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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today,
what I want to do is to continue where we were
last time with option pricing. As I promised you
last time, having gone through the history
of option pricing and the special role
that MIT played, today I actually want to
do some option pricing. I want to show you a simple
but extraordinarily powerful model for actually coming up
with a theoretical pricing formula for options,
and frankly, all derivative securities. So we're gonna actually do that
in the space of about a half an hour, and then we're
going to conclude. And I want to turn, then,
to the next lecture, which is on risk and return. I want to now, after we
finish option pricing, take on the challenge of
trying to understand risk in a much more concrete way
than we've done up until now. OK, so let's turn to
lecture 10 and 11. And I'd like you to
take a look at slide 16. OK, this will be the first
model of option pricing that any of you have ever seen. You've all heard
of Black-Scholes. We talked a bit
about it last time. Frankly, this is a simpler
version of option pricing that ultimately can
actually be used to derive the Black-Scholes
formula as well. But the reason I
love this model is because it is so simple that
with only basic high school algebra, you can actually
work out all of the analytics. So all of you
already have the math that it takes to
implement this formula, and even to derive the formula. But the underlying economics
is extraordinarily deep, and so it's a
wonderful way of sort of getting a handle on how these
very complex formulas work. So here's what
we're going to do. We're going to simplify the
problem in the following way. We're going to use-- the framework, by
the way, is called the binomial
option-pricing model that was derived by our very own
John Cox, Steve Ross, and Mark Rubenstein of UC Berkeley. And although this
is a simpler version of option pricing than
Black and Scholes, it turns out that on the street,
this is used much more commonly than the Black-Scholes formula. So let me show you how it works. We're going to start with a very
simple framework of one period option pricing,
meaning we're going to focus on a stock that
survives for two periods-- this period and the next period. And then we're going to consider
the pricing of an option on that stock that
expires next period. We're going to figure out
what the price is this period. So we've got a stock
XYZ, and let's suppose the current stock price is S0. And let's suppose that
we have a call option on this stock with
a strike price of K, and where the option
expires tomorrow. And so tomorrow's
value of the option is simply equal to C1, which is
the maximum of tomorrow's stock price minus the strike, or
0, the bigger of those two. That's the payoff
for the call option. And the question that
we want to attack is, what is the
option's price today? In other words, what is C0? So we draw a timeline,
as I've told you, for every one of these problems. Draw a timeline just so
that there's no confusion. So tomorrow, the stock price
is going to be worth S1, and the option price
is just equal to C1, which is the payoff,
since it expires tomorrow. And the payoff is just the
maximum of S1 minus K and 0. And the object of
our focus is to try to figure out what the value
of the option is today. And so I'm going to argue
that if we can figure out what it is today,
based upon this, then we can actually generalize
it in a very natural way to figure out what the price
is for any number of periods in the future. So how do we do that? Well, we first have to make
an assumption about how the stock price behaves. As I mentioned
last time, we need to say something about the
dynamics of stock prices, and remember that Bachelier,
that French mathematician that came up with a rudimentary
version of an option pricing formula in 1900, he
developed the mathematics for Brownian motion,
or a random walk, for the particular stock price. So you have to assume
something about how the stock price moves. So what we're going to do
in a simplified version is to assume that the stock
price tomorrow is a coin flip. It's a Bernoulli trial. That's the technical term. So it either goes up or down. And if it goes up, it goes
up by a gross amount u. So the value of
the stock tomorrow, S1, is going to be equal
to u multiplied by S0. So if it goes up
by 10% tomorrow, then u is equal to 1.1. Or it can go down by a
factor of d tomorrow, and so if it goes down
by 10%, then d is 0.9. So we're going to
simply assert that this is the statistical
behavior of stock prices. Now, granted, this is a very,
very strong simplification, but bear with me. After I derive the simple
version of the pricing formula, I'm going to show
you how to make it much, much more complex. And the additional
complexity will be really simple to
achieve once we understand this very basic version. Now, the probability
of going up or down is not 50/50-- doesn't
have to be 50/50. So I'm going to assert that it's
equal to some probability of p and 1 minus p. So it either goes up by p or
goes down with probability 1 minus p, and the amount
that it goes up or down is given by u and d. And I'm going to assert
that u is greater than d. Question? AUDIENCE: So u and d
are not the changes, but you're just assuming that,
in your case, [INAUDIBLE].. PROFESSOR: Sorry, that they're-- AUDIENCE: It's not necessary
to always use [INAUDIBLE] d, or can you assume-- PROFESSOR: Yeah, I'm assuming
as a matter of normalization that u is greater than d. It doesn't matter. I mean, one thing has to
be bigger than the other, so I just may as well assume
that u is greater than d. Yeah, question? AUDIENCE: Is it necessary
to add up to 1 [INAUDIBLE]?? PROFESSOR: u and do don't
have to add up to anything. That's right. p and 1 minus p always
add up to 1, right. So for example, if this
is a growth stock that's really doing well, then u may
be 1.1, 10%, and d may be 0.99. So in other words, when it
goes down, it goes down by 1%. When it goes up,
it goes up by 10%. So on average, when
you multiply by p, 1 minus p, depending
on what they are, you can get a stock that's
got a positive drift. If, on the other
hand, you've got a stock that's
declining in value, then it may end up that d much
smaller than u, and 1 minus p is bigger than p,
which means that you're more likely to be going
down than you are going up. So it's pretty general. Yeah? AUDIENCE: u is bigger than 1 and
d is lower than 1, or you could [INAUDIBLE]? PROFESSOR: It
doesn't have to be. But typically you would
think that in the up state, it's going to be bigger
than 1, and the down state, it will be less than 1. But it doesn't have to be. What I'm going to normalize it
to be is u is greater than d. And later on, we may make some
other economic assumptions that I'll come to that will
