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visit MIT OpenCourseWare at ocw.mit.edu. ANDREW LO: I hope you all had
a good Columbus Day weekend. The stock market certainly did. Any questions from last time? No? OK. So what I want to do
today is to continue talking about futures
and forward contracts. Today we're going to finish
up on these interesting instruments, with a
couple of examples, and then with a specific method
for pricing forward and futures contracts. So let me refresh your memory,
it's been a week, so I know. So we're going to go back and
look at a specific futures contract. And I'm going to
take this contract and then try to talk a bit about
how you might use contracts like this in hedging
your risks, as well as in making certain kinds
of bets, if you will. So remember that this
contract is a contract that was issued on July 27, 2007-- so it was the middle
of the summer-- for oil to be
delivered in December. And there's a specific
date in December where all oil futures contracts
of this type will settle-- that is will come
to maturity-- where the date is going to
be specified in advance and everybody knows it. And so in July, when
you buy this contract at a price of $75.06 per
barrel, and each contract is for 1,000 barrels. When you quote
"buy the contract," what that means is that you
are agreeing today, July 27-- you are agreeing
today, that come December you are going to
buy 1,000 barrels at a price of $75.06 per barrel. So that's the agreement. And the party who is selling you
the contract, the counterparty, is agreeing to provide
you with that oil at that price in December. So the futures price is $75.06. And as we said last
time, the current market price on July 27, 2007,
that's called the spot price. The spot price may be higher or
lower than the futures price, depending on what
expectations are for what's going to happen with
oil over the subsequent six month period. Now the initial margin,
as I mentioned last time, was $4,050. The maintenance
margin, the margin that you need to maintain. So that if the initial
margin goes down in value, you have to actually put
money back into your brokerage account, your margin account. And if you fall below
that $3,000 threshold, you'll get a phone call, which
is known as a margin call. Several weeks ago,
somebody told me that they've been getting
lots of phone calls all from the same person,
a person called margin. And you know that can
happen when markets go awry. Now again no cash
changes hands today, because the value of the
contract when it's struck is a zero NPV transaction. And how do you
know it's zero NPV? Again, because if it's
positive for one party, it's coming from
the other party. So which party would
you like to be? You'd like to be the party
receiving that positive NPV, nobody wants to be the party
that is losing the NPV. So the futures price
will adjust, in order to make it zero NPV. In fact, that's what we mean
when we say that it's zero NPV. It is the futures
price that makes it so. So I'll give me an example. If it turns out that somebody
suggests a futures price of $60 a barrel on that day. Lots and lots of
people are going to want to buy that
contract, because that's a good deal relative to
where oil really should be. And that means lots of people
are going to want to buy, but nobody's going to want
to sell at that price of $60. So if everybody wants to buy
and nobody wants to sell, what has to happen to the price? Exactly, it goes up. And it keeps going up
until the number of buyers equals the number of sellers. That's the point at which
it's a zero NPV transaction. Now let's take a look
at what the payoff is of such a contract on day zero,
in this case July 27, 2007, the contract is worth nothing. But if the futures
price moves tomorrow, then the contract could
actually have value. And a diagram of how that
works is something like this. If the futures price-- not the spot-- the spot
obviously will move also. But I'm talking about
the futures price, because the futures
contract is specified so that every day's worth of
gains or losses in the futures contract relative
to its price, you will have to either get paid,
or you will have to pay. So this blue line
shows you the payoff if you're holding a
long position in one of these contracts-- if
you bought a contract. If you sold the contract,
then your payoff diagram is the dotted line. Now the blue line basically
says that if the futures price goes above $75.06,
then you make money. If the futures price goes
below $75.06, you lose money. So when you buy a
contract like this, it is as if you
actually bought the oil. But you haven't
really bought the oil, all you've done is
to buy the right and obligation to purchase
the oil in the future. Let me let me give you
another example that will make this even more concrete. The only way to
understand this-- because this is not a natural
security for most of us-- stocks and bonds you
might find natural, futures contracts are
weird in that they have zero investment today,
and so they're worthless today. But they're not worthless
after the initial date when you enter into that agreement. So let's do an
example Yesterday, you bought 10 December live
cattle contracts on the Chicago Mercantile Exchange at a
price of $0.7455 per pound. OK, so you basically
bought some cows. And the contract size
is 40,000 pounds of cow. I don't know how
much cow that is, but even if you're on the Atkins
diet that's plenty of cow. [LAUGHTER] And so what you
have though in this contract is not the cows, but rather
you have the obligation to buy the cows in December for
a price of $0.7455 per pound, and there is 40,000
pounds of it. So the value of your
position is the size of the contract, multiplied
by the futures price, multiplied by the
number of contracts. So it's $298,200. That's the size
of your position, or sometimes that's also
called the notional size. You've heard that term over
the last few weeks-- notional. Well, this is an
example of a notional. So you don't actually
own $298,200 of anything, because of course, we've said
that the contract is zero NPV when you enter into it. All you've done is to agree
to buy 40,000 pounds of cow in December at a
particular price. So the idea is that you
control the notional amount of $298,200, and what
you do specifically get is the profits and
losses from that notional. So let's do an example. That was the position yesterday. No money changed hands. You got some initial margin
that you had to put down, but that's really your money
in a brokerage account. You're not giving it to anybody. It's safety money,
it's collateral. Now today what happens? Let's suppose that
today the futures price closes at $0.7435. All right, it's just gone
down by 2/10 of a cent. The value of cattle
has gone down. Your holding long
this cattle contract, maybe you're a restaurateur,
you have a chain of steakhouses, and so you need to buy
cattle on a regular basis. So the price of
cattle just went down. Did you make money
or lose money? Yeah, you lost, if you're long. On the other hand, if
you're a cattle farmer and you sold the contract,
you did a good thing, because you locked in
the price of $0.7455, and now the price went down. So let's calculate what the
value of the notional size of the position is. It's $297,400. That yields a loss of $800. So you know what happens today? Today, your broker will
deduct $800 from your account, from your margin account,
and take that $800 and put it into the
cattle farmer's account. So now he has the $800. Now, if the day
after, if tomorrow, it turns out that the
price of cattle goes up by 2/10 of a cent,
it goes back to $0.7455. You know what happens? You get $800 back. Now the cattle farmer loses that
$800 and gives it back to you. You see this way you
always settle up every day. So if for some reason the cattle
farmer ends up going bankrupt, and isn't able to deliver
any cattle to you, then you're at out at most
