Financial theory and models

The multi-period model

Tribes and the filtering of information

Conditional expectations and Martingales
Securities, dividends, and trading gains
State price deflator
Various characterizations of no arbitrage

Frank Hansen
Department of Economics
Copenhagen University
2022
Fundamentals of the multi-period model

We shall formulate the framework for the resolution of uncertainty over

time in a general model suggested by Arrow, Debreu and others.

It is referred to as the standard model, and it constitutes the foundation

of modern state pricing theory. There are T periods in the model and
T + 1 dates labelled t = 0, 1, · · · , T .

Any system of subsets of Ω is naturally equipped with a partial order

structure. If A ⊆ B we simply write A ≤ B, and say that A is dominated
by B.

If the true state of nature is in A and A ≤ B, then the true state of

nature is also in B. If therefore A is true and A ≤ B, then B is also true.

The events known at time t are represented by a so-called tribe (or

algebra) Ft of subsets of Ω, to be defined next.

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A tribe of events
A system F of subsets of Ω is called a tribe (or algebra) if it satisfies
the following conditions:
(i) The empty set ∅ and the whole state space Ω are in F.
(ii) The intersection A ∩ B ∈ F for A, B ∈ F.
(iii) The complement Ω\A ∈ F for A ∈ F.
Since the union A ∪ B can be written (verify this) as

A ∪ B = Ω\[(Ω\A) ∩ (Ω\B)],

if follows from (2) and (3) that also A ∪ B ∈ F for any events A, B in a
tribe F.
We notice that A ∩ B is the largest set contained in both A and B. It is
denoted A ∧ B and is called the minorant event of A and B.
Likewise, A ∪ B is the smallest set that contains both A and B. It is
denoted A ∨ B and is called the majorant event of A and B.
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The filtering of information

Since the events known at time t are represented by a tribe of subsets

of Ω, it follows that the minorant and majorant events of known events
are also known.
The convention is that Ft ⊆ Fs for t ≤ s meaning that information,
once revealed, is never forgotten.
Furthermore, F0 is chosen as the diffuse tribe {∅, Ω} meaning that
there is no information about the future state of the world at time 0,
while FT is chosen as the discrete tribe {A | A ⊆ Ω} meaning that the
true state of nature is fully revealed at time T .
The filtration F = {F0 , . . . , FT } represents how information is revealed
through time.
An adapted process is a sequence X = (X0 , X1 , . . . , XT ) of stochastic
variables on Ω such that Xt is Ft -measurable for t = 0, 1, . . . , T . This
means, for now, that the outcome of Xt is determined at time t.

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Conditional expectations
Let t = 0, 1, . . . , T .
There is a map Et that to any FT -measurable function x : Ω → R
assigns the unique Ft -measurable function Et [x] : Ω → R such that

E[xy ] = E[Et [x]y ]

for every Ft -measurable function y : Ω → R.

The conditional expectations are linear and surjective maps satisfying

Et [1] = 1 (unital)
Et [xy ] = Et [x]y for Ft -measurable y
Et [x] ≥ 0 for x ≥ 0 (positive)
Et [x] > 0 for x > 0 (faithful)
Et2 [x] = Et [Et [x]] = Et [x] (idempotent)

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Martingales
The second property expresses that Et is an orthogonal projection of
the vector space of FT -measurable functions onto the subspace of
Ft -measurable function with respect to the inner product

(x | y ) = E[xy ].

For t ≤ s we finally obtain

Es [Et [x]] = Et [Es [x]] = Et [x]

for every FT -measurable function x.

An adapted process X = (X0 , X1 , . . . , XT ) is said to be a martingale if

Xt = Et [Xs ] for t ≤ s.

If the weather is a Martingale, then the weather today is the same as

the expectation of the weather tomorrow.
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Securities

A security is a pair (δ, S) of adapted processes such that

δ = (δ0 , δ1 , . . . , δT ) is a dividend process specifying the dividend
paid per unit of the security at each time.
S = (S0 , S1 , . . . , ST ) is a price process specifying the price per
unit of the security at each time.
Note that S indicates the price of the security, ex dividend.

