Prof: The subject of
today's lecture is hedging. So this is what hedge funds do.
It's what almost everyone on
Wall Street does nowadays, at least to some extent,
say half the people on Wall Street nowadays.
It was hardly done at all in
the past. So first of all,
just to mention what a hedge fund is, a hedge fund is a firm
that manages money. And why is it different from
any other firm that manages money?
Well, the definition of hedge
fund basically has three parts. One is that hedge funds hedge.
Now I'll say what that is in a
second. Secondly, hedge funds use
borrowed money in addition to their investors' money to buy
assets. So they do what's called
leveraging, which is going to be an important subject for the
last few lectures of the course. And thirdly,
they charge their investors very high fees.
That's basically the definition
of a hedge fund, because they're supposed to be
so good at what they do, they can charge high fees and
still get the investors. So what is hedging?
Hedging is the idea that you
want to cancel out some of your risks.
So for instance,
you might know that a company's not going to default,
whereas the rest of the market thinks it is going to default.
So you know the company's worth
more than the rest of the market thinks, so you might be tempted
to buy the company. But if the interest rate were
to change, say go way up and the company's
paying, has constant cash payments,
if the interest rate goes way up and the company has constant
cash payments, you could end up losing money
anyway, because the present value's
going to go down, just because the interest rate
has gone up. So what a hedger would do is,
the hedger would say, "Look, I'm relying on my
expertise as an evaluator to realize there's no default risk.
I want to bet that there's no
default risk in this company. It's not going to default.
I don't want to bet on which
way interest rates are going to go,
because I don't know which way interest rates are going to go,
so I want to hedge myself against that."
So to take-- sorry,
I'm just starting here. Hello.
Sorry, hard to talk when
everyone else is talking. So what does hedging a risk
means? It means no matter which way
the risk factor goes, you're going to still end up
with the same amount of money. So the first person to do this
and call himself a hedge fund was somebody named Jones in the
1940s and he was a stock picker who would try to find the best
possible stock to buy. So before him,
people would say, "Okay, Ford is a great
auto company. We're going to buy Ford."
The trouble is that Ford may in
fact turn out to be the best auto company,
but because the whole economy collapsed,
it may be that Ford collapsed with the rest of the economy,
even though it did better than all the other auto companies.
So what Jones said is,
"I'm not going to just buy Ford.
What I'm going to do is,
I'm going to buy Ford and I'm going to sell General Motors,
so that way I'm going to be betting not on whether Ford is
better than General Motors, and in addition,
whether the whole economy's going to go up.
I only want to bet on whether
Ford is better than General Motors."
So if the whole economy goes
down, Ford and General Motors will go down together.
So what you own in Ford,
you won't own something as valuable any more,
but what you owe by having sold General Motors short also won't
be as valuable. So what you've bought and what
you've sold will cancel each other out and you won't have
lost any money. On the other hand,
if Ford ends up doing better than General Motors,
you'll have gained more with Ford than you'll have lost with
General Motors. So whether the economy goes up
or down, if the gap between Ford and
General Motors stays the same, you won't lose or win money,
so you're hedged against general changes in the economy.
You're only going to make or
lose money depending on the spread between Ford and General
Motors. So you've canceled out the
economy wide risk and you're only holding one risk,
which you really think that you've understood.
So that's what hedging is.
So I'm going to now show you
that hedging is actually a slightly delicate thing.
It's quite complicated and
we're going to learn the technique of dynamic hedging.
Now why is dynamic hedging so
important? Dynamic hedging is so important
because there are so many things that can happen in the future,
and as I said, for example,
if you hold a mortgage, you may think the mortgage is
more valuable, because you know people aren't
going to default, and the rest of the economy
thinks they're going to default. On the other hand,
the value of the mortgage will change with the interest rate,
so you want to protect yourself against the interest rate
change. But it's not just the
interest rate change. Over ten years or 30 years,
the life of the mortgage, there are going to be a huge
number of interest rate changes. There'll be a whole path of
interest rate changes, and the mortgage will be paying
different cash flows along the whole path.
In over 30 years,
even if you think of only one change a day,
there's an exponentially growing set of changes,
so there's an exponential number of paths.
You'll never be able to
equalize your cash flows, you would think,
over every single path. There are just too many of them.
