Transcript Here - UT Mathematics

University of Texas – Department of Mathematics
SATURDAY MORNING MATH GROUP
Presents: “Bulls, Bears, and Mathematicians”
By Mike Tehranchi
Disclaimer:
This talk will not teach
you how to predict the
stock market or get rich
quick.
What is a stock?
Owning stock in a
company indicates
ownership of the assets
and the future earnings of
that company.
A company’s stock is
divided into many small
pieces called shares.
Example: Microsoft (MSFT)
The total market value of Microsoft’s 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!)
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.
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 don’t have to meet or negotiate prices.
Nowadays, you can buy and sell stock over the internet.
Bulls
Price of one share of Microsoft
Jan 1996 to Dec 1999
70
60
50
40
30
20
10
0
Jan-96
Jul-96
Jan-97
Jul-97
Jan-98
Jul-98
Jan-99
Jul-99
Bears
Price of one share of Microsoft
Jan 2000 to Nov 2003
70
60
50
40
30
20
10
0
Jan-00
Jul-00
Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Mathematicians
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
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).
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.
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.
In fact, since all faces are equally likely to occur
we have
1
P{ X  1}  P{ X  2}  ...  P{ X  6} 
6
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.
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:
105
96
20
P{Y  0} 
, P{Y  1} 
, P{Y  2} 
221
221
221
Let Z be a random variable. For every real
number z we have the inequality
0  P{Z  z}  1
Notice that if a random variable Z takes
exactly n values z1 , z2 ,..., z n
P{Z  z1}  P{Z  z2 }  ...  P{Z  zn }  1
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
E[Z ]  z1P{Z  z1}  z2 P{Z  z2}  ...  zn P{Z  zn }
Example: Let X be the number showing on
one roll of a die.
1
1
1 7
E[ X ]  1  2   ...  6  
6
6
6 2
Caution: Don’t 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.
Expected value and stocks
There are so many factors affecting the value of
the assets and earnings of a company that it
. sense to assume that the value is random.
makes
Let Y be a random variable corresponding to the
value of one share of a company. Suppose, we
know the distribution of Y:
1
3
P{Y  5}  , P{Y  8} 
4
4
How much should you pay for a share of stock in this
company?
A reasonable answer would be the expected value of Y.
1
3
1
E[Y ]  5   8   7
4
4
4
The following activity explores this method of
calculating a fair price for a risky pay out.
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.
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.
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
Game 2:Let’s 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.
Here are some examples:
Player #1: T,H
$22=$4.
Player #2 calculates:
Player #1: T,T,T,T,H
$52=$25
Player #2 calculates:
Game 3:Let’s 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.
Here are some examples:
Player #1: T,H
$22=$4.
Player #2 calculates:
Player #1: T,T,T,T,H
$25=$32
Player #2 calculates:
What’s 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 n1 tails in a row followed by a head is 1/2n
Using the formula for expected value, the
average winnings in playing Game 1 many
times should be
1
1
1
1
1   2  2  3  3  ...  n  n  ...  2
2
2
2
2
Now let’s look at Game 2. We can
calculate the expected value of the
winnings in a similar way:
1
1
1
1
2
2
2
1   2  2  3  3  ...  n  n  ...  6
2
2
2
2
2
Finally, let’s look at Game 3. Once more
calculate the expected value of the
winnings:
1 2 1 3 1
1
n
2   2  2  2  3  ...  2  n  ...
2
2
2
2
1
=1+1+1+ … = INFINITY!
What’s going on?
There are at least four explanations for this
paradoxical result:
1. 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!
2. 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 E[N] = 2. It’s
tempting to think
E[2 ]  2
N
E[ N ]
4
But this is false. What’s true is
E[2 ]  2
N
E[ N ]
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 don’t want to pay
for those really rare events that could make you
ridiculously rich, since it wouldn’t make you that
much happier than if you were just extremely rich.
4. There’s 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!
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.
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.
Let’s 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.
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.
What if the price of the stock instead drops to $4?
It wouldn’t make sense to pay $6 for something
you can buy for $4, so in this case, you don’t
exercise your option. In other words, the option is
worthless in this case.
How much would you pay?
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!
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?
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. That’s a free lunch!
The price of the option
should be such that there
are no free lunches!
Let’s 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.
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 > 5a + pb
4a > 5a + pb
There is no solution to the two inequalities if
and only if p = 1/3.
We’ve 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.
log(S(t)/S(0))
3
2.5
2
1.5
1
0.5
0
-0.5
1986
1988
1990
1992
1994
1996
1998
2000
2002
Central Limit Theorem
Let X1, X2, … Xn be n independent random
variables with the same distribution. Then the
random variable
X 1  X 2  ...  X n  nE[ X ]
Z
n
is approximately normal if n is large.
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
1
P{Z  z} 
 2

z


exp 
1
2
  dx
x 2

Daily returns for MSFT
from Apr-86 to Nov-03
2000
1800
1600
1400
1200
1000
800
600
400
200
0
-0.045
-0.0357
-0.0264
-0.0171
-0.0078
0.0015
0.0108
0.0201
0.0294
0.0388
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
P
P  2 2  2 P
 rx

x
 rP
2
t
x 2
x
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?