Advances in Risk Management

TARAS BELETSKI AND RALF KORN 185

which implies (with the notation of Korn (1997a)):

̃I(y)=(U′)−^1 (y)=cI(T)−½y

leading to

X(y)=E(H(T) ̃I(yH(T)))=E(cH(T)I(T))−½yE(H(T)^2 )

Y(x)=(X)−^1 (x)=2(E(cH(T)I(T))−x)/E(H(T)^2 )

As in Korn (1997a) we then obtain:

Proposition 4 In the case ofx<x ̃the optimal final wealth for problem

(9.37) is given by:

B∗=I(T)E(B)−(E(B)I(0)−x)

H(T)

E(H(T)^2 )

(9.44)

with a minimal quadratic hedging error of:

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

(E(B)I(0)−x)^2

E(H(T)^2 )

(9.45)

Proof: The form ofB∗follows from the main result of Korn (1997a) as we
there have:

B∗= ̃I(Y(x)H(T))=E(B)I(T)−E(E(B)H(T)I(T)−x)

H(T)

E(H(T)^2 )

UsingE(H(T)I(T))=I(0) andB∗instead ofBin (9.38), simplifying the result-
ing expression leads directly to the minimal hedging error as given in
(9.45).

Remark(i) Note that the hedging error above consists of two different com-
ponents. First, there is the unavoidable error termVar(B)E(I(T)^2 ) which only
vanishes if the height of the premium (more precisely, the part not depend-
ing on price changes due to inflation) is exactly foreseeable. The remaining
part

(E(B)I(0))−x)^2

E(H(T)^2 )

is non-negative, but can vanish depending on the amount of moneyxavail-
able for hedging activities (for example, forx=E(B)I(0)) which also shows
that Propositions 3 and 4 are consistent for exactly this choice ofx).