Extended One-Factor Short-Rate Models

Transcription

1 CHAPTER 5 Exended One-Facor Shor-Rae Model 5.1. Ho Le Model Definiion 5.1 (Ho Le model). In he Ho Le model, he hor rae i aumed o aify he ochaic differenial equaion dr() =θ()d + σdw (), σ>0, θ i deerminiic, and W i a Brownian moion under he rik-neural meaure. Theorem 5.2 (Ho Le model). In he Ho Le model, we have he following formula: r() =r()+ E(r() F()) = r()+ θ(u)du + σ(w () W ()), θ(u)du and V(r() F()) = σ 2 ( ), P (, T )=A(, T )e r()(t ), σ 2 T A(, T )=exp 6 (T )3 (T u)θ(u)du, dp (, T )=r()p (, T )d σ(t )P (, T )dw (), 1 d P (, T ) = σ2 (T ) 2 r() σ(t ) d + dw (), P (, T ) P (, T ) dw T () =dw ()+σ(t )d, dr() = [ θ() σ 2 (T ) ] d + σdw T (), f(, T )=r() σ2 2 (T )2 + θ(u)du and df(, T )=σdw T (), ( ) 1 df (; T,S)=σ F (; T,S)+ (S T )dw S (), τ(t,s) ZBC(, T, S, K) =P (, S)Φ(h) KP(, T )Φ(h σ), 25

2 26 5. EXTENDED ONE-FACTOR SHORT-RATE MODELS ZBP(, T, S, K) =KP(, T )Φ( h + σ) P (, S)Φ( h), σ = σ(s T ) T and h = 1 σ ( ) P (, S) ln + σ P (, T )K 2, β Cap(, T,N,K)=N [P (, T i 1 )Φ( h i + σ i ) (1 + τ i K)P (, T i )Φ( h i )], Flr(, T,N,K)=N i=α+1 β i=α+1 [(1 + τ i K)P (, T i )Φ(h i ) P (, T i 1 )Φ(h i σ i )], σ i = σ(t i T i 1 ) T i 1 and h i = 1 σ i ln ( ) P (, Ti ) + σ i P (, T i 1 )K 2. Theorem 5.3 (Calibraion in he Ho Le model). IfheHo Lemodelicalibraed o a given inere rae rucure f M (0,): 0, i.e., f(0,)=f M (0,) for all 0, hen θ() = fm (0,) + σ 2 for all 0. Theorem 5.4 (Zero-coupon bond price in he calibraed Ho Le model). If he Ho Le model i calibraed o a given inere rae rucure f M (0,): 0, hen P (, T )=e r()(t ) P M (0,T) (T P M (0,) exp )f M (0,) σ2 (T )2, 2 P M (0,)=exp 0 f M (0,u)du for all Hull Whie Model (Exended Vaicek Model) Definiion 5.5 (Shor-rae dynamic in he Hull Whie model). In he Hull Whie model, he hor rae i aumed o aify he ochaic differenial equaion dr() =k(θ() r())d + σdw (), k, σ > 0, θ i deerminiic, and W i a Brownian moion under he rikneural meaure.

3 5.2. HULL WHITE MODEL (EXTENDED VASICEK MODEL) 27 Remark 5.6 (Hull Whie model). The Hull Whie model i alo called he exended Vaicek model or he G++ model and can be conidered, more generally, wih he conan k and σ replaced by deerminiic funcion. Theorem 5.7 (Shor rae in he Hull Whie model). Le 0 T. The hor rae in he Hull Whie model i given by r() =r()e k( ) + k θ(u)e k( u) du + σ and i, condiionally on F(), normally diribued wih e k( u) dw (u) and E(r() F()) = r()e k( ) + k θ(u)e k( u) du ( V(r() F()) = σ2 1 e ( )). Remark 5.8 (Shor rae in he Hull Whie model). A in he Vaicek model, he hor rae r() in he exended Vaicek model, for each ime, can be negaive wih poiive probabiliy, namely, wih probabiliy Φ r(0)e k + k 0 θ(u)e k( u) du, σ 2 (1 e ) which i ofen negligible in pracice. On he oher hand, he hor rae in he Vaicek model i mean revering provided ϕ = lim k θ(u)e k( u) du exi, and hen 0 E(r()) ϕ a. Theorem 5.9 (Zero-coupon bond in he Hull Whie model). In he Hull Whie model, he price of a zero-coupon bond wih mauriy T a ime [0,T] i given by P (, T )=Ā(, T )e r()b(,t ), Ā(, T )=A(, T )exp k θ(u)b(u, T )du and A and B are a in he Vaicek model, Theorem 4.4 wih θ =0.

