he shor-run, hese commodiy prices end o rever o he shor-run equilibrium price; in he long-run, hey end o rever o a long-run rend. The paper also suggess ha i is imporan o include a ime rend in modeling he sochasic process of commodiy prices. 25

26 Reference Bakshi, G., C. Cao, Z. Chen. Empirical Performance of Alernaive Opion Pricing Models. Journal of Finance 52, , Bandi, Federico & Thoug Nguyen. On he Funcional Esimaion of Jump- Diffusion Models. Journal of Economerics, 116: , Brazee, Richard & David Newman. Observaions on Recen Fores Economics Research on Risk and Uncerainy. Journal of Fores Economics, 2: , Deng, Shijie. Sochasic Models of Energy Commodiy Prices and Their Applicaions: Mean-reversion wih Jumps and Spikes. Working Paper Dixi, A.K. & R.S. Pindyck. Invesmen under Uncerainy (Princeon, NJ, USA: Princeon Universiy Press), Duffie, D., J. Pan, K. Singleon. Transform Analysis and Asse Pricing for Affine Jump Diffusions. Economerica 68, , Fan, Jiangqing & Qiwei Yao. Efficien Esimaion of Condiional Variance Funcions in Sochasic Regression. Biomerika, 85: Fan, Jianqing & Chunming Zhang. A Re-examinaion of Sanon s Diffusion Esimaions wih Applicaions o Financial Model Validaion. Working Paper Insley, M.C.. A Real Opions Approach o he Valuaion of a Foresry Invesmen. Journal of Environmenal Economics and Managemen, 3: ,

27 Jiang, George J. & John L. Knigh. Parameric versus Nonparameric Esimaion of Diffusion Processes A Mone Carlo Comparison. Working Paper Johannes, Michael. The Saisical and Economic Role of Jumps in Coninuous-Time Ineres Rae Models. Journal of Finance. Forhcoming Pindyck, Rober. The Long-Run Evoluion of Energy Prices. The Energy Journal, 20:1-27,1999. Pindyck, Rober. The Dynamics of Commodiy Spo and Fuures Markes: A Primer. The Energy Journal, 22(3), 1-29, 2001 Schwarz, E.S. The Sochasic Behavior of Commodiy Prices: Implicaions for Valuaion and Hedging. Journal of Finance, 52: , Schwarz, E.S. & J.E. Smih. Shor-Term Variaions and Long-Term Dynamics in Commodiy Prices. Managemen Science, 46: , Saphores, J.-D., Lynda Khalaf, & Denis Pelleier. On Jump and Arch Effecs in Naural Resource Prices: An Applicaion o Sumpage Prices from Pacific Norhwes Naional Foress. Paper presened a he annual meeings of he Canadian Economics Associaion, June Sanon, Richard. A Nonparameric Model of Term Srucure Dynamics and he Marke Price of Ineres Rae Risk. The Journal of Finance. 52: , Trigeorgis, Lenos. Real Opions Managerial Flexibiliy and Sraegy in Resource Allocaion. (Cambridge, Massachuses, USA: The MIT Press)

28 Appendix I Some researchers poined ou ha non-parameric esimaion of price volailiy migh cause spurious non-lineariy. (See Fan and Zhang, 2001). This appendix ess he hypohesis ha he S shape of he volailiy curves of real prices of Canadian Lumber and US Lumber can be explained only by spurious non-lineariy caused by non-parameric esimaion a 90% confidence. Firs, we esimae he price process using he mean revering model. We hen use he Mone Carlo mehod o generae 500 simulaions of he esimaed mean-revering model wih same size, and use non-parameric mehod o esimae he 0.05, 0.5 and 0.95 quaniles of he simulaions. If he esimaion of real daa by non-parameric mehod is wihin he esimaions of he 0.05 and 0.95 quaniles 24 of he simulaions, hen a 90% confidence, he non-lineariy of he esimaion is spurious. However, if all or par of he esimaion of real daa is no wihin he esimaions of he 0.05 and 0.95 quaniles of he simulaions, hen a 90% confidence, he S shape of he esimaion canno be explained only by spurious non-lineariy caused by non-parameric mehod. The esimaions of he real prices of Canadian sofwood lumber and US sofwood lumber using a non-parameric mehod and he mean-revering model, as well as he quaniles of simulaion esimaions are shown in Figure A-1 and Figure A-2, respecively. We can see ha in some areas he esimaed curve using a non-parameric mehod breaks he boundaries he 5% and 95% quaniles of he simulaion esimaions. I seems o indicae 24 The 0.05 and 0.95 quaniles of esimaion by non-parameric mehod is similar o he 90% confidence inervals of esimaions by parameric mehod. 28

