Advances in Risk Management

184 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

approach to portfolio optimization (see Korn (1997b)) is still valid but cannot
ensure a non-negative final wealth process. Let us first consider the case of:

9.4.1 Investment in bond, stock and inflation

In this setting we have a complete market. If the investor can afford it then
he will use a trading strategy that will minimize the utility functionU(x)of
(9.39) pathwise, for example, his final wealth should satisfy

Xπ(T)=cI(T) (9.40)

which – by completeness of the market – the investor can exactly attain with
an initial wealth of

x ̃=cE(H(T)I(T))=I(0)E(B) (9.41)

whereH(T) is given by:

H(T)=exp(−(rN+½‖θ‖^2 )T−θ′W(T)),

θ:=σ−^1

((

μ

rN−rR+ν

)

−rN 1

)

(9.42)

In the case ofx≥x ̃it is not necessary for the investor to use more money than
x ̃for hedging activities as this would lead to a deviation from the optimal
final wealth of (4.7). We therefore have proved:

Proposition 3 Letx ̃be defined as in (9.41) and assumex≥x ̃. Then,

the optimal final wealth for the hedging problem (9.37) is given by rela-

tion (9.40), the optimal hedging strategy is to buy c units of the inflation

index (respectively a suitable inflation product delivering the same final

payment) and to hold it until maturity T. The corresponding minimal

quadratic hedging error equals:

Var(B)E(I(T)^2 ) (9.43)

The remaining moneyx−x ̃can be used for different purposes.

Proof: In light of the arguments preceding Proposition 3 only (9.43) has to be
shown, but this follows directly from the final wealth of the form of relation
(9.40) into the utility function (9.38).
In the case ofx<x ̃we are in the situation of Korn (1997a) and can use the
martingale approach. Note first that we have:

U′(x)= 2 c·I(T)− 2 x