All right.
Good morning.
The subject of today's
lecture is options.
And I think, maybe I'd better
first define what an option
is, before I move to say
anything about them.
Because some of you may not have
encountered them, because
they're not part of everyday
life for most people, although
they are, in a sense.
I'll get back to that.
Let me just define
the term here.
[SIDE CONVERSATION]
PROFESSOR ROBERT SHILLER: So
there's two kinds of options.
There's a call and a put, OK?
A call option is an option to
buy something at a specified
price, and the price is called
the ''exercise price'' or
''strike price.'' Those
are synonyms.
And a put option is the right
to sell something at the
specified exercise price.
And it has another term it has
to be specified, and that's
the exercise date.
OK.
Options go back thousands
of years.
It must have happened before we
have any recorded records.
If you're thinking of buying
something from someone, but
you don't want to put up the
money today, you go to some
lawyer, and say, write
up a contract.
I want to buy an option
to buy this thing.
So, for example, if you are
thinking of building a
building on land that is owned
now by a farmer, but you're
not ready to do it.
You may be thinking about it.
You can go to the farmer and
say, I'd like to buy that
corner of that acre there.
I'd like to have the
right to buy it.
I'll pay you now
for the right.
And you get a lawyer and you
write up a contract.
And that's an option.
You have an option to buy at the
exercise price until the
exercise date.
Now, in modern terminology,
we have two kinds,
American and European.
It doesn't refer to geography,
those two terms. The terms
refer, instead, to when
you can exercise.
So, the American option is
better than the European
option for the buyer, because
the American option can be
exercised at any time until
the exercise date.
Whereas the European option can
be exercised only on the
exercise date.
But you see the American option
has to be better, or
not worse than --
I don't know if it's strictly
better -- but not worse than
the European option, because
you have more options.
I think, we've defined
what they are.
Do you understand well
enough what they are?
They occur naturally in life.
I remember, Avinash Dixit was
writing about options and he
said, well, when you're dating
someone and you know the
person will marry you, you have
an option that you can
exercise at any time by
agreeing to marry.
Now, one of the theorems in
option theory is, you usually
don't want to exercise
a call option early.
And so, Dixit would say, well,
maybe that's why a lot of
people have trouble
getting married.
They don't want to exercise
their option early.
What we'll see is that options
have option value.
They give you a choice, and so
there's something there.
When you exercise an option,
that is, when you actually buy
the thing, or in the case of
a put, sell the thing, then
you're losing the choice.
So, you've given up something.
Of course, you have to also
exercise eventually, if things
are going to make sense.
Usually, when we talk about
options, we're talking now
about options to buy a share
of stock, or 100 shares of
stock, and that's the
usual example.
But they occur all
over the place.
Let me mention some other
examples of options.
The usual story is
the stock option.
You go to your broker and you
say, I'd like to buy an option
to buy 100 shares
of Microsoft.
I don't want to buy Microsoft, I
want to buy an option to buy
Microsoft, which happens to be
cheaper, by the way, usually.
Usually, it costs more money to
actually buy the thing than
to buy the right to buy the
thing at another price.
We'll get back to that.
But in a sense, let's think
about this, stocks themselves
are options in a sense, with
a zero exercise price.
Maybe, I'll have to get
back and explain, but
what I mean is --
let me get back and explain
that in a minute.
But let me go ahead to other
examples, mortgages.
An ordinary home mortgage has
an option characteristic to
it, in the sense that, if the
price of your home drops a
lot, you can just walk away from
the mortgage, and say,
I'm out of here.
It's like not exercising
an option.
It's analogous.
Or I can choose to prepay a
mortgage early, and that's
like exercising an option.
So, option pricing gets into
all sorts of things.
OK.
I thought I should say something
about the purposes
of options, before I move on to
try to discuss what their
properties and pricing are,
which is the main subject of
this lecture.
I can give two different
justifications for options.
Why do we have options?
Some people cynically think
that options are
just gambling vehicles.
It's another way to gamble.
You can go to the casino,
you can play poker,
or you can buy options.
Well, I think for some people
that's just what it is, the
volatile risky investments
that can make
you a lot of money.
But I think, they have a basic
purpose, or purposes.
First of all, theoretical.
If we were trying to design the
ideal financial system,
what would we do?
Some people thought of ideal
economic systems without
reference to finance,
like Karl Marx --
I come back to him -- the great
communist, who thought
that we would have an ideal
communist state and there'd be
no financial markets.
When they actually tried it
and they tried to do it, I
think they gradually realized
that not having any financial
markets makes our
entrepreneurship, our
management of enterprises,
kind of blind.
We can't see, where we're going
because there's no prices.
We don't know what anything
is worth.
There was an old joke that the
communist countries survived
only because they had prices
from capitalist
countries to rely on.
Otherwise, they don't know
anything about values or
profits, right?
So, we need prices.
Many people have written about
this, but I mentioned, in
1964, Kenneth Arrow, who is an
economic theorist, wrote a
classic paper, in which he
argued that, unless we have
prices for all states of nature,
there's a sense in
which the economic system
is inefficient.
