Volatility Structures of Forward Rates and the Dynamics of the Term Structure* Peter Ritchken and L. Sanakarasubramanian

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1 Volailiy Sucues o Fowad Raes and he Dynamics o he em Sucue* ypesee: RH 1s poo: 1 Oc nd poo: 3d poo: Pee Richken and L. Sanakaasubamanian Case Wesen Reseve Univesiy; Bea Seans & Co Fo geneal volailiy sucues o owad aes, he evoluion o inees aes may no be Makovian and he enie pah may be necessay o capue he dynamics o he em sucue. his chape ideniies condiions on he volailiy sucue o owad aes ha pemi he dynamics o he em sucue o be epesened by a wo-dimensional sae vaiable Makov pocess. he pemissible se o volailiy sucues ha accomplishes his goal is shown o be quie lage and includes many sochasic sucues. In geneal, analyical chaaceisaion o he eminal disibuions o he wo sae vaiables is unlikely, and numeical pocedues ae equied o value claims. Eicien simulaion algoihms using conol vaiaes ae developed o pice claims agains he em sucue. his chape deals wih he picing o coningen claims when inees aes ae sochasic. he mehodology used incopoaes all cuen inomaion in he yield cuve. his appoach, pioneeed by Ho and Lee (1986) and signiicanly genealised by Heah, Jaow and Moon (199), elies upon makes being dynamically complee wih coninuous ading oppouniies. Analogous o he Black Scholes model, whee peeences ae embedded ino he sock pice and he volailiy is exogenously povided, in his appoach peeences ae embedded ino he obsevable em sucue, and he volailiy uncion o owad aes is exogenously speciied. Heah, Jaow and Moon (199) show ha o peclude abiage oppouniies among bonds o dieen mauiies, he di and volailiy ems in he evoluion o all owad aes mus be elaed o a common make pice o inees ae isk. Fuhe, hey ideniy a unique maingale measue ha can be used o pice all inees ae claims. Unounaely, he compuaion o pices is complex because he evoluion o he em sucue unde he maingale measue is usually no Makovian wih espec o a inie-dimensioned sae space. 1 he pah dependence is also made appaen in a *his chape was is published in Mahemaical Finance 5(1), (1995), pp. 55 7, and is epined wih he pemission o Blackwell Publishes. I is a subsanial evision o an ealie pape eniled On Coningen Claim Valuaion in a Sochasic Inees Rae Economy, June We ae paiculaly gaeul o he commens by an anonymous eeee and he associae edio o Mahemaical Finance. HE NEW INERES RAE MODELS

2 elaed pape by Heah, Jaow and Moon (199) whee hey use a binomial appoximaion o povide an alenaive deivaion o hei main esuls. Due o pah dependency he laice usually gows exponenially wih he numbe o ime peiods. he pah dependence causes complicaions even i simulaion models ae used o compue he pices o Euopean claims. In paicula, since he enie em sucue mus be manipulaed along each simulaed pah, simulaions ae compuaionally inensive, and he build-up o eos due o disceeness may be sevee. Many single-aco models o he em sucue esolve he complicaions inoduced by pah dependence by imposing suicien sucue on he poblem so ha he evoluion o he em sucue is pahindependen. In conas, his chape ideniies condiions which pemi he em sucue s dynamics o be capued by wo vaiables ahe han he enie em sucue. In paicula, we ideniy condiions on he volailiy sucue o owad aes ha pemis he em sucue s dynamics o be epesened by a wo-dimensional suicien saisic. Unlike he geneal Heah, Jaow and Moon (HJM) models, ou models hus lead o a wo-sae-dimensional Makov epesenaion o he em sucue. he pemissible se o volailiy sucues ha accomplishes his goal is shown o be quie lage and includes many sochasic and deeminisic sucues. Fo example, he volailiy sucue can be speciied in such a way ha he esuling spo ae dynamics is geneaed by a squae oo pocess. In geneal, an analyical chaaceisaion o he eminal disibuions o he wo sae vaiables is unlikely, and hence numeical pocedues ae equied. We illusae how simulaion echniques, combined wih conol vaiables, can be used o esablish eicien picing algoihms o all ypes o claims agains he em sucue. he chape poceeds as ollows. In he nex secion we ideniy he pah dependence ha is implici in models o he em sucue in which he volailiy sucue o owad aes is abiay. We hen show ha i he volailiy sucue is caeully cuailed, bond pices can be expessed in ems o he spo ae and an addiional saisic which capues all he elevan hisoy o he em sucue dynamics. he esuling spo ae pocess induced by he volailiy consain is developed. his pocess is a wosae-vaiable Makovian pocess and has he desiable popey o incopoaing ino is dynamics all he inomaion povided om he em sucue. When he volailiy sucues ae deeminisic hen simple analyical soluions o picing inees ae claims ae available. Fo all ohe cases numeical pocedues ae equied. Founaely, since he maingale measue which is elevan o picing is Makovian, models can be esablished which do no equie he enie em sucue o be manipulaed along each pah. he hid secion discusses simulaion mechanisms and povides examples o picing inees-sensiive opions, when he volailiy sucue o owad aes is non-deeminisic. he convegence o simulaed esuls is subsanially acceleaed by using appopiae conol vaiables. he valuaion o claims ha depend on pices o isky asses as well as on yields dawn om he em sucue is also consideed. he ouh secion summaises he chape. Pah dependence and volailiy sucues Fowad aes ae assumed o ollow a diusion pocess o he om d(, ) = µ (, ) d+ (, ) dw( ); given (, ) (1) Hee µ (, ) and (, ) ae he di and volailiy paamees which could depend on he level o he em sucue isel, and δω(τ) is he Wiene incemen. Inegaing (1) yields he elaion beween cuen owad aes and hei values a ime. In paicula, (, ) = (, ) + µ ( s, ) ds+ ( s, ) dw( s) o () RISK BOOKS

