Advances in Risk Management

TARAS BELETSKI AND RALF KORN 177

Trautmann (1999)). To understand the qualitative behavior of the optimal
strategy we look at the corresponding portfolio process:

π 1 (t)=

φ 1 (t)BIL(t,I(t))

X(t)

=

λσI−rR

(1−γ)σ^2 I

BIL(t,I(t))

ψ 1 (t)I(t)

=

λσI−rR

(1−γ)σI^2

θ(t,I(t)). (9.22)

By comparing relations (9.20) and (9.5) and using

ψ 1 (t)I(t)=BIL(t,I(t))−e−rN(T−t)F(1−(d(t)−σI

√

T−t)) (9.23)

we obtain the following relations:

θ(t,I(t))>1, θ(t,I(t))→

{

+∞, forI(t)→ 0

1, forI(t)→+∞

(9.24)

for example, the absolute value of the optimal portfolio process for the port-
folio problem (OP1) is always bigger than the one of the corresponding basic
problem (P). To interpret this, we look at the following two

Special cases

a) In the special case of an inflation-linked zero coupon bond, for exam-
ple, for

Ci=0, i=1,...,n

we obtain

ψ 1 (t)=

Fexp(−rR(T−t))N(d(t))

I(t 0 )

(9.25)

BIL(t,I(t))=F

(

exp(−rN(T−t))+

CI(t,I(t))

I(t 0 )

)

(9.26)

which would lead to the same limiting behavior of the quotientθ(t,I(t)) as
in relation (9.24).
In the special case of an inflation-linked bondwithout deflation protection,
for example, a usual inflation-linked coupon bond where the final payment
of the notional has the form:

F

I(T)

I(t 0 )

(9.6*)