4 28 5. EXTENDED ONE-FACTOR SHORT-RATE MODELS Theorem 5.10 (Forward rae in he Hull Whie model). In he Hull Whie model, he inananeou forward inere rae wih mauriy T i given by f(, T )=k θ(u)e k(t u) du σ2 2 B2 (, T )+r()e k(t ). Theorem 5.11 (Calibraion in he Hull Whie model). IfheHull Whiemodel i calibraed o a given inere rae rucure f M (0,): 0, hen θ() =f M (0,)+ 1 k f M (0,) + σ2 2 ( 1 e ) for all 0. Theorem 5.12 (Zero-coupon bond in he calibraed Hull Whie model). If he Hull Whie model i calibraed o a given inere rae rucure, hen P (, T )=e r()b(,t ) P M (0,T) B(, P M (0,) exp T )f M (0,) σ2 ( 1 e ) B 2 (, T ). 4k Theorem 5.13 (Opion on a zero-coupon bond in he Hull Whie model). In he Hull Whie model, he price of a European call opion wih rike K and mauriy T and wrien on a zero-coupon bond wih mauriy S a ime [0,T] i given by ZBC(, T, S, K) =P (, S)Φ(h) KP(, T )Φ(h σ), σ and h are a in he Vaicek model, Theorem 4.9. ZBP(, T, S, K) =KP(, T )Φ( h + σ) P (, S)Φ( h). Theorem 5.14 (Cap and floor in he Hull Whie model). In he Hull Whie model, he price of a cap wih noional value N, capraek, and he e of ime T, i given by β Cap(, T,N,K)=N [P (, T i 1 )Φ( h i + σ i ) (1 + τ i K)P (, T i )Φ( h i )], i=α+1 while he price of a floor wih noional value N, floorraek, and he e of ime T, i given by β Flr(, T,N,K)=N [(1 + τ i K)P (, T i )Φ(h i ) P (, T i 1 )Φ(h i σ i )], i=α+1 σ i and h i are a in he Vaicek model, Theorem 4.10.

5 5.3. BLACK KARASINSKI MODEL Black Karainki Model Definiion 5.15 (Black Karainki model). In he Black Karainki model, he horraeigivenby r() =e y() wih dy() =k(θ() y())d + σdw (), k, σ > 0, θ i deerminiic, and W i a Brownian moion under he rikneural meaure. Remark 5.16 (Black Karainki model). The Black Karainki model i alo called he exended exponenial Vaicek model and can be conidered, more generally, wih he conan k and σ replaced by deerminiic funcion. Theorem 5.17 (Shor rae in he Black Karainki model). The hor rae in he Black Karainki model aifie he ochaic differenial equaion ) dr() = (kθ()+ σ2 2 k ln(r()) r()d + σr()dw (). Le 0 T.Thenri given by r() =exp ln(r())e k( ) + k e k( u) θ(u)du + σ e k( u) dw (u) and i, condiionally on F(), lognormally diribued wih E(r() F()) =exp ln(r())e k( ) + k and V(r() F()) = exp 2ln(r())e k( ) + e k( u) θ(u)du + (1 σ2 e ( )) 4k σ 2 ( exp 1 e ( ))[ σ 2 exp e k( u) θ(u)du ( 1 e ( )) ] 1. Remark 5.18 (Shor rae in he Black Karainki model). Since he hor rae r in he Black Karainki model i lognormally diribued, i i alway poiive. A diadvanage i ha P (, T ) canno be calculaed explicily. An advanage of he Black Karainki model i ha r i alway mean revering provided ϕ = lim k θ(u)e k( u) du 0

