DYNAMIC ECONOMETRIC MODELS Vol. 8 Nicolaus Copernicus University Toruń 2008

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1 DYNAMIC ECONOMETRIC MODELS Vol. 8 Ncolaus Coperncus Unversy Toruń 2008 Por Fszeder Ncolaus Coperncus Unversy n Toruń Julusz Preś Szczecn Unversy of Technology Prcng of Weaher Opons for Berln Quoed on he Chcago Mercanle Exchange 1. Inroducon A presen he global weaher dervaves marke s developng very fas. Only n he recen perod (Aprl March 2008) he noonal value of all weaher conracs reached over 32 bllon USD. As a resul of such a dynamc ncrease on hs marke he problem of approprae weaher opons prcng appears more ofen. Usually n hese suaons, he Black-Scholes formula s used. Unforunaely, many observers and weaher marke parcpans have noced ha hs approach canno be appled because of he dfferen naure of weaher underlyng 1. I mus be added, ha he unque and complex feaures of weaher ndces made mpossble unl now o creae any complee and unversal procedure for prcng hs class of nsrumens. For hs reason many dfferen approaches of prcng weaher dervaves have been proposed. The mos popular are: hsorcal burn analyss, ndex modellng and daly modellng. Among hese, he las one seems o have he greaes poenal n creang one precse approach of prcng all weaher conracs 2. Therefore n hs paper we concenrae on daly modellng as an approach whch makes use of models of sochasc processes. Moreover hs work we analyse no only daly me seres, bu also monhly values of weaher ndces. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 1 See Dschel (1998). 2 Brx, Jewson and Zehmann (2005).

2 164 Por Fszeder, Julusz Preś The man goal of hs work s o presen he approach of prcng weaher opons usng he ARFIMA-FIGARCH model 3. I has been shown ha hs model neglecs seasonally changng auocorrelaon, whch leads o sgnfcan devaons n opon prcng. We propose n he paper wo prcng models whch s he auhors' conrbuon. Frs he model ARFIMA-FIGARCH used for daly daa (see Moreno, 2003 for analyss of emperaure) has been exended by ncludng seasonaly n varance wha urned ou o be sascally sgnfcan. The second model bases on monhly values of emperaure ndces, whch s a knd of compromse beween smple prcng approaches lke hsorcal burn analyss and more advanced mehods, lke economerc daly modellng. In secon 2 we descrbe all he sochasc models ha we used n our research. In hs par we propose a small mprovemen ha allows he ncluson of seasonaly n varance. Secon 3 conans he descrpon of all sascal feaures of seleced me seres and he resuls of esmaon for analysed models. In he nex secon he analysed models are used n an emprcal example for prcng Chcago Mercanle Exchange s weaher opons for Berln. Secon 5 conans conclusons. 2. Models of Sochasc Processes In he process of ar emperaure we can usually se apar such dsncve feaures lke seasonaly, rend as a resul of global warmng and urbanzaon, long memory, whch s ypcal for clmae processes (see Hurs, 1951, Kwakowsk and Osewalsk, 2002), and random flucuaons. Addonally, we can fnd seasonaly n varance. Volaly of ar emperaure s hgher n wner monhs and smaller n summer monhs. Hence all he above feaures ough o be ncluded n modellng of ar emperaure. The models proposed n hs work are an exended verson of he ARFIMA- FIGARCH model and allow he ncluson of seasonaly n mean and varance. The ARFIMA model, as a generalzaon of he ARIMA model, allows he effecve descrpon of shor and long memory n mean. The FIGARCH model, as a generalzaon of he GARCH model, allows smlar feaures o be descrbed n varance. The exended model ARFIMA ( P, d1, Q) - FIGARCH ( p, d2, q) can be presened n he followng form: d 1 ϕ ( L)(1 L) ( y μ ) = ϑ( L) ε, ε ψ ~ D (0, h ), (1) r m / 2 = 0 = 1 1 μ γ + ( θ cosω + λ snω ), (2) = Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 3 Ths model s used by praconers on he fnancal marke and exss n popular applcaon SWS 6.0 made by Speedwell Weaher Dervaves Ld.

