The Valuation of Temperature Derivatives: The Case for Taiwan

Transcription

1 he Valuaion of emperaure Derivaives: he Case for aiwan Chuang-Chang Chang, Sharon S. Yang, zu-yu Huang 3, Jr-Wei Huang 4 Absrac his paper exends he valuaion model proposed by Cao and Wei (004) o price emperaure derivaives in aiwan. In addiion, we consider he discree ime series model proposed by Campbell and Diebold (005) for modeling he sine funcion in Cao and Wei(004)'s valuaion model. he Campbell and Diebold s (005) ime series model describes he emperaure characerisics by using a Fourier series. I considers no only he condiional mean of emperaure dynamics bu also akes ino accoun he condiional variance of emperaure dynamics. he equilibrium pricing model proposed by Cao and Wei (004) considers a join process for he aggregae dividend and he emperaure o discuss he significance of he marke price of emperaure risk. Besides, since he Gamma class of disribuions include many imporan disribuions, eiher as special or limiing cases or hrough simple ransformaion. herefore, his paper does no resric he emperaure disurbance o follow he normal disribuion, insead, we consider he disribuion of he emperaure disurbance by Gamma ransformaion. In he numerical analysis, we illusrae wih he unique daase of emperaure in aiwan for modeling he emperaure dynamics. he price of he emperaure derivaes of HDDs (Heaing Degree Days) or CDDs (Cooling Degree Deparmen of Finance, Naional Cenral Universiy, No. 300, Jung-da Rd., Jung-Li, aiwan 30, R.O.C. el: ex Fax: ccchang@cc.ncu.edu.w. Corresponding auhor. Deparmen of Finance, Naional Cenral Universiy, No. 300, Jung-da Rd., Jung-Li, aiwan 30, R.O.C. el: ex Naional Cenral Universiy, syang@ncu.edu.w. 3 Naional Cenral Universiy. funnyhju@homail.com. 4 Naional Cenral Universiy @cc.ncu.edu.w.

2 Days) in aiwan marke is evaluaed. JEL classificaion: C5, G3 Keywords: emperaure Derivaives, Equilibrium Pricing Model, HDD, CDD, Risk Aversion.. Inroducion Weaher condiions direcly or indirecly affec he profis of various indusrial secors. For insance, he oupus of he agriculure, he earnings of he power indusry, and he sales quaniies of reail businesses, ec. he unexpeced weaher someimes causes serious losses in many indusries, in order o hedge hese uncerain facors affeced by weaher condiions, weaher derivaives have been developed and raded in he CME since 999. In recen years, he abnormal climae makes he marke for weaher derivaives growing seadily. Among all he weaher derivaives, he emperaure derivaives accouns for he larges proporion of all ransacions. Alhough weaher derivaives have been geing more and more imporan, here is no ye a sandardized and effecive valuaion model for emperaure derivaives. A number of consideraions make pricing weaher derivaives more difficully han pricing radiional derivaives. Firs, because he underlying indices (weaher) are no radable, we canno use he radiional arbirage-free pricing mehod o valuing weaher derivaives. Second, alhough he liquidiy of weaher derivaive markes has improved, he compleeness in weaher derivaive markes is no as good as in radiional derivaive markes. hus, he Black-Scholes formula is no suiable for pricing weaher derivaives. hird, he underlying indices, such as HDDs (Heaing Degree Days) or CDDs (Cooling Degree Days), have complex probabiliy disribuion, so i is hard o derive analyic soluions which conform o he real world. his paper focuses on he emperaure derivaives, and here we briefly inroduce he emperaure derivaives. here are four basic elemens in a emperaure conrac. he firs elemen is he underlying variable. Mos emperaure derivaives are based on he accumulaion of he daily

3 HDD or CDD for he conrac period and he daily HDD and CDD are calculaed as he following: HDD CDD max{ base emperaure daily average emperaure,0} max{ daily average emperaure base emperaure,0} where he daily average emperaure is defined as he average of he daily maximum and minimum emperaures. he second is he conrac periods for HDD and CDD. he HDD season usually covers November o March and he CDD season usually May o Sepember. hird, he emperaure observaion saion which records he daily emperaure for a paricular ciy should be specific. Finally, he ick size for each HDD or CDD should be provided clearly. In his sudy, we will exend Cao and Wei s (004) equilibrium pricing model wih he join process of he emperaure and he aggregae dividend. In he emperaure process, Campbell and Diebold (005) focused on condiional mean dynamics, wih conribuions coming from rend, seasonal and cyclical componens, and also allow condiional variance dynamics, wih conribuions coming from boh seasonal and cyclical componens, In his paper, we will subsiue he discree ime series model proposed by Campbell and Diebold (005) for he sine funcion model proposed by Cao and Wei (004) and considering more general siuaion, we do no resric he emperaure disurbance o follow normal disribuion. Since many disribuion could be ransformed by Gamma disribuion including normal disribuion. herefore, according o he esimaed resul of emperaure variable, we se he disribuion of he emperaure disurbance by Gamma ransformaion. Furhermore, we analyzed he marke price of risk under hree differen levels of lagged-correlaions. he paper proceeds as follows. In Secion we describe he seings of hose models we use, including he uiliy funcion of he represenaive invesor, he daily emperaure behavior, and he aggregae dividends process. In Secion 3 we provide he valuaion formulas of he exend Cao and Wei (004) model by ransformed Gamma disribuions (Viiello and 3

