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ANDREW LO: What I want to
talk about is option pricing.
But given that there's
the midterm coming up,
what I'd like to do
is to actually skip
the more technical part today.
Today, what I was
going to do was
to describe a method
for pricing options,
a particular
option-pricing formula.
Now, we have a course, 15.437,
on options and futures.
And that's really what I would
recommend for those of you who
are interested in derivatives.
But we really can't
let you leave MIT
without understanding a little
bit about the basics of option
pricing.
And it's such a beautiful
argument that it's important,
I think, for all of you
to see it at least once.
But since I'd like you to focus
on it and really absorb it,
and I suspect that most
of you are thinking
about the mid-term, I'd rather
postpone that till Monday,
and then talk today about the
very basics of option payoff
diagrams, which is
relatively straightforward.
And then give you a little bit
of a history of option pricing,
and tell you a bit
about how it came about.
And ultimately,
where the literature
fits within the grand
scheme of things.
So last time, if you recall,
we talked about options
as insurance.
And we went through a very
simple set of examples,
where I described the
put option as really
being parallel to insurance in
all of these different terms.
But the differences are that
a put option, first of all,
can be used early.
So you don't have to wait
until you have an accident
or wait until it expires.
You can decide at
any point in time
that you want to exercise it.
Also, unlike
insurance contracts,
options can be bought and
sold in organized exchanges.
So you can buy a put option.
You can sell a put option.
And then finally, dividends
have an impact on options.
And so most options have
dividend protection,
in the sense that if
there's a dividend paid,
then the strike price will
be adjusted accordingly.
Now, it's important
to understand
the differences between an
option and an underlying.
Because they really have
some very, very important
distinctions, in terms
of their payoffs.
So the way that we
try to emphasize that
is by looking at a diagram
that graphs the option
value as a function of the
underlying parameters that
influence the option.
And the most important
parameter is, of course,
the underlying price
of the stock or asset
on which the option is written.
So this is an example
of a payoff diagram
that plots the value of
the option at maturity
for a call option on
an underlying stock.
And the x-axis is the
price of the stock.
And the y-axis is
the value or price
of the option on the date
of maturity or exercise.
So let's suppose that the option
has a strike price of $20.
That gives the
holder of the option
the right to purchase the stock
for $20 at the maturity date.
So it's a call option,
meaning it gives you the right
to call away or buy the stock.
And the strike
price is set at $20.
Now, if the actual price
of the stock is below $20,
you're never going to
want to call the option.
Rather, you're never going
to want to call the stock.
You're never going to want
to exercise the call option.
Because if you did,
you'd be buying something
for $20 that would be
worth less than $20.
So if the true stock price
is anything less than $20,
this option, at expiration,
is worth nothing to you.
You would never use it.
Now, it's critical to understand
that this payoff diagram is
the value at maturity.
Prior to maturity, if the
value of the underlying stock
is less than $20, the option
could still have value.
Typically it will have value.
Because there's always a chance
that the stock price goes
above $20 at the maturity date.
So let's be clear that this
is the value of the call
option at maturity date.
And if it turns
out that the stock
price is greater than $20,
then the option has value.
And the value increases,
dollar for dollar,
with the stock price above $20.
So the slope of this
line is 45 degrees.
It literally goes up in lockstep
with the underlying stock
price.
To be clear, if the
stock price is $25
and you get to buy it
for $20, the option,
that right to buy
for $20 is worth $5.
Because the stock
is really worth $25.
So the way you can see that is
you can buy the stock for $20,
with this piece of
paper that you own.
And then you can turn
around and sell that stock
on the open market for $25.
So you've made that $5 profit.
The important thing about
this diagram, the blue line,
is that the upside is unlimited.
But the downside is
very much limited, at 0.
OK?
So this is an example
of a security that
has an asymmetric
payoff, asymmetric.
The upside is not the
same as the downside.
Remember the payoff of a stock,
or of a futures contract.
It's symmetric.
It's that straight line.
Here, this is not
a straight line.
It's kinked at the
strike price, K. That's
a very important feature.
Now, it looks like,
from this diagram,
this call option is one
of these propositions
that you hear on late-night TV,
make a $1 million with no money
down.
Like, there's no way to lose.
How could that possibly be?
How could we have come up with
a security that has no downside?
Wouldn't everybody want one?
Yeah?
AUDIENCE: Well, it
has value [INAUDIBLE].
ANDREW LO: Exactly.
Yeah, there's no free lunch.
So of course, everybody
wants it if it's free.
But of course, it's not free.
So you have to pay for it.
You have to pay something
today in order to get access
to this asymmetric payoff.
So the net payoff, that is, if
you were buying the call option
and paying a certain
amount of money,
then the net payoff
to you would be
given by the dotted line,
which is the blue line.
But you subtract from it
the value of the premium
that you pay.
It's called an option premium.
But it's just the price of the
option whenever you bought it.
And then if you want to
take into account the time
value of money, you should take
the future value of that price
that you paid when
you bought the option.
So if you bought the option
in the beginning of the month
and it expires at
the end of the month,
you've paid something at
the beginning of the month.
If you want to find
your net payoff,
you could either, at
the maturity date,
subtract from the
blue line the value
of what you paid multiplied
by the one-month interest rate
factor, so that you subtract
time t dollars from time t
dollars.
Or you can do a present value,
where you take the payoff
and you move it back to
the beginning of time.
Typically what we
do is we actually
ignore the time value of
money, just because it's
a month's worth of interest.
And people don't really
worry about that too much.
Yeah?