tell you a little bit more about what u and d are. Now, if it's true that
the stock price can only take on two values
tomorrow, then it stands to reason that the option
can only take on two values tomorrow. And those are the two values. It's going to be Cu and Cd. Cu is where the stock price
goes up to u times S0. Therefore, the option's going
to be worth u S0 minus K, 0, maximum of those two. And similarly, if it turns out
that the stock price goes down tomorrow, then the option
is worth this tomorrow. Two values for
the stock tomorrow implies two values for
the option tomorrow. Any questions about that? OK, so having said
that, we can now proceed to ask the question,
given this simple framework, what should the option
price today depend on? It's going to be a function
of a bunch of parameters. So what should it depend upon? Well, the parameters that
are given are these-- the stock price today,
the strike price, u and d, p, and the interest rate
between today and tomorrow. Those are the only
parameters that we have. These are it. This is everything. It's going to turn out that
with the simple framework that I've put down, we will be
able to derive a closed-form analytical expression for what
the option price has to be today-- C0. I'm going to do that for
you in just a minute. But it's going to turn out
that that option pricing formula, that f
of stuff, is going to depend on all of these
parameters except for one. One of these parameters
is going to drop out. In other words, one of these
parameters is redundant. And anybody want to
take a guess as to what that parameter might be? What parameter do
you think might not matter for pricing an option? Yeah, Terry. AUDIENCE: The
interest rate, the r? PROFESSOR: The interest rate. Well, that's a good guess,
but that's not the case. That's what I
would have guessed, because that seems
to be the thing that should matter the least,
given how important all of these other parameters are. Anybody want to
take another guess? Yeah, Ken. AUDIENCE: Today's stock price. PROFESSOR: Today's stock price. That's another good guess. [LAUGHTER] Although that's not correct,
because in both the case of the interest rate
and today's stock price, you could ask the question,
suppose the stock price were at $1,000 versus $10. That would matter, wouldn't it? Or if the interest rate
were at 20% versus 1%, that should matter,
shouldn't it? And it does. In fact, if you look at every
single one of these parameters, none of them looks like
they're unnecessary. It looks like all of
them are required. Yeah, John. AUDIENCE: [INAUDIBLE] PROFESSOR: Well, the strike
price remains the same, but the thing is that the
question is whether or not the value of the option
depends on the strike price. And if the option is,
for example, in the money or out of the money, you
would expect that that would make a big difference. AUDIENCE: [INAUDIBLE] PROFESSOR: Right. In fact, let me
tell you that there is no good answer to this,
because all of these parameters look like they belong. But I want to tell you
that one of them will not. One of them will not be in here. And this is going to be a
major source of both confusion and illumination for
what really depends on-- what option pricing
really depends on. All right, so let me just show
you how we're going to do this. Let me illustrate
to you the method. And we're going to do
this in the exact same way that we've priced virtually
everything under the sun. We're going to use an
arbitrage argument. I'm going to
construct a portfolio that will have the identical
payoff to the option, and therefore if the portfolio
has the exact same cash flows as the option,
then the cost of constructing
that portfolio has to be the price of the option. Moreover, if it's
not, you're going to be very happy, because
that will mean that there is an arbitrage opportunity. That is, there's
money to be made. If this theory
fails, then you're going to be able to get rich
beyond your wildest dreams. So we're hoping for
a violation of this. So let's see how we do that. I want you to now forget
about the option for a moment, and I want you to
imagine that at time 0, we construct a portfolio
consisting of stocks and riskless bonds,
in particular delta shares of stocks and B
dollars of riskless bonds-- riskless in terms of default. And the total cost of this
portfolio today, time 0, is simply equal to the
price per share times the number of shares of stock,
so that's S0 times delta, plus the value of
the bonds that I'm buying-- the market
value of the bonds that I'm buying
today, or selling. So B could be a positive
or negative number. Delta could be a positive
or a negative number. And that's my cost
today, time 0. Now, I want to
look at the payoff tomorrow for this portfolio. So V1 is the payoff
for the portfolio. That's what it's worth tomorrow. And V1 is going to be given
by the value of the stocks and the value of the bonds. Now, the stocks
are going to be-- there are two possibilities. Either the stock goes up
or the stock goes down. And if it goes up, it'll
be worth u S0 times delta, and if it goes down it'll
be worth d S0 times delta. I don't know whether it'll go up
or down, but whatever it does, this is the value tomorrow. Now, what about
my bond portfolio? Well, I bought B
bonds, and r now is the gross rate of return,
the gross interest rate, so it's a number
like 1.03 or 1.05. And the reason that
I'm switching notation is I'm following the notation
used originally by Cox, Ross, and Rubenstein, so I
apologize for the kind of cognitive dissonance
that this may generate. But this r, the way that Cox,
Ross, and Rubenstein wrote it, was meant to be a
gross rate of return. So you'll never see a 1
plus r, because this-- in their framework, because
this already contains the 1. So just keep that in
the back of your mind, and make a note of that. r
is the gross interest rate. It's 1 plus the
net interest rate, so it's a number like 1.03
for a 3% rate of return. Actually nowadays,
it should be more like 1.01 for short-term
interest rates, or less. Now, you'll notice
that whether or not stocks go up or
down has no impact on your riskless borrowing. You're going to get r
times B no matter what, or you're going to
owe r times B if B was a negative number, in both
cases, because it's riskless. It has nothing to do
with whether or not stocks go up or down. OK, now here's what
I want you to do. I want you to select a specific
amount of stocks and bonds at date zero in order
to make two things true. I want you to select delta and
B so as to satisfy these two equations. I want you to pick delta and
B so that in the up state, you get Cu and in the
down state you get Cd. Now, what are Cu and Cd? Remember what they are? They're the value of the
call option in the up state and the down state. And you know that in advance. You know what those
two possibilities are. You don't know which
one's going to occur, but you know that if the up
