Advances in Risk Management

182 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

the inflation index if we would compare the actual amount of units of both
products we optimally have to hold in both portfolio problems.

Remarks

(a) Looking at the explicit form of the optimal fraction of wealth in the
inflation-linked product as given in relation (9.31), we realize that unless
we have a very strong opinion for a very high inflation rate (for example,
the excess return of the inflation should be higher than the real interest
rate) it would be optimal to sell inflation-linked bonds short.

(b) In the more general case of an inflation-linked bond with deflation pro-
tection and a non-vanishing correlation between the inflation index and
the stock price, explicit calculation shows that we obtain the following
optimal portfolio processes:

[

φ 1 (t)

φ 2 (t)

]

=(ψ(t)′)−^1 ̄ξ(t)

=

1

1 −γ

X(t)

     

b−rN

σ 12 P 1 (t)

−

λσI−rR

σI^2 P 1 (t)

·

σ 2 σI

σ^21

λσI−rR

σI^2 Iψ 22 (t)

·

(

1 +

σ^22

σ^21

)

−

b−rN

σ 12 I(t)ψ 22 (t)

·

σ 2 σI

σI^2

     

(9.32)

withX(t) denoting the optimal wealth process of both the basic portfolio
problem (P) and the just considered optimal option portfolio problem and
where the replication strategies are given by:

ψ(t)=





10

0

∑n

i:ti>t

Ci

I(t 0 )

exp(−rR(ti−t))+

Fexp(−rR(T−t))N(d(t))

I(t 0 )



 (9.33)

Hence, the moral of both remarks is that the type of risk-averse investors
we are considering are typically selling inflation-linked bond products
short. However, there should be market participants which are interested
in purchasing such inflation-linked products as otherwise there is no use
in offering them at all. Therefore, in the next section we are presenting a
situation where those products are needed.

9.4 HEDGING WITH INFLATION-LINKED PRODUCTS

As we have seen in the last section, a risk-averse investor is usually not
attracted by inflation linked products of the type we considered here. The