Download Chapter 8 Heath–Jarrow–Morton (HJM) Methodology

Chapter 8
Heath–Jarrow–Morton (HJM)
Methodology
→ original article by Heath, Jarrow and Morton [10], MR[20](Chapter 13.1),
Z[28](Chapter 4.4), etc.
As we have seen in the previous section, short rate models are not always flexible enough to calibrating them to the observed initial yield curve.
Heath, Jarrow and Morton (HJM, 1992) [10] proposed a new framework for
modelling the entire forward curve directly.
The setup is as in Chapter 6. We fix a stochastic basis (Ω, F, (Ft )t≥0 , P),
where P is considered as objective probability measure, and a d-dimensional
Brownian motion W .
8.1
Forward Curve Movements
We assume that we are given an R-valued and Rd -valued stochastic process
α(t, T ) and σ(t, T ) = (σ1 (t, T ), . . . , σd (t, T )), respectively, with two indices,
t, T , such that
HJM.1 for every ω, the maps (t, T ) 7→ α(t, T, ω) and (t, T ) 7→ σ(t, T, ω) are
continuous for 0 ≤ t ≤ T ,
HJM.2 for every T , the processes α(t, T ) and σ(t, T ), 0 ≤ t ≤ T , are
adapted,
RT RT
HJM.3 0 0 E |α(s, t)| ds dt < ∞ for all T ,
97
98
HJM.4
CHAPTER 8. HJM METHODOLOGY
RT RT
0
0
E kσ(s, t)k2 ds dt < ∞ for all T .
For a given continuous initial forward curve T 7→ f (0, T ) it is then assumed
that, for every T , the forward rate process f (·, T ) follows the Itô dynamics
Z t
Z t
f (t, T ) = f (0, T ) +
α(s, T ) ds +
σ(s, T ) dW (s), t ≤ T.
(8.1)
0
0
This is a very general setup. The only substantive economic restrictions
are the continuous sample paths assumption for the forward rate process,
and the finite number, d, of random drivers W1 , . . . , Wd .
Notice that the integrals in (8.1) are well defined by HJM.1–HJM.4.
Moreover, it follows from Corollary 8.4.2 that the short rate process
Z t
Z t
r(t) = f (t, t) = f (0, t) +
α(s, t) ds +
σ(s, t) dW (s)
0
0
has a predictable version (again denoted by r(t)) and, by Schwarz’ inequality
and the Itô isometry,
Z T
Z T
Z TZ t
E |r(t)| dt ≤
|f (0, t)| dt +
E |α(s, t)| ds dt
0
0
+T
1
2
ÃZ
0
T
E
0
≤
Z
T
0
+T
µZ
0
|f (0, t)| dt +
1
2
µZ
t
T
Z
0
0
Rt
Hence the savings account B(t) = e
about the zero-coupon bond prices.
0
T
Z
0
¶2 ! 21
σ(s, t) dW (s) dt
0
T
Z
0
T
E |α(s, t)| ds dt
¶ 21
E kσ(s, t)k ds dt
< ∞.
r(s)ds
2
is well defined. More can be said
Lemma 8.1.1.
every maturity
T , the zero-coupon bond price process
³ For
´
RT
P (t, T ) = exp − t f (t, u) du , 0 ≤ t ≤ T , is an Itô process of the form
P (t, T ) = P (0, T ) +
Z
0
t
P (s, T ) (r(s) + b(s, T )) ds +
Z
0
t
P (s, T )a(s, T ) dW (s)
(8.2)
8.1. FORWARD CURVE MOVEMENTS
99
where
a(s, T ) := −
b(s, T ) := −
Z
T
σ(s, u) du,
s
Z
s
T
1
α(s, u) du + ka(s, T )k2 .
2
Proof. Using the classical Fubini Theorem and Theorem 8.4.1 for stochastic
integrals twice, we calculate
Z
T
logP (t, T ) = −
f (t, u) du
t
Z T
Z TZ t
Z TZ t
=−
f (0, u) du −
α(s, u) ds du −
σ(s, u) dW (s) du
t
t
0
t
0
Z T
Z tZ T
Z tZ T
=−
f (0, u) du −
α(s, u) du ds −
σ(s, u) du dW (s)
t
0
t
0
t
Z T
Z tZ T
Z tZ T
=−
f (0, u) du −
α(s, u) du ds −
σ(s, u) du dW (s)
0
0
s
0
s
Z t
Z tZ t
Z tZ t
+
f (0, u) du +
α(s, u) du ds +
σ(s, u) du dW (s)
0
0
s
0
s
¶
Z t
Z T
Z tµ
1
2
a(s, T ) dW (s)
=−
f (0, u) du +
b(s, T ) − ka(s, T )k ds +
2
0
0
0
¶
Z tµ
Z u
Z u
+
f (0, u) du +
α(s, u) ds +
σ(s, u) dW (s) du
0
0
0
|
{z
}
Z tµ
=r(u)
¶
1
2
r(s) + b(s, T ) − ka(s, T )k ds
2
= log P (0, T ) +
0
Z t
+
a(s, T ) dW (s),
0
Rt
Ru
and we have used the fact that 0 σ(s, u)1{s≤u} dW (s) = 0 σ(s, u) dW (s).
