# Statistics in Finance ```Probability
702341 QA in Finance/ Ch 3 Probability in Finance
Probability
 Probability is a measure of the possibility of an event
happening
 Measure on a scale between zero and one
 Probability has a substantial role to play in financial
analysis as the outcomes of financial decisions are
uncertain
e.g. Fluctuation in share prices
702341 QA in Finance/ Ch 3 Probability in Finance
The classical approach to probability
 The range of possible uncertain outcomes is known
and equally likely
 EXPERIMENT, SAMPLE, EVENT
consider the tossing of a fair coin:
the range is limited to two
the tossing of the coin is the experiment
the two possible outcomes refer to the sample space
the outcome whether it is head or tail is the event
702341 QA in Finance/ Ch 3 Probability in Finance
The classical approach to probability
P(A) 
No. of outcomes associated with the event
Total number of outcomes
702341 QA in Finance/ Ch 3 Probability in Finance
The empirical approach to probability
 In finance, we cannot rely on the exactness of a
process to determine the probabilities
 Consider ‘the return of financial assets’
 The range is unlimited
 In such situations the probability of a given outcome
Z, P(Z), is
No. of Z occurences
P(Z) 
No. of exp eriments
702341 QA in Finance/ Ch 3 Probability in Finance
The empirical approach to probability
 E.g. Consider a sample of 100 daily movements in a
share price. Assume that of the 100 absolute
movements, five movements were 0.5 Baht each, 15
were 1 Baht each, 20 were 1.5 Baht each, 30 were 2
Baht each, 20 were 2.5 Baht and 10 were 3 Baht each
702341 QA in Finance/ Ch 3 Probability in Finance
Basic rules of probability
 These rules are :
 the addition rule concerned with A or B happening
 the multiplication rule concerned with A and B occurring
 Which of these rules is applicable will depend on
whether the combined events are
 INDEPENDENT or MUTUALLY EXCLUSIVE ???
702341 QA in Finance/ Ch 3 Probability in Finance
Mutually exclusive
 Two events cannot occur together
Sample space = {1,2,3,4,5,6}
A is the event that the face of die shows odd number:
A = {1,3,5}
B is the event that the face of die is even number:
B = {2,4,6}
AΛ B = { } = &Oslash;
A and B is MUTUALLY EXCLUSIVE
702341 QA in Finance/ Ch 3 Probability in Finance
Mutually exclusive

A
B
702341 QA in Finance/ Ch 3 Probability in Finance
The addition rule applied to nonmutually exclusive events
 P(A or B) = P(A) + P(B) – P(A and B)

A
B
 Assume that the FTSE 100 index may rise with a probability of 0.55 and
fall with the probability of 0.45. Also assume that a particular time interval
the S&amp;P index may rise with a probability of 0.35 and fall with a
probability of 0.65. There is also a probability of 0.3 that both indices rise
together. What is the probability of wither the FTSE 100 index or the S&amp;P
500 index rising
702341 QA in Finance/ Ch 3 Probability in Finance
The multiplication rule applied to nonindependent events
 P(A and B) = P(A) * P(B A)
 P(B A) is the conditional probability of B occurring
given that A has occurred
 Suppose the probability of the recession is 25% and
long-term bond yields have an 80% chance of
declining during a recession What is the probability
that a recession will occur and bond yields will
decline?
702341 QA in Finance/ Ch 3 Probability in Finance
Bayes’ theorem
 Manipulation of the general multiplication rule
 The probability of the updated event An can be
updated to P(A|B) if Scenario B is known to have
occurred by using the following relationships
P( Ak / B) 
P( Ak ).P( B / Ak )
iN
 ( P( A ).P( B / A ))
i 1
i
i
702341 QA in Finance/ Ch 3 Probability in Finance
Bayes’ theorem
 Suppose the economy is in an uptrend three out of
every four years (75%). Furthermore, when the
economy is in an uptrend, the stock market advances
80% of the time. Conversely, the economy declines
one out of every four years (25%), and the stock
market declines 70% of the time when the economy is
in a recession.
702341 QA in Finance/ Ch 3 Probability in Finance
Random variable
Random Variable
 A variable that behave in an uncertain manner
 As this behavior is uncertain we can only assign
probabilities to the possible values of these variables.
 Thus the random variable is defined by its probability
distribution and possible outcomes.
 Two types of random variable: discrete and continuous
702341 QA in Finance/ Ch 3 Probability in Finance
Discrete probability
distribution
 Variables that have only a finite number of possible
outcomes
 For example …a six-sided die is thrown
Possibilities r=1
Probability that Z=r

