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PROFESSOR: I want to
pick up exactly where

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I left off last time.


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When I was talking about
various sins one can

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commit with statistics.


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And I had been talking about
the sin of data enhancement,

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where the basic idea there is,
you take a piece of data, and

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you read much more into
it than it implies.

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In particular, a very common
thing people do with data

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is they extrapolate.


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I'd given you a
couple of examples.

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In the real world, it's often
not desirable to say that I

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have a point here, and a point
here, therefore the next

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point will surely be here.


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And we can just extrapolate
in a straight line.

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We before saw some examples
where I had an algorithm to

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generate points, and we fit a
curve to it, used the curve to

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predict future points, and
discovered it was

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nowhere close.


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Unfortunately, we often see
people do this sort of thing.

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One of my favorite stories is,
William Ruckelshaus, who was

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head of the Environmental
Protection Agency in

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the early 1970s.


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And he had a press conference,
spoke about the increased use

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of cars, and the decreased
amount of carpooling.

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He was trying to get people to
carpool, since at the time

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carpooling was on the way down,
and I now quote, "each car

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entering the central city,
sorry, in 1960," he said, "each

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car entering the central
city had 1.7 people in it.

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By 1970. this had dropped
to less than 1.2.

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If present trends continue, by
1980, more than 1 out of every

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10 cars entering the city will
have no driver." Amazingly

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enough, the press reported this
as a straight story, and talked

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about how we would be
dramatically dropping.

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Of course, as it happened,
it didn't occur.

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But it's just an example of,
how much trouble you can

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get into by extrapolating.


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The final sin I want to talk
about is probably the most

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common, and it's called the
Texas sharpshooter fallacy.

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Now before I get into that, are
any of you here from Texas?

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All right, you're
going to be offended.

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Let me think, OK, anybody
here from Oklahoma?

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You'll like it.


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I'll dump on Oklahoma, it
will be much better then.

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We'll talk about the Oklahoma
sharpshooter fallacy.

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We won't talk about the
BCS rankings, though.

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So the idea here is a
pretty simple one.

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This is a famous marksman who
fires his gun randomly at the

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side of a barn, has a bunch of
holes in it, then goes and

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takes a can of paint and draws
bullseyes around all the places

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his bullets happened to hit.


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And people walk by the barn
and say, God, he is good.

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So obviously, not a good
thing, but amazingly easy

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to fall into this trap.


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So here's another example.


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In August of 2001, a paper
which people took seriously

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appeared in a moderately
serious journal called

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The New Scientist.


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And it announced that
researchers in Scotland had

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proven that anorexics are
likely to have been

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born in June.


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I'm sure you all knew that.


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How did how did
they prove this?

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Or demonstrate this?


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They studied 446 women.


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Each of whom had been
diagnosed anorexic.

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And they observed that
about 30 percent more than

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average were born in June.


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Now, since the monthly average
of births, if you divide this

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by 12, it's about 37,
that tells us that 48

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were born in June.


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So at first sight, this seems
significant, and in fact if you

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run tests, and ask what's the
likelihood of that many more

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being born in 1 month, you'll
find that it's quite unlikely.

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In fact, you'll find the
probability of this happening

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is only about 3 percent, of it
happening just by accident.

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What's wrong with
the logic here?

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Yes?


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STUDENT: They only studied
diagnosed anorexics.

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PROFESSOR: No, because they
were only interested in the

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question of when are anorexics
born, so it made sense

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to only study those.


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Now maybe you're right, that we
could study that, in fact, more

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people are born in June period.


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That could be true.


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This would be one of the
fallacies we looked

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at before, right?


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That there's a lurking variable
which is just that people are

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more likely to be born in June.


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So that's certainly
a possibility.

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What else?


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What else is the flaw?


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Where's the flaw in this logic?


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Well, what did they do?


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They participated in the
Oklahoma sharpshooter fallacy.

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What they did is, they looked
at 12 months, they took the

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months with the most births in
it, which happened to be

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June, and calculated the
probability of 3 percent.

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00:07:29 --> 00:07:34
They didn't start with the
hypothesis that it was June.

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00:07:34 --> 00:07:36
They started with 12 months,
and then they drew a

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bullseye around June.


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So the right question to ask
is, what's the probability, not

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that June had 48 babies, but
that at least one of the

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12 months had 48 babies.


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That probability is a lot to
higher than 3 percent, right?

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In fact, it's about 30 percent.


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So what we see is, again
perfectly reasonable

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statistical techniques,
but not looking at

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things in the right way.


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And answering the
wrong question.

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That make sense to everybody?


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00:08:29 --> 00:08:31
And you can see why people can
fall into this trap, right?

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It was a perfectly sensible,
seemingly sensible argument.

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00:08:36 --> 00:08:45
So the moral of this particular
thing is, be very careful about

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looking at your data, drawing a
conclusion, and then saying how

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00:08:49 --> 00:08:53
probable was that
to have occurred?

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00:08:53 --> 00:08:56
Because again, you're probably,
or maybe, drawing the

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00:08:56 --> 00:08:59
bullseye around something
that's already there.

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Now if they had taken another
set of 446 anorexics, and again

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June was the month, then
there would be some

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credibility in it.


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Because they would have started
with the hypothesis, not that

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there existed a month, but that
June was particularly likely.

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But then they would have to
also check and make sure that

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June isn't just a popular
month to be born, as

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was suggested earlier.


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All right, I could go on
and on with this sort of

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thing, it's kind of fun.


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But I won't.


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Instead I'm going to
torture you with yet

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one more simulation.


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You may be tempted at this
point to just zone out.

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Try not to.


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And as an added incentive for
you to pay attention, I'm

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going to warn you that this
particular simulation will

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appear in the final,
or a variant of it.

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And what we'll be doing is,
early next week we'll be

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00:10:02 --> 00:10:06
distributing code, which we'll
ask you to study, about two or

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00:10:06 --> 00:10:11
three pages of code, and then
on the final we'll be asking

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00:10:11 --> 00:10:13
you questions about the code.


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Not that you have to
memorize it, we'll

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give you a copy of it.


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But you should understand
it before you walk in

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to take the final.


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Because there will not be time
to look at that code for the

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first time during the quiz, and
figure out what it's doing.

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OK, so let's look at it.


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I should also warn you that
this code includes some Python

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concepts, at least one,
that you have not yet seen.

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We'll see it briefly today.