4 Poon, 00). In Secion 4 we show he Mone Carlo simulaion mehod. Finally, in he las secion we conclude.. he Model he Maringale mehod and he equilibrium asse pricing model are wo main caegories of pricing approaches for weaher derivaives. he Maringale mehod ypically applies he principle of non-arbirage o price he derivaives by he radable underlying asse. he mehod is sill adaped o evaluaing he weaher derivaives by obaining he marke price of emperaure risk while emperaure is non-radable. Alon e al. (00) valuaed he weaher derivaive by using he maringale esimaion funcion proposed by Bibby & Sψrensen (995). In general, using he equilibrium asse pricing model needs o know he uiliy funcion of he represenaive invesor. Following Cao and Wei (004), we applied he equilibrium asse pricing model. In a discree framework, according o he Lucas s (978) pure exchange economy, he fundamenal uncerainies are driven by he aggregae dividend { } and he weaher condiion { W }. Aggregae dividend variable can be regarded as aggregae oupu or dividends of he marke porfolio; he weaher condiion could be emperaure, rainfall, snowfall, or he number of yphoons. In our model, he join dynamic combined wih emperaure, W, and aggregae dividend,, is exogenous process on a given probabiliy space (, F, P). he filraion, F (, W ; {0,,,..., }), assembles he infiniely represenaive invesor s informaion a ime. Under he sandard equilibrium condiions, he oal consumpion is equal o he aggregae dividend, and he coningen claim value a ime wih a payoff q a a fuure ime is given by: X (, ) E[ U'(, ) q] (0, ) () U '(, ) where U '( ) is he marginal uiliy funcion of he represenaive invesor. 4

5 he value a ime of coningen claim could be evaluaed by () as long as he agen s preference and he dividend process deermined by he relaion of he oal consumpion and emperaure facors are specified. hus, we have o specialize hese variables of he above seup. he followings are he descripions of hese variables.. he Model of emperaure Variable Alhough weaher forecasing is crucial o he weaher derivaives valuaion, he mos consideraion of hedging he weaher risk is he unpredicable componen of weaher flucuaions. Acually, large weaher flucuaions would generae lile weaher risk if firms could forecas precisely, so hey do no demand a large hedging insrumens when facing weaher flucuaions. In order o assess he possible hedging quaniy agains weaher risk, and o conceive he fi hedging sraegies, i is necessary o deermine he amoun of weaher noise exiss so ha weaher derivaives can eliminae i, and ha needs a weaher model. here have been numerous sudies ha have proposed he mehod of forecasing emperaure. Dischel (998) used a simple sochasic Brownian moion for emperaure forecas. Dornier and Queruel (000) exended Dischel s (998) model, hey all based on a Brownian Moion process. Considine (000) fied he disribuion of HDD hisorical values and valuaed opions by muliplying he payou of he opion and he produc of he probabiliy disribuion. Alon e al. (00) used a sine funcion o model he average hisorical emperaures. Brody e al. (00) observe ha emperaure dynamics exhibi long-range emporal dependencies and applied a fracional Brownian moion o drive an Ornsein-Uhlenbeck process. Campbell and Diebold (005) described he emperaure dynamic process by a ime series model, hey considered o no only he condiional mean bu also he condiional variance in daily emperaure behavior. hey also used he emperaure dynamic process o consruc he fuure disribuion of underlying indices. In order o price he emperaure derivaives, i is essenial o ake accoun of he 5

6 condiional variance behavior and he disribuion of he underlying indices in he valuaion. hree characers should be capured when we describe he emperaure dynamics. Firs, he seasonaliy is mos significan in boh he mean and he variance. Second, he rends refleced he greenhouse effec is relevan bu is likely minor. Finally, he cycle is he sor of persisen and saionary dynamics apar from seasonaliy and rend. We exend Cao and Wei (004) model bu replace he emperaure model by adoping Campbell and Diebold s (005) ime series model for capuring he cyclical volailiy. he emperaure process, { W }, proposed by Campbell and Diebold (005) is shown as follows: W rend Seasonal W ~(0,), () l -l l L iid where rend 0, P d () d () Seasonal [ cos( p ) sin( p )], c, p s, p p Q R S d () d () c, q cos q s, qsin q r r r s s q r s d ( ),,...,365, represening he daes hroughou year. he Fourier series describe he seasonaliy wih a smooh paern, i also reduce he number of parameers which be esimaed and enhance he numerical sabiliy. I increases he forecasing abiliy o consider he GARCH effec ino he emperaure process o reflec he persisen in variance.. Esimaed Resuls of emperaure Process We demonsrae wih he unique emperaure daa in aiwan. he emperaure daa is obained from he aipei Weaher Saion of he Cenral Weaher Bureau in aiwan. We employ he daily maximum and minimum emperaure daa measured in degrees Celsius beween //970 and 3//0. We remove February 9 from every leap year in our 6