3 3 Since he spo ae a ime, (), is given by (, ), we obain ha he dynamics o he spo ae ae hen obained as d d( ) = d(, ) + du (, u ) d ( ) = (, ) = (, ) + µ ( s, ) ds+ ( s, ) dw( s) u (3) (4) Le P(, ) be he pice a dae o a pue discoun bond ha maues a ime. By deiniion, he bond pice is given by P(, ) = exp (, s) ds (5) Since bond pices depend on owad aes, as evidenced by (5), he di and volailiy sucue o bond euns mus be elaed o he di and volailiy sucue o owad aes. In paicula, le dp(, ) = µ (, ) d+ (, ) dw( ) P P P(, ) Usual abiage agumens hen lead o (6) and µ (, ) ( ) P λ ( ) = (, ) [ ] µ (, ) = (, ) λ( ) (, ) P P (7) (8) whee P (, ) = ( s, ) ds Hee λ() is he make pice o inees ae isk a ime, and is pehaps sochasic, bu is he same o all bonds egadless o mauiy. Equaion (1) can now be wien as (9) [ ] + d(, ) = (, ) λ( ) (, ) d (, ) dw( ) P (1) Equaion (1) makes explici he elaion beween he di componen o he evoluion o owad aes, he volailiy sucue o owad aes, and he make pice o isk, which mus be saisied o peclude dynamic abiage oppouniies. Equivalenly, inegaing (1) we obain [ P ] + (, ) = (, ) + ( u, ) λ( u) ( u, ) du ( u, ) dw( u) Using (6) and (11) leads o he elaion beween owad pices and uue bond pices: P(, ) P(, ) = exp ( us, ) ( us, ) duds ( us, ) ( u) du+ dwu ( ) ds P [ λ ] P(, ) (1) his om o he bond picing equaion is given in Heah, Jaow and Moon (199). As hey emphasise, his geneal om is quie disinc om he adiional lieaue. In he adiional lieaue, bond pices ae assumed o be uncions o some sae vaiables (11) HE NEW INERES RAE MODELS