Itô’s formula now implies (8.2) (→ exercise).
We write Z(t, T ) =
P (t,T )
B(t)
for the discounted bond price processes.
100
CHAPTER 8. HJM METHODOLOGY
Corollary 8.1.2. We have, for 0 ≤ t ≤ T ,
Z t
Z t
Z(s, T )a(s, T ) dW (s).
Z(s, T )b(s, T ) ds +
Z(t, T ) = P (0, T ) +
0
0
Proof. Itô’s formula (→ exercise).
8.2
Absence of Arbitrage
In this section we investigate the restrictions on the dynamics (8.1) under
the assumption of no arbitrage. In what follows we let γ ∈ L be such that
E(γ · W ) is a uniformly integrable martingale with E∞ (γ · W ) > 0. Girsanov’s
Change of Measure Theorem 6.2.2 then implies that
dQ
= E∞ (γ · W )
dP
defines an equivalent probability measure Q ∼ P, and
Z t
γ(s) ds
W̃ (t) := W (t) −
0
is a Q-Brownian motion. We call Q an ELMM for the bond market if the
(t,T )
discounted bond price processes, Z(t, T ) = PB(t)
, 0 ≤ t ≤ T , are Q-local
martingales, for all T .
Theorem 8.2.1 (HJM Drift Condition). Q is an ELMM if and only if
b(t, T ) = −a(t, T ) · γ(t) ∀T
dt ⊗ dP − a.s.
(8.3)
In this case, the Q-dynamics of the forward rates f (t, T ), 0 ≤ t ≤ T , are of
the form
¶
Z tµ
Z T
Z t
f (t, T ) = f (0, T ) +
σ(s, T ) ·
σ(s, u) du ds +
σ(s, T ) dW̃ (s).
0
s
0
|
{z
}
HJM drift
(8.4)
Proof. In view of Corollary 8.1.2 we find that
dZ(t, T ) = Z(t, T ) (b(t, T ) + a(t, T ) · γ(t)) dt + Z(t, T )a(t, T ) dW̃ (t).
101
8.2. ABSENCE OF ARBITRAGE
Hence Z(t, T ), 0 ≤ t ≤ T , is a Q-local martingale if and only if b(t, T ) =
−a(t, T ) · γ(t) dt ⊗ dP-a.s. Since a(t, T ) and b(t, T ) are continuous in T , we
deduce that Q is an ELMM if and only if (8.3) holds.
Differentiating both sides of (8.3) in T yields
−α(t, T ) + σ(t, T ) ·
Z
t
T
σ(t, u) du = σ(t, T ) · γ(t) ∀T
dt ⊗ dP − a.s.
Inserting this in (8.1) gives (8.4).
Remark 8.2.2. It follows from (8.2) and (8.3) that
dP (t, T ) = P (t, T ) (r(t) + a(t, T ) · (−γ(t))) dt + P (t, T )a(t, T ) dW (t).
Whence the interpretation of −γ as the market price of risk for the bond
market.
The striking feature of the HJM framework is that the distribution of
f (t, T ) and P (t, T ) under Q only depends on the volatility process σ(t, T )
(and not on the P-drift α(t, T )). Hence option pricing only depends on σ.
This situation is similar to the Black–Scholes stock price model.
We can give sufficient conditions for Z(t, T ) to be a true Q-martingale.
Corollary 8.2.3. Suppose that (8.3) holds. Then Q is an EMM if either
1. the Novikov condition
·
µ Z T
¶¸
1
2
EQ exp
kσ(t, T )k dt
<∞
2 0
∀T
(8.5)
holds; OR
2. the forward rates are positive: f (t, T ) ≥ 0 ∀t ≤ T .
Proof. We have Z(t, T ) = P (0, T )Et (σ(·, T ) · W̃ ). Hence the Novikov condition (8.5) is sufficient for Z(t, T ) to be a Q-martingale (see [24, Proposition
(1.26), Chapter IV]).
