Financial Asset Trading By Insiders

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Financial Asset Trading By Insiders

Mathematical Models for Profit Maximization in Discrete Time

by Srichandra Masabathula

Submitted in Partial Fulfillment for Bachelor of Arts degree with College Honors at

Knox College, Galesburg, IL 61401, USA

9:00 A.M., 3rd March, 2016

College Honors Committee:

Chair: Dr. Kevin J. Hastings, Professor, Dept. of Mathematics, Knox College

Member: Dr. Andrew Leahy, Associate Professor, Dept. of Mathematics, Knox College

Member: Dr. Jonathan Powers, Assistant Professor, Dept. of Economics, Knox College

External Examiner: Dr. Liming Feng, Associate Professor, Dept. of Industrial and Enterprise

Systems Engineering, University of Illinois-Urbana Champaign


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Financial Asset Trading by Insiders

Srichandra Masabathula

3rd March, 2016


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Contents

Acknowledgements 1

1. Introduction 5

2. Theory 9

2.1. Probability Spaces, Algebras and σ -Algebras………………………………………… 11

2.2. Measurability, Random Variables, and Independence……………………………….. 13

2.3. Conditional Expectation……………………………………………………………… 17

2.4. Bivariate Normal Distribution………………………………………………………... 19

3. Model 21

3.1. Description…………………………………………………………………………… 23

3.2. Players………………………………………………………………………………... 23

3.3. The Trading Process………………………………………………………………….. 24

3.4. Notation and Assumptions…………………………………………………………… 24

3.5. Model Types………………………………………………………………………….. 28

4. Single Period Model 31

4.1. Single-Period Model………………………………………………………………… . 33

5. Kyle’s Multiple Period Model 41

5.1. Initial Values of Parameters………………………………………………………… .. 43

5.2. Determining Parameters and Expected Profits Recursively………………………….. 44

5.3. Parameter Computation and Sensitivity……………………………………………… 53

6. Martingale Model 59

6.1. Concept……………………………………………………………………………….. 61

6.2.

Initial Values of Parameters………………………………………………………… ... 626.3. Recursive Relationships………………………………………………………………. 63

6.4. Parameter Computation and Sensitivity………………………………………………. 71

Appendix 79

Bibliography 93


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Acknowledgements


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Working on this Honors project has been an extremely rewarding and a learning experience for me. This thesis, which

is a product of extensive research, hardwork, determination and curiosity, was possible through support and collaboration of

many. I have worked on this thesis to my heart’s content and the way it has turned out is certainly something I am proud of.

However, this undertaking of mine would be largely incomplete if I do not make mention of all the people who have played

a role in shaping this project

First and foremost, I would like to convey my heartfelt gratitude to Dr. Kevin Hastings, who served as my adviser for

the project, and also as the Chair of the College Honors Committee. He is majorly responsible for giving this project a sense

of direction, as it would not even have come close to completion without his continuous guidance and mentorship. I am

grateful for his patience in putting up with me and his constant words of encouragement kept me spirited throughout to cross

the finish line. His exclamation of “If we can get through all of this, we have an Honors!” towards the final days of the

project was momentous and will be cherished by me for years to come.

I would like to convey my sincere thanks to Dr. Andrew Leahy and Dr. Jonathan Powers for serving on the College

Honors committee and for their valuable critique at various stages of the project. I would also like to thank Dr. Liming Feng ,

from University of Illinois, Urbana-Champaign for his timely review and comments as the external examiner.

I would like to thank the Mathematics and Economics departments at Knox College for giving me the academic tools

and skill-sets required to pursue this project. The courses I took in these departments provided me with the prerequisite

knowledge to undertake the project. In fact, it was in Prof. Jonathan Powers’ Industrial Organization class that the prospect

of studying insider trading first originated. He is responsible for formally introducing me to Game Theory, which is an

important aspect of my research. When he discussed games with private information in the class, I wanted to delve deeper

into it to understand how key economic concepts are supported by complex mathematics. Under the guidance of Dr. Hast-

ings, I eventually arrived at the related topic of mathematical models of insider trading to pursue as an Honors project.

I would like to convey my appreciation to the Knox College Library for its plethora of academic resources and for providing me with a personal office space in the Research Tower in Seymour Library. I would also like to thank librarian

Ms. Anne Giffey for her assistance with my research.

I would also like to thank Dean Lori Schroeder for giving me the opportunity to pursue this project under the Honors

Program of Knox College. Ms. Nancy Fennig deserves a special mention for providing administrative support for printing

and binding copies. She also provided logistical support for arranging the Honors defense.

Last but certainly not the least, I am ever grateful to my family and friends for their good cheer that has helped me

maintain my enthusiasm for researching and writing on this topic. Words cannot describe how instrumental the fun times

spent with them were in eep ng up my sp r ts to comp ete this pro ect.

Srichandra Masabathula

Knox College ‘16

March 3,

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Chapter 1

Introduction

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The idea of knowing something others do not is often appealing. I was drawn to the study of insider trading through my

interest in finance and mathematical modeling. The study of insider trading is one way of undestanding how the financial

markets can be manipulated for profit maximization motives. This project aims to quantitatively analyze the worth of the

private information of an insider through the profit she may earn on an investment. Insider trading can take place over any

financial asset, most commonly securities or stocks. The project will evaluate various facets of insider trading through

robust mathematical models that will provide useful results for maximizing the return on investing in the underlying asset.

Over the years, I have learnt a key aspect of producing quality research papers. More than knowing what to write in a

research paper, a more important question to ask is what not to include. In this regard, I would like to mention that my

learning through this honors project in this field of financial mathematics is much more than what is illustrated in this paper.

In addition to Albert Kyle’s [1] pioneering work in the field, I have also studied various other models on insider trading by

prominent academicians. Rene Caldentey and Ennio Stacchetti discuss an interesting model in their paper titled ‘Insider

Trading with a Random Deadline’ [2]. As the name suggests, the model considers cases where the time during which the

insider possesses her information is random. They also consider the value of the underlying asset to be modeled as a brown-

ian motion. Another interesting model is presented by Kerry Back and Shmuel Baruch in their paper titled ‘Information in

Securities: Kyle meets Glosten and Milgrom’ [7]. This paper considers a case where the market makers are also competitive

and their role is not just limited to consummating the trades by the traders to bring the market to equilibrium. They compare

Kyle’s version of single strategy to models in which the insider plays a mixed strategy. Kyle himself looks at more than one

kind of market situation. He discusses sequential auctions and continuous auctions to expand the scope of the subject.

Insider trading is also closely related to game theory. There are a number of game theoretic models that involve private

information and study the behaviour of the players to the market conditions and the information that they have. Examples of

such models are the Bayesian games that are based on the concept Bayesian-Nash equilibrium. As is evident by now, the

topic of insider trading brings together concepts from mathematics, finance and economics. With the wealth of work that has

been done in the field, this project is yet another contribution to better understand the profit maximization models of insider

trading.

With regard to the legalities of insider trading, there is no defending of the fact that insider trading is unethical. The

purpose of this project is to understand how insider trading may actually work and how an insider may be able to take

advantage of her information. The project will be helpful in understanding to what extent the insider can maximize her

profits given her private information. It will allow us to analyze the conditions under which profits are greater. In essence,

we will be able to understand the limits of the private information on the insider’s profits.

The remainder of the paper is organized as follows:

In Chapter 2, we will study some important theoretical concepts in Probability that will be useful in the study of insider

trading. We will start off gaining some understanding of probability spaces, algebras and σ-algebras. This will help usunderstand the different components of the insider trading model and the kinds of properties they may have. We will thenmove on to the study the concepts of measurability and random variables which will further delve into how individual

components can be treated in a probability space. Subsequently, we will study independence to understand how different

random variables interact with each other. Next, we will study conditional expectation which will prove to be invaluable in

the insider trading models that we will study in this paper. We will finish by refreshing our knowledge of multivariate

normal distributions which will make occasional appearances in this paper. The main ideas and concepts presented in this

chapter are borrowed from various undergraduate and graduate textbooks. The most used textbooks are Chung [8], Dineen

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[5], Hastings [4], and Roman [3]. The theorems and propositions discussed in this chapter have been modified to fit their

usage later in the paper.

In Chapter 3, we introduce the concept of insider trading and also discuss the different kinds of traders. We will discuss

the different assumptions that we will make in the model and also try to get used to the notation that will be used when

analyzing the model in this chapter. This chapter is meant to give the reader a sense of what insider trading is, how it works,

and what market conditions provide an environment for insiders to take advantage of their private information. The key

concept of market efficiency will also be introduced in this chapter, which will be useful in ensuring linear equilibrium for

the financial markets. We will end with a brief discussion of the two models that we will study in this paper - one being

Kyle’s [1] model and the other one being our novel extension of Kyle’s model.