3 Prcng of Weaher Opons for Berln Quoed on he Chcago Mercanle Exchange 165 m / 2 d 2 2 ε = α 0 + ( κ cosω + τ sn ω) + [1 = 1 φ ( L)(1 L) β ( L)] ν, (3) s where L s he backshf operaor ( L ε = ε s ), ϕ ( ) = 1 ϕ L, Q ϑ ( ) = 1+ ϑ L, ) = 1 L = 1 q p φ ( L φ L, L ) = = 1 = 1 L L = 1 β ( β, ν ε h, all roos = 2 of polynomals ϕ ( L) = 0 and φ ( L) = 0 are beyond un crcle, 1< d 1 < 0. 5, 0 < d 2 < 1, ω = 2π, m = 365. m In order o oban posve varance h whou mposng addonal resrcons on he parameers n he equaon for varance, a logarhmc formula of varance can be appled ( ν = ε 2 ln h ). The seasonal componen n equaon (2) can be presened n a dfferen han harmonc way: r 12 γ + k = 0 k= 1 μ = c m, (4) k where m k are dummy varables ha mean successve monhs (eg. m 1 = 1 for January and m 1 = 0 for oher monhs). In a smlar way we can descrbe a seasonal componen for varance: 12 d 2 2 ε = α0 + δ kmk + [1 k = 1 φ ( L)(1 L) β ( L)] ν. (5) 3. Modellng Tme Seres of Ar Temperaure and Weaher Indces Hsorcal daa for Berln were obaned from Deuscher Weerdens and hey were colleced over he perod from January 1, 1948 o December 31, 2004 (20808 daly observaons). Dsrbuons of daly values of emperaures are dfferen n specfc monhs (perods) of a year. Research n hs area has shown ha n wner monhs here s lef asymmery whle n summer monhs asymmery s posve. The hypohess abou normaly of dsrbuon, usng a varey of ess, has been reeced for specfc monhs and also for he enre year 4. The resuls of sascal ess ndcae he exsence of he followng properes of daly average emperaure: ncreasng lnear rend, seasonaly boh n mean and varance, auocorrelaon (hgher n wner monhs), long memory n mean and he ARCH effec. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House P 4 Owng o he lmed sze of hs paper some resuls of performed ess and esmaons had o be omed.

4 166 Por Fszeder, Julusz Preś Apar from he daly average of ar emperaure, also monhly emperaure ndces (HDD and CAT) were analysed. These ndces can be calculaed usng he followng formulae: m HDD = max(0, 18 C ), (6) = 1 m y = 1 y CAT =, (7) where y - average daly emperaure, m- number of days n any gven perod. The hypohess abou normaly of dsrbuon was reeced only for he HDD ndex for he monhs of March and December. In he curren research on prcng weaher conracs, only he dsrbuon of he cumulaed monhly ndces (HDD and ohers) s analysed. In hs paper we rea he monhly me seres of hese ndces as a realzaon of sochasc processes and explore her feaures. The performed ess ndcae he exsence of he followng feaures n monhly ndces HDD and CAT: lnear rend, seasonaly n mean and n varance, long memory n mean (weaker han for daly me seres). The model descrbed n equaons (1-3) was seleced o explan he average daly emperaure. The long memory has been descrbed by ARFIMA model (d 1 = (0.0283)). The shor-erm dependences n he me seres of emperaure have been explaned by auoregressve and movng average pars wh lags equal o wo ( P = 2 and Q = 2 ). In he equaon for condonal varance he parameer d 2 urned ou o be nsgnfcan, hence he process of volaly has no long memory. The GARCH(1,1) model wh annual perodcal flucuaons and normal condonal dsrbuon urned ou o be suffcen o descrbe varance of emperaure. For monhly values of HDD and CAT wo models wh dfferen parameersaon of seasonaly were consdered. If he model conssed of a harmonc seasonal componen and lnear rend, he error erm for he HDD ndex was bes descrbed by he AR(1) model. For he seasonal model wh dummy varables and lnear rend (equaon 4), he error erm was bes descrbed by ARFIMA(0,d 1,0). For he CAT ndex, for any parameersaon of seasonaly, lnear rend and long memory were observed. In boh cases, he ARFIMA model (0,d 1,0) urned ou o be he bes. For all ndces uncondonal varance of random erm n all models was varable. The ARCH effec was no presen. The fnal parameersaon (values P, Q, p, q, r and he number of pars of rgonomerc polynomals) were seleced by he Schwarz creron, akng no accoun resuls of proper dagnosc ess. The analysed models were verfed n erms of he average mean and varance usng Mone Carlo smulaons. The resuls are presened n Table 1. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House