29 ha he S shape of he volailiy curve canno be explained only by spurious nonlineariy. Appendix II This appendix demonsraes ha boh he sochasic volailiy model and Schwarz and Smih s wo-facor model indicae he posiive relaionship beween price volailiy and price. Thus, hey canno explain he S shape of he volailiy curve. The simple sochasic volailiy model: dp = B P d + σ P dz (I.1) σ in he diffusion erm follows Brownian moion σ = κ d + υ (I.2) dw where B, κ and υ are consan and W is a sandard one- dimensional Brownian moion. The second order condiional momen of price incremen in equaion (I.1) can be wrien as M ( P = P0 ) = E[( dp ( P P = P0 ) d) ] = E[( σ P P = P0 ) ] = υ P0 µ (I.3) Equaion (I.3) indicaes posiive relaionship beween price and he second order condiional momen, which implies posiive relaionship beween price and price volailiy. Schwarz and Smih (2000) developed a wo-facor model of commodiy prices ha allows mean-reversion in shor-erm prices and uncerainy in he long-erm 29

30 equilibrium level o which prices rever. Le denoe he spo price of a commodiy a ime. Schwarz and Smih decompose he spo price ino wo sochasic facors as S ln( S ) = χ + ξ (I.4) where χ is referred o as he shor-erm deviaion in prices and ξ he long-erm equilibrium price level. In heir parameric model, he shor-erm deviaions, defined as he difference beween spo and equilibrium prices, are expeced o rever oward zero following an Ornsein- Uhlenbeck process: dχ = κ χ d + dz (I.5) σ χ χ The long-erm equilibrium price is assumed o evolve according o geomeric Brownian moion: ξ = µ ξ d + σ ξ dzξ (I.6) d In equaion (I.5) and (I.6), dz and dz are correlaed incremens of sandard Brownian χ ξ moion processes wih dz χ dzξ = ρ χξ d, and κ, σ χ, µ ξ, σ ξ are consans. The expecaion of logarihm of price incremen in equaion (I.4) can be wrien as: E( d ln( S )) = E( dχ + dξ ) = E( κ χ d + σ dz = κ χ d + µ d ξ χ χ + µ d + σ dz ) ξ ξ ξ (I.7) The variance of logarihm of price incremen in equaion (I.4) can be obained by: Var( d ln( S = E[(( κ χ d + σ dz = E[( σ dz 2 χ χ 2 ξ )) = E[( d ln( S χ = σ + σ + 2 ρ σ σ ξ χξ χ + σ dz ) ξ χ 2 )) E( d ln( S ))) χ ] ξ + µ d + σ dz ξ ξ 2 ] ξ 2 ) ( κ χ d + µ d)) ] ξ (I.8) 30

31 Equaion (I.8) indicaes ha he variance of logarihm of price incremen is a consan. Since d ln( S ) = ds / S, we have ds = S d ln( S ). Thus, he second order condiional momen of price incremen can be expressed by: M 2 ( S = S 2 0 = S 0 ) = S 2 2 ( σ + σ χ ξ 2 0 Var( d ln( S + 2 ρ χξ χ ) S σ σ ) ξ = S 0 ) (I.9) Equaion (I.9) indicaes ha, according o Schwarz and Smih s model, over a long period, variance of price incremen will be expeced o increase as he price rises, which also implies a posiive relaionship beween price volailiy and price. 31

32 Figure 2-1: Individual Producer s Supply Curve Price Pe P* Pc Ps 0 Q* Producion Figure 3-1: Effecs of Small Increase in Ne Demand Price Pe P 2 C P ( dn 2 ) B P1 P* P ** P ( dn 1 ) Pc dn 1 0 Change of Invenory 32

