Advances in Risk Management

174 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

Of, course to use it we first have to set up the relevant optimization problem.
For this, we assume that an investor is allowed to trade into a riskless bond
with price given by:

dP 0 (t)=P 0 (t)rNdt, P 0 (0)= 1 (9.8)

In addition, he can invest intonfurther risky securities which might be
stocks, inflation linked bonds, or more general derivatives on inflation
and/or stocks. For a given initial wealth ofxand an admissible portfo-
lio processπ(.)∈A(x) (to be specified later) including the above investment
possibilities letXπ(t) denote the corresponding wealth process. Then, the
investor’s task will be to maximize the expected utility from final wealth,
for example, he tries to solve the portfolio problem (P):

sup

π(.)∈A(x)

E(U(Xπ(T))) (P)

whereU(x) is a utility function (for example, a strictly concave, monoton-
ically increasing and differentiable function). To simplify matters we will
assume that besides the riskless bond above the investor can invest into a
risky stock with price dynamics given by:

dP 1 (t)=P 1 (t)(bdt+σIdW 1 (t)+σ 2 dW 2 (t)), P 1 (0)=p 1 (9.9)

and in some inflation linked product where the inflation dynamics are
given by:

dI(t)=I(t)((rN−rR+λσI)dt+σIdW 2 (t)), I(0)=i (9.10)

with (W 1 ,W 2 ) denoting a two-dimensional Brownian motion. If in such a
situation we assume that it is possible to trade in two derivatives on the
stock and on the inflation with price processes given by:

f(i)(t,P 1 (t),I(t)), f(i)∈C1, 2([0,T)×(0,∞)^2 ), i=1, 2 (9.11)

then the relevant result from Korn and Trautmann (1999) can be formulated
as (where we will omit a proof here as it would be totally similar to the one
given in the Korn and Trautmann (1999) for the case of optimal investment
in stock derivatives).

Theorem 1 Under the assumption that thedelta-matrix(t)=(ij(t)),

i,j=1, 2 with

=

(

f

(1)

p 1 (t,P 1 (t),I(t)) f

(1)

I (t,P^1 (t),I(t))

fp(2) 1 (t,P 1 (t),I(t)) fI(2)(t,P 1 (t),I(t))

)

(9.12)

(with the subscripts denoting the corresponding partial derivatives) is

regular for all t∈[0, T) then the option portfolio problem (P 1 )

max

φ(.)∈B(x)

E(U(Xφ(T))) (OP)