Cointegrated Commodity Pricing Model

Transcript

1 Coinegraed Commodiy Pricing Model Kasushi Nakajima and Kazuhiko Ohashi Firs draf: December 20, 2008 This draf: April 9, 2009 Absrac In his paper, we propose a commodiy pricing model ha exends Gibson-Schwarz wo-facor model o incorporae he effec of linear relaions among commodiy prices, which include co-inegraion under cerain condiions. We derive fuures and call opion pricing formulae, and show ha unlike Duan and Pliska (2004), he linear relaions among commodiy prices, or he error correcion erm, should affec he commodiy derivaive prices. Using crude oil and heaing oil marke daa, we esimae he proposed model. The resul suggess ha here is co-inegraion among hese commodiy prices, and ha is effec on derivaive prices should no be ignored empirically. Keywords: co-inegraion, commodiy prices, convenience yield, energy. Graduae School of Inernaional Corporae Sraegy, Hiosubashi Universiy, Naional Cener of Sciences, 2--2 Hiosubashi, Chiyoda-ku, Tokyo, , Japan. E- mail:knakajima@s.ics.hi-u.ac.jp Graduae School of Inernaional Corporae Sraegy, Hiosubashi Universiy, Naional Cener of Sciences, 2--2 Hiosubashi, Chiyoda-ku, Tokyo, , Japan. E- mail:kohashi@ics.hi-u.ac.jp

2 2 Inroducion Economy is full of equilibrium relaions and co-movemens. They are, for example, purchasing power pariy, covered or uncovered ineres rae pariy, spo-forward relaions, money demand equaion, consumpion spending model, relaions among commodiy prices and so on. Alhough hese relaions are widely known, i does no seem ha hey are sufficienly uilized in finance, especially in he area of derivaive valuaions. These relaions are modeled as co-inegraion which was firs implicily used by Davidson, Hendry, Srba, and Yeo (978) and esablished by Engle and Granger (987). Co-inegraion refers o propery ha holds among wo or more non-saionary ime series variables. Tha is, if cerain linear combinaions of several non-saionary variables become saionary, hese variables are said o be co-inegraed. I is inerpreed as long erm relaionship or equilibrium among variables. This is because co-inegraed variables are ied o each oher o keep cerain linear combinaions saionary, and hence end o move ogeher. Thus, i is naural o consider wheher and how such comovemens among co-inegraed variables affec prices of heir derivaives. While academic papers ha analyze co-inegraion relaionship among economic variables are abound, researches on derivaive pricing wih coinegraion are limied. In our bes knowledge, Duan and Pliska (2004) was he firs o incorporae co-inegraion effec in derivaive pricing. They focused on socks and priced heir opions under he assumpion ermed he local risk-neural valuaion relaionship, which by definiion implies ha he drif erms of sock reurns are equal o he risk-free rae under he riskneural probabiliy. In his seing, hey concluded ha co-inegraion affecs opion prices only when volailiies are sochasic. Commodiy prices, however, behave differenly from sock prices. They are srongly affeced by producion and invenory condiions, and end o deviae emporarily from he prices ha would hold wihou hose effecs. These characerisics are recognized from he heory of sorage by Kaldor (939) and Working (948). To incorporae such emporary deviaions, convenience yield is inroduced, which is one of he crucial ingrediens in commodiy pricing models. For example, Gibson and Schwarz (990, 993) proposed a wo-facor model wih spo commodiy prices and mean revering convenience yields, and priced commodiy fuures and opions. Schwarz (997) invesigaed hree differen (one-, wo-, and hree-facor) models including he Gibson-Schwarz wo-facor model using he daa of crude oil,

3 3 gold, and copper prices, and analyzed heir long erm hedging sraegies. Smih and Schwarz (2000) have modeled commodiy dynamics in a differen seing using long and shor erm facors and finds ha he model is equivalen wih he Gibson-Schwarz model. There are many oher models ha generalize he above including Milersen and Schwaz (998), Nielsen and Schwarz (2004), Casassus and Collin-Dufresne (2005). When convenience yield exiss, he drif of commodiy prices may deviae from he risk-free rae even under he risk-neural probabiliy Thus, in he sandard commodiy pricing models, Duan and Pliska (2004) s risk-neural valuaion framework does no hold, and heir resuls canno be direcly applied o commodiy derivaive pricing. This is he reason why we need o exend Duan and Pliska (2004) s framework and invesigae commodiy pricing wih co-inegraion. For his purpose, we generalize he Gibson-Schwarz wo-facor model by explicily incorporaing linear relaions among commodiy prices, which include co-inegraion under cerain condiions. More specifically, we formulae a commodiy price model in which he emporary deviaion of drif erms from he risk-free rae under he risk-neural probabiliy is described by convenience yield and linear relaions among commodiy prices, or error correcion erms under appropriae condiions. Since in he preceding papers, such emporary deviaion has been modeled as a whole by convenience yield, his paper can also be regarded as a proposal of a new model for specifying a par of he emporary deviaion in erms of linear relaions among prices ha include co-inegraion. I is imporan o noice ha few preceding papers on commodiies have incorporaed co-inegraion in heir derivaive pricing models. One excepion is he research by Dempser, e al. (2008) who analyzed spread opions on wo commodiy prices where heir spread was assumed o be saionary. However, hey did no explicily model he spo prices, and direcly modeled he spread insead. This approach made heir model simple, bu did no enable us o price fuures and opions on each commodiies, no heir spread, whose prices were co-inegraed. Therefore, his paper is he firs o invesigae he effec of linear relaions among prices ha include co-inegraion on derivaive pricing of commodiies for which Duan and Pliska (2004) s risk-neural valuaion does no hold, and o evaluae derivaives on commodiies prices, no heir spreads. More precisely, we formulae he Gibson-Schwarz wo-facor model wih linear relaions among commodiy prices, or co-inegraion under cerain condiions,

4 4 and obain he analyic formula for commodiy fuures and opions. We also invesigae empirically he effec of such linear relaions on derivaives prices using he daa of crude oil and heaing oil boh raded in NYMEX. The res of his paper is organized as follows: In secion 2, we model commodiy spo prices and convenience yields wih co-inegraion using error correcion erm in he drif of spo prices. We also invesigae he relaion beween our model and Gibson-Schwarz model, and derive he closed-form pricing formulae of fuures and call opions. In secion 3, we show he sae equaion and observaion equaion for he Kalman filer, and conduc empirical analysis using crude oil and heaing oil. Secion 4 gives discussions, and secion 5 concludes. 2 The Model 2. Gibson-Schwarz wo-facor wih Co-inegraion (GSC) Model We propose a model ha exends Gibson-Schwarz wo-facor model, hereafer he GS model, (Gibson and Schwarz (990), Schwarz (997)) o explicily incorporae linear relaions among commodiy prices, or co-inegraion under cerain condiions. We adop he coninuous-ime specificaion of coinegraed sysems shown by Duan and Pliska (2004). As usual in commodiy price models, we sar wih describing he behavior of spo prices and convenience yields under he risk-neural probabiliy. Assume ha here are n commodiies whose spo prices and convenience yields follows ds i () = S i ()(r δ i () b i z())d S i ()σ Si dw Si (), i =,..., n() dδ i () = κ i (ˆα i δ i ())d σ δi dw δi (), i =,..., n. (2) under he risk-neural probabiliy. Here, r is risk-free ineres rae which is assumed o be consan. b i, σ Si, κ i, ˆα i, and σ δi are consan coefficiens. W () = [W S ()...W Sn ()W δ ()...W δn ()] is he 2n dimensional Brownian moion under he risk-neural probabiliy wih dw Si ()dw Sj () = ρ Si S j d, dw Si ()dw δj () = ρ Si δ j d, dw δi ()dw δj () = ρ δi δ j d i, j =,..., n.