state occurs, it'll be Cu, and if the down state
occurs, it'll be Cd. So I want you to find
two numbers, delta and B, that make those
two equations true. Can you always do that? How do you know you
can always do that? OK, can you always
find a delta and a B to make those two
relationships true? Yeah. AUDIENCE: You have two
equations and two variables. PROFESSOR: Ah, you have
two linear equations in two unknowns. And from basic high
school algebra, you know that unless those
two linear equations are multiples of each other,
you can always find one-- exactly one solution
that satisfies those two equations in two unknowns. Kind of a handy feature
about linear equations. So as long as these two
equations are said to be linearly independent-- that's
a fancy way of saying that they're actually two
different equations, they're not multiples
of each other-- as long as these two
equations are not multiples of each
other, you can always find two numbers, delta
and B, to make that true. And here they are. Those are the two numbers,
delta star and B star. I solved the
equation for you, not that you couldn't
do it on your own, but for convenience,
there it is. So let me tell you
what we've done. We've put together a portfolio
of stocks and bonds at date 0 such that at time 1, the
value of this portfolio is always equal to
the value of the call option in no matter what state
of the world actually occurs. Well, by the principle
of arbitrage, what this tells us is that
the cost of putting together this portfolio that replicates
the call option's cash flows, the value of that
portfolio at date 0 must equal the price
of a call option. So we're done. The solution of what is the
call option price at date 0, it's given by this
formula right here. There it is-- a
closed-form solution. Now, before we beat
up on it and say, gee, there's only
two possibilities, life is more
complicated than that, and also there's only
one period, let's not-- let's not beat up
on it just yet. Let's take a look to see
whether or not this makes sense and whether we
agree and understand that if, in fact, the
assumptions are true, that this is indeed
the price of an option. Because this is a pretty
remarkable formula. It's a remarkable formula
for its simplicity, and for the fact
that we actually have been able to
derive it explicitly. Now, the other amazing
thing is that there is a missing parameter here. Now you see what the
missing parameter is. What this formula
doesn't depend on is the probability of the
thing going up or down. Now, that's astonishing. It's astonishing because what
it says is that you and I, we can disagree on whether
General Electric is going to go up tomorrow
or down tomorrow, and yet we still
are going to agree on what the value of a
General Electric call option is tomorrow. That's a remarkable
fact, and it has to do with a very
deep, deep phenomenon going on in option pricing,
which is that option pricing is all about pricing the relative
magnitude of the security relative to the stock price. And once we understand the basic
features of the stock price, like whether or not
it can go up or down by u or d, that's more important
than the actual probabilities of u and d. So this expression-- and when
you fill in for Cu and Cd, you can plug in for that
maximum of u S0 minus k, 0, you'll see that there's
no p in there as well. So any questions about this? Yeah. AUDIENCE: You just
said that we could disagree on what we-- if the
stock will go up or down, [INAUDIBLE]. PROFESSOR: Yeah, the probability
of it going up or down, right. AUDIENCE: And what if we
disagree on the actual number? PROFESSOR: Then we will
disagree on the option price. So we have to agree
on the u and the d. AUDIENCE: But if we-- PROFESSOR: Sorry--
yeah, the u and the d. AUDIENCE: [INAUDIBLE] we will
disagree the option price, or that's why this
market is possible, because someone will
think it will go up-- PROFESSOR: No, no,
it's not the reason that the market
will be possible. The possibility of
the market actually does depend on
whether or not there's a demand for this
particular kind of payoff. But that doesn't necessarily
hinge on the u or the d. In other words, we can
agree on the u and the d, but it turns out that you
think that the price is going to go up, therefore, you want to
have that kind of a call option bet. I think the price
is going to go down, so I'm happy to sell it to
you, because I think I'm going to get a good deal on it. So we disagree on the p. You think that there's a high p. I think it's a low p. That's what drives the market. And the beauty of
this particular setup is that it tells you that you
can actually agree on a price, but you have very
different reasons for engaging in the transaction. And then you will have markets
for this particular security. Now, I want to go through
and look at this formula and try to understand it. First of all, we see that this
formula is a weighted average of the value of the
call option in the up state and the down state. It's a weighted average. And this part inside
the bracket you can think of as a weighted
average of the outcome. But then you discount it back
to the 0th period using the one period interest rate. And again, remember
this is not meant to be a perpetuity
kind of expression. This r is a gross
interest rate, so it is equivalent to our
old 1 over 1 plus r, where the r that we used
is the net interest rate. Here, because of the Cox,
Ross, and Rubenstein notation, this is meant to be the
gross interest rate. So this looks like
a present value, because whatever is
inside the bracket, you can think of as
some weighted average of the value at date 1, and then
this brings it back to date 0. But now let's look at
the weighted average. The weights r minus d
and u minus d, those-- it turns out that this
plus this adds up to 1, so indeed, it is a
weighted average. When you multiply by
theta and 1 minus theta, the weights add up to 1. You're basically taking
a weighted average. But I want to argue
that it's more than just a simple weighted average. I'm gonna argue that
these weights are always non-negative. So in fact, this looks like
not just a weighted average, this looks an awful lot like
a kind of an expected value, like a probability
weighted average. This looks like a probability. It's not a
probability, but I want to argue that this number
is always non-negative, and they add up to 1. So when you've got two
numbers that are not negative and they add up to 1,
you can interpret them as a probability. Now, what's the argument for
why this number is always going to be non-negative? The condition that's
required for these numbers to be non-negative is
that the interest rate r, the risk-free
rate, is strictly contained in between u and d. So you've got u here, d here. r has to be in the middle. And when that's the
case, then you've got these things looking
