The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 420 Wednesday, February 4, 2004 1:08 PM

420 The Mathematics of Financial Modeling and Investment Management

Examples

To illustrate the above we now proceed to detail the calculations for the

previous example of three assets, three dates, and four states. Let’s first

write the equations for the risk-free asset:

1 pω

1 = ---------- ∑ -----------------π ()ωRts

πAkt ω ∈A

kt

PAkt)

s

(

,

1  p 1

1 = ----------------------------π 21 R 12

p 2

()R 12 



(), + ------------------π 22 ,

πA 11 p 1 + p 2 p 1 + p 2 

1  p 3

1 = ----------------------------π 23 R 12

p 4

()R 12 



(), + ------------------π 24 ,

πA

21

p 3 + p 4 p 3 + p 4 

1

1 = ----------[p 1 π 2 () 1 R 02 , + p 2 π 2 () 2 R 02 , + p 3 π 2 () 3 R 02 , + p 4 π 2 () 4 R 02 , ]

πA 10

πA

11

= π 1 () 1 = π 1     () 2

πA 21 = π 1 () 3 = π 1   () 4

πA 10 = π 0 () 1 = π 0   () 2 = π 0 () 3 = π 0 () 4

We can now rewrite the pricing relationships for the other risky

assets as follows:

At date 2, prices are zero: S 2 i = 0.

At date 1, the relationship

d 2

i

S 1

i

= E 1    -----------

R 12 ,

holds. In fact, we can write the following: