The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 421 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 421

Si A = Si 1 () 1 = Si 1 () 2

11 ,

1

= --------------[PA( , , ()d

i

A 11 )π 21 2 () 1 + PA( 22 , A 11 ()d

i

12 , )π 22 2 ()^2 ]

π 1 () 2

1  p 1 d

i 1

2 d

i 2

= --------------------------π^2 () p^2 ()

2 ()^1 R 12 , --------------+ ------------------π 2 ()^2 R 12 , --------------

π 11 p 1 + p 2 R 12 , p 1 + p 2 R 12 , 

d

i

2 ()^1 d 2 ()

i

2

= QA( 12 , A 11 , )--------------+ QA( 22 , A 11 , )--------------

R 12 , R 12 ,

q d 2

i

() q d 2

i

1 1 2 ()^2

= ------------------ --------------+ ------------------ --------------

q 1 + q 2 R 12 , q 1 + q 

2 R 12 ,

Si A 21 , = S 1 i () 3 = Si 1 () 4

di 2 () 3 d 2 i () 4

= QA( 32 , A 11 , )--------------+ QA( 42 , A 11 , )--------------

R 12 , R 12 ,

q 3 d 2 i() 3 q 4 di 2 () 4

= ------------------ --------------+ ------------------ --------------

q 3 + q 4 R 12 , q 3 + q 4 R 12 ,

At date 0, the relationship

d 1 i d 2 i

S 0 i = E 0 -----------+ -----------

R 01 , R 02 ,

holds. In fact we can write the following:

S

i i

1 2

i

3

i

= S 0 ()= S 0

i

A 10 , ()= S 0 ()= S 0 ()^4



p 1 [π 11 1 + ()d 2

i

()] 



()d

i

1 () π 21 1



()d

i

2 + ()d

i

2



= ----------^1   + p^2 [π^121 () π^222 ()]^

π  + p 3 [π 1 ()^3 d 1 () π 23 ()]

i 3 + ()di 3

A (^10) ^2 
 + p 4 [π 1 () 4 d 1 i () π 4 + 2 () 4 di 2 () 4 ]