5 Prcng of Weaher Opons for Berln Quoed on he Chcago Mercanle Exchange 167 Table 1. Resuls of models verfcaon for mean and varance based on Mone Carlo smulaons 5 Type of model Parameers Model 1 (daly) Model 2 (monhly) Model 3 (monhly) January HDD June CAT January HDD June CAT January HDD June CAT Mean Varance Δ for mean * Δ for varance * Model 2 ncludes harmonc srucure of seasonaly, whereas n model 3 hs componen consss of dummy varables. The asersk ndcaes ha he null hypohess abou he equaly of he expeced value (varance) of he ndex assumng ha he analysed model s rue wh expeced value (varance) for populaon was reeced a he 5% level. Tes sascs Δ for mean (varance) were calculaed as absolue values of he dfferences beween means (varances) calculaed for generaed daa and sample. Mean and varance values for he January HDD ndex were and respecvely, whle for he June CAT ndex hese parameers were and Comparsons of he examned models have also been performed usng oher crera, lke adused deermnaon coeffcen or roo mean squared error (Table 2). Table 2. Evaluaon of qualy of models Type of model January - HDD June - CAT RMSE Ad. R² RMSE Ad. R² Model 1(daly) Model 2 (monhly wh harmonc seasonaly) Model 3 (monhly wh dummy varables) The obaned resuls show ha despe he overesmaon of varance durng summer monhs, he model consruced for daly observaons of emperaure has he bes ably o descrbe monhly values of HDD and CAT ndces. The effec of he varance overesmaon s caused by omng seasonal varably of auocorrelaon of emperaure (see Fgure 1). Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 5 For he seleced model me seres of emperaure wh lengh 30 or 31 observaons were generaed and aferwards ndces were calculaed. For models of ndces, he values of ndces were generaed drecly. The complee procedure of verfcaon s gven n Caballero, Jewson and Brx (2002).

6 168 Por Fszeder, Julusz Preś Fgure 1. Auocorrelaon funcon for average daly emperaure n dfferen perods of he year. 4. Prcng Weaher Opons The models presened above have been appled n prcng wo dsnc monhly opon conracs (call) wh cap value 6. Underlyng for hese opons were ndces: HDD for January 2004 and CAT for June Specfcaon of hese conracs s shown n Table 3. Table 3. Specfcaon of examned CME weaher opons for Berln Name Conrac 1 Conrac 2 Type of opon Call Call Index HDD Berln CAT Berln Perod of lfe January 2004 June 2004 Srke value 600 HDD 550 HDD Tck value GBP GBP Maxmum payou (cap) GBP GBP In order o compare prcng approaches, hree mehods menoned above: hsorcal burn analyss (HBA), ndex modellng (IM) and daly modellng (DM) were used. The las mehod ncludes all analysed models. The resuls are gven n Table 4. In he case of January s conracs a clearly lower evaluaon was obaned usng he model wh harmonc seasonaly (monhly model 2), for whch he mean from he smulaon urned ou o be lower by 14 pons han he mean from he sample. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 6 An opon conrac wh cap value can be obaned by buyng a call opon and sellng he call opon wh he same expraon dae bu wh dfferen srke prces.