one day's worth of movements. And that's one of the reasons
why futures markets and futures brokers are so careful
about closing down accounts that don't meet
their margin requirements. It's because they don't want
to have credit risk lingering, growing, and unknown. The first moment that you
do not make a margin call-- you do not deposit the requisite
margin-- the first time that happens, they
have the right, which they exercise always,
to close down your position. You're out of the game,
and that's the end. So it reduces dramatically,
the amount of credit risk that either
counterparty has. I don't have to trust you
that three months from now you're going to actually have
40,000 pounds of cow for me. All I need to do is to make
sure that this contract settles every day. And the uncertainty then
gets resolved day by day, but your credit risk is very
well managed, and mine is too. So this is a very
important innovation. It's very different from a
forward contract, in the sense that forward contracts contain
enormous amounts of credit risk right. Because once we
enter into a forward, that's just like a
futures contract, but the difference is that we
don't exchange any money ever until the settlement date. And by that point you could
be so far out of the money, you could be so
far in debt to me, as well as to other
creditors, that you just can't afford to pay. And so I'm stuck with
this piece of paper that says you're going to give
me 40,000 pounds of cattle, and you can't even afford
to buy me a steak dinner. That's a problem. So this futures exchange
is a beautiful thing. It reduces credit risk. It also encourages liquidity,
it encourages trading. Why? Because at any point in time,
on any given day between now and December, if
you decide that you want to get out of the
restaurant business and you don't want
this contract any more, you can get out of it. Poof. You just get out of it by
doing an opposite transaction. So if you bought a
contract for December, you know what you do when
you want to get out of it? You sell a contract
for December. You literally sell. So it's actually
duplicated transaction, but it's of the opposite sign. And so they cancel out, because
you're going to get delivery, and you will provide delivery,
and those will cancel out. Yeah, Justin. AUDIENCE: The price of oil
has been going down lately. So let's say I had a
long position in oil, and then I found out that
I was going to really lose half of that money,
and I decided just to forego my margin. What else would I have to pay? ANDREW LO: Well,
first of all, you are liable for all of the
losses, not just the margin. So the margin
account is not meant to be a non-recourse loan. They will go after your assets. Now you could
declare bankruptcy, personal bankruptcy, and get
protection under Chapter 7. But that will hurt
your credit ratings and all other nasty things will
happen to you if that occurs. AUDIENCE: So when you're
saying that they close out your account, when
your margins down. So they close it out,
but if your losses are higher than your
margin was anyway, so you're still liable for those
in addition to [INAUDIBLE].. ANDREW LO: So you
make a good point. Is it generally possible
that your losses are greater than the amount of margin? So in that case, who gets
left holding the bag? You know who gets
left holding the bag? That blue box in the middle, the
Futures Clearing corporation. But the reason that they
establish a particular level of margin is to be able
to ensure that that's a very unlikely event. And it goes back to what
are the likely daily swings in the futures price. If you put enough
margin in your account, so that I can be sure
that 99% of the time you can cover the daily swing,
then I don't have to worry. Now, of course, if we had a day
like last Friday, or on Monday, you know that's
pretty outrageous. That's one of the reasons why
a number of futures exchanges have increased
their margin levels. It's because the daily swings
have gotten much bigger. But as long as they can
cover the one day movement, they don't have to go after
your home or your other assets. Yeah, Dennis. AUDIENCE: You said if I
bought a contract now, then I just have to sit
on the same contract. What happens if I bought at
a $1.00, it's at $0.50 now, I can't sell at a $1.00. ANDREW LO: Oh, no. You certainly cannot
sell at the $1.00, you have to sell at $0.50. Says AUDIENCE: So I'm not
really out of the position. ANDREW LO: You are
out of the position, because you don't
have a claim, or you don't have a commitment to enter
into that trade in December. That's what I mean when I say
you're out of the position. You also happen to be out
of money in your example. [LAUGHTER] In other
words, you lost $0.50. That's not coming back. But what it means
to sell is that you bought a contract that
says in December you're going to buy 40,000 of cattle-- you're committed to doing that. Now if on the other
hand, the next day you decide you want to get
out of that commitment-- the way to get out
of it is not to try to contact the
counterparty and say, would you mind
canceling my trade. The way to do it is to simply
sell a commitment at 40,000 pounds of cattle for December. So your commitment to
buy and your commitment to sell, basically
cancel each other out. So on settlement date, the
Futures Clearing corporation will net out all of
these buys and sells, and the net amount
will be transacted between the providers
of the cattle and the buyers of the cattle. AUDIENCE: So this
means that there's no physical delivery then. ANDREW LO: That's right. So that's an example where
if you bought and sold, then you would be netted
out and you would not have a physical delivery. Yeah, [INAUDIBLE]. AUDIENCE: So if
the margin is just a fund for exchanging
commodities, what does the Futures Clearing
corp-- what do they make? Is it a percentage of-- how do they make money? ANDREW LO: Well, their job
is really not to make money, but to create an
exchange for its members. So many exchanges
are not for profit. Some of them are for
profit, but the objective of the Futures
Clearing corporation is really not to
make a lot of money. What they're trying to do
is just create a market and let people who want
to trade with each other, trade freely and efficiently. They will charge perhaps
a small transaction fee, that you have
to pay in order to support the operations. But they're not trying to make
tons of profits off of that necessarily. Now they may be trying
to make profits off of other activities, but the
objective of the Clearing Corporation itself is not
to make tons of profit, It's really just to provide
a stable environment where people can transact. And in some cases, the
members of the exchange own the Clearing Corporation. So it's their own
dollars that support the actual physical operations
of the organization. AUDIENCE: And just
going back to that point you had about no
physical delivery. Two or three weeks
ago, the price of oil spiked,
[INAUDIBLE] I think the way I read in the
papers, was people were trying to sell it, not
buy it, because otherwise they would get physical delivery. So do you recall that? ANDREW LO: I recall the spike. I certainly don't recall the
logic about physical delivery. I mean it could be that
there are a number of people who are long the oil
contracts that basically want it to be cash settled. And the way that they
have it cash settled at a particular point in
time, before settlement date, is they close out
their positions. And so by closing
out their positions, they basically
reverse the trade, and that would actually
push down the oil price. So maybe the reverse argument,
a lot of short sellers were trying to argue that
oil was going to go down, and they wanted to cover their
position, so they bought. In any case, you don't have
to take physical delivery if you specify to your broker
that you want all of this to be cash settled. Yeah? AUDIENCE: Two part question. Would you say that the