Chapter 4 is mainly a demonstration of a simple version of the two models that will be discussed in this paper. It

discusses the single period model which is common to both models and helps in developing a basic understanding of how an

insider trading model works. It encompasses all aspects of the model and presents results that can easily be replicated and

transferred to other models.

Chapter 5 is where we discuss Kyle’s multiple period model in detail. Though this chapter is a replication of Kyle’s

multiple period model, it may be looked as our interpretation of Kyle’s model. The notation used is different in this model

and a lot of the information that is not included in his paper has been filled in here. This ensures that the topic discussed is

much easier to understand at the undergraduate level. Understanding Kyle’s model in such detail is important to us because

this model can form the basis for a number of other models. In other words, other models can be developed as extensions to

this model by altering a few assumptions and adding more parameters. One such extension is discussed in Chapter 6. For

this reason, Kyle’s model has proved to be one of the seminal works in the field. We will start with gaining an understand-

ing of the parameters and the variables that we will use in the chapter and also getting familiar with the properties they hold.

We will then carry out the method of backward induction that will allow us to calculate the insider’s decisions and profits at

each time period. Following the backward induction process, we will show some important computations using the resultsfrom the backward induction process. We will finish with an evaluation of the results using actual numerical values of the

parameters to see how much profit an insider can make. This will help us observe how sensitive the profits of the insider are

to the parameters used in the model. To carry out this evaluation, we have written a Mathematica program than generates

these results based on the conditions in the model. The Mathematica program is shown in the Appendix.

Chapter 6, as mentioned above, describes a novel extension to Kyle’s multiple period model. The format of the chapter

is mostly similar to that of Chapter 5. The first part of the chapter explains the key differences between Kyle’s model and

our extended model. It discusses the new parameters and assumptions and also explains the theoretical concept of Martin-

gale processes that is used in this chapter. We then move on to evaluating the model step by step to understand the decisions

insiders should take at each time interval. In this chapter, we will carry out an evaluation of the results from the backwardinduction process. The numerical evaluation in this model will be done in a similar way as in the previous chapter. The

purpose of this numerical evaluation largely to see how the profits of the insider in this model are sensitive to the new

parameters in this model. The Mathematica program that generates required results for this model is also shown in the

Appen x.

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Chapter 2

Theory

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Probability Spaces, Algebras and σ-Algebras2.1

Algebras and σ-Algebras

Algebras are simply collections of subsets of a set. They have some specific properties that they satisfy, and can be

thought of as basic building blocks of the models of insider trading that we will be studying in the later chapters. Events

involving prices of the asset, or the quantities traded by the traders in the market, can all be gathered together as collections

of sets.

Definition 2.1.1

Let Ω be a nonempty set. A collection of subsets of Ω is called an algebra of sets (or just an algebra) if it satisfiesthe following properties:

1 Empty set is in :∅∈

2

is closed under comp ementat on

A1 ∈⇒ A1c ∈

3) is closed under finite unions A1, A2, ..., An ∈

⇒ A1 ⋃ A2 ⋃ ... ⋃ An ∈4 is closed under finite intersections

A1, A2, ..., An ∈⇒ A1 ⋂ A2 ⋂ ... ⋂ An ∈

It may be noted that by DeMorgan’s laws, 4) follows from 2) and 3).

The o ow ng examp e will e p illustrate the concept of a ge ras:

Example 2.1.2

Let Ω = {1, 2, 3, 4, 5} and = ∅, {2, 3}, {1, 4, 5}, Ω.In order to show that is an a ge ra on Ω, we must prove that the propert es of an a ge ra discussed are satisfied:

Part 1: Empty Set is in

Since we are given that ∅ is in , 1) is trivially satisfied.

Part 2: is closed under complementation

We have,

∅c = Ω and Ωc = ∅{2, 3}c = {1, 4, 5} and {1, 4, 5}c = {2, 3}

Hence, 2 is satisfied.

Part 3: is closed under finite unions

The collections of unions of sets involving ∅ would just be the individual sets in , and unions of sets with Ω would

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just be Ω. Therefore, these unions are all subsets of . The only non-trivial case for collections of unions would be theunion of {2, 3} and {1, 4, 5}. For this case, we observe,

{2, 3} ⋃ {1, 4, 5} = Ω∈Hence, 3 is satisfied. Thus, we have shows that is an a ge ra on Ω.

The algebras that we just studied are finite subsets of a set. In the case that a set may have infinite subsets, such as

po nts in a line, we need a collection broader than what the a ge ras can offer. The σ-a ge ras come in an y in this respect.Definition 2.1.3

A σ-a ge ra on a set Ω is a collection of subsets of Ω if it satisfies the o ow ng propert es:1) Ω is a part of

Ω∈2) is closed under complementation

A1

∈⇒ A1

c

∈3) is closed under countable unions∞

n=1

An = { x : x ∈ An for some n}

4) is closed under countable intersections∞

n=1

An = { x : x ∈ An for all n}

The elements of are nothing but subsets of Ω, and are known as -measurable sets. An event A can be considered both as a subset of Ω, or as a point in . The pair (Ω, ) is called a measurable space.Probability Spaces

Probability spaces enable us to mathematically model phenomena consisting of randomly occurring states. A probabil-

ity space, also known as a probability triple, consists of three parts: a sample space, a set of events , and an assignment of probabilities to the events. The probabilistic models of insider trading that we will study in this paper will have “events”

such as the past and current prices of the asset, and will assign a probability measure to each of the events, or collection of

events. The formal definition of a probability space is given below:

Definition 2.1.4

A probability space is a triple (Ω, , P ) where Ω is a set (the sample space), is a σ-algebra on Ω and P , the probabil-ity measure, is a mapping from

into [0, 1] such that

P (Ω) = 1and if ( An)n=1

∞ is any sequence of pairwise disjoint events in , then

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P ∞

n=1

An = n = 1

∞ P ( An) (2.1)

Measurability, Random Variables, and Independence2.2

Measurability and Random Variables

There are some specific σ-algebras on the real line, one of which is the Borel σ-algebra. Being familiar with the idea ofBorel algebras will help us understand the concept of measurability that we will require in our study.

Definition 2.2.1

The Borel σ-algebra on , ℬ(), is the σ-algebra generated by, i.e. the smallest σ-algebra containing, the openintervals in . Subsets of which belong to ℬ() are called Borel sets.

The Borel σ-algebra on [a, b], a closed interval in , is defined as B ⋂ [a, b] : B ∈ℬ()

The Borel algebra can also be generated by the closed intervals. The Borel σ-algebra is also the smallest σ-algebracontaining all intervals. We can understand this better through the following example:

Dineen [5]Example 2.2.2

Every countable subset A = ( xn)n=1∞ of is a Borel set. It is evident that every one point subset { x} of is a closed

interval, { x} = [ x, x], and hence,

A = n=1∞

[ xn, xn] ∈ℱ (C ) = ℬ()

In advanced analysis texts it is shown that there exist subsets of which are not Borel subsets.Definition 2.2.3

If (Ω, , P ) is a probability space, a mapping X : Ω→ is called a random variable. X is called -measurable if X -1( B) ∈ for every Borel subset B ⊂.Definition 2.2.4

The σ-algebra generated by a random variable X is the smallest σ-algebra with respect to which X is measurable.This

σ-algebra is usually denoted by

σ( X ). It can be shown that

σ( X ) consists of all sets of the form X -1( B), where B is

a Borel set.

Definition 2.2.5

Let X be a random variable on Ω. Then X defines an algebra on Ω whose elements are the inverse images of the subsetsof im( X ), that is,

X = {{ X ∈ A} A ⊆ im( X )}

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Note:

If is a random variable, we denote the inverse image of a set B under by -1( B). We also use the following notation

occas ona y for or nary functions:

{ X ∈ B}For a spec c state x, instead of wr t ng -1( x). we denote it y:

{ X = x}

Further, the set of distinct values { x1, ..., xn} (not necessarily a finite set) of is called the image of and is denoted by

im( X ).

A special and useful type of random variable is the indicator function.

Example 2.2.6

Let B

⊆Ω. Then, the indicator function is such that

I B = 1 ω ∈ B0 ω ∉ B

In particular, if B ∈ where is a σ-algebra, then X = I B is measurable with respect to .Indicator functions and their propert es will be useful in our stu y. Below are some characterisitics of the indicator:

1) I Bc + I B = 1

2) I A⋃ B = I A + I B - I A⋂ B3) I A⋂ B = I A I B

Proposition 2.2.7

If c is a real number and X and Y are -measurable random variables defined on Ω, then X + Y , X - Y , X ·Y , and c X are -measurable.Proof

We will only prove this proposition for the case of the sum of two random variables. The other proofs follow in similar

fashion.

