Chaper 34 Brace-Gaarek-Musiela mdel 34. Review f HJM under risk-neural IP where f ( T Frward rae a ime fr brrwing a ime T df ( T ( T ( T d + ( T dw ( ( T The ineres rae is r( f (. The bnd prices saisfy B( T IE " ( ; ( ; ( u du r(u du F( f ( u du db( T r( B( T d ; ( T {z } B( T dw ( vlailiy f T -mauriy bnd. T implemen HJM, yu specify a funcin A simple chice we wuld like use is ( T T ( T f( T where > is he cnsan vlailiy f he frward rae. This is n pssible because i leads # ( T df ( T 2 f ( T f ( u du f ( u du! d + f( T dw ( 335
336 and Heah, Jarrw and Mrn shw ha sluins his equain lde befre T. The prblem wih he abve equain is ha he d erm grws like he square f he frward rae. T see wha prblem his causes, cnsider he similar deerminisic rdinary differenial equain where f ( c>. We have This sluin ldes a c. f ( f 2 ( ; d d f ( ; f ( + f ( f ( f 2 ( du ; f ( ; f ( f ( c ; c c ; ; c c 34.2 Brace-Gaarek-Musiela mdel New variables Curren ime Time mauriy T ; Frward raes Bnd prices r( f ( + r( f ( r( (2. r( f ( + (2.2 T D( B( + (2.3 (u v ; du dv D( ; ; ; + f ( v dv f ( + u du B( + ;r( D( (2.4 T
CHAPTER 34. Brace-Gaarek-Musiela mdel 337 We will nw wrie ( ( T ; raher han ( T. In his nain, he HJM mdel is df ( T ( ( d + ( dw ( (2.5 db( T r(b( T d ; ( B( T dw ( (2.6 where ( ( u du (2.7 ( ( (2.8 We nw derive he differenials f r( and D(, analgus (2.5 and (2.6 We have Als, dr( df ( + {z } + f ( + d T differenial applies nly firs argumen (2.5,(2.2 ( ( d + ( dw ( + r( d (2.8 i hr( + 2 ( ( 2 d + ( dw ( (2.9 dd( db( + {z } + B( + d T differenial applies nly firs argumen (2.6,(2.4 r( B( + d ; ( B( + dw ( ; r( D( d (2. [r( ; r( ] D( d ; ( D( dw ( (2. 34.3 LIBOR Fix > (say, year. $ D( invesed a ime in a ( + -mauriy bnd grws $ a 4 ime +. L( is defined be he crrespnding rae f simple ineres +L( D( ( + L( ( D( nr L( r( u du ;
338 34.4 Frward LIBOR > is sill fixed. A ime, agree inves $ D( + D( D( + D( a ime +, wih payback f $ a ime + +. Can d his a ime by shring bnds mauring a ime + and ging lng ne bnd mauring a ime + +. The value f his prfli a ime is ; D( + D( +D( + D( The frward LIBOR L( is defined be he simple (frward ineres rae fr his invesmen D( + D( Cnnecin wih frward raes ( + L( +L( D( D( + L( R f; g n R + ; ( + nr + ; (4. s ( + r( + r( f ( + r( lim # L( nr + ( + nr + ; ; > fixed r( is he cninuusly cmpunded rae. L( is he simple rae ver a perid f durain. We cann have a lg-nrmal mdel fr r( because sluins lde as we saw in Secin 34.. Fr fixed psiive, wecan have a lg-nrmal mdel fr L(. (4.2 34.5 The dynamics f L( We wan chse (, appearing in (2.5 s ha dl( ( d + L( ( dw (
CHAPTER 34. Brace-Gaarek-Musiela mdel 339 fr sme (. This is he BGM mdel, and is a subclass f HJM mdels, crrespnding paricular chices f (. Recall (2.9 dr( i hr( u+ 2 u ( ( u 2 d + ( u dw ( Therefre, d +! + d (5. + i + hr( u+ 2 u ( ( u 2 du d + ( u du dw ( i hr( + ; r( + 2 ( ( + 2 ; 2 ( ( 2 d +[ ( + ; ( ] dw ( and dl( (4 d 2 nr 4 + ( + (4., (5. d ; 3 5 + ( + + 2 + d [+L( ] (5.2 [r( + ; r( + 2 ( ( + 2 ; 2 ( ( 2 ] d +[ ( + ; ( ] dw ( + 2 [ ( + ; ( ] 2 d [ + L( ] [r( + ; r( ] d + ( + [ ( + ; ( ] d +[ ( + ; ( ] dw (! 2