You really need the price of
everything, including the
price of some possibility.
In a sense, that's what options
are giving you, the
existence of options
is giving you.
So, Steven Ross, who used to
teach here at Yale, a friend
of mine, lives here in New
Haven, in 1976, in The
Quarterly Journal of Economics,
wrote a classic
paper about options, showing
that, in a sense, they
complete the state space.
They create prices for
everything that affects
decision-making.
I'm not going to get into the
technicalities of the paper,
but I wanted to start with a
theoretical justification for
options, so you'll see
why we're doing this.
I don't want this to come across
as a lecture, on how
you can gamble in the
options market.
This is about making things work
right for the economic
system, improving
human welfare.
But a lot of people
don't get that.
That's why Karl Marx
was so successful.
It seems too abstract.
What does this options
market do for us?
Let me just go back to the
example I started out with.
You're a construction firm,
and you're thinking of
building something, a new
supermarket, where people can
buy their food.
And you note that there's a pair
of expressways crossing
somewhere, and you think, that's
the perfect place to
build a mega supermarket,
because everyone can
get there by car.
And there's a lot of land,
I can build a big
parking lot, perfect.
But before you think further,
you go to buy an option on the
land, right?
So, you knock on the door at the
farmhouse, and there's a
farmer with all these acres, and
you say, I'm thinking of
building a mega supermarket
here.
I'd like to buy an option
on your farm.
You learn something
at that moment.
You might learn that the farmer
says, I've already sold
an option, so I can't do it.
You could try to talk to the
person I sold it to, and see
if you can buy it from him.
Or the farmer might say, I've
had three other offers, and
I'm raising my price
to some millions
and millions of dollars.
Then, you have second thoughts
about doing it.
You see what I'm saying,
that the price
discovery is in there?
It's making things happen
differently.
You're learning something.
The farmer is learning
something.
You are learning something from
the options market, and
ultimately it decides where
that supermarket will go.
So, that's the theoretical
purpose of options.
I wanted to talk, also, about a
behavioral purpose of options.
It's a little fuzzier about the
actual benefits of options
from this standpoint.
The behavioral theory of
options says that --
Very many different aspects
of human behavior tie into
options, but I would say it
has something to do with
attention anomalies
and salience.
Psychologists talk about this,
that people make mistakes very
commonly in what they pay
attention to, what strikes the
fancy of their imagination.
Salience is something
psychologists also talk about.
Salient events are events that
tend to attract attention,
tend to be remembered.
Now, when you think of options,
a lot of options are
what are called incentive
options, OK?
And when you get your first job,
you may discover this.
They'll give you options
to buy shares in the
company you work for.
Why do they do that?
I think it's because of certain
human behavioral
traits that I mention
here, your
attention and your salience.
It's not necessarily very
expensive for a company to
give you options to buy shares
in the company, but it puts
you in a situation, where you
start to pay attention to the
value of the company.
It becomes salient for you, and
you start hoping that the
price of the company will go up,
because you have options
to buy it, at a strike price.
You hope that the company's
price per share goes above
your strike price,
because then your
options are worth something.
They're in the money.
So, it may change your
motivation and your morale at
work, or sense of identity
with the company.
All these sorts of
things figure in.
That's why we have incentive
options.
They can also give you
peace of mind.
Insurance is actually related to
options in the sense that,
when you buy insurance on your
house, it's like buying a put
option on your house, although
it may be not directly
connected to the home's
value, right?
When you buy an insurance policy
in your house, and the
house burns down, you collect
on the insurance policy.
Well, the price of your
house fell to zero.
If you had bought a put option
on the house, it would do the
same thing, right?
You would have an option to
sell it at a high price on
something that's
now worthless.
So, insurance is like options,
and insurance gives
you peace of mind.
So, people think in certain
repetitive patterns, and one
of them is, that I would like to
not worry about something.
So, I can get peace of mind,
if I have a put option on
something that I might otherwise
worry about.
All right, maybe that's enough
of an introduction, but I'm
giving you both theoretical
reasons for options and
behavioral reasons.
I think of them as basically
inevitable.
You may have people advising
you not to bother
with options markets.
That might be right for you, in
a sense, but I think that
they're always going to be with
us, and so it's something
that we have to understand.
I have a newspaper clipping
that I cut out.
I've been teaching this course
for over 20 years, so
sometimes I don't update
my newspaper clippings.
I have a newspaper clipping from
the options page that I
made in 2002, OK?
So, that's nine years ago.
But I can't update it anymore,
because newspapers don't print
option prices anymore.
So, I could go on some
electronic trade account and
get an updated option page.
But why don't we just stick with
The Wall Street Journal?
This is a clipping from The
Wall Street Journal, April
2002, when they used to have
an options page, OK?
I just picked America Online.
I don't know why.
It's an interesting company.
You remember America Online,
a web presence?
It used to be bigger
than it is now.
And actually, in 2000, America
Online merged with Time
Warner, OK?
So, we actually have
two different rows
corresponding --
Forget Ace Limited, the second
row says AOL.TW.
