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CHAPTER 7
Heath–Jarrow–Morton Framework
7.1. Heath–Jarrow–Morton Model
Definition 7.1 (Forward-rate dynamics in the HJM model). In the Heath– Jarrow–Morton model, briefly HJM model, the instantaneous forward interest rate with maturity T is assumed to satisfy the stochastic differential equation
df (t, T ) = α(t, T )dt + σ(t, T )dW (t),
where α and σ are adapted and W is a Brownian motion under the risk-neutral measure.
Theorem 7.2 (Bond-price dynamics in the HJM model). In the HJM model, the price of a zero-coupon bond with maturity T satisfies the stochastic differential equation
dP (t, T ) =
r(t) + A(t, T ) +^12 Σ^2 (t, T )
P (t, T )dt + Σ(t, T )P (t, T )dW (t),
where
A(t, T ) = −
∫ T
t^ α(t, u)du^ and^ Σ(t, T^ ) =^ −
∫ T
t^ σ(t, u)du. Theorem 7.3 (Bond-price dynamics implying HJM model). If the price of a zero-coupon bond with maturity T satisfies the stochastic differential equation
dP (t, T ) = m(t, T )P (t, T )dt + v(t, T )P (t, T )dW (t),
where m and v are adapted, then the forward-rate dynamics are as in the HJM model with
α(t, T ) = v(t, T )vT (t, T ) − mT (t, T ) and σ(t, T ) = −vT (t, T ). 43
44 7. HEATH–JARROW–MORTON FRAMEWORK
Theorem 7.4 (Drift restriction in the HJM model). In the HJM model, we necessarily have
A(t, T ) = − 12 Σ^2 (t, T ) and α(t, T ) = σ(t, T )
∫ T
t^ σ(t, u)du. Theorem 7.5 (Bond-price dynamics in the HJM model). In the HJM model, the price of a zero-coupon bond with maturity T satisfies the stochastic differential equations
dP (t, T ) = r(t)P (t, T )dt + Σ(t, T )P (t, T )dW (t)
and
d (^) P (t, T^1 ) = Σ
(^2) (t, T ) − r(t) P (t, T ) dt^ −^
Σ(t, T ) P (t, T ) dW^ (t). Theorem 7.6 (T -forward measure dynamics of the forward rate in the HJM model). Under the T -forward measure QT^ , the instantaneous forward interest rate with maturity T in the HJM model satisfies
df (t, T ) = σ(t, T )dW T^ (t),
where the QT^ -Brownian motion W T^ is defined by
dW T^ (t) = dW (t) − Σ(t, T )dt.
Theorem 7.7 (Forward-rate dynamics in the HJM model). In the HJM model, the simply-compounded forward interest rate for the period [T, S] satisfies the sto- chastic differential equation
dF (t; T, S) =
F (t; T, S) + (^) τ (T, S^1 )
(Σ(t, T ) − Σ(t, S)) dW S^ (t).
Theorem 7.8 (Zero-coupon bond in the HJM model). Let 0 ≤ t ≤ T ≤ S. In the HJM model, the price of a zero-coupon bond with maturity S at time T is given by
P (T, S) = P P^ ((t, St, T )) eZ^ ,
where
Z = − (^12)
∫ T
t
(Σ (^2) (u, S) − Σ (^2) (u, T )) (^) du +^ ∫^ T t
(Σ(u, S) − Σ(u, T )) dW (u)
= − (^12)
∫ T
t
(Σ(u, S) − Σ(u, T ))^2 du +
∫ T
t
(Σ(u, S) − Σ(u, T )) dW T^ (u).
46 7. HEATH–JARROW–MORTON FRAMEWORK
Theorem 7.14 (Zero-coupon bond in an HJM model with separable volatility). In an HJM model with separable volatility, the price of a zero-coupon bond with maturity T at time t ∈ [0, T ] is given by
P (t, T ) = P P^ (0 (0, T, t^ ) )exp
f (0, t)B(t, T ) − 12 φ(t)B^2 (t, T )
e−r(t)B(t,T^ ),
where φ(t) =
∫ (^) t 0
σ^2 (u, t)du and B(t, T ) = (^) η(^1 t)
∫ T
t
η(u)du.
