The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 409 Wednesday, February 4, 2004 1:08 PM

409

i

2

Arbitrage Pricing: Finite-State Models

SA = S 1 i () 1 = Si 1 () 2

11 ,

1

= ------------[PA 12 A 11 ()d 2

i

)π 21 () 1 + PA( 22 , A 11 )π 2 () 2 d 2

i

( , , , ()]

πA

11 ,

1

= ------------ ------------------π 21 1

p 2

()d 2

i

()

p 1

()d 2

i

()+ ------------------π 22 2

πA 11 , p 1 + p 2 p 1 + p 2

i

3

i

= S 1 4

i

SA 21 , ()= S 1 ()

1

= ------------[PA( 32 , A 21 , )π 2 () 3 di 2 () 3 + PA( 42 , A 21 , )π 2 () 4 d 2 i () 4 ]

πA 21 ,

= ------------^1 ------------------π

23 3

p 4

()d 2 i ()

p 3

()d 2 i ()+ ------------------π 24 4

πA 21 , p 3 + p 4 p 3 + p 4

i i + π

21 1

i + π

SA 10 = 22 2







p 1 [πA

11

dA

11

()d 2 i ()]+ p 2 [πA

11

dA

11

()d 2 i ()]

, , , , ,

i i 

+ π 23 3 A 12 dA 12 ()d 2

i

+ p 3 [πA 12 dA 12 ()d 2 ()]

i

, , ()]+ p^4 [π , , + π^244



These equations illustrate how to compute the state-price deflator

knowing prices, payoffs, and probabilities. They form a homogeneous sys-

tem of linear equations in π 2 (1), π 2 (2), π 2 (3), π 2 (4), πA 11 , , πA 21 , , πA 10 ,.

p 1 di 2 ()π 21 2 () SAi

11

(p 1 + p 2 )π

,

1 ()+ p 2 d 2 i ()π 22 – = 0

, A 11

p 3 d 2 i ()π 3 2 () 3 + p 4 d 4 i ()π 4 2 () 4 – Si A 21 , (p 3 + p 4 )πA 21 , = 0

p 1 d 2 i ()π 1 2 () 1 + p 2 di 2 ()π 2 2 () 2 + p 3 di 2 ()π 3 2 () 3 + p 4 di 4 ()π 4 2 () 4

+ (p i

1 + p 2 )

i

dA 11 ,πA 11 , + (p 3 + p 4 )dA 23 πA 23 – SA π = 0

i

, , 10 , A 10 ,

Substituting, we obtain

p 1 d 2 i ()π 1 2 () 1 + p 2 d 2 i ()π 2 2 () 2 – Si A 11 , (p 1 + p 2 )πA 11 , = 0