VALUATION OF INTEREST RATE OPTIONS IN A TWO-FACTOR MODEL OF THE TERM STRUCTURE OF INTEREST RATES CONSTANTIN MELLIOS

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2 VLUION OF INERES RE OPIONS IN WO-FCOR MOEL OF E ERM SRUCURE OF INERES RES CONSNIN MELLIOS Pof d ion

3 Valaion of In Ra Opion in a wo-fao Modl of h m S of In Ra Rémé No pénon n modèl d la pa m d ax d'inéê à dx vaiabl d'éa : l ax d'inéê inanané a val moynn obvé n o péiod. L hoix d fa fondé l éla d mpiiq d la gamm d ax n d palli l faibl héoiq d modèl xian. an l ad d modèl no évalon l opion obligaion ona fowad f d'obligaion. ba W pn a wo-fao modl of h m of in a in whih boh h ho a and i ho m man a amd o b ohai. h hoi of h wo fao i bad on mpiial vidn and i o mdy h hoial wakn of xiing modl. In hi famwok w vala opion on bond pi and opion on bond fowad and f ona.

4 Valaion of In Ra Opion in a wo-fao Modl of h m S of In Ra. Inodion wo main appoah hav bn followd in h aadmi lia o modl h m of in a. Whav h appoah modl y o xplain om fa of h bhavio of h m and o idnify h o of ik aoiad wih h dynami of h yild v. In h fi appoah modl dvlopd among oh by Vaik 977 ohan 978 nnan and Shwaz Coadon 98 Shaf and Shwaz 984 Cox Ingoll and Ro 985 a b and Longaff and Shwaz 99 xplain h volion of h n hogh h ohai volion of on o wo a vaiabl h po in a and h long-m in a o h volailiy of in a. In h modl h ohai po fo h a vaiabl i ih pifid xognoly no-abiag m modl o dmind ndognoly fom ampion on invo pfn and hnologi inmpoal m modl. wo-fao modl ovom h limiaion impod by ingl-fao modl in ha hy allow h n on bond of all maii o b impfly olad. hy alo allow h modl o fi many obvd hap of h m. lhogh h modl a fl fo valing in a niiv laim in a onin way hy qi ampion abo invo

5 3 pfn and h ling m do no mah h iniial yild v. In h ond appoah modl a pfn-f and onin wih h n m of in a. hy fall ino h agoi. In h fi on h modl imply modlling h dynami of dion bond pi. o and L 986 w h fi o bild a modl by ing a binomial o dib h volion of dion bond pi ha povid a pf fi o h n yild v. h modl of h ond agoy blong o h la of ah Jaow and Moon 99 hnfoh h JM modl modl. h onino-im JM modl bqnly gnali h o-l modl. hy l h fowad a follow a ohai po govnd by a fini nmb of o of nainy. In h hid on modl pify h ho-a po lik ll and Whi hy xnd h Vaik and CIR modl by ling h paam in h ohai po of h inanano a o b dminii fnion of im in od o mah h modl o h iniial m. In hi pap a wo-fao modl of h m i popod in whih boh h ho a and i ho m man a amd o b ohai. hi ampion i bad on mpiial di whih hav invigad in a bhavio. Chan al. 99 and 995 hav imad vaio onino-im modl of h ho a. hy hav fond only wak vidn ha h ho a v o a long m man val. hi wak vidn gg ha h ho a i man ving abo a ho m

6 4 man. hfo w ppo ha h ho a v o a ho n man whih follow ilf a man-ving po. h ampion of ohai man i alo ppod by mpiial vidn. In Oob 979 h Fdal Rv Fd hangd i monay poliy. Paon and Sn 99 hav hown ha h man paam of h ho a po a diffn in h piod bfo and af hi da. Rnly gim-wihing modl of in a a dvlopd fo inan amilon 988 Cai 994 ay 996 and i and Wbb 997. In h modl a po in a po an hif andomly bwn gim of low man a o high man a.g. h Fd xpimn of 979 o 98. i and Wbb giv an onomi inpaion o a boad la of wofao modl inlding ohai man modl. In paila h dynami bhavio of h man i lad o fial and monay onomi polii. hi pap aim a piing Eopan-yp in a oningn laim nd a wo-fao modl of h m of in a. h hoi of h fao h ho a and ho m man i bad on mpiial vidn and ah of h fao i amd o follow an Onin-Uhlnbk O-U haf po. Oh ohai pifiaion fo h man hav bn popod. Fo xampl nnan and Shwaz 979 wo fao o dib movmn of h m : h ho a and h yild o maiy on a ppal opon bond. ogan 993 dmona ha hi modl admi abiag opponii. Longaff and Shwaz 99 in a gnal qilibim famwok bild a wo-

7 5 fao modl whih an b inpd a a andom volailiy pifiaion ba h volailiy of h inanano in a i a fnion of h wo fao. hy obain lod-fom fomla fo dion bond opion whih dpnd on invo' pfn. Chn 994 dvlop a h-fao modl of h m of in a. In hi modl h n ho a h ho m man and h n volailiy of h ho a follow a qa-oo po. Chn obain a gnal fomla fo valing in a divaiv ha qi h ompaion of high-dimnional ingal. wo-fao modl dvlopd fo inan by Rihad 978 who ag ha h inanano in a i h m of h al a of in and h inflaion a. Chn and So 993 dompo h inanano in a ino wo npifid fao ah of whih follow a qa oo po. ommon haaii of h wo modl i ha h i lil hoial ppo o h hoi of h fao. No ha h modl of h fi appoah mniond abov a all pfn-dpndn. In h famwok of h ond appoah w div impl fomla fo in a oningn laim. h maind of h pap i oganid a follow. In h ond ion h wo-fao modl i pnd and h fndamnal paial diffnial qaion PE fo h dion bond pi i divd. Sion h and fo a dvod o h piing of opion on bond on bond fowad ona and bond f ona. h fifh ion off om mak and onlion. n appndix inld all poof.

8 6. h modl In hi ion w dib a dynami man in a modl and div h dion bond pi. h inanano in a i amd o follow a man-ving po of h fom : d d dw pn h ho m man of h ho a whih follow an O-U po givn by : b d d dw wh a b and a poiiv onan. dw and dw a indpndn Win po nd h hioial pobabiliy ma P. h O-U po wa fi d by Vaik 977 o modl h m. h ho a ha a ndny o v o a andom ho m man val. h la follow a ohai diffnial qaion SE whih ha a man ving dif plling owad i onan long m man. In hi minal wok aion-kp 979 and aion- Plika 98 hav hown ha in od o avoid abiag opponii fo any non dividnd paying a hon a

9 7 nmai h xi a niq ompl mak pobabiliy ik-nal ma Q qivaln o h hioial pobabiliy P h ha h laiv pi of any iy i a Q-maingal. Und Q h xpaion of h inanano n of any finanial iy qal h ikl a. In hi famwok h n val of a finanial laim i qal o h ondiional xpaion nd h ik-nal pobabiliy of h diond final payoff. L λ and λ b fixd al-vald onan ha an b inpd a h mak pi of ik aoiad wih and ~ pivly. fin dw dw λd and ~ dw dw λ d. Und Q aoding o ianov' ~ ~ hom dw and dw a andad ownian moion and and aify h SE': ~ d d dw ~ d dw [ λ ] [ b λ ] d ' h mak pi a < of a dion bond dliving on monay ni a maiy i nod. Wihin hi famwok ing h andad abiag agmn in nnan and Shwaz 979 h pi of a dion bond wih wo a vaiabl aifi h following PE :

