Long Term Spread Option Valuation and Hedging

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Long Term Spread Option Valuation and Hedging"

Transcription

1 Long Term Spread Opion Valuaion and Hedging M.A.H. Demper, Elena Medova and Ke Tang Cenre for Financial Reearch, Judge Buine School, Univeriy of Cambridge, Trumpingon Sree, Cambridge CB 1AG & Cambridge Syem Aociae Limied 5-7 Porugal Place, Cambridge CB5 8AF Abrac Thi paper inveigae he valuaion and hedging of pread opion on wo commodiy price which in he long run are coinegraed. For long erm opion pricing he pread beween he wo price hould herefore be modelled direcly. Thi approach offer ignifican advanage relaive o he radiional muli-facor pread opion pricing model ince he correlaion beween wo ae reurn i nooriouly hard o model. In hi paper, we propoe one and wo facor model for po pread procee under boh he ri-neural and mare meaure. We develop pricing and hedging formulae for opion on po and fuure pread. Two example of pread opion in energy mare he crac pread beween heaing oil and WTI crude oil and he locaion pread beween Bren blend and WTI crude oil are analyzed o illurae he reul. JEL claificaion: G1 Key word: pread opion, coinegraion, mean-reverion, opion pricing, energy mare Correponding auhor, addre: Judge Buine School, Univeriy of Cambridge, Trumpingon Sree, Cambridge CB 1AG, Tel: +44 (0) , Fax: +44 (0) , 1

2 1. Inroducion A pread opion i an opion wrien on he difference (pread) of wo underlying ae price S 1 and S repecively. We conider European opion wih payoff he greaer or leer of S (T)-S 1 (T)-K and 0 a mauriy T and rie price K and focu on pread in he commodiy (epecially energy) mare (boh for po and fuure). In hee mare pread opion are uually baed on difference beween price of he ame commodiy a wo differen locaion (locaion pread) or ime (calendar pread), beween he price of inpu and oupu (producion pread) or beween he price of differen grade of he ame commodiy (qualiy pread). The New Yor Mercanile Exchange (NYMEX) alo offer radable opion on he heaing oil/ crude oil or gaoline/crude oil pread (crac pread). I i naural o model he pread by modelling each ae eparaely. Margrabe (1978) wa he fir o rea pread opion and gave an analyical oluion for rie price zero (he exchange opion). Wilcox (1990) and Carmona and Durrleman (003) ue Bachelier (1900) formula o analyically price pread opion auming he underlying price follow arihmeic Brownian moion. I i more difficul o value a pread opion if he wo underlying price follow geomeric Brownian moion. Variou numerical echnique have been propoed o price uch an opion. Rubinein (1991) value he pread opion in erm of a double inegral. Demper and Hong (000) ue he fa Fourier ranform o evaluae hi inegral numerically. Carmona and Durrleman (003) offer a good review of pread opion pricing. Many reearcher have modelled a pread opion by modelling he wo underlying ae price in he ri neural meaure a 1 ds = rs d+ σ S dw ds = rs d+ σ S dw W1 W = ρ. Ed d d (1) The correlaion ρ play a ubanial rôle in valuing a pread opion; rading a pread 1 Boldface i ued hroughou o denoe random eniie here condiional on S 1 and S having realized value S 1 and S a ime which i uppreed for impliciy of noaion.

3 opion i equivalen o rading he correlaion beween he wo ae reurn. However, Kir (1995), Mbanefo (1997) and Alexander (1999) have uggeed ha reurn correlaion i very volaile in energy mare. Thu auming a conan correlaion beween wo ae a in (1) i inappropriae for modelling. Bu here i anoher longer erm relaionhip beween wo ae price ermed coinegraion which ha been lile udied by ae pricing reearcher. If a coinegraion relaionhip exi beween wo ae price he pread hould be modelled direcly for long erm opion pricing. Thi i he opic of hi paper which i organized a follow. Secion give a brief review of price coinegraion and he principal aiical e for coinegraion and for he mean reverion of pread. Secion 3 propoe one and wo facor model of he underlying pread proce in he ri-neural and mare meaure and how how o calibrae hee model. Secion 4 preen opion pricing formulae for opion on po and fuure pread. Secion 5 provide wo example in energy mare which illurae he heoreical wor and Secion 6 conclude.. Coinegraed price and mean reverion of he pread The value of a pread opion i deermined by he dynamic relaionhip beween wo underlying ae price and he correlaion of he correponding reurn ime erie i commonly underood and widely ued. Coinegraion i a mehod for reaing he long-run equilibrium relaionhip beween wo ae price generaed by mare force and behavioural rule. Engle and Granger (1987) formalized he idea of inegraed variable haring an equilibrium relaion which urn ou o be eiher aionary or o have a lower degree of inegraion han he original erie. They ued he erm coinegraion o ignify co-movemen among rending variable which could be exploied o e for he exience of equilibrium relaionhip wihin he framewor of fully dynamic mare. 3

4 The parameer in equaion (1) can be calibraed uing reurn daa bu he coinegraion relaionhip mu be inveigaed wih price daa (Hamilon, 1994). In general, he reurn correlaion i imporan for hor erm price relaionhip and he price coinegraion for heir long run counerpar. If wo ae price are coinegraed (1) i only ueful for hor erm valuaion even when he correlaion beween heir reurn i nown exacly. Since we wih o rea long erm pread opion pricing we hall inveigae he coinegraion (long erm equilibrium) relaionhip beween ae price. Fir we briefly explain he economic reaon why uch a long-run equilibrium exi beween price of he ame commodiy a wo differen locaion, price of inpu and oupu and price of differen grade of he ame commodiy. The law of one price (or purchaing power pariy) implie ha coinegraion exi for price of he ame commodiy a differen locaion. Due o mare fricion (rading co, hipping co, ec.) he ame good may have differen price bu he mipricing canno go beyond a hrehold wihou allowing mare arbirage (Samuelon, 1964). Inpu (raw maerial) and oupu (produc) price hould alo be coinegraed becaue hey direcly deermine upply and demand for manufacuring firm. There alo exi an equilibrium involving a hrehold beween he price of a commodiy of differen grade ince hey are ubiue for each oher. If uch long-erm equilibria hold for hee hree pair of price coinegraion relaionhip hould be deeced in he empirical daa. Duan and Plia (003) model he log-price raher han he price coinegraion beween wo US mare indice. In equiy mare inveor are concerned wih index reurn raher han level o hi may be a good choice. However from he economic argumen i follow ha he pread beween wo po commodiy price reflec deviaion from a general (poibly growh) equilibrium, e.g. he profi of producing (producion pread), hipping (locaion pread) or wiching (qualiy pread). Conien wih Jure and Yang (006) we model here price coinegraion. Alhough he model of hi paper are applicable o calendar pread we will no rea hem here. 4

5 In empirical analyi economi uually ue equaion () and (3) o decribe he coinegraion relaionhip: S = c + ds + ε () 1 ε - ε = ωε + u, (3) -1-1 where S 1 and S are he wo ae price and u i a Gauian diurbance. Engle and Granger (1987) demonrae ha a coinegraion model i he ame a an error correcion model, i.e. he error erm ε in () mu be mean-revering (3). Thu a imple way o e he coinegraion relaionhip i o e wheher ω i a ignificanly negaive number in equaion (3), i.e. wheher he pread proce i mean-revering (Dicy-Fuller, 1979). Equaion () can be een a he dynamic equilibrium of an economic yem. When S 1 and S deviae from he long-run equilibrium relaionhip hey rever bac o i in he fuure. For boh locaion and qualiy pread S 1 and S hould ideally follow he ame rend, i.e. d hould be equal o 1. However for producion pread uch a he par pread (he pread beween he elecriciy price and he ga price) d may no be exacly 1. Uually 3/4 of a ga conrac i equivalen o 1 elecriciy conrac o ha inveor rade a 1 elecriciy / 3/4 ga pread which repreen he profi of elecriciy plan (Carmona and Durrleman, 003). Since gaoline and heaing oil are coinegraed ubiue, he d value could be 1 for boh he heaing oil/crude oil pread and he heaing oil/gaoline pread (Girma and Paulon, 1999). For our hree pread of inere locaion, producion and qualiy d hould be 1. Leing x denoe he pread beween wo coinegraed po price S 1 and S i follow from () and (3) in hi cae ha x -x -1 =c - c -1 -ω(c -1 -x -1 )+ u, (4) i.e., he pread of he wo underlying ae i mean-revering. No maer wha he naure of he underlying S 1 and S procee he pread beween hem can behave quie differenly from heir individual behaviour. Thi ugge modelling he pread 5

