Implementation of the Black, Derman and Toy Model

Transcription

1 Implemenaion of he lack Deman and oy Model Semina Financial Engineeing o.univ.-pof. D. Engelbe J. Dockne Univesiy of Vienna Summe em 003 Chisoph Klose Li Chang Yuan

2 Implemenaion of he lack Deman and oy Model Page Conens of his pape Conens Inoducion o em Sucue Models em Sucue Equaion fo Coninuous ime Oveview - asic Pocesses of One-Faco Models he lack Deman and oy Model (D) Chaaceisics Modeling of an aificial Sho-Rae Pocess... 8 Valuing Opions on easuy onds he D-Model and Realiy Implemenaion and Applicaion of he D-Model Lis of Symbols and Abbeviaions Refeences Suggesions fo fuhe eading... 19

3 Implemenaion of he lack Deman and oy Model Page 3 1. Inoducion o em Sucue Models Inees ae deivaives ae insumens ha ae in some way coningen on inees aes (bonds swaps o jus simple loans ha sa a a fuue poin in ime). Such secuiies ae exemely impoan because almos evey financial ansacion is exposed o inees ae isk and inees ae deivaives povide he means fo conolling his isk. In addiion same as wih ohe deivaive secuiies inees ae deivaives may also be used o enhance he pefomance of invesmen pofolios. he ineesing and cucial quesion is wha ae hese insumens woh on he make and how can hey be piced. In analogy o sock opions inees ae deivaives depend on hei undelying i.e. he inees ae. ond pices depend on he movemen of inees aes so do bond opions. he quesion of picing coningen claims on inees aes comes down o he quesion of how he undelying can be modeled. Fuue values ae unceain bu wih he help of sochasic models i is possible o ge infomaion abou possible inees aes. he ool needed is he em sucue he evoluion of spo aes ove ime. One has o conside boh he em sucue of inees aes and he em sucue of inees ae volailiies. em sucue models also known as yield cuve models descibe he pobabilisic behavio of all aes. hey ae moe complicaed han models used o descibe a sock pice o an exchange ae. his is because hey ae concened wih movemens in an enie yield cuve no wih changes o a single vaiable. As ime passes he individual inees aes in he em sucue change. In addiion he shape of he cuve iself is liable o change. We have o disinguish beween equilibium models and no-abiage models. In an equilibium model he iniial em sucue is an oupu fom he model in a noabiage model i is an inpu o he model. Equilibium models usually sa wih he assumpion abou economic vaiables and deive a pocess fo he sho-em isk-fee ae. 1 hey hen exploe wha he pocess implies fo bond pices and opion pices. he disadvanage of equilibium models is ha hey do no auomaically fi oday s em sucue. No-abiage models ae designed o be exacly consisen wih oday s em sucue. he idea is based on he isk-neual picing fomula when a bond is valued ove a single peiod of ime wih binominal laice. 1 When () is he only souce of unceainy fo he em sucue he sho ae is modeled by One-Faco Models.

4 Implemenaion of he lack Deman and oy Model Page 4. em Sucue Equaion fo Coninuous ime In ou pape we pefe o use discee ime models because he daa available is always in discee fom and easie o compue. he coninues ime models do no povide us wih useful soluions in he D wold neveheless hey ae he fundamenal and olde pa of inees ae deivaives. Coninuous ime models ae moe anspaen and give a bee undesanding of he D model heefoe we hee give a sho geneal inoducion ino his maeial. We assume ha he following em sucue equaion 3 is given: wih P + P µ P 1 ( ) + ( ) () = 0 P = ; P P σ P = ; P P P = We have a second ode paial diffeenial equaion. I is of couse of gea impoance ha he paial diffeenial equaion above leads o soluions fo inees deivaives. No evey Io pocess can solve he em sucue above. I uns ou ha some Io pocesses which ae affine em sucue models lead o a soluion. he zeo bond pice in ime o mauiy has he following fom: P A( ) ( ) ( ) () = he dif and volailiy have he geneal fom: e whee < µ σ ( ) g() () + h() ( ) c() () + d() If he pice P() above is given we can deive P P and P as he following: P P A( ) ( ) ( ) = = ( ) e = P P P A( = = e ) ( ) ( ) ( ) = P he coninuous ime models can lead o closed fom mahemaical soluions. u in he D case we canno find such a soluion. 3 Rudolf (000) pp.38-40