But dynamic hedging says you
don't have to equalize over every path from the very
beginning. That is, you don't have to hold
a portfolio at the very beginning that equalizes your
cash flows by the end of every path.
You only have to equalize your
cash flow one step at a time and then dynamically change your
hedge, so it's a much easier thing to
hedge than it sounds at first glance.
So I want to illustrate that,
because you can't know what that means yet until we see an
example. So let's start with a very easy
thing to understand, the World Series,
where we started before. So you have some expertise,
in fact, there are other people besides you who have the same
expertise, these other bookies. You all know the Yankees have a
60 percent chance of winning every game against the Phillies,
and let's just assume for simplicity that the World Series
is only going to take a couple of weeks.
The interest rate is 0 over
such a short period of time. You don't care whether you get
the money now or at the end of 2 weeks, just so long as you get
it. You can borrow or lend at 0
interest over those short number of days.
And secondly,
you know that the probability of winning is 60 percent.
Now suppose that some naive
person comes to you and says, "I'm a Phillies fan.
I want to bet at even odds that
the Phillies are going to win for 100 dollars."
So you're going to now get a
payoff that's -100 dollars at the end of game 7,
assuming I counted right. So this is the start.
This would be game 1 here,
so it's 1,2, 3,4, 5,6, 7.
So by the end of game 7,
if you played it to the end, you would have made 100 dollars
if the Yankees won a majority of the games,
and you'd have lost 100 dollars if the Yankees didn't win a
majority of the games. So how much is that worth to
you? Well obviously it's worth a lot
more than 20 dollars, because if they only played 1
game and the Yankees had a 60 percent chance of winning,
the expected payoff for the Yankees,
that bet for 1 game would be 20 dollars,
because you know in 1 game a .6 chance of 100 and a .4 chance of
- 100, that gives you an expectation
of 20. 60 - 40 is 20.
However, we're playing a seven
game series. The whole point of playing a
long series is that the Yankees are more likely to win.
The better team is more likely
to win, and in fact,
we know that by doing this backward induction calculation,
we know that the expected winnings for the Yankees is 42
dollars. However, the Yankees--you could
lose 100 dollars. If the Phillies just got lucky,
you could lose as well as win 100 dollars.
You don't want to face that
risk, so what could you do? Well, the simplest thing to do
is to go to another bookie and say,
"I want to bet on the World Series,"
with this other bookie. So if the bookie is willing to
make a bet with you for the whole series,
so the bookie has to be pretty sophisticated and do the same
calculation, the bookie is going to let you
bet however you want to at the right odds.
So what would you do?
So here you can win 100.
There are only two outcomes.
You can win 100 or you can lose
100. Now the odds of winning the
series, I forgot to say, we figured this out before.
Oh dear, what were the odds?
Hope I have another one of
these World Series things. Okay, so what are the odds that
the Yankees are going to win the World Series?
We calculated this before.
Do you remember what it was?
Well, we know some probability
P times 100 some probability (1 - P), -100 gives you 42.
So P times 100 (1 - P) times -
100 = 42 approximately. That's 42, right?
So we know that therefore,
2P times 100 = 142, so P = 142 over 100 times 2,
which equals about .71. Okay, so this other bookie
would have calculated the odds are 71 percent that the Yankees
are going to win the World Series.
We calculated this a few
classes ago. The Yankees have a 71 percent
chance of winning the World Series.
So if you went to another
bookie who could do this calculation,
the other bookie would be willing to give you odds of 71
percent Yankees versus 39 percent [correction:
29 percent] Phillies,
so you could just unload this whole bet and get 42 dollars for
sure. In fact, what would you bet
with the other guy? What would you bet with the
other bookie? What would you do if you could
bet on the whole World Series? Somebody suggested that.
He was sitting over there at
the end of the last class. What would you do if you could
bet on the whole World Series with the other bookie?
What bet would you make?
You know that the bet is giving
you 42 dollars on expectation. The trouble is,
some of the time it's giving you 100, some of the time it's
giving you -100. You don't want to fix this risk.
You want to end up with 42
dollars for sure. Now how can you do that?
Well, if you did -68
[correction: -58] here, that would = 42,
and if you did 142 here, that would also = 42.
So could you figure out a way
of betting with the other bookies 142 versus 58?
You win 142 if the Phillies
win, and you lose 58 if the Yankees win?
Student:
> Prof: That's not 42 by
the way. Okay, would another bookie be
willing to give you this bet? What's the answer?
