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INTRODUCTION TO FINANCIAL ECONOMICS

Gordan Zˇ itkovic´

Department of Mathematics University of Texas at Austin

Summer School in Mathematical Finance, July-August 2009

This version: July 28, 2009

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FINANCIAL ECONOMICS

These lectures are about an oversimpliﬁed view that many mathematicians have of ﬁnancial economics. The name of the game is transfer of wealth either in time or across states of the world. Example 1. When you spend $95 on a bond which pays $100 in a year

from now, you have effectively exchanged (transferred) 95 today-dollars for 100 a-year-from-now-dollars.

Example 2. Consider a share of a stock of a publicly traded food com-

pany which is currently trading at $100 and you know that the price of the same stock in a year from now will fall (say to $80) if there is a drought, and rise (say to $110) otherwise. In a simpliﬁed world where only two possible futures can arise (drought and no drought), this stock allows you to transfer 100 today-dollars into an uncertain amount of money, which can be described as 80 a-year-from-now-dollars-if-a-drought-occurs and 110 a-year-from-now-dollars-if-there-is-no-drought.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FINANCIAL ECONOMICS

The useful thing - and this is the central service the ﬁnancial section provides - is that you can also do exactly the opposite: you can get $95 right now in exchange for a future payment of $100. That is how you can afford to buy a car and enjoy it now (or a house, but let’s not go there) and use your future income to pay for it. Analogously, you could short-sell the stock, i.e., get $100 today for a promise to pay either $80 or $110 in a year from now, contingent on the state of the world (drought or no drought). The main purpose of these lectures is to describe a mathematical formalism built precisely with the purpose of understanding various ways wealth can be transferred.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FINANCIAL ECONOMICS

The goal here is to cover only the minimum amount of material for a student to be able to go on and learn about the basic notions of mathematical ﬁnance in a subsequent course, we keep our models as simple as possible. A single consumption good is posited and the future is multi-period but ﬁnite. Similarly, there is only a ﬁnite number of states of the world. Mathematically, we work exclusively in a ﬁnite-dimensional Euclidean space, but we interpret its elements in different ways and give them different names, depending on the role they play. This way, we keep conceptually different but formally equivalent objects separated and we pave the way to the inﬁnite-dimensional case, where vectors come in many ﬂavors and varieties.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

THE WORLD

We model the world as a tree-like structure that describes the possible ways the future can unfold. There are two qualitatively different dimensions in which wealth can be transferred - through time and through uncertainty. For the set of time instances, we take T = {0, 1, . . . , T}, for some T ∈ N. The uncertainty is modeled by a nonempty ﬁnite set Ω which we calls the set of states of the world. Things become interesting when we start to incorporate the fact that we learn more and more about the true state of the world as time marches on, i.e., when we model the gradual resolution of uncertainty.

ALGEBRAS

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

Deﬁnition. A family F of subsets of Ω is called an algebra if

1. ∅ ∈ F, 2. if A ∈ F then Ac ∈ F , and 3. if A, B ∈ F then A ∪ B ∈ F .

The notion you are going to hear about more often is a σ-algebra. A σ-algebra is a generalization of the concept of an algebra which is useful only if the number of states of the world is inﬁnite. Otherwise, the two concepts coincide, so we have absolutely no reason to complicate our lives with the extra σ.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = INFORMATION

You can picture information as the ability to answer questions (more information gives you a better score on the test . . . ), and the lack of information as ignorance.

In our world, all the questions can be phrased in terms of the elements of the state-space Ω. Remember - Ω contains all the possible futures of our world and the knowledge of the exact ω ∈ Ω (the “true state of the world”) amounts to the knowledge of everything.

So, the ultimate question would be “What is the true ω?”, and the ability to answer that would promote you immediately to the level of a Supreme Being. In order for our theory to be of any use to us mortals, we have to allow for some ignorance, and consider questions like

“Is the true ω an element of A?”,

(1)

where A is a subset of Ω.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = INFORMATION II

The state of our knowledge can be described by the set of all questions you know the answer to. Since all questions about the true state of the world can be phrased as questions about sets of states, the proper mathematical description of our current knowledge is nothing but the collection of all those A for which we know the answer to (1).