6 direcly uing an Ornein-Uhlenbec proce for long-erm opion evaluaion becaue he coinegraion relaionhip ha ubanial influence in he long run. Such an approach give a lea hree advanage over alernaive a i: 1) avoid modelling he correlaion beween he wo ae reurn, ) cache he long-run equilibrium relaionhip beween he wo ae price and 3) yield an analyical oluion for pread opion. For example, Jure and Yang (006) employ an Ornein-Uhlenbec proce o model he pread beween Siamee win equiie 3 and preen an opimal ae-allocaion raegy for pread holder..1 Coinegraion e To e he coinegraion of wo ae price, we fir need o e wheher each generae a uni roo ime erie. In an efficien mare ae price ingly will uually generae uni roo (independen incremen) ime erie becaue he curren price hould no provide forecaing power for fuure price. If wo ae price procee are uni roo bu he pread proce i no, here exi a coinegraion relaionhip beween he price and he pread will no deviae ouide economically deermined bound. The augmened Dicey-Fuller (ADF) e may be ued o chec for uni roo in ae price ime erie. The ADF aiic ue an ordinary lea quare (OLS) auo regreion S S = δ + δ S + δ ( S S ) + η (5) i+ 1 i i 1 i= 1 p o e for uni roo, where S i he ae price a ime, δ i, i=0,,p, are conan and η i a Gauian diurbance. If he coefficien δ 1 i negaive and exceed he criical value in Fuller (1976) hen he null-hypohei ha he erie ha no uni roo i rejeced. We can ue an exenion of (4) correponding o (5) o e he coinegraion 3 For deail on Siamee win ee Froo and Dabora (1999). 6

7 relaionhip: S = c + d S + ε 1 ε - ε = χ + χ ε + χ ( ε ε ) + u i+ 1 i i 1 i= 1 p (6) When χ 1 i ignificanly negaive he hypohei ha coinegraion exi beween he wo underlying ae price procee S 1 and S i acceped (Hamilon, 1994).. Mare meaure mean reverion e In Secion 3 we will ee ha he mean-revering propery of he po pread can be deeced by examining he mean-reverion of fuure pread wih a conan ime o mauriy. Empirically we eimae F ( + Δ, + Δ + τ ) F(, + τ ) = α + β F(, + τ ) + ε, (7) where F(,+τ) i he fuure pread of mauriy +τ oberved a, Δ i he ampling ime inerval and ε i a random diurbance. If β i ignificanly negaive hen he po pread i deemed o be mean-revering and a coinegraion relaionhip i aen o exi beween he wo underlying ae price. Thi mehod examine he evidence for mean-reverion in he mare meaure uing hiorical fuure price daa..3 Ri-neural meaure mean-reverion e We can ue (ex ane) mare daa analyi o e wheher inveor expec he fuure pread o rever in he ri-neural meaure. Thi mehodology focue on relaion beween pread level and he pread erm rucure lope defined a he change acro he mauriie of fuure pread. A negaive relaionhip beween he po pread level (or hor-erm fuure pread level) and he fuure pread erm rucure lope how ha ri neural inveor expec mean-reverion in he po pread. Indeed, ince each fuure price equal he rading dae expecaion of he delivery dae po price in he ri-neural meaure he curren erm rucure of he fuure pread reveal where inveor expec he po pread o be in fuure. Deecing an invere relaionhip beween curren pread level and fuure lope uppor a negaive relaionhip in he ri neural meaure beween he curren pread level and i fuure 7

8 movemen. Beembinder e al (1995) aemp o dicover ex ane mean reverion in commodiy po price. To dicover he negaive relaionhip in our cae we eimae x x = ς + γ x + ε, (8) L S S where x L and x S are repecively long-end and hor-end pread level in he fuure pread erm rucure and ε i a noie erm. If γ i ignificanly negaive here i evidence ha he po pread i ex ane mean-revering in he ri-neural meaure..4 Coninuou ime conequence Regreion model (7) and (8) allow he empirical examinaion of he mean-revering properie of he po pread in repecively he mare and ri-neural meaure. Moreover by eing wheher he pread proce wih d:=1 i mean-revering we can examine wheher hi ideal coinegraion relaionhip hold in (). If eing indicae mean-reverion hen a coinegraion relaionhip may be uppoed and he pread proce can be modelled direcly. The underlying coninuou ime po pread proce x hould hen follow he coninuou ime verion of equaion (4) in he mare meaure: dx = ( ψ ( ) x ) d + σdw, (9) where i he mean reverion peed, ψ() i a funcion of phyical ime, σ i a conan volailiy and W i a Wiener proce. The oluion of (9) ha he propery ha he variance of he pread x will no blow up aympoically in ime and i uncondiional variance i aionary. The radiional wo price pread proce model doe no poe hee properie. By differencing he wo equaion in (1) we obain he pread of he wo conrac price a where d( S S ) = r( S S ) d+σ ( S, S, ρ) dw, (10) σ( S, S, ρ) = σ S + σ S + σ σ S S ρ i he inananeou volailiy of he pread a ime. Thu he price pread proce 8

9 doe no mean-rever in he ri neural meaure and he andard deviaion of he pread increae over ime and blow up aympoically. We do no ee hi for pread beween coinegraed commodiy price in hiorical mare daa (Villar and Jouz, 006). Commodiy and equiy pread procee are differen however. In he ri neural meaure any radable equiy porfolio wihou dividend paymen - including pread - hould grow a he ri-free rae. Thu non-dividend paying oc pread will no be mean-revering in hi meaure. However in he cae of phyical commodiie, epecially hoe commodiie which canno be ored, he ri-neural drif mu encompa ome form of ochaic convenience yield. We ee hi empirically in he mean-revering properie of ome commodiy price (Schwarz, 1997). Becaue of convenience yield pread procee for commodiie can be quie differen from hoe for equiie. 3. Modelling he Spread Proce We have een ha we can model he pread proce direcly uing an Ornein-Uhlenbec proce if a coinegraion relaionhip exi beween he wo underlying ae price. If we wih o price coningen claim on he pread we mu re-pecify (9) in he ri neural meaure. We now preen direc model of he po and fuure pread procee wih one and wo facor. 3.1 One facor model Fir conider a one-facor model of he po pread in he ri neural meaure pecified by dx = ( θ x ) d+ σ dw, (11) where θ i a conan which repreen he long-run mean of he pread proce. For impliciy we aume ha θ doe no depend on ime. Solving (11) given he aring ime v and pread poiion x v we obain 9

10 ( v) ( v) x = xve + θ[1 e ] + e σe dw (1) which follow a normal diribuion a ime wih mean a x e e ( v) ( v) = v + θ[1 ] (13) v and andard deviaion b ( v) 1 e = σ. (14) Thu b σ a o ha he pread andard deviaion end o a conan aympoically. Define F(,T,x ) a he fuure pread ( he pread of wo fuure price) of mauriy T oberved in he mare a ime when he po pread i x. In he ri neural meaure he po pread proce x mu aify he no arbirage condiion E[ x x ] = F(,T,x ), (15) T i.e. in he abence of arbirage he condiional expecaion wih repec o he po pread x of he ou-urn po pread a T in he ri-neural meaure i he fuure pread oberved a ime <T. Thi mu hold becaue i i cole o ener a fuure pread (long one fuure and hor he oher). From (1) and he no arbirage conrain (15) we have FT x = xe + e (16) T ( ) T ( ) (, ; ) θ[1 ] From Io lemma i follow ha he fuure pread F(,T;x ) wih fixed mauriy dae T aifie ( ) d F F F T (,T;x ) = d + d = e σ d x x W, (17) i.e. he fuure pread i a maringale and i volailiy decay exponenially in ime o mauriy (T-). 3. Two facor model 10

11 The mean revering po pread in (11) mainly reflec he hor o medium erm properie of he fuure pread (Gabillon, 1995) wih he volailiy of he fuure pread decaying exponenially wih ime o mauriy. Thi i a erm rucure which i no flexible enough o mach he volailiy erm rucure oberved in he mare. The po pread uually ha an eimaed rong mean-reverion peed o ha he one facor model eimaed volailiie of fuure pread wih mauriie longer han or 3 year are quie cloe o zero while he oberved pread normally have quie noiceable volailiy. Thu anoher facor y i needed o reflec he long-end movemen of he fuure pread erm rucure which relae o movemen of fundamenal uch a orage or hipping co change beween he wo commodiie. For impliciy and conien wih mare equilibrium we will aume ha he long-run mean of he y proce i zero in he ri neural meaure. In he ri neural meaure he underlying po pread proce x and he long-run facor y follow dx = ( θ + y x ) d+ σ dw dy = yd+ σ dw EdWW d = ρd, (18) where he laen facor y i a 0 mean-revering proce (if i poiive) repreening he long end of he pread erm rucure. We can inerpre he dynamic of he po pread pecified by (18) a revering o a ochaic long run mean θ+y. Since he volailiy of he long end pread i uually much maller han ha of he hor end σ hould be quie mall. Since fundamenal (e.g. orage co) have lower peed of adjumen, we can expec o be much maller han. Obviouly he long-end movemen of he pread hould no be a priori conrained o be mean-revering and i could be Wiener proce lie in ome circumance. Given σ, he maller he value he cloer he y proce i o a Wiener proce. Gabillon (1995) conidered a imilar model o (17) for oil fuure conrac price. Appendix 1 how ha he oluion of (18) in he ri neural meaure i 11

