Advances in Risk Management

TARAS BELETSKI AND RALF KORN 175

with

Xφ(t)=φ 0 (t)P 0 (t)+φ 1 (t)f(1)(t,P 1 (t),I(t))

+φ 2 (t)f(2)(t,P 1 (t),I(t)) (9.13)

(and B(x) the set of all admissible trading strategies in the bond and the

two derivatives for an initial wealth of x) admits the following solution:

(a) The optimal final wealth B* coincides with the optimal final wealth

in the corresponding basic portfolio problem (P) where the investor is

assumed to be able to trade the stock and the inflation index.

(b) Letξ(t)=(ξ 0 (t),ξ 1 (t),ξ 2 (t)) denote the optimal trading strategy of the
corresponding basic portfolio problem (P). Then the optimal trading
strategy for the option portfolio problemφ(t)=(φ 0 (t),φ 1 (t),φ 2 (t)) is
given by:

φ ̄(t)=((t)′)−^1 ·ξ ̄(t),

φ 0 (t)=

(

X(t)−

∑d

i= 1

φi(t)f(i)(t,P 1 (t),...,Pd(t))

)

P 0 (t)

(9.14)

withφ ̄(t)=(φ 1 (t),φ 2 (t))′and ̄ξ(t)=(ξ 1 (t),ξ 2 (t)).

Equipped with this result we are now able to solve various particular port-

folio problems related to inflation explicitly. But first we recall examples of

explicit solutions of the basic portfolio problem (P):

Step 1: Solving the basic portfolio problem (P). In this step we will treat the

inflation index as a tradable good and solve the portfolio problem (P) for

the choices of

U(x)∈{ln (x), 1/γxγ} withγ< 1 (9.15)

With the notation of

σ=

(

σ 1 σ 2

0 σI

)

(9.16)

the solutions of the basic portfolio problem (P) for these choices of the utility

functions are well-known and are given by:

π∗(t)=

1

1 −γ

(σσ′)−^1

(

b−rN

λσI−rR

)

(9.17)