5 5 We assume ha he commodiy prices are relaed linearly hrough z() = µ z a 0 a i ln S i () (3) i= where µ z, a 0, and a i s are consan. Assume ha ln S i are co-inegraed, hen by rearranging he equaion as ln S () = ( µ z a 0 n i=2 a i ln S i () z())/a, if a 0, z() can be inerpreed as an error correcion erm, a i as co-inegraion vecors, and b i as adjusmen speed of he error correcion erm. Using Io s lemma, he dynamics of z() is dz() = b(m z())d b = b i a i i= a i δ i ()d i= m = a 0 n i= a ir 2 n i= a iσ 2 S i b a i σ Si dw Si () (4) i= (5) The se of equaions (), (2), and (3) corresponds o he GS model wih one excepion ha a linear relaion z() among prices affecs he drif erms. z() represens he error correcion erm of co-inegraion among commodiy prices when hey are co-inegraed. Thus, we call his model he Gibson- Schwarz 2-facor wih co-inegraion model, hereafer he GSC model. I is worh menioning ha while he GSC model bases is specificaion of co-inegraion on ha of Duan and Pliska (2004), heir price dynamics, especially he drif erms, under he risk-neural probabiliies are differen. This difference comes from he characerisics of underlying asses. For socks, which Duan and Pliska (2004) focused on, i is naural o assume ha he drif erms of reurns should be equal o he risk-free rae under he riskneural probabiliy. On he oher hand, for commodiies, i is sandard o assume ha he drif erms may deviae emporarily from he risk-free rae Alhough we rea he case where here is only one linear relaion among prices i.e., he case wih one dimensional z(), we can exend he model o have several linear relaions, or muli-dimensional z().

6 6 even under he risk-neural probabiliy by reflecing invenory and producion condiions. The GSC model assumes ha such deviaion is described by convenience yield and he erm z(). Consequenly, in he GSC model, Duan and Pliska (2004) s risk-neuralizaion mehod does no hold, and linear relaions of prices, or co-inegraion under cerain condiions, affec derivaive prices even hough volailiies σ Si are consan. 2.2 Relaion beween GSC and GS models Since he GSC model specifies a par of emporary deviaion of drifs from he risk-free rae by he linear relaion z() and he res by convenience yield, while he GS model (and oher usual commodiy price models) specifies he whole emporary deviaion by convenience yield, i is ineresing o invesigae he relaion beween he GSC and GS models. To see his, le us fix i =. To disinguish he GSC and GS models, we assume he GS model described by ds () = S ()(r δ ())d S ()σ S dw S () (6) dδ () = κ (α δ ())d σ δ dw δ (). (7) where δ is he convenience yield in he GS model. Now, if b = 0 in he GSC model, hen δ = δ, and he GSC model clearly coincides wih he GS model. On he oher hand, if b 0, hen for boh models o describe exacly he same price process, we mus have δ () = δ i () b z(). Is dynamics is given by dδ () = dδ () b dz() = κ ˆα b bm b bz() (b a κ )δ () b a i δ i ()d (σ δ b a σ S )dw S () b a i σ Si dw Si () By he uniqueness of drif and volailiy erms, we have κ α κ δ () = κ ˆα b bm b bz() (b a κ )δ () b σ δ = σ δ b a σ S 0 = b a i σ Si i = 2,..., n i=2 i=2 a i δ i () i=2

7 7 Focusing on he drif erm and he corresponding coefficiens of z(), δ i () and he consan erm, we have κ α = κ ˆα b bm κ = b a κ κ b = b b b a i = 0. The forh equaion leads o a i = 0(i = 2,..., n) and he second and hird equaions imply κ = b = b a κ κ = 0 α = b m. Hence, he long erm mean of convenience yield δ shoulod be b m and he adjusmen speed of convenience yield is b a. Thus, we obain ha if b 0, for he GSC and GS models describe exacly he same price process, i should hold ha a i = 0(i = 2,..., n) and dδ () = σ δ dw δ () z() = µ z a 0 a ln S () dz() = b(m z())d a δ ()d a σ S dw S () b = b a m = a 0 a r a 2 σs 2. b We can now summarize he relaion beween he GSC and GS models as follows:. If b = 0, hen he GSC and GS models coincide. 2. If b 0 and a i = 0(i = 2,..., n), hen for he GSC and GS models o describe exacly he same price process, he following should hold. dδ () = κ (α δ ())d σ δ dw δ () κ = b a α = b m σ δ = σ δ b a σ S

8 8 and he GSC convenience yield δ is dδ () = σ δ dw δ () wih error erm z() as z() = µ z a 0 a ln S () 3. If b 0 and a leas one of a i (i = 2,..., n) is no 0, hen he GSC and GS models canno describe he same price process. The GSC model includes he GS model as a special case, and in he hird case above, uilizing he GS model migh cause misspecificaion if he GSC model is correc. 2.3 Fuures and Opion Prices for GSC model We can derive he fuures and European call opion prices on commodiy i in he closed forms. Noe ha under he assumpions above, he spo price

9 9 of commodiy i is calculaed as S i (T ) = S i () expx i (, T ) (8) ( ) X i (, T ) = r b i m σ2 i 2 ˆα b i a j ˆα j i (T ) b b i(m z()) ( e b(t ) ) b b i a j ˆα j b 2 (e b(t ) ) (ˆα i δ i ()) ( e κ i(t ) ) κ i b i a j δ j () b(b κ j ) (eb(t ) e κ j(t ) ) b i a j ˆα j b(b κ i ) (eb(t ) e κ j(t ) ) σ Si (W Si (T ) W Si ()) κ i σ δi (W δi (T ) W δi ()) b i a j b σ S j (W Sj (T ) W Sj ()) b i a T j e b(t s) σ Sj dw Sj (s) b e κ i(t s) σ δi dw δi (s) T T κ i T b i a j (ˆα j δ j ()) bκ j ( e κ i(t ) ) b i a j bκ j σ δj (W Si (T ) W Si ()) b i a j e κ j(t s) σ δj dw δj (s) bκ j b i a j b(b κ j ) (eb(t s) e κ j(t s) )σ δj dw δj (s). Denoe by E[ ] he expecaion under he risk-neural probabiliy. Using risk neuraliy and propery of momen generaing funcion, we obain he fuures price of commodiy i as follows (cf. Cox, e al., 98). Proposiion 2.. Assuming (), (2), and (3), he fuures price of commodiy i wih mauriy is given by G i (, T ) = E [S i (T )] = S i () exp µ Xi (, T ) σ2 X i (, T ) 2

10 0 where µ Xi (, T ) = E [X i (, T )] ( and = r b i m σ2 i 2 ˆα i b i(m z()) ( e b(t ) ) b b i a j ˆα j b 2 (e b(t ) ) b i a j ˆα j b (ˆα i δ i ()) ( e κ i(t ) ) κ i ) (T ) b i a j δ j () b(b κ j ) (eb(t ) e κ j(t ) ) b i a j ˆα j b(b κ i ) (eb(t ) e κ j(t ) ) b i a j (ˆα j δ j ()) bκ j ( e κ j(t ) )