Suppose now there are n securities with dividend processes

δ = (δ 1 , . . . , δ n ) and price processes S = (S 1 , . . . , S n ).

A trading strategy θ is an n-tuple θ = (θ1 , . . . , θn ) of adapted processes

such that
θt = (θt1 , . . . , θtn )
represents the portfolio of securities held after trading at time
t = 0, 1, . . . , T .

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Trading gains
We consider now a trading strategy θ = (θ1 , . . . , θn ).

The dividend process δ θ = (δ0θ , δ1θ , . . . , δTθ ) generated by θ is defined by

(
− θ0 · S0 t =0
δtθ =
θt−1 · (St + δt ) − θt · St t = 1, . . . , T



 − (θ01 S01 + · · · + θ0n S0n ) t =0

n
= X
i


 θt−1 (Sti + δti ) − θti Sti t = 1, . . . , T .
i=1

δtθ calculates the trading gains, cum dividend, obtained by holding the
corresponding portfolio of securities from t − 1 to t.

Note that the dividend process is adapted.

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Arbitrage

Let L denote the vector space of n-dimensional adapted processes,

and let Θ denote the vector space of trading strategies. The market
space
M = {δ θ | θ ∈ Θ}
is a linear subspace of L. Indeed,

δ θ1 + δ θ2 = δ θ1 +θ2 and λδ θ = δ λθ

for θ, θ1 , θ2 ∈ Θ and λ ∈ R.
A dividend process is non-vanishing if δt 6= 0 for at least one t.

Definition
A trading strategy θ ∈ Θ is called an arbitrage, if the dividend process
δ θ is non-vanishing and δtθ ≥ 0 for t = 0, 1, . . . , T .

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First characterization of no arbitrage

A linear map F : L → R is called a linear functional. It is said to be

strictly positive if F (x) > 0 for x ∈ L with x > 0.

Proposition
The dividend-price pair (δ, S) admits no arbitrage, if and only if there
exists a strictly positive linear map F : L → R such that F (δ θ ) = 0 for
every trading strategy θ ∈ Θ.

Proof: Let L+ denote the convex cone of non-negative processes in L.

Suppose that there is no arbitrage, then M ∩ L+ = {0}.
Note that M is a subspace of L and that L+ is convex. By a theorem
from Linear Algebra there is a non-vanishing linear functional F on L
that vanishes on M. Possibly by changing sign we may assume that F
is positive on L+ \{0} and hence strictly positive.
The converse is obvious. 

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State price deflator

Consider a dividend-price pair (δ, S).

A one-dimensional strictly positive adapted process π is said to be a

state price deflator if ST = (0, . . . , 0) and
 
T
1 X
Sti = Et  δji πj 
πt
j=t+1

for i = 1, . . . , n and t = 0, 1, . . . , T − 1.

Note that security prices vanish at time T when the model terminates.

The formula expresses that the deflated price πt Sti of the ith security at
time t equals the expected value at time t of the sum of future deflated
trading gains.

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Characterization in terms of trading strategies

A one-dimensional strictly positive adapted process π is a state-price

deflator if and only if ST = (0, . . . , 0) and
 
T
1 X
θt · S t = Et  δjθ πj  t = 0, 1, . . . , T − 1
πt
j=t+1

for every trading strategy θ.

The formula expresses that the deflated price πt θt · St of the portfolio at

time t equals the expected value at time t of the sum of future deflated
trading gains.

Proof: Let π be a state-price deflator and θ a trading strategy. We

calculate the expected value at time t of future deflated trading gains:

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The expectation of future deflated dividends
   