AUDIENCE: Can you
make some inferences
about the future
price of the stock
by looking at the
price of the option?
ANDREW LO: Yes,
absolutely, you can.
And we're going to show
you how to do that when
I give you the asset
pricing formula for it.
But you're absolutely right.
By looking at the
option, that gives you
information about
what's going on.
Just like when I tell you
for crisis management,
if you look at
T-bills today, you
get a sense of how much demand
there is for cash, putting
money in your mattress.
By looking at
options, you actually
get a sense of where markets
are going to be going.
So after I give you
a pricing formula,
next time, I'm going to show
you the prices of options.
In particular,
we're going to look
at the price of a put option on
the S&P 500 for the next month
and for the next two months.
And you're going to find a
very, very big difference
in those two.
That's telling you something
about where the market thinks
volatility is going
in the S&P 500
over the next couple of months.
So yes, there'll be all
sorts of wonderful things
you'll be able to tell
by looking at the prices.
But in order to do
that, we do have
to understand how
these payoffs work.
So getting back
to this diagram--
I want to make sure
everybody is with me now--
this dotted line shows
you your net payoff
and a net of the price you
paid for this particular call
option.
And the neat thing
about this net payoff
is that it then describes to
you the fact that this is not
a surefire way to make
money and not lose any.
You might lose money, because
you paid something upfront
for the call option.
And so the only way you're
going to come out ahead
is if the stock price
actually exceeds--
not this point, but actually
something like this point.
So the stock price has to go up
by a little bit more than $20
in order for you to
make money, net of what
it cost you to buy that option.
Now, I want you to
go back and think
about the difference between an
option and a futures contract.
Remember a futures contract
we said was no money down,
0 NPV when you get
into the futures.
That's not true
with a call option.
A call option is actually worth
a positive amount of money
on day one.
So if you want a
call option, you've
actually got to pay for it.
And then there's an issue about
whether you'll make money.
Because it depends on
whether the stock price
exceeds this point.
It's got to exceed not only the
strike price, but the amount
that you paid for that option.
Any questions about that?
Or is that pretty clear?
So this is important.
So ask now if you
don't quite get it.
Because if you don't
get this, you're
going to get
confused by what I'm
going to say in a few minutes.
Let me give you another example,
just to really fix ideas.
Let's do the put option case.
Now, the put option allows me to
sell the stock for, let's say,
$20, or before
the exercise date.
So with a put option, am I
going to hope if I buy a put--
so I buy the right to
sell the stock at $20.
That's a little bit
hard to keep track of.
I'm buying a piece of paper
that gives me the right
to sell the stock for $20.
If I own that piece of
paper, this put option,
am I going to wish that the
stock price goes up or down?
AUDIENCE: Down.
AUDIENCE: Down.
ANDREW LO: Down.
I'm only going to get
paid on the put option
if the stock price
goes below $20.
Because then I have
something valuable, right?
If it goes below $20, I get
to sell the stock for $20.
So I make the difference
between what it's worth and $20.
If the stock price
is at $20 or above,
then my put option
expires, worthless.
I'm not going to use it.
Because it would be foolish
for me to sell something
for $20, when I can sell
on the open market for $25.
So the payoff diagram is
exactly the opposite of this.
In fact, it looks like this.
So now the blue line is
the payoff of the option
itself, the gross payoff.
$20 and above, it's worthless.
But $20 and below, this
is the 45-degree line.
But unlike the call option,
my upside for the put option
is limited.
Is limited to what?
AUDIENCE: 0.
ANDREW LO: Right, 0 or whatever
that is, $10, in this case.
If the stock price goes
to $0, then my put option
is worth a maximum of $10.
So the upside is bounded
by that $10 limit.
And that's the gross upside.
If I look at the net, I
subtract how much I pay.
And then I get the dotted line.
Any questions about
the put options payoff?
Now, just to fix ideas, let
me go back to the call option
and show the difference
between the stock return
versus the call option return.
If you take a look at
the call option, again,
it's going to look
like this when
you subtract from it the price.
But the stock is going to
look like that line there.
Meaning that the stock
return is linear.
But the option
return is non-linear.
And this is one of the most
important and subtle ideas
with this instrument.
Up until now, all
of the instruments
that we've looked at stocks,
bonds, futures, and forwards,
their payoffs have
been relatively simple,
in the sense that they're
straight lines if you
plot the underlying
price and their payoffs.
This is the first time we have
analyzed the security that has
a bizarre structure like this.
And you might think
it's straightforward
because well, you understand the
contractual terms of an option.
But from a risk-and-return
perspective,
it's actually quite a
bit more complicated
than most people
would appreciate.
One of the reasons that we are
in a current financial crisis
today is because of the
complexity of the securities
that have been created.
And the complexities are
really along the lines
of these non-linearities.
As I mentioned to you,
insurance is a put option.
So you can actually use the
theory of option pricing
to value insurance contracts,
like credit default swaps.
In fact, the payoff of
a credit default swap
is not that different from
something that looks like this.
And what that means is that
a portfolio of credit default
swaps does not behave
like a portfolio of stocks
or a portfolio of bonds.
They have very
important differences,
both in terms of their risk,
and in terms of their return.
In this case, you can see
the risk of a put option
is bounded above.
The upside is bounded above.
The downside is bounded.
But the call option is unbounded
above, in terms of its upside.
Bounded below, in
terms of its risk.
What if, now, you
decided you were going
to sell somebody a call option?
Or you were going to
short a call option?
Can anybody guess what the
payoff would look like?
Yeah?