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This is on purpose, because one
of the things I hope you have

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learned to do this semester, is
look up things you don't know,

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and figure out what they do.


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What they mean.


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Because we obviously can't, in
any course, or even any set of

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00:11:03 --> 00:11:05
courses, tell you everything
you'll ever want

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00:11:05 --> 00:11:07
to know in life.


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So intentionally, we've seeded
some things in this program

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that will be unfamiliar, so
during the time you're studying

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00:11:13 --> 00:11:18
the program, get online, look
it up, figure out what they do.

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If you have trouble, we will
be having office hours, where

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you can go and get some help.


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But the TAs will expect you
to have at least tried to

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figure it out yourself.


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00:11:27 --> 00:11:27
Yeah?


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00:11:27 --> 00:11:30
STUDENT: Will the
final be open note?

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00:11:30 --> 00:11:32
PROFESSOR: Final will be
open book, open notes,

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just like the quizzes.


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It will be the first two hours
of the allotted time, we won't

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go the whole 3 hours, OK?


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So it won't be hugely
longer than the quizzes.

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It would be a
little bit longer.

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00:11:49 --> 00:11:56
And again, very much in the
same style of the quizzes.

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00:11:56 --> 00:11:59
All right, let's look at this.


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Let's assume that you've won
the lottery, and have serious

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00:12:03 --> 00:12:09
money that you foolishly wish
to invest in the stock market.

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There are two basic strategies
to choose from, in investing.

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You can either have what's
called an indexed portfolio,

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00:12:27 --> 00:12:41
or a managed portfolio.


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00:12:41 --> 00:12:45
Indexed portfolios, you
basically say, I want to own

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all of the stocks that there
are, and if the stock market

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00:12:49 --> 00:12:51
goes up, I make money, if
the stock market goes

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00:12:51 --> 00:12:53
down, I lose money.


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00:12:53 --> 00:12:55
I'm not going to be thinking
I'm clever, and can pick

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00:12:55 --> 00:12:57
winners and losers, I'm
just betting on the

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00:12:57 --> 00:13:00
market as a whole.


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00:13:00 --> 00:13:03
They're attractive, in that
a, they don't require

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00:13:03 --> 00:13:05
a lot of thought.


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00:13:05 --> 00:13:09
And b, they have what's called
a low expense ratio, since

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00:13:09 --> 00:13:11
they're easy to implement, you
don't pay anyone to be

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00:13:11 --> 00:13:13
brilliant to implement
if for you.

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00:13:13 --> 00:13:17
So they're very low fees.


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A managed portfolio, you find
somebody you think is really

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00:13:21 --> 00:13:26
smart, and you pay them a fair
amount of money, and in return

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00:13:26 --> 00:13:30
they assert that they will pick
winners for you, and in fact,

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00:13:30 --> 00:13:33
you will outperform
the stock market.

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00:13:33 --> 00:13:36
And if it goes up 6 percent,
well you'll go up 10 percent

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00:13:36 --> 00:13:40
or more, and if it goes down,
don't worry, I'm so smart

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00:13:40 --> 00:13:45
your stocks won't go down.


207
00:13:45 --> 00:13:47
There's a lot of debate
about which is the

208
00:13:47 --> 00:13:52
better of these two.


209
00:13:52 --> 00:13:55
And so now we're going to try
and see if we can write a

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00:13:55 --> 00:14:01
simulation that will give us
some insight as to which of

211
00:14:01 --> 00:14:06
these might be better or worse.


212
00:14:06 --> 00:14:10
All right, so that's
the basic problem.

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00:14:10 --> 00:14:16
Now, as we know, and by the way
we're not going to write a

214
00:14:16 --> 00:14:18
perfect simulation here,
because we're going to try and

215
00:14:18 --> 00:14:21
do it in 40 minutes,
or 30 minutes.

216
00:14:21 --> 00:14:24
And it would take at least an
hour do a perfect simulation

217
00:14:24 --> 00:14:26
of the stock market.


218
00:14:26 --> 00:14:29
All right.


219
00:14:29 --> 00:14:33
First thing we need to do is
have some sort of a theory.

220
00:14:33 --> 00:14:36
When we did the spring, we had
this theory of Hooke's Law that

221
00:14:36 --> 00:14:40
told us something, and we built
a simulation, or built some

222
00:14:40 --> 00:14:42
tools around that theory.


223
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Now we need to think about a
model of the stock market.

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And the model we're going
to use is based on what's

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called the Efficient
Market Hypothesis.

226
00:15:00 --> 00:15:05
So the moral here, again, is
whenever you're doing an

227
00:15:05 --> 00:15:07
implementation of a simulation,
you do need to have some

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00:15:07 --> 00:15:12
underlying theory
about the model.

229
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What this model asserts
is that markets are

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informationally efficient.


231
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That is to say, current prices
reflect all publicly known

232
00:15:36 --> 00:15:43
information about each stock,
and therefore are unbiased.

233
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That if people thought that the
stock was underpriced, well

234
00:15:47 --> 00:15:49
people would buy more of it in
the price would have

235
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risen already.


236
00:15:51 --> 00:15:53
If people thought the stock was
overpriced, well, people would

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00:15:53 --> 00:15:56
have tried to sell it, and
it would have come down.

238
00:15:56 --> 00:15:59
So this is a very popular
theory, believed by

239
00:15:59 --> 00:16:05
many famous economists
today, and in the past.

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00:16:05 --> 00:16:09
And says, OK, that effectively
means that the market

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00:16:09 --> 00:16:13
is memoryless.


242
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OK, that it doesn't matter
what the price of the

243
00:16:15 --> 00:16:18
stock was yesterday.


244
00:16:18 --> 00:16:22
Today, it's priced given the
best-known information, and

245
00:16:22 --> 00:16:29
so tomorrow it's equally
likely to go up or down.

246
00:16:29 --> 00:16:32
Relative to the whole
market, right?

247
00:16:32 --> 00:16:36
It's well known that over
periods of multiple

248
00:16:36 --> 00:16:39
decades, the market has a
tendency to go up.

249
00:16:39 --> 00:16:42
And so there's an upward bias
to the stock market, contrary

250
00:16:42 --> 00:16:46
to what you may have
seen recently.

251
00:16:46 --> 00:16:49
But that no particular stock
is more or less likely to

252
00:16:49 --> 00:16:53
outperform the market, because
all the information is

253
00:16:53 --> 00:16:56
incorporated in the price.