12 x ( v) ( v) y v ( v) ( v) = xe v + θ[1 e ] + [ e e ] σ ( u) ( u) ( ) + + W v v [ e e ] d ( u) e σ dw( ). (19) Define: σ ( v) A1 : = [1 e ] 1 1 A e e e ( v) ( ) ( )( ) : ( [1 ] [1 v + v = + ] [1 ]) ( + ) ( ) σσ 1 1 A e e ( + )( v) ( v) 3 : = ( [1 ] [1 ]). + σ (0) Then he andard deviaion of x a ime i and var(x) i b = A + A + ρ A (1) 1 3 σ ρσσ (1 + / ) ( + ) σ + + () aympoically if i no zero. Thi i again becaue boh he x and y procee are mean-revering. If i zero, A become ' 1 ( v) ( v) A = {( v) + [1 e ] [1 e ]} σ. (3) In hi cae he andard deviaion of he pread will grow wih ime and blow up aympoically. However, ince σ i uually quie mall, he peed of growh of he andard deviaion i alo quie mall wheher or no i zero. Thi i conien wih he noion menioned in Mbanefo (1997) ha he pread andard deviaion grow much more lowly han i underlying wo leg. In ummary he mean-revering wo facor model cover wo cae of long-end movemen: 1) a aionary y facor and ) a Brownian moion y facor. From he no arbirage conrain (15) and (19), we have y FTx xe e e e (, ; ) T ( ) [1 T ( ) ] [ ( T ) T ( ) = + θ + ] 1. (4) From Io lemma i follow ha he ri neural fuure pread F(,T) proce wih

13 fixed mauriy dae T aifie (, ) T ( ) [ ( T ) d T e d e e T ( F = σ W+ ) ] σdw (5) and i hu a maringale. I volailiy i compoed of wo par: he fir relaing o he one facor model and he econd o he long run facor y. A a reul, if i much maller han he volailiy of he long-erm fuure pread will end o decay only lowly o zero over ime. 3.3 Spread proce in he mare meaure We will need a ri-adjued verion in order o calibrae he model preened above o mare daa. If we pecify a ri premium proce for x and y hen he drif par of our model can incorporae hee ri premia in he mare meaure (Duffie, 1988). Previou udie aume conan ri premia when modelling Ornein-Uhlenbec procee (ee, e.g. Hull & Whie (1990) and Schwarz (1997)) and o will we. Thu, he ingle-facor model for he po pread proce in he mare meaure follow dx = [ ( θ x ) + λ] d+ σdw, (6) where λ i he ri premium. The wo-facor model in he mare meaure follow dx = [ ( θ + y x ) + λ] d+ σdw dy = ( y + λ ) d+ σ dw EdWW d = ρd, (7) where λ and λ are he ri premia of he x and y procee repecively. Again uing Io lemma on he ri adjued verion of (16) and (4) wih (6) and (7) we obain he fuure pread proce in he mare meaure for boh model. For he one facor model he fuure pread wih a fixed mauriy dae T follow T ( ) T ( ) d (, T) = λe d+ e σd F W. (8) For he wo facor model (8) become 13

14 λ F( ) = λ + [ ] + T ( ) ( T ) T ( ) T ( ) d,t e d e e d e d + ( T ) ( T ) [ e e ] σ dw. σ W (9) From (8) and (9) he fuure pread wih fixed mauriy dae are no mean-revering in he mare meaure. Alhough he model preened above are for po pread, i i no eay o oberve direcly he po price of a commodiy and inveor ypically ue he neare mauriy fuure price o repreen he po price (Clewlow and Sricland, 1999). Bu ince he fuure pread wih fixed mauriy dae i no mean-revering for our model hi mae i difficul o eimae he mean-reverion parameer of he po pread. However we will now how ha he fuure pread wih conan ime o mauriy τ : = T i mean-revering in our model which can be ued o deermine he mean-reverion peed of he po pread. Uing Io Lemma we find (ee Appendix ) ha he proce for he fuure pread wih a conan ime o mauriy can be pecified in he mare meaure a follow. One facor model: τ λe τ df(, + τ) = [ θ + F(, + τ)] d+ e σdw. (30) Two facor model: τ τ λe df(, + τ) = [ θ + ye + + φλ F(, + τ)] d+ σ3dw, (31) where e φ : = τ e τ and σ : = e σ + θ σ + ρθe σσ are conan. τ τ 3 From (30) and (31) a fuure pread wih conan ime o mauriy i mean-revering in boh he one and wo facor model wih he ame mean-reverion peed a he po proce. Noe ha when τ 0, (30) and (31) converge o (8) and (9) repecively. 3.4 Calibraion 14

15 In energy mare conrac mauriie are uually ordered by monhly dae. For example, crude oil fuure conrac raded on he NYMEX are ordered by 30 conecuive fuure monh and hen by quarer up o 7 year. In order o chec he mean-revering propery of he po pread we can ue he 1monh (ime o mauriy) fuure pread wih a monhly obervaion inerval for daa. We ue maximum-lielihood eimaion (MLE) on he panel daa of fuure pread curve in he piri of Chen and Sco (1993) and Pearon and Sun (1994). Thi mehod i commonly ued in fixed income yield curve modelling (e.g. Duffie and Singleon, 1997; Dai, Singleon and Yang, 006). Recenly hi echnique ha alo been ued in he eimaion of convenience yield curve model (Caau and Collin-Dufrene, 005). Since he ae variable are no direcly oberved in our daa e, he Chen and Sco approach pecifie hee laen variable by olving expreion for ome ecuriie which are arbirarily aumed o be priced wihou error in he mare. The remaining ecuriie are aumed o be priced wih meauremen error. To illurae he mehod le F(,T 1 ) o F(,T 5 ) repreen 5 fuure pread available o deermine he model parameer. For he one facor model we uppoe he fir fuure pread a ime i oberved wihou pricing error bu F(,T ) o F(,T 5 ) are priced wih error. Thu he model eimaion equaion are FT (, ) = C+ Dx FT (, ) = C+ Dx+ u FT (, ) = C+ Dx+ u FT (, ) = C+ Dx+ u FT (, ) = C+ Dx+ u, (3) where C i := θ ( i ) [1 T e ], D i := T ( i ) e and u o u 5 are join normally diribued pricing error. The log-lielihood funcion for all fuure pread a ime i given by L : = lnd + lnl + lnl, (33) e 1 where ln L i he log lielihood of he ae variable x (aen a he one monh fuure pread F(,T 1 )) a ime and ln L e i he log lielihood of he oher ecuriie F(,T ) o F(,T 5 ) wih 15

16 1 1 1 ( x xm) ln L : = ln( π ) ln( V) V Δ 1 e V : = σ Δ λ Δ xm : = x 1e + ( θ + )(1 e ) e ' 1 L : = ln( π ) ln( Ω ) uω u and denoe he (1 monh) obervaion inerval. In (33) D 1 i he coefficien Ti ( ) e in he affine ranformaion (16) from x o F(,T i ) and hu he Jacobian of hi ranformaion i 1/D 1. Since he fir one monh fuure pread i priced wihou error i log-lielihood i deermined by he log-lielihood of he ae variable ln L adjued by he Jacobian muliplier 1/D 1. In (34) V i he variance of he ae variable condiional on x -1, x m i he mean of x condiional on x -1 and Ω i he covariance marix for u. The oal log lielihood i deermine he parameer of he one facor model. L (34), which i maximized o For he wo facor model he correponding expreion are FT (, ) = C+ Dx+ Ey FT (, ) = C+ Dx+ Ey FT (, ) = C+ Dx+ Ey+ u FT (, ) = C+ Dx+ Ey+ u FT (, ) = C+ Dx+ Ey+ u, (35) where C i := ( i ) [1 T θ e ], D i := T ( i ) e and E i := ( Ti ) ( Ti ) [ e e ]. Defining J : D E 1 1 = D E he log-lielihood for he wo facor model i e L : = ln J + lnl + lnl, (36) where 16