That's America Online Time
Warner, the merged company,
and then below that, they have
America Online itself.
These were options that were
issued before the merger, and
they apparently are being
exercised in terms of the same
AOL Time Warner stock.
AOL, by the way, was spun off
by Time Warner last year, so
they had a divorce.
They were married in 2000.
They were divorced in 2010.
So, you can get back to
it, AOL options, now.
So, anyway, it shows
the price of the
share at $21.85 a share.
So, you take any of these rows,
and it shows you, for
various strike prices, what
the options prices are.
So, let's go to the top row.
A strike price of $20.00,
expiring in May of 2002, which
is one month into the future.
Remember, it's April
2002 right now.
The volume is the number of
options that were traded
yesterday, and the $2.55 up
there is the price of a call
option, the last price of the
option to be traded yesterday.
This is the morning paper.
It's reporting on yesterday
morning's prices [correction:
yesterday's prices
at closing].
And then, there's put
options traded.
A lot more puts were
traded on that day.
There were 2000 put options
traded on that day in April
2002, and the last price of
the put option was $0.85.
For $0.85, you could buy the
right to sell a share of AOL
Time Warner at $20.00, OK?
And similarly, you could
buy the right to buy it
at $20.00 for $2.55.
So, these are different
strike prices and
different exercise dates.
This one --
I can reach it --
is to buy it at, if it's a call,
$25.00 strike price,
costs you $0.45 to buy that.
But if you want to buy a put,
it costs you $3.60.
And we want to try to understand
these prices, OK?
That's the purpose here.
So, let me say one thing more
before I get into that.
This is presented for the
potential buyers, OK?
These are options prices.
There's also the seller
of the option.
They're called the writer
of the option.
I gave you an example before,
when I talked about the farmer
and you thinking of building
a supermarket.
So, you are the buyer of the
option and the farmer is the
writer of the option.
The farmer is writing
the option to you.
You could also consider buying
an option from someone else,
who's not even the
farmer, right?
It could be some speculator.
You don't have to go
to the farmer.
You can go to somebody else and
say, I'd like to buy an
option on that farm
over there.
And someone would say, sure,
I'll sell you an option on it.
And then I'm good for it.
That means I have to go and buy
it at whatever price from
the farmer.
Maybe that's not such
a good idea.
He might sense my urgency
to buy it.
But if it's a stock, someone
can write an option, who
doesn't even own the stock.
And so, that's called a naked
seller of an option, OK?
Neither the buyer nor the
seller ever have to
trade in the stock.
This is a market by itself.
You could buy an option, and
then you could sell it as an
option without ever
exercising it.
The writer could write an
option, and then buy an option
to cancel it out later, and
then, essentially, get out of
that contract.
So, the option becomes a market
of its own, where
prices of options start to
look like an independent
market, and this is called
a derivatives market.
There's an underlying stock
price, but this is a
derivative of the stock price.
The first options exchange was
the Chicago Board Options
Exchange, which came
in in 1973.
Before that, options were
traded, but they were traded
through brokers and they didn't
have the same presence.
You didn't see all these options
prices in newspapers.
It's when they opened the market
for options, that the
options trading became
a big thing.
So, options markets are
relatively new, if you
consider '73 new.
You weren't born then.
It's not really that long ago.
Since then, there are many more
options exchanges, but
CBOE is the first one.
They're now all over
the world.
And we also have options
on futures.
And so, futures exchanges now
routinely trade options on
their futures contracts.
So, that's a derivative on a
derivative, but it's done.
So, let me draw a simple picture
of option pricing.
So, this is the stock price
and this is the
option price, OK?
And I'm going to mark here,
the exercise price.
Let's look at the exercise
date, the last day.
The option is about to expire,
and this is your last chance
to buy the stock.
Then, it doesn't matter, on
that day, whether it's an
American or European option.
They're both the same
on the last day.
What is the price of the option
as a function of the
stock price?
Well, if the stock price is less
than the exercise price,
the option is worthless,
right?
It will not be exercised.
You won't exercise an option
to buy it for more than you
could just buy it on
the market, right?
STUDENT: You have
to say ''call''.
PROFESSOR ROBERT SHILLER:
Did I not say call?
Yes, I'll put it up here.
We're talking about
call options.
Thank you.
But if it's above the exercise
price, this is a 45 degree
angle, that's a line with the
slope of one, the option
prices rises with the stock.
In fact, it just equals the
stock price minus the exercise
price, right?
So, this region, we say, is
''out of the money.'' The
option is out of the money,
when its prices
[clarification: the stock
price], for a call, is less
than the exercise price.
Here, it's ''in the money.''
I'll put it up
here, in the money.
And then, on the exercise date,
it will always equal the
stock price minus the
exercise price.
So, it's very simple.
Now, one confusion that's often
made: I gave the example
of building a shopping center
or a supermarket on a farm.
Now, someone might think that
you buy an option on it, so
that you can think about it and
make up your mind later.
Well, in a sense, you
could do that.
But the thing is, you will
exercise the option whether or
not you build the shopping mall
or this supermarket, if
it's in the money, right?