5 Implemenaion of he lack Deman and oy Model Page 5 ( ) ( ) P A e A P P A = = = ) ( ) ( ) ( Afe subsiuing hese hee vaiables ino he em sucue equaion we ge ( ) ( ) [ ] () ( ) ( ) 0 1 = + + σ µ A Now we can also subsiue µ and σ ino he equaion above and ge ( ) ( ) ( ) ( ) () d h A + () ( ) ( ) ( ) ( ) () 0 1 = + + c g he equaion above holds fo all and unde he following condiion: ( ) ( ) ( ) ( ) () 0 = + d h A ( ) () ( ) ( ) () 0 1 = + + c g he bounday condiions fo solving his zeo bond ae: ( ) 1 = P ( ) 0 = A ( ) 0 = he eason is ha we know he pice of he zeo bond a mauiy so A() and () mus be zeo. hese wo equaions above lead o he Riccai-Poblem fo solving wo non sochasic paial diffeenial equaions. A geneal soluion mehod fo solving he em sucue equaion wih an affine inees model could be o 1. Compae he coefficiens ( ) µ and σ wih he sochasic sho ae pocess o idenify he coefficiens g() h() c() and d(). Subsiue he coefficiens ino he wo Riccai equaions 3. Solve he wo second ode paial diffeenial equaions o ge A() and ()

6 Implemenaion of he lack Deman and oy Model Page 6 3. Oveview - asic Pocesses of One-Faco Models able I: Some basic single-faco models in coninuous ime 4 [ ()] () Vasicek(1977) d = Φ a d + σ dz [Equilibium model Sho ae model] d = a(b-) d + σ dz [ ()] () Cox Ingesoll Ross (1985) d = Φ a d + σ dz [Equilibium model Sho ae model] d = a(b-) d + c dz Ho Lee (1986) [Fis No-Abiage model] d [ F ( 0 ) + σ ] d + dz () = σ d = θ() d + σ dz lack Deman oy (1990) [No-Abiage model lognomal sho ae model] Hull Whie (1990) [No-Abiage model] lack Kaasinski (1991) Heah Jaow Moon (199) [enown as a bidge beween all em sucue models; Fowad ae model] d () + () () = a d σ dz d ln = θ() d + σ dz [ Φ() a() () ] d + () dz () d = σ d = [θ() - a] d + σ dz [ Φ() a() ln () ] d + () dz () d ln = σ df d ln = (θ - a ln ) d + σ dz ( ) = µ ( ) + ( ) ( ) d σ F F dz 4 See Rudolf (000) p.64 and Clewlow Sickland (1998)