7 Prcng of Weaher Opons for Berln Quoed on he Chcago Mercanle Exchange 169 Table 4. Valuaons of weaher opons usng dfferen approaches Valuaon mehods Prce of conrac 1 [GBP] Prce of conrac 2 [GBP] Relave prce of conrac 1 as a percenage of maxmum payou [% lm] Relave prce of conrac 2 as a percenage of maxmum payou [% lm] HBA IM DM model MM model MM model The dfferen resuls were obaned n prcng June s conrac for he CAT ndex. For he frs wo mehods he obaned valuaons were below 10% of he maxmum payou. The mehods based on me seres modellng esmaed he prce of hs conrac beween 13.43% and 26.72% of he maxmum payou. The valuaon receved on he bass of daly modellng s srongly overesmaed because of he clearly overesmaed level of varance for June (Table 1). The second model clearly overesmaed he mean value and ha s why he prce of he opon was overvalued. The mos relable prcng seems o be he one obaned from he las model (13.43%). Unforunaely, n hs case he esmaon s error wll be hgher, because he analysed model provded an nferor f o he daa. Besdes, he model for monhly values does no ake no accoun he varable number of days n a monh and canno be used drecly o esmae he prce of weaher conracs for all lenghs of me. Daly modellng of emperaure allows he beer usage of hsorcal daa. If one wans o prce a monhly weaher conrac usng hsorcal burn analyss or ndex modellng approaches one can use he daa only from he expraon perod. For example for 10 years of daa he esmaon wll be based on 10 hsorcal values. Applcaon of daly modellng allows he use of 3652 observaons n he prcng process. Therefore hs approach s poenally beer. Only poenally, because assumes ha he model s correc and able o descrbe all properes of weaher me seres. Hence, he rsk of usng any model may be an mporan facor n dervave prcng 7. The bayesan model poolng gves possbly o formally nclude specfcaon uncerany n sascal nference (see for example Osewalsk, 2001). 5. Conclusons The models consruced for monhly observaons of ndces do no explan volaly of emperaure ndces n a suffcen way. The model consruced for daly observaons of emperaure s he bes n descrbng emperaure ndces. Ths model descrbes mos of he mporan feaures. Despe he applcaon of Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 7 See, for example, Jauga (2007).

8 170 Por Fszeder, Julusz Preś such an advanced model, s no able o descrbe he real emperaure process compleely, because neglecs seasonal varably of auocorrelaon, whch causes over- or underesmaons of varance n some perods. Caballero and Jewson (2003) sugges as an alernave usng he SAROMA model. Ths model consders seasonaly n he auocorrelaon funcon, bu he number of parameers ha need o be used n hs model, makes he esmaon process dffcul. Besdes, he SAROMA model requres many observaons n order o avod a spurous explanaon of daa. In addon, hs model oms long memory. The oher soluon could be o nclude n he ARFIMA-FIGARCH model a se of me-varyng parameers of long memory n mean (and perhaps also n varance). Unforunaely, he specfcaon of hs model wll be more complex han he one presened n hs paper. References Brx, A., Jewson, S., Zehmann, Ch. (2005), Weaher Dervave Valuaon, Unversy Press, Cambrdge. Caballero, R., Jewson, S. Brx, A. (2002), Long Memory n Surface Ar Temperaure: Deecon, Modellng and Applcaon o Weaher Dervave Valuaon, Clmae Research, 21, Caballero, R., Jewson, S. (2003), Seasonaly n he Sascs of Surface Ar Temperaure and he Prcng of Weaher Dervaves, Meeorologcal Applcaons, 10, (4), Dschel, R. (1998), Black-Scholes Won Do, Energy Power and Rsk Managemen, 10, Hurs, H. E. (1951), Long Term Sorage Capacy of Reservors, Transacons of Amercan Socey of Cvl Engneers, 116, Jauga, K. (2007), (red.) Zarządzane ryzykem (Rsk Managemen), PWN, Warszawa. Kwakowsk, J., Osewalsk, J. (2002), Modele ARFIMA: podsawowe własnośc analza bayesowska, (ARFIMA Models: he Man Properes and Bayesan Analyss), Przegląd Saysyczny, (Sascal Survey), 50, 2, Moreno, M. (2003), Weaher Dervaves Hedgng and Swap Ilqudy, Weaher Rsk Managemen Assocaon. Osewalsk, J. (2001), Ekonomera bayesowska w zasosowanach (Bayesan Economercs n Applcaons), Cracow Unversy of Economcs, Kraków. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House