11 σx 2 i (, T ) = E [(X i (, T ) µ Xi (, T )) 2 ] ( = σs 2 i σ2 δ i 2σ S i δ i 2b i a j σ κ 2 δi δ i κ i bκ i κ j j 2b i a j σ Sj δ i bκ i 2b i a j σ Si S j b j,k= j,k= σ2 δ i ( e 2κ i(t ) ) 2κ 3 i j,k= j,k= b 2 i a j a k σ δj δ k b 2 κ i κ k b 2 i a j a k σ Sj S k b 2 ) (T ) b 2 i a j a k σ δj δ k b 2 κ j κ k (κ j κ k ) ( e (κ jκ k )(T ) ) b 2 i a j a k σ δj δ k b 2 (b κ j )(b κ k ) 2b i a j σ Si δ j bκ j j,k= 2b i a j σ δi δ j bκ i κ j (κ i κ j ) ( e (κ iκ j )(T ) ) 2b 2 i a j a k σ Sj δ k b 2 κ k 2b ( e2b(t ) ) κ j b ( e (κ j b)(t ) ) κ k b ( e (κ k b)(t ) ) ( e (κ jκ k )(T ) ) κ j κ k b 2 i a j a k σ Sj S k ( e 2b(T ) ) 2b 3 j,k= 2b 2 i a j a k σ Sk δ j b 2 (b κ j ) 2b ( e2b(t ) ) κ j b ( e (κ j b)(t ) ) j,k= ( ) 2 σ2 δ i κ 3 i σ S i δ i κ 2 i ( bi a j σ δi δ 2 j bκ i κ 2 j ( e κ j(t ) ) b i a j σ δi δ j bκ 2 i κ j b ia j σ Si δ j bκ 2 j k= b i a j σ Sj δ i bκ 2 i b 2 i a j a k σ δj δ k b 2 κ 2 j κ k ( e κ i(t ) ) k= b 2 i a j a k σ Sk δ j b 2 κ 2 j )

12 2 ( bi a j σ δi δ 2 j bκ i (b κ j ) b ia j σ Si δ j b(b κ j ) b 2 i a j a k σ Sk δ j b 2 (b κ j ) k= ( b i a j σ Sj δ j κ i b 2 k= b 2 i a j a k σ δj δ k b 2 κ k (b κ j ) ) b ( eb(t ) ) κ j ( e κ j(t ) ) j,k= b 2 i a j a k σ Sj δ k κ k b 3 ( e b(t ) ) 2b i a j σ δi δ j bκ i (b κ j ) 2b i a j σ Sj δ i bκ i (κ i b) ( e (κ i b)(t ) ) j,k= j,k= 2b 2 i a j a k σ δj δ k b 2 κ k (b κ j ) κ i b ( e (κ i b)(t ) ) b i a j σ Si S j b 2 j,k= b 2 i a j a k σ Sj S k b 3 2 ) ( e (κ iκ j )(T ) κ i κ j κ k b ( e (κ k b)(t ) ) ( e (κ jκ k )(T ) ) κ j κ k 2b 2 i a j a k σ Sj δ k b 2 κ k (κ k b) ( e (κ k b)(t ) ) Applying he usual mehod by Harrison and Kreps (979) or Harrison and Pliska (98), we now obain he European call opion price as follows: Proposiion 2.2. Assuming (), (2), and (3), he European call opion price of commodiy i wih mauriy is given by C i (, T ) = e r(t ) E [(S i (T ) K) ] = S i ()e r(t )µ X i (,T ) σ 2 X (,T ) i 2 Φ(d i ) Ke r(t ) Φ(d i2 ) ( ) ln Si () µ K Xi (, T ) σx 2 i (, T ) d i = σ Xi (, T ) d i2 = d i σ Xi (, T ). Noice ha in boh fuures and call opion pricing formulae, a i and b i, he coefficiens for he error erm z() appear hrough µ Xi and σ Xi. This implies he fuures and opion prices of commodiies depend on co-inegraion. Recall also ha he volailiies σ Si of underlying commodiy prices are assumed o

13 3 be consan. Thus, as long as he assumpions made on he GSC model are correc, co-inegraion should affec derivaive prices even hough he volailiies of underlying commodiy prices are consan. The nex quesion is how much co-inegraion affecs derivaive prices. In he following secion, we empirically invesigae his poin. 3 Empirical Analysis 3. The Dynamics of Commodiy Spo Prices, Convenience Yields, and Error Term under Naural Probabiliy Since neiher commodiy spo prices nor convenience yields are observable, we have o esimae he parameers using he Kalman filer 2 wih heir fuures prices. Since we have modeled commodiy spo prices and convenience yields under he risk-neural probabiliy, we need o specify he sdes of commodiy spo prices, convenience yields, and he linear relaion erm z() under he naural probabiliy o esimae he model. For his purpose, we have o formulae he marke price of risk which ransforms he risk-neural probabiliy o he naural probabiliy. Le us assume ha Brownian moions under he risk-neural probabiliy W () and Brownian moions under he naural probabiliy W P () saisfies 2 For he Kalman Filer, see Hamilon (994).

14 4 W () = W P () 0 θ(s, δ(s), z(s))ds W () = [ W S () W Sn () W δ () W δn () ] W P () = [ WS P () WS P n () Wδ P () Wδ P n () ] θ(s, δ(s), z(s)) = ˆβ 0 ˆβ δi δ i () ˆβ z z() ˆβ 0 = ˆβ S 0. ˆβ Sn0 ˆβ δ 0. ˆβ δn 0, ˆβ δi = i= ˆβ S δ i. ˆβ Snδ i ˆβ δ δ i. ˆβ δn δ i, ˆβ z = ˆβ S z. ˆβ Sn z 0. θ is he marke price of risk which is assumed o be linear in δ (),..., δ n (), and z() 3. Since his assumpion includes he case where marke price of risk o be consan, i is a generalized form of he GS model. The consequence of his assumpion can be seen in he following sdes. d ln S i () = (β Si 0 β Si δ i δ i () β Si zz i ())d σ Si dw P S i () (9) β Si 0 = r σ2 S i 2 σ S i ˆβSi 0, β Si δ i = σ Si ˆβδi, β Si z = b i σ Si ˆβSi z dδ i () = (β δi 0 δ i ())d σ δi dw P δ i () (0) β δi 0 = κ i ˆα i σ δi ˆβδi 0, = κ i σ δi ˆβδi, dz() = (β z0 β zδj δ j () β zz z())d β z0 = bm 0, a i σ Si dws P i () () i= a i σ Si ˆβSi 0, β zδi = a i a i σ Si ˆβSi δ i, β zz = b i= a i σ Si ˆβSi z Noe ha convenience yields do no depend on commodiy prices or he erm z(). On he oher hand, he drif erm of z() depends no only on z() bu also on convenience yields. 3 Noe ha his assumpion saisfies he condion for Gisanov heorem. See Lipser and Shiryaev (2000), secion 6.2, example 3 (b). i=

15 5 Each log commodiy price depends on is corresponding convenience yield and he linear relaion erm z(). Mos of he exising models assume ha he drif erms of commodiy spo prices may deviae from he risk-free rae, and ha he convenience yields represen such deviaion. While convenience yields are usually explained by he heory of sorage, here may be oher causes, such as ransacion coss or impacs from oher commodiy prices, ha make he drif erms o deviae emporarily from he risk-free rae. In he GSC model, we use he concep of co-inegraion, and enhance he GS model o incorporae by he erm z() hese oher elemens for deviaion. In oher words, his model can be inerpreed as formulaing he drif erm of a commodiy spo price in erms of hree componens; he risk-free rae, convenience yield, and he linear relaion erm z(). Furhermore, he impac from oher commodiy prices and convenience yields are passed hrough he erm z() o each commodiy prices. In appendices, we derive he soluion for sdes (9), (0), and () and show he sae equaion, observaion equaion, he Kalman filer, forecass, and he maximum likelihood for his model. We also provide a generalized sae equaion which assumes ha he marke price of risk is linear wih ln S (),..., ln S n (), δ (),..., δ n (), and z(). 3.2 Daa We use WTI and heaing oil daily closing pricies raded a NYMEX from January 2, 990 o February 27, Five fuures conracs labeled Mauriy, Mauriy 3, Mauriy 5, Mauriy 7, Mauriy 9, are used in he esimaion. Mauriy sands for he conrac closes o mauriy, Mauriy 3 sands for he hird closes mauriy, and so on. Time o mauriy corresponding o hese prices are also used. We fixed he risk-free rae o be 4%. The basic saisics for hese daa are described in Table. Since he mauriy daes are fixed, he ime o mauriy changes as ime progresses. Comparing WTI crude oil wih heaing oil, we can see ha he sandard deviaion of heaing oil is higher because he average price of heaing oil is higher han ha of crude oil. The mean mauriy and is sandard deviaion are quie close o each oher. Also, noe ha he correlaion beween WTI fuures price and heaing oil are

16 6 Figure : WTI and heaing oil daily closing price from January 2, 990 o February 27, Solid line and dashed line are he price of crude oil and heaing oil, respecively.

