Ineres Rae Dervaves: More Advanced s Chaper 4 4. The Two-Facor Hull-Whe (Equaon 4., page 57) [ θ() ] σ 4. dx = u ax d dz du = bud σdz where x = f () r and he correlaon beween dz and dz s ρ The shor rae revers o a level dependen on u, and u self s mean reverng Analyc Resuls Bond prces and European opons on zero-coupon bonds can be calculaed analycally when f(r) = r 4.3 Opons on Coupon-Bearng Bonds We canno use he same procedure for opons on coupon-bearng bonds as we do n he case of one-facor models If we mae he approxmae assumpon ha he coupon-bearng bond prce s lognormal, we can use Blac s model The approprae volaly s calculaed from he volales of and correlaons beween he underlyng zero-coupon bond prces 4.4 Volaly Srucures In he one-facor Ho-Lee or Hull-Whe model he forward rae S.D.s are eher consan or declne exponenally. All forward raes are nsananeously perfecly correlaed In he wo-facor model many dfferen forward rae S.D. paerns and correlaon srucures can be obaned 4.5 4.6 Example Gvng Humped Volaly Srucure (Fgure 4., page 57) a=, b=0., σ =0.0, σ =0.065, ρ=0.6.40.0.00 0.80 0.60 0.40 0.0 0.00 0 5 0
4.7 Transformaon of he General dx = [ θ() u ax] d σ dz du = bud σ dz where x = f ( r) and he correlaon beween dz and dz s ρ We defne y = x u ( b a) so ha dy = [ θ() ay] d σ3dz3 du = bud σ dz 4.8 Transformaon of he General connued σ ρσ σ σ3 = σ ( b a) b a The correlaon beween dz dz3 s ρσ σ ( b a) σ 3 and Aracve Feaures of he I s Marov so ha a recombnng 3- dmensonal ree can be consruced The volaly srucure s saonary Volaly and correlaon paerns smlar o hose n he real world can be ncorporaed no he model 4.9 HJM : Noaon P(,T ): prce a me of a dscoun bond wh prncpal of $ maurng a T Ω : vecor of pas and presen values of neres raes and bond prces a me ha are relevan for deermnng bond prce volales a ha me v(,t,ω ): volaly of P(,T) 4.0 Noaon connued 4. ng Bond Prces 4. ƒ(,t,t ): forward rae as seen a for he perod beween T and T F(,T): nsananeous forward rae as seen a for a conrac maurng a T r(): shor-erm rs-free neres rae a dz(): Wener process drvng erm srucure movemens dpt (, ) = rptd () (, ) vt (,, Ω ) PTdz (, ) () We can choose any v funcon provdng v (,, Ω ) = 0 for all
ng Forward Raes Equaon 4.7, page 575) 4.3 Tree For a General 4.4 df(, T) = m(, T, Ω) d s(, T, Ω) dz() We mus have mt (,, Ω) = st (,, Ω) s (, τ, Ω) dτ Smlar resuls hold when here s more han one facor T A non-recombnng ree means ha he process for r s non-marov 4.5 4.6 The LIBOR Mare Noaon The LIBOR mare model s a model consruced n erms of he forward raes underlyng caple prces : h rese dae F (): forward rae beween mes and m (): ndex for nex rese dae a me ς (): volaly of F () a me v (): volaly of P(, ) a me δ : Volaly Srucure 4.7 In Theory he Λ s can be deermned from Cap Prces 4.8 We assume a saonary volaly srucure where he volaly of F () depends only on he number of accrual perods beween he nex rese dae and of m() ] [.e., s a funcon only Defne Λ as he volaly of F ( ) when m( ) = If σ s he volaly for he (, provdes a perfec f o cap prces we mus have σ = Ths allows he Λ' s o be deermned nducvely = Λ δ ) caple.if he model
4.9 4.0 Example 4. (Page 579) If Blac volales for he frs hree caples are 4%, %, and 0%, hen Λ 0 =4.00% Λ =9.80% Λ =5.3% Example 4. (Page 579) n 3 4 5 σ n (%) Λ n- (%) 5.50 8.5 7.9 7.74 7.7 5.50 0.64 7. 7. 5.5 n 6 7 8 9 0 σ n (%) Λ n- (%) 6.79 6.30 6.0 5.76 5.54 4.5.98 3.8 3.60 3.40 The Process for F n a One- Facor LIBOR Mare df = Κ Λ m ( ) F dz The drf depends on he world chosen In a world ha s forward rs - neural wh respec o P (, ), he drf s zero 4. Rollng Forward Rs- Neuraly (Equaon 4.6, page 579) I s ofen convenen o choose a world ha s always FRN wr a bond maurng a he nex rese dae. In hs case, we can dscoun from o a he δ rae observed a me. The process for F s 4. df F F = δ Λ Λ δ F j= m() m() m() d Λ m() dz The LIBOR Mare and HJM 4.3 4.4 Mone Carlo Implemenaon of BGM Cap (Equaon 4.8, page 580) In he lm as he me beween reses ends o zero, he LIBOR mare model wh rollng forward rs neuraly becomes he HJM model n he radonal rs-neural world We assume no change o he drf beween rese daes so ha F( j ) = F( j)exp j= δf ( j) ΛjΛj Λ j jl δ Λ δ j ε δ j j
Mulfacor Versons of BGM BGM can be exended so ha here are several componens o he volaly A facor analyss can be used o deermne how he volaly of F s spl no componens 4.5 4.6 Rache Caps, Scy Caps, and Flex Caps A plan vanlla cap depends only on one forward rae. Is prce s no dependen on he number of facors. Rache caps, scy caps, and flex caps depend on he jon dsrbuon of wo or more forward raes. Ther prces end o ncrease wh he number of facors 4.7 Valung European Opons n he LIBOR Mare There s a good analyc approxmaon ha can be used o value European swap opons n he LIBOR mare model. See pages 58 o 584. 4.8 Calbrang he LIBOR Mare In heory he LMM can be exacly calbraed o cap prces as descrbed earler In pracce we proceed as for he one-facor models n Chaper 3 and mnmze a funcon of he form n = ( U V ) P where U s he mare prce of he h calbrang nsrumen, V s he model prce of he h calbrang nsrumen and P s a funcon ha penalzes bg changes or curvaure n a and σ Types of Morgage-Baced Secures (MBSs) 4.9 Opon-Adjused Spread (OAS) 4.30 Pass-Through Collaeralzed Morgage Oblgaon (CMO) Ineres Only (IO) Prncpal Only (PO) To calculae he OAS for an neres rae dervave we value assumng ha he nal yeld curve s he Treasury curve a spread We use an erave procedure o calculae he spread ha maes he dervave s model prce = mare prce. Ths s he OAS.