credit risk involved in a forward contract is
somewhat similar to the credit risk in credit default swaps? And if so, is there something
analogous to a credit default swap that's similar
to a futures contract? ANDREW LO: So the answer
to your first question is yes, because a credit default
swap contract is basically a kind of a forward contract. It does involve
intermediate payments, but if it turns out that the
credit changes dramatically, those intermediate
payments either may be too much or not enough
to cover the underlying value of the contract. And after you strike
a credit default swap, it will take on value. As for an exchange,
what a wonderful idea. That is exactly what's
being proposed now. It hasn't been
done yet, but there have been a number of
proposals to set up exactly a structure like this
for credit default swaps. In order to do that, you have
to standardize those contracts, and you have to be able to do
the paperwork in a relatively efficient manner. And so that's actually
being discussed, debated, and I think that there's
a proposal by the Chicago Mercantile Exchange
to start doing that. If you do start doing that, you
will see that market growing even bigger than it is
today, and at the same time, the risks are actually
going to decrease. Because with daily settlement
of credit default swaps, just like with futures
contracts, all you need is one day margin in order to
eliminate 99% of the problems. AUDIENCE: [INAUDIBLE] ANDREW LO: So let me--
that's a good question. Let me now talk about
how to price futures and we'll take in
interest rates explicitly. So the question is what
determines either a forward or a futures price? You now know what a
futures price is, right? It's the price at which
you're willing to do a future transaction. What determines that price? We say supply and
demand and the market, but is there some logic that
we can give to this process? And the answer is
yes, we're going to use the exact same
argument that we use for pricing everything else. We're going to come up with
two identical cash flows. And two assets that have
identical cash flows have to have the same what? AUDIENCE: Price. ANDREW LO: Price, value, right. So for now, I'm going to
actually ignore the difference between futures and forwards. The only difference is the
back and forth amount of money that we give to each
other, and therefore, the accumulated interest
or foregone interest that we pay when we put
our money back and forth into each other's accounts. So let me abstract
from that and-- you know if you're interested,
you can actually see the derivation of
that in recitation. I want to focus on
the bigger question of how these things are priced
with respect to other prices. So let me start
with some notation. I've got a particular
contract, let's say a futures or a forward contract. And I've also got the
spot price of the asset at a point in time. So I'm going to let St
denote my spot price. I'm going to let F
of little t big T determine the forward price. And H of little t big
T, the futures price. And for now, I'm
going to just assume that H and F are pretty close. Now notice that
when I write down a futures price or
a forward price, I've got two sub-indexes. I've got little t
and big T. The reason I need two is that for every
forward or futures contract, there are two dates you
need to worry about. The date at which you are
pricing the contract, namely today, and the settlement date. So you need to have
those two indexes. So right away we know
that this contract is a little bit more
complicated than say a stock, where there is no
settlement date. So I want to go back
to a comment that was made by one of you when
we first started talking about futures and forwards. And the comment was why
go to the trouble of using these contracts,
when you could just buy the asset itself
and hold onto it. If you need oil in
December, in order to make sure you
have oil in December, why don't you just
buy it in October and hold it for two months. Then you have it in December. And in fact, that's
exactly what we're going to do to figure out
what the appropriate price is of the specific futures
or forward contract. So here we go. I'm going to do my
exact same analysis that I've done
many times before, when we tried to price
bonds, and stocks, and other basic securities. The left hand
column here is going to be the cash flows
associated with a typical forward contract. So a forward
contract is one way. You enter into the contract,
let's say at date zero. And you pay nothing
for the contract right, this is a zero NPV transaction. And you are long the
forward contract, with the forward price F of 0,T. The only cash flow that
occurs with a forward contract is on settlement date. And on settlement date,
you've agreed to pay F of 0,T for delivery of whatever it
is that you bought the forward contract on. So the only cash flow that
comes out of a forward contract is this F right here. Everybody see that? Nothing up my sleeve, it's
very simple calculation. Now, I want you to look at
the right hand column, which is going to be less simple. The right hand
column, I want you to imagine doing the following. I want you to imagine buying
the commodity at date zero. However, I don't want
you to use any money. I want you to buy it
with no money down. That's the start of a
scam, it sounds like it, but I promise you it's not. So the way you're going
to buy the commodity is you have to pay the
price, the spot price. And the spot prices
is S sub 0 You don't have S sub 0, so borrow it. Now, I'm going to
abstract from credit risk, which I know is on
everybody's minds today. But let's suppose that
you're all good credits, so I'm not worried about loaning
you the money at the risk free rate. So now you've borrowed
S sub 0 dollars, and then you spent it right
away buying the asset. So as of date zero, in the right
hand column, you own the asset. Now you have to wait T
periods, and while you wait you may have some costs. For example, if the asset
that you bought is gasoline, well you've got to store
it in just the right way. You probably don't
want to put it next to your furnace
in the basement. You probably want to put it
in a cool place, isolated, and so on and so forth. On the other hand, if what
you bought is pork bellies, you probably want to put that
in a freezer compartment, as opposed to in your garage. So you might have to
pay costs for storing. And at the end of
that time T, you have to pay interest
on your loan. So you borrowed S
sub 0 dollars, you don't get that for free, you
got to pay interest on it. This is a question
about interest, so you've got to pay
interest on that money. And so you have to pay
back at this point T-- you have to pay back
the money you borrowed-- S sub 0, 1 plus R
to the capital T, plus whatever your
storage costs are. But I'm going to allow
that having the asset around might be
kind of convenient. There might be a benefit to
having the asset around-- a convenience yield. Maybe if you need to use it
sooner, you have it there. And having it there saves you a
little bit of trouble in order to be able to get
whatever it is you need to get done with
that underlying asset. So I'm going to deduct from
my cumulative storage costs any convenience yield-- that's future speak for
any kind of benefits that you get from holding
onto the physical asset. So your net storage
costs are given here-- that's what you pay at
the end of T periods. I argue that these two
cash flows give you the exact same
value of the asset. In other words,
in both cases you happen to have the
asset at the time T. So these two contracts
have to have the same value because they offer the
same set of cash flows, in terms of the
underlying commodity. You get the commodity
in both cases. So another way of thinking
about it is if your objective is to have 40,000 pounds of cattle
in December, both of these will get you to the