4 4 and explici omulae linking bond pices o hese sae vaiables ae hen esablished. Fo example, in many single-aco models, he sae vaiable is he spo ae and bond pices ae developed as a uncion o he spo ae. In he above em-sucue-consained appoach, bond pices may no be Makovian wih espec o a inie se o sae vaiables. Indeed, he above single-aco epesenaion o he em sucue highlighs he ac ha bond pices a dae depend on pices a an ealie dae, on he enie pah aken om dae o dae, and o couse on he evoluion o he make pice o isk. Fom a valuaion pespecive, (11) and (1) ae o lile use. I would be desiable o obain a bond picing equaion ha depends on a ew sae vaiables, common acoss all mauiies. I would also be desiable o he equaion o be peeence ee in he sense ha he make pices o isk do no appea. o accomplish his goal, we impose uhe esicions on he pocess by which owad aes ae geneaed. his poin maks ou depaue om Heah, Jaow and Moon (199). Wih he goal o obaining a poxy o he pah o inees aes ha is elevan o picing all bonds, ewie (11) as [ P ] = = (, ) (, ) ( u, ) λ( u) ( u, ) du ( u, ) dw( u) Λ(, ) Λ(, ) elecs he pah dependence as he weighed sum o all Bownian disubances ealised om ime o ime. his pah dependence can be capued by a single saisic common acoss all mauiies,, wihou imposing any addiional esicions on he iniial em sucue, (, ), o on he sucue o he make pice o isk, λ(), poviding a common weighing scheme exiss o owad aes o all mauiies. Howeve, i a unique weighing scheme is o exiss o all owad aes, i mus be he case ha he weighing uncion is independen o. his in un implies ha Equivalenly, ( u, ) ( x, ) dx ( u, ) = u [, ] ( x, ) dx ( u, ) = ( u, ) whee he new quaniy k(, ) is well deined o all >. O couse (u, ) is ully deemined a dae u. his implies ha he igh-hand side is also ully deemined a dae u. In geneal, boh he numeao and he denominao o k(, ) ae only known a a lae dae, bu om he above, hei aio mus be independen o he pah aken om he ealie dae u. Hence, k(, ) mus be deeminisic. Subsiuing =uleads o ( u, ) = ( uk ) ( u, ) whee (u) = (u, u). Similaly, (u, ) = (u)k(u, ). Hence ( x, ) dx = ( u, ) k(, ) ( x, ) dx k( u, ) = k( u, ) k(, ) [ u, ] (13) wih k(, ) 1 u, = [ ] I (, ) is dieeniable wih espec o, he k(, ) is dieeniable wih espec o as well. Le g(, ) = lnk(, ). hen g( u, ) = g( u, ) + g(, ) [ u, ] RISK BOOKS

5 5 Dieeniaing wih espec o yields = g( u, ) g(, ) his implies ha he above deivaive is independen o u. Le κ() = g(u, ). hen, by deiniion, g( u, ) g( u, u) = [ g( u, x) ] dx = ( x) dx x κ u u Fuhe, g(u, u) =. Hence he above equaion educes o k( u, ) = exp ( x) dx κ u (14) Equaion (14) is a necessay condiion ha mus be saisied by he volailiy sucue o owad aes i he pah dependence is o be capued by a single saisic. I uns ou ha his is suicien as well, and in he esuling models all bond pices can be expessed in ems o wo sae vaiables. hese esuls ae summaised below. PROPOSIION 1 (a) I he volailiies o owad aes ae dieeniable wih espec o hei mauiy daes, hen o any iniial em sucue, a necessay and suicien condiion o he pices o all inees coningen claims a ime o be compleely deemined by a wo-sae-vaiable Makov pocess is ha he volailiy sucue o owad aes saisy (, ) = ( )exp κ( x) dx (15) (b) Unde he esicion imposed by (15), he pice o a bond, a any uue dae, can be epesened in ems o is owad pice a dae, he sho inees ae a dae, and he pah o inees aes as { [ ]} P(, ) P(, ) = exp (, ) ( ) (, ) (, ) ( ) 1 β φ + β P(, ) (16) whee PROOF See he Appendix. (, ) P β (, ) = = (, ) φ( ) = ( s, ) ds e κ ( x) dx du Equaion (16) ideniies he wo sae vaiables as he spo inees ae, (), and he inegaed vaiance aco, φ(). Given hese wo values a dae, he enie em sucue can be econsuced. Fuhe, unde he owad ae volailiy esicion, we can expess he evoluion o he wo sae vaiables, () and φ(), in ems o hei cuen values and he level o he owad ae cuve a an ealie dae. Speciically, subsiuing in (4) om (15) and (16) leads o d ( ) = µ ( ) d + ( ) dw ( ) (17) HE NEW INERES RAE MODELS

