The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 407 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 407

partition at time zero is formed by the event {1 + 2 + 3 + 4}. At time 1,

the set of states is partitioned into two mutually exclusive events, {1 + 2}

or {3 + 4}. At time 2 the partition is formed by all individual states.

Note that this is a particular example; different partitions would be log-

ically admissible.

Exhibit 14.1 represents the above structure. Each security is character-

ized by a price process and a payoff process adapted to the information

structure. Each process is a collection of three discrete random variables

indexed with the time indexes 0,1,2. Each discrete random variable is a 4-

vector as it assumes as many values as states. However, as processes are

adapted, they must assume the same value on each partition of the infor-

mation structure. Note also that payoffs are zero at date zero and prices

are zero at date 2. Therefore, in this example, we can put together these

vectors in two 3×4 matrices for each security as follows

Si

0 ()S 1

i () 0 0 d

1

i ()d

1 1 1 2 ()

i 1

i 2 i

Sti ω

2 2

dti ω

0 d 1 i ()d 2 () 2

{ () } ≡

S 0 ()S 1 i () 0

i ; { () } ≡

S 0 ()S 1

i

() 0 0 d 1

i

3 3 () 3 d 2 ()

i

3

i

S 0 ()S 1

i

() 0 0 d 1

i

4 4 () 4 d 2 ()

i

4

The following relationships hold:

Si i i i

0 ()= S 0

i ()= S

0 ()= S 0 ()= ; S 1 ()= S 1

1 2 i 3 i 4 S i ()= ;

A 10 ,^12 SA 11 ,

EXHIBIT 14.1 An Information Structure with Four States and Three Dates