T
X T
X
Et  δjθ πj  = Et  θj−1 · δj πj + (θj−1 − θj ) · Sj πj 
j=t+1 j=t+1
 
T
X T
h X i
= Et  θj−1 · δj πj + (θj−1 − θj ) · Ej δl πl 
j=t+1 l=j+1
 
T
X T
X l−1
X
= Et  θj−1 · δj πj + (θj−1 − θj ) · δl πl 
j=t+1 l=t+2 j=t+1
 
T
X T
X
= Et  θj−1 · δj πj + (θt − θl−1 ) · δl πl 
j=t+1 l=t+2
T
" #
X
= Et θt · δt+1 πt+1 + θt · δl πl = θt · πt St .
l=t+2
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We inserted the definition of the dividend process.
We used that π is a state price deflator.
We used Et Ej = Et for j > t and changed the order of summations.
We canceled terms in the last sum.
We continued to cancel terms across the sums.
In the last equality we used the definition of a state price deflator.
Conversely, to a fixed i ∈ {1, . . . , n} we may choose a trading strategy
θ by setting θtl = δli for l = 1, . . . , n and t = 0, 1, . . . , T . Then δtθ = δti for
t = 1, . . . , T and
   
T T
1 X 1 X
Sti = θt · St = Et  δjθ πj  = Et  δji πj 
πt πt
j=t+1 j=t+1

for t = 0, 1, . . . , T − 1. 

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Second characterization of no arbitrage

Theorem
The dividend-price pair (δ, S) admits no arbitrage, if and only if there is
a state-price deflator.

Proof: We equip L with the inner product

" T #
X
(x, y ) = E xt yt x, y ∈ L
t=0

and notice that L becomes a Hilbert space.

Suppose there is no arbitrage. Then there is a strictly positive linear
functional F : L → R such that F (δ θ ) = 0 for every trading strategy θ.
A linear functional on a vector space is given by the inner product and
a vector. This is called Riesz’ representation theorem.

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Assuming no arbitrage I

There exists thus an adapted process π in L such that

" T #
X
F (x) = (x, π) = E xt πt ∀x ∈ L,
t=0

and since F is strictly positive, we obtain that π is strictly positive.

Fix t = 0, 1, . . . , T and i = 1, . . . , n and let y be an arbitrary
Ft -measurable function. We define
(
0 0≤j <t
θji =
y t ≤j ≤T

for j = i and set θjl = 0 for l 6= i.

Note that θ is an adapted process and thus a trading strategy.

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Assuming no arbitrage II
The corresponding dividend process is given by


 0 0≤j <t

δjθ = −ySti j =t

 y δi

t +1≤j ≤T
j

and
T
" #
X
θ
F (δ ) = E δtθ πt = 0.
t=0

Inserting the dividend process we obtain

   
XT  T
X 
E −ySti πt + y δji πj  = E y − Sti πt + δji πj  = 0.
j=t+1 j=t+1

Since y is an arbitrary Ft -measurable function the assertion follows.

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Assuming a state price deflator

If conversely π is a state-price deflator, then we use the price formula

for trading strategies, put t = 0, and obtain
 
X T
θ0 · S0 = E  δjθ πj  π0−1 .
j=1

Since δ0θ = −θ0 · S0 we conclude that

 
XT
E δjθ πj  = 0
j=0

for every trading strategy θ. Since π is strictly positive, there are no

arbitrage possibilities. 

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Riskless borrowing

There is short time riskless borrowing at time t < T , if there exists a

Ft -measurable portfolio θt such that

θt · (St+1 + δt+1 ) = I.

The short term discount factor dt is defined by setting dt = θt · St .

We may extend the portfolio θt to a trading strategy θ by setting

θs = (0, . . . , 0) for s 6= t. The corresponding dividend process δ θ has
only two payments:
An investment of dt at time t, that is δtθ = −dt
θ
A riskless return of one unit at time t + 1, that is δt+1 = 1.
If in addition there is no arbitrage in the economy, we realize that the
short term discount factor dt = θt · St is uniquely defined.

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The role over strategy

Assume there is riskless borrowing at any time t < T and consider two
different times t < s ≤ T .
Then there exists a trading strategy (called the roll over strategy) such
that the only non-vanishing trading gains are an investment of one unit
at time t and the return of ρ−1
t,s at time s where

ρt,s = dt · · · ds−1

is the discount factor from t to s.

In the absence of arbitrage we may apply the price formula for
portfolios to the roll over trading strategy described above and obtain

πt = Et [ρ−1
t,s πs ]

valid for any state-price deflator π.