7 Implemenaion of he lack Deman and oy Model Page 7 4. he lack Deman and oy Model (D) 4.1. Chaaceisics he em sucue model developed in 1990 by Fische lack Emanuel Deman and William oy is a yield-based model which has poved popula wih paciiones fo valuing inees ae deivaives such as caps and swapions ec. he lack Deman and oy model (hencefoh D model) is a one-faco sho-ae (no-abiage) model all secuiy pices and aes depend only on a single faco he sho ae he annualized one-peiod inees ae. he cuen sucue of long aes (yields on zeo-coupon easuy bonds) fo vaious mauiies and hei esimaed volailiies ae used o consuc a ee of possible fuue sho aes. his ee can hen be used o value inees-ae-sensiive secuiies. Seveal assumpions ae made fo he model o hold: Changes in all bond yields ae pefecly coelaed. Expeced euns on all secuiies ove one peiod ae equal. he sho aes ae log-nomally disibued hee exiss no axes o ansacion coss. As wih he oiginal Ho and Lee model he model is developed algoihmically descibing he evoluion of he em sucue in a discee-ime binominal laice famewok. Alhough he algoihmic consucion is ahe opaque wih egad o is assumpions abou he evoluion of he sho ae seveal auhos have shown ha he implied coninuous ime limi of he D model as we ake he limi of he size of he ime sep o zeo is given by he following sochasic diffeenial equaion 5 : σ ( ) d ln ( ) = θ ( ) ln ( ) d + σ ( ) dz σ ( ) his epesenaion of he model allows o undesand he assumpion implici in he model. he D model incopoaes wo independen funcions of ime θ() and σ() chosen so ha he model fis he em sucue of spo inees aes and he em sucue of spo ae volailiies. In conas o he Ho and Lee and Hull and Whie model in he D epesenaion he sho aes ae log-nomally disibued; wih he esuling advanage ha inees aes canno become negaive. An unfounae consequence of he model is ha fo ceain specificaions of he volailiy funcion σ() 5 See Clewlow and Sickland (1998) p.1

8 Implemenaion of he lack Deman and oy Model Page 8 he sho ae can be mean-fleeing ahe han mean-eveing. I is popula among paciiones paly fo he simpliciy of is calibaion and paly because of is saighfowad analyic esuls. he model fuhemoe has he advanage ha he volailiy uni is a pecenage confiming wih he make convenions. 4.. Modeling of an aificial Sho-Rae Pocess In his chape we descibe a model of inees aes ha can be used o value any inees-ae-sensiive secuiy. In explaining how i woks we concenae on valuing opions on easuy bonds. We wan o examine how he model woks in an imaginay wold in which changes in all bond yields ae pefecly coelaed expeced euns on all secuiies ove one peiod ae equal sho aes a any ime ae lognomally disibued and hee ae neihe axes no ading coss. We can value a zeo bond of any mauiy (poviding ou ee of fuue sho aes goes ou fa enough) using backwad inducion. We simply sa wih he secuiy s face value a mauiy and find he pice a each node ealie by discouning fuue pices using he sho ae a ha node. he em sucue of inees aes is quoed in yields ahe han pices. oday s annual yield y of he N-zeo in ems of is pice S is given by he y ha saisfies he following equaion (1). Similaly he yields (up and down) one yea fom now coesponding o he pices S u and S d ae given by equaion () S = (1) S N u d = () N 1 (1 + y) (1 + y ) u d We wan o find he sho aes ha assue ha he model s em sucue maches oday s make em sucue. Using able II we can illusae how o find sho aes one peiod in he fuue. 6 Mauiy (ys) Yield (%) Yield Volailiy (%) able II A Sample em Sucue 6 Example is aken fom lack Deman and oy (1990) pp.33-39

9 Implemenaion of he lack Deman and oy Model Page 9 he unknown fuue sho aes u and d should be such ha he pice and volailiy of ou wo-yea zeo bond mach he pice and volailiy given in able II. We know ha oday s sho ae is 10 pe cen. Suppose we make a guess fo u and d assuming ha u =14.3 and d =9.79 see Figue I a. A wo-yea zeo has a pice of EUR 100 a all nodes a he end of he second peiod no mae wha sho ae pevails. We can find he one-yea pices by discouning he expeced wo yea pices by u and d ; we ge pices of EUR and EUR see Figue I b. Using equaion () we find ha yields of 14.3 and 9.79 pe cen indeed coespond o hese pices. (e.g =100/1+y u so y u =14.3%). oday s pice is given by equaion (1) by discouning he expeced one-yea ou pice by oday s sho ae i.e. (0.5(87.47)+0.5(91.08))/1.1= We can ge oday s yield fo he wo yea zeo y using equaion (1) wih oday s pice as S. Figue I Finding ou he One-Yea Sho Raes (using a wo-yea zeo) (a) Rae ee (Guessed values) (b) Pice ee (Discouning he pice a mauiy wih u and d o ge he pices in =1; hen discouning hese using oday s sho ae in equaion (1)) 9.79 (c) Yield ee Using equaion o calculae he yields in =1 fo oday : 81.16=100/(1+y) he volailiy of his wo yea yield is defined as he naual logaihm of he aio of he ln one-yea yields : σ = = 19%. Wih he one-yea sho aes we have guessed he wo-yea zeo bond s yield and yield volailiy mach hose in he obseved em sucue of able II. his means ha ou guesses fo u and d wee igh. Had hey been wong we would have found he coec ones by ial and eo. So an iniial sho ae of 10 pe cen followed by equally pobable one-yea sho aes of 14.3 and 9.79 pe cen guaanee ha ou model maches he fis wo yeas of he em sucue.

