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Hull-White and the LMM Framework
Hard
·
25 min read
Derivatives Pricing
Interest Rate Models
Hull-White
Libor Market Model
Term Structure
1
Article
2
Quiz
Quick Quiz
1.
In Hull-White (
d
r
=
(
θ
(
t
)
−
a
r
)
d
t
+
σ
d
W
dr=(\theta(t)-ar)\,dt+\sigma\,dW
d
r
=
(
θ
(
t
)
−
a
r
)
d
t
+
σ
d
W
),
θ
(
t
)
\theta(t)
θ
(
t
)
is fixed by exact fit to the initial curve. For a curve flat at
r
0
r_0
r
0
,
θ
(
t
)
=
\theta(t)=
θ
(
t
)
=
a
r
0
a r_0
a
r
0
(the mean-reversion level only)
a
r
0
+
σ
2
2
a
(
1
−
e
−
2
a
t
)
a r_0+\frac{\sigma^2}{2a}(1-e^{-2at})
a
r
0
+
2
a
σ
2
(
1
−
e
−
2
a
t
)
r
0
r_0
r
0
(just the flat rate itself)
0 (no drift adjustment is needed for a flat curve)
2.
A caplet paying
δ
max
(
L
(
T
0
;
T
0
,
T
1
)
−
K
,
0
)
\delta\max(L(T_0;T_0,T_1)-K,0)
δ
max
(
L
(
T
0
;
T
0
,
T
1
)
−
K
,
0
)
at
T
1
T_1
T
1
can be written as a put on a zero-coupon bond. The effective bond strike is:
X
=
K
X=K
X
=
K
(the same as the rate strike)
X
=
e
−
K
T
1
X=e^{-KT_1}
X
=
e
−
K
T
1
(continuous discounting at the strike)
X
=
1
1
+
δ
K
X=\dfrac{1}{1+\delta K}
X
=
1
+
δ
K
1
X
=
δ
K
X=\delta K
X
=
δ
K
(accrual times strike)
3.
Under the spot Libor measure, the drift of
L
i
(
t
)
L_i(t)
L
i
(
t
)
is a sum over forward rates
L
j
L_j
L
j
,
j
=
β
(
t
)
,
…
,
i
j=\beta(t),\dots,i
j
=
β
(
t
)
,
…
,
i
. Its source is:
A risk premium investors demand for holding the Libor rate
A Girsanov change of numeraire to one common numeraire
A numerical artefact of the simulation discretisation
The Itô correction when moving from log to level representation of
L
i
L_i
L
i
4.
The Libor Market Model recovers Black's caplet formula exactly for each individual caplet, with no approximation.
True
False
5.
The HJM no-arbitrage condition says that under the risk-neutral measure the drift of the forward rate
f
(
t
,
T
)
f(t,T)
f
(
t
,
T
)
equals:
The risk-neutral short rate
r
t
r_t
r
t
itself, independent of
T
T
T
r
t
f
(
t
,
T
)
r_t\,f(t,T)
r
t
f
(
t
,
T
)
— proportional to the forward-rate level
0 for all
T
T
T
— driftless, as under the
T
T
T
-forward measure
σ
(
t
,
T
)
⊤
∫
t
T
σ
(
t
,
u
)
d
u
\boldsymbol{\sigma}(t,T)^\top\int_t^T \boldsymbol{\sigma}(t,u)\,du
σ
(
t
,
T
)
⊤
∫
t
T
σ
(
t
,
u
)
d
u
6.
Why is the LMM generally not amenable to PDE-based pricing, unlike Hull-White?
It is not Markov in a low, finite-dimensional state
The LMM uses discrete tenors, which PDEs cannot handle
The LMM has no closed-form characteristic function
The LMM requires negative rates, which PDE solvers cannot represent
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