33 Figure 3-2: Effecs of Large Increase in Ne Demand Price B 2 ( dn) P LP P 2 3 ( dn) P LP A P 3 D C Pe P(dN) P** P* Pc dn 1 0 Change of Invenory 33

34 Figure 3-3: Effecs of Operaing Opions on Price Volailiy Price Volailiy D D1 D 2 C E A2 B A 1 A 0 Pc P* Pe Price Figure 3-4: Effecs of Small Decrease in Ne Demand Price Pe P ( dn 1 ) P* P 1 P 3 ( dn) B P** C P3 Pc 0 N1 Change of Invenory 34

35 Figure 3-5: Effecs of Large Decrease in Ne Demand Price P(dN) Pe P* 3 ( dn) P LN Pc P*** L M P LN 2 ( dn) P 4 J P 2 K 0 Change of Invenory 35

36 Figure 3-6: Monhly Time Series Daa - Real Price of Canadian Sofwood Lumber 25 Figure 3-7: Monhly Time Series Daa - Real Price of US Lumber Source: Saisics Canada Monhly nominal Price Index for Canadian sofwood lumber: Cansim II Series V (Jan Dec. 2003) Monhly Canadian Consumer Price Index: Cansim II Series V Source: US Bureau of Labor Saisics Monhly Nominal Price Index of US Lumber: Producer Price Index WPU081 (Jan 1947-Dec.2003) Monhly US Consumer Price Index: All Iems ( = 100) 36

37 Figure 3-8: Comparison of Esimaions - Real Price of Canadian Sofwood Lumber 27 (A) Esimaions of Drif (B) Esimaions of Diffusion 27 Mean Revering Model: dp = η ( P P ) d + σ P dz In Non-parameric esimaion: bandwidh h = 6* n 2/9 * sd( P), where n is he number of observaions. 37

38 Figure 3-9: Comparison of Esimaions Real Price of US Lumber 28 (A) Esimaions of Drif (B) Esimaions of Diffusion 28 Mean Revering Model: dp = η ( P P ) d + σ P dz In Non-parameric esimaion: bandwidh h = 6* n 2/9 * sd( P), where n is he number of observaions. 38

39 Figure 4-1: Monhly Time Series Daa - Real Price of US Coon 29 Figure 4-2: Par of Time Series Daa - Real Price of Canadian Sofwood Lumber Source: US Bureau of Labor Saisics Nominal price index of US Coon: Producer Price Index WPU (Jan 1947-Dec.2003) Monhly US Consumer Price Index: All Iems ( = 100) 30 Source: Saisics Canada Monhly nominal Price Index for Canadian sofwood lumber: Cansim II Series V (Jan Dec. 2003) Monhly Canadian Consumer Price Index: Cansim II Series V

40 Figure 5-1: Enry and Exi Threshold Prices as Funcions of Price Volailiy 31 Price Enry Threshold Price Curve Exi Threshold Price Curve O Price Volailiy Figure 5-2: Impac of Misspecificaion on Opion Evaluaion dp D E C A B O * PL PL * PH PH Price 31 Copy from Dixi & Pindyck (1994) p

41 Figure 5-3: Esimaion of Drif Real Price of US Coon 32 (A) Esimaion of Drif Wihou De-Trending (B) Esimaion of Drif Wih De-Trending 32 Model I: dp = ( A + B P ) d + σ P dz Mean Revering Model: dp = η ( P P ) d + σ P dz In Non-parameric esimaion: bandwidh h = 6* n 2/9 * sd( P), where n is he number of observaions. 41

42 Figure 5-4: Esimaion of Diffusion Real Price of US Coon Model I: dp = ( A + B P ) d + σ P dz Mean Revering Model: dp = η ( P P ) d + σ P dz In Non-parameric esimaion: bandwidh h = 6* n 2/9 * sd( P), where n is he number of observaions. 42

43 Figure A-1 Simulaion Esimaions on Real Price of Canadian Sofwood Lumber Figure A-1 Simulaion Esimaions on Real Price of US Sofwood Lumber 43