7 sample o keep 365 days per year. We obain 30,66 observaions for modeling he emperaure dynamics. Figure describes he daily average emperaure series for he las hiry years a aipei Weaher Saion. he ime series graph reveals srong seasonaliy bu shows ha he rend is no exremely visible in he daily average emperaure. Figure : he Figure displays a ime Series Plo of Daily Average emperaure, Based on he emperaure model in Equaion(), we esimae he parameers using he ordinary leas squares mehod. We use he Akaike and Schwarz informaion crieria o selec he opimal order of Fourier series and he opimal number of lags. I urns ou ha P, Q, R, S and L 3 describe he emperaure daa he bes. he resuls of esimaion are shown in able. Addiionally, we assume ha he error disribuion is he Generalized Error Disribuion. Figure presens he residuals of ordinary leas squares for equaion (). he Jarque-Bera saisics show ha he hypohesis of normal disribuion is rejeced. 7

8 ( ) able -: Ordinary Leas Squares Esimaed Resuls of Condiional Mean c, s, c, s, ( ) ( ) ( ) ( ) ( ) ( ) (0.0088) ( ) able -: Ordinary Leas Squares Esimaed Resuls of Condiional Variance c, s, c, s, ( ) ( ) ( ) (0.0338) ( ) ( ) ( ),000,600, Series: Sandardized Residuals Sample /04/970 //0 Observaions 537 Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Figure : Hisograms and Descripive Saisics for Residuals of Ordinary Leas Squares for Equaion ()..3 he Aggregae dividend process Cao and Wei (004) exended Lucas (978) equilibrium asse-pricing model under he pure exchange economy where he fundamenal uncerainies are driven by he aggregae dividend and he weaher condiion. Furhermore, heir model considered he mean-reversion in he rae of aggregae dividend change suggesed Marsh and Meron (987). In Cao and Wei (004), hey illusraed: 8

9 he derivaive prices are normally wihin one percen of he risk-neural prices if he conemporaneous correlaion of he emperaure process and he aggregae dividend process was considered only. However, aking ino accoun he lagged correlaion make marke price of risk become imporan. herefore, he residual of heir aggregae dividend model ook ino accoun he lagged correlaions of emperaure residual. he aggregae dividend model is shown as follows: ln ln, (3) [ mm], 0 m iid iid ~ N(0,) ~(0,) i, -,..., - m i where measures he speed of mean reversion, and describes he randomness due o all facors excep for he emperaure, and i are innovaions of he emperaure process as defined in equaion (). In he ligh of his consrucion, he conemporaneous correlaion beween emperaure and aggregae dividend is ; he coefficien of emperaure lagged erms capure he lagged effecs on he aggregae dividend. Based on he ineviabiliy and assumpion, variance of is m j ( m) is bounded. When represens a fuure ime, he condiional j m [ ],which can be explained by hree pars:he firs par, j - j all facors excep for he emperaure conribue emperaure conribues conribue m j j emperaure process.) - ; he second par, he conemporaneous ; he final par, he lagged erms of he emperaure.( If 0, j 0, j, hen he aggregae dividend is independen he 9

10 .4 he uiliy funcion of he represenaive invesor Following he lieraure generaed by Cao and Wei (004), we consider he uiliy funcion wih consan relaive aversion: where (4) - (, ) 0, 0 U e is he rae of ime preference, and is he risk parameer. According o he models we describe above, nex secion we will use hese models o illusrae he framework of HDD/CDD derivaives. 3. he Valuaion of emperaure Derivaives Before conducing he valuaion of HDD/CDD derivaives, an assumpion for he emperaure disurbances should we consider. Figure illusraes he emperaure residual is no normal disribuion, so we should no se he disurbance follows he normal disribuion. Furhermore, he graph of residual is similar wih he ransformaion of he sandard Gumbel disribuion. herefore, we assume ha he ransformed emperaure disurbances follow he sandard Gumbel disribuion. he following funcion is he sandard Gumbel densiy funcion: -z f( z) exp{- z e }, z Adoping equaion (4), we can derive he marginal uiliy funcion is - U'(, ) e, hen a coningen claim, X, can be illusraed by X e E q (5) ( ) (, ) [ ] Nex, we will use equaion (5) wih he model in Secion and he assumpion of disurbances o price he pure discoun bond and oher HDD/CDD derivaives under he Gamma ranslaed disribuion. 0

11 3. Discoun Facor Le q, he coningen claim a ime can be regarded as he value of a pure discoun bond a ime wih mauriy. We denoed i by D, following close-formed formula: and derive he D e E - ( - ) (, ) [ ] ( i) - ( - ) i B(, ) e A(, ) e C(, ), 0 (6) where for m A (, ) exp m i i ( i j) ˆ m j i m i i j m mi i j i j B (, ) ji j i ji im j0 m mi i j ji C (, ) ji j i ji im j0 (7) (8) (9) and for m A (, ) exp i i ( i j) ˆ m j i m i i j B (, ) -i -- i j j i j0 i ji C (, ) j i j0 (0) () () where 0 and ˆ l(0 l m ) are he realized error erms for he emperaure -