The nice thing about that set is that - for purely logical reasons it has lots of structure. You are probably already guessing that I am talking about the fact that set - let us call it F - has to be an algebra. Let’s see why:

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = INFORMATION III

First of all, I always know that the true ω is an element of Ω, so Ω ∈ F. Then, if I happen to know how to answer the question Is the true ω in A?, I will necessarily also know how to answer the question “Is the true ω in Ac?”. The second answer is just the opposite of the ﬁrst. Equivalently, A ∈ F implies Ac ∈ F. Finally, let A and B be two sets with the property that I know how to answer the questions “Is the true ω in A?” and “Is the true ω in B?”. Then I clearly know that the answer to the question “Is the true ω in A ∪ B?” is “No” if I answered “No” to each of the two questions above. One the other hand, it is going to be “Yes” if I answered “Yes” to at least one of them. Therefore A, B ∈ F implies A ∪ B ∈ F.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = PARTITION

As we shall see in the following problem, the seemingly complicated notion of an algebra is really equivalent to the simpler notion of a partition. Deﬁnition. A partition of a set Ω is a family P of non-empty subsets of Ω such that A ∩ B = ∅ whenever A = B and ∪A∈P A = Ω. Problem 1. Let Ω be a nonempty ﬁnite set, and let A be the family of all

algebras on Ω, and let Π be the family of all partitions of Ω. Construct a mapping F : Π → A as follows: for a partition P = {A1, A2, . . . , Ak} of Ω, let F(P) = A be the family consisting of the empty set and all possible unions of elements of P.

1. Show that so deﬁned family A is an algebra, and 2. Show that the mapping F is one-to-one and onto.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = PARTITION II

Problem 2.[Just for fun, not important for the sequel]

For n ∈ N, let an be the number of different algebras on Ω when Ω has exactly n elements. Show that

1. a1 = 1, a2 = 2, a3 = 5, and that the following recursion holds

Xn an+1 =

k=0

! n ak, k

where a0 = 1 by deﬁnition, and

2. the exponential generating function for the sequence {an}n∈N is f (x) =

eex−1, i.e., that

X ∞ xn

ex −1

an n! = e .

n=0

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

A MODELING EXAMPLE

Three candidates (let call them Candidate 1, Candidate 2 and Candidate 3) participate in a presidential election; Candidates 1 and 2 are women and Candidate 3 is a man. Since we are only interested in the winner of the election, we model the possible states of the world by the elements of the set Ω = {ω1, ω2, ω3} where, as expected, ωi means that Candidate i wins the election. We wake up on the day following the election without knowing how the election went; we do know that there one of the candidates has been elected, but that is all. Our knowledge can be modeled by the algebra

A0 = {∅, Ω},

or, equivalently the trivial partition

P0 = {Ω},

as the only questions we know how to answer are “Is the new president either Candidate 1, Candidate 2 and Candidate 3?” (the answer is “yes”) and “Is the new president neither Candidate 1, Candidate 2 nor Candidate 3?” (the answer if “no”).

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

A MODELING EXAMPLE II

We turn on the radio and hear about how the new president (elect) won by a small margin, and how she has a difﬁcult time ahead of her. Now we know more about the true state of the world; we know that ω3 is not the true state of the world because Candidate 3 is a man. We still don’t know which of the two women candidates won. Our state of the knowledge can be described by the algebra

A1 = {∅, Ω, {ω1, ω2}, {ω3}, Ω},

or the partition

P1 = {{ω1, ω2}, {ω3}},

since we know how to answer more questions now. For example, we know that the answer to the question “Is the new president either Candidate 1 or Candidate 2?” - the answer is “yes”.

We listen to the radio some more and the name of the winning Candidate is mentioned - it is Candidate 2. We know everything now and our information corresponds to the algebra A2

A2 = {∅, {ω1}, {ω2}, {ω3}, {ω1, ω2}, {ω1, ω3}, {ω2, ω3}, Ω},

which consists of all subsets of Ω, with the corresponding partition being

P = {{ω1}, {ω2}, {ω3}}.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FILTRATIONS

The example above not only illustrates the connection between algebras (partitions) and the amount of knowledge or information, it also shows how we learn. By observing more facts, we can answer more questions, and our algebra grows. This typically happens over time, so, in order to describe the evolution of our knowledge, we introduce the following important concept: Deﬁnition. A ﬁltration is a ﬁnite sequence A1, A2, . . . AT of algebras on Ω (indexed by the time set T = {1, 2, . . . , T}) such that

A1 ⊆ A2 ⊆ · · · ⊆ AT .