The answer is yes.
How do you know the other
bookie would be willing to give you this bet?
What is P times 42 (1 - P)
times 42? It's 42.
What's P times 100 (1 - P)
times -100. It's also 42.
So therefore,
what's P times this (1 - P) times that?
Student: 0.
Prof: 0.
So at the odds,
P and 1 - P, this bet is perfectly fair.
So the other bookie would say
that's the odds, 71 percent to 39 [29]
percent. That's exactly what these odds
are. So if P times this (1 - P)
times that is 42, and P times this (1 - P) times
that, that's obviously 42, must be P times this (1 - P)
times that is 0. In other words,
at the odds, P and 1 - P,
this is a fair bet, so the other bookie would give
you those odds. So that's what's hedging is in
its simplest form. We haven't had to do anything
dynamic. Some guy is willing to bet on
the World Series with you, so you know that he's done
something wrong in his calculations.
He's got the wrong P,
so you can take advantage of the situation.
You know that P is higher than
he does. On the other hand,
there's still God in the background and luck,
which might make the Phillies win, and so you don't want to
subject yourself to that risk. So what do you do?
You take the bet,
the advantageous bet, but you put together with the
advantageous bet a hedging bet, where you're betting on the
wrong team, the team you think is going to
lose. But you're betting on the wrong
team, but this time,
at fair odds on the wrong team, and so you're transforming your
risky payoff, albeit with very favorable
odds--P we just said was .71-- you're taking your risky payoff
and turning it into a safe payoff of 42,
no matter what. That's the essence of hedging.
In order to do that,
you might have to bet the opposite way that you think is
going to turn out. So there's principle number one.
So let me pause and see if
there are any questions for that, about that.
The idea is that there's
something you know that you can take advantage of,
but life is more complicated than knowing one thing,
namely what P is. Life also involves all kinds of
other things that might happen. You want to insulate yourself
from those other things and concentrate entirely on what you
know about P and 1 - P and therefore assure yourself of a
bet of 42, no matter what.
And you're taking advantage of
the fact that other people will also be able to bet at P with
you. This one outsider doesn't
understand that all the bookies are willing to take odds of P,
he's offering odds of 1 half, which is crazy.
It's a big gap, 71,29 to P.
So you can take 42 of his
dollars for sure. Okay, so now let's go to--okay,
this is hedging. It's also an arbitrage.
This is--hedging created a pure
arbitrage. In this case,
it's done even better than in my example with Ford,
because you've taken now a bet and transformed it with an
expectation you're going to win, into a bet where you can't
possibly lose. Okay, now suppose that these
other brokers weren't as clever as you.
Suppose that the other
brokers--okay, so that's hedging.
Now we're going to do dynamic
hedging. Suppose the other brokers,
they're not as smart as you. They don't know how to build
these trees. They can't do backward
induction. They just know the odds are 60
percent that the Yankees are going to win any game against
the Phillies. Now what would you do?
Okay, you can't go to another
broker and say, "I want to bet on the
whole 7 game series." The guy's going to say,
"It's just too complicated for me.
I'm a simple man.
I make a simple living.
I just do 1 game bets.
I'll let you bet any game you
want. You tell me what you want to
bet, we'll bet one game at a time, 60/40 odds.
The whole series,
it's much more complicated for me."
So is there anything you can do
now? What can you do now?
So now you have to do something
more complicated, which is called dynamic
hedging. So we started to talk about
this last time, so what would you do?
You only now have the
opportunity of protecting yourself against bad luck by
betting one game at a time with these other brokers.
So what should you do?
Well, let's look at this
picture here. You know that this bet of (
100, -100) by doing backward induction is worth 42.
So if you were a trader,
a hedge fund manager, you would be marking your
position at 42. You're expecting now--you've
just made a trade which you know on average is going to make you
42 dollars. Of course, if you had bad luck,
you could end up losing 100 dollars,
which means you're going to go out of business,
you're going to be fired, your name's going to be in the
newspaper, you're going to be probably
sued by your investors. Or you can--we haven't been
sued by any investors. Or you might make 100 dollars.
Now you don't want to face that.
You want to hedge yourself,
protect yourself, so you're going to make the
money for sure. Now some people don't hedge.