12 variable. Proof: see Appendix. 3. Derivaives Now we assume ha a HDD forward conrac wih a ick size of N$ and a srike price K and he accumulaion of heaing degree days beween daes and. hen, by equaions (4) and equaion (5), we can obain he value a ime of he HDD forward conrac is U'(, ) fhdd (,,, K) E [ HDD(, ) K] U'(, ) E [ (, ) ] ( ) e HDD K (3) ake he forward price ime, F (, HDD, ), o equal he srike price K such ha f (, HDD,, ) 0 K, ha is, U'(, ) E [ HDD(, ) K] 0. U'(, ) (4) We can ge E U '(, )( HDDi) E ( HDD ) i i i FHDD (,, ) K E U'(, ) E (5) Nex, we consider a European HDD opion wih srike price X and he accumulaion period from o. For a HDD call, he expired payoff is max HDDi - X,0, and he i call price can be expressed as

13 - ( - ) - CHDD (,,, X ) e E max HDD -,0 i X i (6) Similarly, a HDD pu price can be expressed by - ( - ) PHDD (,,, X ) e E max X - HDD,0 i i (7) he CDD derivaives are analogously expressed by replacing he above noaion HDD wih CDD. I is oo difficul o obain closed-form soluions for he above pricing formula wihou furher resricions of he dividend and emperaure variables. herefore, Mone Carlo simulaions are resored and he numerical resul will be analyzed in Secion 4. Nex, we have o illusrae he main reason for deermining he price change before simulaing and analyzing. 3.3 Marke price of risk We reference he perspecive of Cao and Wei (004) for decomposing he derivaives price o wo pars: he expeced fuure spo value and he marke price of risk, i.e., he risk premium. he expeced fuure spo value is he value of fuure payoff discouned by he discoun facor. On he oher hand, he risk premium implies he unpredicable emperaure risk, ha is, he correlaion beween he aggregae dividend and he emperaure deermines he risk premium in he emperaure derivaives value. Wih HDD derivaives as examples, we decompose he valuaion of derivaives o clarify he relaionship beween he value and he risk premium. 3

14 4 (,, ), = i i HDD i i i i i i E HDD F E Cov HDD E E HDD E Cov E HDD,, ( ) i i i F HDD i HDD E E HDD (8),,max -,0 (,,, ) (, ) max -,0 (, ) max -,0 ( i i HDD i i i C HDD i Cov HDD X C X D E HDD X E D E HDD X ) (9),,max -,0 (,,, ) (, ) max -,0 (, ) max -,0 ( i i HDD i i i P HDD i Cov X HDD P X D E X HDD E D E X HDD ) (0) where (,) Cov.. is he covariance and is he risk premium of each derivaive. hrough he above equaions, we can observe ha he weaher condiions affec he discoun facor and risk premium which are he major elemens of derivaive price. Similarly, he formula for CDD derivaives can be obained by he same mehod. Due o he negaive correlaion beween HDD and he emperaure, he risk premiums in (8) and (9) are negaive, and in

15 (0) is posiive when 0 and 0 i i. he CDD derivaives are reverse. he summary for differen cases is shown in able. able : he Relaion beween Correlaions and Risk Premium,, F, HDD C, HDD P, CDD 0, i 0 i 0, i 0 i Negaive Posiive,, F, CDD C, CDD P, HDD Posiive Negaive Furhermore, we can observe ha once he emperaure and he aggregae dividend are compleely independen, he derivaives price would equal o he risk neural value. In oher words, if he join processes are compleely independen, i.e., 0, i 0, i, hen he value a ime of emperaure derivaives is he fuure payoff discouned by he risk-free rae. 4. Simulaion and Analysis 4. he Forecas Abiliies In his secion, we adop he Mone Carlo simulaion o generae he emperaure variable alone and also generae he join process of he emperaure and he aggregae dividend. hrough simulaion, we compare he forecas abiliies under he Gamma ranslaed disribuion wih he normal disribuion and analyze he price change under differen scenarios of various parameers. able 3 shows he models of he forecas abiliies under differen emperaure disurbance assumpion. he acual daa period is from 0// o 0//3, and he indices for comparing forecas abiliies are he roo mean square error (RMSE), he mean absolue error (MAE), and he mean absolue percenage error (MAPE). 5

