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Math 53 – Financial Mathematics
February 7, 2007
A. Random Variables
Let S be a sample space. Subsets A  S are called events and have probabilities P(A).
A random variable X is a function X : S 
possible values of X; for example,
. We make probability statements about
P  X  a  means P s  S | X (s)  a .
Every random variable has a cumulative distribution function (also called a cdf,
or just a distribution function) FX :  [0,1] , defined by
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FX
FX (a)  P( X  a)
for every A 
.
Note that knowing FX allows us to compute the probability of X being in any interval.
We have:
P( X  a)
 1  FX (a)
P(a  X  b)  FX (b)  FX (a)
P( X  a)
 lim F (t )
t a
That last construction is common enough that we abbreviate it:
F (a  )  lim F (t )
t a
and now:
P(a  X )  a)
 1  FX (a  )
P ( a  X  b)
 FX (b)  FX (a  )
P( X  a)
 FX (a )  FX (a  )
etc. Effectively, FX tells us everything we need to know to compute probabilities
of events defined by X. (It doesn’t tell us much about events defined by X along
with other random variables; for that we will eventually need joint distributions.)
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If there happens to be a function f X :
 [0, ) such that
b
P(a  X  b)   f X ( x)dx
a
for every a and b, then X is called a continuous random variable and fX is
called its density function (or probability density function, or just pdf). Wherever
fX is continuous, it is equal to the derivative of FX. Note that if X is continuous,
the probability of any single value of X is zero: FX(a-) = FX(a) for every a, so
P(X=a) = 0 for every a.
If there happens to be a function pX such that
P ( a  X  b) 

a  X b
p X ( x)
for every a and b, then X is called a discrete random variable and pX is called
its probability mass function, or just pmf. Not every random variable is discrete
or continuous, but every random variable has a cdf (FX).
B. Normal random variables
A random variable X is called standard normal if it is continuous and its density
function is given by
f X ( x) 
1 
exp  x 2 
2
2 
1
fX
for every x. More generally, it is called
normal with mean  and standard deviation 
if its density function is given by
f X ( x) 
 1  x   2 
exp  
.
 2    
2


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(Summarize properties of normal random variables: If X, Y are normal, so are aX and
X + Y, hence aX + bY; say what their parameters would be.)
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C. Stochastic Processes.
A stochastic process (or just a process) is a collection of random variables  X (t ) for
certain values of t. In this class we will consider only non-negative values t ≥ 0.
The process is a continuous-time process if all non-negative values of t are
allowed; that is, if there is a random variable X(t) for every t with t ≥ 0. It is a
discrete-time process if, for example, t must be a non-negative integer; that is, if
the process consists only of the random variables X(0), X(1), X(2), X(3), …. It is
also a discrete-time process if t must be an integer multiple of some time-step ,
so that the random variables are X(0), X(), X(2), X(3), ….
The variables in a stochastic process are usually very closely related to each other.
Typically, they are measurements of some actual physical process taken at various
times t.
D. The Standard Model: The Discrete Case
The standard model of a stock price is represented by a few closely-related stochastic
processes. The model can be presented by a continuous-time or discrete-time
processes. We’ll consider the discrete-time version first.
The processes used to represent a single company’s share price are:
t = time (for now, in days)
S(t) = closing price of the stock on day t, for t = 0, 1, 2, 3, …
We take S(0) to be a known constant. (That is, it’s a random variable, but with all
of it’s probability concentrated at a single point.) Sometimes we will write it as
S0 or even s0 to emphasize that it is a known constant. Also, we assume that S(t)
is always positive (or more precisely: P( X(t) > 0 ) = 1 ).
L(t) = ln ( S(t) )
for every t
(so that also, S(t) = eL(t) for every t)
A(t) =
S (t )  S (t  1)
for every t = 1, 2, 3,…
S (t  1)
R(t) = L(t) – L(t-1) for every t = 1, 2, 3,…
(“arithmetic daily return”)
(“logarithmic daily return”)
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E. Brownian Motion and Geometric Brownian Motion
A process {L(t)} is called a Brownian motion process with parameters  and  if the
random variables R(t) defined by R(t) = L(t) – L(t-1) are independent and
identically distributed, and each R(t) is normally distributed with mean  and
standard deviation  (hence variance 2).
A process {S(t)} is called a geometric Brownian motion process ( GBM ) if it is
related to a Brownian motion process {L(t)} by L(t) = ln ( S(t) ) for all t.
Theorem. Let {L(t)} be a Brownian motion process, and let s < t. Define
R( s, t ) = L(t) – L(s).
Then the random variable R( s, t ) is normally distributed with mean (t-s) and
variance 2(t-s) (and hence standard deviation  t  s ).
Proof.
R( s, t ) = R( s+1 ) + R( s+2 ) + … + R( t ).
There are exactly (t-s) random variables on the right, and each of them is normal
with mean  and variance 2. Also, they are independent. Therefore, their sum is
also normally distributed, and has the mean and variance indicated in the theorem.
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The standard model says that the log-stock-price process L(t) is Brownian motion, with
a known constant value of L(0) = L0. Accordingly, the stock-price process itself,
S(t), is geometric Brownian motion, with known constant S(0) = S0.
F. The Continuous Case
G. Expected Values and Variances
H. Understanding the Parameters
I. Statistics of A and R
J. Statistics of L and S
K. Predictions with the Standard Model
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