6 6 and whee ( ) dφ( ) = ( ) κ( ) φ( ) d (18) and µ ( ) = κ ( ) d φ λ [ (, ) ( ) ]+ ( ) + ( ) ( ) + d (, ) ( ) = (, ) Equaion (17) descibes he evoluion o he spo inees ae by a wo-sae Makovian pocess. his is in shap conas o he geneal HJM models in which he inees ae pocess canno be descibed by a inie-sae Makov pocess. By consaining he volailiy sucue o owad aes we ae able o capue he dynamics o he enie em sucue by a wo-sae Makovian epesenaion. I he volailiy sucue is no o he om in (15), hen such a epesenaion is no possible, and eihe ull pah dependence (as in he geneal HJM models) will be pesen, o addiional assumpions on volailiies and invesmen behaviou will be necessay o obain a Makovian epesenaion. he class o volailiy sucues admied by (15) is quie lage. In paicula, no esicions ae placed on he volailiy o he spo inees ae. Indeed, he spo ae volailiy a dae could depend on he ull se o inomaion available a dae. As an example o a easible epesenaion, conside he special case o (15) whee and κ ( ) (, ) = ( ) e Valuaion o claims unde he esiced volailiy sucue Heah, Jaow and Moon (199) develop he maingale measues unde which claims on he em sucue may be piced as expecaions o hei eminal payos elaive o a money make und. In pacice, compuing hese expecaions is exemely diicul, excep o he simples o cases whee he volailiy sucue is deeminisic. Fo he geneal case, compuing he eminal em sucues unde he isk-neualised pocess equies consucing he enie em sucue a evey poin in ime. his buden apidly becomes compuaionally expensive, bu is necessay o valuing all ypes o inees ae claims. In conas, models unde he esiced volailiy sucue ideniied in his chape ae wo-sae Makovian. As a esul, all he inomaion conained in he em sucue is capued by wo vaiables. Hence, o consuc he eminal em sucue, any algoihm need only keep ack o hese wo vaiables. In his secion we develop a discee appoximaion algoihm o value Euopean claims on he em sucue. Fuhe, we show how he peomance o he algoihm can be signiicanly enhanced by using con ( ) ( ) = [ ] γ (19) () Noice ha o γ =he volailiy sucue becomes deeminisic. In his case he spo inees ae becomes he only sae vaiable, and he above poposiion educes o a saemen on necessay and suicien condiions ha pemi he em sucue o be Makovian wih espec o he spo inees ae. he condiions have peviously been deived by Cavehill (1994) and by Hull and Whie (1993). he above poposiion genealises hei esuls by consideing a lage class o volailiy sucues ha pemi a wo-sae Makovian epesenaion. Fo γ =.5 he spo ae volailiy is squae oo and owad aes ae linked o spo aes via an exponenially dampened uncion. In geneal, he speciicaion given in (19) and () leads o models o he em sucue ha ae compleely descibed by hee paamees, namely, κ and γ. RISK BOOKS

7 7 ol vaiae echniques, in conjuncion wih picing elaionships developed when inees aes have deeminisic volailiies. o make maes speciic, we shall esic aenion o he case whee he sucue o he spo ae volailiy depends on he level o he spo ae and κ( ) is a consan. Speciically, we assume ha he volailiy sucue is given by (19) and (). Wih hese esicions he inees ae pocess given in (17) educes o γ d d( ) = [ (, ) ( ) ]+ ( ) + ( ) [ ( ) ] + d (, κ φ λ ) d + [ ] γ ( ) dw( ) (1) he volailiy o he esuling spo ae dynamics is simila o ha o models nesed in he sucue [ ] + [ ] d( ) = κ µ ( ) d ( ) dw he main dieence, howeve, is ha he di em depends on inomaion om a pas em sucue as well as on he pah saisic, φ(), which is descibed as γ γ κ ( ) φ( ) ( u) e u = [ ] du () Now, le g() epesen he dae value o a claim having a eminal payou a dae s ha is ully deemined by yields dawn om he em sucue. Using sandad abiage agumens he maingale measue unde which all claims ae piced is obained by seing λ() = in (1). In paicula, ollowing Heah, Jaow and Moon we obain g s g(, s) E ( ) = M( s) (3) whee M(s) is he value o a money und ha is iniialised wih US$1. a dae and olls ove a he cuen iskless ae o eun. he expecaion in he above equaion is aken unde he join isk-neualised pocess and d d( ) = κ( ) [ (, ) ( ) ]+ φ( ) + (, ) d+ ( ) dw( ) d ( ) dφ( ) = ( ) κ( ) φ( ) d (4) (5) In geneal, because o pah dependency, i is unlikely ha analyical expessions o he eminal join disibuion o he sae vaiables (s) and φ(s) can be obained. o esablish numeical appoximaions o he em sucue, we begin by paiioning he ineval [, s] ino n equal ime incemens o widh. Ove he discee inevals le a γ j κ ( ) ( ij, ) = [ i ( ) ] e, j i he appoximaing pah saisic and change in inees ae ae a( ) and [ a ( )], especively, whee φ a j 1 = [ ] a κ ( ) ( j) ( i) e i = j i (6) HE NEW INERES RAE MODELS