Suppose, you changed your mind,
and I don't want to
build the supermarket.
But I'm sitting on an option
that I bought, to buy his land
for a price, which is less than
the market price for it.
Of course, I'll buy it.
So, you're going to buy it,
whether you build the shopping
center or not.
You always exercise the option,
if it's in the money
on the last day.
That's the assumption.
I mean you could not, I suppose,
if you like the
farmer and you want
to be a nice guy.
I don't know.
But usually, what it is, it's a
non-linear relation between
the stock price and
the derivative.
So, the derivative is a broken
straight line function of the
stock price.
Whereas all the portfolios
we construct, are linear.
They're straight lines.
They don't have a
break in them.
So, the option creates a break
in the function of
the stock price --
and this is why Ross emphasized
that options price
something very different,
that's not priced
in the regular --
no portfolio shows you this
broken straight line relation.
Now, I wanted to then
talk about a put.
What is a put?
Let me erase, where it says
in and out of the money.
I'll show it.
I'll do this with a dashed line,
so that you'll see which
one is which --
I'm leaving the call line up.
With a put, a put is out
of the money up here --
I can't really show
it too well --
if the stock price is above the
exercise price, because
you're selling now.
And it's in the money, if the
stock price is less than the
exercise price --
I didn't draw that very well.
That's supposed to be
a 45 degree line.
That's a 45 degree angle, has
a slope of minus one, right?
That's on the exercise date.
Now, it's interesting that
there's a pretty simple
pattern here between
puts and calls.
What if I buy one call and I
short one put, all right?
Or write a put, writing a put
and shorting a put are the
same thing, all right?
What does that portfolio
look like?
Well, if I put that portfolio
together, I want to have plus
one call minus one
put, all right?
My portfolio relation to the
stock price is going to look
like that, right?
It's just going to be
a straight line.
So, the value of my portfolio
is equal to the stock price
minus the exercise price.
Simple as that.
And my portfolio can be negative
now, because I've
shorted something.
I can have a negative
portfolio value.
That's very simple,
can you see that?
This leads us to the put-call
parity equation.
If a put minus a call
[correction: a call minus a
put] is the same thing as the
stock minus the exercise
price, then the prices should
add up too, right?
So, put-call parity --
there's different ways
of writing this.
But it says that the stock price
equals the call price
minus put price plus exercise
price on the last day, on the
exercise day, right?
It's simple.
This is put-call parity
on the exercise date.
Now, let's think about some day
before the exercise date.
Well, you know this is
going to happen on
the exercise date.
So, at any date before the
exercise date, the same thing
should hold, except that
we've got to make
this the present value.
Present discounted value
of the exercise price.
And also we have to add in, in
case there is any dividends
paid between now and the
exercise date, plus the
present discounted value of
dividends paid between now and
the exercise date.
Because the stock gets
that, and the option
holders don't, OK?
So, that's called the put-call
parity relation.
And now I can cross out
''exercise date.'' This should
hold on all dates.
Because if it didn't hold, there
would be an arbitrage,
or profit opportunity.
So, it should hold on this
page, except for minor
failures to hold.
It should hold approximately
on this page.
And let me give you
one example.
See, if it holds.
Let's consider the one
that I can reach.
OK, oh, this is the
stock price.
So, what do I have?
The biggest thing here is the
strike price, exercise price.
So, we want to do --
we'll do this line, $25.00 plus
$0.45 minus $3.50, and
I'm assuming there's
no dividend paid
between now and May.
It comes out very
close to $21.85.
I can't do the arithmetic
in my head.
It may not hold exactly, because
these prices may not
all have been quoted at exactly
the same time, and
there's some transactional
costs that limit this.
Do you see that?
So, because of the put-call
parity relation, The Wall
Street Journal didn't even
bother to put the put prices
in, because you can get
one from the other.
But they do put them in, just
because people like to see
them, and some people might be
trying to profit from the
put-call parity arbitrage.
But for our purposes, we only
have to do call pricing.
Once we've got call pricing,
we've got put prices.
So, I just use the put-call
parity relation and I get put
price prices.
So, now let's think about
how you would price puts
[correction: calls].
The price of a put
[correction: call],
we know what it is on the
exercise date, right?
I'm going to forget
the dashed lines.
There's no dashed lines
here anymore.
We're just talking about
call prices.
All right, so this shows
the price of a put on
the last day --
of a call on the last day.
Now, what about an
earlier day?
[clarification: The following
argument about price bounds
solely applies to
call options.
It also abstracts from dividend
payments of the
underlying stock.]
Well, the price of a call can
never be negative, right?
So, the call price has to
be above this line.
It can never be worth less than
the stock price minus the
exercise price, even before
the exercise date.
And also, it can't be
worth more than
the stock price itself.
I'll draw a 45 degree line
from the origin.
That's supposed to be
parallels of that.
It's obvious that the call price
has to be above this
broken straight line, but
not too far above it.
Above this broken straight line,
representing the price
as a function of the stock
price on the last day.