exact same point. Both of these costs you
nothing on date zero. And therefore, if
they cost you nothing, and they give you the
same outcome at the end, they've got to sell
for the same price. So this has to equal this. That's it. That's the simple argument. And the counter argument or
proof that this has to be true is-- let's assume it's not. Let's assume that this is
a lot bigger than this. Well, if this is bigger than
this, then what should you do? What? AUDIENCE: [INAUDIBLE] ANDREW LO: Right. Which one? Which one? Sell the forward
contract, and then buy this thing,
whatever it is, do this. Now what if it's the reverse? What if this is
bigger than this? Then buy the forward, and
then do the opposite of this. Flip it around. Short sell the asset if you
can, and then take the money and lend it out at interest
rate r, and dot dot dot, you follow the logic. So that gives us a relationship
between the forward price and other stuff. And what is the other stuff? The forward price
has to be related to the spot price, the interest
rate, the time to settlement, and any other weird things
about the commodity that may affect the value of it. Like the storage costs or
the convenience yield-- you've got to factor that in. So this is the relationship
that tells you how to price a forward contract. Now a futures contract
is almost like a forward. The only difference is
the interest differential on a daily basis, where you
actually are moving money back and forth into our accounts. But the cumulative
sum is going to end up being approximately the same. So for the purposes
of this class, I'm going to assert that this
is approximately the same. In fact, you can
show that there's another relationship that
looks at the interest rate per period. And it's a little bit
more complicated, but not much more complicated. You can see that in your
textbook, if you're interested. But for now, I want to just
focus on this relationship. This relationship tells us how
to price futures and forwards. And now if I divide by 1 plus r,
f to the T, then what I've got is that the forward price
divided by the interest rate, that calculates the current
value of that forward price, has got to equal the spot
price plus the present value of the net storage costs. This is the relationship
that we've been looking for, and you guys have been
struggling for the last couple of lectures. You've been asking well,
gee, doesn't the interest rate belong in there, and
what about having the asset, wouldn't it be nice to have
it, and so on and so forth. All of those considerations
are summed up in this one expression. A very nice expression. Very intuitive. What you pay at date T, when you
take the present value of it, that has to be equal to what
the thing is worth today plus any benefits for
having the thing, as opposed to not having the thing
between now and settlement. That's it. Now this is for
the very beginning when you strike the contract. What about at an arbitrary
point in time between 0 and T? Well, all of these arguments
work exactly the same way when you're looking at two dates
t and T, as opposed to 0 and T. So the relationship
that I showed you, it's a little bit
more complicated now, because you've got to take into
account the fact that the time to settlement is
not capital T, it's capital T minus
where you are today. But that's the only change. Other than that,
everything is the same. And you have to make
sure that you accumulate the future value of all
the net storage costs, so that you actually move
all of the costs to the end, and then you bring
it back to time T. Now, let's take
this out for a spin. Let's see how this works. Let's take a look at gold. Gold is easy to store. There's no storage costs really. I mean gold is relatively
compact, a little heavy, so you're going to have to lift
it and put it in your vault, as some of you, I'm
sure, are doing nowadays. [LAUGHTER] But the bottom
line is that the storage costs are negligible. There are no dividends. Gold does not pay out dividends. There are no real
benefits either, there is no convenience yield. It's not like you need
a little piece of gold every once in a while
for your pleasure, and so you want to scrape
that off and enjoy it. [LAUGHTER] It just
sort of sits there. So if that's the
case, you factor that into that relationship
that I showed you, and that last term, the PV of
net storage costs is nothing. And so the relationship
is really simple. The forward slash
futures price today is just equal to
what the current spot price is multiplied by 1 plus
the risk free rate of interest between today and
a settlement date. If this relationship
is violated-- when you look at gold
futures, and gold spot, and you see that this
relationship was violated, that's a sign that
there's an arbitrage. You can make money off of that. So that really is the way
to make a million dollars with no money down-- is to try to find violations
of this arbitrage relationship. It's going to be hard, because
there are a lot of people that are looking at it all the time. And so when there is an
inequality of some sort, it's probably not
going to be very big, and it probably
won't last very long. But to the person
who found it first, they might actually be able
to make a little bit off of that discrepancy, by
either buying or selling gold, and transacting these
markets quickly. Let me do another example. So this is gold. What about gasoline? Gasoline it turns out, is
very different from gold. First of all, it's a pain
in the neck to store safely. So if you don't want to be
blown up in the middle-- and this is what I really mean
by blowing up right, gasoline-- you want to prevent
that from happening, you've got to pay
a storage cost. On the other hand, there
is a convenience yield. If you've got the
gasoline, you can actually use it along the way. You have to replenish
it, in order to get the same level
of stock at the end, but it's convenient
that you have it, instead of having to go get it. Because getting it
involves trouble and costs. So that's the convenience yield. So if you factor that in, then
what you get is the futures or forward price is equal to 1
plus r,f plus a plus a storage cost per period, minus a
convenience yield per period, and then raised to
the T minus t power, multiplied by the
current spot price. If this is violated,
then you're going to want to do one of two things. Either you're going to want
to buy your own gasoline and store it, or you're going
to want to short it and do the opposite. After Hurricane
Katrina hit, we had violations from this for a
period of time, which suggested that it was actually
worthwhile for you to go out and build your
own storage facilities, because the storage
facilities were destroyed. Now it's only if you
had the technology to build those storage
facilities that you could actually profit from it. But there are periods of time
where market dislocation can occur, and the discrepancy
between futures prices and spot prices-- that gives you
valuable information about what's happening in
markets, and in some cases, in non-financial contexts,
like commodities. Whether there's a shortage
or whether there's a glut. Weather impacts
these commodities. And so by looking
at this relationship you gather very
valuable information. Here's another example. Another example is financials. I'm going to take this
example as the last one that I want to
focus on, because I want to now talk about how to
use this for your own purposes. A financial future
is a futures contract on an index like the S&P 500. So there, all of those
contracts are cash settled, there is no physical delivery. Although, you can easily
imagine a situation where you could have
physical delivery. Somebody literally
delivers 500 shares of stocks, 500 stocks with
a certain number of shares for each, in order
to get the S&P 500. But that's a pain,
and that defeats the purpose of the
futures market, which is to try to
