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Quiz: SVI Parametrisation
Quick Quiz
1.
For raw SVI
w
(
k
)
=
a
+
b
[
ρ
(
k
−
m
)
+
(
k
−
m
)
2
+
σ
2
]
w(k)=a+b[\rho(k-m)+\sqrt{(k-m)^2+\sigma^2}]
w
(
k
)
=
a
+
b
[
ρ
(
k
−
m
)
+
(
k
−
m
)
2
+
σ
2
]
, what are the asymptotic slopes of
w
(
k
)
w(k)
w
(
k
)
as
k
→
+
∞
k\to+\infty
k
→
+
∞
and
k
→
−
∞
k\to-\infty
k
→
−
∞
?
b
ρ
b\rho
b
ρ
for both wings, since
ρ
\rho
ρ
sets the asymmetry
b
b
b
for both wings, since
(
k
−
m
)
2
+
σ
2
≈
∣
k
−
m
∣
\sqrt{(k-m)^2+\sigma^2}\approx|k-m|
(
k
−
m
)
2
+
σ
2
≈
∣
k
−
m
∣
b
(
1
+
ρ
)
b(1+\rho)
b
(
1
+
ρ
)
as
k
→
+
∞
k\to+\infty
k
→
+
∞
and
b
(
1
−
ρ
)
b(1-\rho)
b
(
1
−
ρ
)
in magnitude as
k
→
−
∞
k\to-\infty
k
→
−
∞
0 for both wings, since
w
(
k
)
w(k)
w
(
k
)
tends to a constant
2.
The no-butterfly condition needs
g
(
k
)
≥
0
g(k)\ge 0
g
(
k
)
≥
0
for all
k
k
k
(with
g
g
g
built from
w
,
w
′
,
w
′
′
w,w',w''
w
,
w
′
,
w
′′
). Why is this tied to the Dupire local variance?
g
(
k
)
≥
0
g(k)\ge0
g
(
k
)
≥
0
is equivalent to the vol smile being convex in
k
k
k
Negative
g
(
k
)
g(k)
g
(
k
)
makes the implied-vol inverter diverge
σ
loc
2
=
∂
T
w
/
g
(
k
)
\sigma_{\text{loc}}^2=\partial_T w/g(k)
σ
loc
2
=
∂
T
w
/
g
(
k
)
, so
g
<
0
g<0
g
<
0
implies arbitrage
Negative
g
(
k
)
g(k)
g
(
k
)
makes some call prices negative
3.
In SSVI, the sufficient condition
θ
t
ϕ
(
θ
t
)
(
1
+
∣
ρ
∞
∣
)
<
4
\theta_t\phi(\theta_t)(1+|\rho_\infty|)<4
θ
t
ϕ
(
θ
t
)
(
1
+
∣
ρ
∞
∣
)
<
4
for global no-butterfly-arbitrage is most directly related to:
Positive semidefiniteness of the calibration Jacobian
The Feller condition for a CIR variance process underlying SSVI
The no-calendar-spread condition across consecutive maturities
Lee's moment formula, which bounds the wing slope of total variance as
∣
k
∣
→
∞
|k|\to\infty
∣
k
∣
→
∞
4.
Fitting each maturity slice independently with raw SVI automatically yields a surface free of calendar-spread arbitrage.
True
False
5.
Why is the jump-wings (JW) parametrisation often preferred over raw SVI for numerical calibration?
JW reduces the number of free parameters from 5 to 3
Its parameters are directly observable, giving better-conditioned fits
It uses a functional form that is easier to differentiate analytically
JW guarantees the calibrated slice is butterfly-arbitrage-free
6.
A 1-year raw-SVI slice has
(
a
=
0.04
,
b
=
0.2
,
ρ
=
−
0.7
,
m
=
0
,
σ
=
0.1
)
(a=0.04,\ b=0.2,\ \rho=-0.7,\ m=0,\ \sigma=0.1)
(
a
=
0.04
,
b
=
0.2
,
ρ
=
−
0.7
,
m
=
0
,
σ
=
0.1
)
. What are the ATM total variance
w
(
0
)
w(0)
w
(
0
)
and ATM implied vol?
w
(
0
)
=
a
+
b
σ
=
0.06
w(0)=a+b\sigma=0.06
w
(
0
)
=
a
+
bσ
=
0.06
;
σ
ATM
=
0.06
/
1
≈
24.5
%
\sigma_{\text{ATM}}=\sqrt{0.06/1}\approx 24.5\%
σ
ATM
=
0.06/1
≈
24.5%
w
(
0
)
=
a
+
b
ρ
σ
=
0.026
w(0)=a+b\rho\sigma=0.026
w
(
0
)
=
a
+
b
ρ
σ
=
0.026
;
σ
ATM
≈
16
%
\sigma_{\text{ATM}}\approx 16\%
σ
ATM
≈
16%
w
(
0
)
=
0.06
w(0)=0.06
w
(
0
)
=
0.06
;
σ
ATM
≈
6
%
\sigma_{\text{ATM}}\approx 6\%
σ
ATM
≈
6%
w
(
0
)
=
a
=
0.04
w(0)=a=0.04
w
(
0
)
=
a
=
0.04
;
σ
ATM
=
0.04
=
20
%
\sigma_{\text{ATM}}=\sqrt{0.04}=20\%
σ
ATM
=
0.04
=
20%
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