Trading Strategies for Sliding, Rollinghorizon, and Consol Bonds


 Benedict Carr
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1 Trading Sraegie for Sliding, Rollinghorizon, 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 coninuouime eup By definiion, a liding bond repreen he price proce of a dicoun bond wih a fixed ime o mauriy Example of meaurevalued rading raegie inroduced by Björk e al [2],[3] which are baed on he price proce of a liding bond are dicued In paricular, a elffinancing raegy which involve holding a any ime one uni of a liding bond i examined he wealh proce of hi raegy i referred o a he rollinghorizon bond In conra o he liding bond, which doe no repreen a radable ecuriy, he rollinghorizon bond or he rollingconol bond may play he role of a fixedincome ecuriy wih infinie lifepan in porfolio managemen problem Key word: dicoun bond, liding bond, rollinghorizon bond, conol, erm rucure model 1 Inroducion Le B, T and for he price a ime of a zerocoupon or dicoun bond which pay one uni of cah a ime T For impliciy, we hall wrie D, T o denoe he price a ime of a T +  mauriy zerocoupon bond, ie, D, T = B, +T Similarly, f, T and for he inananeou forward rae prevailing a ime for he fuure infinieimal ime period [T,T + dt ], alo we wrie r, T =f, +T In financial inerpreaion D,T repreen he price proce of a liding bond The erm liding bond emphaize he fac ha he ime o mauriy, T, i fixed hroughou hough arbirary, o ha we are inereed in he dynamic of he proce D, T when he acual ime,, pae I i inuiively clear ha he proce D,T doe no repreen he wealh of a elffinancing rading raegy Le u menion in hi regard ha in conra o he dicreeime eup, in which he concep of a elffinancing propery i raher obviou, he noion of a coninuouly rebalanced elffinancing rading raegy i quie echnical I hinge eiher on he noion of he Iô inegral wih repec o emimaringale ee, eg, he eminal paper by Harrion and Plika [8] or more generally on he noion of he inegral of a meaurevalued proce wih repec o a given family of emimaringale The laer concep wa formally inroduced and examined by Björk e al [2] ee alo [3] in he conex of coninuouime modelling of bond marke I i worhwhile o poin ou ha he price D,T doe no repreen he value proce of he ocalled rolling bond Typically, by a rolling bond one mean a rading raegy which aume ha: a a any inan, he whole wealh i inveed in dicoun bond which maure a a given fuure dae T, b a he mauriy dae T, he whole wealh i reinveed in dicoun bond of ome mauriy U>T,and o forh Generally peaking, he wealh proce of a dicreely adjued rolling bond i uniquely deermined by hree facor: he iniial endowmen, he e of reviion and mauriy dae <T 1 <T 2 <, and, of coure, by a given collecion of bond price BT j,t j+1,j=1, 2, 1
2 2 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond I i obviou ha he value of he rolling bond manifely repreen he wealh proce of a elffinancing rading raegy Therefore, in any arbiragefree marke model, he dicouned value of he rolling bond necearily follow a local maringale under he po maringale meaure P A cuomary, we aume ha P i a maringale meaure for he bond marke which correpond o he choice of he implied aving accoun a he numeraire ae A hown in Secion 3, in a fairly general coninuouime eup, he dynamic of D,T under P are governed by he expreion ee Goldy e al [6] or Brace e al [4] for relaed reul in he HJM framework dd, T =D, T r r, T d + dm, where r = f, =r, i he horerm rae of inere prevailing a ime, and M follow a local maringale under P The formula above uppor our iniial gue ha he proce D,Tdoe no repreen he wealh proce of a elffinancing rading raegy, o ha i canno be conidered a he price proce of a radable ecuriy The naural queion which arie in hi conex, i he form of a elffinancing rading raegy in which he oal wealh in coninuouly reinveed in bond wih a fixed horizon dae We hall referohewealhproce,u, T, of uch a raegy a he rollinghorizon bond proce We hall examine oher example of dynamic porfolio which involve inveing in bond wih fixed horizon dae The problem of a coninuouly rebalanced rading in a conol bond i alo addreed Thi reearch wa moivaed by he work of Bielecki and Plika [1] devoed o he rik eniive dynamic ae managemen More recenly, hey made an aemp o exend heir previou reul o he cae of financial model wih uncerain inere rae peronal communicaion 2 Dicreeime Cae A already menioned, he liding bond proce D, T =B, + T doe no correpond o he wealh of a elffinancing raegy for he definiion of a elffinancing raegy in a dicreeime eup, we refer he reader o [8] or ChapIII in [1] Neverhele, we hall check ha in a dicreeime eup i i no difficul o conruc a elffinancing rading raegy which aume, in paricular, holding a any inan one dicoun bond of mauriy + T Thi raegy, which i baed on a uiably adjued number of forward rae agreemen, involve alo an addiional cah componen If here i abence of arbirage beween all bond wih differen mauriy dae and cah ha i, when all forward rae are nonnegaive hi componen repreen a equence of nonnegaive cahflow 21 Sliding Bond Le u denoe by ˆr, T = ˆf, + T he imple forward rae prevailing a ime for rikle borrowing or lending over he fuure period [ + T, + T +1] In oher word, we aume ha a pary enering a forward rae agreemen a ime, i eniled o receive 1 + ˆr, T dollar a ime + T +1, for each dollar depoied a ime + T A imple noarbirage argumen how ha ˆr, T aifie 1+ˆr, T = B, + T B, + T +1 = D, T D, T +1 1 Le ˆr =ˆr, and for he inere rae which prevail a ime over he nex period ha i, he dicreeime horerm rae A uual, he aving accoun ˆB i inroduced by eing 1 ˆB = 1 + ˆr u 2 u= We hall focu on a paricular elffinancing rading raegy whoe wealh proce i inended o mimic he price of a liding bond We aume ha a any ime hi porfolio involve dicoun bond of mauriy + T To explain how he porfolio i rebalanced a any dae, le u aume ha a