like probabilities. Now, the question is, is
that a reasonable assumption? Is it reasonable to
assume that d is less than r is less than u? Can anybody give
me some intuition for why that makes
economic sense? It has nothing to
do with mathematics. The mathematics couldn't care
less as to whether or not that inequality held. Brian? AUDIENCE: If the downside
was less than the rate, then you'd just automatically
buy the security. PROFESSOR: Right. AUDIENCE: And if the upside was
less than the risk-free rate, they'd you'd just go
into the risk-free bills. PROFESSOR: Right. That's exactly right. That's a very important
economic insight. Let me go through that slowly. So Brian, you said if r is
less than the downside, then what happens in that case? AUDIENCE: Then you'd want to
buy the stock, the security. PROFESSOR: If the stock in
its worst possible state offers more than
T-bills, why would you ever want to buy T-bills? In fact, you wouldn't. And if that were
true, then what would happen to the price of T-bills? The price of T-bills-- AUDIENCE: It would go down. PROFESSOR: It would go to 0. Nobody would hold it, and
therefore the value of it would go to zero. It would not exist any longer. So if we're going
to assume that there exists riskless borrowing,
that can't be true. We can't have r over here. Now, what about the
other side, Brian? What happens if r is over here? What did you say? AUDIENCE: Then you'd want
to go into the risk-free. PROFESSOR: Right. You would never hold
the stock, because even in the best possible
world for the stock, you would not be able to get
as good a return as T-bills, in which case the value of
the stock would go to zero, and therefore there'd be no
more stocks in the economy. The only situation where you
can have stocks and T-bills coexisting in this
simple world-- the only case where that's true
is if this inequality held. That's the economics of
this pricing formula. It has nothing to do with math. It's the economics. And the economics tells
you that these things have to be non-negative. That's good, because
that suggests that the price of the
call option at date 0 can never be negative,
because these guys, Cu and Cd, are non-negative, and 1
over r is non-negative. So if it turns out that the
weights can never be negative, then you know that
you've got something that really is a pricing formula. You're never going to
punch in some numbers and get out a formula that says
this thing is worth minus 2. But more importantly,
it suggests that there is a
probability interpretation. But the probability is not
the mathematical probability that matters. It is the economic probability. And there is a term for
this particular probability. This is known as the
risk-neutral probabilities of the particular economy
that we've created. And it turns out that
these probabilities can be used to price
not just options, but anything under the sun. So there's a very,
very important property and very deep property
that we can't go into here, but you'll cover in 15 437,
about the so-called risk neutral probabilities. But now we've got
a formula here. This is a bona fide
pricing formula. And the beauty of
it is that if it is violated-- if it is violated
but the assumptions are correct, then there is
a way to create a money machine, an arbitrage, either
by buying the cheap stuff and shorting the
expensive, or vice versa, in the case where the
signs are flipped. So here's the argument. Suppose that C is greater than
V. Then here's the arbitrage. Suppose it's less,
and then you basically construct the
opposite arbitrage. Therefore the cost of the option
has to be equal to the value that we computed. Yeah, [INAUDIBLE]. AUDIENCE: Is this function
sensitive [INAUDIBLE] PROFESSOR: Well, I
mean, you tell me. It's a convex combination
of these two things. So in that sense-- AUDIENCE: [INAUDIBLE] PROFESSOR: That's
right, exactly. Yeah. AUDIENCE: And this is
not going to [INAUDIBLE] PROFESSOR: Yeah. AUDIENCE: The u and d are
determined by the market-- PROFESSOR: No. AUDIENCE: [INAUDIBLE] PROFESSOR: No. That's a modeling assumption. So in advance, we
agree what u and d are. Now in a minute, I'm
going to start relaxing all of these assumptions. But before we do
that, I want make sure we all agree on what this says. Yeah. AUDIENCE: So the p is
missing, as you said. PROFESSOR: Yes. AUDIENCE: But isn't that-- isn't that embedded in
Cu and Cd, because you rely on a market
price for Cu and Cd? PROFESSOR: No, there is no
market price for Cu and Cd. Let's go back and take a look
at what Cu and the Cd are. That's not a market price. This is not a market price. Cu is basically the outcome
of u times S0 minus K, and Cd is the outcome
of d S0 minus K, 0. That's not a market price. We have to agree in advance on
what the possible outcomes are. But once we agree
on those outcomes, everything follows from that. There's no market price here. The only market price is S0. That is the market price. That is determined today. But fortunately, that
market price we observe. We can see it. Now, there is a link
between the market price, u and d, p, because if the stock
price today is worth $20, and tomorrow we say that
there are two possibilities, either it's $30 or $10,
then that tells you that p sort of has to
be somewhere around 0.5. We may disagree. I may think it's 0.55. You may think it's
0.45, whatever. But when we aggregate
all of our expectations, we come up with $20
for the stock today. So it's all related. It's in there. But we don't need to make
an assumption explicitly for what p is. That is the power of this kind
of option pricing approach. [INAUDIBLE] AUDIENCE: [INAUDIBLE] the reason
you get the market here is we agree on everything
except that I think that the higher [INAUDIBLE]. PROFESSOR: Yeah. AUDIENCE: --it will
go up, and you think it will go down [INAUDIBLE]. PROFESSOR: Right. AUDIENCE: What if
we fundamentally don't agree on u and d? PROFESSOR: Oh, then
we have a problem. We need to assume a particular
u and d that we can agree on. So let me turn to that now. Let me turn to the
extension of this. So what I've derived is a
one-period pricing model-- very, very simple. It turns out that you can do
a multiperiod pricing model. And this multiperiod
generalization is given by this. What is that multiperiod
generalization? Basically you have-- let me see
if I have the diagram here-- the multiperiod
generalization is simply that you now have a
bunch of possibilities, and you are figuring out
what the price of the option is at date 0 when it pays
off at date capital N, or lowercase n in this case-- n periods. And you can use exactly the same
arbitrage argument that I just showed you, but it's
a little bit more complicated now because
you've got multiple branches. But it's still, at every
step of the way, a binomial or a Bernoulli trial. And so in a multiperiod setting,
you get a binomial tree. Now, the reason that this