10 Implemenaion of he lack Deman and oy Model Page 10 Valuing Opions on easuy onds We can use he model o value a bond opion. u befoe we can value easuy bond opions we need o find he fuue pices of he bond a vaious nodes on he ee. Fo ou example shown in Figue II and III we conside a easuy wih a 10 pe cen coupon a face value of EUR 100 and hee yeas lef o mauiy. he applicaion of he D model povided us wih he sho aes which ae shown in Figue II a. Figue II Valuaion of a pofolio of hee zeoes (a) D Sho Raes (b) One-Yea Zeo (c) wo-yea Zeo (d) hee-yea Zeo 110 Fo convenience we can conside his 10 pe cen hee-yea easuy as a pofolio of hee zeo-coupon bonds 7. See Figue II panels (b c d). he ee shown in panel (a) was buil he same way as explained in he pevious secion of his pape o value all zeo s accoding o oday s yield cuve. Figue III a below shows he pice of he 10 pe cen easuy as he sum of he pesen values of he zeoes EUR he ee in Figue III b shows he hee-yea easuy pices obained afe subacing EUR 10 of accued inees on each coupon dae. Figue III hee-yea easuy Values (a) Pesen Value of Pofolio (Fig. II b+c+d = 3-yea-easuy) (b) Pice (Pesen Value of 3-yeaeasuy less accued inees) a one-yea zeo wih face-value of EUR 10 a wo-yea zeo wih a EUR 10 face-value and a hee-yea zeo wih a EUR 110 face-value

11 Implemenaion of he lack Deman and oy Model Page 11 Having found a pice fo he easuy EUR We can easily value opions on his secuiy e.g. a wo-yea Euopean call and pu suck a EUR 95. he pocess is shown in Figue IV. A expiaion he diffeence beween he bond s pice and he sike pice is calculaed o ge he calls (vice vesa- he pus) value. Is he opion ou of he money is value is zeo. o ge he possible values one-yea befoe expiaion we discoun wih 0.5(1.69) + 0.5(5.) he sho ae fo ha peiod. Fo example = % Figue IV wo-yea Euopean Opions on a hee-yea easuy (a) Pice (b) Rae ee 1.77 (* ou of he money) * (c) Call Value ee max(0; S-X) *.89 0* 0* (d) Pu Value ee max(0; X-S) We have pice Euopean-syle opions by finding hei values a any node as he discouned expeced value one sep in he fuue. Ameican-syle opions can be valued in a simila manne wih lile exa effo. In he beginning of his chape we oulined he appoach aken by lack Deman and oy in hei oiginal publicaion 8 of hei model. hey indicae a ial and eo pocedue wihou poviding any hins o faciliae he guessing game. Jamishidian 9 (1991) impoved he implemenaion of he oiginal D model by implemening saeconingen pices and fowad inducion o consuc he sho inees ae ee. jeksund and Sensland 10 developed an alenaive o Jamshidian s fowad inducion mehod which poves o be even moe efficien. Wih he help of wo fomulas hey povide a closed fom soluion o he calibaion poblem. Given he iniial yield and volailiy cuves a bond pice ee is modeled ha helps o appoximae he sho inees ae ee. he idea is o use infomaion fom he calculaed ee o adjus inpu which is used o geneae a new ee by he wo appoximaion fomulas. Avoiding ieaion pocedues hey assue shoe compuaion imes and accuae esuls. 8 lack Deman and oy A One-Faco Model of Inees Raes and Is Applicaion o easuy ond Opions Financial Analyss Jounal Jan-Feb 1990 pp Jamshidian Fowad Inducion and Consucion of Yield Cuve Diffusion Models Jounal of Fixed Income June 1991 pp jeksund Sensland Implemenaion of he lack-deman-oy Model Jounal of Fixed Income Vol.6() Sep 1996 pp.66-75