254
00:16:56 --> 00:17:00
And that leads to a notion
of being able to model

255
00:17:00 --> 00:17:04
the market, how?


256
00:17:04 --> 00:17:08
How would you model individual
stocks if you believe

257
00:17:08 --> 00:17:14
this hypothesis?


258
00:17:14 --> 00:17:15
Somebody?


259
00:17:15 --> 00:17:16
What's going to happen?


260
00:17:16 --> 00:17:18
STUDENT: Random walk.


261
00:17:18 --> 00:17:20
PROFESSOR: Yes, exactly right.


262
00:17:20 --> 00:17:23
So we would model it
as a random walk.

263
00:17:23 --> 00:17:26
In fact, there's a very famous
book called A Random Walk Down

264
00:17:26 --> 00:17:30
Wall Street, that was one of
the first to make

265
00:17:30 --> 00:17:34
this hypothesis.


266
00:17:34 --> 00:17:38
Now later, we may decide to
abandon this model, but for

267
00:17:38 --> 00:17:43
the moment let's accept that.


268
00:17:43 --> 00:17:53
And let's think about how we're
going to build the simulation.

269
00:17:53 --> 00:17:58
Whenever I think about how to
build an interesting program,

270
00:17:58 --> 00:18:01
and I hope whenever you think
about it, the first thing I

271
00:18:01 --> 00:18:05
think about is, what are the
classes I might want to

272
00:18:05 --> 00:18:08
have, what are the types?


273
00:18:08 --> 00:18:12
And it seems pretty obvious
that at least two of the

274
00:18:12 --> 00:18:18
things I'm going to want
are stock and market.

275
00:18:18 --> 00:18:23
After all, I'm going to try and
build a simulation of the stock

276
00:18:23 --> 00:18:26
market, so I might as well have
the notion of a market, and

277
00:18:26 --> 00:18:31
probably the notion of a stock.


278
00:18:31 --> 00:18:35
Which should I implement first?


279
00:18:35 --> 00:18:40
Well, my usual style of
programming would be to

280
00:18:40 --> 00:18:43
implement the one that's
lowest down in the

281
00:18:43 --> 00:18:48
hierarchy, near the bottom.


282
00:18:48 --> 00:18:50
I won't be able to show you
what a market does unless I

283
00:18:50 --> 00:18:55
have stocks, but I can look at
what an individual stock does

284
00:18:55 --> 00:18:58
without having a market.


285
00:18:58 --> 00:19:00
So why do I implement
this first?

286
00:19:00 --> 00:19:03
Because it will be
easier to unit test.

287
00:19:03 --> 00:19:10
I can build class stock, and I
can test class stock, before

288
00:19:10 --> 00:19:16
I have a class market.


289
00:19:16 --> 00:19:30
So now let's look at it.


290
00:19:30 --> 00:19:38
Clean up the desktop
a little bit.

291
00:19:38 --> 00:19:42
This is similar to, but not
identical to, what you

292
00:19:42 --> 00:19:48
have in your handout.


293
00:19:48 --> 00:19:53
All right, so there's
class stock.

294
00:19:53 --> 00:19:58
And I'm going to initialize
it, create them, with

295
00:19:58 --> 00:19:59
an opening price.


296
00:19:59 --> 00:20:02
When a stock is first
listed in the market, it

297
00:20:02 --> 00:20:06
comes with some price.


298
00:20:06 --> 00:20:10
I'm gonna keep as part of each
stock, it's history of prices,

299
00:20:10 --> 00:20:13
which we can initialize, well,
I've initialized it as empty,

300
00:20:13 --> 00:20:17
but that's probably the
wrong thing, right?

301
00:20:17 --> 00:20:26
I probably should have had it
being the, starting here,

302
00:20:26 --> 00:20:32
right, the opening price.


303
00:20:32 --> 00:20:34
Now comes an interesting part.


304
00:20:34 --> 00:20:38
Self dot distribution.


305
00:20:38 --> 00:20:46
Well, I lied to you a little
bit in my description of

306
00:20:46 --> 00:20:53
what it meant to have the
Efficient Market Hypothesis.

307
00:20:53 --> 00:20:59
I said that no stock is likely
to outperform the market or

308
00:20:59 --> 00:21:02
underperform the market.


309
00:21:02 --> 00:21:06
But it's not quite true,
because typically what they

310
00:21:06 --> 00:21:21
actually do that, is they
say it's adjusted for risk.

311
00:21:21 --> 00:21:27
It's clear that some stocks are
more volatile than others.

312
00:21:27 --> 00:21:31
If you will buy stock in an
electrical utility which has a

313
00:21:31 --> 00:21:34
guaranteed revenue stream,
because no matter how bad the

314
00:21:34 --> 00:21:39
economy gets, a lot of people
still use electricity, you

315
00:21:39 --> 00:21:43
don't expect it to
fluctuate a lot.

316
00:21:43 --> 00:21:50
If you buy stock in a high tech
company, that sells things on

317
00:21:50 --> 00:21:55
the internet, you might expect
it to fluctuate enormously.

318
00:21:55 --> 00:21:59
Or if you buy stock in a
retailer, you might expect

319
00:21:59 --> 00:22:04
it to go up or down more
dramatically with the economy,

320
00:22:04 --> 00:22:09
and so in fact there is a
notion of risk, and I'm not

321
00:22:09 --> 00:22:12
going to do this in this
simulation, but usually people

322
00:22:12 --> 00:22:15
have to be paid to take risk.


323
00:22:15 --> 00:22:18
And so it's usually the case
that you can get a higher

324
00:22:18 --> 00:22:20
return if you're willing
to take more risk.

325
00:22:20 --> 00:22:27
We might or might not have
time to come back to that.

326
00:22:27 --> 00:22:32
But more generally, the point
is, that each stock actually

327
00:22:32 --> 00:22:36
behaves a little
bit differently.

328
00:22:36 --> 00:22:39
There's a distribution
of how it would move.

329
00:22:39 --> 00:22:45
So even if, on average, the
stock is expected to not move

330
00:22:45 --> 00:22:51
at all from where it starts,
some stocks will be expected to

331
00:22:51 --> 00:22:56
just trundle along without much
change, not very volatile.

332
00:22:56 --> 00:23:02
And other stocks might jump
up and down a lot because

333
00:23:02 --> 00:23:03
they're very volatile.


334
00:23:03 --> 00:23:07
Even if the expected value
is the same, they'd

335
00:23:07 --> 00:23:10
move around a lot.