make things simple and to make it more efficient. So a stock index future
is really a pure bet on an underlying index. And it gives you, the
investor or the hedger, a way to get exposure or get out
of exposure of that underlying in a very direct way. Now in this case, there's
no real convenience yield, but there is a
dividend that gets paid by the particular
set of securities. So if you're holding
the S&P 500 portfolio, then you're going to be
getting paid dividends for individual stocks
in that portfolio. And so you'd want to factor
in in your futures arbitrage relationship the fact that
you're getting a benefit, like a convenience yield,
that you have to subtract off of this relationship. So you don't have
a cost of storage, because this is a
financial futures, but you do have a convenience
yield, in terms of a payment if you held the physical
shares of the S&P 500. So that's the difference
between futures. Futures, you don't
get that dividend, so you got to take that out in
order to do the calculation. That tells you what the futures
price is relative to the spot. So now if I give you an
exam question that says, today's spot price
is such and such, and the risk free interest
rate over the next three month period is such and
such, you should be able to tell me what the no
arbitrage futures price should be today. Or vice versa, if I told you
what the futures price is, and I told you what
the interest rate is, you should be able to
infer from that what the spot price is going to be. On October 19, 1987, the morning
before the New York Stock Exchange opened, there
was a very big discrepancy between the spot price
and the futures price. That discrepancy caused
arbitrageurs to rub their hands and say, oh my god, this
is Christmas coming early, I'm going to take
advantage of this. And so what they
ended up doing-- it turned out because
the relationship was violated in one specific way-- they ended up buying the
futures and shorting the stocks. That was the beginning of
the October, 1987 crash, that within a day dropped
the market by about 20%. Nowadays, that's no big
deal, we're used to that. [LAUGHTER] But back then,
it was really something. So now that you have examples
of how these prices are determined, let me take this out
for a different kind of a spin. I want to show you how you
use one of these things. And the way I'm going to do
that is with the S&P 500. What I skipped were
more numerical examples that I would encourage you
to go through on your own. But this is an example
that's important, so I want to take you
through it carefully, make sure everybody understands. Suppose that you've
got $1 million to invest in the stock
market, and you've decided that you want to
invest it in the S&P 500. You don't want to invest it in
any other individual stocks. You want a broadly
diversified investment, and the S&P looks like
a pretty good thing. So there are several
ways of doing this, I'm going to focus just on two. One of them is you
could put your money in 500 different stocks. And you have to spend a little
bit of time figuring out what the proportions
are, because if you want to replicate the S&P, the
S&P is a value weighted index, it's not equal weighted. It's weighted by
market capitalization. So you've got to actually
go through and figure out how big each company
the S&P is, and then calculate those weights. And then you've got to give
this order to your broker, and $1 million dollars
isn't what it used to be, so I suspect that
that would generate some pretty tiny trades. You got 500
securities, and you've got a bunch of different
odd lot trades. Good luck finding
a broker that's willing to do it at
a reasonable price. It's a pretty long list. Or you can buy a
futures contract. In particular, you
can buy a contract on the S&P 500 futures. So I want to go through and
show you what that involves. Now let's take your $1
million and let's deposit it at the Futures
Brokerage account. So the money is sitting there,
earning whatever interest they pay you on that account. Which is not much, it's probably
akin to a money market return. So what you do is you want
to buy futures contracts and you want to
have the equivalent exposure of $1 million
invested in the S&P 500. Now the way that the S&P
500 futures contract works, is that the value of the
contract, the notional amount of the contract, is
250 times the index. Whatever the index is worth,
they just make up a number, like I don't know
250, and multiply that by the value of the index. And they say that is what your
exposure is for one contract. So what is that? Let's suppose the the S&P
index is now at a 1,000. So the value of the futures
contract is 250 times that and that's
going to be $250,000. In order for you to have
the equivalent of $1 million in the S&P, you need
four of those contracts. Four times a notional of
250 is equal to $1 million. Now what does this say? This says that you
are essentially agreeing that you're going
to buy the S&P 500 whenever it settles. But you're not really
buying the S&P 500, you're buying a pure bet that
is equivalent to 250 times the S&P 500. So let's take a look
at what that means. Suppose that the S&P index
fluctuates, bounces around, then it turns out that you'll
see that your cash portfolio-- the portfolio fluctuations
if you had put $1 million into the S&P directly-- it fluctuates in
exactly the same way that your futures
portfolio fluctuate. If the S&P goes down to
900, the notional value of your portfolio with
four contracts is $900,000. So you've actually
lost $100,000, and that's going to be
deducted from your account. If on the other hand,
the S&P goes up by 100, then your cash portfolio
will be worth $1,100,000, and your futures portfolio
will be worth the same. You will now get $100,000
deposited into your account. By holding this
futures contract, it's as if you were actually
invested in the S&P. What you're getting is
the daily fluctuations. But you actually don't
own the security, you simply agreed to buy this
so-called index on the maturity date. And by doing so, and
because that contract value is so closely tied
to the S&P 500 index, it moves in lockstep
with the cash portfolio. Any questions about this? Yeah. AUDIENCE: Does someone
own those shares behind you or it's just-- ANDREW LO: No. AUDIENCE: --an agreement that
we're going to wait on this-- ANDREW LO: Exactly. Right. So you and I, we're just going
to agree, we're going to bet. We're going to bet
and we're going to agree on a particular
price for S&P 500 three months from now. And if it goes up, and I bought
the contract, then I win. If it goes down, then you sold
the contract, then you win. But it's a pure bet
between you and me. AUDIENCE: In the
middle is the Futures-- ANDREW LO: The Futures
Clearing corporation sits in the middle to make sure
that you and I don't run away. AUDIENCE: Why do they do this? ANDREW LO: Why do they do this? AUDIENCE: I mean why do they
[INAUDIBLE] days of sunlight. ANDREW LO: Well, first of
all, in some cases they do. So for example, you
could buy a contract on the number of degree days
of a certain amount in Florida. Now why would you
want to do that? It turns out that one of
the largest crops of oranges are grown in Florida. And it turns out that the
output of oranges groves is very closely
tied to temperature. So if it goes up to 39
degrees or below 32 degrees, you can actually have very
different kind of crops. And so you can bet on
that, and at some point you can actually trade on it. I don't know if you
can trade on it now, but there are markets for
some of the wildest things. And the reason that
you have these markets is because when two mutually
consenting adults have opposite views and they
want to express them, then you want to be able
to let them do that, and allow them to basically
either hedge their risks, or take on risks that
they're able to do. So this is an example of that. You're an investor. You want to buy