is such a powerful extension is that nowhere have I
specified what a period is. I just said it's a period,
today versus tomorrow. But it could be today versus
three minutes from now, or three femtoseconds from
now, or three years from now. I haven't specified. So if you say we
can't agree on a u and a d, fine, let's not
agree on a u and a d. Let's agree that between now
and five minutes from now, there are 256 possible
outcomes for the stock price. Do you agree on that? You think we can agree on that? Is that something
that's easy to agree on? Well, if that's
the case, then all I need to do is to have
enough steps between now and five minutes from now
to have 256 possibilities. And by the way, I chose
that number specifically as a power of 2 because with
these kinds of branches, it's actually very
easy to be able to get that kind of a tree,
with that many branches. So now you see that
the u and the d, that's not relevant,
because we can make it as small as you would like. If you would like to have
it really, really fine, I can get it down to double
precision, 32 decimal places, by basically taking one
period to be a millisecond. And this binomial
option pricing formula will apply exactly
in the same way. It turns out that when you
let the number of periods go to infinity, and
at the same time, you control the u
and the d and make them smaller and smaller
and smaller and smaller so as to be able to get a tree
that is reasonably realistic, you know what you get? You get the
Black-Scholes formula. The pricing formula that
you get is a solution to this parabolic
partial differential equation with the following
boundary conditions. And so using the simple binomial
two-step kind of process, when you let it go to
infinity and you shrink the
probabilities and the u and the d to make it
more and more refined, you get the
Black-Scholes formula. This is something
that Black and Scholes never, never contemplated. So this is a completely
different approach that allows you to reach the
exact same conclusion, which is a startling one. Now, as I told you
at the beginning, when people apply option pricing
formulas, most of the time they do not do this. They do not solve
the heat equation. What they do is that. They do a binomial tree. The reason is because in
order to solve these PDEs, except in a very, very small
number of textbook example cases, you can't solve
this analytically anyway. You can't get a formula. You have to solve
it numerically. And so if you're going to
go to the trouble of solving these differential
equations numerically, you may as well just do
the binomial option pricing formula, because that's
numerical as well. And it's a lot simpler
computationally to be able to do that binomial tree. By the way, for those
of you computing fans who like to think
about parallel processing, these kinds of binomials
trees are extraordinarily easy to parallelize. So if you thought about
the old days, where you had a connection
machine that was developed by
Danny Hillis, you had 64,000 processors
in parallel. You can actually
make use of that by implementing a binomial tree. Nowadays we've got
grid computing. The most recent advance is to
be able to use both hardware and software to do
distributed computing. The binomial tree
is ideally suited for being able to do that. So you can evaluate
extraordinarily complex derivatives very, very quickly
using this kind of a framework. So you're not giving up
a lot by the u and the d, because we can make the u
and the d as fine as possible so that ultimately we would
all say, yeah, enough. I agree, all right,
leave me alone. I don't want any
more binomial trees. This is complicated enough. 256 of them over a
five-minute interval is enough for all
practical purposes. Yeah. AUDIENCE: [INAUDIBLE] if
that cannot be solved, why was it so important? PROFESSOR: Oh, no,
this can be solved. The solution of this equation
is the Black-Scholes formula. What I said cannot be solved
is when you have a more complicated security. So for example,
the option pricing formula that we looked at with
the simple plain vanilla call and put option, that's
relatively straightforward. But think about something like
a mortgage-backed security that has all sorts of conversion
features and knockout features, and other types of
legal restrictions, as well as certain
rights and requirements. Then it's not so easy. It looks much more complicated. For example, this
particular coefficient that multiplies this
second derivative ends up being a highly
non-linear function, not just a quadratic. Or this piece here becomes
a nonlinear function, or the boundary conditions
are kind of weird. In that case, you can't
solve it analytically. You have to use numerical
methods to solve it. AUDIENCE: [INAUDIBLE] PROFESSOR: This is just
the arbitrage condition that says that the solution C
will give you a null arbitrage price for the call option. So the equivalent of this PDE,
partial differential equation, is-- go back-- is this, the
simultaneous equation up there and down here,
and then this expression that says that the
price of the option has to be given by this
particular portfolio. That's what the PDE looks
like in continuous time, or when you have an infinite
number of time steps. So it is not-- that's absolutely
a good question, because this is solvable. But very quickly, when you
change the terms of a contract, it turns out that it's
very hard to model. Yes. AUDIENCE: Question
about the random walk. PROFESSOR: Yes. AUDIENCE: Can you just
briefly mention how that feeds into the final answer, and how
it will change things if it's-- PROFESSOR: Well, the random walk
hypothesis is implicit in here, because I've got a coin toss. And the coin toss is
independent period by period. If the coin toss
is not independent, then that's the wrong formula. In other words, you don't have
a simple binomial distribution if you don't have
IID coin tosses. The random walk is
basically the assumption of IID coin tosses--
independently and identically distributed. That's what IID stands for-- IID coin tosses. So that's where the
Bachelier assumption came in. In order for Bachelier to
derive the heat equation, or some variant of
the heat equation, he was implicitly assuming
that what happens in one period for the stock has no bearing
on what happens in next period. If stock prices are
correlated over time, then these formulas do not work. You need a different
kind of formula. It's actually not that far off. You can derive an expression
for an option pricing formula with correlated returns. In fact, professor Wang
and I published a paper, I think it's maybe
close to 10 years ago, where we worked out that case. But up until then,
most people assumed that stock prices
are not correlated, so the Brownian motion or random
walk idea fit in very nicely with this binomial. If they're correlated, then you
no longer have IID Bernoulli trials, you have a