16 According o he forecas abiliy, he Gamma ranslaed disribuion is beer han he normal one. I s very reasonable and convincible since he Gumbel disribuion is a special case of exreme value disribuion and i is more capable for capuring he abnormaliy of variaion of emperaure. able 3: he Forecas Abiliies under Differen emperaure Disurbance Assumpion RMSE MAE MAPE Normal Gamma ransformaion y y RMSE,MAE,MAPE. N N N Noe: Where N N N y y y y y Nex, following he seing of he Cao and Wei s (004) model, we would like o se he various parameers for differen scenarios. Firs, we se rae of ime preference a.4%, which is he reurn on he one-year erm deposi of he Bank of aiwan as he risk-free rae. Second, according o previous empirical researches, Shiller (983) esimaed o be and Marsh and Meron (987) esimaed o be 0.945, we se, he mean reversion parameer of he dividend process, a 0.9. hird, in he Cao and Wei model, he aggregae dividend variable is subsiued by alernaive macroeconomic variables such as GDP or aggregae consumpion; moreover, heir esimaed resul showed ha he number of lagged error erm is 30. Because he daa ses of macroeconomic variables in aiwan are low frequen, we do no esimae he equaion (3) bu adop he Cao and Wei s seup for he aggregae dividend model. Furhermore, we consider hree case of he number of lagged error erm in (3) for simulaion: (i) he conemporaneous correlaion only, (ii) he 5 lagged error erms, and (iii) he 30 lagged error erms. Fourh, Cao and Wei have exhibied ha he conemporaneous correlaion,, is resriced in he range from -0.5 o 0.5 o reflec he influence degree of 6

17 he emperaure variaions for aggregae dividend in he real world. We choose -0.5, -0.5, 0.5, and 0.5 for he analysis. For simpliciy, he coefficiens, i i, are calculaed by he simple geomeric decay funcion of, which is he same as he Cao and Wei s (004) mehod. Here we se, i i o have he same sign for keeping unchanged across seasons. Given he conemporaneous correlaion, i is calculaed as i q i wih 0q. he 30 decay muliplier is chosen by 30 q so ha he lagged effec gradually fade away. Under his srucure, he porion of he dividend variance conribued by he emperaure variaions is: - m - j m j j j. Wih 0.5 and 0.5, he proporions are 5.56% and 3.64%, respecively. I seems ha one-sevenh of he GDP is deemed o be weaher sensiive, he level of correlaion, 0.5, almos be similar o realiy. Fifh, under he above srucure of i and, we find by seing he overall variance of he dividend process, m - j j, be similar o he variance in sock marke. Here we adop he daily variance, 0.054%, of he aiwan Sock Exchange Capializaion Weighed Sock Index (AIEX) in he period from 00/9/ o 0//30 o represen he sock marke variance. Sixh, he risk aversion parameer is also a major facor of derivaive price, so we will also se o be a comparaive saic variable. Mos empirical research have exhibi ha should range beween 0 and -.0, we will ake ino accoun and 0 for comparison. However, some sudies sugges ha he equiy premium puzzle should be consider, so we will addiionally consider 40 o 7

18 reflec he case of he equiy premium puzzle. In each risk aversion level, we will examine wo correlaion scenarios, posiive and negaive. Sevenh, because he growh rae in dividend process and he iniial are only affec he mean of he dividend process insead of variance, given he volailiy parameer and he risk aversion, we se and by e R (, )( ) D(, ) so ha he risk-free ineres rae, R (, ) is kep a.4%. hrough his seup, we could ensure ha he comparisons and analyses are under he same baseline in differen scenarios. Finally, he parameers of emperaure process are se by able. We choose January, 0 as he valuaion dae, which is he following dae afer he las sample dae. For breviy, we only repor HDD conracs in he period from November, 0 o March 3, 03 due o CDD conracs are he reverse of HDD conracs. he values of HDD derivaives are calculaed as he average of 50,000 realizaions. 4. he Resuls of Simulaion Now we will compare he risk premiums of derivaives under differen lagged correlaed number, correlaion levels, and risk aversions. For each derivaive, he risk premium,, which is defined in secion 3, will exhibi by he percenage difference beween he derivaive values (excluding he discoun facor) and he risk-neural value. Firs, we observe he HDD forward conrac. able 4 shows he simulaed resuls of HDD forward price under he Gamma ranslaed disribuion and he normal disribuion, where he values in parenheses are he simulaed resuls under he normal disribuion. able 4: Risk Premium in Forward Prices: Lagged Correlaions Risk-neural forward (669.85) a a a a a a a 0.5 zero-lagged Forward Price

19 correlaion ( ) (670.6) ( ) (67.00) ( ) ( ) Risk Premium % 0.087% -0.3% 0.7% % 0.678% (-0.04%) (0.056%) (-0.57%) (0.74%) (-0.478%) (0.774%) Forward Price lagged correlaions (669.3 ) (67.08 ) ( ) ( ) (65.73 ) ( ) Risk Premium -0.68% 0.7% -0.95%.04% % 4.78% (-0.093%) (0.84%) (-0.739%) (0.67%) (-.557%) (.68%) Forward Price lagged correlaions ( ) (67.38 ) ( ) ( ) (65.74 ) ( ) Risk Premium -0.63% 0.46% -0.98%.3% % 4.909% (-0.0%) (0.9%) (-0.845%) (0.69%) (-.639%) (.769%) a 0.5 Forward Price zero-lagged correlaion ( ) ( ) ( ) (67.50 ) ( ) ( ) Risk Premium % 0.5% % 0.409% -.36%.47% (-0.05%) (0.080%) (-0.86%) (0.47%) (-0.805%) (0.856%) Forward Price lagged correlaions ( ) (67.39 ) ( ) ( ) ( ) ( ) Risk Premium % 0.340% -.390%.53% -4.83% 6.755% (-0.%) (0.30%) (-0.977%) (.073%) (-3.64%) (4.05%) Forward Price lagged correlaions (667.9) ( ) ( ) (677.8 ) ( ) ( ) Risk Premium -0.34% 0.369% -.54%.566% % 6.997% (-0.90%) (0.80%) (-0.988%) (.094%) (-3.779%) (4.55%) Noe:. he risk-neural forward is calculaed by 0, i 0, i. he forward price is calculaed by excluding he discoun facor. 9

