Risk Analysis in Finance and Insurance, 2nd Edition

106 Risk Analysis in Finance and Insurance

Now we introduce the standard basic notions related to a (B, S)-market.

If processes β =(β

t

)

t≤T

and γ =(γ

t

)

t≤T

are adapted to filtration F,then

π =(π

t

)

t≤T

:= (β

t

,γ

t

)

t≤T

is called a portfolio or strategy on a (B, S)-market.

The capital (or value)ofstrategyπ is given by

X

π

t

= β

t

B

t

+ γ

t

S

t

.

A contingent claim f

T

with the repayment date T is defined to be a F

T

-

measurable non-negative random variable. We say that a strategy π is self-

financing if

dX

π

t

= β

t

dB

t

+ γ

t

dS

t

.

A self-financing strategy π is called a perfect hedge for a contingent claim

f

T

if

X

π

T

≥ f

T

(a.s.).

We say that a strategy π replicates f

T

if

X

π

T

= f

T

(a.s.).

Ahedgeπ

∗

is