10 8 [ λ ] [ b λ ] 0 3 bj o h maiy ondiion. Sbip of dno paial divaiv wih p o h a vaiabl and im. h olion o qaion 3 i givn by: Q E xp d F wh E Q i h xpaion nd h qivaln pobabiliy ma wih p o h ik-adjd po ' and ' fo h inanano in a and h ho m man. F i h infomaion availabl a da. W h andad paaion of vaiabl mhod and onid a dion bond pi fnion of h fom : C xp 4 wh and C a fnion of im o maiy - ha a amd o b wi oninoly diffniabl. h fnion aify h minal ondiion: 0 0 C0 0. iffniaing 4 wih p o and yild : - C - and. Sbiing

11 9 h paial divaiv ino h PE 3 and aanging m w obain : [ ] [ ] 3' 0 C b λ λ h lf-hand id of 3' i lina in and. Colling m in and m indpndn of and giv h following diffnial qaion OE bj o h minal ondiion : a - 0 hi qaion ha h olion : - - b 0 h olion of i givn by h following xpion: * λ C b Ingaing fo C yild:

12 0. and wh : 4 4 * * * * λ λ λ b b b b b C h modl povid a lod-fom olion fo dion bond pi whih a fnion of h in a and i ohai ho m man. No ha h bond pi dpnd on h mak pi of ik and hfo i no abiag-f. i h daion whih ma h niiviy of a bond o hang in h lvl of in a. I i idnial o ha obaind by Vaik' modl. i a ma ha a h niiviy of a bond o hang in h ho n man of in a. If 0 h ond fao h ho man diappa. Fhmo if b onan h abov olion d o Vaik' fomla. h ling bond pi aifi h maiy ondiion. h yild o maiy givn by h following qaion :

13 Ln R C i lina in and. a of i dpndn on wo fao h yild v an aain mo omplx and alii hap han i poibl fo on-fao modl of h m. Fhmo diffn maiy dion bond pi a impfly olad a popy whih i onin wih aliy. Sin h dion bond and h opion on h dion bond a oningn laim on h am fao and h am abiag agmn an apply o h ii hi PE a idnial. hy diff only by hi bonday ondiion. a l h val of h opion i a fnion of h mak pi of ik and dpnd on h invo' pfn. Inad of olving a PE fo ah kind of opion whih i im onming and qi dio all in h following ion w an alnaiv mhod piond by Jamhidian 989 o obain pfn-f piing fomla fo h in-a-niiv opion. W mak of h fowad-nal pobabiliy ma qivaln o h ik-nal ma. Und hi nw ma in abn of abiag opponii h fowad pi of any finanial a i a maingal having h am vaian a nd h hioial pobabiliy. In hi famwok in a opion a vald by abiag indpndnly of invo' pfn.

14 3. Opion on dion bond L C b h pi a da of a Eopan all win on a dion bond pi of maiy wih ik pi K and xpiy da wh. h minal val of h all a da i: C max[ - K O]. Und h ik-nal pobabiliy Q h n val of h opion i givn by: [ K0] F Q C E xp d max 5 On ha o omp h xpaion of a pod of andom vaiabl whih i in gnal diffil. owv hi xpaion n o o b qal o h xpaion nd h fowad-nal pobabiliy ma Q of i minal payoff mliplid by h xpaion ompd nd h ik-nal ma Q of h dion fao. hn h olion o qaion 5 divd in h appndix i:

15 3. Ln wh: 6 d d K d d N K d N C hi fomla i imila o h Jamhidian' on-fao modl h only diffn bwn h wo modl i ha in o modl h bond pi dpnd on wo fao 4. v a h ndlying a and i h nmai aoiad wih h fowad-nal pobabiliy ma. 4. Opion on dion bond fowad ona Fowad and f ion 5 ona hmlv hav no val. h nlik piing opion i i lvan o div h fowad o h f pi b no o dmin h val of h ona. I follow fom a impl abiag agmn ha in

16 4 abn of abiag opponii h fowad pi a im fo a fowad ona maing a da win on a dion bond of maiy da wh i givn by: /. L C b h val of a Eopan all win on a bond fowad ona of maiy wih ik pi K and xpiy da wh. h minal val of hi all a da i: [ 0] C max K wh h ponial gain - K m b diond bak fom o in i a fowad pi. hfo a da w hav : C Q E xp max[ 0] d K F h olion o whih i hown in h appndix o wi :