AUDIENCE: If you're going
to sell a call [INAUDIBLE]
agreement that's your payoff.
ANDREW LO: Yeah.
AUDIENCE: As long
as the stock doesn't
go above the [INAUDIBLE] so
someone can call you out.
Then it's [INAUDIBLE].
ANDREW LO: So what is
it going to look like,
in terms of the diagram?
How would I have to change this?
AUDIENCE: It would
be flat, say, $2.
And then it would go down.
ANDREW LO: That's right.
It would be a mirror image
of the blue line, where
you reflect it along the
x-axis, it would go this way.
And what that means is that
your downside is unlimited.
But your upside is very limited.
Now, why would anybody
want to do that?
That seems like a terrible deal.
Well, the difference
is that you are now
getting paid to do that.
In other words, if
you flip this image--
let me draw it here.
If you now have a call
option that you've shorted,
you go down here.
This is $20.
You will get paid
for doing this.
Meaning if you look
at your net return,
it's going to look like this.
So that means that as long
as the stock price stays
below a little bit
extra than $20,
you will actually get
to keep that premium.
But if the stock price goes
up, your losses are unbounded.
That's different.
That's a different
payoff structure
than what we're used to with
traditional instruments.
You can do all sorts
of calculations.
Long Call looks like that.
Long Put looks like that.
Shorter Call looks like this.
And Shorting a Put
looks like that.
And once you take all of these
things and put them together,
you can mix and match and get
some really interesting payoff
types of structures.
So let me give you an example.
This is just payoff
tables that will show you
when you get paid what.
So this is a very
helpful exercise for you
to go through, just to
verify that these graphs are,
in fact, what they should be.
So I would ask you to go
through this on your own.
And there are all
sorts of trade-offs
that you can
implement by looking
at these various
different payoffs
and putting them together.
For example, you can
buy a stock and a put,
or buying a call with one
strike and selling a call
with another, or buying a call
and a put with the same strike.
Each of these
portfolios of options
gives you a different
kind of a payoff diagram.
And as a result, it allows you
to make bets on market events
that you otherwise wouldn't
be able to make a bet on.
So let me give you
an example of this.
Let's see.
Let's do something
like, oh, I don't know.
How about a call and a put?
Suppose you decide
to buy a call,
and you buy a put with the
exact same strike price.
So buying a call at
a strike price of $50
will give you that left diagram.
And then buying a put with
the strike price of $50
will give you the right diagram.
And your payoff for
those two, at maturity,
is going to look like a V.
Now, in fact, you have
to subtract how much you
paid in order to do this.
So your net payoff will
be this V, shifted down.
Sorry, I didn't do the
interactive graphics here.
But it's going to look like
this, where what I've done
is I've subtracted the
amount of money it cost you
to buy the put and the call.
Yeah, question.
AUDIENCE: In this particular
example, really all
you're saying is, except for a
short range in the stock price
around the strike price,
you will always make money.
ANDREW LO: That's right.
AUDIENCE: So is
there a reason why
you wouldn't do a
lot of this, if you
know that there is
some movement that's
going to happen in the stock?
ANDREW LO: So the question
is, how much does it
cost you to do that?
When you say small,
it's all relative.
You've got to find out
exactly what that is.
The smaller the range
is, the more expensive
it'll be for you
to actually buy it.
So there's a trade-off.
It's all a matter of
how much you pay for it.
But before I go
there, let me just
make sure everybody
understands what
this payoff is accomplishing.
What are you doing
when you are buying
a portfolio with a payoff
diagram that looks like this?
What you're doing is
saying that you're
going to make lots
of money if the stock
price goes way up or way down.
The only way you're not
going to make money,
if the stock is not
doing a whole lot,
if it's staying around here.
OK.
So this is an example
where you are making a bet.
Not that markets
are going to go up,
not that markets are
going to go down,
but that markets are
going to be wild.
That is, you're making
a bet on volatility.
Which may seem like a
pretty good bet nowadays.
But the problem is that
there's a difference
between this diagram
and this diagram.
And what is the difference?
What determines how big or
small this little tiny area is,
where you don't make any money?
What determines that is how
much you have to subtract
and how far this V
gets shifted down.
And you know what?
Right now, it's
shifted down a lot.
In other words, it costs a
lot to buy a put and a call.
Why does it cost a lot?
Because volatility is very high.
And when you're buying a
put, you're buying insurance.
It's very, very expensive
now to buy insurance.
Because we're in the
middle of a hurricane.
And that's probably
the worst time for you
to buy hurricane insurance,
is when you're actually
in the middle of a hurricane.
So what that means is, that this
thing has shifted down a lot.
So that means that you have to
have really, really volatile
markets in order to make money.
So it's shifted down enough
so that supply equals demand,
as you would expect.
So there's no free lunch
going on out there.
It's priced fairly.
Now, even though
it's priced fairly,
if it turns out that you're
the kind of person that really
doesn't like a lot of
risk and you believe
there's going to be
tons more volatility
coming, then for you,
it's worth it to do it.
For somebody else who
doesn't believe that there's
going to be a lot more
volatility coming,
it's worth it to be on the
other side of that trade.
By the way, if I'm on the
other side of the trade,
what does my payoff
diagram look like then?
If I'm selling a put and a
call, what will it look like?
AUDIENCE: The opposite.
ANDREW LO: Yeah,
exactly, the opposite.
We're going to flip it, flip
this thing against the x-axis.
So it'll be an upside-down
V. But because we're
shorting puts and calls,
we get money upfront.