And the closer you get to the
last day, the closer the
options price will get
to that curve.
So, on some day before the
exercise date, the call option
price will probably look
something like that, right?
It's above the broken straight
line because of option value.
So, think it this way, suppose
an option is out of
the money today --
well, you can see out of
the money options.
For a call, this is out
of the money, right?
Because its stock price is
$21.85, but I've got an option
to buy it for $25.00.
All right, that's going to be
worthless, unless the option
price [correction: stock price]
goes up before it expires.
So, it's only worth something,
because there's a chance that
it will be worth something
on the exercise date.
And what are people paying
for that chance?
$0.45, not much.
Why are they paying so little?
Well, you can say intuitively,
it's because it's pretty far.
$21.85 is pretty far from
$25.00, and this option only
has a month to go.
What's the chance that the price
will go up that much?
Well, there is a chance,
but it's not that big.
So, I'm only willing
to pay $0.45 to buy
an option like that.
So, we're somewhere like
here on that row
that I've shown you.
The reason you don't want to
exercise an option early is,
because, if you exercise it
early, your value drops down
to the broken straight
line, right?
It's always worth more than
the broken straight line
indicates before the
exercise date.
So, if you want to get your
money out, sell the option.
Don't exercise it early.
So, that's why the distinction
between European and American
options is not as big or as
important as you might think,
at first. [clarification:
American call options should
indeed not be exercised early.
However, there are circumstances
under which it
is optimal to exercise an
American put option early.]
So, we can just price European
options, and then we can infer
what other options would
be, what put
options would be worth.
Let's now talk about
pricing of options.
And the main pricing equation
that we're going to use is the
Black-Scholes Option
Pricing equation.
But, before that, I wanted to
just give you a simple story
of options pricing, just
to give you some
idea, how it works.
And then I'm going to not
actually derive the
Black-Scholes formula, but I'm
going to show it to you.
I'm going to tell you a simple
story, just to give some
intuitive feel about the
pricing of options.
And to simplify the story, I'm
going to tell a story about a
world, in which there's only
two possible prices for the
underlying stock.
That makes it binomial.
There's only two things that can
happen, and you can either
be high or low, all right?
So, let me get my notation.
I'm going to use S as the
stock price, all right?
I'm going to assume that the
stock price, that's today --
this is also a simple world in
that there's only one day.
The option expires tomorrow.
There's only one more price
we're going to see.
So, the stock is either going
to go up or down.
So, u is equal to one plus the
fraction that it goes up.
u stands for up.
And d is down, is one plus
the fraction down.
So, that means that stock price
either becomes Su, which
means it goes up by a fraction,
multiple u, or it is
Sd, which means it goes
down by a multiple d.
And that's all we know, OK?
But now we have a call option:
Call C the price of the call.
We're going to try to
derive what that is.
But we know, from our broken
straight line analysis, we
know what C sub u is, the price
if the stock goes up.
And we know what C sub d is,
it's the price if down, OK?
So, suppose the option has
exercise price E, all right.
Do you understand this world?
Simple story.
Now, what I want to do is
consider a portfolio of both
the stock and the option,
that is riskless.
I'm going to buy a number of
options equal to H. H is the
hedge ratio, which is the number
of shares purchased per
option sold.
So, I'm going to sell a call
option to hedge the stock
price, to reduce the risk
of the stock price, OK?
And so, hedge ratio is shares
purchased over options.
Each option is to buy
one share, OK?
So, what I'm going to
do is, write one
call and buy H shares.
So, let me erase this and
start over again.
I'm on my way to deriving the
options price for you --
a little bit of math.
So, I'm going to write one call
and buy H shares, OK.
If the stock goes up, if we
discover we're in an up world
next period, my portfolio
is worth uHS
minus C sub u, right?
Because the share price goes
from S to uS, and I've got H
shares, and I've written
a call, so I have
to pay C sub u.
If it's down, then my
portfolio is dHS
minus C sub d, OK?
This is simple enough?
Now, what I want to do is
eliminate all risk.
So, that means I want to choose
H, so that these two
numbers are the same.
And if I do that, I've
got a riskless
investment, all right?
So, set these equal
to each other.
And that implies something about
H. We can drive what H
is, if I just put these two
equal to each other and solve
for H. And I get H equals C sub
u minus C sub d, all over
u minus d times S, OK?
So, I've been able to put
together a portfolio of the
stock and the option
that has zero risk.
If I do this, if I hold this
amount of shares in my
portfolio, I've got a
riskless portfolio.
So, that means that the riskless
portfolio has to earn
the riskless rate, right?
It's the same thing as a
riskless rate [correction:
same thing as a riskless
investment], so it has to earn
that [clarification: earn
the riskless rate].
If I can erase this now,
I'm almost there,
through option pricing.
The option pricing then says
that, since I'm derived what H
is, the portfolio has to be
worth one plus the interest
rate times what I put in, which
is HS minus C. And that
has to equal the value of the
portfolio at the end, which is
either uHS minus C sub u,
or dHS minus C sub d,
the same thing, OK?