stocks, but you don't want to buy 500 little
stocks one by one. You want to get the
exposure right away. Now of course, there's
another way to do this, you can put it in a mutual fund. But the problem
with the mutual fund is that it only gets priced once
a day, whereas this thing gets priced every second of the day
when the futures exchange is open. Of course, nowadays, you can
buy an ETF, an Exchange Traded Fund. So that's another way
of getting exposure. But the S&P futures was
around long before ETFs and allowed people to do all
sorts of hedging transactions. Now I'm going to give you a
second example that I think will make it a little
bit more clear, and actually will answer
a question that was asked, I think two lectures ago. When I first started
this lecture, I said that maybe a
company would only want to hedge 25% of its risk. And somebody asked well,
what does that mean 25%? And I said, I'll
answer that question. Well, so I'm going to
answer that question now. So suppose now as a
different example, you have a diversified
portfolio of large cap stocks worth $5 million. So you already own the
stocks, and it's currently worth $5 million, but you
don't have any confidence that the market is going
to stay where it is. You think it's going to go down. And so you want to
hedge some of that risk. You don't want to
hedge all of it, because you do have faith that
over time markets will do well, but you just want
to be able to dampen a little bit of the downward
spiral if it does occur. So you might consider selling
25% of your portfolio. Getting rid of 25% of it
and putting that in cash. That's one way to do it. But the problem as you
know is that it's not that easy to sell
25% of 500 stocks, because you have to again, slice
the portfolio, stock by stock. You're going to have a trade
list of 500 stocks, which comprise 25% of your portfolio. So it's a pain. But here's an
easier way to do it. You can short sell
five S&P contracts. And I'm arguing that that
will do the exact same as if you just liquidated
25% of your portfolio. Now let's see if that's right. So let's go through
the exact same table. The cash portfolio--
let's see what happens to the cash portfolio if
the S&P goes up or down by 100 points. If it goes up by 100 points,
then you've made money. You've got $5.5 million. If it goes down by a 100
points, you've lost money. You've lost to $4.5 million. Now let's see what happens
if you don't do anything with the cash portfolio,
but you simply short sell five S&P futures contracts. If you do that then obviously
if the S&P doesn't change, then nothing happens
to your portfolio. But if the S&P goes
up, then you're going to make some money. Sorry. So yeah, if the
S&P goes up, you're going to lose money in the sense
that what's going to happen is that your short positions
are going to cost you $125,000. How did I get $125,000? Anybody work through
the math for me? The S&P 500 goes
up by a 100 points. The futures price goes up
by 250 times 100 points. My position, I've got
five of these contracts, I've just lost
$25,000 per contract. I've got five of these
contracts, I lost $125,000. Now what about the downside? The downside if the
S&P goes down by 100, then the price goes
down by $25,000. I'm short, so I make
$25,000 per contract. I've got five contracts,
I've made $125,000. So look what happens. In this case, when
the S&P goes up, I don't make as much, because
my hedge works against me. On the other hand,
when the S&P goes down, I don't lose as much, because
the hedge is working for me. Because I've only taken
out 25% of my portfolio with this hedge, it's
dampening, but not eliminating that kind of fluctuation. Yeah? AUDIENCE: I think that this
an obvious question, but why do you do that, versus
just putting it in cash. Because you can make the
argument that if you had 25%, and had it earning interest,
and so you'd still be up too. ANDREW LO: Well,
that's the argument that I gave earlier,
which is that you'd have to sell 25% of your portfolio. This is a way of doing it. And not only that, if
you did it this way, it would be a lot cheaper
to implement in the sense that you don't have to
do 500 transactions, you do one transaction. So the transactions
cost is a lot cheaper, and it's also easier
to keep track of. You don't have to figure
out what the price of 500 securities are. You've got the price of just
one security to worry about. Yeah. AUDIENCE: And I
think also you're not losing out on
what you could've had in cash in
terms of interest, because that interest is
factored in to the futures. ANDREW LO: That's right. Remember we used that
interest equation so all the foregone
interest is in there. OK, so the meaning of
I want to hedge 25% means I'm going to use
the futures contract, so that the notional exposure
is 25% of the current value of my portfolio. So if you're Merck
pharmaceutical company that has a certain percentage
of their revenues in foreign denominated
currencies, you can hedge half of the
risk of those exchange rate fluctuations by taking half
of the revenue stream-- let's say it's $10 billion-- and buying or selling, depending
on which way you're going, the amount of futures
or forwards to get rid of that exposure. Yeah. AUDIENCE: In this example, we
put our million in the margin account, but we only
should put as much as [INTERPOSING VOICES]. ANDREW LO: That's right. You don't have to put $1
million in the margin account, because typically
the margin is going to be something like
in this case 7% or 8% of the notional exposure. So you could take the rest of
that money and go to Las Vegas if you like. Although, some would say this
is better than Las Vegas. Yeah. AUDIENCE: This is
the main reason why we buy futures and not ETFs. You can leverage your
bet as much as you want. ANDREW LO: That's right with
an ETF, if you want $1 million of exposure, you got to put
$1 million into the ETF. With the futures contract,
if you want to put $1 million of exposure on, you need 7%. And the reason is
obvious, it's because of that daily mark to market. So ETFs have not killed
the futures market, but it does provide another
vehicle for retail investors who may not want the leverage,
who may not need to leverage, to not have to worry
about the leverage. This leverage-- leverage is a
scary thing, as I said before. This is the chain saw that
you don't want to be giving your eight-year-old as a toy. Because when prices
move quickly, you're going to have very
big swings in the underlying value of your margin account. So if you've got only
7% margin in an account, think about it, that means that
if the prices go down by 7%, you are wiped out. Your entire margin
account is gone. When futures brokers
take your money, they assume that you
know what you're doing. And so they assume that the
margin that you're putting down is margin that you
can afford to lose, and that you understand
that what you're getting is much bigger exposure
that presumably is either for speculative
purposes, in which case you won't over leverage, or for
hedging purposes, in which case you've got some other assets
that are counterbalancing these swings. Like in this case. You know obviously,
when you look at the fluctuations
in your positions, they are extraordinarily
big relative to the margin. Let's do a quick back of
the envelope calculation. Let me tell you what I mean. Suppose that you
put 5% margin down. You buy a contract,
put 5% margin down, and let's suppose that the
price of the futures contract drops by 2.5%. What is the rate of return
on the amount of money you've put down as margin, if
that's your initial investment? You can think about
it as an investment, because that's the only
way a futures broker will let you buy a contract. If you put down $100,000 and
the futures price goes down by $50,000, what's the rate
of return on your investment? Yeah, it's minus 50%,
that's a big move. That's a huge move in a day. So when you deal
