TARAS BELETSKI AND RALF KORN 171and the money market account will be set up and solved in section 9.3.
Finally, in section 9.4 we show how inflation-linked products can be used to
hedge inflation dependent claims.
9.2 Modeling the evolution of an inflation index
The harmonized consumer price index (HCPI) is designed to measure infla-
tion in the countries of the European monetary union. It is an average over
11 country-based inflation indices. As it is therefore an official index it is a
natural candidate for options to be written on its future value or for linking
bond payments to it. However, to value these contracts we need a model for
the evolution of an inflation index over time.
To develop such a model we will base our considerations on macro-
economic foundations as there should of course be relations between
(different kinds of) interest rates and inflation. The most prominent rule
in this area is the so-called Fisher equation (Fisher, 1930). It states that the
nominal interest rate is the sum of the real interest rate and the expected
inflation:
rN(t)=rR(t)+E[i(t)] (9.1)whererN(t) is the nominal interest rate for the bond maturing at timet,E[i(t)]
is the expected (simple) inflation rate for the time horizontandrR(t)isthe
real interest rate for the bond with maturityt, which corresponds to the
growth of real purchasing power in the case of investment with the nominal
interest raterN(t). In the special case of constant real interest rates, thus
the nominal interest rates follow the movements of expected inflation rate
(a fact empirically supported byAng and Bekaert (2003) and Nielsen (2003)).
By interpreting the relative instantaneous change:
dI(t)
I(t)of the inflation indexI(t) as the (instantaneous) rate of inflationi(t) the Fisher
equation suggests the following model of a generalized geometric Brownian
motion for the evolution of the inflation index:
dI(t)=I(t)((rN(t)−rR(t))dt+σIdWI(t)), I(0)=i (9.2)where nowrN(t),rR(t) are interpreted as the relevant instantaneous rates.
Even more, we assume that this equation holds in equilibrium, for example,
equation (9.1) is valid with respect to the risk-neutral pricing measure (the
modeling of the evolution of the inflation index under a subjective mea-
sure can be done by including an additional drift rate such as for example,
λσI∈IR; see also sections 9.3 and 9.4). According to the specification of the