So, I've already derived what
H is, and I substituted into
that, and I solved for C. So,
substitute H in and solve for
C, and we get the
call price, OK?
It's a little bit complicated,
but the call option price has
to equal one plus r minus d,
all over u minus d, times C
sub u over one plus r, plus u
minus one minus r, all over u
minus d, times C sub d
all over one plus r.
And I'll put a box around that
because that's our option
price formula, OK?
Did you follow all that?
This is derived --
This option price formula
was derived from a
no arbitrage condition.
Arbitrage, in finance, means
riskless profit opportunity.
And the no arbitrage condition
says, it's never possible to
make more than the riskless rate
risklessly, all right?
If I could, suppose I had some
way -- suppose the riskless
rate is 5%, and I can make 6%
risklessly, then I will borrow
at the riskless rate and put
it into the 6% opportunity.
And I'll do that until
kingdom come.
There's no limit to how
much I'll do that.
I'll do it forever.
It's too much of a profit
opportunity to ever happen.
One of the most powerful
insights of theoretical
finance is, that the
no arbitrage
condition should hold.
It's like saying, there are no
$10 bills on the pavement.
When you walk down the street
and you see a $10 bill lying
there on the street, your first
thought ought to be, are
my eyes deceiving me?
Because somebody else
would have picked it
up if it were there.
How can it be there?
I once actually had
that experience.
I was walking down the
street in New York.
It was actually a $5 bill.
It was just lying there
in the street.
And so, I reached down to pick
it up, and then, suddenly, it
disappeared.
And it was people on one of the
stoops of one of these New
York townhouses playing
a game.
They'd tied a string
to a $5 bill.
And they would leave it on the
street, and watch people reach
for it, and they'd
snatch it away.
That's the only time in my life
I ever saw a $5 bill on
the pavement.
And so, it's a pretty good
assumption that, if you see
one, it isn't real.
And that's all this is saying,
that if the option price
didn't follow this formula,
something would be wrong.
And so, it had better followed
this formula.
Now, that is the basic
core option theory.
Now, the interesting thing about
this theory is, I didn't
use the probability of up and
the probability of down.
So somebody says, wait a minute,
my whole intuition
about options is: I'd buy
an option, because it
might be in the money.
When I was just describing this
here, this is $0.45, I
said, that's not much,
because it probably
won't exceed $25.00.
It's so far below it.
So, it seems like the options
should really be fundamentally
tied to the probability
of success.
But it's not here at all.
There's no probability in it.
You saw me derive it.
Was I tricking you?
Well, I wasn't.
I don't play tricks.
This is absolutely right.
You don't need to know the
probability that it's in the
money to price an option,
because you can price it out
of pure no-arbitrage
conditions.
So, that leads me then to the
famous formula for options
pricing, the Black-Scholes
Option Pricing Formula, which
looks completely different
from that.
But it's a kindred, because it
relies on the same theory.
And there it is.
This was derived in the late
70's, or maybe the early 70's,
by Fisher Black, who was at MIT
at the time, I think, but
later went to Goldman Sachs, and
Myron Scholes, who is now
in San Francisco,
doing very well.
I see him at our Chicago
Mercantile Exchange meetings.
Fisher Black passed away.
It doesn't have the probability
that the option is
in the money, either, but it
looks totally different from
the formula that I
wrote over there.
The call price is equal to the
share price, S, times N of d
sub 1, where d sub 1 is this
equation, minus e to the minus
r, the interest rate, times time
to maturity, T, times the
exercise price times N of d sub
2, where this is d sub 2.
And the N function is the
cumulative normal
distribution function.
I'm not going to derive all
that, because it involves
what's called the calculus
of variations.
I don't think most of you
have learned that.
In ordinary calculus, we have
what's called differentials,
dy, dx, et cetera.
Those are fixed numbers
in ordinary calculus.
In the mid 20th century,
mathematicians, notably the
Japanese mathematician Ito,
developed a random version of
calculus, where dx and dy
are random variables.
That's called the stochastic
calculus.
But I'm not going to use that.
I'm not going to derive this.
But you can see how to price an
option using Black-Scholes.
But Black-Scholes is derived,
again, by the no-arbitrage
condition and it doesn't
have the probability.
Oh, the other variable that's
significant here is sigma,
which is the standard deviation
of the change in the
stock price.
So, once we put that in, someone
could say, well,
probabilities are getting in
through the back door, because
this is really a probability
weighted sum of the changes in
stock prices.
Well, probability is not really
in here at all, but
maybe there's something like
standard deviation, even in
this equation.
Because we had C sub u and C sub
d, and that would give you
some sense of the variability.
[clarification: In the binomial
asset pricing model,
u and d give you some sense
of the variability of the
underlying stock price,
analogous to sigma in the
Black-Scholes formula.]
I'm going to leave this
equation just
for you to look at.
But what it does do is, it shows
the option price as a
nice curvilinear relationship,
just like the
one I drew by hand.
Which then, as time to exercise
goes down, as we get
close to the exercise date,
that curve eventually
coincides with the broken
straight line.