17 1 1 1 ( x x ) 1 ( y y ) ln L : ln( ) ln( V ) ln( V ) m m = π x y Vx Vy e V : = + { [1 e ] + [1 e ] [1 e } V x y Δ 1 1 Δ 1 Δ ( + ) Δ σ + ( ) 1 e : = σ ( ρσ σ 1 1 e + ( + ) Δ Δ + [ (1 ) (1 )] Δ e σ : Δ Δ 1 1 ( Δ ) (1 Δ Δ ) ( Δ m ) ( ) λ y : = e y + (1 e ) m Δ Δ 1 ) λ λ x = x e + y e e + e e e λ Δ + ( θ + )(1 e ) e ' 1 L : = ln( π ) ln( Ω ) uω u. (37) In preliminary udy we found ha he eimaed correlaion ρ beween he long end and he hor end wa inignifican. Thi mae economic ene in ha he long end movemen are low and driven by fundamenal while he hor-erm movemen which are random, fa and driven by mare rading aciviie. Tha innovaion in he long and hor run hould be uncorrelaed ha been ued o analyze he long-run and hor-run componen of oc price (Fama and French, 1988). Rouledge, Seppi and Spa (000) aer ha he long run movemen of commodiy fuure price hould have zero correlaion wih he hor-run movemen becaue he phyical invenory can regenerae or renew in oc ou period (Corollary 1., p.1304). Thu if he ime o mauriy of he fuure daa are long enough hee correlaion eimae hould be zero. Bu due o daa availabiliy hee eimae may no be eimaed a inignificanly differen from zero and hen he fir ri facor doe no aborb enough of he hor-erm price movemen (Schwarz and Smih, 000). In our wo facor model we will aume he correlaion ρ o be zero. 4. Spread opion pricing and hedging If he underlying ae price follow a Gauian proce he European call and pu 17

18 price wih mauriy T on hi ae can be calculaed repecively a c = B b ( a exp[ π K) b ] + B( a a K) Φ( K ) b (38) p = B b ( a exp[ π K) b ] B( a a K) Φ( K ), (39) b where B i he price of a dicoun bond, a and b are repecively he mean and andard deviaion of he underlying a mauriy, and K i he rie price of he opion (ee Appendix 3). We have een ha he pread diribuion a ime T follow a normal diribuion in boh one facor and wo facor model o ha equaion (38) and (39) can be ued o price he pread opion. 4.1 Pricing wihin fuure price mauriie Opion on he po pread Since he fuure pread i he expecaion of he fuure po pread in he ri-neural meaure he mean of he underlying ae a opion mauriy T can be obained a curren ime a a = F(, T ). (40) Equaion (14) and (1) how b for he one and wo facor model repecively a conan given an iniial ime and a fixed mauriy dae T. A pread opion value wih mauriy T depend on F(,T) hrough (40) inveor can uilize fuure of he ame mauriy dae o hedge. The dela of call and pu on he pread are given repecively by c a K Δ c = = BΦ( ) (41) a b p a K Δ p = = BΦ( ). (4) a b Since a pread can be een a long one ae and hor he oher imulaneouly he dela hedge yield an equal volume hedge, i.e. long and hor he ame value of 18

19 commodiy fuure conrac. No maer how many facor are deemed o drive he fuure price, only he correponding mauriy fuure conrac are uilized o hedge he pread opion providing hey are available in he mare. Opion on he fuure pread Define he opion mauriy a R, he fuure mauriy a T>R and he curren ime a. The fuure pread i a maringale o ha i mean in he ri neural meaure i a [ (, )] (, ) = EQ F R T = F T. (43) I andard deviaion in he one facor model i b ( T R) ( T ) e e = σ (44) and in he wo facor model i b = A + A + ρ A, (45) F F F 1 3 where F σ ( T R) ( T ) A1 = [ e e ] σ 1 1 A e e e e F ( T R) ( T ) T ( R) T ( ) = { [ ] + [ ] ( ) e ( + ) e ( + )( T R) ( + )( T ) [ ]} σσ 1 1 A e e e e F ( + )( T R) ( + )( T ) T ( R) T ( 3 = { [ ] [ ) ]}. + (46) Since R and T are nown uing (38) and (39) b i a conan in boh model and (38) and (39) can be ued o price call and pu opion on he pread a. Again inveor can ue fuure conac wih mauriy T o hedge hee opion poiion wih equal volume hedge given by (41) and (4) repecively. The hedge-raio difference beween hedging a po and a fuure pread opion arie from he difference in he b 19

20 value (compare (14) and (1) wih (44) and (45)). If an opion (on he po pread) ha a mauriy longer han he correponding fuure raded in he mare (which i common for many real opion), inveor canno ue hee mehod o price and hedge he opion. We udy hi iuaion nex. 4. Pricing beyond fuure price mauriie Suppoe a ime we now he value of he ae variable x in he one facor model or x and y in he wo facor model, hen we can foreca he mean pread a ime T> uing hee ae variable. For he one facor model a xe e T ( ) T ( ) = + θ[1 ] (47) and for he wo facor model T ( ) T ( y ) [1 ] [ ( T ) T ( a ) = xe + θ e + e e ]. (48) The andard deviaion b will be he ame a hoe in previou ecion. If no fuure conrac of long enough mauriy are available, inveor mu ue everal hor erm fuure o hedge he long erm opion, i.e. hey need o hedge he individual facor underlying he fuure conrac. There i quie a large lieraure on how o ue hor-erm fuure o hedge long-erm one, e.g. Brennan and Crew (1997), Neuberger (1998) and Hilliard (1999). For he one facor model he call dela on he laen po pread i c a Δ x = =Δc a x e T ( ), (49) where c i he price of a call opion, Δ c i given by (41) and T i he opion mauriy. If he fuure pread F(,T 1 ) i uilized o hedge hi opion poiion he hedge raio i c x Δ = =Δ e e =Δe x F T ( ) T ( 1 ) T ( T1) F c c. (50) 0

21 Similarly he dela for pu on he po pread i p x Δ = =Δ e e =Δ e x F T ( ) T ( 1 ) T ( T1) F p p, (51) where Δ p i given by (4). Applying he wo facor model, inveor mu ue wo horer erm fuure F(,T 1 ) and F(,T ) o hedge long erm opion. Ideally T 1 hould be hor (e.g. 1 monh) and T he longe fuure mauriy available in he mare. The call dela on he laen wo facor x and y are repecively c a Δ x = =Δc a x e T ( ) (5) and c a ( T [ ) T ( Δ ) y = =Δ c e e ] a y. (53) Suppoe he fuure pread F(,T 1 ) and F(,T ) are uilized o hedge he opion poiion. Then o obain he dela neural hedge raio n 1 and n one mu olve Δ + ne + ne = 0 x T ( 1 ) T ( ) 1 Δ + n [ e e ] + n [ e e ] = 0. y ( T1 ) ( T1 ) ( T ) ( T ) 1 (54) Since hi paper focue on long erm opion valuaion and hedging we hould dicu he opion mauriie appropriae for he ue of our valuaion model. The anwer i relaed o he mean-reverion peed. The decay half life ln/ of he mean-revering pread proce can be ued o repreen i mean-reverion rengh. We propoe ha if an opion ime o mauriy i longer han hi half decay ime, our mehodology (boh one facor and wo facor model) i appropriae o evaluae he pread opion. Alo, we expec ha he radiional pread opion model (1) will over-value hee longer erm opion becaue of he variance blow-up phenomenon previouly dicued. Mbanefo (1997) noed ha long-erm pread opion (longer 1

22 han 90 day) will be overvalued if mean-reverion of he pread i no conidered. We noe ha jump and ochaic volailiy are no imporan in deermining heoreical or empirical long erm pread opion price (Bae, 1996; Pan, 003) which allow model of hi paper o remain parimoniou. 5. Example 5.1 Crac pread: Heaing oil/ WTI crude oil (CSHC) 4 The crac pread beween heaing oil and WTI crude oil (heaing oil crac pread) repreen he profi from refining heaing oil from crude oil, i.e. he price of heaing oil minu he price of crude oil. We have een in Secion ha a relaive deviaion beween he (equilibrium) inpu and oupu price relaionhip could exi for hor period of ime, bu a prolonged large deviaion will lead o he producion of more end produc unil he oupu and inpu price are nearer he long-erm equilibrium relaionhip. Thu we expec he heaing oil crac pread o be mean-revering. Daa The daa for modelling crac pread coni of NYMEX daily fuure price of WTI crude oil (CL) and heaing oil (HO) from January 1984 o January 005. The ime o mauriy of hee fuure range from 1 monh o more han year. In order o e for uni roo, a ingle monhly daa poin i colleced on he fir day of each monh by aing he price of he fuure conrac wih one monh ime o mauriy. For example, if he rading day i 1 February 000 hen he fuure conrac aen for he ime erie i he 1 March 000 conrac. We alo creae a long-end crac pread wih ime o mauriy 1 year. The mehodology i exacly he ame a wih he 1 monh ime erie, bu due o daa unavailabiliy we only ue daa from January 1989 o January 005 o conruc he long-end crac pread. 4 Abbreviaion ued in hi ecion are NYMEX rading code.