They say when you hedge,
you're betting against yourself, and something bad can
happen. The hedge might go negative,
right? It may be that the Yankees win,
just as you think, but because you've hedged,
you're not going to get 100 dollars.
You're only going to get 42
dollars. So the essence of hedging is
you give something up on the upside to get something on the
downside. So people don't like hedging
because they're giving something up.
Okay, so you're giving
something up. The point is,
you're getting something back on the downside,
and you're making it so you're locking in your profit.
So hedging is a good thing.
So people who don't hedge I
think they're just very simple minded.
So there are a lot of people
who still don't hedge, but anyway, there's a
revolution in financial thought in the last 30 years,
anticipated by Ford [correction: Jones],
but it didn't really get--people didn't really catch
on until the '70s that you should hedge.
So how can you hedge this if
you can only bet 1 time, 1 game at a time?
That was the question we dealt
with last time. What should you do?
Well, you see the principles
we've learned so far tell us what to do.
You started here thinking on
average you're going to make 42 dollars.
You should be marking your
position correctly that it's worth 42 dollars.
Now if you win the first game,
you're at a big advantage. The Yankees are up a game and
by backward induction we know that then your position will be
worth 64 dollars. If you lose,
your position will be worth 8 [correction: 9]
dollars. Now what is 60 percent of 64
plus 40 percent of 9? It's 42.
60 percent of 64 is 38.4.
So you're at 42 and you could
go with 60 percent probability to 60-- I can't even remember
the number--64, or down to .9,40 percent.
Sorry, I'm getting senile,
9 with 40 percent probability. So 60 percent times that is
38.4 40 percent of 9 is 3.6 and the two add up to 42.
So that's why--so to say the
value's worth 42, how did you get the 42?
We were doing backward
induction. We took 60 percent of that 40
percent of that and we got 42. So if you're marking yourself
to market, after the first game of the
series, you'll have to admit to the
world that you're either now worth 64 dollars,
or you've gotten crushed, and you're down to 8
[correction: 9] dollars.
That already might get you
fired. Going from 42 to 8 [correction:
9], that's a pretty drastic loss.
So even the first day,
if you have to mark to market, you're going to be revealing to
the world that you've screwed up and now your position's only 8
[correction: 9]. You've told all your investors,
you've done this brilliant thing.
They're up 42 dollars.
The next day,
you're going to have to tell them, "Well,
I'm sorry, you're only up 8 [correction: 9]
dollars." They're going to be very upset.
So what would you do?
You know what to do.
What would you do?
Student: Hedge.
Prof: Hedge.
Good answer.
How would you hedge?
Student:
> Prof: Right.
So what do you say?
This is 9, so you'd go ( 33,
-22). So that's equal to 42 and
that's also equal to 42. Can you do this?
Is it fair odds to do this?
Well, yes, it has to be fair
odds to do this, because 60 percent of 64 40
percent of 9 is indeed 42. 60 percent of 42 40 percent of
42 obviously is 42. So therefore the difference,
60 percent of this 40 percent of that must be 0,
so in other words, it's fair odds.
And sure enough,
this is obviously 3 to 2 odds, right?
60/40 means you're betting at 3
to 2 odds. That's 3 to 2 odds.
So it is a fair bet,
so the other bookie who's willing to bet at 3 to 2 odds is
going to give you this bet. So again, you think the Yankees
are going to win, you make a bet with the naive
Philly fan, you're betting the Yankees are going to win.
You go to your broker friend
and you bet against your Yankees.
You're going to lose 22 dollars
if the Yankees win, and win 33 if the Phillies win.
You're betting against
yourself, but of course you're betting against yourself to a
smaller extent than you're betting against the other guy,
and therefore you still locked in 42 dollars.
So now we don't have to go any
further. It would only be confusing to
go further. I usually go further,
so it would just be complicated.
If you can do something once,
you can do it many times. So by making the right hedging
bet, you can lock in a profit up here and up here.
But that means you can lock it
in going forward to the very end, because you can keep
repeating that hedge over and over again.
Maybe I'd better go one step
further. You don't really believe me.
So what could you do going from
here to here? Well, the average of here and
here is about 9. You've collected 33 dollars,
don't forget, from your hedging bet.
So although this says 9,
you've got the 33 in the background, so it's 42.
But it's 9.
So what you want to do,
this is 9, you just want to transform this and this into 9.