12 Implemenaion of he lack Deman and oy Model Page he D-Model and Realiy he D model offes in compaison o he Ho-Lee model moe flexibiliy. In he case of consan volailiy he expeced yield of he Ho-Lee model moves exacly paallel bu he D model allows moe complex changes in he yield-cuve shape. Figue V sho ae 11 (00) (up) A(up) (up) (down) A(down) (down) =0 =1 he sho ae can be calculaed by ( ) A( down) ( up) ( down) A up ( ) A( down) ( up) ( down) ( up) ( down) ( up) ( down) A up = Fom he equaion above (igh side) we can easily deive he sensiiviy of ond pices of diffeen mauiies o changes in he sho aes. 1 he sensiiviy of he sho aes is songly dependen on he shape of he yield cuve. Upwad sloping em sucue end o poduce an elasiciy above 1. I mus be sessed ha using D which is a one-faco model does no mean ha he yield cuve is foced o move paallel. he cucial poin is ha only one souce of unceainy is allowed o affec he diffeen aes. In conas o linealy independen aes a one faco model implies ha all aes ae pefecly coelaed. Of couse aes wih diffeen mauiy ae no pefecly coelaed. One faco models ae bually simplifying eal life. So wo (o hee) faco models would be a bee choice o mach he aes See Rebonao (00) p.65 1 See Rebonao (00) p Fancke (000) p.11-14

13 Implemenaion of he lack Deman and oy Model Page 13 Mainly hee advanages using one faco models ahe han wo o hee faco models can be menioned: 1. I is easie o implemen. I akes much less compue ime 3. I is much easie o calibae he ease of calibaion o caps is one of he advanages in he case of he D model. I is consideed by many paciiones o oupefom all ohe one-faco models. he D model suffes fom wo impoan disadvanages 14 : Subsanial inabiliy o handle condiions whee he impac of a second faco could be of elevance because of he one-faco model Inabiliy o specify he volailiy of yields of diffeen mauiies independenly of fuue volailiy of he sho ae An exac mach of he volailiies of yields of diffeen mauiies should no be expeced and even if acually obseved should be egaded as a lile moe han fouious. Anohe disadvanage of he D model is ha he fundamenal idea of a sho ae pocess ha follows a mean evesion does no hold unde he following cicumsance. If he coninuous ime D isk neual sho ae pocess has he fom: d ln whee: { } d + () dz() () θ () + f [ ψ () ln () ] = σ u ().. is he median of he sho ae disibuion a ime θ f ψ () u = ln lnσ = () () = lnu() () Fo consan volailiy ( σ = cons ) f = 0 he D model does no display any mean evesion. Fuhemoe we have o noe ha volailiy em sucues ae no necessay he same fo diffeen makes. he D model is fo example used in he US- and he Euopean make whee cap volailiies ae declining ahe smoohly possibly afe an iniial hump. his is no he case fo he Japanese cap volailiies which decease apidly wih opion mauiy. See Figue VI. 14 See Rebonao (00) p.68

