Advances in Risk Management

TARAS BELETSKI AND RALF KORN 183

only exception is if he is pretty sure that there will be a tendency for a
huge increase of the inflation index expressed via a high risk premium. On
the other side, there are companies that have to hedge inflation to do their
original business. Typical such candidates for buying the above inflation-
linked products are (usually non-life) insurance companies who are facing
a risk process due to their insurance business that is closely linked to the
inflationasthevalueoftheinsuredgoodsareobviouslyrelatedtoit. Tostudy
one of their relevant problems we consider a financial market consisting of a
riskless bond, a stock and some inflation-linked product where we assume
the following price processes (for simplicity we assume the inflation index
to be tradable which – as seen – is equivalent to the assumption that an
inflation-linked bond without deflation protection is traded):

dP 0 (t)=P 0 (t)rNdt, P 0 (0)= 1 (9.34)

dP 1 (t)=P 1 (t)(μdt+σ 11 dW 1 (t)), P 1 (0)=p 1 (9.35)

dI(t)=I(t)((rN−rR+ν)dt+σ 21 dW 1 (t)+σ 22 dW 2 (t)) I(0)=i (9.36)

We assume that an insurer wants to hedge a payment that is inflation related
(for example, payments arising from a car insurance) and has the form,

B ̃=B·I(T)

whereBis independent of the Brownian motionWand represents the value
of the insurance premium that has to be paid out if the insurance case would
happen today. Further,Bcan be interpreted as the total outcome of the
risk process of the insurance company until the time horizon. The hedging
problem we are considering is

minπ E(B ̃−Xπ(T))^2 (9.37)

where the utility criterion can be simplified to

E(B ̃^2 )− 2 E(B)E(I(T)Xπ(T))+E(Xπ(T)^2 ) (9.38)

for example, it is sufficient to consider the utility maximization problem
with the following (random) utility function:

U(x)=U(x;I(T))=−x^2 + 2 cx·I(T) (9.39)

with a suitable constantc. Note that this utility function – although strictly
concave – is not strictly increasing, but has an invertible derivative. As
in Korn (1997) one can show that the usual procedure of the martingale