Advances in Risk Management

178 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

we obtain:

ψ 1 (t)=

∑

i:ti>t

Ci

I(t 0 )

exp(−rR(ti−t))+

Fexp(−rR(T−t))

I(t 0 )

(9.27)

BIL(t,I(t))=

∑

i:ti>t

Ci

I(t)

I(t 0 )

exp(−rR(ti−t))

+F

I(t)

I(t 0 )

exp(−rR(T−t)) (9.28)

leading to:

π 1 (t)=

λσI−rR

(1−γ)σI^2

BIL(t,I(t))

ψ 1 (t)I(t)

=

λσI−rR

(1−γ)σI^2

(9.29)

for example, we have the same optimal portfolio process as in the basic prob-
lem (P). This is not surprising as the inflation-linked bond without deflation
protection is simply a linear product with regard to the inflation index which
therefore can be identified as a tradeable good.
If we now put the insights from the two special cases together then we
note that the higher absolute value of the optimal portfolio process for (OP1)
compared to (P) has its reason in the protection against deflation.As for small
values of the inflation index the total payment of the inflation linked bond
is typically dominated by the final payment, the price of the inflation linked
bond then behaves more like a nominal bond. To mimic the optimal stock
position, therefore more and more units of the inflation linked bond have to
be sold short.
The following numerical example illustrates the above discussion. In
Figure 9.1 the simulated path of the inflation index is presented for the
time intervalt=[0, 30], where the time unit represents one year. The infla-
tion index is assumed to follow the geometric Brownian motion of (9.10)
with the following parameters: nominal and real interest rates,rN=0.07
andrR=0.05 respectively, market price of riskλ=0.3, volatilityσI=0.20 on
the yearly basis andI(0)=100 (the seemingly high value ofλσIis chosen for
demonstrational purposes to obtain positive values for the optimal fractions
of inflation products later on).
In Figure 9.2 we present the optimal portfolio processes for three different
portfolio problems, each characterized by the structure of the inflation-
linked bond available as investment opportunity. These different structures
are an inflation-linked zero coupon bond with deflation protection (9.26), an
inflation-linked bond without deflation protection (9.28) and an inflation-
linked bond with deflation protection (9.5). The inflation-linked bonds are
assumed to have the following characteristics: face valueF=100, date of
maturityT=30, coupon paymentsCi=10 and coupon datesti=i, where