17 7 Table : Saisics of Daa. Fuures Conrac Mean price Mean mauriy (Sandard deviaion) (Sandard deviaion) WTI crude oil Mauriy 34.4 (23.89) 0.0 (0.04) Mauriy (24.5) 0.35 (0.04) Mauriy (24.30) 0.59 (0.04) Mauriy (24.40) 0.83 (0.04) Mauriy (24.45).08 (0.04) Heaing oil Mauriy (68.24) 0.09 (0.04) Mauriy (69.33) 0.34 (0.04) Mauriy (70.06) 0.58 (0.04) Mauriy (70.45) 0.83 (0.04) Mauriy (70.40).07 (0.04) 3.3 Esimaion Resuls Now, we esimae he model using he Kalman filer. In able 2, we repor he esimaed parameers wih sandard errors. As we can see, he esimaed co-inegraion vecors are (4.90, -7.65) and he adjusmen speed are 0.8 and 0.30 respecively. These values are significan, which suggess ha here exiss co-inegraion and hence ha he GS model is misspecified. The values of b, b 2 measures o wha degree co-inegraion affecs he spo prices. I is suggesed ha heaing oil price is affeced by co-inegraion much more han crude oil price. Noe ha a 0 is significan, which means ha he erm z() includes ime drif. Boh ˆα in GSC model are esimaed as 0. Which signifies ha here were no long erm means of convenience yields. However, in he GS model ˆα are significanly posiive. Also, noice ha κs in he GS and he GSC model are differen. These are caused by he erm z(), which sugges ha he long erm mean have been replaced by z(). Le us urn o he volailiy parameers. Crude oil and heaing oil spo prices have posiive correlaion. The corresponding spo price and conve-

18 8 Table 2: Parameers esimaes and sandard errors in paranhesis. Daa are WTI and heaing oil daily closing pricies raded a NYMEX from January 2, 990 o February 27, GS crude oil GS heaing oil GSC Volailiy Prameers σ S ( ) ( ) σ S ( ) ( ) δ S ( ) ( ) δ S ( ) ( ) ρ S S ( ) ρ S δ ( ) (0.0345) ρ S δ ( ) ρ S2 δ (0.0054) ρ S2 δ ( ) (0.0298) ρ δ δ ( ) Covenience yield parameers κ.6475 ( ) ( ) κ ( ) ( ) ˆα ( ) ( ) ˆα ( ) ( ) Coinegraion parameers b ( ) b ( ) a ( ) a (0.0307) a ( ) Marke price of risk parameers ˆβ S ( ) ( ) ˆβ S ( ) (0.3620) ˆβ δ ( ) ( ) ˆβ δ (0.3800) ( ) R(, ) ( ) ( ) R(2, 2) ( ) ( ) R(3, 3) ( ) ( ) R(4, 4) ( ) ( ) R(5, 5) ( ) ( ) R(6, 6) ( ) (0.0000) R(7, 7) ( ) ( ) R(8, 8) ( ) ( ) R(9, 9) ( ) ( ) R(0, 0) ( ) ( ) Log-likelihood sample size

19 9 nience yield have relaively high posiive correlaion, which is consisen wih he GS model, however heir correlaions are more lower. Moreover, Crude oil spo price and heaing oil convenience yield have no correlaion. We can also see ha he correlain for heaing oil spo price and crude oil convenience yield are relaively low. I is inuiive ha spo price and convenience yield among differen commodiies should no be srongly correlaed. Volailiy of spo price and convenience yield are differen wih he GS model and he GSC model have larger values. The larger value of volailiy of convenience yield is caused by a i, b i and ρ. Table 3: RMSE (roo mean square error) and ME (mean error) for each fuures. Conracs RMSE ME Models GS GSC GS GSC Crude oil Mauriy Mauriy Mauriy Mauriy Mauriy Heaing oil Mauriy Mauriy Mauriy Mauriy Mauriy Table 3 shows roo mean square error (RMSE) and mean error (ME) of he model. Boh model have small values which means ha he models are well fied and i is hard o decide which have beer performance.

20 20 4 Hedging Fuures In his secion, we implemen he GSC and GS model for hedging long erm fuures 4 conracs, where we call i as arge fuures, using shor erm fuures. Since he GS model have wo sochasic variables, we need wo fuures which has differen mauriies o hedge and he weigh can be calculaed by solving he following sysem of equaions 5. Φw = φ (2) ] Φ = w = [ w [ Gi (,T ) S i G i (,T 2 ) S i G i (,T ) δ i G i (,T 2 ) δ i w 2 ] φ = [ G i(,t ) S i G i (,T ) δ i where w i are weighs for fuures wih mauriy T i and T is he mauriy of arge fuures. For he GSC model which has S i, δ i, and z as he sochasic variables, we need hree fuures o hedge and now he sysem of equaions for (2) are Φ = ] G i (,T ) S i G i (,T 2 ) S i G i (,T 3 ) S i G i (,T ) δ i G i (,T ) z G i (,T 2 ) δ i G i (,T 2 ) z G i (,T 3 ) δ i G i (,T 3 ) z w = [ w w 2 w 3 ] φ = [ G i(,t ) S i G i (,T ) δ i G i (,T ) z ]. To calculae he hedging porfolio, we need he values of sae variables S i (), δ i (), and z(). There are wo mehods o calculae sae variables. One way is o use he Kalman filers, which we call Kalman filer mehod. Anoher way is o calculae sae variables by solving he observaion equaion which only needs fuures price and esimaed parameers. This mehod will be called as simulaneous equaion mehod. We implemened boh mehod. Hedging error raio is calculaed by dividing hedging error value by he arge fuures 4 Recall ha we are assuming risk-free rae as consan which implies ha fuures and forwards should be equally valued. 5 See Brennan and Crew (995) and Schwarz (997) for hedging long erm forwards using shor erm fuures.

21 2 price of each hedging sar period. The hedging error value is he oal of difference beween arge fuures price and he value of hedge porfolio. We hedge fuures which maures in 9 monhs and 0 years wih fuures which maures in, 3, 5 monhs. Since we can no calculae he hedging error precisely wih long erm fuures ha are no raded in he marke such as 0-year fuures, we also hedge 9-monh fuures for precise evaluaion of he hedging error. For 0-years fuures, we calculae he hedging error by using he heoraical price of 0-year fuures. The oal hedging period is from January 2, 990 o February 27, We roll he fuures 3 business days before i maures and each hedging period is roughly monh. The hedging weigh and he hedging error are calculaed daily. The performance of hedging simulaion for 9-monh fuures are indicaed in able 4 and figure 2. While he hedging error raio for crude oil are relaively small, he performance for heaing oil are no good. This is rue for boh GSC and GS model. We also see ha he Kalman filer mehod is raher beer han simulaneous equaion mehod. Figure 3 shows he weighs of fuures in he hedging porfolio which sae variables are calculaed by Kalman filers. I indicaes ha long erm fuures are hold posiively and shor erm fuures are hold negaively or raher moderaely han longer erms. Since he arge fuures are long erm, his resul is consisen. Table 4: Performance of hedging 9-monh fuures. Kalman filer indicaes ha he sae variables are calculaed by using Kalman filers. Simulaneous indicaes ha he sae variables are calculaed by solving he observaion equaion. Conracs Mean of hedging error raio Mehod GS GSC Crude oil Kalman filer Simulaneous Heaing oil Kalman filer Simulaneous In able 5 and figure 4, we show he performance of hedging simulaion for 0-year fuures. Obviously, he hedging error raio is poor han he 9-monh

22 22 Figure 2: Performance of hedging 9-monh fuures. Solid line and dashed line indicaes hedge performance of WTI crude oil and heaing oil, respecively.