Now, I wanted to tell you about
implied volatility.
This equation can be used
either of two ways.
The most normal way to do it,
to use this equation, is to
get the price that you think
is the right price for an
option, to decide whether I'm
paying too much or too little
for an option.
So with this formula, I can
plug in all the numbers.
To use this formula, I have to
know what the stock price is.
That's S. I have to know what
the exercise price is.
And I have to know what the
time to maturity --
these are all specified by the
stock price and the contract.
I have to know with the
interest rate is.
And if I also have some idea of
the standard deviation of
the change in the stock
price, then I can
get an option price.
But I can also turn it around.
If I already know what the
option is selling for in the
market, I can infer what the
implied sigma is, right?
Because all the other
numbers in the
Black-Scholes formula are clear.
They're in the newspaper,
or they're
in the option contract.
There's this one hard to pin
down variable, what is the
variability of the
stock price?
And so, what people often use
the Black-Scholes formula to
is, to invert it and calculate
the implied volatility of
stock prices.
So, when call option prices are
high, why are they high
relative to other times?
Well, it must be that
people think --
I'm going back to the old
interpretation, that the
probability of exercise
is high, right?
If an out of the money call is
valuable, it must be people
think that sigma is high.
So, let's actually solve
for how high that is.
I can't actually solve
this equation.
I have to do it numerically.
But I can calculate, for any
call price, given the stock
price, exercise price, time to
maturity, and interest rate.
I can calculate what volatility
would imply that
stock price.
And so, that's where we are
with Black-Scholes.
So, implied volatility is the
options market's opinion as to
how variable the stock market
will be between now and the
exercise date.
So, one thing we can do is
compute implied volatility.
And I have that here
on this chart here.
What I have here, from 1986 to
the present, with the blue
line, is the VIX, V-I-X , which
is computed now by the
Chicago Board Options
Exchange.
When the CBOE was founded,
they didn't
know how to do this.
Black and Scholes invented their
equation in response to
the founding of the CBOE.
And now, the CBOE publishes
the VIX.
And that's where I got this,
off their website cboe.com.
And so, they have computed,
based on the front month, the
near options, what the options
market thought the volatility
of the stock market was.
That's the blue line.
And you can see, it had a lot
of changes over time.
That means that options prices
were revealing something about
the volatility of the
stock market.
Now, the blue line is
from the Chicago
Board Options Exchange.
What I did, and I calculated
this myself, the orange line
is the standard deviation of
actual stock prices over the
preceding year, of monthly
changes, annualized.
That's actual volatility.
But it's actual past
volatility.
Let's make it clear,
what this is.
What the VIX is, is the sigma in
the Black-Scholes equation.
But it is, in effect, the
market's expected standard
deviation of stock prices.
And to get it more precise, it's
the standard deviation of
the S&P 500 Stock Price Index
for one month, multiplied by
the square root of 12, because
they want to annualize it.
It's for the next month.
Why do they multiply it by
the square root of 12?
Well, that's because, remember
the square root rule.
These stock prices are
essentially independent of
each other month to month, so
the standard deviation of the
sum of 12 months is going to be
a square root of 12 times a
standard deviation
of one month.
And this is in percent
per year.
So, that means that the implied
volatility in 1986 was 20%.
And then, it shot way up to
60%, unimaginably quick.
That might be the record high,
I can't quite tell from here.
Remember, I told you the
story of the 1987
stock market crash?
The stock market fell
over 22% in one day.
Well, actually, on the S&P,
it was only 20%, but
a lot in one day.
It really spooked the
options markets.
So, the call option prices went
way up, thinking that
there's some big volatility
here.
We don't know, which
way it'll be next.
Maybe it will be up, maybe
it will be down.
It pushed the implied
volatility, temporarily, up to
a huge level.
It came right back down.
My actual volatility, I
calculated this for each day
as the volatility of
the market over
the preceding year.
Well, since I put October 1987
in my formula, I got a jump up
in actual volatility, but not
at all as big as the options
market did.
See, the option market is
looking ahead and I have no
way to look ahead, other than to
look at the options market.
So, to get my actual volatility,
I was obliged to
look at volatility in the past,
and it went up because
of the 1987 volatility,
but not so much.
So what this means is,
that, in 1987,
people really panicked.
They thought something
is really going on
in the stock market.
They didn't know what it was and
they were really worried,
and that's why we see this spike
in implied volatility.
There's a couple other spikes
that I've noted, the Asian
financial crisis occurred
in the mid 1990's.
Now, that is something that was
primarily Asian, but it
got people anxious over
here as well.
You know, Korea, Taiwan,
Indonesia, Hong Kong, these
countries had huge turmoil.
But it came over here in the
form of a sudden spike in
expected volatility.
People thought, things could
really happen here.
So, all the option prices
got more valuable.
And then there's this spike.
This is the one that
you remember.
This is the financial crisis
that occurred in
the last few years.
Notably, it peaks in the fall
of 2008, which was the real
crisis, when Lehman Brothers
collapsed, and it created a
crisis all over the world.
There was a sharp and sudden
terrible event.