Markov chain, and you have to use Markov
pricing in order to be able to get this formula. If you're interested in this,
I urge you to take 15 437, because that's where we go
into it in much more depth. Yeah. AUDIENCE: [INAUDIBLE] use the
binomial coin to value options, and we see a range of prices. PROFESSOR: Yeah. AUDIENCE: So how do
we approach that? Do we take some kind of average? Is this common, or do we receive
a specific u and a d each time? I mean, I imagine it could
be a range [INAUDIBLE].. PROFESSOR: No, no. So the way that you
would apply this is that you would,
first of all, pick the number of periods that
are appropriate to the problem at hand. So if you have got
an option that's expiring in three
months, then typically, if you did it on a daily
basis or an hourly basis, that would be more than enough. And then you would assume that
there would be a u and a d in order to match the
approximate outcomes that you would expect. And then out of that, you
would actually get a number. So this, this C0, when you plug
in all of these parameters, you actually get a
number, like $30.25. That's the price of the option. And of course if you
change the parameters, you change the strike price,
the interest rate changes, the u and the d changes, that will
change the value of the option price as well. AUDIENCE: [INAUDIBLE]
every now and then, [INAUDIBLE] to receive a
range from you, and a range-- PROFESSOR: No, no, no. What you do is you start
off with an assumption for what u and d exactly are. Not a range, but actually if
it goes up, it goes up by 1.05. If it goes down it
goes down by 0.92. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Oh, well,
it varies depending on the particular instrument
that you're trying to price. So-- well, no, what I mean
is options on what stock? So in other words, with any
kind of option pricing formula, you actually have to
calibrate these parameters. So you have to figure out
what the interest rate is, and then typically what
is done is you assume a particular grid, and then use
a u and a d that will capture all the elements of that grid. So for example,
let's assume that u is 25 basis points plus 1, and
d is 1 minus 25 basis points. So that means you can capture
stock price movements that go up by 25 basis
points or down, and you assume a
number of n in order to get that tree to
be as fine as you would like for the particular
time that you're pricing it at. So in other words, if I use 25
basis points and n equal to 1, that means that I can
capture a situation where, at maturity, the stock price
goes up or down by 25 basis points. If I now go four
periods, then I can capture a situation where
the stock price goes up by 1% or down by 1% in
25-basis-point increments. And if I want more
refinements, then I keep going, let n get bigger and
bigger and bigger. And then whatever that is,
that final number of nodes will be the possible
stock price values. AUDIENCE: [INAUDIBLE] historical
data on the specific stock to-- PROFESSOR: You would
use historical data. You would use historical--
because the way you calibrate this is you can show
that the expected value-- so the expected
value of S1 is just equal to the probability
of u S0 plus 1 minus probability of d S0. So you've got the
expected value. Calculate the variance
of S1, and you'll get another expression
with u and d and p, and then you simply
use historical data to match the parameters
and pick them so that they give you a
reasonable approximation to reality. AUDIENCE: [INAUDIBLE]
doesn't continue to behave as the history-- PROFESSOR: Yes. AUDIENCE: --so the options-- PROFESSOR: Yeah. AUDIENCE: --don't match. PROFESSOR: Absolutely. That's always the
case, isn't it? In other words, if
you don't have IID, you're going to get a problem. But remember, it doesn't
depend upon the p. And so in that sense, if
there's a change in p, as long as the u and
the d are appropriate, you'll still be able to capture
the value of the option. Question. AUDIENCE: I'm
trying to figure out the analogy with
the [INAUDIBLE].. PROFESSOR: Yeah. AUDIENCE: So I understand
how it works in temperature. What would be here
that [INAUDIBLE].. PROFESSOR: Let me-- let me
not talk about that now, because I suspect that
while you may be interested and a couple of other
people, we probably don't have everybody
being physicists here. So we'll talk about
that afterwards. And also, that's
something that, again, in 437, they may touch upon. But I want to keep moving
along, because this is already more complicated than
the nature of what I want to cover in this course. So let me get back
to you on that, but we can talk
about it afterwards. Any other questions about this? Yeah. AUDIENCE: I have a
question about volatility, and how it is going to
play in the equation. Like for example, I
have two scenarios. They all, in three months,
could go up or down by u or d. But the volatility of those to
scenarios vary dramatically. PROFESSOR: Right. AUDIENCE: So how does-- PROFESSOR: How does
volatility enter into this. That's a good question. Well, what do you
think volatility is captured by in this
simple Bernoulli trial? AUDIENCE: The difference
between u and d. PROFESSOR: Exactly, exactly. Volatility is a measure of
the spread between u and d. Holding other things equal-- by that, I mean holding the
current stock price equal, holding the
probability p equal-- so fixing that, as I increase
the spread between u and d, I'm increasing the volatility. And if there's one thing
that we see that matters is the spread between u and d. So if the spread between
you and d increases, that actually will have
an impact on this formula, and you have to work
out the effects, which is a very easy thing to do. You can even do this
in a spreadsheet. But you can show that as
the volatility increases, the value of the call option
is actually increasing. So take a look at
that, and you'll see that it behaves the way
that we think it should. OK, other questions? OK, well, so I'll
leave I'll leave it at this point, which is to
say that the derivatives literature is huge. And it has really spawned a
number of different not only securities, but also
different methods for hedging and managing
your portfolios, to the point where really,
derivatives are everywhere. And there are some examples
that I've given you here, but this is an area
which is considered rocket science because
of the analytics that are so demanding. So this is a natural area
for students here at MIT to be involved in, but it's
certainly not the only area. And ultimately, what's important
about derivatives is not just the pricing and the hedging,
but rather the application. So the fact that we
spend a fair bit of time at the beginning of
this lecture talking about payoff diagrams, that
wasn't just for completeness. That really is one of the
most important aspects, is how you use options
in order to tailor the kinds of risks