The knowledge we accumulate over time is typically more about values of certain quantities (like sports statistics), and less about presidential candidates.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

AN EXAMPLE

Suppose that the batting average B of a certain baseball player is modeled in the following, simpliﬁed, way for the period of the next 2 years. At time t = 0 (now) his batting average is .250, i.e., B0 = .250. Next year (corresponding to t = 1), it will either go up to B1 = .300, down to B1 = .220, or player will leave professional baseball and so that B1 = .000. After that, the evolution continues: after B1 = .300, three possibilities can occur B2 = .330, B2 = .300 and B2 = .275. Similarly, if B1 = .220, either B2 = .275 or B2 = .200. Finally, if B1 = .000, the player is retired and B2 = .000, as well. A possible mathematical model for this situation can be built on a state space which consists of 6 states of the world Ω = {ω1, ω2, ω3, ω4, ω5, ω6}, where each state of the world corresponds to a particular path the future can take. For example, in ω1, the batting averages are B0 = .250, B1 = .300, B2 = .330, while in ω4, B0 = .250, B1 = .220 and B2 = .275.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

AN EXAMPLE II

Things will be much clearer from the picture:

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

AN EXAMPLE III

At time 0, the information available to us is minimal, the only questions we can answer are the trivial ones “ Is the true ω in Ω?” and “ Is the true ω in ∅?” , and this is encoded in the algebra {Ω, ∅}.

A year after that, we already know a little bit more, having observed the player’s batting average for the last year. We can distinguish between ω1 and ω5, for example. We still do not know what will happen the day after, so that we cannot tell between ω1, ω2 and ω3, or ω4 and ω5. Therefore, our information partition is {{ω1, ω2, ω3}, {ω4, ω5}, {ω6}} and the corresponding algebra A1 is (I am doing this only once!)

A1 = {∅, {ω1, ω2, ω3}, {ω4, ω5}, {ω6}, {ω1, ω2, ω3, ω4, ω5}, {ω1, ω2, ω3, ω6}, {ω4, ω5, ω6}, Ω}.

It is interesting to note that algebra A1 has something to say about the batting average a year from now, but only in the special case when B2 = .000. If that special case occurs - let us call it injury - then we do not need to wait until year 2 to learn what the batting average is going to be. It is going to remain .000. Finally, after 2 years, we know exactly what ω occurred and the algebra A2 consists of all subsets of Ω.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

RANDOM VARIABLES

Let us think for a while how we acquired the extra information on each new day. We have learned it through the quantities B0, B1 and B2 as their values gradually revealed themselves to us. Deﬁnition. Any function B : Ω → R is called a random variable. Deﬁnition. A random variable B is said to be measurable with respect to an algebra A (denoted by B ∈ mA) if it is constant on each member of the partition P corresponding to A. Deﬁnition. A random variable B is said to generate the algebra A (denoted by A = A(B)) if

B ∈ mA, and B ∈ mB, for any algebra B = A with A ⊂ B.

Does B2 generate the algebra A2 in the previous example? How would you give a deﬁnition of an algebra generated by two random variables?

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FILTRATIONS AND PROCESSES

Remember that T = {0, 1, . . . , T} is the time set. Deﬁnition. A ﬁnite sequence {Bk}k∈T , of random variables is called a (stochastic) process. The algebras A0, A1 and A2 in previous example are interpreted as the amounts of information available to us agent at days 0, 1 and 2 respectively. They were generated by the accumulated values of the process {Bk}k∈T (for T = 2). Deﬁnition. A ﬁnite sequence {Ak}k∈T of algebras is called a ﬁltration if A0 ⊆ A1 ⊆ · · · ⊆ AT . Deﬁnition. A process {Bk}k∈T is said to be adapted to the ﬁltration {Ak}k∈T , if Bk ∈ mAk, for k ∈ T . Deﬁnition. A ﬁltration {Ak}k∈T is said to be generated by the process {Bk}k∈T , if Ak = A(B0, B1, . . . , Bk), for k ∈ T .

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

THE INFORMATION TREE

A process {Bk}k∈T can be thought of as a mapping B : T ×Ω → R. It is, additionally, adapted to {Ak}k∈T , if and only if B(t, ω) = B(t, ω ) for all t ∈ T and ω and ω belong to the same element of the partition (corresponding to) At. In the language of “abstract nonsense”, the mapping B factorizes through a certain quotient set of T × Ω. More precisely . . . Deﬁnition. The set N , whose elements are called nodes is the quotient set (the set of all equivalence classes) of the product T × Ω with respect to the equivalence relation ∼ which is deﬁned as

(t, ω) ∼ (s, ω ) if and only if t = s and ω and ω belong to the same partition element in At.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

Gordan Zˇ itkovic´

Department of Mathematics University of Texas at Austin

Summer School in Mathematical Finance, July-August 2009

This version: July 28, 2009

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FINANCIAL ECONOMICS

These lectures are about an oversimpliﬁed view that many mathematicians have of ﬁnancial economics. The name of the game is transfer of wealth either in time or across states of the world. Example 1. When you spend $95 on a bond which pays $100 in a year

from now, you have effectively exchanged (transferred) 95 today-dollars for 100 a-year-from-now-dollars.