To do this, we need a further simplification. It turns out that the measurability condition X -1( B) ∈ need only bechecked on a system of generating Borel sets, such as rays of the form (-∞, q). So we will prove:

{ω∈Ω : X (ω) + Y (ω) < q} = ( X + Y )-1 ((-∞, q)) ∈for any q

∈.

If ω∈Ω and X (ω) + Y (ω) < q, then X (ω) - q < -Y (ω) and there exists a rational number p such that X (ω) - q


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Hence,

{ω∈Ω : X (ω) + Y (ω) < q} = p∈

{ω : X (ω) < q + p} ⋂ {ω : Y (ω) < - p}

We know that X and Y are measurable and is countable. From the definition of σ-algebras, it also follows that for aset An, if ( An)n=1∞ is a sequence in , then n=1∞ An ∈. Thus, we have,

{ω∈Ω : X (ω) + Y (ω) < q} ∈Hence, we have proved that ( X + Y )-1 ((-∞, q)) ∈.

The idea of random variables is vital to the proposed study of insider trading. In our study, we will be using the concept

of random variables for values of assets and the quantities of the asset traded by the traders in the market. We will try to

predict the values of their outcomes based on other random variables that they are conditioned on.

We will find that the results discussed in Proposition 2.2.7 are extremely useful in our model of insider trading as we

will have a number of cases where we will have to deal with sums of random variables. For example, we will be concernedwith the sums of the quantities traded by different kinds of traders in the market.

Independence

An important concept for our study is the independence of events, or random variables. As mentioned, random vari-

ables are key to our study and we will have situations where the outcome of one event is not affected by the other. This

information is useful not only for understanding how insider trading works, but also helps greatly in simplifying the calcula-

tions by enabling us to use the properties that independent events bring with them.

Definition 2.2.8

If (Ω, , P ) is a probability space, the events A ∈ and B ∈ of (Ω, P ) are independent if P A ⋂ B = P ( A) · P ( B) (2.2)

The events A1, ..., Am are independent if for any sub collection Ai1 , ..., Aim of these events

P Ai1 ⋂ ... ⋂ Aim = P ( Ai1 ) ⋯ P ( Aim )(2.3)

The following properties are straightforward to prove from the definition of independence.

Proposition 2.2.9

Let (Ω, , P ) denote a probability space and let A, B, and C belong to Part 1:If P ( A) > 0, then A and B are independent if and only if P ( B A) = P ( B)

Part 2:

If A and B are independent, then A and Bc are independent.

Part 3:

If and B are disjoint and both are independent of C , then ⋃ B is independent of .

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Independence of events can also be extended to collections of events. Independence with respect to σ-algebras may bedefined as follows:

Definition 2.2.10

Let (Ω, , P ) denote a probability space and let 1 and 2 denote σ-algebras on Ω with 1 ⊂ and 2 ⊂. We saythat 1 and 2 are n epen ent σ-a ge ras if every ∈1 is n epen ent of every B ∈2.

The o ow ng examp e with a small samp e space will e p us understand n epen ence better:

Example 2.2.11

Let Ω = {1, 2, 3, 4, 5}.Let A = {1, 2}, B = {3, 4}, C = {5}, D = {2, 3}, E = {1, 4, 5} be subsets of Ω.Let there be two σ-algebras 1 = ∅, A, B, C , Ω and 2 = ∅, D, E , Ω contained in Ω.Let,

P ({1}) = P ({4}) =1

3

P ({2}) = P ({3}) =1

6

({5}) =1

12

We will show that the σ-algebras 1 and 2 are independent of each other. By Proposition 2.2.9 Part 2, we know thatif A and D are independent, then A and Dc = E are also independent. Further, by Proposition 2.2.9 Part 3, if A and B are

disjoint and both are independent of D, then A ⋃ B is independent of D. Using these two parts of Proposition 2.2.9, it issufficient to show that A and B are independent of D.

First, we show that A and D are independent. We have,

P ( A) = P ({1, 2}) =1

3+

1

6=

1

2

P ( D) = P ({2, 3}) =1

6+

1

6=

1

3

For and D to be independent, we need to show that P A ⋂ D = P ( A) · P ( D). P A ⋂ D = P {1, 2} ⋂ {2, 3}

= P ({2})

=1

6= P ( A) · P ( D)

For B and D to be independent, we need to show that P B ⋂ D = P ( B) · P ( D). So, P ( B) · P ( D) = P {3, 4} ⋂ {2, 3}

= P ({3})

=1

6

=1

2·1

3

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= P ( B) · P ( D)

Hence, we have shown that σ-algebras 1 and 2 are independent.F na y, σ-a ge ras generate y random variables hold the n epen ence factor in the o ow ng manner:

Definition 2.2.12

We say that random variables and on the probability space (Ω, , P ) are independent if the σ-algebras theygenerate, X and Y , are independent.

If and are independent random variables on (Ω, , P ) and and are events, then -1( A) and Y -1( B) are indepen-dent events. Hence,

P X -1( A) ⋂ Y -1( B) = P X -1( A) · P Y -1( B) (2.4)Proposition 2.2.13

If X is independent of , i.e., σ ( X ) is independent of , then X and I A are independent random variables, where A ∈.Proof

We are given that X is independent of , and that A ∈. In order to show that X and I A are independent randomvariables, we will have to show that,

P [ X (ω) ∈ B, I A(ω) ∈ {1}] = P [ X (ω) ∈ B] · P [ I A(ω) ∈ {1}]and,

P [ X (ω) ∈ B, I A(ω) ∈ {0}] = P [ X (ω) ∈ B] · P [ I A(ω) ∈ {0}]Therefore, we have,

P [ X (ω) ∈ B, I A (ω) ∈ {1}] = P [ X (ω) ∈ B, ω ∈ A]= P X (ω) ∈ B ⋂ A= P [ X (ω) ∈ B] · P [ A]= P [ X (ω) ∈ B] · P [ I A(ω) ∈ {1}]

and,

P [ X (ω) ∈ B, I A (ω) ∈ {0}] = P [ X (ω) ∈ B, ω ∉ A]= P X (ω) ∈ B ⋂ Ac= P [ X (ω) ∈ B] · P [ Ac]= P [ X (ω) ∈ B] · P [ I A(ω) ∈ {0}]

Conditional Expectation2.3

Conditional expectation is probably the most important concept of probability theory that will use in our study. The

random variables that we will use for the prices of assets and the quantities of the asset traded certainly depend on some

market conditions. Conditional expectation allows us to carry out computations taking into account the known information

and the factors on which the random variables are conditioned on. For example, the price of the asset in the next time

interval would be conditioned on the past and current prices of the asset. We need conditional expectation to probabilisti-

ca y forecast the pr ce of the asset in the next time per o g ven the past and current pr ces.

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Basic Definitions

Definition 2.3.1

Let be a random variable on Ω, ℱ , P and let be a σ-algebra on Ω such that ⊂ℱ . The conditional expectationY

= E

[ X

] is an -measurable random variable such that, E [Y · I A] = E [ X · I A], (2.5)

i.e.,

E [ E [ X ] · I A]] = E [ X · I A] (2.6)for all A ∈. Moreover, it can be shown that the defining equation (2.5) is equivalent to the condition that for all -measurable random variables Z ,

E [Y · Z ] = E [ X · Z ] (2.7)

When is a random variable we let E [ X Y ] = E [ X Y ]. We call E [ X ] and E [ X Y ] the conditional expectations ofg ven

and respect ve y.

Conditional expectation also carries some important properties that are vital in the computations that we will have to

undertake in the following chapters. These properties are key to the profit maximization agenda of the insider and their

importance cannot be stressed enough.

Properties

Proposition 2.3.2

Let and Y denote random variables on the probability space Ω, ℱ , P and let and ℋ denote σ-algebras on Ω where ℋ⊂⊂ℱ .

Part 1: Linearity of Conditional Expectation

E [a X + b Y ] = a E [ X ] + b E [Y ]Part 2: Taking out what is known

If Y is -measurable, then E [ X ·Y ] = Y · E [ X ]

Part 3: Independence drops out

E [ X ] = E [ X ]Part 4: Tower Law

E

E

( X

)

ℋ= E

X

ℋProof

Part 1:

We will show that a E [ X ] + b E [Y ] satisfies the defining conditions for E [a X + b Y ]. We know that E [a X ] and E [b Y ] are both -measurable. So, if A ∈, then

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E [(a E [ X ] + b E [Y ]) I A] = E [(a E [ X ]) I A + (b E [Y ]) I A]= E [(a E [ X ]) I A] + E [(b E [Y ]) I A]= a E [ E [ X ] · I A] + b E [ E [Y ] · I A]= a E [ X · I A] + b E [Y · I A]

= E [a X · I A] + E [b Y · I A]

= E [(a X + b Y ) I A]

Hence, the near ty property for conditional expectat on is satisfied.