336
00:23:10 --> 00:23:12
So how can we model
this kind of thing?

337
00:23:12 --> 00:23:16
Well, we've already looked
at the basic notion.

338
00:23:16 --> 00:23:19
Last time we looked at the
notion, last lecture we looked

339
00:23:19 --> 00:23:25
at the idea of a distribution.


340
00:23:25 --> 00:23:29
And when we do a simulation,
we're pulling the samples

341
00:23:29 --> 00:23:33
from some distribution.


342
00:23:33 --> 00:23:39
It could be normal, everything,
that would be a Gaussian, where

343
00:23:39 --> 00:23:45
if you recall there was a mean,
and a standard deviation, and

344
00:23:45 --> 00:23:49
most values were going to
be close to the mean.

345
00:23:49 --> 00:23:52
Especially if there is a
small standard deviation.

346
00:23:52 --> 00:23:56
If there's a large standard
deviation it would be spread.

347
00:23:56 --> 00:23:59
Or it could be uniform,
where every value was

348
00:23:59 --> 00:24:00
equally probable.


349
00:24:00 --> 00:24:03
We also looked at exponential.


350
00:24:03 --> 00:24:07
So we're going to assign to
each stock, when we create

351
00:24:07 --> 00:24:11
it, a distribution.


352
00:24:11 --> 00:24:17
Some way of visualizing, or
thinking about, where we draw

353
00:24:17 --> 00:24:20
the price changes from.


354
00:24:20 --> 00:24:26
This gets us into a new
linguistic concept, which

355
00:24:26 --> 00:24:29
we'll see down here.


356
00:24:29 --> 00:24:33
You don't have this particular
code on your handout, you do

357
00:24:33 --> 00:24:36
have a code that uses
the same concept.

358
00:24:36 --> 00:24:39
So here's my unit
test procedure.

359
00:24:39 --> 00:24:42
And here's where I'm going
to create distributions.

360
00:24:42 --> 00:24:45
And I'm going to look at two.


361
00:24:45 --> 00:24:49
A random-- a uniform,
and a Gaussian.

362
00:24:49 --> 00:24:56
What lambda that does, it
creates on the fly a function,

363
00:24:56 --> 00:24:59
as the program runs.


364
00:24:59 --> 00:25:03
That I can then pass around.


365
00:25:03 --> 00:25:08
So here, I'm going to look at
the thing random dot uniform,

366
00:25:08 --> 00:25:13
for example, between minus
volatility and plus volatility.

367
00:25:13 --> 00:25:16
So ignoring the lambda,
what do we expect random

368
00:25:16 --> 00:25:19
dot uniform to do?


369
00:25:19 --> 00:25:25
It has equally likely, in the
range from minus volatility

370
00:25:25 --> 00:25:35
to plus volatility, it will
return any value in here.

371
00:25:35 --> 00:25:39
But notice the previous
line, where I am

372
00:25:39 --> 00:25:42
computing volatility.


373
00:25:42 --> 00:25:47
If I wanted every stock to have
the same volatility, I could

374
00:25:47 --> 00:25:52
just do that, if you will, at
the time I wrote my program.

375
00:25:52 --> 00:25:57
But here I want it to be
determined, chosen at run time.

376
00:25:57 --> 00:26:03
So first, I choose a volatility
randomly, from some

377
00:26:03 --> 00:26:06
distribution of possible
volatilities from 0 to,

378
00:26:06 --> 00:26:12
in this case, 0.2.


379
00:26:12 --> 00:26:17
Think of this as the
percentage move per day.

380
00:26:17 --> 00:26:22
So 2/10 of a percent,
would be the move here.

381
00:26:22 --> 00:26:28
And then I'll create this
function, this distribution d

382
00:26:28 --> 00:26:33
1, which will, whenever I call
it, give me a random, a

383
00:26:33 --> 00:26:37
uniformly selected value
between minus and

384
00:26:37 --> 00:26:42
plus volatility.


385
00:26:42 --> 00:26:48
Then when I create the
stock, here, I can pass

386
00:26:48 --> 00:26:54
it in, pass in d 1.


387
00:26:54 --> 00:26:55
OK, it's a new concept.


388
00:26:55 --> 00:26:58
I don't expect you'll all
immediately grab it, but you

389
00:26:58 --> 00:27:05
will need to understand it
before the quiz comes along.

390
00:27:05 --> 00:27:08
And then I could also do a
Gaussian one here, with the

391
00:27:08 --> 00:27:11
mean of 0 and the standard
deviation of volatility

392
00:27:11 --> 00:27:13
divided by 2.


393
00:27:13 --> 00:27:15
Where do these
parameters come from?

394
00:27:15 --> 00:27:17
I made them up out
of whole cloth.

395
00:27:17 --> 00:27:20
Later we'll talk about how
1 could think about them

396
00:27:20 --> 00:27:24
more intelligently.


397
00:27:24 --> 00:27:29
Now what do I do with that?


398
00:27:29 --> 00:27:31
All right, we'll see
that in a minute.

399
00:27:31 --> 00:27:38
But people understand what
the basic idea here is.

400
00:27:38 --> 00:27:44
Now, I can set the
price of a stock.

401
00:27:44 --> 00:27:47
And when I do that, I'll
append it to history.

402
00:27:47 --> 00:27:50
I can, oh, these have
got some remnants which

403
00:27:50 --> 00:27:52
we really don't need.


404
00:27:52 --> 00:27:58
I'll get rid of this which is
just an uninteresting thing.

405
00:27:58 --> 00:28:05
And let's look at make move.


406
00:28:05 --> 00:28:06
Because this is the
interesting thing.

407
00:28:06 --> 00:28:11
Make move is what we call to
change the price of a stock,

408
00:28:11 --> 00:28:14
at the beginning or end
of a day if you will.

409
00:28:14 --> 00:28:18
So the first thing it does, is
it says, if self dot price is

410
00:28:18 --> 00:28:23
0, I'm just going to return.


411
00:28:23 --> 00:28:26
This is not the right
thing to do, by the way.

412
00:28:26 --> 00:28:30
Again, there are
some bugs in here.

413
00:28:30 --> 00:28:33
You won't find these bugs
in your handout, right?

414
00:28:33 --> 00:28:35
Code is different
in the handout.

415
00:28:35 --> 00:28:39
But I wanted to show these to
you so we could think about.