So the upside-down V is going
to be pushed up over the x-axis.
So it's going to look like
the mirror image of this.
And as long as
stock prices are not
more volatile than this
range, we will make money.
But our downside is
unlimited in both directions.
So you got to be
really confident
that you know that markets
aren't going to be any more
volatile than they are now.
Now, if you were Warren Buffett,
and you bought Goldman Sachs
three, four weeks
ago, and you thought
it was a great deal
then, well, you
would have lost money by now.
Warren Buffett has lost money.
On the other hand,
as you all know,
Warren Buffett doesn't it make
investments for the short term.
He's thinking about
this investment
as a 10, 20-year investment.
And over 10 or 20
years, I suspect
it will be a very good deal.
But if you're looking
at what's going on
over the next few
weeks, the question
is, do you believe
that markets will be
less volatile or more volatile?
If you do, you're
going to be on one side
or the other of that trade.
The point is that
this now allows
us to make bets on volatility.
Whereas before, with
a futures or a forward
or a stock or a
bond, you only could
bet on it going
up or going down,
or mispricings
because certain kinds
of arbitrage relationships
have been violated.
This is the first
time that we've
been able to make a
bet on wild swings.
And that's a really
amazing thing.
It's an extraordinary innovation
to be able to do that.
It allows individuals to
engage in kinds of side bets
that they otherwise
wouldn't be able to.
And more importantly, it
allows other individuals
to insure against certain
kinds of eventualities
that they'd never be able to do.
Now you can buy insurance
against volatility,
which is a pretty remarkable
thing to be able to do.
OK, so that's just one
example of an option strategy,
a very simple one.
There are other examples
that I've given you here.
For example, this is
kind of a fun one.
This is two calls.
That should be a
minus sign, sorry.
Call1 minus Call2.
So you basically
buy a call option,
and you short another one
at different strike prices.
And so what this allows you to
do, this is really interesting.
This gives you upside
from 50 up until 60.
So you buy a call at 50.
You're short a call at 60.
So that means you're going to
get upside between 50 and 60.
And then nothing after that
and nothing before that.
Now, this seems like a
really ridiculous strategy
to engage in.
Why would you want to
cut off your upside?
Because with a call, if
you just bought a call,
you'd basically get
all of the upside.
Right?
Why would you ever
want to do this?
Anybody tell me what
the logic for that is?
Yeah.
AUDIENCE: Because it's cheaper.
ANDREW LO: Exactly.
It's cheaper.
It's cheaper because when
you short the call at 60,
you're getting money today.
So that helps you
finance the call at 50.
It's cheaper, but
it's not a free lunch.
What you're getting in
exchange for that extra premium
is you're giving up any profits
above and beyond the stock
price going above 60.
So you're giving up
the unbounded upside.
And you're bounding it at 60.
But the benefit of
giving up that upside
is that you now have some money
to reduce the cost of getting
that call at 50.
OK?
And so you might use
this if you think,
well, I suspect that the stock
has got some room to grow.
I think it will bounce
around between 50 and 60.
But I can't possibly
see the stock
ever being worth more than 60.
So I'm happy to give up that
upside to other people who
are more optimistic than me,
and get some money for it
and help me to finance my
purchase of the stock at 50.
So it's cheaper.
That's the bottom line.
Another way of
looking at it is, you
have to move this whole diagram
down by how much it costs.
It turns out you move it down
by less than if it were just
the pure call option by itself.
So the way you can think of
it is you buy the call option.
You move it down by that much.
And then you sell the other
call and you move it up
by the amount of that 60 call.
So you can do this.
You can do this in reverse.
You can bet on the
downside, in that way.
You can do something
that's called
a butterfly spread, where
basically it looks like this.
So the payoff if it stays
within a range, you get paid.
But if it's really volatile,
then you don't get paid.
So you're willing to give
up the upside on both ends
because you think the stock
is going to be self-contained.
You're betting against
volatility increasing,
and you're using
the ability to get
rid of those unbounded gains
to finance the positions.
And it turns out that with
these kinds of payoffs,
you can prove mathematically
that it's possible to generate
any other payoff in the world.
There is a mathematical
result that's
actually related to this Taylor
approximation and Fourier
expansion that says that any
possible security that you can
come up with can be approximated
by a sequence of calls
and puts.
That's a really powerful idea.
But in fact, from a
practical purpose,
you don't even need to
use anything that fancy.
If you have just a very small
number of calls and puts,
you can put together
extraordinarily complex payoff
diagrams that will get you
whatever kind of a risk profile
you're looking for.
That's the power
of option pricing.
OK, any questions about
these payoff diagrams?
I would urge you to work
through a few examples just
to make sure you
really understand them.
Because it's a easy thing to
think that you understand.
But unless you're forced
to go through the exercise
and draw these
diagrams, you won't
have an appreciation
for how to do them
and how important they are.
Yeah, question.
AUDIENCE: I'm
curious for a while,
is there any
implicit volatility,
I mean implicit in the price of
an option and a call and a put,
is respective volatility
of the market.
But how much inside that
price is it actually
generating volatility itself?
Do you see what I'm saying?
The price, the call
could also be a motor
of volatility in the market.
ANDREW LO: Yes. .
That's a great question.
Let me repeat it.
In fact, that question
was asked shortly
after Black and Scholes
came up with their formula.
It created the whole
literature, which
was started by our very own
former Dean, Dick Schmalensee.
He wrote a paper
with a fellow named--
I think it's Robert Trippi,
small Schmalensee and Trippi.