0  PX j  1
1
2
3
4
5
6
1/6
1/6
1/6
1/6
1/6 1/6
PX  1
j
702341 QA in Finance/ Ch 3 Probability in Finance
Discrete probability
distribution
Event: Toss two coins
T
T
T
Count the number of tails
Probability Distribution
Values
Probability
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
T
702341 QA in Finance/ Ch 3 Probability in Finance
Continuous probability
distribution
 Variables that can be subdivided into an infinite
number of subunits for measurement
 For example …speed, asset returns
 Consider: a movement in an asset price from 105 to
109 will give a return of … %
702341 QA in Finance/ Ch 3 Probability in Finance
Continuous probability
distribution
 To overcome this practical problem, we must define
our continuous random variable by integrating what
is know as a probability density function (pdf)

f ( X )dX  1
702341 QA in Finance/ Ch 3 Probability in Finance
Expected value of a random
discrete variable
 Expected value (the mean)
 Weighted average of the probability distribution
  E  X    X jP X j 
j
 e.g.: Toss 2 coins, count the number of tails,
compute
   X jP X j 
j
  0  2.5  1.5    2 .25   1
702341 QA in Finance/ Ch 3 Probability in Finance
Expected value of a random
discrete variable
 Expected value (the mean)
– Weight average squared deviation about the mean
2

  E  X       X j    P  X j 


2
2
– e.g. Toss two coins, count number of tails,
compute variance
   X j    P X j 
2
2
  0  1 .25   1  1 .5    2  1 .25   .5
2
2
2
702341 QA in Finance/ Ch 3 Probability in Finance
Computing the Mean for Investment
Returns
Return per \$1,000 for two types of investments
P(XiYi)
Economic condition
Investment
Dow Jones fund X Growth Stock Y
.2
Recession
-\$100
-\$200
.5
Stable Economy
+ 100
+ 50
.3
Expanding Economy
+ 250
+ 350
E  X    X   100.2  100.5   250.3  \$105
E Y   Y   200.2  50.5  350 .3  \$90
702341 QA in Finance/ Ch 3 Probability in Finance
Computing the Mean for
Investment Returns
Return per \$1,000 for two types of investments
Investment
Economic condition Dow Jones fund X Growth Stock Y
P(XiYi)
.2
Recession
-\$100
-\$200
.5
Stable Economy
+ 100
+ 50
.3
Expanding Economy
+ 250
+ 350
   100  105  .2   100  105  .5    250  105  .3 
2
2
X
2
2
 X  121.35
 14, 725
   200  90  .2    50  90  .5    350  90  .3
2
2
Y
 37,900
2
 Y  194.68
2
702341 QA in Finance/ Ch 3 Probability in Finance
Probability Distribution
Important probability distributions in finance
Discrete:
 BINOMIAL
 POISSON
Continuous:
 NORMAL
 LOG NORMAL
702341 QA in Finance/ Ch 3 Probability in Finance
Binomial probability distribution
 Only two possible outcomes can be taken on by the
variable in a given time period or a given event.
e.g. getting head is success while getting tail is failure
 For each of a succession of trials the probability of
two outcome is constant
e.g. Probability of getting a tail is the same each time
we toss the coin
702341 QA in Finance/ Ch 3 Probability in Finance
Binomial probability distribution
 Each binomial trial is identical
e.g. 15 tosses of a coin; ten light bulbs taken from a
warehouse
 Each trial is independent
the outcome of one trial does not affect the outcome
of the other
702341 QA in Finance/ Ch 3 Probability in Finance
Binomial probability distribution
Su2
j=2
Sud = Sdu
j=1
Sd2
j=0
Su
S
Sd
J = number of success
702341 QA in Finance/ Ch 3 Probability in Finance
Binomial probability distribution
The probability of achieving each outcome depends on:
1. the probability of achieving a success
2. the total number of ways of achieving that outcome
e.g. consider the case of j = 1 (Sdu = Sud)
1. each way has a probability of 0.25
2. there are two ways to achieving an outcome
702341 QA in Finance/ Ch 3 Probability in Finance
Binomial probability distribution
Combination rule
n!
n X
X
P X  
p 1  p 
X ! n  X  !
P  X  : probability of X successes given n and p
X : number of &quot;successes&quot; in sample  X  0,1,
, n
p : the probability of each &quot;success&quot;
n : sample size
or the number of binomial trials
702341 QA in Finance/ Ch 3 Probability in Finance
A binomial tree of asset prices
 The most common application of the binomial
distribution in finance is ‘security price change’
 It is assumed that over the next small interval of time
security price will wither rise (‘a success’) or fall (‘a
failure’) by a given amount
 The binomial distribution is an assumption in some
option pricing models
702341 QA in Finance/ Ch 3 Probability in Finance
A binomial tree of asset prices
3 stages in developing the expected value of asset price:
 create a binomial lattice
 determine the probabilities of each outcome
 multiply each possible outcome by the appropriate
probability and sum the products to arrive at the
expected value
702341 QA in Finance/ Ch 3 Probability in Finance
A binomial tree of asset prices
In each of the time period the asset
may rise with probability of 0.5, or
it may fall with a probability of 0.5
Su2
Su
Sud = Sdu
S=50
Sd
suppose: u= 1.10, d = 1/1.10
T0
Sd2
T1
T2
702341 QA in Finance/ Ch 3 Probability in Finance
A binomial tree of asset prices
Su2
(60.50)
Su
(55)
Sud = Sdu
(50)
S=50
Su
(45.45)
T0
T1
Sd2
(41.32)
T2
702341 QA in Finance/ Ch 3 Probability in Finance
A binomial tree of asset prices
The expected value is calculated as:
(60.50 x 0.25) + (50.0 x 0.50) + (41.32 x 0.25)
= 50.46
The variance is
(60.50-50.46)2 x 0.25 + (50.0-50.46)2 x 0.50 +
(41.32-50.46)2 x 0.25
= 46.18
702341 QA in Finance/ Ch 3 Probability in Finance
The Poisson distribution
 Discrete events in an interval
 The probability of One Success in an interval is
stable
 The probability of More than One Success in this
interval is 0
 e.g. number of customers arriving in 15 minutes
 e.g. information which causes market price to move
arrive at a rate of 10 pieces per minute
702341 QA in Finance/ Ch 3 Probability in Finance
The Poisson distribution
e 
P X  
X!
P  X  : probability of X &quot;successes&quot; given 