23 23 Figure 3: Weighs of fuures for hedging 9-monh fuures. For GS model, solid line and dashed line indicaes hedging weighs of -monh fuures and 3-monh fuures, respecively. For GSC model, solid line, dashed line, and doed line indicaes hedging weighs of -monh fuures, 3-monh fuures, and 5-monh fuures, respecively.

24 24 fuures. However, noe ha his hedging error raio is calculaed by using heoraical price. The hedging weigh which sae variables are calculaed by Kalman filers for his simulaion are shown in figure 5. Again we can see ha he longer erm fuures are posiive and shorer erm fuures are negaive or relaively small. Table 5: Performance of hedging 0-year fuures. Kalman filer indicaes ha he sae variables are calculaed by using Kalman filers. Simulaneous indicaes ha he sae variables are calculaed by solving he observaion equaion. Conracs Mean of hedging error raio Mehod GS GSC Crude oil Kalman filer Simulaneous Heaing oil Kalman filer Simulaneous Discussion 5. Relaions among fuures prices I should be noed ha he linear relaions among commodiy spo prices canno be replaced by hose among heir fuures prices. Namely, here is no sraighforward way o ransform he linear relaions among unobservable spo prices ino he linear relaions among observable fuures prices. Indeed, from (3) and he pricing formula for fuures, we have z() = µ z a 0 i= ( ) a i ln G i (, T ) a i µ Xi (, T ) σ2 X i (, T ) 2 Since here are µ Xi (, T ) and σx 2 i (, T ) which conains a i in a non-linear fashion in he equaion, we can no apply linear esimaion for a i such as (3).

25 25 Figure 4: Performance of hedging 0-year fuures. Solid line and dashed line indicaes hedge performance of WTI crude oil and heaing oil, respecively.

26 26 Figure 5: Weighs of fuures for hedging 0-year fuures. For GS model, solid line and dashed line indicaes hedging weighs of -monh fuures and 3-monh fuures, respecively. For GSC model, solid line, dashed line, and doed line indicaes hedging weighs of -monh fuures, 3-monh fuures, and 5-monh fuures, respecively.

27 27 Things are more complicaed wih b i. Le us see he dynamics of fuures price. dg i (, T ) = G i (r δ i() b i z())s i () G i κ i (ˆα i δ i ()) G i S i δ i ( b(m z()) a j δ j ()) G i z 2 σ2 S i Si 2 () 2 G i 2 G i Si 2 2 σ2 δ i 2 G i a δi 2 j a k σ Sj S 2 k z 2 j,k= σ Si δ i S i () 2 G i a j σ Si S S i δ j S i () 2 G i i S i z 2 G i a j σ δi S j d δ i z σ Si S i () G i S i dw Si () σ δi G i δ i dw δi () = σ Si S i () G i S i dw Si () σ δi G i δ i dw δi () = σ Si S i ()( ˆβ Si 0 ˆβ Si δ i δ i () ˆβ Si zz()) G i S i d σ δi ( ˆβ δi 0 ˆ δ i ()) G i d δ i a j σ Sj ( ˆβ Sj 0 ˆβ Sj δ j δ j () ˆβ Sj zz()) G i z d σ Si S i () G i S i dw P S i () σ δi G i δ i dw P δ i () a j σ Sj G i z dw S j () a j σ Sj G i z dw S j () a j σ Sj G i z dw P S j () a j σ Sj ˆβSj z where we used maringale propery for G i (, T ) = E [S i (T )] in he second equaliy and changed he measure o naural probabiliy in he las equaliy. Comparing his equaion o (), he adjusmen coefficiens is 0 under riskneural probabiliy and σ Si ˆβSi z G i S i n G i. S i This means ha () z he adjusmen coefficiens for fuures have nohing o do wih b i which are

28 28 adjusmen coefficiens for spo prices. Furhermore, we emphasize ha he linear relaion is no observable in he GSC model. There are wo aspec of his unobservabliy. Firs, i is modeled as spo prices which is no observable. If we model he linear relaion using fuures price, he advanage of he model will be he observabiliy of he price which allows us o use easy regression analysis and avoid using echnical Kalman filer. Second, we modeled he linear relaion under riskneural probabiliy which is no observable from hisorical daa ha should move under naural probabiliy. While a i does no change wih probabiliies, b i does as we have seen in he above equaion. The adjusmen coefficiens are changed by he marke price of risks and i implies ha if co-inegraion exis, he effecs of error erm on spo prices under naural probabiliy and under risk-neural probabiliy will be differen. Thus, i may be ineresing o model he linear relaions among observable fuures prices insead of unobservable spo prices under naural probabiliy, and analyze he effecs on spo prices and oher derivaives. 5.2 Condiions for co-inegraion Alhough he GSC model was conceived by he concep of co-inegraion or VECM model, he model need no o be co-inegraed. The only condiions needed is he exisence and uniqueness of srong soluion of sdes () and (2). However, if we assume he GSC model o be co-inegraed, we can use co-inegraion ess ha are widely used. Le us discreize he sdes (), (2), and (4). y() = c Φ 0 y( ) ε() (3) where ln S () ln S ( ). ln S n () ln S n ( ) y() = δ (). δ n () z()

29 29 and oher coefficiens are given in he appendix. If we assume ha any z ha saisfies I n Φ 0 z = 0 hen z >, hen y is saionary 6 which implies co-inegraion in discree ime. However, noe ha his condiion is srong. We do no need ha convenience yields o be saionary, bu if we exclude convenience yields from he condiion, ln S i () may no be I() since here are convenience yields in he drif erms. The difficuly sems from he correlaion of convenience yield wih oher variables. Anoher condion may be uilized which is proposed by Come (999) for coninuous auoregressions bu i is also srong because of he exisence of convenience yield Muli-dimensional z() In his paper we have assumed ha here are only one linear relaion which is represened by he erm z(). This can be relaxed o h(< n) differen linear relaions [z ()... z h ()] ha can be formalized as, h ds i () = S i ()(r δ i () b ij z j ())d S i ()σ Si dw Si (), dδ i () = κ i (ˆα i δ i ())d σ δi dw δi (), i =,..., n z j () = µ z a 0j a ij ln S i () j =,..., h. i= i =,..., n Wih a sraigh forward argumen we can derive he fuures and call opion formulae. We can also exend he assumpion on marke price of risk and formalize he sae and observaion equaion for Kalman filers. The difficuly of his model is ha here are many parameers o consider when implemening he model o hisorical daa. The parameers needed o be esimaed would be n( 2n) parameers for volailiies and correlaions, 2n parameers for convenience yields (ˆα, κ), h(2n ) parameers for linear relaions (a 0j, a ij, b ij ), 2n( n) nh parameers for marke price of risks (ˆβ), and oher parameers which depends on he number of commodiies and fuures mauriy daa used for covariance marix R in he observaion 6 See Hamilon (994) for saionariy of VAR processes. 7 See also Bergsrom (990) for saionary condion of coninuous auoregressions.