6 30 5. EXTENDED ONE-FACTOR SHORT-RATE MODELS exi, and hen E(r() F()) exp ) (ϕ + σ2 4k a and V(r() F()) exp )[ (2ϕ + σ2 exp ( σ 2 ) ] 1 a Deerminiic-Shif Exended Model Definiion 5.19 (Shor rae in a deerminiic-hif exended model). In a deerminiic-hif exended model, he hor rae i given by r() = x()+ ϕ() wih dx() = μ(, x())d + σ(, x())dw (), ϕ, μ, σ are deerminiic funcion and W i a Brownian moion under he rik-neural meaure. The ochaic differenial equaion for x i called he reference model, and price of zero-coupon bond and forward inere rae in he reference model are denoed by Px REF (, T )andfx REF (, T ), repecively. Theorem 5.20 (Zero-coupon bond in a deerminiic-hif exended model). In a deerminiic-hif exended model, he price of a zero-coupon bond wih mauriy T a ime [0,T] i given by ( ) T P (, T )=exp ϕ(u)du Pr ϕ REF (, T ). Theorem 5.21 (Forward rae in a deerminiic-hif exended model). In a deerminiic-hif exended model, he inananeou forward inere rae wih mauriy T i given by f(, T )=ϕ(t )+f REF r ϕ (, T ). Theorem 5.22 (Calibraion in a deerminiic-hif exended model). If a deerminiic-hif exended model i calibraed o a given inere rae rucure f M (0,): 0, hen ϕ() =f M (0,) f REF r ϕ (0,) for all 0.

7 5.5. EXTENDED CIR MODEL 31 Theorem 5.23 (Zero-coupon bond in a calibraed deerminiic-hif exended model). If a deerminiic-hif exended model i calibraed o a given inere rae rucure, hen P (, T )= P M (0,T) P M (0,) P REF r ϕ (0,) P REF r ϕ (0,T) P REF r ϕ (, T ). Theorem 5.24(Opiononazero-couponbondinadeerminiic-hifexended model). In a deerminiic-hif exended model, he price of a European call opion wih rike K and mauriy T and wrien on a zero-coupon bond wih mauriy S a ime [0,T] i given by ( ) S ZBC(, T, S, K) =exp ϕ(u)du ZBC REF r ϕ (, T, S, K ), ( ) S K = K exp ϕ(u)du. T 5.5. Exended CIR Model Definiion 5.25 (Shor rae in he exended CIR model). In he exended CIR model, he hor rae i given by r() =x()+ϕ() wih dx() =k(θ x())d + σ x()dw (), k, σ, θ > 0andW i a Brownian moion under he rik-neural meaure. Remark 5.26 (Exended CIR model). The exended CIR model i alo called he CIR++ model and can be conidered, more generally, wih he conan k and σ replaced by deerminiic funcion. Theorem 5.27 (Zero-coupon bond in he CIR++ model). In he CIR++ model, he price of a zero-coupon bond wih mauriy T a ime [0,T] i given by P (, T )=Ā(, T )e r()b(,t ), Ā(, T )=A(, T )exp ϕ()b(, T ) and A and B are a in he CIR model, Theorem ϕ(u)du

8 32 5. EXTENDED ONE-FACTOR SHORT-RATE MODELS Theorem 5.28 (Forward rae in he CIR++ model). In he CIR++ model, he inananeou forward inere rae wih mauriy T i given by f(, T )=ϕ(t ) ϕ()b T (, T )+kθb(, T )+r()b T (, T ), B i a in he CIR model, Theorem Theorem 5.29 (Calibraion in he CIR++ model). If he CIR++ model i calibraed o a given inere rae rucure f M (0,): 0, hen ϕ() =f M (0,)+ϕ(0)B T (0,T) kθb(0,t) r(0)b T (0,T) for all 0, B i a in he CIR model, Theorem Theorem 5.30 (Zero-coupon bond in he CIR++ model). If he CIR++ model i calibraed o a given inere rae rucure, hen P (, T )= P M (0,T) P M (0,) A(0,)A(, T ) e (r(0) ϕ(0))(b(0,t ) B(0,))+ϕ()B(,T ) e r()b(,t ), A(0,T) A and B are a in he CIR model, Theorem Exended Affine Term-Srucure Model Theorem 5.31 (Exended affine erm-rucure model). Aume he reference model i an affine erm-rucure model, i.e., P REF r (, T )=A(, T )e r()b(,t ). If hi model i exended according o Definiion 5.19 by uing he deerminiic hif ϕ, hen we have he following formula: P (, T )=Ā(, T )e r()b(,t ), Ā(, T )=A(, T )exp ϕ()b(, T ) ϕ(u)du, f(, T )=ϕ(t) ϕ()b T (, T ) A T (, T ) A(, T ) + r()b T (, T ), and if he exended model i calibraed o a given inere rae rucure, hen ϕ() =f M (0,)+(ϕ(0) r(0))b T (0,)+ A T (0,) A(0,), P (, T )= P M (0,T) P M (0,) A(0,)A(, T ) e (r(0) ϕ(0))(b(0,t ) B(0,))+ϕ()B(,T ) e r()b(,t ). A(0,T)