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Implied Vol Surfaces and Smile Dynamics
Hard
·
25 min read
Derivatives Pricing
Implied Volatility
Local Volatility
SABR
Vol Surface
1
Article
2
Quiz
Quick Quiz
1.
By Breeden-Litzenberger, which condition on the call-price surface is equivalent to a non-negative risk-neutral density?
∂
C
/
∂
T
≥
0
\partial C/\partial T\ge 0
∂
C
/
∂
T
≥
0
(calendar-spread condition)
∂
C
/
∂
K
≤
0
\partial C/\partial K\le 0
∂
C
/
∂
K
≤
0
(call-spread monotonicity)
∂
2
C
/
∂
K
2
≥
0
\partial^2 C/\partial K^2\ge 0
∂
2
C
/
∂
K
2
≥
0
(butterfly positivity)
∂
2
C
/
∂
T
2
≥
0
\partial^2 C/\partial T^2\ge 0
∂
2
C
/
∂
T
2
≥
0
(convexity in maturity)
2.
In Dupire's formula
σ
L
2
(
K
,
T
)
=
∂
T
C
+
(
r
−
q
)
K
∂
K
C
+
q
C
1
2
K
2
∂
K
K
C
\sigma_L^2(K,T)=\dfrac{\partial_T C+(r-q)K\partial_K C+qC}{\tfrac12 K^2\partial_{KK}C}
σ
L
2
(
K
,
T
)
=
2
1
K
2
∂
K
K
C
∂
T
C
+
(
r
−
q
)
K
∂
K
C
+
q
C
, the denominator
1
2
K
2
∂
K
K
C
\tfrac12 K^2\partial_{KK}C
2
1
K
2
∂
K
K
C
is proportional to:
The vega of the option at strike
K
K
K
The forward price at maturity
T
T
T
The risk-neutral density of
S
T
S_T
S
T
at
K
K
K
The slope of the implied-vol smile
3.
In SABR, which parameter primarily controls the slope (skew) of the implied-vol smile at a fixed backbone?
ν
\nu
ν
(vol of vol)
α
\alpha
α
(initial volatility level)
ρ
\rho
ρ
(forward-vol correlation)
β
\beta
β
(CEV exponent)
4.
Under a local-volatility model, the future implied-vol smile (as seen from today) tends to flatten relative to today's smile as the spot moves forward.
True
False
5.
Roger Lee's moment formula bounds the wing behaviour of the smile. As
k
=
ln
(
K
/
F
)
→
+
∞
k=\ln(K/F)\to+\infty
k
=
ln
(
K
/
F
)
→
+
∞
, the upper bound on
σ
^
2
(
k
,
T
)
T
/
∣
k
∣
\hat\sigma^2(k,T)\,T/|k|
σ
^
2
(
k
,
T
)
T
/∣
k
∣
is:
0
4
2
1
6.
Which best describes the difference in smile dynamics between SABR and Heston?
SABR is used only for equities and Heston only for interest rates
Heston gives better short-maturity skew; SABR better captures long-maturity term structure
SABR gives better short-maturity skew; Heston better captures long-maturity term structure
Both produce identical smile dynamics by no-arbitrage
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