and return profiles that you'd like to have. And now that you know
how to price them, you can have a
very clear sense of whether or not they
are appropriate from a risk-return tradeoff. But they are very
different, as you can see, from the
securities that we've done. However, having
done it, having now priced options and
other derivatives, which are really relatively
straightforward extensions, we've now been able
to price virtually 99% of all the securities
that you would ever run into. We've done stocks. We've done bonds. We've done futures,
forwards, and now options, so there really isn't any other
kind of financial security out there that you could
possibly come across that you don't know how to price. You may not realize it
yet, and the purpose of the second half
of the course is to introduce risk and show you
how to use all of these methods to price all of the
other securities that you will come
into contact with. And then, of course, in 402
and other finance courses, you'll see that
much more closely. So for example, a
revolving credit agreement, a sinking fund debt issue,
a credit default swap, an interest rate swap-- all of these securities are
mixtures of the securities that we've seen till now. And the pricing method
in all of these cases is exactly the same, which
is identify the cash flows, come up with another portfolio
that has the same cash flows, but where you know
how to construct it, therefore the price
of that security has to be equal to
the price of the thing that you're trying to value. That's the basic principle in
virtually all financial pricing applications. So once you understand
these concepts, you can literally price
anything under the sun, and all you need between now
and then is practice, practice, practice in doing that. All right, so that wraps up
the lecture on derivatives. And now I want to turn
to risk and reward, because up until
now, we've really talked about risk
in an indirect way, and I want to talk about it
in a much more direct fashion by looking at measures of risk. So what I want to
do now is to turn to a little bit of
statistical background to talk about risk and return. I want to motivate
it first, and then give you the measures
that we're going to use for capturing
risk and return, and then apply it to stocks,
and get a sense of what kinds of anomalies are out there
that we should be aware of. And then I'm going to
take these measures, and then tell you how to
come up with the one number that I've had to put off for
the first half of the semester, which is the cost of capital-- the required rate of return, the
risk-adjusted rate of return. We are now going
to get to a point where we can actually
identify what that number is, and how to make that
risk adjustment. So that's where we're going. Now, to give you a quick summary
of where we are, as I told you, we've priced all of these
different securities. But underlying all
of these prices is a kind of a net present
value calculation where we're taking some kind of a
payoff or expected payoff and discounting it
at a particular rate, and we need to figure out what
that appropriate rate of return is. I've said before that
that rate of return is determined by
the marketplace. But what we want to know is how. How does the market do that? Because unless we understand
a little bit better what that mechanism is, we
won't be in a position to be able to say that the
particular market that we're using is either working
very well or completely out to lunch and crazy. So we need to deconstruct
the process by which the market gets to that. In order to do that, we
have to go back even farther and peel back the onion
and ask the question, how do people measure risk,
and how do they engage in risk taking behavior? So we have to do a little
bit more work in figuring out these different
kinds of measures, and then talking explicitly
about how individuals actually incorporate that into
their world view. Along the way, we're going
to ask questions like, is the market
efficient, and how do we measure the performance
of portfolio managers? This past year, the
typical portfolio manager has lost about 30% to 40%. That's a pretty
devastating kind of return. And in that environment, if you
found a portfolio manager that ended up losing you 10%,
you might think, gee, that's pretty good. Does that really make sense? Is it ever the case that we
want to congratulate a portfolio manager for losing money for us? We have to answer that
question in the context of how you figure out what an
appropriate or fair rate of return is. So that's what we're
going to be doing. Now, to do that,
I need to develop a little bit of new notation. And so the notation that
I'm going to develop is to talk about
returns that are inclusive of any kind of
distributions, like dividends. So when I talk about
the returns of equities, I'm going to be talking
explicitly about a return that includes the dividend. And so the concept
that we're going to be working on,
for the most part, for the next half of this course
is the expected rate of return. We obviously will be talking
about realized returns, but from a portfolio
management perspective, we're going to be focusing not
just on what happened this year or what happened
last year, but we're going to be focusing on
the average rate of return that we would expect over the
course of the next five years. We're going to be looking
at excess returns, which is in excess of the net
risk-free rate, little rf. And what we refer to as
a risk premium is simply the average rate of
return of a risky security minus the risk-free rate. So the excess return you can
think of as a realization of that risk premium. But on average over a
long period of time, the number that we're going
to be concerned with most is this risk premium number,
the average rate of return minus the risk-free rate. Over the course of the
last 100 years or so, US equity markets have
provided an average rate of return minus the risk-free
rate on the order of 7%. That's pretty good, but
that's a long-run average. The realized excess
rate of return this year is horrible, so
I'm not even going to talk about what that
number is, but it's bad. But do you see the difference
between this year's rate of return versus the
long-run average? And we can talk
about both of them, but we're going to use
different techniques for each. So the technique for talking
about the statistical aspects of returns will be from
the language of statistics. We're going to talk about
the expected rate of return. I'm going to use the Greek
letter mu to denote that. We're gonna also talk about
the riskiness of returns, which I'm going to use the variance
and the standard deviation to proxy for. So the variance is
simply the expected value of the squared excess return. That gives you a sense of the
fluctuations around the mean. And the standard deviation
is the square root of the variance. And we use the standard