If f (t, T ) ≥ 0, then 0 ≤ P (t, T ) ≤ 1 and B(t) ≥ 1. Hence 0 ≤ Z(t, T ) ≤ 1.
But a uniformly bounded local martingale is a true martingale.
102
CHAPTER 8. HJM METHODOLOGY
8.3
Short Rate Dynamics
What is the interplay between the short rate models of the last chapter and
the present HJM framework? Let us consider the simplest HJM model: a
constant σ(t, T ) ≡ σ > 0. Suppose that Q is an ELMM. Then (8.4) implies
µ
¶
t
2
f (t, T ) = f (0, T ) + σ t T −
+ σ W̃ (t).
2
Hence for the short rates
σ 2 t2
+ σ W̃ (t).
2
This is just the Ho–Lee model of Section 7.5.4.
In general, we have the following
r(t) = f (t, t) = f (0, t) +
Proposition 8.3.1. Suppose that f (0, T ), α(t, T ) and σ(t, T ) are differenRT
tiable in T with 0 |∂u f (0, u)| du < ∞ and such that HJM.1–HJM.4 are
satisfied when α(t, T ) and σ(t, T ) are replaced by ∂T α(t, T ) and ∂T σ(t, T ),
respectively.
Then the short rate process is an Itô process of the form
Z t
Z t
σ(u, u) dW (u)
(8.6)
ζ(u) du +
r(t) = r(0) +
0
0
where
ζ(u) := α(u, u) + ∂u f (0, u) +
Z
u
∂u α(s, u) ds +
0
Z
u
∂u σ(s, u) dW (s).
0
Proof. Recall first that
r(t) = f (t, t) = f (0, t) +
Z
t
α(s, t) ds +
Z
t
σ(s, t) dW (s).
0
0
Applying the Fubini Theorem 8.4.1 to the stochastic integral gives
Z t
Z t
Z t
σ(s, t) dW (s) =
σ(s, s) dW (s) +
(σ(s, t) − σ(s, s)) dW (s)
0
0
0
Z t
Z tZ t
=
σ(s, s) dW (s) +
∂u σ(s, u) du dW (s)
0
0
s
Z t
Z tZ u
=
σ(s, s) dW (s) +
∂u σ(s, u) dW (s) du.
0
0
0
103
8.4. FUBINI’S THEOREM
Moreover, from the classical Fubini Theorem we deduce, similarly,
Z
t
α(s, t) ds =
0
Z
t
α(s, s) ds +
0
and finally
f (0, t) = r(0) +
Z tZ
0
Z
u
∂u α(s, u) ds du,
0
t
∂u f (0, u) du.
0
Combining these formulas, we obtain (8.6).
8.4
Fubini’s Theorem
In this section we prove Fubini’s Theorem for stochastic integrals. For the
classical version of Fubini’s Theorem, we refer to the standard textbooks in
integration theory.
In what follows we let A denote a closed convex subset in [0, T ]2 and
c
A = [0, T ]2 \ A its complement, e.g. A = {(t, s) ∈ [0, T ]2 | t ≤ s}.
Theorem 8.4.1 (Fubini’s Theorem for stochastic integrals). Consider
the Rd -valued stochastic process φ = φ(t, s) with two indices, 0 ≤ t, s ≤ T ,
satisfying the following properties:
1. for every ω, the map (t, s) 7→ φ(t, s, ω) is continuous on the interior of
A and Ac
2. for every s ∈ [0, T ], the process φ(t, s), 0 ≤ t ≤ T , is adapted,
3.
RT RT
0
0
E kφ(t, s)k2 dt ds < ∞.
RT
Then the stochastic process ψ(s) := 0 φ(t, s) dW (t) has a B[0, T ] ⊗ FT RT
measurable version (denoted again by ψ), and λ(t) := 0 φ(t, s) ds is piecewise continuous.
RT
RT
Moreover, 0 ψ(s) ds = 0 λ(t) dW (t), that is,
Z
0
T
µZ
0
T
¶
Z
φ(t, s) dW (t) ds =
0
T
µZ
0
T
¶
φ(t, s) ds dW (t).
(8.7)
104
CHAPTER 8. HJM METHODOLOGY
Proof. We may assume that kφk ≤ N , otherwise we replace φ by φ1{kφk≤N }
and let N → ∞. Notice that τ (s, ω) := inf{t | kφ(t, s, ω)k > N } ∧ T is
B[0, T ] ⊗ FT -measurable and a stopping time for every fixed s (→ exercise).
That λ is piecewise continuous and adapted follows from the classical
Fubini Theorem (→ exercise).