X
X : number of &quot;successes&quot; per unit
 : expected (average) number of &quot;successes&quot;
e : 2.71828 (base of natural logs)
702341 QA in Finance/ Ch 3 Probability in Finance
The Poisson distribution
ex. Find the probability of 4 customers arriving in 3
minutes when the mean is 3.6.
e3.6 3.64
P X  
 .1912
4!
ex. information which causes market price to move
arrive at a rate of 10 pieces per minute. Find the
probability of only eight pieces of information
arriving in the next minute ???
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution





Most important continuous probability distribution
Bell shaped
f(X)
Symmetrical
Mean, median and
mode are equal
X

f ( X )dX  1

Mean
Median
Mode
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution
f X  
1

e
1
2
2
X





2 2
f  X  : density of random variable X
  3.14159;
e  2.71828
 : population mean
 : population standard deviation
X : value of random variable    X   
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution
 Probability is the area under the curve!
P c  X  d   ?
f(X)
c
d
X
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution
 There are an infinite number of normal distributions
 By varying the parameters  and &micro;, we obtain
different normal distributions
 An infinite number of normal distributions means an
infinite number of tables to look up !!!
702341 QA in Finance/ Ch 3 Probability in Finance
Standardizing example
Z
X 

6.2  5

 0.12
10
Standardized
Normal Distribution
Normal Distribution
  10
 5
Z 1
6.2
X
Z  0
0.12
Z
702341 QA in Finance/ Ch 3 Probability in Finance
P  2.9  X  7.1  .1664
Z
X 

2.9  5

 .21
10
Z
X 

7.1  5

 .21
10
Standardized
Normal Distribution
Normal Distribution
  10
Z 1
.0832
.0832
2.9
 5
7.1
X
0.21
Z  0
0.21
Z
702341 QA in Finance/ Ch 3 Probability in Finance
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
Z 1
.02
.5832
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
0
Z = 0.21
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution
Example:
We wish to know the probability of a given asset,
which is assumed to have normally distributed returns,
providing a return of between 4.9% and 5%. The
mean of the return on that asset to date is 4%, and the
standard deviation is 1%
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution
Example:
The earnings of a company are expected to be \$4.00
per share, with a standard deviation of \$40. Assuming
earnings per share are a continuous random variable
that is normally distributed, calculate the probability
of actual EPS will be \$3.90 or higher.
702341 QA in Finance/ Ch 3 Probability in Finance
The normal distribution
Example:
The earnings of a company are expected to be \$4.00
per share, with a standard deviation of \$40. Assuming
earnings per share are a continuous random variable
that is normally distributed, calculate the probability
of actual EPS will be between \$3.60 and \$4.40.
702341 QA in Finance/ Ch 3 Probability in Finance
The lognormal distribution
 Modern portfolio theory assumes that investment
return are normally distributed random variable.
Is that true ?


702341 QA in Finance/ Ch 3 Probability in Finance
The lognormal distribution
 However, this is not true, investment return can only
take on values between -100% and   % which are
not symmetrically distributed, but skewed.
100%
E (r )

702341 QA in Finance/ Ch 3 Probability in Finance
The lognormal distribution
 Mathematical trick !!!
Distribution of Vt/Vo is Lognormal
Distribution of ln(Vt/Vo) is Normal
 Vt 
P ln 
 V0 
 Vt 
P 
 V0 
0
Vt /V0

ln( Vt / V0 )
702341 QA in Finance/ Ch 3 Probability in Finance
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