17 5 [ ]. Ln wh: 7 d d K d d KN d N C h fomla obaind i minin of ha of lak 976 wih ohai in a. No ha whn h maiy da of h fowad ona oinid wih ha of h opion hn h opion on h po and h opion on h fowad hav idnial val. Indd a da h fowad pi i qal o h po bond pi. 5. Opion on dion bond f ona hown by Cox Ingoll and Ro 98 h f pi may diff fom h fowad pi d o h daily making o mak. h no-abiag f pi a im of a

18 6 f ona maing a im win on a dion bond ha ma a da wh i qal o : Q E F Sin h f ona i amd o b makd o mak oninoly and hn o hav alway zo-val h f pi nd h ik-nal pobabiliy ma Q i a maingal. a h f pi a maiy i qal o h po pi i impli ha h f pi a any da i givn by h ik-nal xpd val of h po bond pi a h f ona maiy. ming ha magin a alld in a onino ah han daily mann h xplii olion o h abov xpaion i hown o h appndix o b :

19 7 xp No ha h f pi do no dpnd on h mak pi of ik and hfo i pfn-f. h f pi diff fom h fowad pi by an adjmn fao ha fl h ovaian bwn h bond pi volailiy and h fowad pi volailiy. h pi a qal only whn h in a and i ho m man a dminii. L C 3 b h pi of a Eopan all wih maiy and ik pi K win on a bond f ona wih maiy wh. h val of hi all a im i givn by : [ ] 0 max xp 3 Q F K d E C 8

20 8 Sin F pi diff fom fowad pi by an adjmn fao w an h am hniq a in ion 4 o vala an xpion fo h pi of a Eopan f opion. Uing a ggion fom Jamhidian 993 and omping fi h fowad pi of h f ona h olion o qaion 8 givn in h appndix i : [ ] Ln wh: 9 3 d d K d d KN d N C γ γ

21 9. wih i dfind in qaion 7.. γ aigh ompaion of h xpion C qaion 6 C qaion 7 and C 3 qaion 9 how ha all h val diff. If h opion and boh h fowad and f ona hav h am maiy i.. hn h opion on h bond and h opion on h fowad ona hav idnial val. Y h opion on h f ha a diffn pi ba in a and h ho m man a ohai. Whn h wo fao in a and i ho n man a dminii h pi of h h opion a qal. nally h opion maiy diff fom ha of h fowad and f ona i.. < *. If h wo fao a dminii hn fowad and f pi a idnial. owv h val of opion on fowad ona i

22 0 diffn fom ha of opion on f ona. hi i d o h making-o- mak mhanim. 6. Conlion In hi pap w hav divd fomla fo Eopan opion on dion bond and on dion bond fowad and f ona. h bond pi dpnd on wo-a vaiabl: h ho a and i ho m man. h hoi of h a vaiabl i bad on mpiial vidn and i o mdy h hoial wakn of xiing modl. h onomi famwok allow on o val any yp of in a divaiv. la h poibl diion fo fh ah an b pd along h lin of hi modl. Fi a pod an b fond o fi h modl o any givn iniial m. Sond h modl an b xndd o an innaional onomy wh boh domi and xhang a a ohai. Finally an mpiial wok will b nay o h impliaion of h modl in h valaion of in a oningn laim. Mahmaial ppndix In od o div h opion pi 6 7 and 9 in h main x h fowad-nal pobabiliy ma h xpion of and and h following l a d :

23 Rl L X i a aian vaiabl N hn : { } [ ] { } [ ] { } fnion. h indiao : xp > > wh LnK N E LnK N E K K X X X h olion o ' and ' hav h fom : >. wh ~ ~ ~ * * dw b dw dw b λ h fowad-nal pobabiliy ma dnod Q j i qivaln o Q fo j > 0. hi hang of ma mak nmai h p dion bond j. Sin < < j j j j d E d E 0 0 / xp and / xp