34 Bu L( 2 nr 4 + ( + ; 3 5 [r( + ; r( ] [ + L( ][r( + ; r( ] Therefre, dl( L( d + [ + L( ][ ( + ; ( ][ ( + d + dw (] Take ( be given by Then ( L( [+L( ][ ( + ; ( ] (5.3 dl( [ L( +( L( ( + ] d + ( L( dw ( (5.4 Ne ha (5.3 is equivalen ( + ( + Plugging his in (5.4 yields " dl( L( +( L( ( + L2 ( 2 ( +L( L( ( +L( (5.3 # d + ( L( dw ( (5.4 34.6 Implemenain f BGM Obain he iniial frward LIBOR curve L( frm marke daa. Chse a frward LIBOR vlailiy funcin (usually nnrandm (
CHAPTER 34. Brace-Gaarek-Musiela mdel 34 Because LIBOR gives n rae infrmain n ime perids smaller han, we mus als chse a parial bnd vlailiy funcin ( < fr mauriies less han frm he curren ime variable. Wih hese funcins, we can fr each 2 [ slve (5.4 bain L( < Plugging he sluin in (5.3, we bain ( fr <2. We hen slve (5.4 bain L( <2 and we cninue recursively. Remark 34. BGM is a special case f HJM wih HJM s ( generaed recursively by (5.3. In BGM, ( is usually aken be nnrandm; he resuling ( is randm. Remark 34.2 (5.4 (equivalenly, (5.4 is a schasic parial differenial equain because f he L( erm. This is n as errible as i firs appears. Reurning he HJM variables and T, se K( T L( T ; Then and (5.4 and (5.4 becme dk( T dl( T ; ; L( T ; d dk( T ( T ; K( T [ ( T ; + d + dw (] K( T ( T ; ( T ; K( T ( T ; d + d + dw ( +K( T (6. Remark 34.3 Frm (5.3 we have If we le #, hen and s We saw befre (eq. 4.2 ha as #, ( L( [ + L( ] ( + ; ( ( L(! ( + ( ( T ; K( T!( T ; L(!r( f ( +
342 s K( T!f ( T Therefre, he limi as # f (6. is given by equain (2.5 df ( T ( T ; [ ( T ; d + dw (] Remark 34.4 Alhugh he d erm in (6. has he erm 2 ( T ;K 2 ( T +K( T his equain d n lde because 2 ( T ; K 2 ( T +K( T 2 ( T ; K 2 ( T K( T 2 ( T ; K( T invlving K 2, sluins 34.7 Bnd prices Le ( nr r(u du Frm (2.6 we have The sluin B( T ( B( T d ( [;r(b( T d + db( T ] ( ; B( T ( T ; dw ( ( his schasic differenial equain is given by B( T (B( T ; (u T ; u dw (u ; 2 This is a maringale, and we can use i swich he frward measure Girsanv s Therem implies ha IP T (A B( T A A (T dip B(T T dip 8A 2F(T (T B( T ( (u T ; u 2 du W T ( W ( + (u T ; u du T is a Brwnian min under IP T.
CHAPTER 34. Brace-Gaarek-Musiela mdel 343 34.8 Frward LIBOR under mre frward measure Frm (6. we have dk( T ( T ; K( T [ ( T ; + d + dw (] ( T ; K( T dw T + ( s and K( T K( T K(T TK( T K( T ( ( (u T ; u dw T + (u ; 2 (u T ; u dw T + (u ; 2 (u T ; u dw T + (u ; 2 2 (u T ; u du 2 (u T ; u du 2 (u T ; u du (8. We assume ha is nnrandm. Then X( (u T ; u dw T + (u ; 2 (u T ; u du (8.2 2 is nrmal wih variance and mean ; 2 2 (. 2 ( 2 (u T ; u du 34.9 Pricing an ineres rae caple Cnsider a flaing rae ineres paymen seled in arrears. A ime T +, he flaing rae ineres paymen due is L(T K(T T he LIBOR a ime T. A caple precs is wner by requiring him pay nly he cap c if K(T T >c. Thus, he value f he caple a ime T + is (K(T T ; c +. We deermine is value a imes T +. Case I T T +. C T + ( IE ( (T + (K(T T ; c+ F( ( F( (K(T T ; c + IE (T + (K(T T ; c + B( T + (9.
