The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 410 Wednesday, February 4, 2004 1:08 PM

410 The Mathematics of Financial Modeling and Investment Management

p 3 di 2 ()π 23 4 () SAi

21

(p 3 + p 4 )π

,

3 ()+ p 4 d 4 i ()π 24 – = 0

, A 21

[(p 1 + p 2 )SAi 11 , + (p 1 + p 2 )di A 11 , ]πA 11 ,

+ [(p 3 + p 4 ) i + (p 3 + p 4 ) i ]πA

21

• i
  ,
  π = 0
  , , ,
  SA
  21
  dA
  21 ,
  SA
  10 A 10

This homogeneous system must admit a strictly positive solution to

yield a state-price deflator. There are seven unknowns. However, as the

system is homogeneous, if nontrivial solutions exist, one of the

unknowns can be arbitrarily fixed, for example πA 10 ,. Therefore, six

independent equations are needed. Each asset provides two conditions,

so a minimum of three assets are needed.

To illustrate the point, we assume that all states (which are also

events in this discrete example) have the same probability 0.25. Thus

the events of the information structure have the following probabilities:

the single event at time zero has probability 1, the two events at time 1

have probability 0.5, and the four events at time 2 coincide with indi-

vidual states and have probability 0.25. Conditional probabilities are

shown in Exhibit 14.2.

For illustration purposes, let’s write the following matrices for pay-

offs for each security at each date in each state:

015 50 08 30 05 38

d 0 5 112

1

i ω d

2

{ () } ≡^015100 ; { i () ω} ≡ 0 8 120 ; { d() } ≡

3

i ω

020 70 015 40 08 42

020110 015140 0 8 130

We will assume that the state-price deflator is the following given pro-

cess:

1 0.8 0.7

{ πt () ω} ≡ 1 0.8 0.75

1 0.9 0.75

1 0.9 0.8

Each price is computed according to the previous equations. For exam-

ple, calculations related to asset 1 are as follows:

S^11121314

2 ()= S 2 ()= S 2 ()= S 2 ()=^0