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.