University of Texas Department of Mathematics

1
University of Texas Department of Mathematics
SATURDAY MORNING MATH GROUP
Presents Bulls, Bears, and Mathematicians
By Mike Tehranchi
2
Disclaimer
This talk will not teach you how to predict the
stock market or get rich quick.
3
What is a stock?
Owning stock in a company indicates ownership of
the assets and the future earnings of that
company. A companys stock is divided into many
small pieces called shares.
4
Example Microsoft (MSFT)
The total market value of Microsofts assets and
potential future earnings is about
272,000,000,000. There are about 11,000,000,000
shares of Microsoft stock available to
buy. Therefore, the price of one share is about
25. (By the way, Bill Gates owns more than a
billion shares of Microsoft stock!)
5
Why study the stock market?
Nearly everything has a price. What make stocks
different than, say, houses? Suppose you want
to sell your house. First you have to find
someone interested in buying the house. Then
you have to negotiate a price that seems fair.
Similarly, suppose you want to buy a house.
First you have to find a potential seller. Then
you have to negotiate a fair price. In both
cases, the procedure takes a lot of time and
money.
6
On the other hand, buying or selling stock in a
company is usually very quick and inexpensive.
Most stock is traded in a stock exchange, so
buyers and sellers dont have to meet or
negotiate prices. Nowadays, you can buy and
sell stock over the internet.
7
(No Transcript)
8
Bulls
9
Price of one share of Microsoft Jan 1996 to Dec
1999
10
Bears
11
Price of one share of MicrosoftJan 2000 to Nov
2003
12
Mathematicians
13
In 1997, Robert Merton and Myron Scholes won the
Nobel Prize in Economics for inventing a method
to calculate the price of a stock option. (More
on this later.)
Merton
Scholes
14
Quantifying risks
When you buy something, you are trading certainty
for uncertainty. For instance, suppose you
buy one share of Microsoft stock for 25. In
this case, you are exchanging money (whose worth
you know for certain) for 1/11 billionth of the
assets and earnings of the Microsoft corporation
(whose worth you can never know for certain).
15
Probability Theory
We can model risk by appealing to probability
theory. Here are some of the ingredients of this
theory A random variable is a function that
assigns a numerical value to the outcome of an
experiment.
16
Examples of random variables
The simplest examples come from gambling. For
instance, roll a standard six sided die, and let
X be the number that is face up. Then X is a
random variable.
17
In fact, since all faces are equally likely to
occur we have
For instance, if you rolled the die very many
times, you would see that the random variable X
takes the value 2 about one-sixth of the time.
18
Here is another example Deal a two card black
jack hand from a standard deck of cards. Let Y
be the number of cards worth ten points (10,
jack, queen, king) in the hand. Then Y is a
random variable. As it turns out, it has the
following distribution
19
Let Z be a random variable. For every real
number z we have the inequality
Notice that if a random variable Z takes exactly
n values
20
The expected value of a random variable Z is the
average of the possible values of Z, weighted by
the probability that it attains those values.
The expected value can be calculated by the
formula
Example Let X be the number showing on one roll
of a die.
21
Caution Dont let the word expected fool you.
The expected value of a random variable is not
necessarily its most likely value. In fact, the
above example shows that it possible that the
random variable never actually equals its
expected value.
The law of large numbers states that if you
repeated an experiment over and over again, the
average of the realizations of the random
variable approaches its expected value.
22
Expected value and stocks
There are so many factors affecting the value of
the assets and earnings of a company that it
makes sense to assume that the value is
random. Let Y be a random variable corresponding
to the value of one share of a company. Suppose,
we know the distribution of Y

• .

23
How much should you pay for a share of stock in
this company? A reasonable answer would be the
expected value of Y.
The following activity explores this method of
calculating a fair price for a risky pay out.
24
The St. Petersburg Paradox Game 1
Get a partner. Determine who is player 1 and
who is player 2. Player 1 flips a penny.
Player 2 writes down the outcome of the flip. H
for heads, T for tails. If the first flip was
an H, then the game is over. And player 1 wins
1.
25
4. If the first flip was a T, then player 1
flips again. If the second flip is an H, then
the game is over. Player 1 wins 2. 5. If
the second flip is a T, then player 1 flips
again. Player 2 should be recording the
outcomes. 6. Player 1 continues flipping the
coin until they get an H. Player 2 counts the
total number of flips. If there are n flips,
then Player 1 wins n.
26
Here are some examples   Player 1 H Player 2
calculates 1.   Player 1 T,H Player 2
calculates 2.   Player 1 T,T,T,T, H Player
2 calculates 5
27
Game 2Lets change the rules
Play the game again, this time with a new rule
6   6. Player 1 continues flipping the coin
until they get an H. Player 2 counts the total
number of flips. If there are n flips, then
Player 1 wins n2.  
28
Here are some examples   Player 1 T,H
Player 2 calculates 224.   Player 1
T,T,T,T,H Player 2 calculates 5225
29
Game 3Lets change the rules, again
Play the game again, this time with another rule
6   6. Player 1 continues flipping the coin
until they get an H. Player 2 counts the total
number of flips. If there are n flips, then
Player 1 wins 2n.  
30
Here are some examples   Player 1 T,H
Player 2 calculates 224.   Player 1
T,T,T,T,H Player 2 calculates 2532
31
Whats the paradox?
The probability of getting H on the first flip is
½. Likewise, the probability of getting T on the
first flip and H on the second flip is ¼.
  Continuing, the probability of Player 1
flipping the sequence T,T,T,T,T,T,H is 1/128. In
general, the probability of Player 1 flipping
n-1 tails in a row followed by a head is 1/2n
32
Using the formula for expected value, the average
winnings in playing Game 1 many times should be
33
Now lets look at Game 2. We can calculate the
expected value of the winnings in a similar way
34
Finally, lets look at Game 3. Once more
calculate the expected value of the winnings
111 INFINITY!
35
Whats going on?