30 30 equaion. If we assume 3 commodiies and wo linear relaions for he model using 3 mauriies of fuures for each commodiy, he number of parameers will be 80 parameers! In order o conduc an realisic empirical analysis, he number of commodiies and linear relaions used have o be very small. 6 Conclusion In his paper, we formulaed a commodiy pricing model ha incorporaed he effec of linear relaions among commodiy prices, which included coinegraion under cerain condiions. We derived fuures and call opion pricing formulae, and showed ha unlike Duan and Pliska (2004), he linear relaions among commodiy prices, or he error correcion erm under appropriae condiions, should affec hose derivaive prices in he sandard seup of commodiy pricing. We emphasize ha he proposed model can be inerpreed as a generalizaion of sandard commodiy models, especially he GS model. This is because we decompose he deviaion of commodiy reurn from he risk-free rae under he risk-neural probabiliy ino wo componens; convenience yield and he linear relaion erm z(). The proposed model can hus describe no only he usual sorage effecs capured by convenience yield, bu also oher causes such as impacs from oher commodiy prices and ransacion coss. Comparing he GS wih he GSC model, i should be noed ha if price of a commodiy is affeced by convenience yields of oher commodiies hrough he erm z(), he GS model is misspecificied as long as he GSC model is correc. In he empirical analysis, we assumed ha he marke price of risk is linear in convenience yield and he erm z(), and ulized he Kalman filer echnique. Using crude oil and heaing oil marke daa, we esimaed he proposed model. The resul suggesed ha here are co-inegraion among hese commodiy prices, and ha is effec on derivaive prices should no be ignored empirically. We also implemen he model o hedging long erm fuures. Finally, i should be noed ha while he linear relaions among spo prices play an imporan role, such spo prices are assumed o be unobservable in he sandard commodiy pricing models including ours. Thus, i would be ineresing o model he linear relaions among observable fuures prices insead of unobservable spo prices, and analyze he effecs of he lin-

31 ear relaion, or co-inegraion under cerain condiions, on derivaives. We lay his opic for fuure sudy. 3

32 32 Appendices Appendix Derivaion of equaion (8) In his secion, we derive (8). This is done by calculaing he erm z(t ). Firs, noe ha T z(s)ds = b (z(t ) z()) m(t ) a j b σ S i (W Si (T ) W Si ()) T a j b δ j(s)ds Since z() is a linear sde i can be derived as z(t ) = e b(t ) z() i= T T e b(t s) ( bm e b(t s) a j σ Sj dw Si (s) = e b(t ) z() m( e b(t ) ) T b(t s) e i= a i δ i (s))ds a i δ i (s))ds i= T e b(t s) a j σ Sj dw Si (s) = e b(t ) z() m( e b(t ) ) a i δ i () (e b(t ) e κ i(t ) ) b κ i= i a i ˆα i b (eb(t ) a i ˆα i ) (e b(t ) e κ i(t ) ) b κ i= i= i T a i (e b(t s) e κ i(t s) )σ Si dw Si (s) i= b κ i T e b(t s) a i σ Si dw Si (s) i=

33 33 where we used T e b(t s) δ i (s)ds = T e b(t s) κ i(s ) δ i () e b(t s) ˆα i e b(t s) κ i(s ) ˆα i ds T s e b(t s) κ i(s u) σ δi dw δi (u)ds = e bt κ i δ i() (e (bκi) e (bκi)t ) b κ i ˆα i b (eb(t ) ) e bt κ ˆα i i (e (bκi) e (bκi)t ) b κ i T e bt κ iu b κ i (e (bκ i)u e (bκ i)t )σ δi dw δi (u).

34 34 The las equaion is a consequence of Fubini s heorem for sochasic inegrals 8. Collecing erms, we have T ( ) T X i (, T ) r σ2 S i 2 δ i(s) b i z(s) ds σ Si dw Si (s) = ( r b i m σ2 i 2 ˆα i b i(m z()) ( e b(t ) ) b b i a j ˆα j b 2 (e b(t ) ) b i a j ˆα j b (ˆα i δ i ()) ( e κ i(t ) ) κ i ) (T ) b i a j δ j () b(b κ j ) (eb(t ) e κ j(t ) ) b i a j ˆα j b(b κ i ) (eb(t ) e κ j(t ) ) σ Si (W Si (T ) W Si ()) κ i σ δi (W δi (T ) W δi ()) b i a j b σ S j (W Sj (T ) W Sj ()) b i a T j e b(t s) σ Sj dw Sj (s) b e κ i(t s) σ δi dw δi (s) T T κ i T b i a j (ˆα j δ j ()) bκ j ( e κ j(t ) ) b i a j bκ j σ δj (W Si (T ) W Si ()) b i a j e κ j(t s) σ δj dw δj (s) bκ j b i a j b(b κ j ) (eb(t s) e κ j(t s) )σ δj dw δj (s). Using properies of sochasic inegrals, we have µ Xi (, T ) and σ Xi (, T ) as in he paper. 8 See Ikeda and Waanabe (989), chaper III, lemma 4. or Heah, Jarrow, and Moron (992), appendix lemma 0., and is corollaries.

35 35 Appendix 2 Proof of Proposiion 2.2 We prove he call opion pricing formula. From Harrison and Kreps (979) or Harrison and Pliska (98), we have C i (, T ) = e r(t ) E [(S i (T ) K) ] = e r(t ) (S i ()e x i K)n(x i µ Xi (, T ), σx 2 i (, T ))dx i D where n(x µ, σ 2 ) is densiy funcion of normal disribuion wih mean µ and variance σ 2 and ( ) K D = x i x i ln S i () The inegral can be calculaed as D where e x n(x µ Xi, σ 2 X i )dx = = = D e x (x µ Xi ) 2 i 2σ X 2 i dx 2πσXi 2πσXi 2πσXi = e µ X i σ = e µ X i σ d i = ln 2 X i 2 2 X i 2 e µ σ X 2 X i i 2 2σ X (x µ i Xi ) σ 4 X i 2σ 2 (x µ X ) 2 i 2σ X i X 2 i D e µ σ X 2 X i i 2 (x µ X σ 2 i X ) 2 i 2σ X 2 i D 2πσXi 2π di D ( ) Si () µ K Xi σx 2 i dx e (x µ Xi σ X 2 ) 2 i 2σ X 2 i dx e y2 2 dy and µ Xi = µ Xi (, T ), σ Xi = σ Xi (, T ) for noaional convenience. Also, D σ Xi e (x i µ Xi ) 2 di2 2σ X 2 i dx i = 2πσXi 2π e y2 2 dy dx

36 36 where Collecing all erms, d i2 = ln ( ) Si () µ K Xi σ Xi C i (, T ) = S i ()e r(t )µ X i (,T ) σ 2 X (,T ) i 2 Φ(d i ) Ke r(t ) Φ(d i2 ) Appendix 3 Soluions for equaions (9), (0), and () ln S i () = ln S i (0) β Si 0 β S i δ i δ i (0) (e β δ i δ i ) β Si δ i β δi 0 β 2 δ i δ i (e β δ i δ i ) β S i δ i β δi 0 β S i zz(0) β zz (e β zz ) β S i zβ z0 (e βzz ) β S i zβ z0 βzz 2 β [ zz βsi zβ zδj β δj δ j δ j (0) β Si zβ zδj β δj 0 (e β δ j δ j ) (e βzz ) β δj δ j (β δj δ j β zz ) β δj δ j β zz β S i zβ zδj β δj 0 β ] S i zβ zδj β δj 0 (e βzz ) β δj δ j β zz β δj δ j βzz 2 β Si δ i σ δi (e β δ i δ ( s) i )dwδ P 0 β i (s) δi δ i ) β Si zβ zδj σ δj ( (e β δ j δ ( s) j ) (e βzz( s) ) dwδ P β δj δ j β zz β δj δ j β j (s) zz 0 0 σ Si W P S i () β Si za j σ Sj β zz (e β zz( s) )dw P S j (s)

37 37 δ i () = e β δ i δ i δ i (0) β δ i 0 (e β δ i δ i ) 0 e β δ i δ i ( s) σ δi dw P δ i (s) z() = e βzz z(0) β z0 (e βzz ) β zz β zδi β δi δ i δ i (0) β zδi β δi 0 (e β δ i δ i e βzz ) β i= δi δ i ( β zz ) β zδ i β δi 0 ( e βzz ) β zz β zδi σ δi (e β δ i δ ( s) i e βzz( s) )dwδ P i= 0 β i (s) zz e βzz( s) a i σ Si dws P i (s) i= 0