Now let 0 = t0 < t1 < · · · < tn = T be a partition of the interval [0, T ],
and define
n−1
X
φn (t, s) :=
φ(ti , s)1(ti ,ti+1 ] (t).
i=0
It is then clear that
Z T
n−1
X
ψn (s) :=
φn (t, s) dW (t) =
φ(ti , s) (W (ti+1 ) − W (ti ))
0
(8.8)
i=0
is B[0, T ] ⊗ FT -measurable. Moreover,
!
Z T
Z T ÃX
n−1
φ(ti , s) (W (ti+1 ) − W (ti )) ds
ψn (s) ds =
0
=
0
i=0
µ
Z
n−1
T
X
i=0
T
=
Z
0
0
µZ
¶
φ(ti , s) ds (W (ti+1 ) − W (ti ))
(8.9)
¶
T
φn (t, s) ds dW (t).
0
From the Itô isometry and dominated convergence we have
¸
·Z T
£
2
2¤
(φn (t, s) − φ(t, s)) dt = 0 ∀s.
lim E (ψn (s) − ψ(s)) = lim E
n→∞
n→∞
0
(8.10)
Let A := {(s, ω) | limn ψn (s, ω) exists}. Then A is B[0, T ] ⊗ FT -measurable
and so is the process
(
b ω) := limn ψn (s, ω), if (s, ω) ∈ A
ψ(s,
(8.11)
0,
otherwise.
b
But in view of (8.10) we have ψ(s) = ψ(s)
a.s. Hence ψ has a B[0, T ] ⊗
F
-measurable
version,
which
we
denote
again
by ψ, so that the integral
R TT
ψ(s) ds is well defined and FT -measurable.
0
105
8.4. FUBINI’S THEOREM
From the Itô isometry and dominated convergence again we then have
E
"µZ
T
ψn (s) ds −
0
Z
=T
T
E
0
·Z
Z
0
T
0
T
ψ(s) ds
¶2 #
≤T
Z
T
0
£
¤
E (ψn (s) − ψ(s))2 ds
¸
(φn (t, s) − φ(t, s)) dt ds → 0 for n → ∞. (8.12)
2
On the other hand,
E
"µZ
0
T
µZ
¶
T
0
φn (t, s) ds dW (t) −
=E
≤ TE
·Z
0
"Z
T
0
T
Z
0
µZ
0
T
¶2 #
λ(t) dW (t)
T
φn (t, s) ds −
0
T
Z
Z
T
φ(t, s) ds
0
¶2
#
dt
¸
(φn (t, s) − φ(t, s)) ds dt → 0 for n → ∞. (8.13)
2
Combining (8.12) and (8.13) with (8.9) proves the theorem.
Corollary 8.4.2. Let φ be as in Theorem 8.4.1. Then the process
Z s
ψ(s) :=
φ(t, s) dW (t), s ∈ [0, T ],
0
has a predictable version.
Proof. This follows by similar arguments as in the proof of Theorem 8.4.1.
Replace (8.8) by the predictable process
ψn (s) :=
Z
0
s
φn (t, s) dW (t) =
n−1
X
i=0
φ(ti ∧ s, s) (W (ti+1 ∧ s) − W (ti ∧ s)) ,
and modify (8.10) and (8.11) accordingly (→ exercise).
106
CHAPTER 8. HJM METHODOLOGY
Chapter 9
Forward Measures
We consider the HJM setup (Chapter 8) and assume there exists an EMM
Q ∼ P of the form dQ/dP = E(γ · W ), as in Section 8.2, under which all
discounted bond price processes
P (t, T )
,
B(t)
t ∈ [0, T ],
are strictly positive martingales.
9.1
T -Bond as Numeraire
Fix T > 0. Since
·
¸
1
1
> 0 and EQ
=1
P (0, T )B(T )
P (0, T )B(T )
we can define an equivalent probability measure QT ∼ Q on FT by
1
dQT
=
.
dQ
P (0, T )B(T )
For t ≤ T we have
¸
· T
dQT
dQ
P (t, T )
|Ft = EQ
| Ft =
.
dQ
dQ
P (0, T )B(t)
This probability measure has already been introduced in Section 7.6. It is
called the T -forward measure.
107
108
CHAPTER 9. FORWARD MEASURES
Lemma 9.1.1. For any S > 0,
P (t, S)
,
P (t, T )
t ∈ [0, S ∧ T ],
is a QT -martingale.
Proof. Let s ≤ t ≤ S ∧ T . Bayes’ rule gives
h
i
P (t,T ) P (t,S)
¸ E
·
|
F
Q
s
P (0,T )B(t) P (t,T )
P (t, S)
| Fs =
EQT
P (s,T )
P (t, T )
P (0,T )B(s)
=
P (s,S)
B(s)
P (s,T )
B(s)
=
P (s, S)
.