• There are at least four explanations for this
  paradoxical result
• We tend to think of very rare events as being
  impossible, and thus ignore them when informally
  calculating the expected value.
• For instance, the probability of flipping 25
  tails and then a head is 1/67108864. However, in
  this case you would win 67108864!

36

• Let N be the random variable corresponding to the
  number of flips in a game. The winnings for Game
  3 would then be 2N.
• For Game 1 we calculated EN 2. Its
  tempting to think

But this is false. Whats true is
37
3. If you could some how measure happiness, the
average person would likely be much happier to
win 1,000 than 1. On the other hand, that same
person would probably be equally happy to win
1,001,000 and 1,000,000. That is, happiness
generally does not increase linearly with wealth.
Thus, in Game 3, you probably dont want to
pay for those really rare events that could make
you ridiculously rich, since it wouldnt make you
that much happier than if you were just extremely
rich.
38

• Theres only about 30 trillion dollars in the
  entire world economy.
• Since 30 trillion is about 245, in the very rare
  event that you flip 44 tails in a row, you would
  win all the money in the world!

39
Stock options
A stock option is a contract that gives the owner
the right, but not the obligation, to buy a given
stock at fixed price some time in the future. An
important question in financial mathematics is,
What is the fair price of a stock
option? Merton and Scholes won the Nobel Prize
for providing an answer.
40
How does an option work? Imagine that the price
of a given stock today is 5. And suppose you
own the option to buy the stock tomorrow for 6.
Lets assume that there are two equally likely
possibilities. One possibility is that the stock
price goes up to 7 tomorrow, and the other
possibility is that the stock price drops down to
4 tomorrow.
41
What if the stock price goes up to 7 tomorrow?
You could exercise your option by buying the
stock at the cheaper price of 6. You can then
sell the stock back at the market price of 7,
pocketing the 1 difference.
42
What if the price of the stock instead drops to
4? It wouldnt make sense to pay 6 for
something you can buy for 4, so in this case,
you dont exercise your option. In other words,
the option is worthless in this case.
43
How much would you pay?
44
A first attempt We could compute the expected
value of the payout of the option by the usual
formula 1(1/2) 0(1/2) 1/2 But this would be
wrong!
45
Suppose that the price of the option was 0.50.
Then there is a free lunch available. Sell
three copies of the option and buy the stock.
You would receive 1.50 from your sales and you
would spend 5 on your purchase, so today you
would be down 3.50. What happens tomorrow?
46
Case 1 The stock price goes up to 7. In this
case, you would have to pay the option holders 1
each, so your total is 7-3 4. Case 2 The
stock price goes down to 4. In this case, you
would not have to pay the option holders
anything, so your total is still 4. You only
spent 3.50, but in both cases you get 4.00 the
next day. Thats a free lunch!
47
The price of the option should be such that there
are no free lunches! Lets try to compute this
price in our example. Let p be the price of the
option, to be determined. Buy a shares of the
stock and sell b copies of the option.
48
You have spent 5a p b dollars. Tomorrow you
will have 7a b dollars in Case 1 and 4a in Case
2. There would be a free lunch if there was a
solution to 7a b gt 5a pb 4a gt 5a
pb There is no solution to the two inequalities
if and only if p 1/3.
49
Weve come up with a price for the stock option
in the simple case where the stock price can only
take three values and moves once a day. But the
real world problem is a lot more involved.
Prices change much more frequently and can take
on quite a large number of possible values.
The solution to the option price problem
involves more technical math in this case, but
the same simple idea applies.
50
log(S(t)/S(0))
51
Central Limit Theorem
Let X1, X2, Xn be n independent random
variables with the same distribution. Then the
random variable
is approximately normal if n is large.
52
A random variable is normal if the histogram of
its values looks like a bell curve. More
precisely, a random variable Z is normal if
53
Daily returns for MSFT from Apr-86 to Nov-03
54
The Black-Scholes partial differential equation
Let P(t,x) be the price of a stock option at
time t when the price of the stock is x dollars.
Then the following equation holds
55
It turns out that the Black-Scholes differential
equation is almost exactly the same as the
equation from classical physics that describes
the distribution of temperature in a
material! Who would have thought that the stock
market would have anything to do with physics?