with margin, you have to be
extraordinarily careful. You have to have very,
very tight risk controls. You have to understand
what the swings can be, and you have to manage that
risk very, very carefully, on an intradaily
basis in some cases, because these futures prices can
swing a lot even within a day. Any other questions? Well, that's it for
futures and forwards. You now know how to price them. You now know how to use
them for hedging purposes. And there are all
sorts of other kinds of futures and forwards--
interest rate, bond, currency, single stock futures now exist. In fact, there are even
futures contracts on the VIX, there's futures contracts
on electricity usage, there's futures contracts on
the presidential election. If you go to the Iowa Electronic
Markets, the University of Iowa, they created
a futures exchange that has two contracts. One that pays $1 if
McCain gets elected, and the other that pays
$1 if Obama gets elected. And by looking at
the prices, you can actually see what the folks
that are trading these futures contracts are thinking, in
terms of who's got the edge. So the futures prices contain an
enormous amount of information. But keep in mind the information
is only as good as you are. By you, I mean the market. If the market is
comprised of knuckleheads, the prices you get will
be knucklehead prices. If the market
contains really smart sharp sophisticated
individuals, you'll get extremely
informative prices. So prices, while they
are the best thing that we have as a
guide for the future, they're clearly not perfect. And there are periods of
time when the market prices are less perfect than others. And as I told you before,
for the next three weeks, finance theory is going to be
on vacation in the US stock market, because all
the uncertainty that has been building up over
the last several years are now focused on
the next three weeks. Markets will be swinging
back and forth pretty wildly, and it's because people
are reacting emotionally, not necessarily with their
full logical capabilities. That's it for
futures and forwards, and now what I'm going
to turn to is options. These are the last
set of securities that I want to go
through with you that are not like the securities
that we've done before. And let me just pull up the
lecture notes for options. I want to start with a little
bit of an introduction for how to motivate options. I think most of you
know what options are. Their name is quite
apropos, because they do give you options. Futures and forwards require
you to engage in a transaction, but options don't. They give you the right,
but not the obligation. So you have the
option of not entering into that final transaction
at settlement date. I'm going to start
with some motivation, then go through some
payoff diagrams, go through options
strategies, and then I'm going to talk very briefly
about valuation of options. I have to talk to you
guys about Black-Scholes. You can't leave MIT without
hearing about Black-Scholes. [LAUGHTER] So I've got to
do a little bit of that. But really the derivation is
quite a bit more sophisticated, and that's why you might
want to take 15.437 Options and Futures, where
the entire course is devoted to these instruments. They are that complex
and that important. So let me first describe
exactly what an option is. An option actually is a
specific example of something that you now know of more
generally as a derivative. A derivative security
gets its name because the value
of the security is derived from yet
another security. It's derivative, as opposed to
I guess fundamental or primary. And examples of derivatives
are warrants versus options. Options are securities
that you can think of as pure bets
between two parties. Warrants are options
that are issued by a company on its own shares. So the net supply
of options is zero, but the net supply of
warrants is not zero, it's issued by companies. And there are two
different kinds of options, calls and puts. A call option is
a piece of paper that says the holder
of this piece of paper is allowed to buy a
security on or possibly before a particular date,
usually called the exercise date or maturity date. And the difference between
being able to exercise early versus exercising at
the maturity only, is the difference between an
American and a European option. An American option is one where
you can exercise it early. And a European option is one
where you can only exercise it on a specific date, the maturity
date or the expiration date. And puts are the
opposite of calls. Instead of giving
you the right to buy, it gives you the right
to sell or to put the stock to somebody else. And the prices at
which you get to either buy in the case of calls,
or sell in the case of puts, is called the strike price
or the exercise price. All right. So I'm going to define a
little bit of notation. Stock prices is S sub t. Strike price is K. Notice that K
does not have a time subscript, because it's fixed at the
time the options are issued and it doesn't change throughout
the life of that option, it's part of the contract terms. And then the call price is C,t. Put price is P,t. And the value of these
contracts at maturity is actually pretty simple. If today a particular stock
is trading at $60 a share, and you purchase an option to
buy that stock at $70 a share in three months, does that
piece of paper have any value? The current price is
$60, this piece of paper gives you the right to buy
it at $70 in three months. Is that worthless? Why not? The price is at $60, you
can get it at $60 today. So why would you want it at
$70 three months from now? Exactly the price may go up. The reason the piece
of paper is not worth zero today is that
there is a chance, no matter how small
you might think it is, there is still a chance that
something wonderful might happen in the next
three months, and then the price will go up to $80. And if it goes up to
$80, you'll be very happy that you have the
right to buy it at $70. How happy will you be? You'll be $10 per share happy. [LAUGHTER] That's what
that expression says. On the expiration date,
you get to buy the shares-- if you're holding a call
option, you get to buy it for K dollars, but in fact the market
has determined that the price at that time is
really S,T dollars. So if you're holding this piece
of paper, this is your profit-- S,T minus K per share. Now if it turns out that
you get to buy it for $60, and it ends up trading
at $40, well then you're not going to
exercise that right. You're going to let
the option expire, and when it expires
it'll be worthless if this number is negative. It can be negative of
course, but you're not obligated to buy it. On the other hand, if this
were a futures contract, you certainly are
obligated to buy it and then you'd get
a negative return. But an option is a
wonderful thing, in that the payoff is never negative. It's either zero or it's S,T
minus K. That's for a call. Now a put option, it's
exactly the reverse. If you get to sell the
stock, then your profit is what you get to
sell it at versus what it's really trading at. And so you actually
hope that it's really trading at a very low price. Because if you get to
sell it at a high price, but it's trading at a low price,
you profit the difference. So the payoff for a
put option is exactly the reverse, maximum
of zero and K minus S. Now, it's very important that
you understand this asymmetry, because that asymmetry
is going to lead to all sorts of interesting
things about these instruments. And before we go and talk
about that kind of asymmetry, I want to give you another
way of looking at options. Which is to look at options as
a kind of insurance contract, because actually all
insurance contracts are a form of an option. So let me give you an example. Suppose that you want to insure
the value of a particular stock that you're holding. You're holding General
Electric and it's trading at $20 a
