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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58 Risk Analysis in Finance and Insurance
The no-arbitrage condition implies the existence of p = p
∗
such that
p
∗
h(0; n, N)+(1− p
∗
) h(1; n, N )=1. (2.9)
Further, there is a δ
∗
> 1 such that
h
−1
(0; n, N)=p
∗
+(1− p
∗
)δ
N−n
∗
, (2.10)
h(1; n, N)=δ
N−n
∗
p
∗
+(1−p
∗
)δ
N−n
∗
−1
,
and
δ
N−n
∗
= h(1; n, N ) h
−1
(0; n, N) .
To verify equalities (2.9)–(2.10), we consider a portfolio π, where one unit is
invested in a zero-coupon bond with the maturity time N,andγ units are
invested in a zero-coupon bond with the maturity time
N. The value of this
portfolio at time n is
X
π
n
= B(n, N)+γB(n,
N)
= B(n − 1,N)
h(ξ
n
; n, N)
B(n − 1,N)
+ γB(n − 1,
N)
h(ξ
n
; n,
N)
B(n − 1,
N)
.
We say that portfolio ...
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