23 To calibrae he one-facor and wo-facor pread model uing (3) and (35), we calculae he monhly fuure pread wih 5 fuure conrac from January 1989 o January 005. The ime ep Δ i choen o be 1 monh and he conrac choen are 1 monh, 6 monh, 9 monh, 1 monh and 15 monh ime o mauriy fuure pread. Uni roo and coinegraion e Fir, we conduc he ADF e on he heaing oil and crude oil price uing he longer ime erie. A noed in Secion 3.4 we can e he mean-reverion of he po pread by examining he 1 monh fuure pread. Figure 1 how he 1 monh fuure price of crude oil and heaing oil. Iner Figure 1 abou here Table 1 how he reul of he ADF e of Secion.1 on 1 monh fuure of crude oil and heaing oil ariing from eimaing (5). In order o rejec he hypohei ha a ime erie ha a uni roo he coefficien δ 1 mu be ignificanly negaive. Iner Table 1 abou here From Table 1, we canno rejec he hypohei ha boh crude oil and heaing oil are uni-roo ime erie (cf. Girma and Paulon (1999) and Alexander (1999)). Nex we e he pread ime erie direcly by eimaing (5) o find a very rong mean-revering peed ignifican a he 1% level which ugge ha coinegraion doe exi in he daa. In oher word, he mean-reverion of he pread doe no appear o be caued by he eparae mean-reverion of he heaing and crude oil price bu by he long-run equilibrium (coinegraion) beween hem which mu be conidered in long-run derivaive pricing of he crac pread. Thi agree wih Girma and Paulon (1999). Hiorical pread only offer u heir mare-meaure characeriic bu a regreion relaing he hor erm and long erm pread can give u he ri neural pread. We eimae he regreion equaion (8) uing he 1 year crac pread a he long-end 3

24 fuure pread and he 1 monh crac pread a he hor-end fuure pread. The reul are given in Table and he 1 year and 1 monh crac pread depiced in Figure. Iner Table abou here In Table he eimae of γ i ignificanly le han zero. Since hi e inveigae he mean-revering propery of pread in he ri-neural meaure, we conclude ha an Ornein-Uhlenbec proce i appropriae o model he po proce. Thu boh he ex ane and ex po e give evidence ha he po crac pread i mean-revering in boh he mare and ri neural meaure. Iner Figure abou here Model calibraion In order o calibrae he (one-facor and wo-facor) model we ue he full daa e and employ equaion (3) and (35). From preliminary udy we noiced ha he heaing oil price how a eaonal paern which i inheried in he crac pread. To eliminae he influence of eaonaliy, we ued an equally weighed porfolio of 1 monh and 6 monh fuure (oppoie eaon) o deermine he x facor and a imilar porfolio of 9 monh and 15 monh fuure o deermine he y facor. We hen performed a MLE opimizaion o calibrae he one facor model. Table 3 li he reul. One can ee ha he andard deviaion of he θ, σ and eimae are quie mall, i.e. θ, and σ can be deermined quie preciely, unlie he λ eimae. However we do no need he mare price of ri λ a inpu o opion pricing. The aympoic eimae of he andard deviaion of he pread (when goe o infiniy in (14) i $.03. Figure 3 how he one monh pread and he laen po pread, which are very cloe o each oher. Iner Table 3 abou here Iner Figure 3 abou here 4

25 Iner Table 4 abou here Similar o he iuaion in he one facor model, he mare price of ri λ and λ canno be preciely eimaed in he wo facor model, bu eimae of all he oher parameer (σ, σ,, and θ) can. The aympoic eimae of he andard deviaion of he pread i $.65, which i higher han ha for he one facor model. Thi i eay o underand, ince he one facor model only coun he hor-erm variance of he pread, while he wo facor model ae accoun of boh he long end (fundamenal) and he hor end (rading aciviie). Alo, auming zero correlaion beween he wo facor, we can examine he raio of he long-end and he hor-end variance A 1 :A in (0) a ime goe o infiniy, which i abou 1:1 in hi example. Thu he aympoic variance of he crac pread i nearly equally conribued by hor-end (fir facor) and long-end (econd facor) movemen. Since hi raio i quie high he econd facor i obviouly imporan in derivaive pricing and hould no be omied. Since he one facor model i need in he wo facor model by aing σ, λ and y v (he aring value of he y facor) o be zero, we can compare he difference in log-lielihood core for each daa e o ee wheher he addiional parameer of he wo-facor model provide a aiically ignifican improvemen in ha model abiliy o explain he oberved daa. The relevan e aiic for hi comparion i he chi-quared lielihood raio e (Hamilon, 1994) wih 3 degree of freedom and he 99 h percenile of hi diribuion i Given ha he log-lielihood core increae by abou 1, he improvemen provided by he wo facor model are quie ignifican. Iner Figure 4 abou here Figure 4 how he ime evoluion of he long and hor facor wih correlaion coefficien , which i no ignifican and hu conien wih he aumpion ha he correlaion beween he wo facor i zero. Opion valuaion on he po pread 5

26 The decay half life i abou half a year for he one facor model o ha when valuing an opion longer han half a year he mehodology (uing one facor or wo facor model) in hi paper i appropriae. On 3 January 005 he HO06N (heaing oil fuure wih mauriy July 006) conrac had a value 44. ($/Barrel); on he ame day he CL06N (crude oil fuure wih mauriy July 006) raded a ($/Barrel). By uing he parameer in Table 3 and 4, Table 5 give he European opion value on he po crac pread wih mauriy July 006 on hi dae. For comparion we alo calculae opion value from a model which ignore he coinegraion effec. We can ue he Blac (1976) drifle GBM model o imulae boh crude and heaing oil fuure price pah, and hu calculae he pread opion value 5. The average correlaion coefficien (over a 0 year period) i 0.89 beween heaing and crude oil. We ue call opion value o compare he differen model; i i hen eay o obain he pu value by pu-call pariy. Iner Table 5 abou here Iner Table 6 abou here From Table 5 we can ee ha he opion value from he one facor model i ypically maller han ha from he wo-facor model and he laer i much maller han ha from he Blac model. Since he Blac model doe no conider mean-reverion (he coinegraion) of he pread, i pread diribuion a mauriy i wider han ha of a coinegraed model and hu yield a larger opion value. Pu imply, a non-coinegraed model ignore he long-run equilibrium beween crude and heaing oil price and hu over-price he opion. Since he wo facor model accoun for long-erm pread movemen, i hould yield a wider pread diribuion a mauriy and hu ha a larger opion value han he one facor model. In Table 6, boh he one facor and wo facor model yield an equal volume hedge bu he Blac model doe no. A i well nown, he le dipere he underlying erminal diribuion, he more 5 Since here i no analyical oluion when he rie i no zero one convenien way o calculae he opion value i by Mone Carlo pah imulaion. 6

27 eniive he opion dela are o he rie price 6. Thu he one-facor model yield he mo eniive dela and he Blac model ha he lea eniive dela among he hree model. 5. Locaion pread: Bren / WTI crude oil (LSBW) We define he LSBW locaion pread a he price of WTI crude oil (CL) minu he price of he Bren blend crude oil (ITCO). WTI i delivered in he USA and Bren in he UK. Daa NYMEX daily fuure price of WTI crude oil were decribed in he previou example. The daily Bren fuure price are from January 1993 o January 005. The ime o mauriy of he Bren fuure conrac range from 1 monh o abou 3 year. A in he previou example, monhly daa i ued o e for he uni roo in Bren oil price. We alo creae a monhly long-end LSBW pread wih ime o mauriy of 1 year ranging from January 1993 o January 005. In order o calibrae he one-facor and wo-facor model, we calculae he monhly fuure pread wih 5 mauriie from January 1993 o January 005 a monhly inerval, i.e. he ime ep Δ in (30) and (31) i 1 monh. The 5 conrac involved are 1 monh, 3 monh, 6 monh, 9 monh and 1 monh fuure pread. Uni roo and coinegraion e A from he previou example we now ha he WTI crude oil price follow a uni roo proce, in hi example we need only conduc he ADF e on Bren crude oil price. Figure 5 how he 1 monh fuure price of WTI crude oil and Bren blend. We again ae he 1 monh fuure price a repreenaive of he po price. Iner Figure 5 abou here 6 The eniiviy i defined a he raio of he change of he dela in he change of he rie price. 7