8 8 and [ ] = + a ( i+ 1 ) a ( i 1 ) a ( i) whee a () = () and a (i) appoximaes he inees ae (i ). hen a [ ( i+ 1) ] = ( ),[ i+ 1] (, i ) a a κ[ (, i ) ( i) ] + φ ( i) + + (, i i) Z() i whee Z(i) is a nomal andom vaiable wih mean zeo and vaiance. Using (6) and () he pah saisic, φ a ( ) can be updaed as ollows: a (7) a [ () i ] a κ a κ φ ( i+ 1) = e φ ( i) + [ 1 e ] (8) κ o invesigae he convegence o (7) exensive simulaions wee peomed ove a wide ange o paamee seings. Speciically, given paicula values o he paamees, κ,, and γ, we simulaed a pah o 8 paiions, updaing evey ime incemen. Using he same sequence o andom numbes we hen consideed updaing less equenly. he pecenage eo in he esuling eminal value o () and φ() was compued o each pah and solely elecs he bias om less equen updaes. en housand pahs wee simulaed, and he is ou cenal sample momens o he maginal disibuion o hese pecenage eos wee compued. able 1 summaises he mean pecenage eo when he numbe o updaes ove he hee-monh peiod anged om 5 o. he able only epos he aveage pecenage eos o he maginal disibuions. he magniudes o he second, hid and ouh momens o he pecenage eos wee negligibly small and ae no epoed. he able illusaes he ac ha he pecenage eos decline wih he numbe o paiions wih no sysemaic bias. Fo he paamees chosen in able 1 i appeas ha he momens o he ue maginal disibuions o () and φ() ae well appoximaed wih 1 updaes. Fo a hee-monh peiod his coesponds o updaing he pah once a day. 3 he analysis was epeaed o dieen ime peiods, and simila esuls wee obained. Indeed, in all ou simulaions he ule o allowing one evision peiod pe day appeaed o povide esuls ha wee no economically dieen om ine paiions. In all ou uue analyses, ou simulaed join disibuions wee based on appoximaely wo evisions pe day. able 1. Convegence o he mean pecenage eo o eminal values o he sae vaiables =. =.5 = 1. N VAR Vol = 1% Vol = 3% Vol = 1% Vol = 3% Vol = 1% Vol = 3% φ φ φ φ he volailiy (Vol) o he sho ae pocess was chosen o be (()). he yield cuve was la a 1%. he ime is hee monhs. Numbes displayed ae he aveage o he pecenage eos compued as 1( N 8 ) 8 and 1(φ N φ 8 ) φ 8 whee N (φ N ) is he inees ae (pah vaiable) geneaed wih N updaes. Fo example, when he mean evesion paamee, κ, equals.5, and he volailiy o he sho ae is se a 3%, hen he mean pecenage eo in compuing he aveage eminal inees ae wih 5 updaes is.9% o is value using 8 updaes. γ RISK BOOKS