20 3. he risk premiums are he percenage differences beween he risk-neural value and he derivaive price. For example, under, 0.5, and 5-lagged correlaions, he price is 0.7% (0.84%) higher han he risk-neural value under he Gamma ranslaed (normal) disribuion. hrough able 4, we can obain he following observaion: (i) i is consisen wih he heoreical resul a able ha he correlaion s sign deermines he sign of risk premium. (ii) given a fixed correlaion, a higher risk aversion brings abou a larger risk premium, hus i makes sense ha hose invesors ask for more reurn. (iii) under he same risk aversion level, a higher correlaion level cause a larger risk premium, which also make inuiive sense. (iv) comparing hree differen case of lagged correlaed number, he risk premium is higher if we consider more lagged correlaion, bu he risk premiums beween 5-lagged correlaion and 30-lagged correlaion are almos he same. (v) he larges risk premium under he Gamma ranslaed (normal) disribuion is 6.997% (4.55%) for 30-lagged (30-lagged) correlaion when -40 and 0.5. (vi) no only he forward prices bu also he risk premiums under he Gamma ranslaed disribuion are higher han under he normal disribuion. Second, we analyze he resuls for call and pu opions. For he purpose of leing he risk-neural call and pu opions be a-he-money and equaliy, under he Gamma ranslaed (normal) disribuion we would se he srike price equaling o he risk-neural forward price, (669.85). Oher seup is he same as he forward price. he resuls of call and pu opions are illusrae in able 5 and able 6, respecively. able 5: Risk Premium in Call Prices: Lagged Correlaions Risk-neural call price (7.3056) a a a a a a a 0.5 zero-lagged Call Price

21 correlaion (7.39 ) (7.47) (6.457 ) (7.400 ) (5.739 ) (9.89 ) Risk Premium %.679% -6.40%.533% % 7.34% (-0.60%) (0.60%) (-3.07%) (0.346%) (-5.737%) (9.470%) Call Price lagged correlaions (7.000 ) (7.588 ) (4.77 ) (9.00 ) (9.40 ) (37.60 ) Risk Premium -.48%.73% %.05% % 6.4% (-.9%) (.035%) (-9.8%) (6.938%) (-9.538%) (36.090%) Call Price lagged correlaions (6.806 ) (7.77 ) (4.686 ) (9.37 ) (9.74 ) ( ) Risk Premium %.97% % 4.344% -37.% 63.64% (-.83%) (.543%) (-9.595%) (7.566%) (-9.780%) (37.653%) a 0.5 Call Price zero-lagged correlaion (7.085 ) (7.47 ) (6.43 ) (8.47 ) (4.67 ) (30.56 ) Risk Premium -.898%.336% % 5.839% -3.78% 7.59% (-0.807%) (0.60%) (-4.57%) (4.67%) (-.9%) (0.437%) Call Price lagged correlaions (6.6 ) (7.833 ) (3.800 ) (3.000 ) (6.404 ) ( ) Risk Premium -.64% 4.86% % 6.5% -45.8% % (-.543%) (.933%) (-.838%) (3.530%) (-39.95%) (58.604%) Call Price lagged correlaions (6.333 ) (8.78 ) (3.563 ) (3.676 ) (6.385 ) ( ) Risk Premium % 4.453% -5.56% 7.37% % 9.9% (-3.56%) (3.56%) (-3.706%) (6.007%) ( %) (60.973%)

22 able 6: Risk Premium in Pu Prices: Lagged Correlaions Risk-neural pu price (7.3056) a a a a a a a 0.5 Pu Price zero-lagged correlaion (7.306 ) (7. ) (7.486 ) (6.57 ) (8.957 ) (4.696 ) Risk Premium 0.5% -.9% 4.34% % 7.096% % (0.000%) (-0.7%) (0.660%) (-3.840%) (6.046%) (-9.558%) Pu Price lagged correlaions (7.6 ) (6.639 ) (9.686 ) (4.743 ) (36.0 ) (9.360 ) Risk Premium 3.533% -.490% 9.46% -0.3% 4.679% % (.9%) (-.44%) (8.77%) (-9.385%) (3.647%) (-9.099%) Pu Price lagged correlaions (7.6) (6.694 ) (8.97 ) (4.89 ) (36.783) (9.09 ) Risk Premium.63% -3.68% 8.587% -0.06% 43.74% -4.59% (.9%) (-.38%) (6.0%) (-9.07%) (34.707%) (-30.09%) a 0.5 Pu Price zero-lagged correlaion (7.556 ) (6.944 ) (8.76 ) (6.48 ) (9.6 ) (4.444 ) Risk Premium.9% -.365% 4.40% % 5.549% -3.9% (0.96%) (-.3%) (3.90%) (-3.873%) (8.484%) (-0.478%) Pu Price lagged correlaions (8. ) (6.306 ) (30.34 ) (3.857 ) ( ) (6.88 ) Risk Premium 3.983% -4.9% 4.00% % 57.89% -56.7% (.950%) (-3.66%) (.09%) (-.69%) (49.378%) ( %)