416
00:28:39 --> 00:28:41
What I'm more interested in
here than in the result

417
00:28:41 --> 00:28:46
of the simulation, is the
process of creating it.

418
00:28:46 --> 00:28:48
So why did I put this here?


419
00:28:48 --> 00:28:53
Why did I say if self dot
price equals 0 return?

420
00:28:53 --> 00:28:57
Because the first time I wrote
the program, I didn't have

421
00:28:57 --> 00:29:01
anything like that here, and a
stock could go to 0

422
00:29:01 --> 00:29:03
and then recover.


423
00:29:03 --> 00:29:05
Or even go to negative values.


424
00:29:05 --> 00:29:09
Well we know stock prices
are never negative.

425
00:29:09 --> 00:29:11
And in fact we know if the
price goes to 0, it's

426
00:29:11 --> 00:29:14
delisted from the exchange.


427
00:29:14 --> 00:29:21
So I said, all right, we better
make a special case of that. it

428
00:29:21 --> 00:29:24
turns out, that this will be a
bug, and I want you to think

429
00:29:24 --> 00:29:29
about why it's wrong for me
to put this check here.

430
00:29:29 --> 00:29:33
The check needs to be somewhere
in the program, but this is

431
00:29:33 --> 00:29:36
not the right place for it.


432
00:29:36 --> 00:29:40
So think about why I
didn't leave it here.

433
00:29:40 --> 00:29:44
OK, then we'll get the old
price, which we're going to try

434
00:29:44 --> 00:29:49
and remember, and now comes
the interesting part.

435
00:29:49 --> 00:29:51
We're going to try and
figure out how the

436
00:29:51 --> 00:29:54
price should change.


437
00:29:54 --> 00:29:59
So I'm first going to compute
something called the base move.

438
00:29:59 --> 00:30:02
Think of this as kind of the
basis from which we'll be

439
00:30:02 --> 00:30:05
computing the actual move.


440
00:30:05 --> 00:30:12
I'll draw something from the
distribution, so this is

441
00:30:12 --> 00:30:15
interesting, I'm now calling
self dot distribution, and

442
00:30:15 --> 00:30:20
remember this will be
different for each stock.

443
00:30:20 --> 00:30:23
It will return me some random
value from either the Gaussian

444
00:30:23 --> 00:30:28
or the normal distribution.


445
00:30:28 --> 00:30:30
With a different volatility for
the stocks because that was

446
00:30:30 --> 00:30:35
also selected randomly,
plus some market bias.

447
00:30:35 --> 00:30:37
Saying, well, the market on
average will go up a little

448
00:30:37 --> 00:30:42
bit, or go down a little bit.


449
00:30:42 --> 00:30:46
And then I'll set the new
price, if you will, self dot

450
00:30:46 --> 00:30:52
price, to self dot price
times 1 plus the base move.

451
00:30:52 --> 00:30:53
So notice what this says.


452
00:30:53 --> 00:31:00
If the base move is 0, then
the price doesn't change.

453
00:31:00 --> 00:31:02
So that makes sense.


454
00:31:02 --> 00:31:04
Interesting question.


455
00:31:04 --> 00:31:13
Why do you think I said self
dot price times 1 plus the base

456
00:31:13 --> 00:31:16
move, rather than just adding
the base move to the stock,

457
00:31:16 --> 00:31:18
price of the stock?


458
00:31:18 --> 00:31:22
Again, the first time I coded
this, I had an addition there

459
00:31:22 --> 00:31:24
instead of a multiplication.


460
00:31:24 --> 00:31:31
What would the ramifications
of an addition there be?

461
00:31:31 --> 00:31:35
That would say, how much the
stock changed is independent

462
00:31:35 --> 00:31:39
of its current price.


463
00:31:39 --> 00:31:43
And when I ran that it, I got
weird results, because we know

464
00:31:43 --> 00:31:49
that a Google priced at, say,
300, is much more likely to

465
00:31:49 --> 00:31:58
move by 10 points in a day than
a stock that's priced at $0.50.

466
00:31:58 --> 00:32:02
So in fact, it is the case, if
you look at data, and by the

467
00:32:02 --> 00:32:04
way, that's the way I ended up
setting a lot of these

468
00:32:04 --> 00:32:07
parameters and playing with it,
was comparing what my

469
00:32:07 --> 00:32:11
simulation said to
historical stock data.

470
00:32:11 --> 00:32:14
And indeed it is the case that
the price of the stock, the

471
00:32:14 --> 00:32:17
move, the amount of move,
tends to be proportional

472
00:32:17 --> 00:32:20
to the price of the stock.


473
00:32:20 --> 00:32:22
Expensive stocks move more.


474
00:32:22 --> 00:32:25
Interestingly enough, the
percentage moves are not much

475
00:32:25 --> 00:32:29
different between cheap
stocks and expensive stocks.

476
00:32:29 --> 00:32:34
And that's why, I ended up
using a multiplicative factor,

477
00:32:34 --> 00:32:47
rather than an additive factor.


478
00:32:47 --> 00:32:50
This is again a general lesson.


479
00:32:50 --> 00:32:53
As you build these kinds of
simulations, or anything like

480
00:32:53 --> 00:32:58
this, you need to think through
whether things should be

481
00:32:58 --> 00:32:59
multiplicative or additive.


482
00:32:59 --> 00:33:03
Because you get very different
results, typically.

483
00:33:03 --> 00:33:07
Multiplicative is what you want
to do if the amount of change

484
00:33:07 --> 00:33:13
is proportional to the current
size, whether it's price or

485
00:33:13 --> 00:33:18
anything else, and additive if
the change is independent of

486
00:33:18 --> 00:33:22
the current value, typically,
is I think the general

487
00:33:22 --> 00:33:27
way to think about it.


488
00:33:27 --> 00:33:31
Now, you'll see this other
kind of peculiar thing.

489
00:33:31 --> 00:33:39
So I've now set the price, and
then I've got this test here.

490
00:33:39 --> 00:33:45
If mo, mo stands for momentum.


491
00:33:45 --> 00:33:53
I'm now exploring the question
of whether or not stock prices

492
00:33:53 --> 00:34:03
are indeed memoryless,
or the stock changes.

493
00:34:03 --> 00:34:09
And the fancy word
for that is Poisson.

494
00:34:09 --> 00:34:14
People often model things as
Poisson processes, which is to

495
00:34:14 --> 00:34:20
say, processes in which past
behavior has no impact on

496
00:34:20 --> 00:34:24
future behavior,
it's memoryless.