24 hn aoding o ianov' hom h po ~ j W W j and ~ j W W j a ownian moion nd Q j. In paila h fowad pi / j i a maingal nd hi ma. ivaion of qaion 7 : all on bond fowad ona On wan o vala : [ ] 0 max xp Q F K d E C L Q h fowad-nal pobabiliy ma qivaln o Q. Und Q h fowad pi i a maingal h ha5 : 3 dw dw d pplying Iô' lmma o yild :

25 3 } dw dw d d xp 0 0 h fowad pi a da of opion xpiy hav lognomal diibion. hfo : ; d d d d Ln N Ln Und Q h all pi C i givn by : [ ] [ ] Q F K E C 0 max [ ] [ ] K Q K Q F KE F E > > Uing h l and aanging m yild qaion 7. ivaion of qaion 6 : all on dion bond

26 4 Ling in qaion 7 yild dily qaion 6 a paila a of 7 whn h opion xpiy da oinid wih ha of h fowad ona : a h fowad pi i qal o h po pi. ivaion of h qaion of h pi of h f ona h f pi i givn by : [ F ] Q E 4 o obain h f pi a a fnion of h fowad pi w fi omp :

27 5 { xp * * b b λ L ~ ~ * dw dw b Y λ

28 6 ~ * dw b Z ivn and Y and Z a nomally diibd wih ondiional man and vaian a follow : [ ] [ ] 6d 6 6b 6a * * Q Q Z Va b Z E Y Va b Y E λ y plgging 5 ino 4 and by ing xpion 6 on an omp h xpaion nd Q ha giv h f pi. ivaion of qaion 9 : all on bond f ona Comping h fowad pi φ of h f ona on an h am hniq a fo all on fowad ona. Following Jamhidian 993 on obain h following xpion fo h fowad pi of h f ona :

29 7 xp xp ov xp xp d d d d F d E Q γ φ fining a nw fowad-nal pobabiliy ma Q qivaln o Q h f ona pi bom a maingal. hn mak ha C 3 an b win : [ ] ] 3 F 0 max xp K d E C Q φ If hi fomla i ompad wih ha giving C i i adily n ha qaion 7 an b applid wih inad of xp γ inad of whih yild dily qaion 9.

30 8 REFERENCES lak F. 976 "h Piing of Commodiy Cona" Jonal of Finanial Eonomi 3 pp lak and Shol 973 "h Piing of Opion and Copoa Liabilii" Jonal of Poliial Eonomy 3 pp nnan M. and Shwaz E. 979 " Conino im ppoah o h Piing of ond" Jonal of anking and Finan 3 pp nnan M. and Shwaz E. 98 "n Eqilibim Modl of ond Piing and a of Mak Effiiny" Jonal of Finanial and Qaniaiv nalyi 7 pp Cai J. 994 " Makov Modl of Unondiional Vaian in RC" Jonal of in and Eonomi Saii pp Chan K. Kaolyi. Longaff F. and Sand. 99 "n Empiial Compaion of lnaiv Modl of h Sho-m In a" h Jonal of Finan 48 pp Chn L. 997 "Sohai Man and Sohai Volailiy : h-fao Modl of h m S and i ppliaion in Piing of In Ra ivaiv" Fdal Rv oad Woking Pap. Chn R. and So L. 993 "Piing In Ra Opion in a wo-fao Cox-Ingoll-Ro Modl of h m S" Rviw of Finanial Sdi 5 pp

