Heath Jarrow Morton Framework

CHAPTER 7 Heah Jarrow Moron Framework 7.1. Heah Jarrow Moron Model Definiion 7.1 (Forward-rae dynamics in he HJM model). In he Heah Jarrow Moron model, brieflyhjm model, he insananeous forward ineres rae wih mauriy T is assumed o saisfy he sochasic differenial equaion df(, T )=α(, T )d + σ(, T )dw (), α σ are adaped W is a Brownian moion under he risk-neural measure. Theorem 7.2 (Bond-price dynamics in he HJM model). In he HJM model, he price of a zero-coupon bond wih mauriy T saisfies he sochasic differenial equaion dp (, T )= ( r()+a(, T )+ 1 ) 2 Σ2 (, T ) P (, T )d +Σ(, T )P (, T )dw (), A(, T )= α(, u)du Σ(, T )= σ(, u)du. Theorem 7.3 (Bond-price dynamics implying HJM model). If he price of a zero-coupon bond wih mauriy T saisfies he sochasic differenial equaion dp (, T )=m(, T )P (, T )d + v(, T )P (, T )dw (), m v are adaped, hen he forward-rae dynamics are as in he HJM model wih α(, T )=v(, T )v T (, T ) m T (, T ) σ(, T )= v T (, T ). 43

44 7. HEATH JARROW MORTON FRAMEWORK Theorem 7.4 (Drif resricion in he HJM model). In he HJM model, we necessarily have A(, T )= 1 2 Σ2 (, T ) α(, T )=σ(, T ) σ(, u)du. Theorem 7.5 (Bond-price dynamics in he HJM model). In he HJM model, he price of a zero-coupon bond wih mauriy T saisfies he sochasic differenial equaions dp (, T )=r()p(, T )d +Σ(, T )P (, T )dw () 1 d P (, T ) = Σ2 (, T ) r() d Σ(, T ) dw (). P (, T ) P (, T ) Theorem 7.6 (T -forward measure dynamics of he forward rae in he HJM model). Under he T -forward measure Q T, he insananeous forward ineres rae wih mauriy T in he HJM model saisfies df(, T )=σ(, T )dw T (), he Q T -Brownian moion W T is defined by dw T () =dw () Σ(, T )d. Theorem 7.7 (Forward-rae dynamics in he HJM model). In he HJM model, he simply-compounded forward ineres rae for he period [T,S] saisfies he sochasic differenial equaion ( ) 1 df (; T,S)= F (; T,S)+ (Σ(, T ) Σ(, S)) dw S (). τ(t,s) Theorem 7.8 (Zero-coupon bond in he HJM model). Le T S. In he HJM model, he price of a zero-coupon bond wih mauriy S a ime T is given by P (, S) P (T,S)= P (, T ) ez, Z = 1 ( Σ 2 (u, S) Σ 2 (u, T ) ) du + (Σ(u, S) Σ(u, T )) dw (u) 2 = 1 2 (Σ(u, S) Σ(u, T )) 2 du + (Σ(u, S) Σ(u, T )) dw T (u).

7.3. RITCHKEN SANKARASUBRAMANIAN MODEL 45 7.2. Gaussian HJM Model Definiion 7.9 (Gaussian HJM Model). A Gaussian HJM model is an HJM model in which σ is a deerminisic funcion. Theorem 7.1 (Opion on a zero-coupon bond in a Gaussian HJM model). In a Gaussian HJM model, he price of a European call opion wih srike K mauriy T wrien on a zero-coupon bond wih mauriy S a ime [,T] is given by ZBC(, T, S, K) =P (, S)Φ(h) KP(, T )Φ(h σ ), σ = (Σ(u, S) Σ(u, T )) 2 du h = 1 ( ) P (, S) σ ln + σ P (, T )K 2. The price of a corresponding pu opion is given by ZBP(, T, S, K) =KP(, T )Φ( h + σ ) P (, S)Φ( h). Definiion 7.11 (Fuures price). The fuures price a ime of an asse whose value a ime T isx(t )isgivenby Fu(, T )=E(X(T ) F()). Theorem 7.12 (Fuures conrac on a zero-coupon bond in a Gaussian HJM model). In a Gaussian HJM model, he price of a fuures conrac wih mauriy T on a zero-coupon bond a ime T wih mauriy S is given by { } P (, S) T FUT(, T, S) = P (, T ) exp Σ(u, T )(Σ(u, T ) Σ(u, S)) du. 7.3. Richken Sankarasubramanian Model Definiion 7.13 (HJM model wih separable volailiy). An HJM model wih separable volailiy is an HJM model in which here exis posiive funcions ξ η such ha σ(, T )=ξ()η(t).