3 UNSW, Repor NoS9711, Sepember 1997 hi verion: June ime we hold one uni of T mauriy bond and no cah Since boh he price B1,TofT mauriy bond and he price B1,T +1ofT + 1mauriy bond are no known a ime, o finance he purchae of T + 1mauriy bond a ime 1, we ener a ime a forward rae agreemen wih he nominal amoun $1, over he period [T,T +1] Combined wih one uni of T mauriy bond, he forward rae agreemen provide he amoun 1 + ˆf,T uni of cah a ime T +1, or equivalenly, 1 + ˆf,TB1,T + 1 uni of cah a ime 1 In hi way, we are able o buy one uni of T +1 mauriy bond a ime 1; in addiion, i remain ˆf,TB1,T + 1 uni of cah a ime 1 in our porfolio 1 Proceeding in an analogou manner, we conruc a elffinancing raegy which involve a ime 2 one uni of T + 2mauriy bond, combined wih ˆf,TB1,T ˆr 1 + ˆf1,T +1B2,T +2=ˆr,TD1,T1 + ˆr 1 +ˆr1,TD2,T uni of cah, and o forh More generally, uch a raegy denoed by φ in wha follow amoun o holding a any dae one uni of + T mauriy zerocoupon bond and A uni of cah, where A = 1 ˆru 1,TDu, T 1 + ˆr v 3 I i eviden ha he wealh of φ aifie V φ =A + D, T for any Le ˆV φ and ˆD,T and for dicouned procee; ie, ˆV φ =V φ and ˆD, 1 T =D, T ˆB Propoiion 21 For a fixed T, le φ be he elffinancing rading raegy which involve a any dae one bond of mauriy + T, combined wih A uni of cah and a uni poiion in a uiable forward rae agreemen If V φ =B,T hen he following aemen are rue i The wealh proce of φ equal V φ =D, T + The dicouned wealh of φ aifie ii For any we have ˆV φ = ˆD, T + ˆB 1 v=u 1 ˆru 1,TDu, T 1 + ˆr v, 4 v=u ˆru 1,T ˆDu, T, 5 D +1,T D, T = ˆr ˆr, T 1+ˆr, T D, T + M, 6 where M i a maringale difference proce under P, ie, E P M F = for every iii If ˆr,T follow a nonnegaive proce hen ˆD,T follow a upermaringale under P Proof Equaliy 4 i a raighforward conequence of 3, combined wih he definiion of he raegy φ Formula 5 follow eaily from 2 and 4 To eablih 6, oberve ha from 5 we obain ˆV +1 φ ˆV φ =1+ˆr, T ˆD +1,T ˆD, T, 7 or equivalenly, 1 + ˆr, T 1 B +1 ˆV+1 φ ˆV φ = D +1,T 1 + ˆr, T ˆr D, T 1 For impliciy, we aume ha ˆf,T i nonnegaive Oherwie, ˆf,TB1,T + 1 repreen he amoun of cah which i borrowed a ime 1 The ame remark applie o any dae
4 4 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond Since φ i elffinancing, he dicouned wealh proce ˆV φ follow a maringale under P, and hu noice ha ˆB +1 i F meaurable 1 + ˆr, T 1 ˆB+1 ˆV+1 φ ˆV φ F = E P Conequenly, D +1,T= 1+ˆr 1+ˆr, T D, T + M, where M i a maringale difference under P For iii, noice ha ince he lefhand ide in 7 alo define a maringale difference, we have 1 + ˆr, T E P ˆD +1,T F ˆD, T = Therefore ˆD,T follow a P upermaringale when ˆr,T follow a nonnegaive proce Le u conider he pecial cae of T =1 Even in hi cae, he raegy examined in Propoiion 21 doe no correpond o a rolling bond, or equivalenly, o a aving accoun a already menioned, in a dicreeime framework rolling over oneperiod bond i equivalen o horerm cah invemen For inance, if we ar wih B, 1 of cah inveed in bond mauring a ime 1, hen we will poe one uni of cah a ime 1 On he oher hand, he wealh a ime 1 of he raegy φ equal 1 + ˆr,TD1, 1 = 1 + ˆf,TB1, 2 = 1 + ˆf,T1 + ˆr 1 1, and hu i repreen a random variable, in general Remark I i worhwhile o noe ha he raegy φ of Propoiion 21 implicily aume ha FRA are available for rade a zero co Indeed, a any ime our raegy involve no only uni poiion in liding bond combined wih he appropriae amoun of cah, bu alo a uni poiion in a judiciouly choen FRA If FRA were no aumed o be among radable ae in our model, he ue of ynheic FRA would of coure reul in an undeirable change of bond poiion in our porfolio φ We may aume, however, ha only bond of all mauriie and cah are radable ae in our economy Thi aumpion force u o modify he amoun of cah hold a any ime combined, a before, wih one uni of + T mauriy bond The amoun of cah Ã held a ime can now be eaily checked o aify 1 Ã = ˆru, T 1Du, T 1 + ˆr v Indeed, we now imply ell a ime 1 a bond of mauriy + T 1 held from he previou dae, and we buy a bond of mauriy + T, hence he incremenal cahflow a ime equal cf 1 B, + T 1 B, + T =D, T 1 D, T =ˆr, T 1D, T We wrie φ o denoe he elffinancing rading raegy obained in hi way, o ha obviouly V φ =D, T +Ã for any Noe ha we have now wo differen bu formally equivalen dynamic porfolio, φ and φ, which boh involve liding bond in an ougoing porfolio a any dae I eem naural o refer o φ a he parially hedged liding bond porfolio; acually, he cah componen A of φ behave in a le uncerain way han he correponding cah componen Ã of φ more preciely, he flow of cah occuring a ime, expreed in uni of he liding bond, follow a predicable proce We hall briefly examine relaionhip beween boh procee, A and Ã Fir, a expeced,2 he proce A Ã i a maringale under he maringale meaure P, when dicouned by he aving accoun ˆB To check hi, noe ha v=u A +1 =1+ˆr A +ˆr, T D +1,T 2 Oherwie, holding a long poiion in φ and a hor poiion in φ or vice vera would be an arbirage opporuniy ¾
