In this section we present some necessary definitions and sketch descriptions of probability and probability distributions that are applied in Chapters Four and Five. Much information on probability, its theory and modeling abounds in the mathematical and statistical literature. In particular, the reader may refer to Feller (1957), Ross (2000), Dineen (2005) and Shreve (2004) for details.
1.5.1 Probability spaces
Let Ω be a nonempty set and letF be a collection of subsets of Ω.
Definition 1.5.1. A σ-algebra (sometimes called a σ-field) is a collection F of subsets of Ω with the following properties:
(i)
φ,Ω∈ F; (1.5.1)
(ii) If
A∈ F, then A0 ∈ F; (1.5.2)
(iii) If
A1, A2, ...∈ F, then
∞
[
k=1
Ak,
∞
\
k=1
Ak∈ F. (1.5.3)
The points in F being subsets of Ω are called F-events orF-measurable sets. A pair (Ω,F) is called ameasurable space.
Definition 1.5.2. If F is a σ−algebra of subsets of Ω, then P : F −→ [0,1] is a probability measure if
(i)
P(φ) = 0 and P(Ω) = 1; (1.5.4)
(ii)
A1, A2, ...∈ F, then P
∞
[
k=1
Ak
!
≤
∞
X
k=1
P(Ak). (1.5.5) It therefore follows that, if A, B ∈ F, then A⊆B implies that P(A)⊆P(B).
The triple (Ω,F,P) is called a probability space. We note here that the probability space is the proper setting for mathematical theory. This means that we must firstly carefully identify an appropriate (Ω,F,P) whenever we try to solve problems.
Having defined the elements in the σ-algebra from a set-theoretic viewpoint, we now consider them as events. Associated with each event is information, which in the financial world, in- creases as time increases. A sample space Ω ={ω1, ω2, ..., ωN}is the set of all possible outcomes of some experiment, ε , while theσ-algebra F represents the events that are observed and can be recorded when the experiment is performed. In other words it is the information we receive upon performing the experiment. Thus after the experiment we can observe whether or not A = {ωj1, ωj2, ..., ωjm;j = 1,2, ..., N} ∈ F occurred. If F1 and F2 are two σ-algebras on Ω, then F1 ⊂ F2 if and only if F2 contains more information than F1.
Definition 1.5.3. Let (Ω,F) be a measurable space.
(i) A discrete filtration on (Ω,F) is an increasing sequence of σ−algebras (F)∞k=1 such that F1 ⊂ F2 ⊂ · · · ⊂ Fi ⊂ · · · ⊂ F.
(ii) A continuous filtration on (Ω,F) is a set of σ−algebras (F)t∈I,where I is an interval in R such that for all t, s∈I, t < s, we have Ft ⊂ Fs ⊂ F.
We callFk (respectivelyFt ) the history up to time k (respectively timet).
1.5.2 Random variables
Unlike the point of time of the impact on the ground of a stone dropped from certain altitude being known before execution of the experiment (Newton’s Laws), quantities of complex systems (such as stocks, commodity prices etc) are nondeterministic. However, their values may be predicted under uncertainties. Contrary to the falling stone, data which cannot be described successfully by a deterministic mechanism can be modeled by random variables.
Definition 1.5.4. Let (Ω,F,P) be a probability space. A random variable is a real-valued function X defined on Ω with the property that for every Borel subset B of R the subset of Ω given by
{X ∈ B} ={ω ∈Ω;X(ω)∈ B}
is in theσ−algebraF.
For properties of random variables the reader is referred to the references listed at the beginning of this section. A random variable X is a numerical quantity the value of which is determined by the random experiment choosing ω ∈ Ω. The properties of probability measures P(B) for every Boral subset B of R . Denoting the distribution measure of X under P by µX, we have for the set of all probabilities,
µX[a, b] =P(ωi :a≤Xω ≤b), −∞< a≤b <∞ (1.5.6) a measure that determines the distribution of X. In other words the distribution is defined by the probabilities of all events which depend upon X.
We can describe the distribution function of a random variable in terms of its cumulative distribution function (cdf)
F(x) = P(X ≤x), x∈R (1.5.7)
The F(x) is monotonically increasing and converges for x−→ −∞to 0 and for x−→ ∞ to 1.
If there is a function,p, such that the probabilities can be computed by means of an integral
P(a≤X ≤b) =
b
Z
a
p(x)dx, (1.5.8)
p is called the probability density or, simply, density of X. When the cumulative function is a primitive of p,
F(x) =P(X ≤x) =
x
Z
−∞
p(y)dy. (1.5.9)
Thus p(x) is a measure of the likelihood that X takes values close tox and the pdf f(x) = ∂
∂xF(x), i.e., φ(x)dx= Φ0(x)dx. (1.5.10) The most important family of distributions with densities is the family of normal distribution.
It is characterised by two parameters the mean, µ, and the variance,σ2.The density is given by fX x;µ, σ2
= 1
√2πσ2 exp (
−1 2
x−µ σ
2)
, (1.5.11)
for −∞< x <∞,−∞< µ <∞, σ2 ≥0,and fZ(z; 0,1) = 1
√2π exp
−z2 2
. (1.5.12)
The distribution with density (1.5.12) is called the standard normal distribution for which the mean and variance of Z are zero and one respectively and that in (1.5.11) shows that X is a normal random variable distributed as X ∼N(µ, σ2).
Closely related to the normal distribution is the log-normal distribution that is very important in modeling commodity prices. LetX be a positive random variable with natural logarithm of which, ln (X) ∼ N(µ, σ2). We say that X is log-normally distributed with parameters µ and σ2. Its cumulative distribution function follows from (1.5.9) as
F(x) =P(lnX ≤x) = Φ
lnX−µ σ
, x >0. (1.5.13) Hence
fX lnx;µ, σ2
= 1
√
2πσ2xexp (
−1 2
lnx−µ σ
2)
. (1.5.14)
Most other probability distributions especially the continuous types are either special cases or derivable from the normal distribution.
1.5.3 Moments of random variables
LetX be a random variable defined on a probability space (Ω,F,P).The first moment known as the mathematical expectation or the mean E(X) of a real random variable X is a measure for the location of the distribution ofX.IfX has a densityp(x),then its expectation is defined as
E(X) =
R∞
−∞xp(x)dx if x is continuous P∞
i=1xip(xi) if xis discrete,
(1.5.15)
where R
ΩX(ω)dP(ω) is a Lebesgue integral. A measure of the dispersion of a random variable X around its mean is given by the variance V(X) as
V(X) =
∞
Z
−∞
(x−E(X))2p(x)dx. (1.5.16)
The log-normally distributed random variable X defined in (1.5.14) has mean
E(X) = eµ+12σ2 (1.5.17)
and variance
V(X) = e2µ+σ2
eσ2 −1
. (1.5.18)
Details of other relevant definitions, theorems and axioms can be found in the references cited earlier.