Motivation
A goal in this course is to price financial derivatives via the
typically involves the following
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
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
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
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.
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 : R → R be two Borel-measurable functions. Then f (X) and g(Y )
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.
MATH4091/7091: Financial calculus Probability background II,probability distribution,probability
formula assignment,probability explained,probability function,financial assignment help calculus
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