5 UNSW, Repor NoS9711, Sepember 1997 hi verion: June and Ã +1 =1+ˆr Ã +ˆr +1,T 1D +1,T=1+ˆr Ã + D +1,T 1 D +1,T Therefore E ˆB 1 P +1 A +1 Ã+1 F = ˆB 1 A Ã, ince in view of he maringale propery of dicouned bond price under P E ˆB 1 P +1 1+ˆr, T D+1,T D+1,T 1 F = ˆB 1 1+ˆr, T D, T +1 D, T =, where he la equaliy follow from 1 Second, i can be eaily checked ha boh raegie lead o he ame dynamic in he coninuouime limi ee Secion 31 below 3 22 Rollinghorizon Bond In conra o he previou ecion, we hall now aume ha he oal wealh of a elffinancing raegy, denoed by ψ, i reinveed a any dae in dicoun bond mauring a ime +T ha i, no cah componen i preen For a fixed T, we wrie V ψ =U, T and we refer o he proce U,T a he dicreeime rollinghorizon bond Aume ha a ime we hold α uni of dicoun bond of mauriy T, o ha he iniial wealh of ψ equal 4 U,T=αB,T A ime 1, he wealh i reinveed a ime 1 in bond mauring a ime T +1, o ha U1,T=αB1,T=αD1,T Conequenly, a ime 2 we have B1,T B1,T +1 D1,T 1 = αd1,t D1,T U2,T=αB2,T +1 D1,T 1 D1,T = αd2,t D1,T 1 D1,T D2,T 1 D2,T Simple inducion argumen how ha for any we have 1 U, T =αb, T + 1 Du, T 1 Du, T = αd, T Du, T 1 Du, T The la formula lead o he following imple reul we may and do aume, wihou lo of generaliy, ha α=1 Propoiion 22 For any fixed T, he wealh proce U,T of he dicreeime rollinghorizon bond aifie U, T =D, T 1+ˆru, T 1, The dicouned wealh Û, T =U, T ˆB 1 Û, T = ˆD, T equal 1+ˆru, T 1, 8 Le u reurn o he liding bond proce D,T The abence of arbirage implie ha D, T +1=B, + T +1=E P B +1,+ T +1 D +1,T F = E P F, 9 1+ˆr 1+ˆr 3 We are indebed o he anonymou referee for hi obervaion, a well a for he uggeion o examine he pure bond/cah raegy φ 4 Of coure, he iniial condiion play no eenial role in wha follow
6 6 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond where P and for he po maringale meaure Combining 9 wih 1, we obain D, T =E P D +1,T 1+ˆr, T 1+ˆr F 1 On he oher hand, ince he dicouned wealh Û,T necearily follow a maringale under P, equaliy 8 give 1+ˆr +1,T 1 D, T =E P D +1,T F 11 1+ˆr To derive 1 from 11 or vice vera, i i enough o how ha or equivalenly, E P D +1,T1 + ˆr, T F = E P D +1,T1 + ˆr +1,T 1 F, E P B +1,+ T F = E P B +1,+ T +1 F 1 + ˆf, + T Dividing boh ide of he la equaliy by 1 + ˆr and uing 9, we obain equaliy B, + T =B, + T ˆf, + T, which coincide wih he definiion of ˆf, + T cf 1 23 Conol Bond By definiion, a dicreeime conol bond known alo a a perpeual bond or perpeuiy iafixed income ecuriy wih infinie lifepan, which pay one uni of cah a any fuure dae Therefore, a any dae, a conol bond may idenified wih an infinie porfolio of uni dicoun bond wih mauriy dae, +1, Conequenly, he conol price D i a ochaic proce given by he formula in wha follow, we hall implicily aume ha he erie which define he conol price converge D = B, T = D, T, T = in paricular, lim T B, T = for every The following propoiion deal wih a elffinancing raegy which involve a conol bond Propoiion 23 Le ψ be a elffinancing rading raegy which involve holding a any dae one uni of a conol bond and A uni of cah If A =hen A = ˆB The conol price D aifie ˆB 1 u T = ˆru 1,TDu, T, 12 T = D +1 =1+ˆr D 1 + N, 13 where N i a maringale difference proce under P The dicouned price ˆD conol bond aifie ˆD +1 = ˆD where ˆN i a maringale difference proce under P = D ˆB 1 of a ˆB 1 + ˆN, 14
7 UNSW, Repor NoS9711, Sepember 1997 hi verion: June Proof Equaliy 12 follow eaily from Propoiion 21 To derive 13, oberve fir ha E P D+1 F =E P B +1,T F =1+ˆr B, T =1+ˆr D 1 T =+1 T =+1 On he oher hand, uing 1, we obain E P D+1 F =E P D +1,T F =1+ˆr T = D ˆr, T 1+ˆr, T 1 D, T T = A comparion of he righhand ide in he formulae above lead o he following equaliy ˆr, T 1+ˆr, T 1 D, T = 1 15 T = We are now in a poiion o eablih 13 Noe fir ha E P A +1 F = ˆB +1 E P +1 ˆB 1 u ˆru 1,TDu, T F T = = 1+ ˆr A + E P ˆr, T D +1,T F = 1+ ˆr A +1+ˆr = 1+ ˆr A +1, T = ˆr, T 1+ˆr, T 1 D, T where he hird equaliy follow from 1, and in he la equaliy we have ued 15 Since obviouly E P V +1 ψ F =E P A +1 + D +1 F =1+ˆr V ψ =1+ˆr A + D, equaliy 13 eaily follow Formula 14 i a raighforward conequence of Rollingconol Bond By a dicreeime rollingconol bond we mean he wealh proce a elffinancing rading raegy which aume ha a any dae he oal wealh i inveed in conol bond We find i convenien, however, o modify lighly he definiion of a conol bond and hu alo of i price proce, namely, we pu 5 D = B, T = D, T T =+1 Thi mean ha he fir paymen from he conol bond ake place a he nex dae Noe ha rivially D = D 1 for any Uing he ame argumen a in he derivaion of 13, we obain T = T =1 E P D +1 F =1+ˆr D B, +1 Le U denoe he wealh proce of a elffinancing rading raegy which involve a any dae only invemen in modified conol bond wih price proce D by convenion, a he iniial dae we ar wih one uni of a modified conol bond The proce U i referred o a he dicreeime rollingconol bond 5 Le u re ha he diincion beween he conol bond and he modified conol bond i irrelevan in a coninuouime eup In a dicreeime framework, he choice of a modified conol bond lead o more inuiive rading raegie ¾
