Advances in Risk Management

180 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

0

0.45

0.5

0.55

0.6

0.65

0.7



( 1

t)

0.75

0.8

0.85

0.9

0.95

51015

t

20 25 30

Zero inf.-bond

Inf.-bond

Inf.-bond with deflation

Figure 9.2Optimal portfolio process of inflation-linked bonds with

different structure characteristics

In order to maintain this optimal pure fraction of the bond and inflation
index (stock) in the portfolio, one has to increase the relative fraction of the
inflation-bond in the portfolio, when the inflation index is getting lower,
because the replicating strategy (9.20) becomes smaller, too.
The other interesting aspect of this example is the asymptotic behavior of
the optimal portfolio processes of inflation-linked bonds with deflation pro-
tection at the maturity dateT. Depending on the level of the inflation index
I(t) the price of the inflation-linked bondBIL(t,I(t)) approaches three differ-
ent asymptotical forms. In the case of inflation, for example, whenI(t)>I(t 0 ),
the asymptote is equal toψ(t)I(t). ForI(t)=I(t 0 ) we haveψ(t)I(t)+F/2
as an asymptote and in the case of deflation, for example,I(t)<I(t 0 ), the
asymptotic function forBIL(t,I(t)) isψ(t)I(t)+F. In these different cases the
asymptotic representations of the optimal portfolio processπ 1 (t)are:

π 1 (t)∼

λσI−rR

(1−γ)σI^2

, t→T, I(t)>I(t 0 )

π 1 (t)∼

λσI−rR

(1−γ)σI^2

(

1 +

F

2 ψ(t)I(t)

)

, t→T, I(t)=I(t 0 )