They wrote a paper on implied
volatilities of options.
So the ideal is that
options are actually
dependent on volatility.
And I'll show you that not
this time, but next time.
I'm going to go through
a pricing model.
And you're going to see
how volatility actually
plays a very concrete role.
So they came up with
a brilliant idea.
Let's take a look
at an option price.
And we know what
the stock price is.
We know what the
strike price is.
We know what the
other parameters are.
Let's ask the question,
given the price of an option,
what is the volatility that
is consistent with that market
price?
Because allowing you
to invert the market
price for the
volatility gives you
information about
what's going on.
It's exactly the
question you asked,
about information implicit
in the market price.
It turns out that that's
done all the time.
And not only is it
done all the time,
but there is now an index that's
been created by the Chicago
Board Options Exchange
called the VIX, which
stands for the
Volatility Implied Index.
What they do is they look
at options on the S&P 500.
And they ask the question,
what is the volatility that
is consistent with the
option price on the S&P
for at-the-money option.
At-the-money means
the strike price
is equal to the current price
of the stock, or the index,
of this case.
And that's an incredibly
important concept.
Because that tells you
something about where
the market sees volatility
going forward, not just looking
backwards.
But today, right now, what does
the market think volatility
should be?
And if you look at the VIX
over the last few weeks,
you're going to be shocked.
We're going to take a look
at it next time, next Monday.
I'm going to do this in class,
where I'll show you what
that volatility looks like.
Historically, the S&P 500 has
had a volatility level of what?
Does anybody know?
What's the typical
stock market volatility?
AUDIENCE: About 15%?
ANDREW LO: 15% to 20%.
Yeah, it's bounced around
there, on an annualized basis.
Last week on an
intradaily basis,
the VIX index, which is
the Implied Volatility,
reached an interdaily high of
89% volatility, for the stock.
And right now, I
don't know what it is.
I haven't checked today.
But my guess is it's
probably 60 to 70.
AUDIENCE: 71%.
ANDREW LO: Is it, what?
AUDIENCE: 71%.
ANDREW LO: 71%.
OK, 71% annual volatility.
Now, that's the forward-looking
implied volatility
for S&P options.
And what that tells
you is that we're
in for some turbulent
times ahead.
If you look at the implied
volatility for the one year
contract, it's going
to be much lower.
Because people are
going to expect
that the volatility of the
S&P, going forward in time,
is going to decline between
now and a year from now.
At least, we hope so.
Otherwise a lot of
people are going
to be needing zan--
zan-- what is it?
Zantac and other kinds
of pharmaceuticals.
OK.
So those are option diagrams.
And I want to mention
one last thing
before we go to the history
of option pricing theory.
I want to mention that one
of the reasons option pricing
theory has been so
important in finance
is because soon after the papers
by Black and Scholes and Merton
were published, it became clear
that everywhere you looked,
there were options to be found.
That is, all other kinds of
financial securities, when
you looked more closely, they
were actually options as well.
So let me give you an example
I said before that stock
prices were not like options.
Well, as an approximation,
that's true.
But in reality, if you look
carefully at what a stock is,
in fact, a stock is an option.
So let me see how that is.
Well, equity, the
equity of a corporation
is a claim on the
corporation's assets.
But if that corporation has any
kind of debt financing, then
actually the equity
holders are second in line.
The bondholders
are first in line.
So the equity only gets
paid after the bondholders
get paid off.
So in particular, if you
think about the maturity
date of the bonds,
then on maturity date,
the value of the equity is
the maximum of either 0,
or the value of
the firm's assets
minus the face
value of the bond,
or what the bond has
to be paid off at.
Because if the bondholders
don't get paid,
then the equity
holders get nothing.
Then the bondholders
get the firm.
All of the assets of the firm
transfer to the bondholders
through bankruptcy proceedings.
At least, that's the theory.
So the value of the equity on
the maturity date for the bonds
is actually the maximum
of 0, V minus B.
Now, that should look
very familiar to you.
That should look like the
payoff of a call option.
Where the strike price
is B, and the value
of the underlying security is V,
the value of the firm's assets.
So what that means is
that equity holders can
be viewed as owning an
option on the firm's assets
with a strike price of B.
And the bondholders look
like they have a put option.
They've shorted a
put on the firm.
But that's leveraged with
a certain amount of debt.
It's a protected levered put,
is the way that people usually
put it.
So the debt is the
minimum of V or B.
You either get the
assets, or you get what
you owed, which is smaller.
And you can show that that's
equivalent to B minus max of 0B
minus V. That looks like
a short put position mixed
in with some borrowing.
And when you add the two, you
see that the value of the firm
is equal to the
value of the debt
and the value of the equity.
The point of this example
is that option pricing
can be used to value the capital
structure of a corporation
as well.
And within the last few
years, a very active part
of the hedge fund
industry has been
devoted to engaging in something
called capital structure
arbitrage.
Capital structure arbitrage says
that this equation has to hold.
But in practice,
there is a discrepancy
with what the market value for D
is, and the market value for E.
And using option pricing theory
and models for credit risk,
hedge funds have been able to
make a play by either buying
a company's equity and shorting
their debt, or buying the debt
and shorting the equity,
whichever is cheaper or more
expensive, and engaging in
what seems like an arbitrage
transaction.
Now, that presupposes
that you've got the credit
calculations done correctly.
So in order to engage
in those kind of trades,
you have to have
superior credit modeling
capabilities, certainly
better than what
rating agencies were doing.
And actually, there were cases
where hedge funds were actively
betting against
rating agency models.