Once you've transformed this
and this into 9, which you can do because 60
percent of this 40 percent of that is 9,
so the other broker--you can do an offloading hedging bet with
another broker at 60/40 odds to get 9 for sure here.
Why is that?
Because 60 percent of this 40
percent of that is 9, so therefore there must be
offsetting-- a bet he'd take that gave you 9
for sure in both cases, because the offsetting bet
would also be valued at 0 according to 60/40 odds.
So you can transform this and
this into 9s but don't forget, you've been paid the 33,
so actually this is 42 and 42. So you can keep 42 and 42 going
forever and by the end, you'll have 42 forever.
So that's what dynamic hedging
is. It's as simple as that.
Yes.
Student:
In order to make money with hedging, wouldn't you have to
have someone who would bet against you with different odds?
Prof: Yes.
So how do you make money in
hedging? You effectively were betting
against the Phillies fan at 50 percent odds,
and you've got your broker who's willing to give you 60
percent odds, and you're exploiting that gap.
So let's do another--excellent
question you're asking. I haven't answered his question
yet. I want to give another example
to answer his question. What is really going on with
the hedging? What am I relying on?
How many brokers do I have to
deal with to hedge? What's going on here?
I've given an example of a
World Series where it's very simple to figure out what to do.
So now I want to give an
example of a bond. It doesn't matter what bond.
I have to confess,
I haven't done this in advance. Here we have a 30-year bond,
8 percent 30-year bond. It's an 8 percent 30-year bond.
So we know what its payment is.
Homeowners are all perfectly
not paying attention, the bond is going to be worth
some amount of money. So here are the interest rates,
I forgot to say. So we have to assume what the
risk is. So in fact, what does this mean?
We know that the risk is 6--the
interest rate starts at 6 percent and it has a 20 percent
volatility. So what does that tell us?
It tells us that the interest
rates could go up or down by this amount.
Now, we've assumed that the
odds that it can go up or down are 50/50.
So we've assumed that everybody
thinks the odds are 50/50 that the interest rate could go up or
down. That's an assumption,
and so now I'm going to make--we've assumed that the
odds are 50/50. Let's assume that everybody is
willing to trade you 50/50 odds. So what does that mean?
This is like your other broker.
So the interest rate is here,
r_0. It can go up to r_0
times up, or it can go down to r_0 times some down.
Up remember,
is--up, by the way, this = r_0 e to the
volatility times e to the drift, and down = r_0 e to
the - volatility e to the drift. It doesn't matter.
r_0 can go up or it
can go down. So U is bigger than 1,
D is less than 1 probably. So we see examples of that.
Now I'm making an assumption
here that the odds are 1 half and 1 half and that everybody
understands and is willing to bet on these odds of 1 half and
1 half. So betting on an interest rate
move is called--that kind of betting on interest rates means
trading interest rate derivatives.
So there are a bunch of ways
that you can trade in the interest rate market to get 1
dollar if the interest rate goes up to this number,
in exchange for losing 1 dollar if it goes down to this number.
And these things are called
swaps and they're called interest rate derivatives and
they're called a whole bunch of other things,
which it's going to take too long to explain.
So let's suppose that you can
go out into the interest rate derivative market and simply
trade a security that'll pay 1 dollar if things go up or you
can buy another security that will pay 1 dollar if things go
down. Now what is the price of this
security that pays a dollar if the interest rates go up?
Let's say they can only go up
or down to one of two values. So what would be the cost if
everyone agrees the odds are 50 percent of getting up here?
What is the price today going
to be of getting 1 dollar if the interest rate goes up?
Student:
X divided by the interest rate. Prof: Excellent.
Who said that?
Perfect.
I didn't expect anyone to say
that so fast, so right.
So the price here,
we could write is .5 divided by (1 r_0),
and this price going down is going to be .5 divided by (1
r_0). How did he figure that out?
Well, the odds are 50 percent
of going up here and 50 percent of going down here,
but don't forget, you're getting the money later
and this is a 1 period interest rate, 1 r_0.
So therefore,
what you'd be willing to pay here is 50 percent of the dollar
that you'll get, discounted by 1 r_0.
What you paid here for that is
50 percent of the dollar you'd be paid, discounted by 1
r_0. So, exactly right.
But now these are like the
World Series, game by game odds.