Part 2:

We are seeking to show that Y · E [ X ] is the proper conditional expectation of E [ X ·Y ]. It suffices to show that forevery -measurable random variable Z ,

E [Y · E [ X ] · Z ] = E [ X Y · Z ]Y · Z is -measurable, thus, us ng Definition 2.3.1,

E [Y · E [ X ] · Z ] = E [ X Y · Z ]Hence, E [ X ·Y ] = Y · E [ X ]

Part 3:

If ∈, then and I A are n epen ent and hence, E [ X · I A] = E [ X ] E [ I A]

= E [ E [ X ] · I A]

Hence, E [ X ] = E [ X ].Part 4:

We are g ven that ℋ ⊂. Us ng Definition 2.3.1, we get for A ∈ℋ and ℋ ⊂ E [ E [( X )] · I A] = E [ X · I A]

= E E X ℋ · I AHence, we have E E [ X ] ℋ = E X ℋ.

Bivariate Normal Distribution2.4

We will also come across the bivariate normal distribution in modeling two key random variables in the study that will

be discussed in later chapters. Having sound knowledge of the density functions and the parameters carried by the random

variables will be extreme y beneficial.

Definition 2.4.1

The pair of random variables ( X , Y ) is said to have the bivariate normal distribution with parameters μ x, μ y, σ x2, σ y2, and ρ if its joint density function is

f ( x, y) =1

2 π σ x σ y 1 - ρ2·Exp

-1

2 1 - ρ2 x - μ x

σ x2- 2 ρ ( x - μ x) ( y - μ y)σ x σ y +

( y - μ y)2σ y2

where, ~ N μ x, σ x2 and Y ~ N μ y, σ y2 (Hastings [4]).

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Theorem 2.4.2

Let ( X , Y ) have the bivariate normal density with parameters μ x, μ y, σ x2, σ y2, and ρ. Then,1) the marginal distribution of is N μ x, σ x2;2) the marginal distribution of Y is N

μ y,

σ y2

;

3) ρ is the correlation between X and Y ;4) if = 0, then X and Y are independent;

5) the conditional density of Y given X = x is normal with conditional mean and variance

μ y x = μ y + ρ σ yσ x ( x - μ x) (2.8)

σ y x2 = σ y21 - ρ2 (2.9)

The results for conditional mean and variance are acquired by completing the square on in the exponent of the bivari-

ate normal density function. (Hastings [4]).

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Chapter 3

Model

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Description3.1

In the two models that will be discussed in this paper, we mainly have one risky asset that is traded in the market. In

essence, a risky asset can be anything that has some monetary value at some point in time, with some degree of volatility. Its

value may change for various reasons depending on the conditions in the market. The most common example of a riskyasset would be a share of stock in a company. The value of a stock changes based on the valuation of the company. It may

experience a large change due to any mergers or acquisitions the company may be involved in, or simply due to daily

functioning of the business, based on the costs it incurs, and the revenues it earns. The volatility of the asset is what makes it

interesting, and allows the owner to make profits. Risky assets in this model must also have the flexibility to be traded with

ease. This is because assets would have to be traded frequently, and are modeled as a sequence of many auctions where the

owners will have the opportunity to buy or sell their assets.

The purpose of studying and preparing models on insider trading is to evaluate the ways in which the insider can take

advantage of her information, given the market conditions. Through this study we will be able to understand the extent to

which the insider can benefit from her private information. We would be able to answer whether the possibilities for an

insider to maximize profits are limitless, or if there is some sort of a cap at a maximum profit. These are interesting ques-

tions that can be answered by developing a thorough understanding of existing models, and also developing our own models

with different conditions.

Players3.2

In the two models of insider trading that we will discuss in this paper, the three distinct roles the players in the market

will have are as follows:

The Insider - The Tactician

The insider is like the protagonist of this model. The ‘insider’ is a term used to describe someone who possesses some

kind of private information about some aspect of the market, which will allow the insider to take advantage of, and maxi-

mize her profits through intelligent investment. The insider can be anyone who has a better idea of the markets, most

commonly through the knowledge acquired by being in an influential position in a financial institution, such as a high-level

emp oyee, CFO, or CEO.

The Uninformed Traders - The Noise

The uninformed traders, or, the noise traders, make their trading decisions based on intuition, personal biases, and their

prediction of what the future price of the asset might be. The noise traders are assumed to trade randomly in a normal

distribution as there is no specific pattern to their decisions. There are usually a large enough number of noise traders to

significantly affect the market prices of the asset at each auction. Due to the large number of noise traders, the market maker

can neither st ngu s the insider from the noise traders, nor set pr ces rect y to counter the insider’s moves.

The Market Maker - The Closer

The market maker is the one who acts after the insider and the noise traders have issued their orders. The trade is

consummated by the market maker based on the respective quantities traded by the insider and the noise traders. The role of

the market maker in this model is to set prices efficiently at each auction to bring the market to equilibrium. A particular

trading platform may have one market maker or it may have more than one market maker. In the case that there are multiple

market makers, the market makers would compe e among themselves to balance the market at a mar et-c ear ng pr ce, or

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more technically, the equilibrium price. In a real sense, the brokers in a financial market act as the market makers. It is

important to understand that market makers do not have to operate with the purpose of making a profit through their trading.

Their job is to close the market. They may derive their income from external sources such as receiving a commission from

the trades or through fixed salaries from the owners. Their monetary reward would not be factored into our model as one of

the market incentives.

The Trading Process3.3

Trading may be described as the process in which the investors of the asset have the opportunity to buy or sell quanti-

ties of the asset at prices set by the market maker. Trading takes place during discrete time intervals at an event called an

auction. At each auction, tra ng takes p ace in two s eps. The two s eps of the tra ng process are as follows:

Step One - Choosing Quantities

The first step involves the insider and the noise traders simultaneously choosing the quantities of the asset they wish to

trade. In essence, they both place their market orders. The insider makes her decision based on her private information about

the asset, the past prices of the asset, and the quantities of the asset she traded in previous auctions. The private information

about the asset may be available in many ways. It could either be the liquidation value of the price of the asset at final time,

or it could be the distribution of the final value of the asset. We will look at the specific models later in this paper.

The noise traders trade random quantities of the asset at the auctions. The quantities traded by the noise traders are

independent of the past quantities traded by the insider and the noise traders. It is important to know that neither the insider

nor the noise traders know the current market price at which the trading will take place. They only choose the quantities they

want to trade based on the information they have at their own disposal.

Step Two - Choosing Prices

In this step, the market maker comes in to set a market-clearing price at which the trading will take place. Since the

market maker sets the price after the insider and the noise traders choose their quantities to be traded, the information

available to the market makers consists of current and past quantities of the asset traded collectively by the insider and noise

traders. In addition, the market maker would also have the past prices of the asset at her disposal as it is public information

by now. The market maker would not be able to differentiate between the quantities traded by the insider and those traded

by the noise traders. Kyle[1] states that, “noise trading provides camouflage which conceals his [the insider’s] trading from

market makers”. In our model, the market maker only requires the aggregate quantities to set an equilibrium price for the

entire market.

It may be noted that the asset is only cashed in by the insider and the noise traders at final time. In all other time inter-

vals in which trading place, the traded quantities are logged in and record is kept of what quantities of the asset each trader

owns. At the end of the last time interval, the insider gets the value of the r s y asset in money.

Notation and Assumptions3.4

Notation

The notations used in this model are given below to facilitate smooth flow of information in the rest of the paper. This

section can also act as a legend for reference.

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The auctions take place during the individual time intervals [0, 1], [1, 2], …, [ N - 1, N ]. The end of each time interval

is when the market maker sets the tra ng pr ces.

A p ctor a representat on of the timeline and the activities that p ace in each time interval is g ven below:

Period 1 Period 2 Period N

0 1 N-1 N

Insider and noise

traders place their

orders

Market maker sets the

price and consummates

the trade

fig (1.)

Here, n refers to any auction in the timeline in genera . N refers to the final time.

V n refers to the value of the underlying risky asset at time n.

n refers to the price at which trading takes place at time n, or the price set by the market maker at time n. 0 takes a

non-random value p0 such that P 0 = p0.

n refers to the quant ty of the r s y asset traded y the insider ur ng time per o n.

Y n refers to the quant ty of the r s y asset traded y all the noise traders ur ng time per o n.