8 8 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond Propoiion 24 The wealh proce U of he rollingconol bond equal U = D D u +1 D u = D 1 D u D u 1, Proof By aumpion, U = D = T =1 B,T A ime 1 we have U 1 = B1,T=D1 = α 1 D 1, T =1 where α 1 = D1 / D 1 Similarly, a ime 2 we obain U2 = α 1 B2,T=α 1 D2 = α 1 α 2 D 2, T =2 where α 2 = D 2 / D 2 In general, we have he following equaliy U = D The aered formula now eaily follow α u = D ¾ D u D u 3 Coninuouime Cae In wha follow, we hall aume ha he inananeou po and forward inere rae exi Our main goal i o examine coninuouime raegie and value procee aociaed wih liding, rolling and conol bond wihou making reference o any paricular model We denoe by f, T he inananeou forward rae prevailing a ime for he fuure infinieimal ime period [T,T +dt ], and we wrie r, T =f, + T The following relaionhip are hu obviou T B, T =exp f, d, f, T = ln B, T 16 T I i naural o aume ha r = f, =r, repreen he horerm rae of inere Therefore he aving accoun proce or accumulaion facor equal B =exp r d =exp r, d 17 We aume hroughou ha he family of bond price B, T, T, i regular meaning ha for every he random variable B, ake value in he pace of coninuou funcion on R +, and he following Aumpion A1 i aified Aumpion A1 Inananeou forward rae f,t follow nonnegaive coninuou adaped procee on he underlying probabiliy pace Ω, F R+, P For any T he proce r, T i coninuou, and lim U T up > r, U r, T = We aume ha he model i arbiragefree; more preciely, we poulae ha here exi a probabiliy meaure P, equivalen o he original probabiliy P, and uch ha dicouned price B, T =B, T B 1 follow local maringale under P When working in he coninuouime eup, we find i convenien o refer o he general framework developed by Björk e al [2] Following [2], we aume ha he bond componen of a rading raegy i repreened by an adaped, ochaic proce µ, which ake value in he pace M = MR + of igned meaure on R + The pace M i endowed wih he oal variaion norm, denoed by V ; he aociaed Borel σfield i denoed
9 UNSW, Repor NoS9711, Sepember 1997 hi verion: June by BM We aume ha a meaurevalued proce µ i predicable, meaning ha he mapping µ :Ω R +, P M, BM i meaurable, where P and for he σfield of predicable ube of Ω R + A any dae, he wealh of a porfolio µ equal V µ =µ B, = B, z dµ z If, in addiion, a raegy φ involve alo a cah componen denoed by η o ha φ =µ, η hen he wealh of φ aifie V φ =µ B, +η B, R + For any fixed T >, a raegy µ = δ +T wih null cah componen amoun o holding a any ime a uni dicoun bond mauring a + T, and hu V µ =D, + T The elffinancing propery of a meaurevalued rading raegy propoed in [2] read a follow: we ay ha φ =µ, η i elffinancing if he wealh proce V φ aifie V φ =V φ+ µ db, + η db for every R +, where he fir inegral i underood in he ene of Björk e al [2], and he econd i he LebegueSielje inegral Le u emphaize ha in order o define he Iô ochaic inegral of a meaurevalued proce wih repec o a given family of emimaringale, one need o impoe uiable regulariy condiion Aumpion 21 and 22 in [2] which are reproduced here for he reader convenience A meaurevalued proce µ i called an elemenary inegrand if i admi he repreenaion n 1 µ = ν I Γ [, 1]ω, + ν i I Γi ] i, i+1]ω,, R +, i=1 where = < 1 << n <, ν i Mand Γ i F i for i =,,n The following Aumpion A2 will be in force from now on Aumpion A2 There exi a e A wih PA = 1 uch ha for any ω A and every µ MR + he realvalued proce µ B, i coninuou There exi a predicable random meaure κdu, d =l dud on U R +, U BR +, where U, U i a Luin pace, and a meaurable mapping p :U Ω R +, U P R +, BR +, uch ha: a pu, ω,, i weakly coninuou, and i a eminorm on M, uch ha pu, ω,, µ µ V, b K =1+κ[,] U < for every >, and for any T> here exi a conan C T uch ha for any opping ime τ T and any elemenary inegrand µ E P µ u dbu, 2 τ C T E P K τ p 2 u, ω,, µ κdu, d We wrie up τ q 2 µ = E P K U U p 2 u, ω,, µ κdu, d and we conider he pace L 2 loc M of predicable meaurevalued procee µ uch ha q µ < for every > I i hown in [2] ha he ochaic inegral µ db, i welldefined for any µ L 2 loc M For furher properie of hi inegral, we refer o [2] Le u only oberve ha in erm of dicouned procee B, T =B, T B 1 and V φ =V φb 1 we have for every R + V φ =µ B, +η and V φ =V + µ db,
10 1 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond Thi mean ha for any meaurevalued proce µ L 2 loc M, if η i choen o aify η = V + µ db, µ B,, R +, 18 hen he raegy φ =µ, η i elffinancing, wih V φ =V The following imple reul will prove ueful Lemma 31 Le µ n be a equence of meaurevalued elemenary inegrand Aume ha here exi a meaurevalued proce µ L 2 loc M uch ha lim q µ n µ =for every R + For any n, le η n be given by he formula η n = V + µ n db, µ n B,, R + Aume ha lim η n = η for every R + Then he raegy φ =µ, η i elffinancing and V φ = lim V φ n for every R + In paricular, he componen η aifie equaliy 18 Proof I i enough o oberve ha in view of our aumpion we have lim All aemen are hu raher obviou µ n B, = ¾ µ B,, R + 31 Sliding Bond In hi ecion, our goal i o derive he dynamic of he liding bond in a general coninuouime framework Propoiion 31 i Le A be a oluion o he following random ordinary differenial equaion ha i, da = r A + r, T D, T d, A R, 19 A = B A + B 1 r, T D, T d 2 Then he rading raegy φ =µ, η, where µ = δ +T and η = A B 1 wih A given by 2 i elffinancing ii The dynamic of D,T are governed by he following expreion where M follow a local maringale under P dd, T = r r, T D, T d + dm, 21 Proof Conider he