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Levenberg-Marquardt for Model Calibration
Hard
·
25 min read
Calibration
Numerical Optimisation
Heston Model
Implied Volatility
1
Article
2
Notebook
3
Quiz
Quick Quiz
1.
In the LM update
(
J
⊤
J
+
λ
D
)
δ
θ
=
−
J
⊤
r
(J^\top J+\lambda D)\,\delta\theta=-J^\top r
(
J
⊤
J
+
λ
D
)
δ
θ
=
−
J
⊤
r
, how does the step behave as
λ
→
∞
\lambda\to\infty
λ
→
∞
?
The step size grows without bound
δ
θ
→
−
1
λ
D
−
1
J
⊤
r
\delta\theta\to-\tfrac{1}{\lambda}D^{-1}J^\top r
δ
θ
→
−
λ
1
D
−
1
J
⊤
r
δ
θ
\delta\theta
δ
θ
becomes the Gauss-Newton direction
The algorithm halts because the system becomes singular
2.
Marquardt's scaling uses
D
=
d
i
a
g
(
J
⊤
J
)
D=\mathrm{diag}(J^\top J)
D
=
diag
(
J
⊤
J
)
instead of
D
=
I
D=I
D
=
I
. Its key advantage is that it:
Guarantees convergence to the global minimum from any start
Forces the damping parameter
λ
\lambda
λ
to decrease monotonically
Makes the LM step invariant to rescaling of individual parameters
Removes the need to compute the Jacobian
J
J
J
3.
The gain ratio
ρ
=
F
(
θ
)
−
F
(
θ
+
δ
θ
)
L
(
θ
)
−
L
(
θ
+
δ
θ
)
\rho=\dfrac{F(\theta)-F(\theta+\delta\theta)}{\mathcal{L}(\theta)-\mathcal{L}(\theta+\delta\theta)}
ρ
=
L
(
θ
)
−
L
(
θ
+
δ
θ
)
F
(
θ
)
−
F
(
θ
+
δ
θ
)
decides step acceptance. What does
ρ
<
0
\rho<0
ρ
<
0
indicate?
The actual objective increased — the proposed step is uphill
The residuals have converged to zero
The Jacobian is degenerate at the current iterate
The quadratic model underestimates the actual decrease in
F
F
F
4.
Satisfying the first-order condition
∥
J
⊤
r
∥
∞
<
ε
\|J^\top r\|_\infty<\varepsilon
∥
J
⊤
r
∥
∞
<
ε
in LM guarantees the calibrated parameters are the global minimum of the objective.
True
False
5.
A central-difference Jacobian has truncation error
O
(
h
2
)
O(h^2)
O
(
h
2
)
plus round-off from finite precision. What determines the optimal step size
h
h
h
?
The optimum is always
h
=
10
−
6
h=10^{-6}
h
=
1
0
−
6
, independent of the function
Smaller
h
h
h
lowers truncation but raises round-off;
h
∗
h^*
h
∗
balances them
Larger
h
h
h
lowers both truncation and round-off error at once
Larger
h
h
h
lowers truncation error but raises round-off from cancellation
6.
In Heston calibration,
κ
\kappa
κ
and
ν
ˉ
\bar\nu
ν
ˉ
(long-run variance) are often poorly identified. The formal characterisation is:
J
⊤
J
J^\top J
J
⊤
J
has near-zero eigenvalues in
κ
\kappa
κ
-
ν
ˉ
\bar{\nu}
ν
ˉ
directions
The constraint
κ
>
0
\kappa>0
κ
>
0
is binding at the optimum
The Jacobian
J
J
J
has an exact column of zeros for
κ
\kappa
κ
The objective is non-differentiable along the
κ
ν
ˉ
\kappa\bar\nu
κ
ν
ˉ
ridge
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