December 2015
Intermediate to advanced
522 pages
20h
English
Content preview from Stochastic Volatility Modeling
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Forward variance models 257
Let us introduce the implied volatility for strike
K
,
bσ
K
. By denition of
bσ
K
:
P
K
Mkt
= P
K
BS
(S
0
, 0; bσ
2
K
T ). (7.76) is thus equal to:
1
2bσ
T
T
Z
∞
0
2
K
2
θ(K)
P
K
BS
(S
0
, 0; bσ
2
K
T ) −P
K
BS
(S
0
, 0;
b
σ
2
T )
dK
The highest value for this expression is obtained by setting
θ(K) = 1
for strikes
such that
bσ
K
>
b
σ
and
θ(K) = 0
otherwise. We thus get our nal expression for
the (sub-optimal) lower bound :
C
b
σ
Mkt
≥
1
2bσ
T
T
Z
∞
0
1
bσ
K
>
b
σ
2
K
2
P
K
BS
(S
0
, 0; bσ
2
K
T ) −P
K
BS
(S
0
, 0;
b
σ
2
T )
dK
(7.77)
This idea can be extended to the case of a call on forward-starting variance – see
[28].
Imagine that implied volatilities all lie above the (volatility) strike of the call
on realized ...
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