deviation simply because that's in
the same units. It's in units of
percent per year, whereas the variance is in units
of percentage points squared per year, so it's
a little easier to deal with the
standard deviation. And those concepts are the
theoretical or population values of the
underlying securities that we're going to look at. We also want to look at
the historical estimates, and the historical
estimates are given by the sample counterparts. So this is the sample mean, the
sample variance, and the sample standard deviation. You should all remember
this from your DMD class. But if not, we'll have the TAs
go over it during recitation. You can also look in the
appendix of Brealey, Myers, and Allen, and they'll provide
a little review about this. Now, there are lots
of other statistics, and the only one that
I'm gonna spend time on is the correlation. There's the median
instead of the mean. You can look at skewness, which
way the distribution leans. But what we're going to
look at in just a little while is correlation,
which is how closely do the returns of two
investments move together. If they move together
a lot, then we say that they're highly
correlated, or co-related. And if they don't
move together a lot, they're not very
highly correlated. And in some cases, if they
move in opposite directions, we say that they're
negatively correlated. So correlation, as most
of you already know, is a statistic that's a
number between minus 1 and 1, or minus 100% and
100%, that measures the degree of association
between these two securities. We're going to be making
use of correlations a lot in the coming couple
of lectures to try to get a sense of whether or not
an investment is going to help you diversify your
overall portfolio, or if an investment
is only going to add to the risks of your portfolio. And you can guess as to how
we're going to measure that. If the new investment is
either zero correlated or negatively correlated
with your current portfolio, that's going to help in terms
of dampening your fluctuations. But if the two investments
move at the same time, that's not only going to
not help, that's going to actually
add to your risks. And you don't want
that, at least not without the proper reward. So that's a brief
preview of how we're going to use these statistics. And you get some
examples here about what correlation looks like. Here I've plotted four
different scatter graphs of the return of one
asset on the x-axis and the return of another
asset on the y-axis. And the dots
represent those pairs of returns for different
assumptions about correlation. So the upper left-hand
scatter graph is a graph where
there's no correlation. The correlation is zero. The scatter graph
on the lower left is where there's very high
positive correlation-- 80% correlation between the two. And the scatter graph
on the lower right is where there's a
negative 50% correlation. So we're going to
use correlation, along with mean and variance,
to try to put together good collections of securities--
in other words, good portfolios of securities. And by doing that, we're going
to show that we can actually construct some very attractive
kinds of investments using relatively simple information. But at the same
time, we're going to use that insight
to then deconstruct how to come up with
the appropriate risk adjustment for cost of
capital calculations. Now, there's a review here
about normal distributions and confidence
intervals, and I'd like you to go over that, either
on your own or with the TAs during recitations. We're going to be using
these kinds of concepts to try to measure
the risk and return of various different
investments. Here's an example of General
Motors' monthly returns. That's a histogram in blue,
and the line, the dark line, is the assumed normal
distribution that has the same mean and variance. And you could see that
it looks like it's sort of a good approximation,
but there are actually little bits of extra probability
stuck out here and stuck out here that don't exactly
correspond to normal. In other words, the
assumption of normality would say that the
probability of getting a return of minus 15%
is relatively low, then getting a return less than
minus 20% is exceedingly low. But the reality is different. There are risks of having much
lower returns in the data. And after this
year, I can tell you that these tails are
going to be fatter. So this is meant to be an
approximation, not reality. The approximation
is what we're going to go over in this course,
and in the very last lecture, I want to tell you how
good that approximation is. And then I'm going to tell
you about a number of courses you might want to take
that focus on getting that last 5% right. So 95% of the distribution
is captured by what I'm going to teach you in this course, but
if you want to get the other 5% right-- and by the way, if
you're going into investments as a profession, it's
all about that 5%-- then you'll want to take
15 433, investments. So with that as
the basic preamble, let me tell you what I'm
going to talk about next time, since we're almost out of time. What we're gonna
do next time is I'm going to talk about
the US stock market. I'm gonna talk about volatility,
about predictability, and then I'm going to talk
a bit about the notion of efficient markets,
and try to describe to you what kinds of
properties we expect from typical investments. And we're actually going
to go through some numbers. I'm gonna show you some
examples of basic statistics for the stock market
that will give you a sense of how things have
behaved over the last 50 years. And what you'll get a sense
of is that in some cases, there is a lot of
predictability. There are certain things
that we can count on. For example, these are
stock market returns from 1946 to 2001. This is monthly
data, monthly returns of the S&P 500 over a
fairly long period of time. And this might sort of look
like a typical person's EKG over the last few weeks. Not surprisingly, there
were periods where we had some pretty bad returns. We're going to see another
one of these things as well over the
more recent period. But when you look at
this thing, you then begin to appreciate that what
we're living through now, while it's bad and
it's scary, it's not at all unusual or
completely unheard of. There are periods
in the stock market where we've seen
really big swings. And by the way,
this is just the US. If I had shown you some
emerging market returns, it would go off the screen. So we're going to talk
about that next time. And out of all of
this chaos, we're going to distill a very
important relationship. We're going to
ultimately come up with a simple linear
equation that shows you how to make that risk
adjustment between the expected return and the underlying
risk of a portfolio. So that's coming up, and
we'll do that on Wednesday. All right, see you then.
