The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 406 Wednesday, February 4, 2004 1:08 PM

406 The Mathematics of Financial Modeling and Investment Management

1   T

(θθθθtSt)A

kt

=---------- ∑ P({}ω Akt) ∑ πj()d

θ

ω



π

ω j ()

A   j =t + 1 

kt ω ∈Akt

 

 

1  pω  T

=---------- ∑ ---------------------------- ∑ πj()dθω



πAkt ω ∈A j =t + 1 

kt







 ∑ pω

ω j ()

 

ω ∈A

kt



It is possible to demonstrate that the payoff-price pair (di

t,St

i )admits

no arbitrage if and only if there is a state-price deflator. These concepts

and formulas generalize those of a one-period setting to a multiperiod

setting.

Given a payoff-price pair (d i

t

i ,S

t) it is possible to compute the state-

price deflator, if it exists, from the previous equations. In fact, it is possi-

ble to write a set of linear equations in the πt, πt – 1 for each period. One

can proceed backward from the period T to period 1 writing a homoge-

neous system of linear equations. As the system is homogeneous, one of

the variables can be arbitrarily fixed; for example, the initial value π 0 can

be assumed equal to 1. If the system admits nontrivial solutions and if all

solutions are strictly positive, then there are state-price deflators.

Examples

To illustrate the above, let’s write down explicitly the previous formulas

for prices, extending the example of the previous section to a two-

period setting. We assume there are three securities and two periods,

that is, three dates (0,1,2) and four states, indicated with the integers

1,2,3,4, so that Ω= {1,2,3,4}. Assume that the information structure is

given by the following partitions of events:

Ii ≡ (I 0 ≡ { A 10 , },I 1 ≡ {A 11 , , A 21 , },I 2 ≡ {A 12 , , A 22 , , A 32 , , A 42 , })

A 10 , ={ 1234 +++ },A 11 , ={ 12 + }, A 21 , ={ 34 + }

A 12 , ={} 1 , A 22 , ={} 2 ,A 32 , ={} 3 ,A 42 , ={} 4

where we use + to indicate logical union, so that, for example, {1 + 2} is

the event formed by states 1 and 2. The interpretation of the above

notation is the following. At time zero the world can be in any possible

state, that is, the securities can take any possible path. Therefore the