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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Advanced Analysis of Financial Risks: Discrete Time Models 99
where
Φ(x)=
1
√
2π
x
−∞
e
−y
2
/2
dy
is a standard normal distribution. Hence, we obtain
E
∗
(S
N
− K)
+
F
n
= S
n
e
δ(N −n)
Φ
ln(S
n
/K)+(N − n)(δ + σ
2
/2)
σ
√
N − n
−K Φ
ln(S
n
/K)+(N − n)(δ −σ
2
/2)
σ
√
N − n
.
Note that for n =0wehave
E
∗
(S
N
− K)
+
e
δN
= S
0
Φ
ln(S
0
/K)+N (δ + σ
2
/2)
σ
√
N
−Ke
−δN
Φ
ln(S
0
/K)+N (δ − σ
2
/2)
σ
√
N
,
which is the discrete version of the Black-Scholes formula foraEuropean
call option.
Now we compute the first expectation from (3.6):
E
∗
(S
N
− K)
+
X
n
F
n−1
= X
n−1
E
∗
e
w
∗
n
(S
N
− K)
+
F
n−1
=X
n−1
E
∗
!
E
∗
e
w
∗
n
(S
n−1
e
δ(N −n+1)
e
w
∗
n
+···+w
∗
N
− K)
+
F
n
F
n−1
"
= X
n−1
E
∗
!
e
w
∗
n
S
n−1
e
δ(N −n+1)
e
w
∗
n
×Φ
ln(S
n−1
/K)+w
∗
n
+(N − n)(δ + σ
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