W n refers to the cumulative expecte pro t of the insider from time n onward that the insider would like to maximize.

πn refers to the total profits earned by the insider from time n onward .Σn-1 refers to the conditional variance of the normally distributed random price variable for the underlying risky asset

g ven the prev ous order information at time n.

σ y2 refers to the variance of the normally distributed random variable for the quantity traded by all the noise traders.In addition to the above letters used to denote key features of the model, we will use the letters from the Greek alphabet

αn, βn, δn, and λn to denote constants that will show up in our calculations. The subscripts also apply to these letters the

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same way as for other letters. These letters from the Greek alphabet with subscripts must be comprehended as constants

appropr ate to that spec c time per o .

Liquidity

The concept of liquidity involves the ease with which a financial asset can be converted to money. In our model,

liquidity refers to the ease of buying and selling assets in the market at short notices. Black [9] explains the concept and itsimportance aptly. He gives some useful insights into market liquidity by pointing that,

The market for a stock is liquid if the following conditions hold:

1. There are always bid and asked prices for the investor who wants to buy or sell small amounts of stock immediately.

2. The difference between the bid and asked prices is always small.

3. An investor who is buying or selling a large amount of stock, in the absence of special information, can expect to do

so over a long period of time at a price not very different, on average, from the current market price.

4. An investor can buy or sell a large block of stock immediately, but at a premium or discount that depends on the size

of the block. The larger the block, the larger the premium or discount.

The liquidation value of the asset is the monetary value of the asset at a given point in time. We assume that the prices

set by the market maker are equal to the liquidation value of the asset. We have already discussed the information the

market maker takes into consideration to set pr ces.

Randomness

In both of our models that we are going to study, there are two essential random variables that will be the driving forces

of the study.

The quantity traded by the noise traders in the nth period Y n is assumed to be a random variable that carries a normal

distribution with mean 0 and variance

σ y2. Since trading involves both buying and selling of the risky asset at each auction,

the mean quantity traded at each auction is 0. There would be some variability in the quantities traded by the noise traders,

and therefore, we assign a variance σ y2 to the distribution. Thus,Y n ~ N 0, σ y2 (3.1)

The most important aspect of this model is the intrinsic value of the risky asset at final time , denoted by V N . The

insider has some information about this value and makes her decisions based on that information, and the existing informa-

tion about the past prices of the asset. The kind of information that the insider has about the asset will be discussed later in

the section. The value of the risky asset at final time is assumed to be a random variable that carries a normal distribution

with mean p0 and variance Σ0. As per condition 3 on market liquidity discussed earlier, due to the large number of buyersand sellers that includes the insider and the noise traders, on average, one can expect the final price of the asset to be the

same as the current price. There would also be some variability in the risky asset attaining a certain value at final time .

Thus,

V N ~ N ( p0, Σ0) (3.2)Linear Model

In addition to assuming that the random variables in the study are all normally distributed, we also make the assumption

that the model displays a linear structure. This means that the prices and quantities are simply linear functions of the known

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values of the prices and quantities.

The prices of the asset at different time periods exhibit the following relationships:

P 1 = P 0 +

λ1( X 1 + Y 1)

2 = P 1 + λ2( X 2 + Y 2).

.

.

n = P n-1 + λn( X n + Y n).

.

.

N -1 = P N -2 + λ N -1( X N -1 + Y N -1) N = P N -1 + λ N ( X N + Y N )

As it can be seen, the prices follow a tractable linear structure in the sense that the price of the asset in the current time

period set by the market, is based on the price of the asset in the previous time period, and the aggregate quantities traded by

the insider and the noise traders. The λn coefficients in each of the linear functions of the prices are constants whose valuescan be derived mat emat ca y when we exercise the models in the o ow ng c apters.

The assumption that the random variables are normally distributed also extends to the linear structure of the quantities

traded by the insider. The quantity traded by the insider at time n, X n, is a function of the intrinsic value of the asset that only

the insider knows and the current price of the asset at which the insider undertook the previous trade.

The quant t es traded y the insider at different time per o s exhibit the o ow ng linear re at ons ps:

1 = β1 (V N - P 0)2 = β2 (V N - P 1).

.

.

n = βn (V N - P n-1).

.

.

N -1 = β N -1 (V N - P N -2) N = β N (V N - P N -1)

The n coefficients in the linear expressions of the quantities are constants that become part of the calculations when the

insider is trying to maximize her expected profit. Their values will be found mathematically when we exercise the insider

trading models in detail in the following chapters. It may be noted that the n’s will always be positive coefficients. This is

because, if V N - P n-1 > 0, the insider would buy quantities of the asset to to take advantage of the low price, making n

positive. Whereas, in the case that V N - P n-1 < 0, the insider would sell quantities of the asset to optimize profit, making n

negat ve. The express on n = βn(V N - P n-1) will on y be true for both cases if n is pos t ve.

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Market Efficiency

Market efficiency is a concept that mainly concerns the market maker as she is the one that sets the market clearing

price on the risky asset at each auction. As discussed before, the market maker sets the price after the insider and the noise

traders have made their decisions on what quantities of the risky asset to trade. Therefore, the market maker sets the price

based on the aggregate quantities of the risky asset traded in the current auction, as well as those in the past auctions. Thus,the market clearing price would be the expectation of the value of the risky asset, conditioned on the aggregate current and

past quantities of the risky asset traded by the insider and the noise traders.

The equ r um pr ce for an auction time n can be calculated as follows:

P n = E [V N ( X n + Y n), ( X n-1, Y n-1), ..., ( X 1 + Y 1)] (3.3)

Model Types3.5

In this paper, we will mainly discuss two kinds of models. One of them is a seminal work on insider trading undertaken by Albert Kyle [1], and the other one is an extension of that model with a few variations. The second model discussed in this

paper is a novel and or g na method of un erstan ng the concept of insider tra ng better.

Kyle’s Model

In the model that Kyle [1] discusses in his paper, he makes the assumption that the insider knows exactly what value the

risky asset will achieve. All the decisions the insider makes thereon are based on the assumption that the exact value of the

risky asset at time is known by the insider. This means that until time is reached, only the insider knows the final value

of the underlying asset before the intrinsic value of the asset is revealed to everyone. In the next chapter we will study the

single period case N = 1 first before proceeding to the multiple period case.

Martingale Model

There is one main variation in our extension of the model described by Kyle. Unlike in Kyle’s model, the insider does

not know the exact value of the intrinsic value of the risky asset at final time N . However, the insider has a better idea of the

normal distribution carried by the random variable of the value of the risky asset at final time N . This distribution is differ-

ent from the known distribution of the asset. In other words, it is different from V N ~ N ( P 0, Σ0). The distribution that theinsider would know may have a different mean, and more importantly, a smaller variance. This puts the insider in a better

position to conjecture what values the asset may achieve at final time N . This information is more valuable than what is

a rea y available in the market.

In essence, we are assuming that there is a martingale process for the random variables V 0, V 1, ... , V N for the values of

the risky asset at different time periods. As we may recall, a martingale process is a stochastic process in which the condi-

tional expectation of the next value, given the current and preceding values, is the current value. Symbolically, a martingale

is a stochastic process U 1, U 2, ..., U n, ..., such that,

E [U n+1 U 1, ... , U n] = U n (3.4)

This assumption of the martingale process models the increments of the intrinsic value of the asset, and allows the

insider to observe the value of the risky asset V n in period n. The increments of the random variables V 0, V 1, ... , V N are

independent and identically distributed (i.i.d) with mean 0 and variance σv2. Therefore,

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(V 1 - V 0), (V 2 - V 1), ... , (V N - V N -1) ~ N 0, σv2 (3.5)

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Chapter 4

Single-Period Model

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Single-Period Model4.1

The single period model will be helpful in understanding how trading works in one time period. It will give us an

insight into how an insider makes her decisions. The rationale behind this model is that the insider and noise traders only

have one opportunity to trade quantities of the asset, and will help us understand how they can make the most of it.

The single period model is common to both types of model that we will study later in this paper. The way the insider

makes her decision on what quantities to trade is the same in the first time interval in both the models. This is because in

Kyle’s model, the insider knows what the intrinsic value of the asset is going to be at final time and in the novel extension to

Kyle’s model in which the intrinsic value follows a martingale process, the insider can observe the next value of the asset.

Essentially, in a single-step model, the private information that the insider possesses is the same for both models.

From now on, for ease of comprehension, we will only refer to the model in this section as the single period version of

Ky e’s model.