Example 2. Consider a share of a stock of a publicly traded food com-

pany which is currently trading at $100 and you know that the price of the same stock in a year from now will fall (say to $80) if there is a drought, and rise (say to $110) otherwise. In a simpliﬁed world where only two possible futures can arise (drought and no drought), this stock allows you to transfer 100 today-dollars into an uncertain amount of money, which can be described as 80 a-year-from-now-dollars-if-a-drought-occurs and 110 a-year-from-now-dollars-if-there-is-no-drought.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FINANCIAL ECONOMICS

The useful thing - and this is the central service the ﬁnancial section provides - is that you can also do exactly the opposite: you can get $95 right now in exchange for a future payment of $100. That is how you can afford to buy a car and enjoy it now (or a house, but let’s not go there) and use your future income to pay for it. Analogously, you could short-sell the stock, i.e., get $100 today for a promise to pay either $80 or $110 in a year from now, contingent on the state of the world (drought or no drought). The main purpose of these lectures is to describe a mathematical formalism built precisely with the purpose of understanding various ways wealth can be transferred.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FINANCIAL ECONOMICS

The goal here is to cover only the minimum amount of material for a student to be able to go on and learn about the basic notions of mathematical ﬁnance in a subsequent course, we keep our models as simple as possible. A single consumption good is posited and the future is multi-period but ﬁnite. Similarly, there is only a ﬁnite number of states of the world. Mathematically, we work exclusively in a ﬁnite-dimensional Euclidean space, but we interpret its elements in different ways and give them different names, depending on the role they play. This way, we keep conceptually different but formally equivalent objects separated and we pave the way to the inﬁnite-dimensional case, where vectors come in many ﬂavors and varieties.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

THE WORLD

We model the world as a tree-like structure that describes the possible ways the future can unfold. There are two qualitatively different dimensions in which wealth can be transferred - through time and through uncertainty. For the set of time instances, we take T = {0, 1, . . . , T}, for some T ∈ N. The uncertainty is modeled by a nonempty ﬁnite set Ω which we calls the set of states of the world. Things become interesting when we start to incorporate the fact that we learn more and more about the true state of the world as time marches on, i.e., when we model the gradual resolution of uncertainty.

ALGEBRAS

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

Deﬁnition. A family F of subsets of Ω is called an algebra if

1. ∅ ∈ F, 2. if A ∈ F then Ac ∈ F , and 3. if A, B ∈ F then A ∪ B ∈ F .

The notion you are going to hear about more often is a σ-algebra. A σ-algebra is a generalization of the concept of an algebra which is useful only if the number of states of the world is inﬁnite. Otherwise, the two concepts coincide, so we have absolutely no reason to complicate our lives with the extra σ.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = INFORMATION

You can picture information as the ability to answer questions (more information gives you a better score on the test . . . ), and the lack of information as ignorance.

In our world, all the questions can be phrased in terms of the elements of the state-space Ω. Remember - Ω contains all the possible futures of our world and the knowledge of the exact ω ∈ Ω (the “true state of the world”) amounts to the knowledge of everything.

So, the ultimate question would be “What is the true ω?”, and the ability to answer that would promote you immediately to the level of a Supreme Being. In order for our theory to be of any use to us mortals, we have to allow for some ignorance, and consider questions like

“Is the true ω an element of A?”,

(1)

where A is a subset of Ω.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = INFORMATION II

The state of our knowledge can be described by the set of all questions you know the answer to. Since all questions about the true state of the world can be phrased as questions about sets of states, the proper mathematical description of our current knowledge is nothing but the collection of all those A for which we know the answer to (1).

