December 2013
Intermediate to advanced
484 pages
14h 54m
English
Content preview from Nonlinear Option Pricing
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370 Calibration of Local Correlation Models to Market Smiles
When one further assumes that there exist two functions a(t, X) and b(t, X)
such that b does not vanish and a + bλ is local in index, then one has
E
ρ
h
D
0t
v
ρ
0
(t, X
t
) −
a(t,X
t
)
b(t,X
t
)
(v
ρ
1
(t, X
t
) − v
ρ
0
(t, X
t
))
I
t
= K
i
E
ρ
[D
0t
|I
t
= K]
+ (a + bλ)(t, K)
E
ρ
h
D
0t
v
ρ
1
(t,X
t
)−v
ρ
0
(t,X
t
)
b(t,X
t
)
I
t
= K
i
E
ρ
[D
0t
|I
t
= K]
= K
2
σ
I
Dup
(t, K)
2
− K
E
ρ
D
0t
r
t
− q
t
− (r
0
t
− q
0
t
)
1
I
t
>K
1
2
∂
2
K
C(t, K)
+
E
ρ
h
D
0t
q
t
− q
0
t
(I
t
− K)
+
i
1
2
∂
2
K
C(t, K)
from which one gets λ(t, X) = λ
(a,b)
(t, X) ≡
f(t,I)−a(t,X)
b(t,X)
with f (t, I) ≡
N
f
(t,I)
D
f
(t,I)
defined by
N
f
(t, K) = K
2
σ
I
Dup
(t, K)
2
− K
E
ρ
(a,b)
D
0t
r
t
− q
t
− (r
0
t
− q
0
t
)
1
I
t
>K
1
2
∂
2
K
C(t, K)
+
E
ρ
(a,b)
h
D
0t
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I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.