497
00:34:24 --> 00:34:29
And in fact, that's what the
Efficient Market Hypothesis

498
00:34:29 --> 00:34:31
purports to say.


499
00:34:31 --> 00:34:35
It says that, since all the
information is in the current

500
00:34:35 --> 00:34:38
price, you don't have to worry
about whether it went up or

501
00:34:38 --> 00:34:45
down yesterday, to decide
what it's going to do today.

502
00:34:45 --> 00:34:48
There are people who don't
believe that, and instead

503
00:34:48 --> 00:34:52
argue that there is this
notion called momentum.

504
00:34:52 --> 00:34:54
These are called
momentum investors.

505
00:34:54 --> 00:34:58
And they say, what's most
likely to happen today, is

506
00:34:58 --> 00:35:00
what happened yesterday.


507
00:35:00 --> 00:35:01
Or more likely.


508
00:35:01 --> 00:35:04
If the stock went up yesterday,
it's more likely to go up

509
00:35:04 --> 00:35:08
today, than if it didn't
go up yesterday.

510
00:35:08 --> 00:35:13
So I wasn't sure which religion
I was willing to believe in, if

511
00:35:13 --> 00:35:17
either, so I added a parameter
called, if you believe in

512
00:35:17 --> 00:35:24
momentum, then you should
change the price by -- And here

513
00:35:24 --> 00:35:28
I just did something taking a
Gaussian times the last change,

514
00:35:28 --> 00:35:32
and, in fact, added it in.


515
00:35:32 --> 00:35:35
So if it went up yesterday, it
will more likely go up today,

516
00:35:35 --> 00:35:39
because I'm throwing in a
positive number, otherwise

517
00:35:39 --> 00:35:41
a negative number.


518
00:35:41 --> 00:35:44
Notice that this is additive.


519
00:35:44 --> 00:35:48
Because it's dealing
with yesterday's price.

520
00:35:48 --> 00:35:51
Change, with the change.


521
00:35:51 --> 00:35:56
OK, so that's why we're
dealing with that.

522
00:35:56 --> 00:35:59
Now, here's where I
should've put in this

523
00:35:59 --> 00:36:03
test that I had up here.


524
00:36:03 --> 00:36:05
Get it out from there.


525
00:36:05 --> 00:36:08
Because what I want to do is,
say if self dot price is less

526
00:36:08 --> 00:36:14
than 0.01, I'm going to set
it to 0, just keep it there.

527
00:36:14 --> 00:36:19
That doesn't solve the problem
we had before though, right?

528
00:36:19 --> 00:36:25
Then I'm going to append
it, and keep the last

529
00:36:25 --> 00:36:28
change for future use.


530
00:36:28 --> 00:36:32
OK, people understand
what's going on here?

531
00:36:32 --> 00:36:35
And then show history is just
going to produce a plot.

532
00:36:35 --> 00:36:37
We've seen that a
million times before.

533
00:36:37 --> 00:36:42
Any questions about this?


534
00:36:42 --> 00:36:45
Well, I have a question?


535
00:36:45 --> 00:36:46
Does it make any sense?


536
00:36:46 --> 00:36:48
Is it going to work at all?


537
00:36:48 --> 00:36:51
So now let's test it.


538
00:36:51 --> 00:37:01
So, I now have this unit test
program called unit test stock.

539
00:37:01 --> 00:37:04
I originally did not make it a
function, I had it in-line, and

540
00:37:04 --> 00:37:07
I realized that was really
stupid, because I wanted

541
00:37:07 --> 00:37:10
to do it a lot of times.


542
00:37:10 --> 00:37:15
So it's got an internal
procedure, internal function,

543
00:37:15 --> 00:37:20
local to the unit test,
that runs the simulation.

544
00:37:20 --> 00:37:25
And it takes the set of stocks
to simulate, a fig, figure

545
00:37:25 --> 00:37:29
number, this is going to print
a bunch of graphs, and I want

546
00:37:29 --> 00:37:32
to say what graph it is,
and whether or not I

547
00:37:32 --> 00:37:35
believe in big mo.


548
00:37:35 --> 00:37:41
It sets the mean to 0, and then
for s in the stocks, it moves

549
00:37:41 --> 00:37:46
it, giving it the bias and
the momentum, then it

550
00:37:46 --> 00:37:49
shows the history.


551
00:37:49 --> 00:37:53
And then computes the mean
of, getting me the mean

552
00:37:53 --> 00:37:54
of all the stocks in it.


553
00:37:54 --> 00:37:57
We've seen this sort of
thing many times before.

554
00:37:57 --> 00:37:59
I've then got some constants.


555
00:37:59 --> 00:38:02
By the way, I want to emphasize
that I've named these constants

556
00:38:02 --> 00:38:04
to make it easier to change.


557
00:38:04 --> 00:38:10
Starting with 20
stocks, 100 days.

558
00:38:10 --> 00:38:15
And then what I do is, I stock
sub 1, stocks 1 will be the

559
00:38:15 --> 00:38:18
empty list, stocks 2
is the empty list.

560
00:38:18 --> 00:38:27
Why do you think I'm
starting with bias of 0?

561
00:38:27 --> 00:38:32
Because, what do you think the
mean should be, if I simulate

562
00:38:32 --> 00:38:35
various things that
the bias of 0?

563
00:38:35 --> 00:38:39
I start $100 as the average
price of the stock,

564
00:38:39 --> 00:38:42
what should the average
price of the stock be?

565
00:38:42 --> 00:38:45
If my code is correct, what
should the average price be,

566
00:38:45 --> 00:38:50
after say, 100 days,
if there's no bias.

567
00:38:50 --> 00:38:51
Pardon?


568
00:38:51 --> 00:38:53
100, exactly.


569
00:38:53 --> 00:38:56
Since there's no upward
or downward bias.

570
00:38:56 --> 00:39:00
They may fluctuate wildly, but
if I look at enough stocks, the

571
00:39:00 --> 00:39:03
average should be
right around 100.

572
00:39:03 --> 00:39:05
I don't know what the
average would be if I

573
00:39:05 --> 00:39:07
chose a different bias.


574
00:39:07 --> 00:39:11
It's a little bit complicated,
so I chose the simplest bias.