28 Table 7 how he reul of ADF e eimaing (5) on he 1 monh fuure of Bren blend. Similar o WTI crude oil, he Bren blend price i alo a uni roo proce (ince δ 1 i poiive), bu he LSBW locaion pread appear o be a mean-revering proce. Thi again ugge he exience of a long-run equilibrium in he daa. To eimae fuure expecaion of he pread, we eimae he regreion equaion (8). A in he previou example, we ue he 1 year and 1 monh LSBW fuure pread. The reul are lied in Table 8. The 1 year and 1 monh LSBW pread evoluion i depiced in Figure 6. Iner Table 7 abou here Iner Figure 6 abou here Iner Table 8 abou here From Table 8 we ee ha he eimae of γ i rongly negaive, o ha he mare appear o expec he po LSBW pread o be mean-revering in he ri-neural meaure. Hence boh he ex ane and ex po analye uppor ha he po LSBW pread i mean-revering o ha mean-reverion hould be accouned for in opion pricing. Model Calibraion We did no find evidence of eaonaliy in he LSBW pread. Hence we ue he 1 monh fuure pread o bac ou he laen po pread facor for he one facor model. For he wo facor model we ue he 1 monh fuure pread o eimae he hor erm x facor and an equally weighed porfolio coniing of he 9 monh and 1 monh fuure pread o eimae he long erm y facor. Table 9 and 10 li he calibraion reul for he one facor and wo facor model. We ee ha he aympoic andard deviaion of he pread are eimaed o be $1.60 and $3.90 repecively for he one and wo facor model. The raio of long-end o he hor-end variance (A 1 :A in (9)) i 5:1 in he wo facor model, i.e. he long end (econd facor) movemen of he pread accoun for much more variance han (fir 8

29 facor) hor end variaion. Similar o he previou example he eimae of he aympoic andard deviaion from he wo facor model i higher han ha from he one facor model. From he Chi-quared (lielihood raio) e he wo facor model i very ignificanly beer han he one facor model in explaining he oberved LSBW pread daa. The laen po pread facor and he wo facor (x and y) are hown in Figure 7 and 8 repecively. The correlaion beween he wo facor i 0.04 which i again conien wih our zero correlaion aumpion. Iner Table 9 abou here Iner Table 10 abou here Iner Figure 7 abou here Iner Figure 8 abou here Opion valuaion on he po pread The decay half life i abou ix monh from he one facor model. Thu o model an opion longer han ix monh he mehod (one facor or wo facor model) in hi paper hould be ued. On he day of 1 December 003 he ITCO06Z (Bren blend crude oil fuure wih mauriy December 006) conrac had a value 4.6 ($/Barrel); on he ame day he CL06Z (WTI crude oil fuure wih mauriy December 006) raded a 5.69 ($/Barrel). By uing he eimaed parameer in Table 9 and 10, Table 11 how he European opion value on he po pread wih mauriy December 006. Since he opion i on he po pread he hedging fuure mauriie hould be he ame a he opion mauriy December 006. The Bren and WTI crude oil conrac price boh follow uni roo procee o we may imulae boh price o calculae he non-coinegraed Blac model pread opion value. Noe ha he average correlaion beween he Bren blend and WTI crude oil i Iner Table 11 abou here 9

30 Table 11 how ha, imilar o he previou example, he opion value of he one facor model i ypically maller han ha of he wo-facor model and he laer i much maller han he Blac model. A before by ignoring coinegraion he Blac model end o over-value he long erm opion. We obain a imilar paern of dela a in he previou example (ee Table 1) and he explanaion for hi remain he ame. Iner Table 1 abou here 6. Concluion In hi paper we have developed pread opion pricing model in which he wo price leg of he pread are coinegraed. Since he coinegraion relaionhip i imporan for he long-run relaionhip beween he wo price pread opion evaluaion hould ae accoun of hi relaionhip if he opion mauriy i long. Auming a coinegraion relaionhip beween he wo underlying ae, we model he pread proce direcly uing he Ornein-Uhlenbec proce, i.e. we model direcly he dynamic deviaion from he long-run equilibrium which canno be pecified correcly by modelling he wo underlying ae eparaely. We fir pecify ri-neural procee for he pread and hen deermine he mare procee by auming conan ri-premia. We alo propoe wo mehod (ex ane and ex po) o e for mean-reverion of he pread proce. Finally we give analyical oluion for he pread opion price and dela. In order o illurae he heory, we udy wo example opion on crac and locaion pread repecively. Boh mare pread procee are found o be mean-revering, which implie ha heir wo price leg are coinegraed. From lielihood raio e he wo facor model i found o be ignificanly beer han he one facor model in explaining he crac and locaion pread daa. The opion value and Gree from our coinegraion model are quie differen from hoe of andard model bu are conien wih he pracical obervaion of Mbanefo (1997). We are currenly woring on Lévy proce verion of our model which may be more appropriae o commodiy mare uch a ga or 30

31 elecriciy and for horer erm opion mauriie generally. Reference Alexander, C., Correlaion and Coinegraion in Energy Mare, in: Kaminy, V. (Ed.), Managing Energy Price Ri, Second Ediion. Ri Publicaion, London, pp Amin, K., Ng, V., Pirrong, S., Valuing Energy Derivaive, in: Managing Energy Price Ri. Ri Publicaion, London, pp Bae, D., Jump and Sochaic Volailiy: Exchange Rae Procee Implici in Deuche Mar Opion. Review of Financial Sudie 9, Beembinder, H., Coughenour, J., Seguin, P., Smoller, M., Mean-Reverion in Equilibrium Ae Price: Evidence from he Fuure Term Srucure. Journal of Finance 50, Blac, F., The Pricing of Commodiy Conrac. Journal of Financial Economic 3, Blac, F., Schole, M., The Pricing of Opion and Corporae Liabiliie. Journal of Poliical Economy 81, Brennan, M., The Pricing of Coningen Claim in Dicree Time Model. Journal of Finance 34, Brennan, M., Crew N., Hedging Long Mauriy Commodiy Commimen wih Shor-Daed Fuure Conrac, in: Demper, M.A.H., Plia, S.R. (Ed.), Mahemaic of Derivaive Securiie. Cambridge Univeriy Pre, Cambridge, pp Carmona, R., Durrleman, V., 003. Pricing and Hedging Spread Opion. SIAM Review 45, Caau, J., Collin-Dufrene, P., 005. Sochaic Convenience Yield Implied from Commodiy Fuure and Inere Rae. Journal of Finance 60, Chen, R., Sco, L., Maximum Lielihood Eimaion for a Mulifacor Equilibrium Model of he Term Srucure of Inere Rae. Journal of Fixed 31

32 Income 3, Clewlow, L., Sricland, C., Valuing Energy Opion in a One Facor Model Fied o Forward Price. Woring Paper, Univeriy of Technology, Sydney, Auralia. Dai, Q., Singleon, K., Yang, W., 006. Regime Shif in a Dynamic Term Srucure Model of US Treaury Bond Yield. Woring Paper, Sanford Univeriy. Demper, M., Hong, S., 000. Pricing Spread Opion wih he Fa Fourier Tranform, in: Geman, H., Madan, D., Plia, S.R., Vor, T. (Ed.), Mahemaical Finance, Bachelier Congre. Springer-Verlag, Berlin, pp Dicy, D., Fuller, W., Diribuion of he Eimae for Auoregreive Time Serie wih a Uni Roo. Journal of he American Saiical Aociaion 74, Duan, J., Plia, S., 004. Opion Valuaion wih Coinegraed Ae Price. Journal of Dynamic and Economic Conrol 8, Duffie, D., Securiy Mare: Sochaic Model. Academic, Boon, MA. Duffie, D., Singleon, K., An Economeric Model of he Term Srucure of Inere Rae Swap Yield. Journal of Finance 5, Engle, R., Granger, C., Coinegraion and Error Correcion: Repreenaion, Eimaion and Teing. Economeric 55, Fama, E., French, K., Permanen and Temporary Componen of Soc Price. Journal of Poliical Economy 96, Froo, K., Dabora, E., How are Soc Price Affeced by he Locaion of Trade. Journal of Financial Economic 53, Fuller, W., Inroducion o Saiical Time Serie. John Wiley & Son, New Yor. Gabillon, J., Analyzing he Forward Curve, in: Managing Energy Price Ri. Ri Publicaion, London, pp Gibon, R., Schwarz, E., Sochaic Convenience Yield and he Pricing of Oil Coningen Claim. Journal of Finance 45,