14 Implemenaion of he lack Deman and oy Model Page 14 Figue VI - Diffeen cap volailiy em sucue fo he Euopean and he Japanese makes I would no be appopiae o use he D model in he Japanese make because of he shap decease in volailiy which would imply ha we have moe infomaion abou a fuue ime peiod han abou an ealie ime peiod which is no coec Implemenaion and Applicaion of he D-Model In his chape we wan o ouline he implemenaion of he D model using a speadshee. o mach ou model o he obseved em sucue we need a speadshee package ha includes an equaion-solving ouine 16. he speadshee equaion akes advanage of he fowad equaion and is an appopiae mehod when he numbe of peiods is no lage. A simple way 17 of wiing ou model is o assume ha he values in he sho ae ( ks ) laice ae of he fom = a ks k e bk s We index he nodes of a sho ae laice accoding o he foma (ks) whee k is he ime (k=01 n) and s is he sae (s=01 k). Hee a k is a measue of he aggegae dif b k epesens he volailiy of he logaihm of he sho ae fom ime k-1 o k. Many paciiones choose o fi he ae sucue only 18 holding he fuue sho ae volailiy consan 19. So in he simples vesion of he model he values of b k ae all equal o one value b. he a k s ae hen assigned so ha he implied em sucue maches he obseved em sucue. 15 Fo fuhe deails see Fancke (000) pp Fo ou demonsaion we used Micosof Excel and is Solve 17 See Luenbege (1998) p Fo a jusificaion of his see Clewlow and Sickland (1998) Secion 7.7. (pp.-3) 19 he convegen coninuous ime limi as shown on page 7 heefoe educes o he following equaion : d ln = θ() d + σ dz. his pocess can be seen as a lognomal vesion of he Ho and Lee model.

15 Implemenaion of he lack Deman and oy Model Page 15 (Sep 1) We ge he daa of a yield cuve and pase i ino Excel. 0 o mach i o ou D model we have o make assumpions concening volailiy. Fo ou case we suppose o have measued he volailiy o be 0.01 pe yea which means ha he sho ae is likely o flucuae abou one pecenage poin duing a yea. (Inpus ae shown wih gey backgound in Figue VII a) (Sep ) We inoduce a ow fo he paamees a k. hese paamees ae consideed vaiable by he pogam. ased on hese paamees he sho ae laice is consuced. We can ene some values close o he obseved spo ae so we can nealy see he developmen of he laices. Wha we have done so fa is shown in Figue VII a. (Sep 3) We use he D model ( = a e ) o consuc ou sho ae laice. ks k b s (Sep 4) Using he sho aes (shown in Figue VII b) we can consuc a new laice fo he elemenay pices/sae pices wih he fowad equaions 1 shown below. (Hee d ks-1 and d ks ae he one peiod discoun facos (deemined fom he sho aes a hose nodes). hee equaions fo he fowad ecusion : - Fo he middle banch : P k 1 s) = 1 [ d P ( k s 1) d P ( k )] 0 ( + k s k s 0 s - Fo going down : P k + 10) = 1 d P ( 0) 0 ( k0 0 k - Fo going up : P k + 1 k + 1) = 1 d P ( k ) 0( k k 0 k (Sep 5) he sum of he elemens in any column gives us he pice of a zeo-coupon bond wih mauiy a ha dae. Fom hese pices he spo aes can be diecly compued. (Figue VII d) (Sep 6) We inoduce one moe ow fo he (squaed) diffeence beween he obseved spo ae and he one ha we jus compued. (Sep 7) Now we un he equaion-solving ouine which adjuss he a values unil he sum of eos is minimized i.e. unil he calculaed spo ae equals he assumed spo ae in he second ow. (Sep 8) We can now use he esuls fo valuing inees ae coningen deivaives such as bond opions caps floos swapions. 0 In ou case we obained he spo ae cuve fom he ÖK websie (hp:// yieldcouse/index.jsp). 1 he mehod of Fowad inducion was fis inoduced by Jamishidian (1991).