23 Pu Price lagged correlaions (8.78 ) (6.389 ) ( ) (4.35 ) (4.705 ) (5.455 ) Risk Premium.48% % 7.53% -5.9% 6.95% % (3.56%) (-3.357%) (.0%) (-.44%) (5.733%) (-43.40%) Noe:. he opion price is calculaed by excluding he discoun facor.. he values in parenheses are he resuls under he normal disribuion. For he call price, he resul is he same as he forward price. As expeced, he resul for he pu price is inverse. I is worh noing ha he risk premiums for opions are larger han for he forward conrac due o he payoff ypes beween hem, linear and non-linear. Under he Gamma ranslaed (normal) disribuion, he larges risk premium, 9.9% (60.973%) for call wih 40 and 0.5 and 6.95% (5.733%) for pu wih 40 and on lagged correlaions. According o he saemens in he able 4 hrough able 6, he risk premium is small if we only consider he conemporaneous correlaion beween he dividend and emperaure processes. Furhermore, he difference of resuls beween 5-lagged correlaions and 30-lagged correlaions is inexplicable. On he oher hand, boh 5-lagged correlaions and 30-lagged correlaions have idenical porions of he variance conribued by he emperaure variaions (wih 0.5 and 0.5, he proporions are abou 5.6% and 3.6%, respecively. ). herefore, we sugges ha he number of lagged erms of correlaion could be Mean Reversion of he Aggregae Dividend and Risk Premium In his paper, we se 0.9 which corresponds o a mean reversion rae of 0.. o see how sensiive he resuls are o his mean reversion parameer, he calculaion in able 5 and able 6 are repeaed by assuming four oher levels of : 0.80, 0.85, 0.90, Noe ha 0.99 roughly correspond o a random walk. In he able 7, i is seen ha higher value 3

24 of, or a lower mean reversion speed, leads o a bigger risk premium in forward and opion values. his makes inuiive sense since a higher means bigger variaions in he aggregae dividends. Furher, he risk premiums for opions are larger han for he forward conrac due o he payoff ypes beween hem, linear and non-linear. o illusrae, for opion, when increase from 0.9 o 0.99, he risk premium increases by more han 0-fold. Wih a near-random walk, he risk premium is more han 0% for all opion values. An obvious conclusion is ha, in deermining he significance of he marke price of risk for he emperaure variable, he degree of mean reversion in he aggregae dividend process mus be carefully deermined. Panel A : Forward Prices able 7: Impac of Mean Reversion in he Dividend Process Mean Risk Neural 0 40 Reversion Forward a a a a a a % 0.03% % 0.540% -.596%.06% % 0.07% -0.53% 0.593% -.87%.85% % 0.46% -0.98%.3% % 4.909% % 0.374% -.973%.83% -6.69% 8.808% %.575% % 8.047% -.368% 5.674% Panel B : Opion Prices Mean Reversion Risk Neural Call (Pu) Price a 0 40 a a a a a 0.8 Call (Pu) % 0.67% -4.80% 6.64% -8.7% 4.4% (.094% ) (-.898% ) (5.335% ) (-4.964% ) (5.553% ) (-9.335% ) 4

25 0.85 Call (Pu) %.678% -6.5% 0.365% -3.78% % (.94%) (-.744%) (6.43%) (-6.64%) (5.55%) (-5.468%) 0.9 Call (Pu) %.97% % 4.344% -37.% 63.64% (.63%) (-3.68%) (8.587%) (-0.06%) (43.74%) (-4.59%) 0.95 Call (Pu) % 4.087% -.606% 6.73% %.66% (3.833% ) (-3.943% ) (0.75% ) (-0.34% ) (9.895% ) (-66.5% ) 0.99 Call (Pu) % 8.48% -6.77%.574% % 450.4% (6.356% ) (-5.48% ) (97.793% ) (-6.898% ) ( % ) (-99.66% ) Noe:. Forward and opion price are all for HDD conracs for he aiwan. Excep for he mean reversion parameer all oher aspecs of he calculaion are he same as in ables 5 and able 6.. a Conclusions his paper combines he equilibrium model proposed by Cao and Wei (004) and he emperaure model proposed by Campbell and Diebold (005). We adop he Gamma ransformaion o se he emperaure disurbance and find ha he model under Gamma ranslaed disribuion is more powerful on he forecas abiliy han under Normal disribuion. Since he daa ses of macroeconomic variables in aiwan are low frequen, we do no esimae he dividend model. We se he parameers of he dividend process by Cao and Wei s (004) seup for dividend model. According o he resuls of he simulaion, we find ha he number of lagged correlaions does no have o consider 30 erms due o he similar resuls of 5-lagged correlaions and 30-lagged correlaions. Furhermore, he marke price of risk is significan under wo levels of lagged erms, 5-lagged correlaions and 30-lagged correlaions, bu is no obvious under he conemporaneous correlaion only. Alhough he resuls are reasonable, he low frequency daa is no perfec. herefore, we sugges ha fuure research could adop he high frequency daa o make he sudy mach 5