38 38 Appendix 4 Sae equaion, observaion equaion, Kalman filers, forecass, and maximum likelihood x = F x C x v where x = [ ln S () ln S n () δ () δ n () z() ] F S S ( ) F S S n ( ) F S δ ( ) F S δ n ( ) F S z( ) F Sn S ( ) F Sn S n ( ) F Sn δ ( ) F Sn δ n ( ) F Sn z( ) F = F δ S ( ) F δ S n ( ) F δ δ ( ) F δ δ n ( ) F δ z( ) F δn S ( ) F δn S n ( ) F δn δ ( ) F δn δ n ( ) F δn z( ) F zs ( ) F zsn ( ) F zδ ( ) F zδn ( ) F zz ( ) C x S ( ). C x C x S n ( ) = Cδ x ( ). Cδ x n ( ) Cz x ( ) Q = Cov(v ) σ S S ( ) σ S S n ( ) σ S δ ( ) σ S δ n ( ) σ S z( ) σ SnS ( ) σ SnSn ( ) σ Snδ ( ) σ Snδn ( ) σ Snz( ) = σ δ S ( ) σ δ S n ( ) σ δ δ ( ) σ δ δ n ( ) σ δ z( ) σ δn S ( ) σ δn S n ( ) σ δn δ ( ) σ δn δ n ( ) σ δn z( ) σ zs ( ) σ zsn ( ) σ zδ ( ) σ zδn ( ) σ zz ( )

39 39 F Si S j () = F Si δ j () =, i = j 0, oherwise β Si δ i β δi (e β δ i δ i ) β S i zβ zδi δ i β zz β Si zβ zδj β δj δ j β zz β δi (e β δ i δ i ) δ i β zz (e βzz ) β δj (e β δ j δ j ) δ j β zz (e βzz ), i = j, oherwise F Si z() = β S z (e βzz ) β zz F δi S j () = 0 e F δi δ j () =, i = j 0, oherwise F zsi () = 0 F zδi () = β zδi (e β δ i δ i e βzz ) β zz F zz () = e βzz CS x i () = β Si 0 β S i δ i β δi 0 (e β βδ 2 δ i δ i ) β S i δ i β δi 0 i δ i β S i zβ z0 (e βzz ) β S i zβ z0 β zz β zz β Si zβ zδj β δj 0 β δj δ j (β δj δ j β zz ) β Si zβ zδj β δj 0 C x δ i () = β δ i 0 (e β δ i δ i ) C x z () = β z0 β zz (e β zz ) β δj δ j β zz β S i zβ zδj β δj 0 β δj δ j β 2 zz β zδ j β δj 0 β δj δ j β zz ( e βzz ) (e β δ j δ j ) (e βzz β δj δ j β zz (e β zz ) β zδj β δj 0 β δj δ j (β δj δ j β zz ) (eβ δ j δ j e β zz )

40 σ Si,S j () = β S i δ i β Sj δ j ρδ i δ j σ δi σ δj β δj δ j β δj δ j ( e β δ j δ j ) k= k,l= β Si δ i β Sj zβ zδk ρ δi δ k σ δi σδ k (β δk δ k β zz ) 40 ( e (β δ iδ i β δj δ ) j ) ( e β δ i δ i ) β δj δ j β δk δ k ( β δk δ k ) ( e(β δ i δ i β δk δ k ) ) ( e β δ i δ i ) ( e β β δk δ k βδ 2 δ k δ k ) k δ k β δk δ k β zz ( β zz ) ( e(β δ i δ i β zz) ) ( e β δ i δ i ) β zz β δi δ i ( e βzz ) βzz 2 β zz β Si δ i β Si za k ρ δi S k σ δi σ Sk ( e (β δ i δ i β zz) ) β δi δ k= i β zz β zz ( e β δ i δ i ) ( e βzz ) β zz β S i δ i ρ δi S j σ δi σ Sj ( e β δ i δ i ) β Si zβ zδk β Sj δ j ρ δk δ j σ δk σ δj (β δk δ k= k β zz )β δj δ j β δk δ k (β δj δ j β δk δ k ) ( e(β δ j δ j β δkδ ) k ) ( e βδ jδ j ) ( e β β δk δ k β δj δ j βδ 2 δ k δ k ) k δ k β δk δ k β zz (β δj δ j β zz ) ( e(β δ j δ j β zz) ) ( e β δ j δ j ) β zz β δj δ j ( e βzz ) βzz 2 β zz β Si zβ zδk β Sj zβ zδl ρ δk δ l σ δk σ δl (β δk δ k β zz )(β δl δ l β zz ) β δk δ k β δl δ l (β δk δ k β δl δ l ) ( e(β δ k δ k β δl δ ) l ) ( e β βδ 2 δ k δ k ) ( e β k δ k β δl δ l β δk δ k βδ 2 δ l δ l ) l δ l β δk δ k β zz (β δk δ k β zz ) ( e(β δ k δ k β zz) ) β δk δ k β δl δ l ( e β βδ 2 δ k δ k ) k δ k β zz

41 ( e βzz ) β δk δ k βzz 2 ( e βzz ) βzzβ 2 δl δ l 2β 3 zz k,l= β δk δ k β zz ( e β β zz βδ 2 δ l δ l ) l δ l β zz β δl δ l ( e 2βzz ) 2 ( e βzz ) βzz 3 β Si zβ zδk β Sj za l ρ δk S l σ δk σ Sl (β δk δ k β zz )β zz β zz β δl δ l (β zz β δl δ l ) ( e(β zzβ δl δ l ) ) β 2 zz ( e β βδ 2 δ k δ k ) ( e βzz ) k δ k β δk δ k β zz ( e 2βzz ) 2 ( e βzz ) 2βzz 2 βzz 2 β zz β Si zβ zδk ρ δk S j σ δk σ Sj β zz β δk δ k (β δk δ k β zz ) ( e(β δ k δ k β zz) ) β δk δ k ( e β β 2 δ k δ k ) k= δ k δ k β Si zβ Sj δ j a k ρ Sk δ j σ Sk σ δj β zz β δj δ k= j ( e β δ j δ j ) β δj δ j k,l= β Si zβ Sj zβ zδl a k ρ Sk δ l σ Sk σ δl β zz (β δl δ l β zz ) 4 ( e β zz β δk δ k βzz 2 β zz ( e (β zzβ δj δ ) j ) ( e βzz ) β zz β δj δ j β zz β δl δ l (β zz β δl δ l ) ( e(βzzβ δl δ l ) ) ( e βzz ) ( e β β δl δ l β zz βδ 2 δ l δ l ) l δ l β δl δ l ( e 2βzz ) 2 ( e βzz ) 2βzz 2 βzz 2 β zz β Si zβ Si za k a l ρ Sk S l σ Sk σ Sl ( e 2βzz ) 2 ( e βzz ) β 2 k,l= zz 2β zz β zz β Si za k ρ Sk S j σ Sk σ Sj ( e βzz ) β zz β zz k= β S j δ j ρ Si δ j σ Si σ δj ( e β δ j δ j ) β δj δ j β δj δ j β Sj zβ zδl ρ Si δ l σ Si σ δl ( e β δ l δ l ) ( e βzz ) β δl δ l= l β zz β δl δ l β zz β Sj za l ρ Si S l σ Si σ Sl ( e βzz ) ρ Si S β zz β j σ Si σ Sj zz l=