They're going to change
depending on where you are in the tree,
because if you go here in the tree,
you've got a higher interest rate, so it's going to be .5
divided by (1 r_U) or something,
which is a bigger number. So the price of betting on 1
dollar-- what you have to pay to get 1
dollar if interest rates go up is going to be more [correction:
less] here than it is here,
say, because you're going to still have 50 percent chance of
getting here, but you're discounting by a
higher interest rate. So it's slightly subtler,
but not much. It's like the broker in the
World Series who game by game, day by day, there's the
interest rate derivative market, and they're willing to trade
you, they're willing to let you bet on which way the interest
rates are going to go. So what have we got now?
We've got this mortgage that's
sitting there. So let's say that you know that
the mortgage is owned by a very rational guy.
The homeowner--he's not the
owner of the mortgage--the homeowners are very rational
people. The market is very foolish.
This is an exaggeration.
The market might think the
homeowner never does anything, so they think the bond is worth
120. You've studied prepayments much
more than the market does. You realize that the homeowners
aren't going to just sit there, dumbly paying whatever they owe
if the interest rates go down. They're going to prepay,
so in fact the mortgage is only worth 98.
So the market thinks it's worth
100. The market understands the
interest rate probabilities and is willing to give interest rate
risk. They agree with you about that.
You don't know anything more
about interest rates than the market does.
What you do know is you know
something about the prepayments that the market doesn't know,
and so you know the bond is really worth 98 instead of 120.
So what should you do?
Let's suppose that the market
is actually treating-- so what the market thinks,
mortgage is = to a 30 year bond--
well, it's not actually a bond because it doesn't have a
principal at the end, so I'm going to have to do a
different sheet here to get it together.
The market thinks mortgage
never prepays. Let's suppose that what you can
trade in the market is these interest rate derivatives.
So what should you do now?
You can trade the mortgage,
you can buy it, you can sell it short,
you can buy or sell these interest rate derivatives.
What would you do?
Let's say you can only buy or
sell 1 unit of the mortgage. There's only 1 mortgage that is
trading, and since you know there's a mistake being made,
you're going to use all of it. So what would you do?
Do you think the mortgage is
overvalued or undervalued? Student: Overvalued.
Prof: It's overvalued,
so therefore what should you do?
Student: Sell it.
Prof: Sell all you're
allowed to sell. I've assumed a little
artificially. All you can do is sell 1 unit
of the mortgage short. So you're going to sell it for
120 dollars. But what does it mean to sell
it short? It means you have to make the
payments that the mortgage is making.
So you know those payments are
only worth 98 dollars? So what does that mean?
It means that you know you're
going to get a profit of 22 dollars.
However, you could get
completely crushed if things don't turn out exactly the way
you thought, so you want to lock in a constant profit for sure.
So you don't want to get a huge
profit more than 20 and no profit in some other cases,
or even a negative profit and a huge profit.
You want your 22 dollars for
sure. So how would you do that?
What would you do?
Yes.
Student:
> Prof: Okay,
so let's be more precise. It's exactly right,
but let's just see what this really means.
So I'm going to copy these
numbers down which I can't see yet.
You have a mortgage that's 98.
You know it's 98 something,
.8 and it could go in the up state to 92.6 or in the down
state it could go to 99.11. Now on the other hand,
the market thinks that this mortgage, this other instrument,
you're selling at the same time something that's worth 120.
You're selling it for 120.
It doesn't matter what the
mistake is the market makes. That's the reason why the
market made its mistake. You don't care why they made
their mistake. You know they're just willing
to give you 120 for it. So you're making a 22 dollar
profit, so what should you do?
If you were marking yourself to
market, what would you do? What would you mark yourself to
market here? Well, you've gotten 122 dollars
so you've marked yourself to market.
Here you'd say you're 21.16
dollars up. Here you'd have to say you're
not up so much money anymore, because you've just lost some
of the value of your mortgage, you've lost 6 bucks.
So you'd have to say you're
only up 15 dollars now. And here you'd have to say you
made a little bit of money, so you're up 22 dollars now.
But you want to lock in this
profit, 21.16 for sure. So what should you do?
If you're marking yourself to
market, you want to turn this money into 21.16,
but more than 21.16. You want to turn this into
21.16 times (1 r_0). You want to turn this into
21.16 times (1 r_0). Because then,
having locked in 21.16 times (1 r_0),
that's no risk and it's got the same present value as this and
so every period, you want to keep locking your
money into 21.16. That's the gap between that and
up there. Maybe 16 wasn't the right
number, by the way. What was this price up here
that they were willing to pay? They were willing to pay 120.58.