rading raegy φ =µ, η inroduced in he aemen of he propoiion, o ha a any inan he wealh of φ equal V φ =D, T +A Our goal i o how ha φ i elffinancing For hi purpoe, we hall examine a uiable equence of rading raegie wih a finie number of reviion Fix > and conider a equence n =<n 1 < < n n = wih he meh max n n i+1 n i which end o a n We reric our aenion o he inerval [, ], and we inroduce a uiable counerpar of a dicreeime elffinancing rading raegy φ inroduced in Propoiion 21 For any naural n we denoe by φ n =µ n,a n hi raegy; in paricular, for any n he wealh a ime of φ n equal D,T+A for every n Furhermore, if a ime n i he wealh of φ n equal V n i φ n =D n i,t+an, where A n n repreen he amoun of cah i n i held a ime n i, hen a he nex rebalancemen dae, n i+1, he wealh of φn i given by he formula V n i+1 φ n =A n n i n exp i+1 n i T + n r u du + D n i+1 i+1,texp f n i,u du T + n i
11 UNSW, Repor NoS9711, Sepember 1997 hi verion: June Since V n i+1 φ n =D n i+1,t+an n i+1, i i clear ha A n n i+1 A n n i = A n n i { n i+1 } { exp r u du 1 + D n T + n i+1,t i+1 n i } exp r n i,u du 1 n T i Conequenly lim An A = lim = lim = n 1 i= n 1 i= A n n A n i+1 n i r n i An n i ru A u + ru, T Du, T du ++r n i,tdn i+1,t n i+1 n i +on i+1 n i More generally, for every [, ]wehave [, ]wehavelim A n = A where A i given by 2 To conclude, i i enough o check ha he equence of elemenary inegrand µ n n 1 µ n = δ T I [, n 1 ]+ δ n i +T I ] n i, n i+1 ], [, ], i=1 aifie he aumpion of Lemma 31 wih µ = δ +T Thi i raher andard and hu we omi he deail we ue he inequaliy pu, ω,, µ µ V For he econd aemen, we noe ha ince µ i elffinancing, i dicouned wealh V µ = V µb 1 necaarily follow a local maringale under he po maringale meaure P, ie, V µ = V µ+ M, where M follow a local maringale under P Equivalenly, we have dv µ =r V µ d + B d M The wealh of µ equal V µ =D, T +A, and hu manifely d D, T +A = r D, T +A d + B d M Taking ino accoun formula 19, we obain 21 Alernaively, we could have ued in he proof of Propoiion 31 he dicreeime raegy φ inroduced in Secion 21 Thi would give u he following relaionhip or equivalenly, V n i+1 φ n =A n n i ¾ n i+1 exp r u du + B n i+1,n i + T, n i V n i+1 φ n =A n n i n exp i+1 n i r u du n + D n T + i+1,texp i+1 T + n i f n i+1,u du Finally, we would ge V n i+1 φ n =A n n i n exp i+1 n i r u du + D n T i+1,texp r n i+1,u du T n i+1 n i Thi would lead o he ame coninuouime dynamic of A We feel, however, ha he ue of FRA make he argumen impler Remark A hown in [2], under mild aumpion, he wealh proce of a elffinancing rading raegy which relie on he coninuou reinvemen of he whole curren wealh in ju mauring
12 12 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond dicoun bond coincide wih he aving accoun proce B The fir aemen may hu be reformulaed a follow: he meaurevalued raegy µ = A δ + δ +T, where A i given by 2 i elffinancing In he HeahJarrowMoron [7] framework, we poulae ha for any mauriy dae T he inananeou forward rae f,t aifie df, T =α, T d + σ, T dw, 22 where α, T andσ, T are adaped ochaic procee, and W i a andard Brownian moion defined on a filered probabiliy pace Ω, F R+, P, where P i he acual probabiliy I i well known ha in he ocalled rikneural world ie, under he po maringale meaure P he dynamic of he price proce of T mauriy bond are given by he following expreion ee Heah e al [7] db, T =B, T r d + b, T dw, 23 where W follow a andard Brownian moion under P, and b, T = T σ, u du Le u recall he following reul ee Appendix A for he proof Theorem 31 In arbiragefree HJM framework, he dynamic of he price proce D,T of he liding bond, under he po maringale meaure P, are dd, T =D, T r r, T d + b, + T dw 24 Taking he la reul for graned, we may formally rederive expreion 19 in he HJM framework Corollary 31 Conider a elffinancing rading raegy φ involving a any ime one uni of + T mauriy bond and A = η B uni of cah Then 19 hold Proof Since V φ =η B + D, T, in view of 24 we ge dv φ = η db + B dη + dd, T r = η db + B dη + B, + T r, T d + b, + T dw On he oher hand, he elffinancing condiion, when formally applied o he meaurevalued raegy φ =µ,η =δ +T,η give dv φ = η db + δ +T db, =η db + db, U U=+T = η db + B, + T r d + B, + T b, + T dw Combining boh repreenaion of dv φ, we obain B dη = D, T r, T d, which i equivalen o 19 ¾ 32 Rollinghorizon Bond Our nex aim i o examine a coninuouime elffinancing raegy which involve, a any dae, only dicoun bond of mauriy + T Such a dynamic porfolio or raher, i wealh proce denoed by U,T i referred o a he rollinghorizon bond
13 UNSW, Repor NoS9711, Sepember 1997 hi verion: June Propoiion 32 Conider a elffinancing rading raegy ψ =C δ +T, which aume ha a any ime all wealh i inveed in dicoun bond of mauriy + T The wealh proce of ψ equal V ψ =C D, T, where C = C exp r, T d, R +, 25 wih iniial condiion C = V D 1,T Furhermore, he proce U, T =V ψ aifie T U, T =U,T+ r U, T d + M, R +, 26 where M follow a local maringale under P Proof We hall make ue of Lemma 31, wih η n = and he equence of dicreely rebalanced porfolio µ n which mimic he dicreeime porfolio examined in Secion 22 We may and do aume, wihou lo of generaliy, ha V = D,T Forafixed and any naural n, we conider a finie number of reviion dae n =<n 1 <<n n = wih meh ending o For dicreely rebalanced approximaion ψ n =µ n, of ψ we have V ψ n =D,T=B,Tand Similarly, V n 1 ψ n =B n 1,T=D n 1,T Bn 1, n + T B n 1,n 1 + T = Dn 1,T Dn 1, n n 1 + T D n 1,T V n 2 ψ n =B n 2,n 1 + T Dn 1,n n 1 + T D n 1,T = D n 2,T Dn 1,n n 1 + T D n 1,T D n 2,n 1 n 2 + T D n 2,T More generally, for any i we have i V n i ψ n =D n i,t Uing 16, we obain j=1 B n j,n j 1 + T B n j,n j + T = D n i,t i j=1 D n j,n j 1 n j + T D n j,t V ψ n =D, T for every [, ] Conequenly, lim V ψ n = n j=1 n j +T exp f n n j 1 +T j, d, lim D, T exp n n j f n j=1 n j 1 j,+ T d = D, T exp r, T d Le u denoe V = C D, T We need o how ha for he equence of meaurevalued elemenary inegrand n 1 µ n = δ T I [, n 1 ]+ V ψ n δ n i +T I ] n i, n i+1 ], [, ], i=1 he aumpion of Lemma 31 are me wih µ = V δ +T Once again hi preen no difficulie Formula 26 i a raighforward conequence of he elffinancing propery of ψ ¾ Remark In he HJM eup, i i poible o how ha du, T =U, T r d + b, + T dw,
14 14 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond ha i, o expre he volailiy coefficien of he rollinghorizon bond in erm of bond volailiie In he pecial cae of he ocalled Gauian HJM model ie, when he coefficien σ in 22 i nonrandom, he volailiy of he rollinghorizon bond follow a deerminiic funcion Thi doe no mean, however, ha he rollinghorizon bond can be eaily ued o hedge andard bond opion Conider, for inance, a European call opion wih expiraion dae U and rike K, wrienona dicoun bond mauring a U +T The value a ime U of he rollinghorizon bond i C U BU, U +T Hence he opion payoff, which equal BU, U + T K +, canno be eaily expreed in erm of he rollinghorizon bond value The following problem arie naurally in he preen conex: given a proce G H, rep find a proce H G, rep uch ha he raegy φ =µ,, where µ equal µ = H δ + G δ +T i elffinancing I appear ha if G or H follow a coninuou emimaringale, he anwer o hi queion i fairly explici, a he following reul how Corollary 32 i Conider he rading raegy φ =µ,, where µ = H δ + G δ +T For any coninuou emimaringale G, if H equal H = B H B 1 D, T dg r, T G d 27 hen φ i elffinancing Converely, given a coninuou emimaringale H, if G equal G = C G C 1 D 1, T dh r H d, 28 where C i given by 25 wih C =1, hen φ i elffinancing ii Le T 1 T 2 be ricly poiive number For a given coninuou emimaringale H, he rading raegy φ =µ,, where µ = G δ +T1 + H δ +T2 i elffinancing if G equal G = C G C 1 D 1, T 1 D, T 2 dh r, T 2 H d Proof Noe ha φ i equivalen o he raegy G ν,h, where G = G C 1,ν = C δ +T and H = H B 1 From Propoiion 32, we know he meaurevalued raegy C δ +T, i elffinancing and i wealh proce equal U, T o ha we may and do conider U,T a he price proce of a primary ecuriy The problem i hu reduced o he andard cae wih only wo primary ecuriie: he rollinghorizon bond and he aving accoun The wealh proce of φ equal V φ = G U, T +H B, 29 and he elffinancing propery read dv φ = G du, T +H db Differeniaing he righhand ide in 29 and comparing wih he la formula, we obain dh = B 1 U, T d G = B 1 C D, T dg C 1 To eablih formula 27 uing Iô produc rule, i i enough o oberve ha dc 1 = r, T C 1 The proof of 28 goe along he imilar line Aerion ii follow eaily from i 33 Conol Bond In a coninuouime framework, i i cuomary ee, for inance, Duffie e al [5] o informally inroduce a conol bond wih uni principal value a a fixedincome ecuriy which pay one uni of cah per uni of ime forever; he principal value of a conol bond i never repaid Such a decripion ¾ d
15 UNSW, Repor NoS9711, Sepember 1997 hi verion: June doe no uniquely pecify a conol bond, however Indeed, i i eay o produce an example of a ecuriy ha pay dividend a a conan rae, bu doe no coincide wih a conol bond The poin i ha he volailiy coefficien in he dynamic of he conol bond price i cloely relaed o he dynamic of dicoun bond Since we do no aume here any paricular model for he bond price, we ake, a ypical in hi noe, an alernae roue which relie on a uiable approximaion of he coninuouime conol bond, which i formally defined hrough equaliy 32 below Recall ha in a dicreeime framework, a conol bond wa repreened by a uniform infinie porfolio of liding bond Therefore, conrucion of elffinancing rading raegie involving a conol bond were raighforward We hall now examine coninuouime counerpar of hee raegie For hi purpoe, le u denoe Ti n = i/n, for any naural n and i =, 1, By definiion, he n h conol bond i repreened by he dynamic porfolio φ n =µ n,, where µ n = n 1 δ +T n i i= The wealh of he porfolio µ n ha i, he price of he n h conol equal D n = V µ n =n 1 D, Ti n, 3 provided ha he erie converge For a fixed naural number K, we inroduce he quaiconol bond by eing φ n,k =µ n,k,, where i= nk µ n,k = n 1 δ +T n i i= The following reul i a raighforward conequence of Propoiion 23, 31, and Fubini heorem Lemma 32 For any naural n and any K>, he raegy ψ n,k =µ n,k,η n,k, where η n,k = n 1 B 1 u i elffinancing Similarly, he raegy ψ n =µ n,η n, where nk i= r, T n i D, T n i d 31 η n = n 1 B 1 i= r, T n i D, T n i d i elffinancing For a fixed K>, we inroduce he proce D K, referred o a he price of a quaiconol bond, by eing K D K = lim Dn,K = D, T dt Similarly, he price proce D of a coninuouime conol bond i formally inroduced by eing D = lim D n = D, T dt, 32 provided ha he la inegral i finie 6 We aume ha lim T D, T = for any For any Borel e A R +, we wrie λ A o denoe he Lebegue meaure rericed o A 6 Since forward rae are nonnegaive, o ha D, i nonincreaing, hi inegral i finie if and only if he erie 3 converge for ome n
