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MATH4091/7091: Financial calculus

Probability Background II

Outline

• Information and σ-algebra
• Independence
• Conditional exception

Motivation

A goal in this course is to price financial derivatives via the risk-neutral probability measure which

typically involves the following abstract pricing formula

price at time t = e−r(T −t)EQ [payoﬀ at time T |Ft] .

Here, the Q is the risk-neutral measure and Ft is some σ-algebra at time t.

In the above formula:

Ft: σ-algebra represents the market information available up to time t.

EQ [payoﬀ at time T |Ft]: conditional expectation given Ft.

We will focus on how σ-algebras can be used model market σ-algebra.

A coin toss example

As an illustrative example, we consider tossing a coins three times.The sample space Ω contains

eight possible outcomes

Ω  = {HHH,HHT,...,TTT}.

If we are told the outcome of the first coin toss only, then the sets

AH = {HHH,HHT,HTH,HTT},              AT = {THH,THT,TTH,TTT}

are resolved by the information. That is, for each of these two sets, once we are told the first coin toss, we know if the true ω is a member. The empty set and Ω are always resolved because, without any information, we know that ω / and ω Ω.

The four sets that are resolved by the first coin toss form the σ-algebra

F1 = {,Ω,AH,AT}.

A coin toss example (cont.)

If we are told the first two coin tosses, the following are resolved

AHH = {HHH, HHT }, AHT = {HT H, HT T }, ATH = {THH,THT}, ATT = {TTH,TTT}.

Of course, all the sets obtained by taking complement union of the above sets are also resolved.

Hence, we have 16 resolved sets which form the σ-algebra

If we are told all three coin tosses, we know the true ω, so every subset of Ω is resolved. They

constitute the σ-algebra

F3 = The set of all subsets of Ω.

Filtration (continuous time)

Let Ω be a non-empty set, and let T be a fixed positive number and assume that for each t [0, T ] there is a σ-algebra Ft. Furthermore, assume that if s t, then Fs Ft. Then the collection of σ-algebras {Ft}t[0,T ] is called a filtration.

Independence of events

Let (Ω, F, P) be a probability space and A, B F be events.

The events A and B are independent if

P(A B) = P(A) × P(B).

When P(B) > 0, the conditional probability of A given B is

P(A|B)=P(A B)/P(B)

Exercise: it can be shown that A and B are independent if and only if P(A|B) = P(A).

Independence of random variables

Two random variables Y : Ω R and Z : Ω R are independent if σ(Y ) and σ(Z) are independent.

A random variable Y : Ω R is independent of a sub-σ-algebra

G F if σ(Y ) is independent of G.

Exercise: Let X, Y be two independent random variables, and let

f, g : RR be two Borel-measurable functions. Then f (X) and g(Y are independent random variables.

Why conditional expectation?

We consider a random variable X defined on a probability (Ω, F, P) and a sub-σ algebra G of F. (Note that G is represents an information set.)

If X is independent of G, then the information in G provides no help in determining the value of X.

If X is G-measurable, then the information in G is suﬃcient to determine the value of X.

In the intermediate case, the can use the information in G to estimate, but not precisely evaluate X.

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