In the single-period model with final time N = 1, we will denote V 1 as the intrinsic value of the asset that only the

insider knows. The initial price, or the current price of the asset is P 0 and price of the asset set by the market maker at the

end of the time interval is P 1. Therefore, given that the quantity traded by the insider is 1, the profit (or loss) that the

insider would make when the asset is cashed in at the end of the time interval would be,

π1 = X 1(V 1 - P 1) (4.1)As mentioned before, the intrinsic value of the asset at final time is modeled as a random variable carrying a normal

distribution with mean 0 and variance Σ0. So,V 1 ~ N ( P 0, Σ0) (4.2)

The quantity traded by the noise traders Y 1 is a random variable with a normal distribution with mean 0 and varianceσ y2. So,

Y 1 ~ N 0, σ y2 (4.3)The price of the asset at final time P 1 is a linear function of the initial price of the asset 0 and the aggregate quantity

traded by the insider and the noise traders. So,

P 1 = P 0 + λ1( X 1 + Y 1) (4.4)A key aspect of the model would be to determine the coefficient λ1.Similarly, the insider decides her quantity of the asset to trade based on her private information, which is the intrinsic

value of the asset at final time, and the initial price of the asset. So,

X 1 = β1(V 1 - P 0) (4.5)Market Efficiency

On the market maker’s side of things, the market maker is challenged with setting the market clearing price to ensure

efficiency of trading of the assets. As discussed before, the price that the market maker sets for the asset final time N = 1 is

conditional on the aggregate quantities traded by the insider and the noise traders. So,

P 1 = E [V 1 ( X 1 + Y 1)] (4.6)

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Expected Profit:

Given the information the insider has, it is natural that the insider would like to maximize her profit. This aspect is

characterized by the expected profit that the insider would like to maximize with respect to the quantity of the asset to be

traded. The expected profit W 1 at the end of the time interval is calculated as the expectation of the profit π1 conditional onthe initial price of the asset, final price of the asset, and the intrinsic value of the asset that the insider knows. P 0 being the

initial value is a known constant p0. So,

W 1 = max X 1

E [π1 P 0, V 1] (4.7)The expanded form would be:

W 1 = max X 1

E [ X 1(V 1 - P 1) P 0, V 1] (4.8)

Substituting the linear expression for P 1 and expanding out gives,

W 1 = max X 1

E [ X 1(V 1 - ( P 0 + λ1( X 1 + Y 1))) P 0, V 1]= max

X 1 E V 1 X 1 - P 0 X 1 - λ1 X 12 - λ1 Y 1 X 1 P 0, V 1

Using linearity of conditional expectation as per Proposition 2.3.2 Part 1, we get,

W 1 = max X 1

E [V 1 X 1 P 0, V 1] - E [ P 0 X 1 P 0, V 1] - E λ1 X 12 P 0, V 1 - E [λ1 Y 1 X 1 P 0, V 1] (4.9)Since the expectations are conditioned on P 0 and V 1, as per Proposition 2.3.2 Part 2, they would be treated as constants in

the calculation of expected profit. Further, since the expected profit is being maximized with respect to the quantity of the

asset traded by the insider, X 1 would also be treated as a constant in the expression. So, we have,

W 1 = max X 1

V 1 X 1 - P 0 X 1 - λ1 X 12 - λ1 E [Y 1] X 1 (4.10)We know that Y 1 ~ N 0, σ y2. So, E [Y 1] = 0. Therefore,

W 1 = max X 1 V 1 X 1 - P 0 X 1 - λ1 X 12 (4.11)

Since we would like to maximize the expected profit with respect to X 1, we differentiate the expression with respect to X 1.

This gives,

W 1' = V 1 - P 0 - 2 λ1 X 1 (4.12)

To make sure that we are maximizing the expected profit, we will also have to check whether the parameters satisfy the

second order condition. The second order condition for the expected profit would be,

W 1' ' = -2 λ1 (4.13)

Since we are trying to find the maximum expected profit, the second order condition must be less than 0. So,

0 > -2 λ1⟹λ1 > 0

Thus, λ1 must satisfy the following second order condition for the expected profit to be the maximum:λ1 > 0 (4.14)

To find the expression for X n that would maximize the expected profit, we set the derivative equal to 0 and solve for X 1.

Doing that, we get,

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0 = V 1 - P 0 - 2 λ1 X 1⟹ X 1 = V 1 - P 0

2 λ1 (4.15)

This expression is the quantity of the asset that the insider must trade in order to maximize her profits. This brings into the

picture the coefficient 1 =1

2 λ1 that accompanies the linear function of the quantity traded by the insider. Therefore,

X 1 = β1(V 1 - P 0) (4.16)Substituting this expression for 1 in Equation (4.15) back into Equation (4.11) before differentiation,

W 1 = V 1V 1 - P 0

2 λ1 - P 0V 1 - P 0

2 λ1 - λ1V 1 - P 0

2 λ12

=V 1 - P 0

2 λ1 (V 1 - P 0) - λ1V 1 - P 0

2 λ12

=(V 1 - P 0)

2

2 λ1-

λ1

(V 1 - P 0)2

(2 λ1)2= (V 1 - P 0)

21

2 λ1 -1

4 λ1=

1

4 λ1 (V 1 - P 0)2

This gives us a closed form expression for the expected profit. This also brings into the picture another one of our coeffi-

cients α1. If we let α1 = 14 λ1 , then we get the following expression for expected profit at the end of the first time interval:

W 1 = α1(V 1 - P 0)2 (4.17)Determining Parameters

In order to know the exact values of 1 and W 1, we need to know the value of λ1. We can try to find a closed formexpression for λ1 using the market efficiency condition that the market makers use to set a market clearing price at finaltime. For market e c ency, we have,

P 1 = E [V 1 ( X 1 + Y 1)]

The right hand side of the expression is nothing but the conditional mean of the bivariate normal distribution of V 1 and

( X 1 + Y 1) from Theorem 2.4.2 Part 5. Therefore,

P 1 = E [V 1] +Cov(V 1, X 1 + Y 1)

Var ( X 1 + Y 1)(( X 1 + Y 1) - E [ X 1 + Y 1]) (4.18)

It will be easier to solve the different components of the expression separately and substitute them back into Equation(4.18). Since V 1 ~ N ( P 0, Σ0),

E [V 1] = P 0

Var (V 1) = Σ0We have Y 1 ~ N 0, σ y2 and X 1 = β1(V 1 - P 0). For the parameters of ( X 1 + Y 1),

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E [ X 1 + Y 1] = E [ β1(V 1 - P 0) + Y 1]= E [ β1(V 1 - P 0)] + E [Y 1]= β1( E [V 1] - E [ P 0]) + 0= β1( P 0 - P 0)= 0

Var ( X 1 + Y 1) = Var ( β1(V 1 - P 0) + Y 1)= β12 Var (V 1 - P 0) + Var (Y 1)= β12 Var (V 1) + Var (Y 1)= β12 Σ0 + σ y2

Now, for the covariance of V 1 and ( X 1 + Y 1), we need to know that V 1 and Y 1 are independent of each other. This is because

the noise traders choose the quantities they want to trade irrespective of what V 1 would be.

Cov(V 1, X 1 + Y 1) = Cov(V 1, X 1) + Cov(V 1, Y 1)

= Cov(V 1, β1(V 1 - P 0)) + 0= β1 Cov(V 1, V 1 - P 0)= β1 Var (V 1)= β1 Σ0

Now we have all the expressions we need to substitute into the market efficiency expression in Equation (4.18). Moving

right along,

P 1 = E [V 1] +Cov(V 1, X 1 + Y 1)

Var ( X 1 + Y 1)(( X 1 + Y 1) - E [ X 1 + Y 1])

= P 0 + β1 Σ0

β12 Σ0 + σ y2( X 1 + Y 1)

Since we have assumed the linear expression P 1 = P 0 + λ1( X 1 + Y 1) for price of the asset, we get,λ1 = β1 Σ0

β1

2

Σ0 +

σ y2 (4.19)

Substituting β1 = 12 λ1 from the expression for 1 above into Equation (4.19) and solving, we get,

λ1 = β1 Σ0 β12 Σ0 + σ y2

⟹λ1 = 1

2 λ1 Σ0 1

2 λ1 2 Σ0 + σ y2

⟹λ1 12

λ1

2

Σ0 + λ1 σ y2 =1

2

λ1

Σ0

⟹ 14 λ1 Σ0 +λ1 σ y

2 =1

2 λ1 Σ0

⟹ λ1 σ y2 =1

4 λ1 Σ0

⟹ λ12 = 14

Σ0σ y2

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⟹ λ1 = 12

Σ0σ y2

1

2

It is clear from the expression for λ1 that it is positive because Σ0 and σ y2 are variances. Thus, λ1 satisfies the second ordercondition in (4.14), and we know that the parameters will give us the maximum expected profit.