And you can see, that actual
volatility shot up to the
highest since 1986, as
well, at that time.
So, implied volatility, you
can't ask easily from this
chart, whether it was
right or wrong.
People were responding to the
information, and the response
felt its way into
options prices.
There's no way to find out, ex
post, whether they were right
to be worried about that.
But they were worried about
these events, and it led to
big jumps in options prices.
Now, I wanted to show the same
chart going back even further,
but I can't do it with options
prices, because I can show
volatility earlier, but I can't
show implied volatility
before around 1986, because the
options markets weren't
developed yet.
But I computed an actual S&P
Composite volatility.
Well, in my chart title,
I said S&P 500.
The Standard and Poor's 500
Stock Price Index technically
starts in 1957, but I've got
what they call the Standard
and Poor's Composite
back to 1871.
And so, these are the actual
moving standard deviations of
stock prices, all the way back
to the beginnings of the stock
market in the U.S. Well, not the
very beginnings, but the
earliest that we can get
consistent data for, on a
monthly basis.
And you can see, this
goes back further
than the other chart.
You can see that the actual
volatility of stock prices,
except for one big event, called
the Great Depression of
the 1930s, has been remarkably
stable, right?
The volatility in the late
20th century, early 21st
century, is just about exactly
the same as the volatility in
the 19th century.
It's interesting, how stable
these patterns are.
There was this one really
anomalous event that just
sticks out, and that is
the Great Depression.
1929 precedes it, it's
somewhere in here.
But somehow people got really
rattled by the 1929 stock
market crash.
And not just in the U.S. This is
U.S. data, but you'll find
this all over the world.
It led to a full decade of
tremendous stock market
volatility around the world,
that has never
been repeated since.
The recent financial crisis
has the second highest
volatility after the
Great Depression.
This isn't long ago.
This is well within
your memories.
Just a few years ago, we had
another huge impact on
volatility.
And as you saw on the preceding
slide, it had a big
impact on implied volatility
as well.
So, I think that we had a near
miss of another depression.
It's really scary what happened
in this crisis.
Also shown here is the first
oil crisis, which we talked
about, in 1974, when oil prices
had been locked into a
pattern because of the
stabilization done by the
Texas Railroad Commission.
But when that broke, and OPEC
first flexed its muscles, it
created a sense of
new reality.
And it caused fear, and it
caused a big spike up in the
volatility of the stock market,
but not quite as big
as the current financial
crisis.
So, this is an interesting
chart to me.
A lot of things I learned from
this chart, and let me
conclude with some thoughts
about this.
But what I learned from this
chart is that, somehow,
financial markets are very
stable for a long time.
So, it would seem like it
wouldn't be that much of an
extrapolation --
when are you people
going to retire?
Did you pick a retirement
date yet?
Well, let's say a half
century from now, ok?
So, that would be 2060?
So, you're going to retire
out here, all right?
Your whole life is in here.
What do you think volatility
is going to do
over that whole period?
Well, judging from the
plot, it's probably
pretty similar, right?
That's not much more history
compared to what
we've already seen.
It's probably just going
to keep doing this.
But there's this risk
of something like
this happening again.
And we saw a near miss here, but
this plot encourages me to
think that maybe outliers, or
fat tails, or black swan
events, are the big disruptors
of economic theory.
Black-Scholes is not a
black swan theory.
It assumes normality of
distributions, and so, it's
not always reliable.
So, this leads me to think that
option pricing theory --
I presented a theory.
The Black-Scholes theory is a
very elegant and very useful
tool, especially useful when
things behave normally.
But I think, one always has to
keep in the back of one's
mind, the risk of sudden major
changes like we've seen here.
So, let me just you
give us some final
thoughts about options.
I launched this lecture
by saying,
they're very important.
And they affect our lives
in many ways.
I've been trying to campaign
for the expansion of our
financial markets.
Working with my colleagues and
the Chicago Mercantile
Exchange, we launched options,
in 2006, on single-family
homes in the United States.
We were hoping that people
would buy put options to
protect themselves against home
price declines, but the
market never took off.
We have, since, seen huge human
suffering because of the
failure of people to
protect themselves
against home price declines.
There were various noises that
were made by people in power,
that suggested that maybe
something could be done.
President Obama proposed
something called Home Price
Protection Program, and it
sounded like an option, a put
option program.
But, actually, it was a much
more subtle program than that.
It was a program to incentivize
mortgage
originators to do workouts on
mortgages, if the mortgages
would default --
if home prices were to fall.
And nothing really
happened with it.
The President can't get things
started, either, not always.
I've been proposing that
mortgages should have put
options on the house
attached to them.
When you buy a house, get
a mortgage, you should
automatically get
a put option.
I've got a new paper on that.
But these are kind
of futuristic
things at the moment.
I'm just saying this at the end,
just to try to impress on
you, what I think is the real
importance of options markets.
People don't manage risks well
in the present world.
Having options or insurance-like
contracts of an
expanded nature will help people
manage their risks
better, and it will make
for a better world.
OK.
I'll see you on Monday.
