April 2011
Intermediate to advanced
328 pages
9h 5m
English
Content preview from Risk Analysis in Finance and Insurance, 2nd Edition
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126 Risk Analysis in Finance and Insurance
4.3 Imperfect hedging and risk measures
Consider the Black-Scholes model (4.6). Let Y
t
≡ Y
π
t
:= X
π
t
/B
t
≥ 0bethe
discounted value of a self-financing portfolio π. The Kolmogorov-Itˆoformula
implies that
dY
t
= φ
t
dW
∗
t
,Y
0
= X
π
0
,
where φ
t
= σγ
t
S
t
/B
t
and dW
∗
t
= dW
t
+ t (μ −r)/σ is a Wiener process with
respect to probability P
∗
, which is defined by its density (4.7).
The set
A = A(x, π, f
T
)=
ω : X
π
T
(x) ≥ f
T
=
ω : Y
π
T
(x) ≥ f
T
/B
T
is called the perfect hedging set for claim f
T
and strategy π with the initial
wealth x.
The theory of perfect hedging that was discussed above allows to find a
hedge with the initial wealth X
0
= E
∗
f
T
/B
T
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