Now that we know the value of λ1, we can easily find the expressions for β1 and α1. We have, β1 = 1

2 λ1=

1

21

2 Σ0σ y2

1

2

=1

Σ0σ y2 1

2

= σ y2

Σ0

1

2

and,

α1 = 14 λ1

=1

41

2 Σ0σ y2

1

2

=1

2 Σ0σ y2 1

2

=1

2

σ y2Σ0

1

2

Thus, we have found the closed form expressions for the coefficients in our model. We found,

λ1 = 12

Σ0σ y2

1

2

(4.20)

β1 =

σ y2

Σ0

1

2

(4.21)

α1 = 12

σ y2Σ0

1

2

(4.22)

It may be observed that we have the expressions for α1, β1, and λ1 in terms of Σ0. However, the market maker sets the price of the asset at final time based on Σ1, which is the variance of the intrinsic value of the asset after the insider and thenoise traders have chosen their quantities. Σ0 is the variance of the intrinsic value of the asset given that the initial price of

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the asset is P 0. Therefore, the optimal profit will make more sense if we establish a relationship between Σ1 and Σ0. To dothis, we will use the conditional variance of the intrinsic value of the asset, given the aggregate quantity traded by insider

and noise traders.

Var (V 1 X 1 + Y 1) = Var (V 1) 1 - ρ2 (4.23)The variance of V 1 given the quantities traded would be Σ1. And we know that variance of V 1 is Σ0. Since we know thecovariance of V 1 and ( X 1 + Y 1), the calculation of 1 - ρ2 should be fairly straightforward.

1 - ρ2 = 1 - Cov(V 1, X 1 + Y 1)Var (V 1) Var ( X 1 + Y 1)

2

= 1 - β1 Σ0

Σ0β12 Σ0 + σ y2

2

= 1 - β12 Σ02

Σ0β12 Σ0 + σ y2= Σ0β1

2

Σ0 + σ y2

- β12

Σ02

Σ0β12 Σ0 + σ y2=

σ y2 β12 Σ0 + σ y2

Thus, go ng back to Equat on 4.23 , we have,

Σ1 = Σ0 σ y2

β12 Σ0 + σ y2 (4.24)

From Equation (4.21), we know that σ y2 = β12 Σ0. By substituting this expression for σ y2 in Equation (4.24),we get a powerful relationship. We get,

Σ1 = Σ0 β12 Σ0

β12 Σ0 + β12 Σ0

⟹ Σ1 = 12

Σ0 (4.25)

UsingEquation (4.25), we can redefine the expressions for λ1, β1, and α1. So,

λ1 = Σ12 σ y2

1

2

β1 = σ y2

2 Σ1

1

2

α1 = 12

σ y22 Σ1

1

2

Hence, we have derived expressions to estimate all the coefficients. This means that we are able to find exactly what

quantity of the asset the insider must trade in order to maximize her profit. We will replicate this process over multiple time

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intervals in the following chapter.

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Chapter 5

Kyle’s Multiple Period Model

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Initial Values of Parameters5.1

The single period model discussed in the previous chapter can be extended to obtain Kyle’s multiple period model. The

multiple period model allows the insider and the noise traders to trade quantities of the asset in each time interval. The

private information available to the insider is the exact value of the asset at the final time. The insider must decide at everytime interval what quantities to trade in order to figure out the best combination to maximize her profits. Just like the single

per o model, we have to make some assumpt ons before we move into the calculations.

Kyle’s model has multiple time periods with final time N . The intrinsic value of the asset is denoted by V N and the

prices of the asset set by the market maker at each time interval are denoted by P 0, P 1, ... P n, ... P N . The insider knows

exactly what V N will be, and this is the only private information that the insider has. We will also assume that at the first

time interval, the expected value of the intrinsic value of the asset is known to all and is equal to P 0. Therefore, in the first

time interval,

V N ~ N ( P 0, Σ0)The asset is only cashed in at final time and the insider trades n quantity of the asset in each time interval. The profit

the insider makes from time n onward is g ven y,

πn = k = n

N

X k (V N - P k ) (5.1)

Therefore, the total return for the insider from investing in the asset is:

π1 = n = 1

N

X n(V N - P n) (5.2)

As usual, the distribution carried by the quantities of the asset traded by the noise traders is,

Y n ~ N

0,

σ y2

(5.3)

Further, the prices set by the market maker and the quantities traded by the insider follow a linear structure and are

characterized by,

P n = P n-1 + λn( X n + Y n) X n = βn(V N - P n-1)

An important breakthrough in the model would be to find closed form expressions for the λn’s and βn’s.Market Efficiency

The market maker will set a price at each time interval which is the expected value of the final intrinsic value of the

asset conditional on the past and current aggregate quantities traded by the insider and the noise traders. Therefore,

P n = E [V N ( X 1 + Y 1), ..., ( X n + Y n)] (5.4)

There is a key insight regarding the final intrinsic value of the asset in Kyle’s multiple period model. As we get closer

to time N , the distribution of the final intrinsic value changes. This is because as we move along time periods, we acquire

more information about the asset and have a better idea of what the final intrinsic value will be. In other words, the mean

and variance of the final intrinsic value V N changes. The conditional distribution of V N for the market maker observed at the

at time n (that is, given ( X 1 + Y 1), ..., ( X n + Y n)) is,

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V N ( X 1+Y 1),...,( X n+Y n) ~ N ( P n, Σn) (5.5)

Determining Parameters and Expected Profits Recursively5.2

The calculation of expected profit in Kyle’s multiple period model is done by backward induction. We start with the

last time interval and calculate the expected profit for the N th time period and work our way backwards for expected profits

along each time interval. The method of backward induction is useful because in the last time interval, the insider has access

to maximum information in terms of the past prices of the asset. Therefore, in the last time interval, the insider knows the

parameters required to make a decision on what quantities to trade, and the profits she will earn.

It may be observed that the expected profit W n at each time interval is cumulative starting from time period n. This

means that while the expected profit in the last time interval represents the profit earned in that particular time period, the

expected profit in the ( N - 1)th time period would be the cumulative expected profit for the ( N - 1)th and the N th time period.

Subsequently, the expected profit in the first time interval is the sum of the profits of all time intervals. In essence, for a

large number of replications, the mean of the sum of expected profits across all time intervals should approach the cumula-

tive expected profit in the first time interval.

At Final Time N

The calculation for expected profit in the last time period works in a similar fashion to the single period model

described in Chapter 4. We will work out the computations with the new parameters as follows:

The expected profit W N that the insider would like to maximize would be the expected value of the profit earned in the

last time period π N conditional on the past prices of the asset, and the known intrinsic value of the asset V N . The insiderwould like to choose a quantity X N to trade that would maximize the expected profit. Therefore,

W N = max X

N

E [π N P 0, P 1, ..., P N -1, V N ]= max

X N E [ X N (V N - P N ) P 0, P 1, ..., P N -1, V N ]

(5.6)

Substituting the linear expression for P N and expanding out gives,

W N = max X N

E [ X N (V N - ( P N -1 + λ N ( X N + Y N ))) P 0, P 1, ..., P N -1, V N ]= max

X N E V N X N - P N -1 X N - λ N X N 2 - λ N Y N X N P 0, P 1, ..., P N -1, V N

Using linearity of conditional expectation as per Proposition 2.3.2 Part 1, we get,

W N = max X N

E [V N X N P 0, P 1, ..., P N -1, V N ] - E [ P N -1 X N P 0, P 1, ..., P N -1, V N ]-

λ N E

X N

2 P 0, P 1, ..., P N -1, V N

-

λ N E [Y N X N P 0, P 1, ..., P N -1, V N ]

(5.7)

Since the conditional quantities include P N -1 and V N , as per Proposition 2.3.2 Part 2, they would be treated as constants in

the calculation of expected profit. In addition, X N being the quantity traded by the insider that maximizes her expected

profit, it would also be treated as a constant before differentiation. Thus,

W N = max X N

V N X N - P N -1 X N - λ N X N 2 - λ N X N E [Y N ] (5.8)We know that Y N ~ N 0, σ y2. So, E [Y N ] = 0. Therefore,

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W N = max X N

V N X N - P N -1 X N - λ N X N 2 (5.9)D erent at ng the expecte pro t to maximize it with respect to N , we get,

W N ' = V N - P N -1 - 2 λ N X N (5.10)

To make sure that we are maximizing the expected profit, we will also have to check whether the parameters satisfy the

second order condition. The second order condition for the expecte pro t would be,

W N ' ' = -2 λ N (5.11)

Since we are try ng to find the maximum expecte pro t, the second order condition must be less than 0. So,

0 > -2 λ N ⟹λ N > 0

Thus, λ N must sat s y the o ow ng second order condition for the expecte pro t to be the maximum:λ N > 0 (5.12)