46 7. HEATH JARROW MORTON FRAMEWORK Theorem 7.14 (Zero-coupon bond in an HJM model wih separable volailiy). In an HJM model wih separable volailiy, he price of a zero-coupon bond wih mauriy T a ime [,T] is given by P (, T )= P (,T) { P (,) exp f(,)b(, T ) 1 } 2 φ()b2 (, T ) e r()b(,t ), φ() = σ 2 (u, )du B(, T )= 1 η(u)du. η() Theorem 7.15 (Shor-rae dynamics in an HJM model wih separable volailiy). In an HJM model wih separable volailiy, he shor rae saisfies he sochasic differenial equaion { } f(,) r() f(,) dr() = + φ() d + dη() η() +ξ()(dη())(dw ()) + σ(, )dw (), φ is as in Theorem 7.14. Corollary 7.16 (Shor-rae dynamics in a Gaussian HJM model wih separable volailiy). In an HJM model wih separable volailiy in which η is deerminisic, he shor rae saisfies he sochasic differenial equaion { } f(,) dr() = f(,) η () η() + () φ()+r()η d + σ(, )dw (), η() φ is as in Theorem 7.14. Theorem 7.17 (Opion on a zero-coupon bond in a Gaussian HJM model wih separable volailiy). In a Gaussian HJM model wih separable volailiy, he price of a European call opion wih srike K mauriy T wrienonazerocoupon bond wih mauriy S a ime [,T] is given by σ = B(T,S) ZBC(, T, S, K) =P (, S)Φ(h) KP(, T )Φ(h σ ), σ 2 (u, T )du h = 1 ( ) P (, S) σ ln + σ P (, T )K 2 wih B as in Theorem 7.14. The price of a corresponding pu opion is given by ZBP(, T, S, K) =KP(, T )Φ( h + σ ) P (, S)Φ( h).

7.3. RITCHKEN SANKARASUBRAMANIAN MODEL 47 Theorem 7.18 (Fuures conrac on a zero-coupon bond in a Gaussian HJM model wih separable volailiy). In a Gaussian HJM model wih separable volailiy, he price of a fuures conrac wih mauriy T on a zero-coupon bond a ime T wih mauriy S is given by { } P (, S) T FUT(, T, S) = P (, T ) exp B(T,S) σ(u, u)σ(u, T )B(u, T )du { ( )( P (, S) S )} s = P (, T ) exp η(u)du η(s) ξ 2 (u)duds. T Definiion 7.19 (Richken Sankarasubramanian model). The Richken Sankarasubramanian model is an HJM model wih separable volailiy for which here exis funcions σ k such ha { } { } ξ() =σ()exp k(u)du η() =exp k(u)du. Theorem 7.2 (Zero-coupon bond in he Richken Sankarasubramanian model). In he Richken Sankarasubramanian model, he price of a zero-coupon bond wih mauriy T a ime [,T] is given by P (, T )= P (,T) { P (,) exp f(,)b(, T ) 1 } 2 φ()b2 (, T ) e r()b(,t ), { } φ() = σ 2 (u)exp 2 k(v)dv du B(, T )= { exp s u } k(u)du ds. Theorem 7.21 (Shor-rae dynamics in he Richken Sankarasubramanian model). In a Richken Sankarasubramanian model in which k is deerminisic posiive, he shor rae saisfies he sochasic differenial equaion ( dr() = k()f(,)+ f(,) ) + φ() k()r() d + σ()dw () wih φ as in Theorem 7.2. Definiion 7.22 (Gaussian HJM model wih exponenially damped volailiy). A Gaussian HJM model wih exponenially damped volailiy is a Richken Sankarasubramanian model in which he funcions σ k are posiive consans.

