Introduction to Financial Mathematics

Introduction to Financial Mathematics

Lecture Notes -- MAP 5601

Department of Mathematics

Florida State University

Fall 2003

Table of Contents

1. Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Elements of Continuous Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Lecture Notes -- MAP 5601 map5601LecNotes.tex i 8/27/2003

1. Finite Probability Spaces

The toss of a coin or the roll of a die results in a finite number of possible outcomes.

We represent these outcomes by a set of outcomes called a sample space. For a coin we

might denote this sample space by {H, T} and for the die {1, 2, 3, 4, 5, 6}. More generally

any convenient symbols may be used to represent outcomes. Along with the sample space

we also specify a probability function, or measure, of the likelihood of each outcome. If

the coin is a fair coin, then heads and tails are equally likely. If we denote the probability

measure by P, then we write P(H) = P(T) = 1

2 . Similarly, if each face of the die is equally

likely we may write P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1

6 .

Defninition 1.1. A finite probability space is a pair (

, P) where

is the sample space set

and P is a probability measure:

If

= {!1, !2, . . . , !n}, then

(i) 0 < P(!i)  1 for all i = 1, . . . , n

(ii)

n Pi=1

P(!i) = 1.

In general, given a set of A, we denote the power set of A by P(A). By definition this

is the set of all subsets of A. For example, if A = {1, 2}, then P(A) = {;, {1}, {2}, {1, 2}}.

Here, as always, ; is the empty set. By additivity, a probability measure on

extends to

P(

) if we set P(;) = 0.

Example. For the toss of a fair die, P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1

6 , while

P (toss is even) = P (toss is odd) = 1

2 and P({2, 4, 6}) = P(2)+P(4)+P(6) = 3 * 1

6 = 1

2 .

The division of {1, 2, . . . , 6} into even and odd, {2, 4, 6} and {1, 3, 5}, is an example of

a partition.

Defninition 1.2. A partition of a set

(of arbitrary cardinality) is a collection of nonempty

disjoint subsets of

whose union is

.

If the outcome of a die toss is even, then it is an element of {2, 4, 6}. In this way

partitions may provide information about outcomes.

Defninition 1.3. Let A be a partition of

. A partition B of

is a refinement of A if every

member of B is a subset of a member of A.

For example B = {{1, 2}, {3}, {4, 5, 6}} is a refinement of {{1, 2, 3}, {4, 5, 6}}. Notice

Lecture Notes -- MAP 5601 map5601LecNotes.tex 1 8/27/2003

that a refinement contains at least as much information as the original partition.

In the language of probability theory, a function on the sample space

is called a

random variable. This is because the value of such a function depends on the random

occurrence of a point of

. However, without this interpretation, a random variable is just

a function.

Given a finite probability space (

, P) and the real-valued random variable X :

! IR

we define the expected value of X or expectation of X to be the weighted probability of

its values.

Definition 1.4. The expectation, E(X), of the random variable X :

! IR is by definition

E(X) =

n Xi=1

X(!i)P(!i)

where

= {!1, !2, . . . , !n}.

We see in this definition an immediate utility of the property

n Pi=1

P(!i) = 1. If X is

identically constant, say X = C, then E(X) = C.

When a partition of

is given, giving more informaiton in general than just

, we

define a conditional expectation.

Definition 1.5. Given a finite probability space (

, P) and a partition of

, A,