16 16 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond Propoiion 33 i For any K>, he raegy φ K =µ K,η K, where µ K = λ [,+K] and η K equal η K = B 1 1 D, K d, R+, 33 i elffinancing The price proce D K of a quaiconol bond aifie dd K = r D K + D, K 1 d + dm, 34 where M follow a local maringale under P ii Trading raegy φ =µ, η, where µ = λ [,+ and η = B 1 d i elffinancing The dynamic of he price proce D of a conol bond are governed by he expreion dd = r D 1 d + dn, 35 where N follow a local maringale under P Proof The proof of i hinge on he approximaion of he raegy φ hrough he equence of elffinancing raegie inroduced in Lemma 32 Since cf 31 η K = lim n 1 B 1 nk i= r, T n i D, T n i d, o eablih 33, i i enough o oberve ha lim I n = lim nk i= = lim n 1 = lim n 1 n 1 r, T n i D, T n i nk i= nk i= T n ru, Ti n i exp r, T dt K D T, Ti n = D T, T dt =1 D, K In view of 33, we have da K =r A K +1 D, K d, where A K = η K B repreen he cah componen a ime Since he elffinancing propery of he limiing raegy φ can be deduced from Lemma 31, we hall concenrae on he derivaion of 34 Since φ i elffinancing, i wealh proce aifie dv φ =r V φ d + dm, where M follow a local maringale under P Therefore and hu we ge da K + D K = r A K +1 D, K d + dd K = r A K + D K d + dm, dd K = r D K + D, K 1 d + dm a required Le u kech he proof of ii Since µ n µ V = for every naural n, Lemma 32 i no direcly applicable, and hu a lighly modified approach i needed Oberve ha for any fixed we have = e z D, z de z = D, z de z D where D, z =e z D, z forz R + For any >, le µ aify d µ z =e z I [,+ dz 36
17 UNSW, Repor NoS9711, Sepember 1997 hi verion: June We conider he porfolio ψ = µ,η, where η = B 1 d and in he rading raegy µ he role of he underlying ecuriie i played by he family of recaled liding bond, 7 and oher properie wih price procee D, T =e T D, T Thi mean ha he wealh of ψ equal V ψ = e z B, z d µ z+η B = D, z dz + η B = D + η B To approximae µ, we ake he following equence of meaurevalued porfolio µ n = e T n i e T n i+1 δ+t n i I i eay o check ha he raegy which combine µ n wih η n, where i= η n = B 1 u i= e T n i e T n i+1 ru, T n i Du, T n i du, i elffinancing Furhermore lim J n = lim = lim i= i= = lim 1 e T n i T n i+1 ru, T n i Du, T n i T n 1 e T n i T n i i+1 ru, T n i exp ru, T dt 1 e T n i T n i+1 D T u, Ti n = i= D T u, T dt =1 Thi mean ha he proce η = lim η n aifie η = B 1 u du, a expeced The proce Ã = η B repreen he amoun of cah which, when held wih one conol bond, give a elffinancing rading raegy Oberve ha dã =r Ã +1 d, o ha he claim ha a conol bond pay one uni of cah per uni of ime i indeed juified Finally, o eablih 35, i i enough o oberve ha dã + D = r Ã +1 d + dd = r Ã + D d + dm ¾ Thi complee he proof Remark Le u reurn once again o he HJM framework Muiela [9] proved in he imehomogeneou Gauian HJM eup ie, when σ, T = σt follow a deerminiic funcion ha he dynamic under he po maringale meaure P of conol price D are governed by he expreion wih dd = r D 1 d + c dw c = D, T b, + T dt, where b, + T = T σu du i he volailiy of he liding bond cf 24 7 Noe ha he definiion of a meaurevalued porfolio may be formally applied o any family of procee, under mild regulariy aumpion
18 18 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond 34 Rollingconol Bond We hall now examine a elffinancing rading raegy which aume ha he oal wealh i coninuouly reinveed in conol bond The wealh proce of hi raegy i referred o a he rollingconol bond; wedenoeibyu Propoiion 34 The raegy which aume holding G uni of conol bond a ime, where G =exp D 1 d, R +, i elffinancing I wealh proce U = G D aifie du = r U d + dm for ome P local maringale M Proof Once again we kech he proof The key obervaion i ha for any we may focu on a equence of dicreely adjued raegie of he paricular form, a decribed in he proof of Propoiion 24 The wealh proce U admi he following aympoic repreenaion for eae of noaion, we aume ha he grid i uniform, ie, n i = i/n U Furhermore, we have lim ln Gn = lim ln n a expeced ¾ Reference = lim D G n =D lim D n i D i=1 n i n = lim n i=1 ln D n i n D n i D i=1 n i n ln D n i n = D 1 d [1] Bielecki, TR, and Plika, SR 1997 Rik eniive dynamic ae managemen Working paper ubmied o J Appl Mah Opim [2] Björk, T, Di Mai, G, Kabanov, Y, and Runggaldier, W 1997 Toward a general heory of bond marke Finance Socha 1, [3] Björk, T, Kabanov, Y, and Runggaldier, W 1997 Bond marke rucure in he preence of marked poin procee Mah Finance 7, [4] Brace, A, G aarek, D, and Muiela, M 1997 The marke model of inere rae dynamic Mah Finance 7, [5] Duffie, D, Ma, J, and Yong, J 1995 Black conol rae conjecure Ann Appl Probab 5, [6] Goldy, B, Muiela, M, and Sondermann, D 1994 Lognormaliy of rae and erm rucure model Preprin, Univeriy of New Souh Wale [7] Heah, D, Jarrow, R, and Moron, A 1992 Bond pricing and he erm rucure of inere rae: A new mehodology for coningen claim valuaion Economerica 6, [8] Harrion, JM, and Plika, SR 1981 Maringale and ochaic inegral in he heory of coninuou rading Sochaic Proce Appl 11, [9] Muiela, M 1993 Sochaic PDE and erm rucure model Journée Inernaionale de Finance, IGRAFFI La Baule [1] Muiela, M and Rukowki, M 1997 Maringale Mehod in Financial Modelling Applicaion of Mahemaic, Vol 36 Springer, Berlin Heidelberg New York [11] Rukowki, M: Spo, forward and fuure Libor rae Working paper, Univeriy of New Souh Wale, 1997 forhcoming in Inerna J Theor Appl Finance