16 Implemenaion of he lack Deman and oy Model Page 16 Figue VII Using a speadshee pogam o implemen he D model (a) Yea Spo ae I a b 000 (b) (c) (d) Sae 9 sho aes (lack Deman and oy) Sae sae pices (hough fowad inducion) P(0k) Spo ae II Eos 7E-14 1E-13 E-1 1E-11 1E-10 5E-10 1E-09 9E-10 5E-10 6E-11 Sum of eos 3E-09 he implemenaion in a Micosof Excel speadshee can be found on a floppy disc aached o his semina pape. (Please enable macos and he add-in Solve.)

17 Implemenaion of he lack Deman and oy Model Page Lis of Symbols and Abbeviaions a mean evesion faco ( ) P value (pice) of he zeo bond a ime wih mauiy ime σ sho ae volailiy µ dif of he sho ae Φ () ime vaious dif in Hull-Whie model d d dz ime ime a mauiy infiniesimal incemen in sho ae infiniesimal incemen of ime infiniesimal incemen in a sandad Wiene Pocess isk neual Maingale measue Φ fowad measue ( ) F fowad ae a ime o ime K S θ() ks a b k s d ks-1 sike o execise pice of a coningen claim asse pice coninuous o simply compounded inees ae ove one ime sep ime dependen dif sho ae in ime k and sae s in he laice aggegae dif volailiy of he logaihm of he sho ae ime in he laice sae in he laice one peiod discoun facos of he fowad equaion d ks

18 Implemenaion of he lack Deman and oy Model Page Refeences Pee jeksund Gunna Sensland Implemenaion of he lack-deman-oy Model Jounal of Fixed Income Vol.6() Sep 1996 pp Fische lack Emanuel Deman and William oy A One-Faco Model of Inees Raes and Is Applicaion o easuy ond Opions Financial Analyss Jounal Jan-Feb 1990 pp Les Clewlow and Chis Sickland Implemening Deivaive Models: Numeical Mehods Wiley 1998 Oo Fancke he Impac of Defaul Risk when Picing Ameican ond Opions when using he Jaow-unbull Appoach Mase hesis Mahemaical Saisics Insiue KH Sockholm 000 John C. Hull Opions Fuues and ohe Deivaives Penice Hall 1997 Fashid Jamshidian Fowad Inducion and Consucion of Yield Cuve Diffusion Models Jounal of Fixed Income June 1991 pp.6-74 David G. Luenbege Invesmen Science Oxfod Univ. Pess 1998 Riccado Rebonao Inees Rae Opion Models Undesanding Analyzing and using Models fo Exoic Inees Rae Opions Wiley 1997 Makus Rudolf Zinssukumodelle Physica Velag 000 Rémi Vignaud Valuaion of inees deivaives wih laice models Mase hesis Depaemen of usiness Sudies Univesiy of Vienna 000

19 Implemenaion of he lack Deman and oy Model Page Suggesions fo fuhe eading.g. ali Kaagozoglu Implemenaion of he D-Model wih Diffeen Volailiy Esimaos Jounal of Fixed Income Mach 1999 pp.4-34 Fische lack Pio Kaasinsky ond and Opion Picing when Sho Raes ae Lognomal Financial Analyss Jounal July-Augus 1991 pp.5-59 John Hull Alan Whie New Ways wih he Yield Cuve Risk Ocobe 1990a John Hull Alan Whie One Faco Inees-Rae Models and he Valuaion of Inees-Rae Deivaive Secuiies Jounal of Financial and Quaniaive Analysis Vol. 8 June 1993 pp John Hull Alan Whie Picing Inees-Rae Deivaive Secuiies Review of Financial Sudies Vol b pp R. Jaow and S. unbull Picing Deivaives on Financial Secuiies subjec o Cedi Risk. he Jounal of Finance Mach 1995 pp R. Jaow D. Lando and unbull A Makov Model fo he em Sucue of Cedi Risk Speads. he eview of financial sudies 1997 Vol. 10 No..