Theorem 7.15 (Short-rate dynamics in an HJM model with separable volatil- ity). In an HJM model with separable volatility, the short rate satisfies the stochastic differential equation
dr(t) =
{ (^) ∂f (0, t) ∂t +^ φ(t)
dt + r(t)^ − η(^ ft)^ (0 , t)dη(t) +ξ(t)(dη(t))(dW (t)) + σ(t, t)dW (t),
where φ is as in Theorem 7.14.
Corollary 7.16 (Short-rate dynamics in a Gaussian HJM model with separable volatility). In an HJM model with separable volatility in which η is deterministic, the short rate satisfies the stochastic differential equation
dr(t) =
{ (^) ∂f (0, t) ∂t −^ f^ (0, t)^
η′(t) η(t) +^ φ(t) +^ r(t)^
η′(t) η(t)
dt + σ(t, t)dW (t),
where φ is as in Theorem 7.14.
Theorem 7.17 (Option on a zero-coupon bond in a Gaussian HJM model with separable volatility). In a Gaussian HJM model with separable volatility, the price of a European call option with strike K and maturity T and written on a zero- coupon bond with maturity S at time t ∈ [0, T ] is given by
ZBC(t, T, S, K) = P (t, S)Φ(h) − KP (t, T )Φ(h − σ∗),
where
σ∗^ = B(T, S)
√∫ T
t^ σ
(^2) (u, T )du and h = 1 σ∗^ ln
( (^) P (t, S) P (t, T )K
∗ 2 with B as in Theorem 7.14. The price of a corresponding put option is given by
ZBP(t, T, S, K) = KP (t, T )Φ(−h + σ∗) − P (t, S)Φ(−h).
7.3. RITCHKEN–SANKARASUBRAMANIAN MODEL 47 Theorem 7.18 (Futures contract on a zero-coupon bond in a Gaussian HJM model with separable volatility). In a Gaussian HJM model with separable volatil- ity, the price of a futures contract with maturity T on a zero-coupon bond at time T with maturity S is given by
FUT(t, T, S) = P P^ ((t, St, T )) exp
−B(T, S)
∫ T
t
σ(u, u)σ(u, T )B(u, T )du
= P P^ ((t, St, T )) exp
(∫ S
T
η(u)du
) (∫ T
t
η(s)
∫ (^) s t
ξ^2 (u)duds
Definition 7.19 (Ritchken–Sankarasubramanian model). The Ritchken– Sankarasubramanian model is an HJM model with separable volatility for which there exist functions σ and k such that
ξ(t) = σ(t) exp
{∫ (^) t 0 k(u)du
and η(t) = exp
∫ (^) t 0 k(u)du
Theorem 7.20 (Zero-coupon bond in the Ritchken–Sankarasubramanian model). In the Ritchken–Sankarasubramanian model, the price of a zero-coupon bond with maturity T at time t ∈ [0, T ] is given by
P (t, T ) = P P^ (0 (0, T, t^ ) )exp
f (0, t)B(t, T ) − 12 φ(t)B^2 (t, T )
e−r(t)B(t,T^ ),
where
φ(t) =
∫ (^) t 0 σ
(^2) (u) exp
∫ (^) t u^ k(v)dv
du
and B(t, T ) =
∫ T
t
exp
∫ (^) s t
k(u)du
ds.
Theorem 7.21 (Short-rate dynamics in the Ritchken–Sankarasubramanian model). In a Ritchken–Sankarasubramanian model in which k is deterministic and positive, the short rate satisfies the stochastic differential equation
dr(t) =
k(t)f (0, t) + ∂f^ (0 ∂t, t )+ φ(t) − k(t)r(t)
dt + σ(t)dW (t)
with φ as in Theorem 7.20.
Definition 7.22 (Gaussian HJM model with exponentially damped volatil- ity). A Gaussian HJM model with exponentially damped volatility is a Ritchken– Sankarasubramanian model in which the functions σ and k are positive constants.