575
00:39:11 --> 00:39:15
Important lesson, so that there
would be some predictability in

576
00:39:15 --> 00:39:19
the results, and I would have
some, if you will, smoke test

577
00:39:19 --> 00:39:22
for knowing whether or not
I was getting, my code

578
00:39:22 --> 00:39:27
seemed to be working.


579
00:39:27 --> 00:39:31
All right, and initially, well,
maybe initially, just to be

580
00:39:31 --> 00:39:39
simple I'm going to start
momentum equal to false.

581
00:39:39 --> 00:39:41
Because, again, it seems
simpler have a model where

582
00:39:41 --> 00:39:43
there's no momentum.


583
00:39:43 --> 00:39:46
I'm looking for the simplest
model possible for the

584
00:39:46 --> 00:39:49
first time I run it.


585
00:39:49 --> 00:39:51
And then we looked at this
little loop before, for i in

586
00:39:51 --> 00:39:55
range number of stocks, I'm
going to create two different

587
00:39:55 --> 00:40:00
lists of stocks, one where the
moves, or distributions, are

588
00:40:00 --> 00:40:04
chosen from a uniform, and the
other where they're Gaussian.

589
00:40:04 --> 00:40:07
Because I'm sort of curious as
to, again, which is the right

590
00:40:07 --> 00:40:14
way to think about
this, all right?

591
00:40:14 --> 00:40:18
And then, I'm going
to just call it.

592
00:40:18 --> 00:40:22
We'll see what we get.


593
00:40:22 --> 00:40:22
So let's do it.


594
00:40:22 --> 00:40:25
Let's hope that all the changes
I mad have not introduced

595
00:40:25 --> 00:40:29
a syntax error.


596
00:40:29 --> 00:40:31
All right, well at least
it did something.

597
00:40:31 --> 00:40:39
Let's see what it did.


598
00:40:39 --> 00:40:44
So the test on the left, you'll
remember, was the one with test

599
00:40:44 --> 00:40:48
one, I believe, was the uniform
distribution, and test

600
00:40:48 --> 00:40:51
two is the Gaussian.


601
00:40:51 --> 00:40:55
So, but let's, what
should we do first?

602
00:40:55 --> 00:41:00
Well, let's do the smoke test
number one: is the mean more

603
00:41:00 --> 00:41:02
or less what we expected?


604
00:41:02 --> 00:41:05
Well, it looks like it's
dead on 100, which was our

605
00:41:05 --> 00:41:09
initial price in test two.


606
00:41:09 --> 00:41:12
And in test one it's a
little bit above 100.

607
00:41:12 --> 00:41:16
But, we didn't do that many
stocks, or that many days,

608
00:41:16 --> 00:41:24
so it's quite plausible
that it's correct.

609
00:41:24 --> 00:41:28
But, just to be sure, not to be
sure, but just to increase my

610
00:41:28 --> 00:41:46
confidence, I'm going
to just run it again.

611
00:41:46 --> 00:41:50
Well, here I'm a little bit
below 100 and in two, and

612
00:41:50 --> 00:41:53
test one a little bit
below 100 as well.

613
00:41:53 --> 00:41:56
You remember last time was
a little bit above 100.

614
00:41:56 --> 00:41:59
I feel pretty good about this,
and in fact I ran it a lot

615
00:41:59 --> 00:42:01
of times in my office.


616
00:42:01 --> 00:42:05
And it just bounces around,
hovering around 100.

617
00:42:05 --> 00:42:06
Course, this is the
wrong way to do it.

618
00:42:06 --> 00:42:09
I should really just put it in
a nice test harness, where I

619
00:42:09 --> 00:42:14
run 100, 200, 1,000 trials, but
I didn't want to bore

620
00:42:14 --> 00:42:15
you with that here.


621
00:42:15 --> 00:42:20
So we'll see that, OK, we
passed the first smoke test.

622
00:42:20 --> 00:42:25
We seem to be where
we expect to be.

623
00:42:25 --> 00:42:30
Well, let's try smoke test two.


624
00:42:30 --> 00:42:32
What else might we want
to see, to see if we got

625
00:42:32 --> 00:42:34
things working properly?


626
00:42:34 --> 00:42:38
Well, I kind of ignored the
notion of bias by making it

627
00:42:38 --> 00:42:49
0, so let's give it
a big bias here.

628
00:42:49 --> 00:43:09
Assuming it will
let me edit it.

629
00:43:09 --> 00:43:11
We just gotta start it
up again, it's the

630
00:43:11 --> 00:43:34
safest thing to do.


631
00:43:34 --> 00:43:38
You wouldn't think I would
have, I don't have -- all

632
00:43:38 --> 00:43:42
right, be that way about it.


633
00:43:42 --> 00:43:46
Fortunately, we've been
through this before.

634
00:43:46 --> 00:43:49
We know if we
relaunch the Finder.

635
00:43:49 --> 00:43:58
Who says Mac OS is flawless?


636
00:43:58 --> 00:44:07
All right, we were down here,
and I was saying, let's try

637
00:44:07 --> 00:44:10
a larger, introduce a bias.


638
00:44:10 --> 00:44:12
Again, we're trying to
see if it does what

639
00:44:12 --> 00:44:13
we think it might do.


640
00:44:13 --> 00:44:17
So what do you think it
should do with a bias?

641
00:44:17 --> 00:44:21
Where should the mean be now?


642
00:44:21 --> 00:44:24
Still around 100?


643
00:44:24 --> 00:44:26
Or higher, right?


644
00:44:26 --> 00:44:29
Because we've now put
in a bias suggesting

645
00:44:29 --> 00:44:31
that it should go up.


646
00:44:31 --> 00:44:33
Oops.


647
00:44:33 --> 00:44:37
It wouldn't have hurt it.


648
00:44:37 --> 00:44:39
All right.


649
00:44:39 --> 00:44:49
So let's run it.


650
00:44:49 --> 00:44:54
Sure enough, for one, we see,
test two, it's a little bit

651
00:44:54 --> 00:45:02
over 100, and for test
one it's way over 100.

652
00:45:02 --> 00:45:11
Well, let's make sure
it's not a fluke.

653
00:45:11 --> 00:45:21
Try it again.


654
00:45:21 --> 00:45:26
So, sure enough, changing
the bias changed the price,

655
00:45:26 --> 00:45:31
and even changed it in
the right direction.

656
00:45:31 --> 00:45:33
So we can feel pretty
comfortable that it's doing

657
00:45:33 --> 00:45:35
something good with that.