To find the expression for N that would maximize the expected profit, we set the derivative equal to 0 and solve for X N .Doing that, we get,

0 = V N - P N -1 - 2 λ N X N ⟹ X N = V N - P N -1

2 λ N (5.13)

Thus, we have obtained an expression for the quantity the insider must trade in the last time interval in order to maximize

her profits. This expression also gives us the first of our β coefficients with β N = 12 λ N . Therefore, the quantity traded by the

insider in the last time interval can be expressed compactly in the linear expression,

X N = β N (V N - P N -1) (5.14)Substituting the expression for X N in Equation (5.13) back into the expression for the expected profit before differentia-

tion in Equation (5.9),

W N = V N V N - P N -1

2 λ N - P N -1V N - P N -1

2 λ N - λ N V N - P N -1

2 λ N 2

=V N - P N -1

2 λ N (V N - P N -1) - λ N V N - P N -1

2 λ N 2

=(V N - P N -1)

2

2 λ N - λ N (V N - P N -1)

2

(2 λ N )2= (V N - P N -1)2

1

2 λ N -1

4 λ N Therefore,

W N =1

4 λ N (V N - P N -1)2

(5.15)

Thus, we have obtained a closed form expression for the expected profit of the insider in the last time interval. The expres-

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sion also introduces the first of our α coefficients with α N = 14 λ N . Therefore, the expected profit in the last time period is,

W N = α N (V N - P N -1)2 + δ N (5.16)Here, we also introduce the δ constant in the expression. Though δ N = 0 and seems unnecessary in the expression

above, it will become relevant as we try to recursively derive expressions for other time periods.

In order to finish the anchoring step in the backward induction, we need to find λ N , which in turn will give us the valuesfor α N and N . Since it depends on Σ N in the same way as the single-step derivation, we will omit the calculations. Thederivation is analogous to the one shown in the single-step model, and allows us to arrive at the following expressions for

λ N , α N , N , and also δ N . It is evident that λ N is positive because Σ N and σ y2 are variances and are therefore positive. Hence,the second order condition is also satisfied, which means that the parameters below give us the maximum expected profit.

λ N = Σ N 2 σ y2

1

2

β N =σ y2

2 Σ N

1

2

α N = 12

σ y22 Σ N

1

2

δ N = 0

General n case

Now, we will attempt to find expressions for the coefficients in other time periods as well. The quadratic expression for

cumulative expected profit can be extended to the general n case, albeit with different parameter values as per the nth time

period as opposed to those in the N th

time period. This gives us,

W n = αn(V N - P n-1)2 + δn (5.17)which is based on the underlying assumption that the quantity traded by the insider in the nth time period is,

X n = βn(V N - P n-1)Lemma 5.2.1

The conditional joint distribution of V n and ( X n + Y n) given the aggregate quantities traded in the past time periods

( X 1 + Y 1), ( X 2 + Y 2), ..., ( X n-1, Y n-1) is a bivariate normal distribution with the following parameters:

E n-1[V n] = P n-1Var n-1(V n) = Σn-1

E n-1[ X n + Y n] = 0Var n-1( X n + Y n) = βn2 Σn-1 + σ y2

Covn-1(V n, X n + Y n) = βn Σn-11 - ρn-12 = σ y

2

βn2 Σn-1 + σ y2

Proof

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We know that V N ( X 1+Y 1),...,( X n+Y n) ~ N ( P n, Σn). So, E n-1[V N ] = P n-1

Var n-1(V N ) = Σn-1As discussed earlier, X n + Y n is the aggregate quantity of the asset traded by the insider and noise traders at time n. We

have Y n ~ N 0, σ y2 and X n = βn(V N - P n-1). So, E n-1[ X n + Y n] = E n-1[ βn(V N - P n-1) + Y n]

= E n-1[ βn(V N - P n-1)] + E n-1[Y n]= βn( E n-1[V N ] - E n-1[ P n-1]) + 0= βn( P n-1 - P n-1)= 0

Var n-1( X n + Y n) = Var n-1( βn(V N - P n-1) + Y n)= βn2 Var n-1(V N - P n-1) + Var n-1(Y n)= βn2 Var n-1(V N ) + Var n-1(Y n)= βn2 Σn-1 + σ y2

For the covariance of V N and ( X n + Y n) in the nth time period (given the information at time (n - 1)), we need to know

that V N and Y n are independent of each other. This is because the noise traders choose the quantities they want to trade

irrespective of what V N would be. So,

Covn-1(V N , X n + Y n) = Covn-1(V N , X n) + Covn-1(V N , Y n)

= Covn-1(V N , βn-1(V N - P n-1)) + 0= βn Covn-1(V N , V N - P n-1)= βn Var n-1(V N )= βn Σn-1

Since we know the covariance of V N and ( X n + Y n), the expression for 1 - ρn-12 can be derived as follows:

1-

ρn-12 =

1-

Covn-1(V N , X n + Y n)

Var n-1(V N ) Var n-1( X n + Y n)

2

= 1 - βn Σn-1

Σn-1βn2 Σn-1 + σ y2

2

= 1 - βn2 Σn-12

Σn-1βn2 Σn-1 + σ y2=

Σn-1βn2 Σn-1 + σ y2 - βn2 Σn-12Σn-1βn2 Σn-1 + σ y2

=σ y2

βn2

Σn-1 + σ y2

In the previous section, we discussed the conditional distribution of V N for the market maker at time n. The market

maker knows the aggregate quantities traded in the nth time interval, i.e., the market maker knows ( X n + Y n). We may recall

that,

V N ( X 1+Y 1),...,( X n+Y n) ~ N ( P n, Σn) (5.18)Using this information and Lemma 5.2.1, we may develop the following two corollaries:

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Corollary 5.2.2

The coefficient λn that facilitates linear pricing in P n = P n-1 + λn( X n + Y n) has the expression,λn = βn Σn-1 βn2 Σn-1 + σ y2 (5.19)

Proof

From (5.18), we get the market efficiency condition such that,

P n = E [V N ( X 1 + Y 1), ..., ( X n + Y n)]

Since we have assumed that V N and Y n carry normal distributions, the expression for market efficiency is the conditional

mean of the bivariate normal distribution in Theorem 2.4.2 Part 5. Therefore,

P n = E [V N ( X 1 + Y 1), ..., ( X n + Y n)]

= E n-1[V N ] +Covn-1(V N , X n + Y n)

Var n-1( X n + Y n)(( X n + Y n) - E n-1[ X n + Y n])

(5.20)

Substituting the results from Lemma 5.2.1 into Equation (5.20),

P n = P n-1 + βn Σn-1

βn2 Σn-1 + σ y2( X n + Y n) (5.21)

Our assumption of the linear expression for P n = P n-1 + λn( X n + Y n) yields,λn = βn Σn-1 βn2 Σn-1 + σ y2

So,

Corollary 5.2.3

The relationship between the conditional variance of V N given the order information up until time n and the conditional

variance of V N given the order information up until time (n - 1) is given by,

Σn = Σn-1 σ y2

βn2 Σn-1 + σ y2 (5.22)

Proof

From (5.18), we have,

Σn = Var (V N X n + Y n)Given the normality of V N and Y n, the conditional variance as per the bivariate normal distribution in Theorem 2.4.2 Part 5

would be,

Σn = Var (V N X n + Y n)= Var n-1(V n) 1 - ρn-12 (5.23)

We know that variance of V N in the nth time interval is Σn-1. From Lemma 5.2.1, we also know 1 - ρn-12. Substituting

these in Equation (5.23), we have,

Σn = Σn-1 σ y2

βn2 Σn-1 + σ y2

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Backward Induction

Now we are ready to start the backward induction process. In order to carry out the backward induction, using the

parameters in the nth time period, we will derive the expressions for the parameters in the (n - 1)th time period.

The expected profits W n-1 include profits from time (n - 1) to . Therefore, in order to find the expression for W n-1, wehave,

n-1 = max X n-1

[ n-1 ( N - n-1) + n 0, 1, ..., n-2, N ] (5.24)

Substituting the expression for W n in Equation (5.17) into Equation (5.24), we get,

W n-1 = max X n-1

E X n-1(V N - P n-1) + αn(V N - P n-1)2 + δn P 0, P 1, ..., P n-2, V N (5.25)Following a similar method as in the computation of expected profit in the previous time interval, we substitute the linear

expression P n-1 = P n-2 + λn-1( X n-1 + Y n-1) into the expression and solve in similar fashion using the same propositions.W n-1 = max

X n-1 E X n-1 (V N - ( P n-2 + λn-1 ( X n-1 + Y n-1))) + αn(V N �

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