Because they felt that
rating agencies had mispriced
some of their ratings based upon
the models that they'd created,
versus the ones the rating
agencies were using.
Now, turns out that when
you look more carefully
at other securities, and even
other kinds of opportunities,
options are there as well.
For example, when I started
here at MIT, 20 years ago,
I remember, distinctly,
some of my senior colleagues
referring to Assistant
Professors as options.
[LAUGHTER]
Now, let me explain.
You know that in academia,
when you start out
as a Assistant
Professor, there's
no guarantee for employment.
You have, typically,
a three-year contract.
And at the end of three
years you either get renewed
or you get fired.
And at the end of
the next three years,
you come up for what's
called a tenure review.
Tenure review means
that they send letters
to 15 of the top people in
your field across the country.
Across the world, actually.
And they base their decision
on whether to give you
lifetime guaranteed employment,
as to whether or not
these 15 people say
that you're the greatest
thing since whatever.
And if you don't get
that kind of review,
then you're asked to leave.
I mean, you have to leave.
There's no choice for
continued employment.
So the idea behind hiring
Assistant Professors
were that each one of them
was viewed as an option.
Meaning that you could
benefit from them for a while.
But if they didn't work out, you
could always get rid of them.
But once you got
tenure, that was it.
There was no longer an option.
So what that suggested,
from a hiring perspective,
is what kind of
Assistant Professor
should you hire if you
believe in option pricing,
as applied to the labor market?
Can you can you
characterize the type of--
Yeah, what was that?
AUDIENCE: Take risks.
ANDREW LO: Take risks.
You want to hire faculty
that are extremely volatile.
Not emotionally, hopefully,
but intellectually.
In other words, because
you get all the upside,
but you don't get any downside.
So what you want to do
is you want to take risk.
You want to take chances on
faculty that may or may not
work out.
And that, in fact,
has been the approach
that we and others have
used in hiring, based
upon this kind of
option pricing analysis.
And it applies to
all sorts of things.
When you think about
getting an education,
you can argue that getting
an education is an option.
You don't have to
use your degree.
You don't have to
use your education.
But you have it.
It's an option.
And so thinking about value in
education, you could actually
use this framework, try
to compute the flexibility
it gives you, in order
to take advantage
of career opportunities.
So there are lots of things
that look like options.
Yeah, Megan?
AUDIENCE: [INAUDIBLE] distressed
debt manager, [INAUDIBLE]?
ANDREW LO: A
distressed debt manager
if they're holding
distressed debt of a company,
they would actually be
holding a short put position.
They're holding the debt.
So they have a
short put position.
So if they wanted
to hedge it, they
can either buy a
put on the assets,
which would then help them
to hedge it out, Yeah, right.
And as I was saying,
there are all sorts
of other examples of options
and derivative securities.
The field has exploded.
There are literally
many, many trillions
of dollars of notional amounts.
Now, again, notional amounts
can be a little bit misleading.
Because you know that
for every option seller,
there's an option buyer.
Options are zero net
investment side bets,
unlike equities,
where companies that
have real assets behind
them issue pieces of paper
called equity.
Options are issued by
the Options Clearing
Corporation for the Chicago
Board Options Exchange.
The Mercantile Exchange
also has options.
There's options
traded everywhere.
In fact, one of the
largest exchanges
is the International
Securities Exchange.
It was started up
by Bill Porter,
the fellow who started e-trade.
And it is the most active
options exchange in the world.
It's all done electronically.
And these options
are pure side bets.
But they're not just for
purposes of gambling.
They're for purposes of hedging
and engaging in insurance,
of the kind that we
talked about before.
So this has really exploded.
And that's why we have an
entire course, 15.437, devoted
to just the pricing of
options and futures.
So we can't, obviously, cover
all of it in this course.
But I want to just give
you a flavor of it.
Let me skip, now,
this next section
on valuation of options.
Because as I said, this is
a little bit more technical.
I want to spend some
time on it and make
sure you all understand it.
And then I will come
back to this on Monday.
When I want to do now is
just to give you a little bit
of a history of option pricing.
Because it's kind of fun.
First of all, in order to
figure out how to price options,
we have to begin
with figuring out
what a particular model would
be for the underlying stock.
In order to price an
option, you actually
have to say something
about how the underlying
security behaves.
So we have to start with that.
And we're going to start in
the very early 16th century,
with probably the first-known
model for asset prices
that ever existed in the world.
And that was developed by
a Italian mathematician
by the name of your
Gerolamo Cardano.
Now, those of you who were
on high school math team,
I suspect you've
heard of Cardano.
Anybody tell me who Cardano was?
No math team geeks here?
All right, Cardano
was, it turns out,
the second person
to have come up
with a solution for
the cubic equation.
You all know what the
quadratic equation is, right?
You know, ax squared
plus bx plus c equals 0.
That's a quadratic equation.
Anybody know what the
solution of that is?
Yeah, what is that?
AUDIENCE: [INAUDIBLE]
Great Great.
All right, You get the
pocket protector award.
[LAUGHTER]
Very good.
It turns out that
there is exactly
the same kind of solution
for the cubic equation.
Of course, nobody
remembers that.
I won't ask you
whether you know that.
You might.
But there is a formula
for the cubic equation.
It turns out that there are no
more formulas beyond the cubic.
So there's something
very special
about the cubic equation.
And this Italian
mathematician, Cardano,
was the first to publish it.
The reason I say that
he's the second person
to come up with it,
is that it turns out
he stole the formula from a
colleague, and a colleague who
had actually come up
with the solution.