48 7. HEATH JARROW MORTON FRAMEWORK Theorem 7.23 (The Gaussian HJM model wih exponenially damped volailiy he Hull Whie model). Suppose r is he shor rae in a Gaussian HJM model wih exponenially damped volailiy. Then r is equal o he shor rae in he corresponding calibraed Hull Whie model. Remark 7.24. Since for a Gaussian HJM model wih exponenially damped volailiy we have σ(, T )=σe k(t ) ) 1 e k(t, B(, T )=, k σ 2 (u, T )du = (1 σ2 e 2k(T )), φ() = σ2 ( 1 e 2k ), 2k 2k we may use Theorem 7.23 o show ha Theorem 7.2 implies Theorem 5.12; Theorem 7.17 implies Theorem 5.13; Theorem 7.18 implies for he Hull Whie model FUT(, T, S) = P (, S) P (, T ) exp ( σ2 2 B(T,S)B2 (, T ) Definiion 7.25 (Gaussian HJM model wih consan volailiy). A Gaussian HJM model wih consan volailiy is a Richken Sankarasubramanian model in which σ is a posiive consan k =. Theorem 7.26 (The Gaussian HJM model wih consan volailiy he Ho Le model). Suppose r is he shor rae in a Gaussian HJM model wih consan volailiy. Then r is equal o he shor rae in he corresponding calibraed Ho Le model. Remark 7.27. Since for a Gaussian HJM model wih consan volailiy we have σ(, T )=σ, B(, T )=T, we may use Theorem 7.26 o show ha ). σ 2 (u, T )du = σ 2 (T ), φ() =σ 2, Theorem 7.2 implies Theorem 5.4; Theorem 7.17 implies he formula for ZBC from Theorem 5.2; Theorem 7.18 implies for he Ho Le model ) P (, S) FUT(, T, S) = ( P (, T ) exp σ2 (S T )(T )2. 2

7.4. MERCURIO MORALEDA MODEL 49 7.4. Mercurio Moraleda Model Definiion 7.28 (Gaussian HJM model wih volailiy depending on ime o mauriy). A Gaussian HJM model wih volailiy depending on ime o mauriy is an HJM model in which here exiss a deerminisic funcion h such ha σ(, T )=h(t ). Theorem 7.29 (Opion on a zero-coupon bond in a Gaussian HJM model wih volailiy depending on ime o mauriy). In a Gaussian HJM model wih volailiy depending on ime o mauriy, he price of a European call opion wih srike K mauriy T wrien on a zero-coupon bond wih mauriy S a ime [,T] is given by τ σ = ZBC(, T, S, K) =P (, S)Φ(h) KP(, T )Φ(h σ ), ( u+μ u h(x)dx) 2 du wih τ = T μ = S T h = 1 ( ) P (, S) σ ln + σ P (, T )K 2. The price of a corresponding pu opion is given by ZBP(, T, S, K) =KP(, T )Φ( h + σ ) P (, S)Φ( h). Theorem 7.3 (Fuures conrac on a zero-coupon bond in a Gaussian HJM model wih volailiy depending on ime o mauriy). In a Gaussian HJM model wih volailiy depending on ime o mauriy, he price of a fuures conrac wih mauriy T on a zero-coupon bond a ime T wih mauriy S is given by { P (, S) τ ( u )( u+μ ) } FUT(, T, S) = P (, T ) exp h(x)dx h(x)dx du wih τ μ as in Theorem 7.29. Definiion 7.31 (Mercurio Moraleda model). The Mercurio Moraleda model is a Gaussian HJM model wih volailiy depending on ime o mauriy for which here exis consans σ, γ, λ > such ha h(x) =σ(1 + γx)e λ 2 x. u

5 7. HEATH JARROW MORTON FRAMEWORK Theorem 7.32 (Opion on a zero-coupon bond in he Mercurio Moraleda model). In he Mercurio Moraleda model, he price of a European call opion wih srike K mauriy T wrien on a zero-coupon bond wih mauriy S a ime [,T] is given by wih σ = ZBC(, T, S, K) =P (, S)Φ(h) KP(, T )Φ(h σ ), 2σ (α λ 2 λ 2 +2αβλ +2β 2 )(1 e λτ ) λβτ(2αλ +2β + βλτ)e λτ 7/2 h = 1 ( ) P (, S) σ ln + σ P (, T )K 2 α =(λ +2γ)(1 e λ 2 μ ) γλμe λ 2 μ, β = γλ(1 e λ 2 μ ) τ μ are as in Theorem 7.29. The price of a corresponding pu opion is given by ZBP(, T, S, K) =KP(, T )Φ( h + σ ) P (, S)Φ( h). Theorem 7.33 (Fuures conrac on a zero-coupon bond in he Mercurio Moraleda model). In he Mercurio Moraleda model, he price of a fuures conrac wih mauriy T on a zero-coupon bond a ime T wih mauriy S is given by ( ) P (, S) 4σ 2 FUT(, T, S) = P (, T ) exp λ 4 z wih z = αα λ 2 + α βλ + αβ λ +2ββ λ 3 (e λτ 1) + α βλ + β αλ +2ββ λ 2 + ββ λ τ 2 e λτ + 2α (αλ +2β) ( ) λ 2 1 e λ 2 τ 2βα λ τe λ 2 τ, α, β, τ, μ are as in Theorem 7.32 α = λ +2γ β = γλ. τe λτ