Advances in Risk Management

TARAS BELETSKI AND RALF KORN 187

with a corresponding minimal quadratic hedging error equalling,

Var(B)E(I(T)^2 )+(E(B))^2 E((I(T)−ˆI(T))^2 ) (9.52)

The remaining moneyx−xˆcan be used for different purposes.

(ii) In the case ofx<xˆthe optimal final wealth for problem (9.38) is given

by:

Bˆ∗=ˆI(T)E(B)−E(E(B)Hˆ(T)ˆI(T)−x)

Hˆ(T)

E(Hˆ(T)^2 )

(9.53)

with a minimal quadratic hedging error of:

Var(B)E(I(T)^2 )+(E(B))^2 E((I(T)−ˆI(T))^2 )+

+

(E(B)E(Hˆ(T)ˆI(T))−x)^2

E(Hˆ(T)^2 )

(9.54)

Proof: The form of the optimal final wealth in both cases follows exactly
as in the situations of Propositions 3 and 4, but now in the reduced market
consisting only of the bond and stock. To show the form (9.52) of the hedging
error, note:

E[BI(T)−E(B)ˆI(T)]^2 =E[BI(T)−E(B)I(T)+E(B)(I(T)−Iˆ(T))]

2

=Var(B)E(I(T)^2 )+(E(B))^2 E((I(T)−ˆI(T))^2 )

+ 2 E(B−E(B))E(I(T)(I(T)−ˆI(T)))

=Var(B)E(I(T)^2 )+(E(B))^2 E((I(T)−ˆI(T))^2 )

With this, the form of the hedging error (9.54) follows as in Proposition 4.

RemarkNote that forσ 11 =0 the results of Proposition 3 and 4 coincide with
those of Proposition 5. The form of both the hedging errors and the optimal
final wealths indicate that they are the results of two succeeding projections.
First,B ̃is projected onto the market that allows for a perfect replication of
inflation (see Propositions 3 and 4) and then to the market that only allows
a partly hedging of inflation via trading stock and bond (see Proposition
5). This becomes most transparent when we are in case (i) of Proposition 5
where the two sums making up the hedging error exactly correspond to this
interpretation.