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Equivalent measures
We have tacitly calculated means with respect to some measure P.
We say that two measures P and Q are equivalent, if they assign zero
probabilities to the same events.

Let P be a probability measure defined on FT and let ξT be a strictly

positive FT -measurable function with mean E P [ξT ] = 1.
The superscript P indicates that we take mean with respect to P.

We may now define a new probability measure

Z
Q(A) = ξT (ω) dP(ω) A ∈ FT
A

and realize that P and Q are equivalent. Note that by this definition

E Q [X ] = E P [ξT X ]

for any function X : Ω → R on the state space.

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Expectations under an equivalent measure

The expectations under the equivalent measure Q are for each

Fj -measurable function Xj given by

EtQ [Xj ] = ξt−1 EtP [ξj Xj ] 0 ≤ t ≤ j ≤ T,

where we set ξs = EsP [ξT ] for s = 0, 1, . . . , T .

Proof: For any Ft -measurable function y we obtain

Xj | y Q = E Q [Xj y ] = E P [Xj y ξT ] = E P EjP [Xj y ξT ]

 

= E P Xj yEjP [ξT ] = E P [Xj y ξj ] = E P EtP [ξj Xj ]y

   

= E P [ξT ξt−1 EtP [ξj Xj ]y ] = E Q [ξt−1 EtP [ξj Xj ]y ]

= ξt−1 EtP [ξj Xj ] | y Q

from which the statement follows. 

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An equivalent Martingale measure
Assume there is riskless borrowing at any time t < T and the security
prices ST = (0, . . . , 0).

We say that a probability measure Q is an equivalent martingale

measure, if Q and P are equivalent and the security prices
 
XT
Sti = EtQ  ρt,j δji  t < T, i = 1, . . . , n,
j=t+1

where EtQ denotes expectations under Q.

Note that the security prices are then expressed as discounted

expected dividends under the equivalent Martingale measure.
Theorem
There is no arbitrage, if and only if there exists an equivalent
Martingale measure.
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Assume no arbitrage
Then there exists a state price deflator π and we set
1 −1
ξT = ρ πT .
π0 0,T
The roll over strategy gave us the formula

πt = EtP [ρ−1
t,s πs ] 0 ≤ t < s ≤ T,

where we now make the dependence of the probability measure P

explicit. Setting t = 0 and s = T we obtain π0 = E P [ρ−1
0,T πT ], thus

1 P −1
E P [ξT ] = E [ρ0,T πT ] = 1.
π0
The measure Z
Q(A) = ξT (ω) dP(ω) A ∈ FT
A
is thus a probability measure.
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Continuation of proof
Since ξT is strictly positive we obtain that P and Q are equivalent.
Applying the roll over strategy again we obtain

πt = EtP [ρ−1 P −1 P
t,T πT ] = ρ0,t Et [ρ0,T πT ] = ρ0,t Et [ξT π0 ] = ξt π0 ρ0,t

for any t < T , where we used that

ρ−1 −1
t,T = ρ0,t ρ0,T and ξt = EtP [ξT ].

Q is an equivalent martingale measure. Indeed,

   
T T
1 X 1 X
Sti = Et  δji πj  = Et  ξj π0 ρ0,j δji 
πt ξt π0 ρ0,t
j=t+1 j=t+1
   
T T
1 Q X X
= E ρ0,j δji  = EtQ  ρt,j δji 
ρ0,t t
j=t+1 j=t+1

for t = 0, . . . , T − 1 and i = 1, . . . , n.
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Assume there is an equivalent Martingale measure
In the presence of a state price deflator we earlier proved that the
deflated price of a portfolio θ at time t equals the expected value of the
sum of future deflated dividends.
We may slightly modify the proof and obtain a similar formular
 
T
1 Q X
θt · St = E ρ0,j δjθ  t = 0, 1, . . . , T − 1
ρ0,t t
j=t+1

in the presence of riskless borrowing and an equivalent Martingale

measure Q. Since δ0θ = −θ0 · S0 we obtain
 
XT
F (δ θ ) = E Q  ρ0,j δjθ  = 0
j=0

for any trading strategy θ, so there can be no arbitrage.

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