344 Case II T. Recall ha IP T + (A A (T + dip 8A 2F(T + where We have ( C T + ( IE B( T + (B( T + ( (T + (K(T T ; c+ F ( 2 3 (B( T + B( T + IE B(T + T + B( T + 6 (K(T T ; c + F( {z } 4(T + B( T + 7 {z } 5 (T + ( B( T + IE T + (K(T T ; c + F( Frm (8. and (8.2 we have K(T TK( T fx(g R where X( is nrmal under IP T + wih variance 2 T ( Furhermre, X( is independen f F(. 2 (u T ; u du and mean ; 2 2 (. C T + ( B( T + IE T + (K( T fx(g;c + F( Se h g(y IE T + (y +i fx(g;c yn ( lg y c + 2 ( ; cn ( lg y c ; 2 ( Then C T + ( B( T + g(k( T T ; (9.2 In he case f cnsan,wehave ( p T ; and (9.2 is called he Black caple frmula.
CHAPTER 34. Brace-Gaarek-Musiela mdel 345 34. Pricing an ineres rae cap Le T T T 2 2 T n n A cap is a series f paymens (K(T k T k ; c + a ime T k+ k n; The value a ime f he cap is he value f all remaining caples, i.e., C( X kt k C Tk ( 34. Calibrain f BGM The ineres rae caple c n L( T a ime T + has ime-zer value where g (defined in he las secin depends n C T + ( B( T + g(k( T 2 (u T ; u du Le us suppse is a deerminisic funcin f is secnd argumen, i.e., Then g depends n ( ( 2 (T ; u du 2 (v dv If we knw he caple price C T + (, we can back u he squared vlailiy R T 2 (v dv. Ifwe knw caple prices C T +( C T +( C Tn+( where T <T <<T n, we can back u 2 (v dv T 2 (v dv 2 (v dv ; 2 (v dv n T n; 2 (v dv (. In his case, we may assume ha is cnsan n each f he inervals ( T (T T (T n; T n
346 and chse hese cnsans make he abve inegrals have he values implied by he caple prices. If we knw caple prices C T + ( fr all T, we can back u R T 2 (v dv and hen differeniae discver 2 ( and ( p 2 ( fr all. T implemen BGM, we need bh (, and ( < Nw ( is he vlailiy a ime f a zer cupn bnd mauring a ime + (see (2.6. Since is small (say year, and <, i is reasnable se 4 ( < We can nw slve (r simulae ge L( r equivalenly, K( T T using he recursive prcedure ulined a he sar f Secin 34.6. 34.2 Lng raes The lng rae is deermined by lng mauriy bnd prices. Le n be a large fixed psiive ineger, s ha n is 2 r 3 years. Then ( n D( n ny k ny k ( k (k; [ + L( (k ; ] where he las equaliy fllws frm (4.. The lng rae is 34.3 Pricing a swap Le T be given, and se n lg D( n n nx k lg[ +L( (k ; ] T T + T 2 T +2 T n T + n
CHAPTER 34. Brace-Gaarek-Musiela mdel 347 The swap is he series f paymens (L(T k ; c a ime T k+ k n; Fr T, he value f he swap is n; X ( IE (T k+ (L(T k ; c Nw s +L(T k We cmpue L(T k k B(T k T k+ B(T k T ; k+ ( IE (T k+ (L(T k ; c ( IE (T k+ IE 2 6 4 F( F ( B(T k T k+ ; ; c F( {z } B(T k T k+ ( (T k B(T k T IE (Tk k+ (T k+ F(T k ( IE (T k+ The value f he swap a ime is n; X ( IE (T (L(T k ; c k+ k n; X k F( ; ( + cb( T k+ B( T k ; ( + cb( T k+ F( [B( T k ; ( + cb( T k+ ] 3 7 F ( 5 ; ( + cb( T k+ B( T ; ( + cb( T +B( T ; ( + cb( T 2 ++ B( T n; ; ( + cb( T n B( T ; cb( T ; cb( T 2 ; ; cb( T n ; B( T n The frward swap rae w T ( a ime fr mauriy T is he value f c which makes he ime- value f he swap equal zer w T ( B( T ; B( T n [B( T ++ B( T n ] In cnras he cap frmula, which depends n he erm srucure mdel and requires esimain f, he swap frmula is generic.