share, and you'd like to make sure that it
never goes below $18 a share. You want to buy insurance that
if it goes below $18 a share, you will get paid $18 a share. Well, the way you do that
is you buy a put option. A put option on General
Electric where the strike price is $18 a share. Because if it goes
below $18 a share, you get to sell General
Electric for that $18. So you'll get the
$18, regardless of whether it goes to $10,
or $5, or who knows what. It turns out that the put option
is exactly like insurance, and let's take a
look and see why. These are the typical terms
of an insurance contract. What's the asset
that you're insuring? General Electric. What's the current asset value? $20 a share. What's the term of the policy? How long do you
have the policy for? It's the time to maturity. What's the maximum coverage? What are you covered for? $18 a share, that's right. That's what you bought
the coverage for, that's what you're going
to get if it goes below. What's the deductible? How much could you lose
before the insurance kicks in? $2 a share, exactly. That's the deductible. And finally, what
does it cost you to buy this insurance,
what's the insurance premium? Exactly, the price of the put. That's it. Beautiful thing. A put option is nothing more
than an insurance contract on the value of a stock. And it's going to
turn out that a call option will be intimately
tied to what a put option is. So every call option can be
converted into a portfolio that includes a put. So all options you can think
of as insurance contracts, but there are a few differences. The difference
between an option is that you can exercise it early. So for example, for
whatever reason, if you decide that you want
to buy General Electric at $18 a share, when it's
trading at $17.50, and you still have
one month to go. But you want to get paid that
$18 now, you can do that. You can't do that with
your car insurance, right? I guess you could, you
could ram it into a post, and I want to get
paid now, so let's-- [LAUGHTER] But that's not
really considered a proper thing to do. So early exercise
is one difference. Second difference
is marketability. If at some point you don't
want the insurance anymore, you can't get rid of it and
give it to somebody else. You can't transfer your auto
insurance to your friend, if you decide you
don't need it anymore. But you can transfer the
insurance policy here. You can sell the
option, you can sell it. And also there
are dividends that are being paid on the stock
that you have to worry about with options, whereas with
a typical insurance contract a car doesn't necessarily
pay dividends. And the reason that's important
is when it pays dividends, the value goes down, and so
you have to make adjustments for that with an option. You have to protect an
option for dividend payments. You don't need to do
that for insurance, because typically you don't
assume that the insurance value, the value of the asset
goes down that much over time. Yep? AUDIENCE: When they
buy the put option, they also eliminated
the chance to enjoy it, from the prices
are going to go up, with the futures we'd
have a higher value. ANDREW LO: Well, no
that's actually not true. With the put
option, it gives you the right to sell the stock. If you buy the stock
and you hold onto it, and you also buy a put,
that protects the downside. But the upside,
that's all yours. Because as the
stock goes up, what happens to the value of the put? AUDIENCE: It's going to zero. ANDREW LO: Exactly,
it stops at zero. So as the stock goes up, the put
doesn't have any value anymore. It becomes worthless,
worth less and less. And on the date of
expiration, if the stock is way above the
value of the strike, then it expires worthless. It doesn't go negative. If it went negative, if
you had a futures position, then you'd be right, you've
actually capped your gains. But this doesn't. See with this you get the
best of both, or so it seems. You get the upside, but
it protects the downside. And as you all probably
know, insurance is not cheap. So it sounds good, but
you've got to pay for this. And so you bet that the price
of a call option or put option is not zero when you strike it. Unlike a futures contract
that's worthless, an option is not
worthless on day zero. It's worth a lot. For example, right now
what's really expensive-- and if you want to check this,
you could take a look for fun. If you want to buy
insurance on the S&P 500-- now we've had a great
rally on Monday, the S&P was up 1,000 points. If you want to buy insurance
on the S&P 500 index, you can do that. There are options on the index. So you might say, OK
let's say that the S&P is at 1,000 today, I would
like to buy protection that over the next
month it doesn't go down by more than 100 points, 10%. So what do you do? You buy a put option on the S&P
with the strike price of what? 900, right. OK, for a month. That's what you want to buy. Go out and calculate that price. You're going to be shocked
at how expensive it is, to get that insurance
for four weeks. Four weeks, that's all. It's really expensive today. I think it's approximately 10
times more expensive today, than it was a year ago. The implied volatility is up by
at least an order of magnitude. So if you want that
insurance, it's available, but you have to pay for it. So the question in all of
these things is is it worth it? In order to decide
whether it's worth it, you've got to do two things. First look into the inner
most workings of your own soul and ask how frightened
you truly are. And the second thing you got
to do is look at the market. And is the market
functioning reasonably well, or is the market reflecting all
of these kinds of crazy fears. In order for us to be able to
talk about it intelligently, we need a way to price it. We need the kind of
logic that I showed you with futures contracts. And we're going
to get that logic. I'm going to show you how
to price these things using a very, very simple model
that is incredibly powerful, but we're not there yet. Before we do that,
I want to make sure you understand what
these contracts can do for you in terms
of changing your payoff profiles of your portfolio. Yeah? AUDIENCE: So wouldn't
European option be similar to a
futures, since you have that you can only
exercise on maturity date? ANDREW LO: Well,
no, that's not what makes it similar to a futures. Because while you cannot
exercise it early, you never have to
exercise it at all. So a European option
gives you only one date where you are able to
exercise, but even on that date you never have to exercise it. With the futures
contract, you have to exercise it on that day. You've made a commitment. AUDIENCE: But it would have
a net present value of zero. ANDREW LO: No, no, it won't,
because still on that date you have a positive
amount of protection. Like the example I gave you. Let's suppose that I
bought a European S&P option for the day after
election day, Wednesday, November 3rd is it. That will have
positive value today. In other words,
I'm going to have to pay money in
order for you guys to sell it to me, because
you're going to be providing me with some protection that if the
wrong thing happens on Tuesday, the world is not going
to blow up on Wednesday. I'm not telling you
what the wrong thing is, I'm neutral in all of this. But that's an example where
that insurance really has value. So you're not going to
give it to me for free, and I'm willing to pay for it. All right, since
we're out of time, I'm going to just leave
you with this diagram that shows you the difference between
a call option and a futures contract. Remember the futures contract
what that looked like-- that was a straight line. Right Exactly. This is not a straight
line, this is kinked-- very kinky security. And so we're going
to talk next time about how to price
kinky securities, and how to combine them,
and engage in even more kinky kinds of payoffs. [LAUGHTER]