658
00:45:35 --> 00:45:37
We could also feel pretty
comfortable that that's

659
00:45:37 --> 00:45:40
probably way too
high a bias, right?

660
00:45:40 --> 00:45:45
We would not expect that the
mean should be over 160, or

661
00:45:45 --> 00:45:49
in one case, 150, after only
100 days trading, right?

662
00:45:49 --> 00:45:53
Things don't typically
go up 50% in 100 days.

663
00:45:53 --> 00:46:03
They go down 50%, but --
All right, so that's good.

664
00:46:03 --> 00:46:06
Oh, let's look at
something else now.

665
00:46:06 --> 00:46:10
Let's go back to where,
a simpler bias here.

666
00:46:10 --> 00:46:13
We'll run it again.


667
00:46:13 --> 00:46:16
And think about, what's the
difference between the

668
00:46:16 --> 00:46:26
Gaussian and the normal?


669
00:46:26 --> 00:46:35
Can we deduce anything
about those?

670
00:46:35 --> 00:46:37
Not, well, let me ask you.


671
00:46:37 --> 00:46:45
What do you think, yes or no?


672
00:46:45 --> 00:46:47
Anybody see anything
interesting here?

673
00:46:47 --> 00:46:47
Yeah?


674
00:46:47 --> 00:46:52
STUDENT: The variance of the
Gaussian seems to be less than

675
00:46:52 --> 00:46:54
the variance of the uniform.


676
00:46:54 --> 00:46:55
PROFESSOR: The variance
of the Gaussian --

677
00:46:55 --> 00:46:59
STUDENT: -- is less.


678
00:46:59 --> 00:47:02
PROFESSOR: So all right, that
appears to be the case here.

679
00:47:02 --> 00:47:04
But let's run it again,
as we've done with

680
00:47:04 --> 00:47:06
all the other tests.


681
00:47:06 --> 00:47:09
So we have a hypothesis.


682
00:47:09 --> 00:47:13
Let's not fall victim to
the Oklahoma sharpshooter.

683
00:47:13 --> 00:47:19
We'll test our hypothesis, or
at least examine it again,

684
00:47:19 --> 00:47:27
see if it's, in some
sense, repeatable.

685
00:47:27 --> 00:47:32
Well, now what do we see?


686
00:47:32 --> 00:47:35
Doesn't seem to be true
this time, right?

687
00:47:35 --> 00:47:37
Not obviously.


688
00:47:37 --> 00:47:42
So, we're not sure about this.


689
00:47:42 --> 00:47:45
So this is something
that we would need to

690
00:47:45 --> 00:47:49
investigate further.


691
00:47:49 --> 00:47:52
And we would need, to have to
look at it, and it's going to

692
00:47:52 --> 00:47:57
be very tricky, by the way, as
to what the right answer is.

693
00:47:57 --> 00:48:04
But if you think about it, it
would not be surprising if the

694
00:48:04 --> 00:48:09
Gaussians, at least, gave us
some surprising, more extreme,

695
00:48:09 --> 00:48:12
results, than the uniform.


696
00:48:12 --> 00:48:16
Because the uniform, as we've
set it up here, is bounded.

697
00:48:16 --> 00:48:22
The minimum and the
maximum is bounded.

698
00:48:22 --> 00:48:26
With the Gaussian,
there's a tail.

699
00:48:26 --> 00:48:29
And you might every once in a
while get this, at least as

700
00:48:29 --> 00:48:34
we've done it in this case,
this large move out at the end.

701
00:48:34 --> 00:48:36
You might not.


702
00:48:36 --> 00:48:40
There's nothing profound about
this, other than the

703
00:48:40 --> 00:48:43
understanding that the details
of how you set these things

704
00:48:43 --> 00:48:47
up can matter a lot.


705
00:48:47 --> 00:48:54
Well, the final thing I want
to look at is momentum.

706
00:48:54 --> 00:49:07
So let's go back, and let's
set mo to true here.

707
00:49:07 --> 00:49:10
Well, doesn't want us to
set mo to true here.

708
00:49:10 --> 00:49:16
Ah, there it does.


709
00:49:16 --> 00:49:19
So, and now let's run it
and see what happens.

710
00:49:19 --> 00:49:25
What do you think
should happen?

711
00:49:25 --> 00:49:26
Anybody?


712
00:49:26 --> 00:49:31
STUDENT: [INAUDIBLE]


713
00:49:31 --> 00:49:36
PROFESSOR: I think
you're right.

714
00:49:36 --> 00:49:41
These ones should curl, see
if I can -- oh, not bad.

715
00:49:41 --> 00:49:48
Let's run it.


716
00:49:48 --> 00:49:53
Well, it's a little hard
to see, but things

717
00:49:53 --> 00:49:56
tend to take off.


718
00:49:56 --> 00:49:59
Because once things started
moving, it tends to

719
00:49:59 --> 00:50:05
move in that direction.


720
00:50:05 --> 00:50:10
All right.


721
00:50:10 --> 00:50:12
How do we go about choosing
these parameters?

722
00:50:12 --> 00:50:14
How do we go about
deciding what to do?

723
00:50:14 --> 00:50:18
Well, we play with it, the way
I've been playing with it, and

724
00:50:18 --> 00:50:22
compare the results to
some set of real data.

725
00:50:22 --> 00:50:26
And then we try and get our
simulation to match the past,

726
00:50:26 --> 00:50:31
and hope that that will
help it predict the future.

727
00:50:31 --> 00:50:33
We're not enough time
to go through all the,

728
00:50:33 --> 00:50:35
to do that a lot.


729
00:50:35 --> 00:50:39
I will be posting code that you
can play with, and I suggest

730
00:50:39 --> 00:50:42
you go through exactly
this kind of exercise.

731
00:50:42 --> 00:50:45
Because this is really
the way that people do

732
00:50:45 --> 00:50:46
develop simulations.


733
00:50:46 --> 00:50:50
They don't, out of whole cloth,
get it right the first time.

734
00:50:50 --> 00:50:54
They build them, they do what
if games, they play with them,

735
00:50:54 --> 00:50:57
and then they try and adjust
them to get them right.

736
00:50:57 --> 00:51:00
The nice thing here is, you can
decide whether you believe

737
00:51:00 --> 00:51:04
momentum and see what it would
mean, or not mean, etc.

738
00:51:04 --> 00:51:06
All right, one more lecture.


739
00:51:06 --> 00:51:09
See you guys next week.


740
00:51:09 --> 00:51:09