The nice thing about that set is that - for purely logical reasons it has lots of structure. You are probably already guessing that I am talking about the fact that set - let us call it F - has to be an algebra. Let’s see why:

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = INFORMATION III

First of all, I always know that the true ω is an element of Ω, so Ω ∈ F. Then, if I happen to know how to answer the question Is the true ω in A?, I will necessarily also know how to answer the question “Is the true ω in Ac?”. The second answer is just the opposite of the ﬁrst. Equivalently, A ∈ F implies Ac ∈ F. Finally, let A and B be two sets with the property that I know how to answer the questions “Is the true ω in A?” and “Is the true ω in B?”. Then I clearly know that the answer to the question “Is the true ω in A ∪ B?” is “No” if I answered “No” to each of the two questions above. One the other hand, it is going to be “Yes” if I answered “Yes” to at least one of them. Therefore A, B ∈ F implies A ∪ B ∈ F.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = PARTITION

As we shall see in the following problem, the seemingly complicated notion of an algebra is really equivalent to the simpler notion of a partition. Deﬁnition. A partition of a set Ω is a family P of non-empty subsets of Ω such that A ∩ B = ∅ whenever A = B and ∪A∈P A = Ω. Problem 1. Let Ω be a nonempty ﬁnite set, and let A be the family of all

algebras on Ω, and let Π be the family of all partitions of Ω. Construct a mapping F : Π → A as follows: for a partition P = {A1, A2, . . . , Ak} of Ω, let F(P) = A be the family consisting of the empty set and all possible unions of elements of P.

1. Show that so deﬁned family A is an algebra, and 2. Show that the mapping F is one-to-one and onto.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

ALGEBRA = PARTITION II

Problem 2.[Just for fun, not important for the sequel]

For n ∈ N, let an be the number of different algebras on Ω when Ω has exactly n elements. Show that

1. a1 = 1, a2 = 2, a3 = 5, and that the following recursion holds

Xn an+1 =

k=0

! n ak, k

where a0 = 1 by deﬁnition, and

2. the exponential generating function for the sequence {an}n∈N is f (x) =

eex−1, i.e., that

X ∞ xn

ex −1

an n! = e .

n=0

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

A MODELING EXAMPLE

Three candidates (let call them Candidate 1, Candidate 2 and Candidate 3) participate in a presidential election; Candidates 1 and 2 are women and Candidate 3 is a man. Since we are only interested in the winner of the election, we model the possible states of the world by the elements of the set Ω = {ω1, ω2, ω3} where, as expected, ωi means that Candidate i wins the election. We wake up on the day following the election without knowing how the election went; we do know that there one of the candidates has been elected, but that is all. Our knowledge can be modeled by the algebra

A0 = {∅, Ω},

or, equivalently the trivial partition

P0 = {Ω},

as the only questions we know how to answer are “Is the new president either Candidate 1, Candidate 2 and Candidate 3?” (the answer is “yes”) and “Is the new president neither Candidate 1, Candidate 2 nor Candidate 3?” (the answer if “no”).

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

A MODELING EXAMPLE II

We turn on the radio and hear about how the new president (elect) won by a small margin, and how she has a difﬁcult time ahead of her. Now we know more about the true state of the world; we know that ω3 is not the true state of the world because Candidate 3 is a man. We still don’t know which of the two women candidates won. Our state of the knowledge can be described by the algebra

A1 = {∅, Ω, {ω1, ω2}, {ω3}, Ω},

or the partition

P1 = {{ω1, ω2}, {ω3}},

since we know how to answer more questions now. For example, we know that the answer to the question “Is the new president either Candidate 1 or Candidate 2?” - the answer is “yes”.

We listen to the radio some more and the name of the winning Candidate is mentioned - it is Candidate 2. We know everything now and our information corresponds to the algebra A2

A2 = {∅, {ω1}, {ω2}, {ω3}, {ω1, ω2}, {ω1, ω3}, {ω2, ω3}, Ω},

which consists of all subsets of Ω, with the corresponding partition being

P = {{ω1}, {ω2}, {ω3}}.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FILTRATIONS

The example above not only illustrates the connection between algebras (partitions) and the amount of knowledge or information, it also shows how we learn. By observing more facts, we can answer more questions, and our algebra grows. This typically happens over time, so, in order to describe the evolution of our knowledge, we introduce the following important concept: Deﬁnition. A ﬁltration is a ﬁnite sequence A1, A2, . . . AT of algebras on Ω (indexed by the time set T = {1, 2, . . . , T}) such that

A1 ⊆ A2 ⊆ · · · ⊆ AT .

