Advances in Risk Management

TARAS BELETSKI AND RALF KORN 173

Ci(before adjustment to the inflation) at timestiis given by:

BIL(t,I(t))=

∑n

i= 1

Ci

I(t)

I(t 0 )

exp

(

−

∫ti

t

rR(s)ds

)

+F

(

exp

(

−

∫T

t

rN(s)ds

)

+

CI(t,I(t))

I(t 0 )

)

(9.7)

wheret 0 ≤t<t 1 ≤ ··· ≤tn=TandCI(t,I(t)) is a fair price of the European

call option on consumer price indexI(T) at timetwith strike priceK=I(t 0 )

and date of maturityT.

Remark(i) Of course one could also link a deflation protection to the single
coupon payments (for example, to pay at least their nominal values). Then,
the price of the so constructed inflation linked bond is given as the sum of
n+1 call options on the inflation index. A special case of the above consid-
ered coupon bond is an inflation linked zero coupon bond with deflation
protection that is obtained by settingCi=0. Also, the necessary modifi-
cations for considering an inflation-linked coupon bond without deflation
protection of the final payment are quite obvious.
(ii) Another possible approach for inflation modeling is the direct mod-
eling of an (instantaneous) inflation rate similar to the short rate approach
of interest rate modeling. We refer to Korn and Kruse (2004) for details on
this method. A similar approach based in stead on the HJM framework is
presented in Jarrow and Yildirim (2003).

9.3 Optimal portfolios with inflation linked products

In this section we will look at various optimization problems including
aspects of inflation. The main problem of investment in the presence of
inflation is of course that inflation itself is not a traded good. However,
inflation-linked bonds or options on an inflation index are traded and can
serve as investment alternatives. As they are derivatives on the underly-
ing inflation index we can use portfolio optimization methods for portfolios
with derivatives as in Korn and Trautmann (1999) or in Kraft (2003). This
approach roughly consists of the following two-step procedure:

Step 1: Solve the optimal portfolio problem (P) as if all the underlyings

(and in particular the inflation index) were tradable (“the basic

portfolio problem”).

Step 2: Replicate the optimal positions of the non-tradable assets via

positions in suitable derivatives and the money market account.