7.4. MERCURIO–MORALEDA MODEL 49 7.4. Mercurio–Moraleda Model Definition 7.28 (Gaussian HJM model with volatility depending on time to maturity). A Gaussian HJM model with volatility depending on time to maturity is an HJM model in which there exists a deterministic function h such that
σ(t, T ) = h(T − t).
Theorem 7.29 (Option on a zero-coupon bond in a Gaussian HJM model with volatility depending on time to maturity). In a Gaussian HJM model with volatility depending on time to maturity, the price of a European call option with strike K and maturity T and written on a zero-coupon bond with maturity S at time t ∈ [0, T ] is given by ZBC(t, T, S, K) = P (t, S)Φ(h) − KP (t, T )Φ(h − σ∗),
where
σ∗^ =
√∫ (^) τ
0
(∫ (^) u+μ u
h(x)dx
du with τ = T − t and μ = S − T
and h = (^) σ^1 ∗ ln
( (^) P (t, S) P (t, T )K
∗
The price of a corresponding put option is given by
ZBP(t, T, S, K) = KP (t, T )Φ(−h + σ∗) − P (t, S)Φ(−h).
Theorem 7.30 (Futures contract on a zero-coupon bond in a Gaussian HJM model with volatility depending on time to maturity). In a Gaussian HJM model with volatility depending on time to maturity, the price of a futures contract with maturity T on a zero-coupon bond at time T with maturity S is given by
FUT(t, T, S) = P P^ ((t, St, T )) exp
{∫ (^) τ 0
(∫ (^) u 0
h(x)dx
) (∫ (^) u+μ u
h(x)dx
du
with τ and μ as in Theorem 7.29.
Definition 7.31 (Mercurio–Moraleda model). The Mercurio–Moraleda model is a Gaussian HJM model with volatility depending on time to maturity for which there exist constants σ, γ, λ > 0 such that
h(x) = σ(1 + γx)e−^ λ^2 x.
50 7. HEATH–JARROW–MORTON FRAMEWORK
Theorem 7.32 (Option on a zero-coupon bond in the Mercurio–Moraleda model). In the Mercurio–Moraleda model, the price of a European call option with strike K and maturity T and written on a zero-coupon bond with maturity S at time t ∈ [0, T ] is given by
ZBC(t, T, S, K) = P (t, S)Φ(h) − KP (t, T )Φ(h − σ∗),
where
σ∗^ = (^) λ^27 σ/ 2
(α^2 λ^2 + 2αβλ + 2β^2 )(1 − e−λτ^ ) − λβτ (2αλ + 2β + βλτ )e−λτ
and h = (^) σ^1 ∗ ln
( (^) P (t, S) P (t, T )K
∗ 2 with α = (λ + 2γ)(1 − e−^ λ^2 μ) − γλμe−^ λ^2 μ, β = γλ(1 − e−^ λ^2 μ)
and τ and μ are as in Theorem 7.29. The price of a corresponding put option is given by ZBP(t, T, S, K) = KP (t, T )Φ(−h + σ∗) − P (t, S)Φ(−h).
Theorem 7.33 (Futures contract on a zero-coupon bond in the Mercu- rio–Moraleda model). In the Mercurio–Moraleda model, the price of a futures con- tract with maturity T on a zero-coupon bond at time T with maturity S is given by FUT(t, T, S) = P P^ ((t, St, T )) exp
( (^4) σ 2 λ^4 z
with
z = αα^0 λ
(^2) + α 0 βλ + αβ 0 λ + 2ββ 0 λ^3 (e
−λτ (^) − 1) + α^0 βλ^ +^ β^0 αλ^ + 2ββ^0 λ^2 τ e
−λτ
- ββ λ 0 τ 2 e−λτ^ +^2 α^0 (αλ λ 2 + 2β)
1 − e−^ λ^2 τ^
− 2 βα λ 0 τ e−^ λ^2 τ^ ,
where α, β, τ, μ are as in Theorem 7.32 and
α 0 = λ + 2γ and β 0 = γλ.