31 9 Coadon. 98 "h Piing of Opion on fal F ond" Jonal of Finanial and Qaniaiv nalyi 7 pp Cox J. Ingoll J.and Ro S. 98 "h Rlaion wn Fowad Pi and F Pi" Jonal of Finanial Eonomi9 pp Cox J. Ingoll J. and Ro S. 985 a "n Inmpoal nal Eqilibim Modl of Pi" Eonomia 53 pp Cox J. Ingoll J. and Ro S. 985 b " hoy of h m S of In Ra" Eonomia 53 pp ohan U. 978 "On h m S of In Ra" Jonal of Finanial Eonomi 6 pp ybvig P. 990 "ond and ond Opion Piing ad on h Cn m S" hnial Rpo Wahingon Univiy. ay S. 996 "Modling h Condiional iibion of In Ra a a Rgim-Swihing Po" Jonal of Finanial Eonomi 4 pp amilon J "Raional Expaion Eonomi nalyi of Chang in Rgim: n Invigaion of h m S of In Ra" Jonal of Eonomi ynami and Conol pp aion M. and Kp. 979 "Maingal and biag in Mlipiod Sii Mak" Jonal of Eonomi hoy 0 pp

32 30 aion M. and Plika S. 98 "Maingal and Sohai Ingal in h hoy of Conino ading" Sohai Po and hi ppliaion 5 pp ah. Jaow R. and Moon. 99 "ond Piing and h m S of In Ra: Nw Mhodology fo Coningn Claim Valaion" Eonomia 60 pp o. and L S "m S Movmn and Piing In Ra Coningn Claim" Jonal of Finan 4 pp ogan M. 993 "Poblm in Cain wo-fao m S Modl" nnal of pplid Pobabiliy 3 pp ll J. and Whi. 990 "Piing In Ra ivaiv Sii" h Rviw of Finanial Sdi 3 pp ll J. and Whi. 993 "On-Fao In-Ra Modl and h Valaion of In-Ra ivaiv Sii" Jonal of Finanial and Qaniaiv nalyi 8 pp Jamhidian F. 989 "n Exa ond Opion Fomla" h Jonal of Finan 44 pp Jamhidian F. 993 "Opion and F Valaion wih minii Volailii" Mahmaial Finan 3 pp Longaff F. and Shwaz E. 99 "In Ra Volailiy and h m S: wo-fao nal Eqilibim Modl" h Jonal of Finan 48 pp Paon N. and Sn.S. 99 "n Empiial Examinaion of h Cox Ingoll and Ro Modl of h m S of In Ra" Woking pap Univiy of oon.

33 3 Rihad S. 978 "n biag Modl of h m S of In Ra" Jonal of Finanial Eonomi 6 pp Shaf S. and Shwaz E. 984 " wo Fao Modl of h m S: n ppoxima nalyial Solion" Jonal of Finanial and Qaniaiv nalyi 9 pp Y.K. 985 "Som Innaional Evidn on h Sohai havio of In Ra" Jonal of Innaional Mony and Finan 4 pp i J. and Wbb N. 997 " Nonlina Modl of h m S of In Ra" Mahmaial Finan 7 pp Vaik O. 977 "n Eqilibim Chaaizaion of h m S" Jonal of Finanial Eonomi 5 pp ybvig 990 a "h ond fao if any in a m modl hold b lad o h diibional fa of in a no addiiv in lvl of in a a i ally amd". h main ida of h ik-nal piing mhod i analogo o ha of h lak and Shol 973 modl. In hi modl h xpaion of h inanano yild of h ndlying a do no aff h val of h opion. h la i a fnion of h ikl a. Sin h pfn paam h dif dop o of h valaion po all a hold an wha a ik-nal invo wold xp h ik-f a. 3 P an b vald in h am mann o hogh h andad all-p paiy.

34 3 4 I i obvio ha h olion givn by h fomla 6 i a lak and Shol 973 yp. h imilaiy bwn hi olion and h lak- Shol ok opion modl ha bn did by Jamhidian h dynami of h bond pi i givn by [ ] [ ] b dw dw d d λ λ wh ~ ~ Uing h olion 4 h inanano a of n of h dion bond an b win a : ~ ~ dw dw d d pplying Iô' lmma o / and ing h fowad-nal pobabiliy ma yild 3.