33 Girma, P., Paulon, A., Ri Arbirage Opporuniie in Peroleum Fuure Spread. Journal of Fuure Mare 19, Hamilon, J., Time Serie Analyi. Princeon Univeriy Pre. Hilliard, J., Analyic Underlying he Meallgeellchaf Hedge: Shor Term Fuure in a Muliperiod Environmen. Review of Finance and Accouning 1, Hilliard, J., Rei, J., Valuaion of Commodiy Fuure and Opion under Sochaic Convenience Yield, Inere Rae, and Jump Diffuion in he Spo. Journal of Financial and Quaniaive Analyi 33, Hull, J., Whie, A., Pricing Inere-Rae Derivaive Securiie. Review of Financial Sudie 3, Johanen, S., Lielihood-baed Inference in Coinegraed Vecor Auo-regreive Model. Oxford Univeriy Pre, Oxford. Jure, J., Yang, H., 006. Profiing from Mean-Reverion: Opimal Sraegie in he Preence of Horizon and Divergence Ri. Woring Paper, Harvard Buine School. Kir, E., Correlaion in he Energy Mare, in: Managing Energy Price Ri. Ri Publicaion, London, pp Margrabe, W., The Value of an Opion o Exchange One Ae for Anoher. Journal of Finance 33, Mbanefo, A., Co-movemen Term Srucure and he Valuaion of Energy Spread Opion, in: Demper, M.A.H., Plia, S.R. (ed.), Mahemaic of Derivaive Securiie. Cambridge Univeriy Pre, Cambridge, pp Meron, R., Opion Pricing when Underlying Soc Reurn Are Diconinuou. Journal of Financial Economic 3, Neuberger, A., Hedging Long-erm Expoure wih Muliple Shor-erm Fuure Conrac. Review of Financial Sudie 1, Pan, J., 00. The Jump-ri Premia Implici in Opion: Evidence from An Inegraed Time-erie Sudy. Journal of Financial Economic 63, Pearon, N., Sun, N., Exploiing he Condiional Deniy in Eimaing he Term 33

34 Srucure: An Applicaion o he Cox, Ingeroll and Ro Model. Journal of Finance 49, Rouledge, B., Seppi, D., Spa, C., 000. Equilibrium Forward Curve for Commodiie. Journal of Finance 55, Rubinein, M., Somewhere Over he Rainbow, RISK 4, Samuelon, P., Theoreical Noe on Trade Problem. Review of Economic and Saiic 46, Schwarz, E., The Sochaic Behaviour of Commodiy Price: Implicaion for Valuaion and Hedging. Journal of Finance 5, Schwarz, E., Smih, J., 000. Shor-erm Variaion and Long-erm Dynamic in Commodiy Price. Managemen Science 46, Shimo, D., Opion on Fuure Spread: Hedging, Speculaion, and Valuaion. Journal of Fuure Mare 14, Villar, J., Jouz, F., 006. The Relaionhip beween Crude Oil and Naural Ga Price. Woring Paper, Deparmen of Economic, The George Wahingon Univeriy. Wilcox, D., Energy Fuure and Opion: Spread Opion in Energy Mare. Goldman Sach & Co., New Yor. 34

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Fortified financial forecasting models: non-linear searching approaches

Fortified financial forecasting models: non-linear searching approaches 0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: non-linear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,

More information

How has globalisation affected inflation dynamics in the United Kingdom?

How has globalisation affected inflation dynamics in the United Kingdom? 292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he

More information

Modeling Energy American Options in the Non-Markovian Approach

Modeling Energy American Options in the Non-Markovian Approach Modeling Energy American Opion in he Non-Markovian Approach Valery Kholodnyi Vienna Auria 06.05.015 VERBUND AG www.verbund.com Ouline Ouline Inroducion Mehodology he Non-Markovian Approach Modeling Energy

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

A Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting

A Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting A Comparaive Sudy of Linear and Nonlinear Model for Aggregae Reail Sale Forecaing G. Peer Zhang Deparmen of Managemen Georgia Sae Univeriy Alana GA 30066 (404) 651-4065 Abrac: The purpoe of hi paper i

More information

Equity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 206-5861

Equity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 206-5861 Equiy Valuaion Uing Muliple Jing Liu Anderon Graduae School of Managemen Univeriy of California a Lo Angele (310) 206-5861 jing.liu@anderon.ucla.edu Doron Niim Columbia Univeriy Graduae School of Buine

More information

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics .4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).

More information

VOLATILITY DYNAMICS OF NYMEX NATURAL GAS FUTURES PRICES

VOLATILITY DYNAMICS OF NYMEX NATURAL GAS FUTURES PRICES VOLATILITY DYNAMICS OF NYMEX NATURAL GAS FUTURES PRICES Hiroaki Suenaga Reearch Fellow School of Economic and Finance Curin Buine School Curin Univeriy of Technology Aaron Smih Aian Profeor Deparmen of

More information

New Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and

New Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and New Evidence on Muual Fund Performance: A Comparion of Alernaive Boorap Mehod David Blake* Trian Caulfield** Chrio Ioannidi*** and Ian Tonk**** June 2014 Abrac Thi paper compare he wo boorap mehod of Koowki

More information

Heat demand forecasting for concrete district heating system

Heat demand forecasting for concrete district heating system Hea demand forecaing for concree diric heaing yem Bronilav Chramcov Abrac Thi paper preen he reul of an inveigaion of a model for hor-erm hea demand forecaing. Foreca of hi hea demand coure i ignifican

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Subsistence Consumption and Rising Saving Rate

Subsistence Consumption and Rising Saving Rate Subience Conumpion and Riing Saving Rae Kenneh S. Lin a, Hiu-Yun Lee b * a Deparmen of Economic, Naional Taiwan Univeriy, Taipei, 00, Taiwan. b Deparmen of Economic, Naional Chung Cheng Univeriy, Chia-Yi,

More information

Calculation of variable annuity market sensitivities using a pathwise methodology

Calculation of variable annuity market sensitivities using a pathwise methodology cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

Trading Strategies for Sliding, Rolling-horizon, and Consol Bonds

Trading Strategies for Sliding, Rolling-horizon, and Consol Bonds Trading Sraegie for Sliding, Rolling-horizon, and Conol Bond MAREK RUTKOWSKI Iniue of Mahemaic, Poliechnika Warzawka, -661 Warzawa, Poland Abrac The ime evoluion of a liding bond i udied in dicree- and

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

Machine Learning in Pairs Trading Strategies

Machine Learning in Pairs Trading Strategies Machine Learning in Pairs Trading Sraegies Yuxing Chen (Joseph) Deparmen of Saisics Sanford Universiy Email: osephc5@sanford.edu Weiluo Ren (David) Deparmen of Mahemaics Sanford Universiy Email: weiluo@sanford.edu

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry

More information

Commodity market modeling and physical trading strategies

Commodity market modeling and physical trading strategies Commodiy mare modeling and physical rading sraegies by Per Einar S. Ellefsen Ingénieur de l Ecole Polyechnique, 8 Submied o he Deparmen of Mechanical Engineering in parial fulfillmen of he requiremens

More information

How Much Can Taxes Help Selfish Routing?

How Much Can Taxes Help Selfish Routing? How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya. Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models

More information

Empirical heuristics for improving Intermittent Demand Forecasting

Empirical heuristics for improving Intermittent Demand Forecasting Empirical heuriic for improving Inermien Demand Forecaing Foio Peropoulo 1,*, Konanino Nikolopoulo 2, Georgio P. Spihouraki 1, Vailio Aimakopoulo 1 1 Forecaing & Sraegy Uni, School of Elecrical and Compuer

More information

Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities

Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for

More information

I S T H E A U S T R A L I A N F O R E X M A R K E T E F F I C I E N T? A T E S T O F T H E F O R W A R D R A T E

I S T H E A U S T R A L I A N F O R E X M A R K E T E F F I C I E N T? A T E S T O F T H E F O R W A R D R A T E I S T H E A U S T R A L I A N F O R E X M A R K E T E F F I C I E N T? A T E S T O F T H E F O R W A R D R A T E U N B I A S E D N E S S H Y P O T H E S I S By D.E. Allen and Paul Taco School o Accouning,

More information

THE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELDS (*)

THE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELDS (*) THE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELDS () Andrés García Miranes a, Javier Población b and Gregorio Serna c ( ) a IES Juan del Enzina, c/ Ramón y Cajal, 4 León, Spain. e-mail:

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in

More information

Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of the Forward Curve in Forecasting Oil Prices Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

More information

Time Series Analysis Using SAS R Part I The Augmented Dickey-Fuller (ADF) Test

Time Series Analysis Using SAS R Part I The Augmented Dickey-Fuller (ADF) Test ABSTRACT Time Series Analysis Using SAS R Par I The Augmened Dickey-Fuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed

More information

Pricing Single Name Credit Derivatives

Pricing Single Name Credit Derivatives Pricing Single Name Credi Derivaives Vladimir Finkelsein 7h Annual CAP Workshop on Mahemaical Finance Columbia Universiy, New York December 1, 2 Ouline Realiies of he CDS marke Pricing Credi Defaul Swaps

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE

More information

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

More information

PROFITS AND POSITION CONTROL: A WEEK OF FX DEALING

PROFITS AND POSITION CONTROL: A WEEK OF FX DEALING PROFITS AND POSITION CONTROL: A WEEK OF FX DEALING Richard K. Lyon U.C. Berkeley and NBER Thi verion: June 1997 Abrac Thi paper examine foreign exchange rading a he dealer level. The dealer we rack average

More information

Can Individual Investors Use Technical Trading Rules to Beat the Asian Markets?