9 9 he simulaion pocedue was used o geneae φ(s), (s), and he money make accoun, M(s), unde he isk-neualised pocess. Fo each pah, he enie em sucue could hen be obained using (16), and he eminal payou o any inees ae claim compued. We simulaed 1, pahs and aveaged he esuling eminal payous o he inees ae claim elaive o he money und. his leads o an appoximaion o he ai value in (3). he convegence o he numeical pices o hei ue values can be signiicanly acceleaed by using conol vaiables. A candidae o he conol vaiable is obained by seing γ o zeo in (19) and (). Fo his case, howeve, φ(s) is deeminisic and he wo-sae model educes o a pah-independen, single-sae-vaiable epesenaion. Speciically, om () we obain [ ] φ ( s) = κ s 1 e κ Fo his speciicaion, inees aes ae nomally disibued and analyical soluions ae available o a wide vaiey o inees ae claims. Fo example, he pice o an s-peiod Euopean call opion wih sike X on a discoun bond wih mauiy dae is given by C(, s) P(, ) N d XP(, s) N d = ( 1) ( ) (9) whee d 1 ln P(, ) XP(, s) = [ ] + ϕ ϕ his model was is developed by Jamshidian (1989). Also, he pice o an s-peiod Euopean yield opion, ha povides he owne wih he opion o buy he yield y(s, ) o X dollas is given by whee and ξ d ϕ = κ = d ϕ 1 κ ( s) κs = [ 1 e ] [ 1 e ] 3 κ C(, s) = P(, s) ξ M( q) q 1 κ ( s) κs [ 1 e ] 1 e 3 ( s) X y ( s, ) ξ ( s) q = ξ and y (s, ) is he owad yield o mauiy viewed om ime, and M( q) = N( x) dx whee N(x) in he cumulaive sandad nomal disibuion. Figues 1 and illusae he ae o convegence o he simulaed pices o hei ue values given in (9) and (3). he conac in Figue 1 is a hee-monh a-he-money call opion on a 15-yea discoun bond, while in Figue he claim is an a-he-money heemonh inees ae call opion on he spo ae. he mean evesion paamee, κ, was se a.5 and he annual volailiy paamee,, was se a %. Noice ha as he iea- q [ ] 1 [ ] (3) HE NEW INERES RAE MODELS

10 1 1. Rae o convegence o he simulaed pice o an a-he-money, hee-monh call opion on a 15-yea discoun bond Convegence o he simulaed pice A-he-money pice Ieaions (housand) he iniial em sucue is la a 1%, and he volailiy sucue o owad aes is given by (19) and () wih κ =.5, =. and γ =. he la hoizonal line is he heoeical value given by (9). Rae o convegence o he simulaed pice o an a-he-money, hee-monh call opion on he spo ae Convegence o he simulaed pice 3.8 A-he-money pice Ieaions (housand) he iniial em sucue is la a 1%, and he volailiy sucue o owad aes is given by (19) and () wih κ =.5, =. and γ =. he la hoizonal line is he heoeical value given by (3) ions incease, he simulaed esuls convege o he analyical, which ae indicaed by hoizonal lines. As can be seen, wih 1, eplicaions he algoihm poduced esuls indisinguishable om hei analyical counepas. Figues 3 and 4 illusae he ae o convegence o he same opion conacs o γ =.5. Since analyical soluions ae available o γ =, conol vaiae pocedues ae RISK BOOKS

11 11 3. Compaison o he ae o convegence o a hee-monh a-he-money call opion on a 15-yea discoun bond wih and wihou conol vaiables 1.76 A-he-money pice ( =.5) Convegence using conol vaiables Ieaions (housand) he iniial em sucue is la a 1%, and he volailiy sucue o owad aes is given by (19) and () wih κ =.5, * =. and γ =. he nealy la line is he value compued using he conol vaiae 4. Compaison o he ae o convegence o a hee-monh a-he-money call opion on he spo inees ae wih and wihou conol vaiables Convegence using conol vaiables 3.8 A-he-money pice ( =.5) Ieaions (housand) he iniial em sucue is la a 1%, and he volailiy sucue o owad aes is given by (19) and () wih κ =.5, * =. and γ =. he nealy la line is he value compued using he conol vaiae used o enhance he speed o convegence. 4 he igues compae he ae o convegence wih and wihou he conol vaiable. In each igue, he nealy hoizonal line epesens he pice o he opion using he γ =model pice as a conol vaiable, while he moe volaile line is he simulaed pice obained wihou using he conol vaiable. he igues sugges ha he pices geneaed using he conol vaiable sabilise by, ieaions. HE NEW INERES RAE MODELS