19 UNSW, Repor NoS9711, Sepember 1997 hi verion: June Appendix A Proof of Theorem 31 I i well known ha in he ocalled rikneural world we have ee, for inance, Heah e al [7] where f, u =f,u+ T α, T = σ, T α, u d + and b, T and for he bond price volailiy ha i, σ, u dw, σ, u du = b, T b, T, 37 T In view of 16, i i clear ha T b, T = σ, u du +T +T ln D, T = f,u du +T α, u d du σ, u dw du Applying he andard and ochaic Fubini heorem o he righhand ide in he la equaliy, we obain ln D, T = +T f,u du Equivalenly, we have +T ln D, T = f,u du + f,u du + +T +T α, u du d α, u du d α, u du d + Since he horerm rae aifie r u = fu, u, i i clear ha and hu r u du = r u = f,u+ f,u du + u α, u d + u α, u du d + +T +T σ, u dw, σ, u du dw σ, u du dw σ, u du dw σ, u du dw, where we have ued once again Fubini heorem The la formula yield in urn ln D, T =lnd,t+ r d I +T wherewewriei o denoe he following inegral To examine I, oberve ha ince I = +T r, T =f, + T =f,+ T + T α, u du d +T σ, u du dw, f, d 38 αu, + T du + σu, + T dw u,
20 2 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond we have r, T d = = +T T f,+ T d + f,u du + Combining he la equaliy wih 38, we find ha I = or equivalenly, I = r, T d r, T d +T u αu, + T du d + αu, + T d du + α, u + T du d +T α, u du d +T +T u σu, + T dw u d σu, + T d dw u σ, u + T du dw, σ, u du dw Upon ubiuion ino previouly eablihed formula for ln D, T, we obain ln D, T =lnd,t+ r r, T d and afer imple manipulaion, we find ha ln D, T =lnd,t+ +T r r, T d +T α, u du d +T α, u du d + σ, u du dw b, + T dw To derive 24, i uffice o oberve ha in view of 37 we have noe ha b, = +T α, u du = 1 2 b, + T 2 Thi end he proof of he propoiion ¾ A horer bu omewha heuriic derivaion of 24 run a follow We noe fir ha +T +T D, T =exp f, u du = B, T exp f, u du = B, T I, T where he emimaringale repreenaion of he proce B,T i given by 23 I i hu enough o deermine he emimaringale decompoiion of he proce I To hi end, conider fir he proce J which equal +T J =lni = f, u du, > T Then +T dj = f, + T d df, u du = f, + T d = f, + T d T +T T +T T α, u d + σ, u dw du +T α, u du d σ, u du dw T = 1 2 b, + T 2 f, + T d + b, + T b, T dw
21 UNSW, Repor NoS9711, Sepember 1997 hi verion: June An applicaion of Iô formula yield di = I 1 2 b, + T 2 f, + T d + b, + T b, T dw b, T b, + T 2 d Conequenly, we ge dd, T = B, T di + I db, T +d I,B,T = D, T 1 2 b, + T 2 f, + T d + b, + T b, T dw b, T b, + T 2 d + r d + b, T dw b, + T b, T b, T d which give 24, afer implificaion 5 Appendix B Le u inroduce he liding yield proce Z,T by eing Z, T = T 1 ln D, T I i eay o check ha he liding yield aifie Z, T = T 1 ru, T r u + bu, u + T 2 du T 1 bu, u + T dw u We are going o conider here only a very paricular cae of a liding yield proce For a general udy of affine erm rucure model, in which a finie family of liding yield play he role of underlying facor, he reader i referred o he paper by DDuffie and RKan 1996 A yieldfacor model of inere rae, Mah Finance 6, A already menioned, we hall now eablih he dynamic of he liding yield for a paricular example of he erm rucure model, namely, he ocalled Vaicek model In hi, one aume ha he horerm inere rae r olve he SDE dr = a br d σdw, 39 where a, b and σ are ricly poiive conan Aume, in addiion, ha he marke price of rik i an affine funcion of he horerm rae ha i, dw = dw λ + λ 1 r d, where λ,λ 1 are conan, and W follow a Brownian moion under he realworld probabiliy P Clearly he horerm rae aifie under P dr = ã br d σdw, where ã = a + λ σ and b = b λ 1 σ I i alo well known ha in he Vaicek model, which can be analyed wihin he HJM framework, he price B, T of a pure dicoun bond i given by he formula B, T =e m,t n,t r, where n, T =b 1 bt 1 e and m, T =b 2 n, T T ab 1 2 σ2 4b 1 σ 2 n 2, T I i alo eay o check ha he bond price volailiy equal b, T =σn, T Oberve ha ln B, T f, T = = m, T n, T + r T T T
22 22 MRukowki: Trading Sraegie for Sliding, RollingHorizon, and Conol Bond Le u denoe g =1 e b Elemenary calculaion how ha m, T T and n, T T Therefore he forward rae r, T equal = gt 1 2 σ2 b 2 gt ab 1 = e bt r, T =f, + T =gt ab σ2 b 2 gt + r e bt Uing 24, we conclude ha he dynamic of D,T under he maringale meaure P are given by he expreion dd, T =gt D, T r ab σ2 b 2 gt d + σb 1 dw Similarly, under he realworld probabiliy P we have dd, T =gt D, T r ãb σ2 b 2 gt d + σb 1 dw Le u finally find he dynamic under P of he yield proce Z, T =Y, + T of he liding bond Since Z, T = T 1 ln D, T, uing Iô formula, we obain dz, T = T 1 gt r ab 1 d + σb 1 dw On he oher hand, we have TZ, T = m, + T +n, + T r and hu The la formula yield r = m, + T +TZ, T n, + T and hu gt r = b 2 gt bt ab 1 2 σ2 1 4 b 2 σ 2 g 2 T +bt Z, T, dz, T = ãt bz, T d σb 1 T 1 gt dw, where ãt =b 2 bt T 1 gt ab 1 2 σ b 2 T 1 σ 2 g 2 T +ab 1 T 1 4 We are in a poiion o formulae he following reul Corollary 51 Aume ha he horerm rae proce r aifie 39 Then he dynamic of he liding yield Z,T are given by he expreion dz, T = ãt bz, T d σt dw, where ãt i given by 4 and σt =σb 1 T 1 gt A imilar reul i valid under he realworld probabiliy, provided ha he marke price for rik i an affine funcion of he hor erm rae r
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