Can Individual Investors Use Technical Trading Rules to Beat the Asian Markets? Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weak-form of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information

Estimating Time-Varying Equity Risk Premium The Japanese Stock Market 1980-2012

Estimating Time-Varying Equity Risk Premium The Japanese Stock Market 1980-2012 Norhfield Asia Research Seminar Hong Kong, November 19, 2013 Esimaing Time-Varying Equiy Risk Premium The Japanese Sock Marke 1980-2012 Ibboson Associaes Japan Presiden Kasunari Yamaguchi, PhD/CFA/CMA

More information

INTRODUCTION TO FORECASTING

INTRODUCTION TO FORECASTING INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren

More information

Efficient Pricing of Energy Derivatives

Efficient Pricing of Energy Derivatives Efficien Pricing of Energy Derivaives Anders B. Trolle EPFL and Swiss Finance Insiue March 1, 2014 Absrac I presen a racable framework, firs developed in Trolle and Schwarz (2009), for pricing energy derivaives

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

The International Investment Position of Jamaica: An Estimation Approach

The International Investment Position of Jamaica: An Estimation Approach WP/04 The Inernaional Invemen Poiion of Jamaica: An Eimaion Approach Dane Docor* Economic Informaion & Publicaion Deparmen Bank of Jamaica Ocober 2004 Abrac Thi paper eek o inroduce he inernaional invemen

More information

Chapter 8 Student Lecture Notes 8-1

Chapter 8 Student Lecture Notes 8-1 Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

Two-Group Designs Independent samples t-test & paired samples t-test. Chapter 10

Two-Group Designs Independent samples t-test & paired samples t-test. Chapter 10 Two-Group Deign Independen ample -e & paired ample -e Chaper 0 Previou e (Ch 7 and 8) Z-e z M N -e (one-ample) M N M = andard error of he mean p. 98-9 Remember: = variance M = eimaed andard error p. -

More information

Variance Swap. by Fabrice Douglas Rouah

Variance Swap. by Fabrice Douglas Rouah Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1 Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

More information

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

More information

Explore the Application of Financial Engineering in the Management of Exchange Rate Risk

Explore the Application of Financial Engineering in the Management of Exchange Rate Risk SHS Web o Conerence 17, 01006 (015) DOI: 10.1051/ hcon/01517 01006 C Owned by he auhor, publihed by EDP Science, 015 Explore he Applicaion o Financial Engineering in he Managemen o Exchange Rae Rik Liu

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Markit Excess Return Credit Indices Guide for price based indices

Markit Excess Return Credit Indices Guide for price based indices Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semi-annual

More information

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100... Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.

More information

An Optimal Strategy of Natural Hedging for. a General Portfolio of Insurance Companies

An Optimal Strategy of Natural Hedging for. a General Portfolio of Insurance Companies An Opimal Sraegy of Naural Hedging for a General Porfolio of Insurance Companies Hong-Chih Huang 1 Chou-Wen Wang 2 De-Chuan Hong 3 ABSTRACT Wih he improvemen of medical and hygienic echniques, life insurers

More information

Stock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S.

Stock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S. Valuaion of Sock Opion Gran Under Muliple Severance Rik GURUPDESH S. PANDHER i an aian profeor in he deparmen of finance a DePaul Univeriy in Chicago, IL. gpandher@depaul.edu GURUPDESH S. PANDHER Execuive

More information

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance

More information

Dividend taxation, share repurchases and the equity trap

Dividend taxation, share repurchases and the equity trap Working Paper 2009:7 Deparmen of Economic Dividend axaion, hare repurchae and he equiy rap Tobia Lindhe and Jan Söderen Deparmen of Economic Working paper 2009:7 Uppala Univeriy May 2009 P.O. Box 53 ISSN

More information

Chapter 9 Bond Prices and Yield

Chapter 9 Bond Prices and Yield Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

The Twin Agency Problems in Corporate Finance - On the basis of Stulz s theory -

The Twin Agency Problems in Corporate Finance - On the basis of Stulz s theory - The Twin Agency Problem in Corporae Finance - On he bai of Sulz heory - Von der Fakulä für Machinenbau, Elekroechnik und Wirchafingenieurween der Brandenburgichen Technichen Univeriä Cobu zur Erlangung

More information

The Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of

The Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of Prof. Harris Dellas Advanced Macroeconomics Winer 2001/01 The Real Business Cycle paradigm The RBC model emphasizes supply (echnology) disurbances as he main source of macroeconomic flucuaions in a world

More information

Data Analysis Toolkit #7: Hypothesis testing, significance, and power Page 1

Data Analysis Toolkit #7: Hypothesis testing, significance, and power Page 1 Daa Analyi Toolki #7: Hypohei eing, ignificance, and power Page 1 The ame baic logic underlie all aiical hypohei eing. Thi oolki illurae he baic concep uing he mo common e, - e for difference beween mean.

More information

Optimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference

Optimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference Opimal Longeviy Hedging Sraegy for Insurance Companies Considering Basis Risk Draf Submission o Longeviy 10 Conference Sharon S. Yang Professor, Deparmen of Finance, Naional Cenral Universiy, Taiwan. E-mail:

More information

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why Did the Demand for Cash Decrease Recently in Korea? Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

More information

OPTIMAL BATCH QUANTITY MODELS FOR A LEAN PRODUCTION SYSTEM WITH REWORK AND SCRAP. A Thesis

OPTIMAL BATCH QUANTITY MODELS FOR A LEAN PRODUCTION SYSTEM WITH REWORK AND SCRAP. A Thesis OTIMAL BATH UANTITY MOELS FOR A LEAN ROUTION SYSTEM WITH REWORK AN SRA A Thei Submied o he Graduae Faculy of he Louiiana Sae Univeriy and Agriculural and Mechanical ollege in parial fulfillmen of he requiremen

More information

LEASING VERSUSBUYING

LEASING VERSUSBUYING LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss

More information

Chapter 1.6 Financial Management

Chapter 1.6 Financial Management Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005 FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

More information

The Chase Problem (Part 2) David C. Arney

The Chase Problem (Part 2) David C. Arney The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

NASDAQ-100 Futures Index SM Methodology

NASDAQ-100 Futures Index SM Methodology NASDAQ-100 Fuures Index SM Mehodology Index Descripion The NASDAQ-100 Fuures Index (The Fuures Index ) is designed o rack he performance of a hypoheical porfolio holding he CME NASDAQ-100 E-mini Index

More information

The Kinetics of the Stock Markets

The Kinetics of the Stock Markets Asia Pacific Managemen Review (00) 7(1), 1-4 The Kineics of he Sock Markes Hsinan Hsu * and Bin-Juin Lin ** (received July 001; revision received Ocober 001;acceped November 001) This paper applies he

More information

Order Flows, Delta Hedging and Exchange Rate Dynamics

Order Flows, Delta Hedging and Exchange Rate Dynamics rder Flows Dela Hedging and Exchange Rae Dynamics Bronka Rzepkowski # Cenre d Eudes rospecives e d Informaions Inernaionales (CEII) ABSTRACT This paper proposes a microsrucure model of he FX opions and

More information

FUTURES AND OPTIONS. Professor Craig Pirrong Spring, 2007

FUTURES AND OPTIONS. Professor Craig Pirrong Spring, 2007 FUTURES AND OPTIONS Professor Craig Pirrong Spring, 2007 Basics of Forwards and Fuures A forward conrac is an agreemen beween a buyer and a seller o ransfer ownership of some asse or commodiy ( he underlying

More information

Price elasticity of demand for crude oil: estimates for 23 countries

Price elasticity of demand for crude oil: estimates for 23 countries Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh

More information