186 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS(ii) The form of the hedging error obtained in Proposition 3 is quite natural
as due to the independence ofBand the capital market,E(B) is the best
forecast for B made up out of the actions at the capital market, andE(I(T)^2 )
represents the minimal possible uncertainty due to inflation.
To demonstrate the effect that the use of the inflation index has on the
hedging error we also have to solve problem (9.37) if we are not allowed to
trade the inflation index (or any other inflation-linked product – besides the
stock, of course). We will reduce this problem to solving again a (modified)
quadratic problem but now in the market that consists only of the bond and
the stock.
9.4.2 Investment in bond and stock
If we can only invest in bond and stock then we cannot hedge the
randomness that is caused byW 2 (.). We therefore introduce:
I(t)=Iˆ(t)exp(−½σ^222 t+σ 22 W 2 (t)) (9.46)and thereby directly obtain that the utility criterion in this reduced market
equals:
E(B ̃^2 )− 2 E(B)E(ˆI(T)Xπ(T))+E(Xπ(T)^2 ) (9.47)for example, we can now solve a conventional utility maximization problem
in the complete market made up of the stock and the bond with:
U(x)=U(x;ˆI(T))=−x^2 + 2 cx·ˆI(T) (9.48)Hˆ(T)=exp(−(rN+½θˆ^2 )T−θˆW 1 (T)),θˆ=(μ−rN)/σ 11 (9.49)xˆ=cE(Hˆ(T)ˆI(T))=I( 0 )exp((
−rR+ν−(μ−rN)σ 21
σ 11)
T)
E(B) (9.50)Exactly the same arguments as in the case (i) lead to:Proposition 5 Letxˆbe defined as in (9.50).(i) Forx≥xˆ, the optimal final wealth for the hedging problem (4.4) if
only investment in bond and stock is allowed is given by:E(B)ˆI(T) (9.51)