December 2013
Intermediate to advanced
484 pages
14h 54m
English
Content preview from Nonlinear Option Pricing
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54 Some Excursions in Option Pricing
Let us set
Φ
T
t
= X
p
t
exp
1
2
p(p − 1)Y
T
t
with Y
T
t
=
R
T
t
ξ
s
t
ds. We write
dξ
T
t
= µ(t, T ) dt + ν(t, T ). dW
t
, ξ
T
t=0
= ξ
T
0
where (µ(t, s), 0 ≤ t ≤ s)
s≥0
and (ν(t, s), 0 ≤ t ≤ s)
s≥0
are two families of
respectively scalar-valued and vector-valued processes. Then
dY
T
t
=
−ξ
t
t
+
Z
T
t
µ(t, s) ds
!
dt +
Z
T
t
ν(t, s) ds
!
. dW
t
and
dΦ
T
t
Φ
T
t
=
1
2
p(p − 1)σ
2
t
dt + pσ
t
dW
0
t
+
1
2
p(p − 1)
−ξ
t
t
+
Z
T
t
µ(t, s) ds
!
dt +
Z
T
t
ν(t, s) ds
!
. dW
t
!
+
1
8
p
2
(p − 1)
2
Z
T
t
ν(t, s) ds
!
2
dt +
1
2
p
2
(p − 1)σ
t
Z
T
t
ν
0
(t, s) ds
By imposing that Φ
T
t
is a driftless process, we obtain the condition for all
T ∈ [t, ∞):
m
T
t
≡
1
2
p(p − 1)σ
2
t
+
1
2
p(p − 1)
−ξ
t
t
+
Z
T
t
µ(t, s) ds
!
+
1
8
p
2
(p − 1)
2
Z
T
t
ν(t, s) ds
!
2
+
1
2
p
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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.