And Cardano heard about
it and said, well,
please tell me what it is.
And the other
person said, I'm not
going to tell you what it is.
Because you're going to just
write it up and claim credit.
And Cardano says no,
no, I promise I won't.
And then the guy says,
all right, here it is.
He told him.
And then Cardano did,
in fact, rip him off.
So it's known as
Cardano's formula,
but it really shouldn't.
And I'm embarrassed to say,
I don't remember the guy
who actually invented it.
But Cardano, in addition
to having come up
with this solution, or
stolen this solution,
Cardano also wrote
a book on gambling.
And this book, which
is titled Liber De Ludo
Aleae, The Laws of
Gambling, he developed
what was the precursor to the
modern mathematical description
of stock prices.
And it was described
in this way.
"The most fundamental
principle of all in gambling
is simply equal conditions, e.g.
of opponents, of bystanders,
of money, of situation,
of the dice box,
and the die itself.
To the extent to which you
depart from that equality,
if it is in your opponent's
favor, you are a fool,
and if in your own,
you are unjust."
It turns out that
what he was describing
was, essentially, a 50/50 bet.
Or what we call a
fair game, or what
is now known as a martingale.
A martingale simply says that
expected winnings and losses
is equal to 0.
Or rather, your expected
wealth next period
is equal to whatever
your wealth is today
if you have a fair game
that you're betting on.
It turns out that
that simple model
developed into what we now
think of as the Random Walk
Hypothesis.
And the Random Walk was
really the fundamental driver
behind the option pricing
model that Black and Scholes
and Merton developed.
Now, the reason the Random
Walk holds a very special place
in the hearts of
financial economist
is because most
economists suffer
from a psychological disorder
that we call physics envy.
We all wish that we had
these three laws that
explains 99% of all behavior.
In fact, economists have
99 laws that explain
maybe 3% of economic behavior.
But there's one example, only
one, in the history of finance,
where an economist actually
came up with an idea
before a physicist.
And that was later
adopted by a physicist.
And the idea I'm talking about
is the Random Walk hypothesis,
or in the continuous time
realm, Brownian motion.
In 1900, a student by the
name of Louis Bachelier
was writing a
dissertation in Paris.
He was a mathematics
PhD student.
But he was writing about
pricing warrants that were
trading on the Paris Bourse.
So it was a finance thesis.
And in order to
solve the problem,
he had to come up with a
mathematical description
for the underlying price.
And he came up with this
notion of what we now
call Brownian motion,
of Random Walk.
And he did it a full three years
before a well-known physicist
published a paper on that.
Anybody know who
that physicist was?
AUDIENCE: Was it Brown?
ANDREW LO: No.
No, Brown was many years before.
And he was a biologist.
Yeah?
AUDIENCE: Einstein.
ANDREW LO: That's right,
Albert Einstein, in 1903,
actually published a paper
on the photoelectric effect
and Brownian motion.
And if you take a look
at what Baschelier did,
he was working with the
French mathematician
by the name of Henri Poincare.
Poincare was a very
well-known mathematician
who was the advisor
to Baschelier,
and who is renowned
now for a variety
of different contributions,
including the theory
of dynamical systems.
Baschelier wrote this thesis
and developed the mathematics
of Brownian motion.
And when he was
looking for a job,
Poincare wrote a letter
of recommendation.
And this is what Poincare
wrote about Baschelier.
He said that "The manner in
which the candidate obtains
the law of Gauss is most
original, and all the more
interesting as the
same reasoning might,
with a few changes, be extended
to the theory of errors.
He develops this
in a chapter which
might at first seem strange.
For he titles it
'Radiation of Probability.'
In effect, the author
resorts to a comparison
with the analytical theory
of the propagation of heat."
Now, remember this is
a thesis on pricing
warrants on the Paris Bourse.
Fourier's reasoning
is applicable almost
without change to this problem.
Which is so different from that
for which it had been created."
And of course, his
adviser, at the end,
always has to complain a
little bit about his student,
as we all do.
So he said, "It is regrettable
that the author did not
develop this part of
his thesis further."
What Poincare was mentioning,
with regard to Fourier,
was the theory of
heat conduction.
In physics, there is
a very standard model
that everybody that goes into
advanced physics will cover.
And that is, how does
heat get conducted
through a solid medium?
And in deriving the
equation that ultimately
is known as the heat
equation, you actually
use the same theory that
Baschelier applied to pricing
warrants on the Paris Bourse.
He gets what's known as a
partial differential equation.
And that's it right there.
That's the equation that
he used in his thesis.
If you look at his thesis,
you'll see it there.
That's the heat equation.
It's the same equation that
explains the conduction of heat
in a solid medium.
But he derives it
for the purpose
of pricing this
financial security.
Now, it turns out that
there was one slight mistake
that Baschelier
made in his thesis.
It was a mathematical error
that, ultimately, didn't really
affect the results.
But it became known.
And when he came
up for tenure, they
wrote to all the various
different big names.
And he was ultimately
turned down for tenure,
because they found this mistake.
And he was blackballed.
So he couldn't get a job
except for a small teaching
college, a women's teaching
college in the south of France.
Which frankly, sounds
pretty good to me.
[LAUGHTER]
But you know, for
him, it was not
the way he would not
want to end his career.
But at the end of
his career, it was
discovered that this
mistake was not as serious.
And people wrote him
a letter saying, gee,
you're a great guy anyway.
Paul Samuelson,
actually, was the person
who discovered
Baschelier's thesis when
he was in Paris at
the Sorbonne, reading
through various
different archives.