P (s, T )
We thus have an entire collection of EMMs now! Each QT corresponds to
a different numeraire, namely the T -bond. Since Q is related to the risk-free
asset, one usually calls Q the risk neutral measure.
T -forward measures give simpler pricing formulas. Indeed, let X be a
T -claim such that
X
∈ L1 (Q, FT ).
(9.1)
B(T )
Its arbitrage price at time t ≤ T is then given by
h RT
i
π(t) = EQ e− t r(s) ds X | Ft .
h R
i
T
To compute π(t) we have to know the joint distribution of exp − t r(s) ds
and X, and integrate with respect to that distribution. Thus we have to
compute a double integral, which in most cases turns out to be rather hard
work. If B(T )/B(t) and X were independent under Q (which is not realistic!
it holds, for instance, if r is deterministic) we would have
π(t) = P (t, T )EQ [X | Ft ] ,
a much nicer formula, since
• we only have to compute the single integral EQ [X | Ft ];
109
9.2. AN EXPECTATION HYPOTHESIS
• the bond price P (t, T ) can be observed at time t and does not have to
be computed.
The good news is that the above formula holds — not under Q though, but
under QT :
Proposition 9.1.2. Let X be a T -claim such that (9.1) holds. Then
EQT [|X|] < ∞
(9.2)
π(t) = P (t, T )EQT [X | Ft ] .
(9.3)
and
Proof. Bayes’s rule yields
EQT [|X|] = EQ
·
¸
|X|
< ∞ (by (9.1)),
P (0, T )B(T )
whence (9.2). And
·
¸
X
π(t) = P (0, T )B(t)EQ
| Ft
P (0, T )B(T )
P (t, T )
= P (0, T )B(t)
E T [X | Ft ]
P (0, T )B(t) Q
= P (t, T )EQT [X | Ft ] ,
which proves (9.3).
9.2
An Expectation Hypothesis
Under the forward measure the expectation hypothesis holds. That is, the
expression of the forward rates f (t, T ) as conditional expectation of the future
short rate r(T ).
To see that, we write W for the driving Q-Brownian motion. The forward
rates then follow the dynamics
¶
Z tµ
Z T
Z t
f (t, T ) = f (0, T ) +
σ(s, T ) ·
σ(s, u) du ds +
σ(s, T ) dW (s).
0
s
0
(9.4)
110
CHAPTER 9. FORWARD MEASURES
The Q-dynamics of the discounted bond price process is
P (t, T )
= P (0, T ) +
B(t)
Z
t
µ Z
−
P (s, T )
B(s)
0
T
s
¶
σ(s, u) du dW (s).
(9.5)
This equation has a unique solution
P (t, T )
= P (0, T )Et
B(t)
µµ Z
−
T
·
¶
¶
σ(·, u) du · W .
We thus have
µµ Z
−
dQT
|F = Et
dQ t
T
·
¶
¶
σ(·, u) du · W .
(9.6)
Girsanov’s theorem applies and
T
W (t) = W (t) +
Z t µZ
0
T
s
¶
σ(s, u) du ds,
t ∈ [0, T ],
is a QT -Brownian motion. Equation (9.4) now reads
f (t, T ) = f (0, T ) +
Z
t
σ(s, T ) dW T (s).
0
Hence, if
EQT
·Z
0
T
2
¸
kσ(s, T )k ds < ∞
then
(f (t, T ))t∈[0,T ]
is a QT -martingale.
Summarizing we have thus proved
Lemma 9.2.1. Under the above assumptions, the expectation hypothesis
holds under the forward measures
f (t, T ) = EQT [r(T ) | Ft ] .
111
9.3. OPTION PRICING IN GAUSSIAN HJM MODELS
9.3
Option Pricing in Gaussian HJM Models
We consider a European call option on an S-bond with expiry date T < S
and strike price K. Its price at time t = 0 (for simplicity only) is
h RT
i
π = EQ e− 0 r(s) ds (P (T, S) − K)+ .
We proceed as in Section 7.6 and decompose
£
¤
£
¤
π = EQ B(T )−1 P (T, S) 1(P (T, S) ≥ K) − KEQ B(T )−1 1(P (T, S) ≥ K)
= P (0, S)QS [P (T, S) ≥ K] − KP (0, T )QT [P (T, S) ≥ K] .
This option pricing formula holds in general.
We already know that
dP (t, T )
= r(t) dt + v(t, T ) dW (t)
P (t, T )
and hence
P (t, T ) = P (0, T ) exp
·Z
t
0
¶ ¸
Z tµ
1
2
r(s) − kv(s, T )k ds
v(s, T ) dW (s) +
2
0
where
We also know that
³
v(t, T ) := −
P (t,T )
P (t,S)
´
t∈[0,T ]
Z
T
σ(t, u) du.
t
is a QS -martingale and
QT -martingale. In fact (→ exercise)
(9.7)
³
P (t,S)
P (t,T )
´
t∈[0,T ]
is a
P (0, T )
P (t, T )
=
P (t, S)
P (0, S)
·Z t
¸
Z
¢
1 t¡
2
2
× exp
σT,S (s) dW (s) −
kv(s, T )k − kv(s, S)k ds
2 0
0
·Z t
¸
Z
1 t
P (0, T )
S
2
σT,S (s) dW (s) −
exp
kσT,S (s)k ds
=
P (0, S)
2 0
0
where
σT,S (s) := v(s, T ) − v(s, S) =
Z
T
S
σ(s, u) du,
(9.8)
112
CHAPTER 9. FORWARD MEASURES
and
P (0, S)
P (t, S)
=
P (t, T )
P (0, T )
¸
· Z t
Z
¢
1 t¡
2
2
kv(s, S)k − kv(s, T )k ds
σT,S (s) dW (s) −
× exp −
2 0
0
¸
· Z t
Z
P (0, S)
1 t
T
2
=
σT,S (s) dW (s) −
kσT,S (s)k ds .
exp −
P (0, T )
2 0
0
Now observe that
·
¸
P (T, T )
1
Q [P (T, S) ≥ K] = Q
≤
P (T, S)
K
·
¸
T
T P (T, S)
Q [P (T, S) ≥ K] = Q
≥K .
P (T, T )
S
S
This suggests to look at those models for which σT,S is deterministic, and
(T,S)
)
hence PP (T,T
and PP (T,T
are log-normally distributed under the respective
(T,S)
)
forward measures.
We thus assume that σ(t, T ) = (σ1 (t, T ), . . . , σd (t, T )) are deterministic functions of t and T , and hence forward rates f (t, T ) are Gaussian distributed.
We obtain the following closed form option price formula.
Proposition 9.3.1. Under the above Gaussian assumption, the option price
is
π = P (0, S)Φ[d1 ] − KP (0, T )Φ[d2 ],
where
log
d1,2 =
h
P (0,S)
KP (0,T )
i
qR
T
0
±
1
2
RT
0
kσT,S (s)k2 ds
,
kσT,S (s)k2 ds
σT,S (s) is given in (9.8) and Φ is the standard Gaussian CDF.
Proof. It is enough to observe that
R
P (0,T )
)
1 T
−
log
+
kσT,S (s)k2 ds
log PP (T,T
(T,S)
P (0,S)
2 0
qR
T
kσT,S (s)k2 ds
0
9.3. OPTION PRICING IN GAUSSIAN HJM MODELS
and
113
R
(T,S)
P (0,S)
1 T
log PP (T,T
−
log
+
kσT,S (s)k2 ds
)
P (0,T )
2 0
qR
T
kσT,S (s)k2 ds
0
are standard Gaussian distributed under QS and QT , respectively.
Of course, the Vasicek option price formula from Section 7.6.1 can now
be obtained as a corollary of Proposition 9.3.1 (→ exercise).
114
CHAPTER 9. FORWARD MEASURES
Chapter 10
Forwards and Futures
→ B[3](Chapter 20), or Hull (2002) [11]
We discuss two common types of term contracts: forwards, which are
mainly traded OTC, and futures, which are actively traded on many exchanges.
The underlying is in both cases a T -claim Y, for some fixed future date T .
This can be an exchange rate, an interest rate, a commodity such as copper,
any traded or non-traded asset, an index, etc.
10.1
Forward Contracts
A forward contract on Y, contracted at t, with time of delivery T > t, and
with the forward price f (t; T, Y) is defined by the following payment scheme:
• at T , the holder of the contract (long position) pays f (t; T, Y) and
receives Y from the underwriter (short position);
• at t, the forward price is chosen such that the present value of the
forward contract is zero, thus
i
h RT
− t r(s) ds
(Y − f (t; T, Y)) | Ft = 0.
EQ e
This is equivalent to
h RT
i
1
− t r(s) ds
f (t; T, Y) =
EQ e
Y | Ft
P (t, T )
= EQT [Y | Ft ] .