26 realiy more closely. 6

27 References Alaon, P., Djehiche, B., and Sillberger, D., 00, On Modelling and Pricing Weaher Derivaives, Applied Mahemaical Finance, 9, -0. Bibby, B. M., and Sørensen, M., 995, Maringale Esimaion Funcions for Discreely Observed Diffusion Processes,, Brody, D. C., Syroka, J., and Zervos, M., 00, Dynamical Pricing of Weaher Derivaives, Quan. Finance,, Campbell, S. D., and Diebold, F. X., 005, Weaher Forecasing for Weaher Derivaives, Journal of he American Saisical Associaion, 00, 6-6. Cao, M., and Wei, J., 004, Weaher Derivaives Valuaion and Marke Price of Weaher Risk, Journal of Fuures Markes, 4, Considine, G., 000, Inroducion o Weaher Derivaives, Weaher Derivaives Group, Aquila Energy. Dischel, B., 998, A Las: A Model for Weaher Risk, Energy Power Risk Mgm,, 0-. Dornier, F. and Queruel, M., 000, Cauion o he Wind, Energy Power Risk Mgm, 3, Lucas, R. E., 978, Asse Prices in an Exchange Economy, Economerica, 46, Marsh,. A., and Meron, R., 987, Dividend Behavior for he Aggregae Sock Marke, Journal of Business, 60, -40. Shiller, R. J., 983, Do Sock Prices Move oo Much o be Jusified by Subsequen Changes in Dividends? Reply, American Economic Review, 73, Viiello, L., and Poon, S. H., 00, General Equilibrium and Preference Free Model for Pricing Opions under ransformed Gamma Disribuion, he Journal of Fuures Markes, 30,

28 Appendix: he Proof of he Value of he Pure Discoun Bond Firs, we illusrae he disribuion of he emperaure residual. According o Figure, we can see i look like a reversal of he Gumbel disribuion, hence we use he Gamma ransformed mehod proposed by Viiello and Poon (00) o explain he densiy funcion of he emperaure residual. heorem: Le hz () is some ransformaion of z. If x in hz () ax has a gamma densiy and h() is a monoonic differeniable funcion, hen he densiy funcion of z is given by f( z) h'( z) h( z) e ( p) p h( z) (A.) where h'( z ) is he firs derivaive of hz () which is gamma disribued, and f () z is a ransformed gamma densiy. For p and se hz ( ) exp( z), hen he variable z follows he sandard Gumbel densiy. Based on he above heorem, we can obain he densiy funcion of he emperaure residual. By he heorem, since he emperaure residual is similar he Gumbel disribuion, we can se he h( ) exp( ) x, where x~ Exp( ). hen we ge he densiy funcion of is: f ( ) e e (A.) and he momen generaing funcion of is: M () E( e ) E[( X)] ( ) (A.3) Second, in he equaion (3), we ierae he process and obain 8

29 ln j ln j j m j j ln j i ji j j i0 (A.4) where 0, and we suppose j has m lagged error erms wihou loss of generaliy. where for m and m j i ji j i0 m i m mi i ( i j) ˆ j i j m j i m ji i j i i j i ji im j0 for m (A.5) m i ji j i0 j i i ( i j) ˆ i j m j i m j i i j i j0 (A.6) Here ˆ l(0 l m) are he realized error erms for he emperaure variable. hird, we use he momen generaing funcion o obain he condiional expecaion of he aggregae dividend a ime. Because j follow i.i.d normal disribuion and he momen generaing funcion of is given by equaion (A.3), we can use he momen generaing funcion o obain he condiional expecaion of he aggregae dividend a ime. Assume ha j i i, j and F is he infiniely represenaive invesor s informaion a ime. 9

30 for m m mi i i j ij E( F) A(, ) Eexp i ji i j i F i i ji im j0 A i ( i) B(, ) (, )exp (, ) C where for m A (, ) exp m i i ( i j) ˆ m j i m i i j m mi i j ij B (, ) ji j i ji im j0 m mi j i i j C (, ) ji j i im ji j0 i i i j E( F) A(, ) Eexp i j i F i i j0 ( i) B(, ) A (, )exp C (, ) i where A (, ) exp i i j B (, ) j i j0 i i j C (, ) j i j0 i i ( i j) ˆ m j i m i i j Finally, we use he above resul o derive he value of he pure discoun bond. By he equaion (5), assume ha q, hen D e E e e E - ( - ) (, ) ( ) ( ) ( i) i B (, ) (, ) e A e C - ( - ) (, ) 30

31 where for m and for m m i i ( i j) (, ) exp ˆ A m j i m i i j m mi i j i j B (, ) ji j i ji im j0 m mi j i ji C (, ) ji j i ji im j0 A (, ) exp i i j B (, ) j i j0 i ji C (, ) j i j0 i i ( i j) ˆ m j i m i i j 3