The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 408 Wednesday, February 4, 2004 1:08 PM

408 The Mathematics of Financial Modeling and Investment Management

3 i 4 Si

S 1 A 21

i ()= S

1 ()= ,

d i i i

1

i ()= d

1 ()= ; d 1

i ()= d

1 1 ()=

i 2 d

A 11 ,^34 dA 21 ,

where, as above, St

i

()ω is the price of security i in state ωat moment t

and dt

i

()ω is the payoff of security i in state ωat time t with the restric-

tion that processes must assume the same value on partitions. This is

because processes are adapted to the information structure so that there

is no anticipation of information. One must not be able to discriminate

at time 0 events that will be revealed at time 1 and so on.

Observe that there is no payoff at time 0 and no price at time 2 and

that the payoffs at time 2 have to be intended as the final liquidation of

the security as in the one-period case. Payoffs at time 1, on the other

hand, are intermediate payments. Note that the number of states is cho-

sen arbitrarily for illustration purposes. Each state of the world repre-

sents a path of prices and payoffs for the set of three securities. To keep

the example simple, we assume that of all the possible paths of prices

and payoffs only four are possible.

The state-price deflator can be represented as follows:

π 0 ()π 1 1 ()π 1 2 () 1

()π 1 ()π 2 2 () 2

{ () ω} ≡

π 02

()π 13 ()

πt

π 03 ()π 23

π 0 ()π 4 1 ()π 4 2 () 4

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

π 1 () 1 = π 1 () 2 π 1 () 3 = π 1 () 4

A probability pωis assigned to each of the four states of the world.

The probability of each event is simply the sum of the probabilities of its

states. We can write down the formula for security prices in this way:

i 1 Si i

A 22 3 SA

i

= S 2 42 4

i ()= = S

2

i ()= = S

2

i ()= = S

2

S i ()= 0

A 12 , ,^2 SA 32 , ,