115
116
CHAPTER 10. FORWARDS AND FUTURES
Examples The forward price at t of
1. a dollar delivered at T is 1;
2. an S-bond delivered at T ≤ S is
P (t,S)
;
P (t,T )
3. any traded asset S delivered at T is
S(t)
.
P (t,T )
The forward price f (s; T, Y) has to be distinguished from the (spot) price
at time s of the forward contract entered at time t ≤ s, which is
h
−
EQ e
10.2
RT
s
r(u) du
i
(Y − f (t; T, Y)) | Fs
h RT
i
= EQ e− s r(u) du Y | Fs − P (t, T )f (t; T, Y).
Futures Contracts
A futures contract on Y with time of delivery T is defined as follows:
• at every t ≤ T , there is a market quoted futures price F (t; T, Y), which
makes the futures contract on Y, if entered at t, equal to zero;
• at T , the holder of the contract (long position) pays F (T ; T, Y) and
receives Y from the underwriter (short position);
• during any time interval (s, t] the holder of the contract receives (or
pays, if negative) the amount F (t; T, Y) − F (s; T, Y) (this is called
marking to market).
So there is a continuous cash-flow between the two parties of a futures contract. They are required to keep a certain amount of money as a safety
margin.
The volumes in which futures are traded are huge. One of the reasons
for this is that in many markets it is difficult to trade (hedge) directly in the
underlying object. This might be an index which includes many different
(illiquid) instruments, or a commodity such as copper, gas or electricity,
etc. Holding a (short position in a) futures does not force you to physically
deliver the underlying object (if you exit the contract before delivery date),
and selling short makes it possible to hedge against the underlying.
117
10.2. FUTURES CONTRACTS
Suppose Y ∈ L1 (Q). Then the futures price process is given by the
Q-martingale
F (t; T, Y) = EQ [Y | Ft ] .
(10.1)
Often, this is just how futures prices are defined . We now give a heuristic
argument for (10.1) based on the above characterization of a futures contract.
First, our model economy is driven by Brownian motion and changes in
a continuous way. Hence there is no reason to believe that futures prices
evolve discontinuously, and we may assume that
F (t) = F (t; T, Y) is a continuous semimartingale (or Itô process).
Now suppose we enter the futures contract at time t < T . We face a continuum of cashflows in the interval (t, T ]. Indeed, let t = t0 < · · · < tN = t
be a partition of [t, T ]. The present value of the corresponding cashflows
F (ti ) − F (ti−1 ) at ti , i = 1, . . . , N , is given by EQ [Σ | Ft ] where
Σ :=
N
X
i=1
1
(F (ti ) − F (ti−1 )) .
B(ti )
But the futures contract has present value zero, hence
EQ [Σ | Ft ] = 0.
This has to hold for any partition (ti ). We can rewrite Σ as
N
X
i=1
N
X
1
(F (ti ) − F (ti−1 )) +
B(ti−1 )
i=1
µ
1
1
−
B(ti ) B(ti−1 )
¶
(F (ti ) − F (ti−1 )) .
If we let the partition become finer and finer this expression converges in
probability towards
À
Z T ¿
Z T
Z T
1
1
1
dF (s) +
d
,F
=
dF (s),
B(s)
B
B(s)
t
t
t
s
since the quadratic variation of 1/B (finite variation) and F (continuous) is
zero. Under the appropriate integrability assumptions (uniform integrability)
we conclude that
¸
·Z T
1
dF (s) | Ft = 0,
EQ
B(s)
t
118
CHAPTER 10. FORWARDS AND FUTURES
and that
M (t) =
Z
0
t
1
dF (s) = EQ
B(s)
·Z
T
0
¸
1
dF (s) | Ft ,
B(s)
t ∈ [0, T ],
is a Q-martingale. If, moreover
·Z T
¸
2
EQ
B(s) dhM, M is = EQ [hF, F iT ] < ∞
0
then
F (t) =
Z
t
B(s) dM (s),
0
t ∈ [0, T ],
is a Q-martingale, which implies (10.1).
10.3
Interest Rate Futures
→ Z[28](Section 5.4)
Interest rate futures contracts may be divided into futures on short term
instruments and futures on coupon bonds. We only consider an example
from the first group.
Eurodollars are deposits of US dollars in institutions outside of the US.
LIBOR is the interbank rate of interest for Eurodollar loans. The Eurodollar
futures contract is tied to the LIBOR. It was introduced by the International
Money Market (IMM) of the Chicago Mercantile Exchange (CME) in 1981,
and is designed to protect its owner from fluctuations in the 3-months (=1/4
years) LIBOR. The maturity (delivery) months are March, June, September
and December.