12 1 I he economy is expanded o include isky asses, claims having eminal values deemined by boh isky asses and yields dawn om he em sucue can be eadily valued. o illusae his, assume ds( ) = µ ( ) d + dv( ), S( ) given (31) s s S( ) Hee µ s () is he insananeous di, s () is he insananeous volailiy, and dv() is he sandad Wiene incemen wih E[dw()dv()] = ρ vw d. Using sandad abiage agumens, he equivalen maingale measue unde which all secuiies ae piced is given by ds( ) = ( ) d+ ( ) dv( ) s S( ) d d d ( ) = κ( )[ (, ) ( )]+ ϕ( ) + (, ) d + ( ) dw ( ) ( ) dφ( ) = ( ) κ( ) φ( ) d As an example, conside he pice o a call opion on a sock when he owad aes ae given by (19) and () and he volailiy o he insananeous euns on he sock ae consan. Fo he special case whee γ =, he pice o a call opion on he sock wih sike X and mauiy s can be compued as E C (, s) S( ) N d XP(, τ ) N d = ( ) ( ) 1 d d 1 = = [ ] + ln S( ) XP(, s) [ ] ln S( ) XP(, s) y y 1 1 y y [ ] = + ( s, ) ρ ( ) ( s, ) d y s p vw s p s [ ] + ρ = s+ κs 3+ 4e e κ s 1 e s 3 κ κ [ ] κs κs vw s κs (3) he convegence o he a-he-money call opion o γ = 1, using he above model as a conol vaiable, is demonsaed in Figue 5. Fo he example pesened, we chose κ =.5, * =., s =., and ρ =.5. he nealy hoizonal line plos he pice geneaed using he γ =pice as a conol vaiable, while he vaiable line is he simulaed value o γ =.5. he above examples illusae how he Makovian popey o he em sucue can be ully exploied o yield simple algoihms o picing claims agains he em sucue. Conclusion Fo geneal volailiy sucues o owad aes, he evoluion o inees aes is no Makovian, and he enie pah is necessay o capue he dynamics o he pocess. Numeical models such as simulaion and laice pocedues o valuing claims have o manipulae he enie em sucue along each pah. In his chape we ideniied necessay and suicien condiions on he volailiy sucue o owad aes ha eliminae he make pice o isk om he bond-picing equaion while allowing pices o be ep- RISK BOOKS

13 13 5. Compaison o he ae o convegence o a hee-monh a-he-money call opion on he isky sock wih and wihou conol vaiables A-he-money call pice ( =.5) Simulaing he call on he asse,6 5,1 7,6 1,1 1,6 15,1 17,6,1,6 5,1 7,6 Ieaions he iniial em sucue is la a 1%, and he volailiy sucue o owad aes is given by (19) and () wih κ =.5, * =. and γ =. he volailiy o he sock s =. and he coelaion ρ =.5. he nealy la line is he value compued using he conol vaiae geneaed using (3) esened by a wo-sae Makov pocess, whee he spo inees ae is one o wo sae vaiables, he second epesening a measue o he hisoy o is evoluion. As a consequence, he buden o caying ull inomaion on he em sucue a each poin is damaically educed. he class o volailiy sucues ha pemi his educion is quie lage and imposes no esicions on he volailiy o spo inees aes. We also showed ha i he volailiy sucue o owad aes does no belong o his class hen pah dependence canno be capued by a wo-dimensional sae vaiable Makovian model. Moe pecisely, o make he pocess Makovian addiional assumpions on volailiies and/o on inveso behaviou will be equied. Given ha he volailiy sucue belongs o he esiced class, simple simulaion models wee esablished o picing inees ae claims. he eiciency o such algoihms wee impoved using conol vaiaes. I emains o uue eseach o ideniy non-deeminisic volailiy sucues belonging o he esiced class ha lead o analyical soluions o he eminal isk-neualised pocess. I also emains o uue eseach o esablish he pah econnecing laice appoximaions o he isk-neualised pocess. Such laices would pemi Ameican claims o be valued. Finally, he analysis pesened hee ocused on a single-aco model o he em sucue. hese esuls can eadily be exended o muliaco economies. In a wo-aco economy, o example, pah dependence will be capued by wo suicien saisics and he em sucue could be made Makovian wih espec o ou sae vaiables. Such exensions, ogehe wih empiical ess ae also he subjec o uue eseach. Appendix PROOF OF PROPOSIION 1 1(a). he necessiy o he esicion has aleady been oulined in he moivaion o he poposiion and will no be epeaed hee. he suiciency obains om he developmen o he bond picing equaion below. HE NEW INERES RAE MODELS