So Paul Samuelson's
responsible for resurrecting
the reputation and the
work of Louis Baschelier.
You can see his thesis now.
It's been republished
and reprinted.
But the point of the
thesis is that by assuming
that the underlying stock
price was a Random Walk,
and by developing the
mathematics of the Random Walk,
he was able to figure out
what the price of an option
was on that stock.
And it turns out
that the pricing
of the option on
the stock reduces
to solving this heat equation.
And that explains why
there are, nowadays,
so many physicists and
mathematicians that
are in finance.
It's because the whole body
of knowledge that comes along
with the physical interpretation
for the heat equation
can be applied, virtually
identically and verbatim,
to the pricing of options and
other derivative securities.
And so very quickly, we can
see that the information that's
contained in these
market prices can
be understood within a
mathematical framework
that we know.
So now going back
to the history,
it turns out that this was
not known in the 1970s.
It wasn't rediscovered by
Paul Samuelson until later on.
The folks that actually
worked on option pricing, that
tried to figure out the
mathematical prices of options
were quite a few.
Kruizenga, who is an MIT PhD
student in the 1950s-- oh,
question?
No?
OK.
In the 1950s, there
was an MIT PhD student
of Paul Samuelson's who tried
to work on this problem.
And he actually has a thesis
titled "Put and Call Options--
A Theoretical and
Market Analysis."
It's actually in
the MIT archives,
if you want to go
take a look at it.
But he didn't quite get it.
He didn't get the
right solution,
because he didn't have
the mathematical machinery
to be able to work out
the final elements of it.
K. Sprenkle, a student
at Yale in 1961,
wrote a thesis under Jim
Tobin and Arthur Okun, titled
"Warrant Prices As Indicators of
Expectations and Preferences,"
and tried to price it as well.
But he wasn't able to come up
with a pricing formula either.
And there were a number
of other attempts
to try to come up with the
appropriate pricing formula,
including attempts
by Samuelson in '65,
where he had to make assumptions
on individual preferences
in order to get a price.
That didn't work out.
And then Samuelson and
Merton in '69, they
tried to come up with
a pricing formula that
was preference free.
And they still couldn't do it.
Along came Black and Scholes.
Fischer Black who, at that
time, was a consultant working
at Arthur D. Little.
He wasn't even an academic.
The Arthur D. Little
building, that's
the building that
is right over there,
the one that they
won't let us tear down.
Because it's supposed to be
an architectural gem of sorts.
That was the Arthur
D. Little Building.
Fischer had his office there.
Myron had his office in the next
building over, Myron Sholes.
And they started talking
about option pricing.
And Fischer came
up with an analysis
that was very much along
the lines of Baschelier.
He basically got this formula,
but he couldn't solve
it because he had never
heard of the heat equation
because his Fischer Black's
background was in computer
science, not in mathematics.
It was ironic, because Fischer
Black actually had a PhD.
Not in economics or finance,
but in applied math.
But he had never taken physics.
So he was doing discrete math.
So he started talking
to Myron Scholes
and as legend would
have it, Myron
took that heat equation, went
over to the math department
here, and asked one of the
mathematics professors,
have ever seen this thing?
And a math person
looked at him and said,
oh yeah, that's
just heat equation.
Here, you solve it like this.
And so Myron apparently took
it back to Fischer Black.
And Fischer said, hmm,
this is interesting.
We can now write a paper.
And they wrote a paper on this.
At the same time, Bob
Merton was working
on another direction
that was trying
to come up with a solution.
Ultimately, he came up
with the same solution.
They didn't know it because they
had actually not communicated
to each other.
But ultimately,
Myron and Fischer,
they sent their
paper to something
like five economics journals.
Every single one of
them rejected the paper
saying this is too specialized.
It's not really economics.
It's not finance.
We don't know what
it is, but go away.
And it was only until
they were able to change
the title of the paper
from option pricing
to the pricing of options
and corporate liabilities
that they finally--
so it was exactly this--
well, I'll show you next time.
They changed it to
start focusing more
on corporate finance.
They ultimately got
their paper published.
It turns out that Merton used
a very different approach
but got to the same point.
And so Black and
Scholes got their paper,
ultimately, accepted
into The JPE.
Merton got his paper
accepted The Bell Journal,
both in the same year.
In fact, Merton got his
paper published first.
But he argued that
the paper should
be delayed because he wanted
Fischer Black and Myron
Scholes have their paper
come out in the same year.
He felt that he derived
so much intuition
for what Black and
Scholes were doing,
that he didn't want to get
there first, because it was not
fair to them.
That was one of the
most extraordinary acts
of professional ethics
in the profession.
Because it was pretty clear to
both of them what was at stake.
This was a huge problem that
took an enormous amount of time
to solve.
And of course, the
rest is history.
They were awarded the Nobel
Prize in 1997, Myron and Bob.
Unfortunately, Fischer Black had
died of cancer the year before.
But it was very clear in
the Nobel address, both
on the participants' part, as
well as the Nobel Committee,
that Black should have
received it as well.
So that's the history and the
heritage of option pricing.
You can see why MIT is
rightly proud of it.
And given that we're out
of time, let me stop here.
And then next time,
what we're going to do
is to take up where we left off,
and focus on the actual pricing
formula.
I'm going to derive it for you.
Not the Black-Scholes formula,
but a simpler version.
And you'll see it, and you'll
be able to take a look at it
and play with it.
We'll go on from there.
OK, I'll see you on Wednesday
for the mid-term exam.