Fix a maturity date T and let L(T ) denote the 3-months LIBOR for the
period [T, T + 1/4], prevailing at T . The market quote of the Eurodollar
futures contract on L(T ) at time t ≤ T is
1 − LF (t, T ) [100 per cent]
where LF (t, T ) is the corresponding futures rate (compare with the example
in Section 4.2.2). As t tends to T , LF (t, T ) tends to L(T ). The futures price,
used for the marking to market, is defined by
1
F (t; T, L(T )) = 1 − LF (t, T ) [Mio. dollars].
4
10.4. FORWARD VS. FUTURES IN A GAUSSIAN SETUP
119
Consequently, a change of 1 basis point (0.01%) in the futures rate LF (t, T )
leads to a cashflow of
106 × 10−4 ×
1
= 25 [dollars].
4
We also see that the final price F (T ; T, L(T )) = 1 − 41 L(T ) = Y is not
P (T, T + 1/4) = 1 − 14 L(T )P (T, T + 1/4) as one might suppose. In fact, the
underlying Y is a synthetic value. At maturity there is no physical delivery.
Instead, settlement is made in cash.
On the other hand, since
1
1 − LF (t, T ) = F (t; T, L(T ))
4
1
= EQ [F (T ; T, L(T )) | Ft ] = 1 − EQ [L(T ) | Ft ] ,
4
we obtain an explicit formula for the futures rate
LF (t, T ) = EQ [L(T ) | Ft ] .
10.4
Forward vs. Futures in a Gaussian Setup
Let S be the price process of a traded asset. Hence the Q-dynamics of S is
of the form
dS(t)
= r(t) dt + ρ(t) dW (t),
S(t)
for some volatility process ρ. Fix a delivery date T . The forward and futures
prices of S for delivery at T are
f (t; T, S(T )) =
S(t)
,
P (t, T )
F (t; T, S(T )) = EQ [S(T ) | Ft ].
Under Gaussian assumption we can establish the relationship between the
two prices.
Proposition 10.4.1. Suppose ρ(t) and v(t, T ) are deterministic functions
in t, where
Z T
v(t, T ) = −
σ(t, u) du
t
120
CHAPTER 10. FORWARDS AND FUTURES
is the volatility of the T -bond (see (9.7)). Then
F (t; T, S(T )) = f (t; T, S(T )) exp
µZ
T
(v(s, T ) − ρ(s)) · v(s, T ) ds
t
¶
for t ≤ T .
Hence, if the instantaneous correlation of dS(t) and dP (t, T ) is negative
dhS, P (·, T )it
= S(t)P (t, T )ρ(t) · v(t, T ) ≤ 0
dt
then the futures price dominates the forward price.
Proof. Write µ(s) := v(s, T ) − ρ(s). It is clear that
¶
µ Z t
Z
1 t
S(0)
2
kµ(s)k ds
f (t; T, S(T )) =
exp −
µ(s) dW (s) −
P (0, T )
2 0
0
¶
µZ t
µ(s) · v(s, T ) ds ,
× exp
0
and hence
¶
µ Z T
Z
1 T
2
µ(s) dW (s) −
kµ(s)k ds
f (T ; T, S(T )) = f (t; T, S(T )) exp −
2 t
t
¶
µZ T
µ(s) · v(s, T ) ds .
× exp
t
By assumption µ(s) is deterministic. Consequently,
·
µ Z
EQ exp −
T
t
1
µ(s) dW (s) −
2
Z
t
T
2
kµ(s)k ds
¶
¸
| Ft = 1
and
F (t; T, S(T )) = EQ [f (T ; T, S(T )) | Ft ]
¶
µZ T
µ(s) · v(s, T ) ds ,
= f (t; T, S(T )) exp
t
as desired.
10.4. FORWARD VS. FUTURES IN A GAUSSIAN SETUP
121
Similarly, one can show (→ exercise)
Lemma 10.4.2. In a Gaussian HJM framework (σ(t, T ) deterministic) we
have the following relations (convexity adjustments) between instantaneous
and simple futures and forward rates
¶
Z Tµ
Z T
f (t, T ) = EQ [r(T ) | Ft ] −
σ(s, T ) ·
σ(s, u) du ds,
t
s
F (t; T, S) = EQ [F (T, S) | Ft ]
³ RT RS
´
RS
P (t, T )
e t ( T σ(s,v) dv· s σ(s,u) du)ds − 1
−
(S − T )P (t, S)
for t ≤ T < S.
Hence, if
σ(s, v) · σ(s, u) ≥ 0 for all s ≤ min(u, v)
then futures rates are always greater than the corresponding forward rates.
122
CHAPTER 10. FORWARDS AND FUTURES