TARAS BELETSKI AND RALF KORN 181π 1 (t)∼λσI−rR
(1−γ)σ^2 I(
1 +F
ψ(t)I(t))
, t→T, I(t)<I(t 0 )This asymptotic behavior of the optimal portfolio processπ 1 (t) shows the
same feature as the inverse dependence between the optimal portfolio pro-
cess and inflation indexI(t). Having a higher probability for deflation, for
example,I(t)<I(t 0 ), the related fraction of the inflation-linked bond, for
example, the optimal portfolio processπ 1 (t), has a higher value compared
to the situation with a lower deflation probability, for example,I(t)>I(t 0 ).
Problem 2: Inflation-linked bond, stock and non-inflation linked bond.
In addition to the setting in Problem 1 the investor can now also invest
into a stock with price given by equation (9.9). For simplicity we now assume
that the final payment has no protection against deflation and that inflation
and the stock price evolution are independent. More precisely, we look at
the problem:
max
φ(.)∈B(x)E(U(Xφ(T))) (OP2)withXφ(t)=φ 0 (t)P 0 (t)+φ 1 (t)P 1 (t)+φ 2 (t)BIL(t) whereBIL(t) is given as in
Equation (9.28). To solve Problem (OP2) we first determine the replication
strategies of the stock (by itself) and the inflation-linked bond as:
ψ 1 (t)= 1ψ 2 (t)=∑
i:ti>tCi
I(t 0 )exp(−rR(ti−t))+Fexp(−rR(T−t))
I(t 0 )(9.30)Application of Theorem 1 then yields the optimal trading strategy in stock
and the inflation-linked bond as:
φ 1 (t)=ξ 1 (t)
ψ 1 (t)=b−rN(t)
(1−γ)σ 12·X(t)
P 1 (t)φ 2 (t)=ξ 2 (t)
ψ 2 (t)=λσI−rR(t)
(1−γ)σI^2·X(t)
ψ 1 (t)I(t)or in terms of the portfolio process as:
π 1 (t)=b−rN
(1−γ)σ^21, π 2 (t)=λσI−rR
(1−γ)σ^2 I(9.31)We thus have the same optimal portfolio process as in the basic problem (P).
This is not a surprise as our traded products are just linear functions of the
ones traded in (P). As the portfolio process only describes the fraction of the
total wealth invested in the different products we should therefore have this
coincidence which of course disappears for the inflation-linked bond and