14 14 1(b). Fom equaion (11), le R( ; ) (, ) (, ) [ P ] = ( s, ) λ( s) ( s, ) ds ( s, ) dw( s) Hee, R(; ) is deined as he dieence beween he owad ae a dae and ha a he oiginal dae. Using he esicion imposed by (16) on (33), we obain (33) R( ; ) + ( s, ) ( s, ) ds= ( s, ) λ( s) ds+ ( s, ) dw( s) p k(, ) (34) Since he igh-hand side o (34) is independen o, ake =o obain which simpliies o R( ; ) R( ; ) + ( s, ) ( s, ) ds= + ( s, ) ( s, ) ds p p k(, ) k(, ) R( ; ) = k(, ) R( ; ) + ( s, ) s s ds [ (, ) (, ) p p ] = k(, ) R( ; ) + ( s, ) ( s, x) dx ds = (, ) R( ; ) + ( s, ) ds k(, x) dx (, ) (, ) = R( ; ) + ( s, ) ds (, ) = (, R ( ; ) (, ) ) (, ) (, ) p (, x) dx (, ) (, ) φ( ) whee φ( ) = ( s, ) ds Equivalenly, [ ] (, ) (, ) (, ) (, ) = (, ) + (, ) his leads o d R ( ; ) (, ) (, ) = (, ) p d (, ) (, ) (, ) φ ( ) (, ) p (, ) φ( ) p (, ) RISK BOOKS

15 15 Fuhe, om (5) we have d P s ds x R ( ; ) (, ) = exp (, ) + (, ) p dx (, ) φ ( ) (, ) (, x) dx which upon simpliicaion yields he esul. Obseve ha he sae vaiable, () which capues he inomaion elaing o he pah o inees aes is independen o bond mauiy. Fuhe, since he wo sae vaiables, () and (), pemi us o compue he pices o all discoun bonds, hey also pemi us o consuc he enie em sucue a ime. his hen allows one o compue he pices o all ohe inees coningen claims. p 1 An excepion o his case is when volailiy o all owad aes ae deeminisic. In his case simpliicaions esul and analyical soluions o ceain claims ae available. Fo example Jamshidian (1989) and Heah, Jaow and Moon (199) have developed em-sucue-consained models o deb opions in a dynamically complee make. unbull and Milne (1991) esablish an equilibium model whee pices ollow diusion pocesses bu ading daes ae simple picing mechanisms o a lage vaiey o inees ae claims. Excepions o his o couse ae he cases whee he volailiy sucue o all owad aes ae deeminisic. 3 Simila esuls held ue ove all paamee seings. Speciically, we esed he models wih κ anging om o 1, anging om.1 o.4 and γ om o 1. 4 Fo a discussion on conol vaiae echniques and hei use in picing opions see Boyle (1977) and Hull and Whie (1988). In ou conol vaiae scheme, we chose he paamee o he volailiy o he spo ae pocess: ie, = *, say, so ha i equalled he iniial volailiy o he inees ae unde he sochasic volailiy sucue. ha is, * = [()] γ. Fo he esuls pesened in Figues 3 and 4, we chose * a % and κ a.5. BIBLIOGRAPHY Boyle, P., 1977, Opions: A Mone Calo Appoach, Jounal o Financial Economics 4, pp Cavehill, A., 1994, When Is he Sho Rae Makovian?, Mahemaical Finance 4, pp. 35 1; epined as Chape o he pesen volume. Heah, D., R. Jaow and A. Moon, 199, Bond Picing and he em Sucue o Inees Raes: A Discee ime Appoximaion, Jounal o Financial and Quaniaive Analysis 5, pp Heah, D., R. Jaow and A. Moon, 199, Bond Picing and he em Sucue o Inees Raes: A New Mehodology o Coningen Claims Valuaion, Economeica 6, pp Ho,., and S. B. Lee, 1986, em Sucue Movemens and he Picing o Inees Rae Coningen Claims, Jounal o Finance 41, pp Hull, J., and A. Whie, 1988, he Use o he Conol Vaiae echnique in Opion Picing, Jounal o Financial and Quaniaive Analysis 3, pp Hull, J., and A. Whie, 1993, Bond Opion Picing Based on a Model o he Evoluion o Bond Pices, Advances in Fuues and Opions Reseach 6, pp Jamshidian, F., 1989, An Exac Bond Opion Fomula, Jounal o Finance 44, pp unbull, S., and F. Milne, 1991, A Simple Appoach o Inees-Rae Opion Picing, he Review o Financial Sudies 4, pp Vasicek, O., 1977, An Equilibium Chaaceizaion o he em Sucue, Jounal o Financial Economics 1, pp HE NEW INERES RAE MODELS