42 42 σ Si,z() = k= β Si δ i β zδk ρ δi δ k σ δi σ δk (β δk δ k β zz ) β δk δ k ( e β δ k δ k ) k= k,l= β Si δ i a k ρ δi S k σ δi σ Sk β Si zβ zδl β zδk ρ δl δ k σ δl σ δk (β δl δ l β zz )(β δk δ k β zz ) β δk δ k ( e (β δ i δ i β δk δ k ) ) ( e (β δ i δ i β zz) ) ( e βzz ) β zz β zz ( e (β δ i δ i β zz) ) ( e βzz ) β zz β zz β δl δ l (β δl δ l β δk δ k ) ( e(β δ l δ l β δk δ ) k ) ( e β δ k δ k ) β δl δ l β δk δ k β δl δ l (β δl δ l β zz ) ( e(β δ l δ l β zz) ) β δl δ l β zz ( e βzz ) ( e β δ k δ k ) β zz β δk δ k ( e βzz ) β 2 zz k,l= β Si zβ zδl a k ρ δl S k σ δl σ Sk β δl δ l β zz β δl δ l β zz ( e βzz ) l,k= β zz (β zz β δk δ k ) ( e(βzzβ δk δ k ) ) 2β 2 zz β Si zβ zδk a l ρ Sl δ k σ Sl σ δk β zz (β δk δ k β zz ) 2β 2 zz ( e 2β zz ) β δl δ l (β δl δ l β zz ) ( e(β δ l δ l β zz) ) ( e 2βzz ) β 2 zz ( e βzz ) β zz β δk δ k ( e (βzzβ δk δ k ) ) ( e β δ k δ k ) ( e 2βzz ) ( e βzz ) β δk δ k 2β zz β zz β Si za l a k ρ Sl S k σ Sl σ Sk ( e 2βzz ) ( e βzz ) 2β zz β zz l,k= k= k= β zz β zδk ρ Si δ k σ Si σ δk β δk δ k β zz a k ρ Si S k σ Si σ Sk β zz ( e β zz ) ( e β δ k δ k ) ( e βzz ) β δk δ k β zz

43 43 σ Si,δ j () = β S i δ i ρ δi δ j σ δi σ δj ( e (β δ iδ i β δj δ ) j ) ( e β δ j δ j ) β δj δ j β δj δ j β Si zβ zδk ρ δk δ j β δk δ k β zz β δk δ k (β δk δ k β δj δ j ) ( e(β δ kδ k β δj δ ) j ) k= ( e β δ j δ j ) β δk δ k β δj δ j β zz (β zz β δj δ j ) ( e(β zzβ δj δ ) j ) ( e β δ j δ j ) β zz β δj δ j k= β Si za k ρ Sk δ j σ Sk σ δj β zz ρ S i δ j σ Si σ δj β δj δ j ( e β δ j δ j ) σ δi δ j () = ρ δ i δ j σ δi σ δj ( e (β δ iδ i β δj δ ) j ) β δj δ j β zδl ρ δi δ σ δi,z() = l σ δi σ δl ( e (β δ i δ i β δl δ ) l ) β δl δ l= l β zz β δl δ l ( e (β δ i δ i β zz) ) β zz σ z,z () = l= k,l= a l ρ δi S l σ δi σ Sl β zz ( e (β δ i δ i β zz) ) β zδk β zδl ρ δk δ l σ δk σ δl (β δk δ k β zz )(β δl δ l β zz ) ( e (β zzβ δj δ ) j ) ( e β δ j δ j ) β zz β δj δ j β δj δ j β δk δ k β δl δ l ( e (β δ k δ k β δl δ l ) ) ( e (β δ k δ k β zz) ) ( e (βzzβ δl δ ) l ) β δk δ k β zz β zz β δl δ l ( e 2βzz ) 2β zz k,l= k,l= k,l= β zδk a l ρ δk S l σ δk σ Sl β δk δ k β zz β zδl a k ρ Sk δ l σ Sk σ δl β δl δ l β zz a k a l ρ Sk S l σ Sk σ Sl 2β zz ( e 2β zz ) ( e (β δ k δ k β zz) ) ( e 2βzz ) β δk δ k β zz 2β zz ( e (βzzβ δl δ ) l ) ( e 2βzz ) β zz β δl δ l 2β zz

44 44 The observaion equaion is where y = H x C y w y = [ ln G (, T, ) ln G (, T,k ) ln G 2 (, T 2, ) ln G n (, T n,k ) ] H GT S H GT S n H GT δ H GT δ n H GT z H = H GTk S H GTk S n H GTk δ H GTk δ n H GTk z H G2T2 S H G2T2 S n H G2T2 δ H G2T2 δ n H G2T2 z H GnTnk S H GnTnk S n H GnTnk δ H GnTnk δ n H GnTnk z C (, T, ) σ2 X (,T, ) 2. C y = C (, T,k ) σ2 X (,T,k ) 2 C 2 (, T 2, ) σ2 X (,T 2, ) 2 2. C n (, T n,k ) σ2 X (,T n,k ) 2 R, 0 R = Cov(w ) =... 0 R nk,nk H GiTi,k S j = H GiTi,k δ j =, i = k 0, oherwise e κ i T i,k κ i b ia i (e bt i,k e κit i,k ) b(bκ i b ia i ( e κit i,k ) ) bκ i, i = j b ia j (e bt i,k e κ j T i,k ) b(bκ j ) b ia j ( e κj Ti,k ) bκ j, oherwise H GiTi,k z = b i b (ebt i,k )

45 45 C i (, T ) = ( r b i m σ2 i 2 ˆα i b im b ( eb(t ) ) b i a j ˆα j b ) (T ) b i a j ˆα j b 2 (e b(t ) ) b i a j ˆα j b(b κ j ) (eb(t ) e κ j(t ) ) ˆα i κ i ( e κ i(t ) ) b i a j ˆα j bκ j ( e κ j(t ) ). Here we assumed ha here are k mauriies for each commodiy fuures which can be generalized o differen mauriies for each commodiy and used he noaion T ij for he mauriy of commodiy i fuures conrac j h closes o mauriy. The forecass and filers are x = F x C x P = F P F Q Σ = H P H R K = P H Σ x = x K (y H x C y ) P = (I K H )P and he maximum log likelihood is L(ϑ) = nmt ln(2π) 2 u = y H x C y. T ln Σ = T = u Σ u

46 Appendix 5 The Generalized sae equaion We generalize he model by assuming ha marke price of risk is linear wih ln S (),..., ln S n (), δ (),..., δ n (), and z(). For ease of noaion, we use he following sdes insead of sdes (), (2), (4). ds i () = S i ()(r δ i () b i z())d S i ()σ S i dw (), 46 i =,..., n(4) dδ i () = κ i (ˆα i δ i ())d σ δ i dw (), i =,..., n (5) dz() = b(m z())d a i δ i ()d σ z dw () (6) σ z = a i σ Si. i= where W () are 2n dimensional sandard Brownian moion and σ S i σ Sj i= = ρ Si S j σ Si σ Sj σ S i σ δj = ρ Si δ j σ Si σ δj. The difference beween equaions (), (2), (4) and he above are ha he Brownian moions are correlaed or no, however he disribuion are he same which makes no difference when modeling wih eiher equaions. We jus need o apply Cholesky decomposiion o he covariance marix in order o generae σ Si and σ δj. These sdes (4), (5), (6) are more easier o derive he soluions in marix form. Le us assume ha Brownian moions under equivalen maringale measure W () and Brownian moions under naural probabiliy W P () saisfies W () = W P () θ(s, S(s), δ(s), z(s)) = ˆβ 0 ˆβ 0 = ˆβ S 0. ˆβ Sn 0 ˆβ δ 0. ˆβ δn0 0, ˆβ Si = θ(s, S(s), δ(s), z(s))ds ˆβ S S i. ˆβ Si ln S i () i= ˆβ Sn S i ˆβ δ S i. ˆβ δns i, ˆβ δi = ˆβ S δ i. ˆβ δi δ i () ˆβ z z() i= ˆβ Sn δ i ˆβ δ δ i. ˆβ δnδ i, ˆβ z = ˆβ S z. ˆβ Sn z ˆβ δ z. ˆβ δnz,