December 2013
Intermediate to advanced
484 pages
14h 54m
English
Content preview from Nonlinear Option Pricing
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6 Option Pricing in a Nutshell
PROOF From the definition of B
t
(F
T
), we assume that there exists an
admissible portfolio ∆ such that
−D
0t
z +
Z
T
t
∆
s
· d
˜
X
s
+ D
0T
F
T
≥ 0 P
hist
− a.s.
As seen previously (see proof of Lemma 1.1 and Exercise 1),
R
t
0
∆
s
· d
˜
X
s
is a
Q-supermartingale, so E
Q
[
R
T
t
∆
s
· d
˜
X
s
|F
t
] ≤ 0 and we deduce that
D
0t
z ≤ E
Q
[D
0T
F
T
|F
t
]
i.e.,
z ≤ E
Q
[D
tT
F
T
|F
t
]
Taking the supremum over z, we get B
t
(F
T
) ≤ E
Q
[D
tT
F
T
|F
t
]. The inequality
E
Q
[D
tT
F
T
|F
t
] ≤ S
t
(F
T
) can be derived similarly.
These bounds can be sharpened by assuming that the market is complete.
In order to define what it means for a market to be complete, we need to
introduce the notion of attainable payoff:
DEFINITION ...
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