Risk Analysis in Finance and Insurance, 2nd Edition

Advanced Analysis of Financial Risks: Discrete Time Models 79

Now we consider one of the simplest models of this type. We refer to it as a

(B

1

,B

2

,S)-market:

ΔB

i

n

= r

i

B

i

n−1

,B

i

0

=1,i=1, 2 ,

ΔS

n

= ρ

n

S

n−1

,S

0

≥ 0 ,

−1 <a<r

1

≤ r

2

<b,

where (ρ

n

) is a sequence of independent random variables (representing prof-

itability or return of asset S)thattakevaluesb and a with probabilities p and

1 − p, respectively.

Assets B

1

and B

2

can be interpreted as saving and credit accounts and S

represents shares. It is natural to assume that r

1

≤ r

2

in order to avoid the

obvious arbitrage opportunity in the market. If r

1

= r

2

,thenB

1

= B

2

and

we arrive to a (B,S)-market.

A strategy (portfolio) π =(π

n

)

n≤N

in a (B

1

,B

2

,S)-market is defined by

three predictable sequences (β

1

n

,β

2

n

,γ

n

)

n≤N

. The values ...