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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Financial Risk Management in the Binomial Model 43
we obtain
E
!
ε
N
−
μ − r
σ
2
N
k=1
(ρ
k
− μ)
S
N
I
{ω: S
N
≥K}
"
= S
0
E
!
ε
N
−
μ −r
σ
2
N
k=1
(ρ
k
− μ)
ε
N
N
k=1
ρ
k
I
{ω: S
N
≥K}
"
= S
0
E
!
ε
N
−
μ −r
σ
2
N
k=1
(ρ
k
− μ)
+
N
k=1
ρ
k
−
μ −r
σ
2
N
k=1
(ρ
k
− μ)ρ
k
I
{ω: S
N
≥K}
"
= S
0
N
k=k
0
N
k
1 −
μ − r
σ
2
(b −μ)+b −
μ −r
σ
2
(b −μ) b
k
p
k
×
1 −
μ −r
σ
2
(a −μ)+a −
μ − r
σ
2
(a − μ) a
N−k
(1 − p)
N−k
= S
0
N
k=k
0
N
k
p
∗
p
(1 + b)
k
p
k
1 − p
∗
1 − p
(1 + a)
N−k
(1 − p)
N−k
= S
0
N
k=k
0
N
k
p
∗
(1 + b)
k
(1 − p
∗
)(1+a)
N−k
= S
0
(1 + r)
N
N
k=k
0
N
k
p
∗
1+b
1+r
k
(1 − p
∗
)
1+a
1+r
N−k
.
Introducing the notation
p :=
1+b
1+r
p
∗
, and B(j, N, p):=
N
k=j
N
k
p
k
(1 − p)
N−k
,
we arrive at the Cox-Ross-Rubinstein formula
C
N
= S
0
B(k
0
,N,p) − K (1 + r)
−N
B(k
0
,N,p
∗
) .
The
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