The knowledge we accumulate over time is typically more about values of certain quantities (like sports statistics), and less about presidential candidates.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

AN EXAMPLE

Suppose that the batting average B of a certain baseball player is modeled in the following, simpliﬁed, way for the period of the next 2 years. At time t = 0 (now) his batting average is .250, i.e., B0 = .250. Next year (corresponding to t = 1), it will either go up to B1 = .300, down to B1 = .220, or player will leave professional baseball and so that B1 = .000. After that, the evolution continues: after B1 = .300, three possibilities can occur B2 = .330, B2 = .300 and B2 = .275. Similarly, if B1 = .220, either B2 = .275 or B2 = .200. Finally, if B1 = .000, the player is retired and B2 = .000, as well. A possible mathematical model for this situation can be built on a state space which consists of 6 states of the world Ω = {ω1, ω2, ω3, ω4, ω5, ω6}, where each state of the world corresponds to a particular path the future can take. For example, in ω1, the batting averages are B0 = .250, B1 = .300, B2 = .330, while in ω4, B0 = .250, B1 = .220 and B2 = .275.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

AN EXAMPLE II

Things will be much clearer from the picture:

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

AN EXAMPLE III

At time 0, the information available to us is minimal, the only questions we can answer are the trivial ones “ Is the true ω in Ω?” and “ Is the true ω in ∅?” , and this is encoded in the algebra {Ω, ∅}.

A year after that, we already know a little bit more, having observed the player’s batting average for the last year. We can distinguish between ω1 and ω5, for example. We still do not know what will happen the day after, so that we cannot tell between ω1, ω2 and ω3, or ω4 and ω5. Therefore, our information partition is {{ω1, ω2, ω3}, {ω4, ω5}, {ω6}} and the corresponding algebra A1 is (I am doing this only once!)

A1 = {∅, {ω1, ω2, ω3}, {ω4, ω5}, {ω6}, {ω1, ω2, ω3, ω4, ω5}, {ω1, ω2, ω3, ω6}, {ω4, ω5, ω6}, Ω}.

It is interesting to note that algebra A1 has something to say about the batting average a year from now, but only in the special case when B2 = .000. If that special case occurs - let us call it injury - then we do not need to wait until year 2 to learn what the batting average is going to be. It is going to remain .000. Finally, after 2 years, we know exactly what ω occurred and the algebra A2 consists of all subsets of Ω.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

RANDOM VARIABLES

Let us think for a while how we acquired the extra information on each new day. We have learned it through the quantities B0, B1 and B2 as their values gradually revealed themselves to us. Deﬁnition. Any function B : Ω → R is called a random variable. Deﬁnition. A random variable B is said to be measurable with respect to an algebra A (denoted by B ∈ mA) if it is constant on each member of the partition P corresponding to A. Deﬁnition. A random variable B is said to generate the algebra A (denoted by A = A(B)) if

B ∈ mA, and B ∈ mB, for any algebra B = A with A ⊂ B.

Does B2 generate the algebra A2 in the previous example? How would you give a deﬁnition of an algebra generated by two random variables?

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

FILTRATIONS AND PROCESSES

Remember that T = {0, 1, . . . , T} is the time set. Deﬁnition. A ﬁnite sequence {Bk}k∈T , of random variables is called a (stochastic) process. The algebras A0, A1 and A2 in previous example are interpreted as the amounts of information available to us agent at days 0, 1 and 2 respectively. They were generated by the accumulated values of the process {Bk}k∈T (for T = 2). Deﬁnition. A ﬁnite sequence {Ak}k∈T of algebras is called a ﬁltration if A0 ⊆ A1 ⊆ · · · ⊆ AT . Deﬁnition. A process {Bk}k∈T is said to be adapted to the ﬁltration {Ak}k∈T , if Bk ∈ mAk, for k ∈ T . Deﬁnition. A ﬁltration {Ak}k∈T is said to be generated by the process {Bk}k∈T , if Ak = A(B0, B1, . . . , Bk), for k ∈ T .

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

THE INFORMATION TREE

A process {Bk}k∈T can be thought of as a mapping B : T ×Ω → R. It is, additionally, adapted to {Ak}k∈T , if and only if B(t, ω) = B(t, ω ) for all t ∈ T and ω and ω belong to the same element of the partition (corresponding to) At. In the language of “abstract nonsense”, the mapping B factorizes through a certain quotient set of T × Ω. More precisely . . . Deﬁnition. The set N , whose elements are called nodes is the quotient set (the set of all equivalence classes) of the product T × Ω with respect to the equivalence relation ∼ which is deﬁned as

(t, ω) ∼ (s, ω ) if and only if t = s and ω and ω belong to the same partition element in At.

GORDAN Zˇ ITKOVIC´

INTRODUCTION TO FINANCIAL ECONOMICS

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