So that's a little bit more.
21.7 or something.
So it's 21.7, that was the gap.
So it's 120.58,
so the gap here is 21.7. So you want to turn this into
21.7 and this into 21.7. So 21.7 here,
you want to ensure that that's where you're marking yourself to
position. You want to know by the end of
the next year, you could turn this into 21.7
discounted up and then keep it discounting till the very end.
So whether you take your money
now, you take your money later, it's always a sure thing with
exactly the same present value. So how would you turn 21.7,
knowing that you're going to run this risk if you're marking
yourself to market, how would you do that?
Well, you know the value of
this this at this interest rate, in the mortgage derivative
market, it's worth 98.84. So you can always transform
this into exactly 91.84. By doing a fair bet you can get
this to go up to 91.84 and this to go down to 91.--oh,
there's also a payment here. Don't forget the payment.
So sorry, we've forgotten a
payment. The mortgage is much subtler
because you've actually gotten 8 dollars here 8,
and this is 8. So we forgot about this.
So you got an 8 dollar coupon
what was left was 92.6. this is 99.1 8.
So actually,
the total value here is 100.6 and the total value here is
107.11. You discount this by this 6
percent interest and you get 98.84.
So I left out a crucial step.
Remember, the cash flows here
are the coupon the value of the mortgage that's left.
So the value of the mortgage
that's left is 92.65, the 8 dollar coupon.
So over here,
you're going to have a value of 100.6.
Down here, you're going to have
a value of 107.11. So how can you--you want that
to equal, locked in forever--so you've just sold this mortgage
for 120, so you've got 21.7 dollars sitting here.
So now what can you do with
this 21.7 dollars? You can cancel your cash flows
so what you owe on the mortgage is always exactly the same
thing, 98.84. So you want to make this 98.84,
so you have to subtract something off from here,
100.6. You want this to be equal
to--let's do it this way. Let's make this 98.--this is a
simpler way of saying it, I think--98.84.
Let's turn this thing,
the present value of this and this is 98.84.
Therefore you know there must
be some fair trade in the derivatives market that turns
this thing and this thing into 98.84 times (1 r_0),
and this into the same thing, 98.84 times (1 r_0).
How do you know you can do this?
Because the present value of
98.84 times (1 r_0) and 98.84 times (1
r_0), 1 half of that 1 half of that,
divided by (1 r_0) is 98.84.
So therefore the present value
of this thing has to be the same as the present value of that
thing, and therefore there must be
some trade in the derivatives market you can make,
where you sell some of these interest rate derivatives and
you buy some of these interest rate derivatives,
which guarantees you 98.84 in both cases.
So that means the present value
of what you have to deliver in cash--
the value of what you'd have to deliver in cash--
is 98.84, still 21.7 dollars, in present value terms,
less than what you've received. What have you done with this
120 dollars? You've invested that at the
same interest rate and so now you've got a profit.
You owe 98.84,
jacked up by the interest rate, but the 120 you've invested.
You've got that 120 the
interest rate. You've always got more that
you're carrying on than the present value of what you owe,
and you keep going to the end of the tree.
When it ends,
you're still going to owe 98.84,
increased by the interest rate a bunch of times,
but then the value of the money you have is going to be 120
increased by the same interest rate,
and so you're going to end up with money for sure.
So then you can just reverse it
all, and get the money 21.7 locked in for sure at the
beginning and owe nothing later in the tree.
So to say it all one more time,
you have to think of yourself as marking to market.
If you're marking to market
correctly, you know what you owe is only worth 98 dollars and
what you've sold it for is 120 dollars.
That's a profit of 21.7.
But you want to make sure you
can maintain the same profit forever,
or to put it another way, if you've bought this bond--
you want to make sure you can lock in your profit forever.
So how would you do this?
There are different ways of
doing it. Let's say you want this 21.7
dollars for sure at the beginning and you don't want to
ever have to make any payments. In the end you want everything
to cancel out. What would you do?
How would you do this?
This number tells you,
you owe in present value terms 98.84 dollars.
So you've gotten 120 dollars.
You could take the 120 dollars,
put 21.7 in your pocket, and now, with this 98 dollars,
you can buy these cash flows. What can you buy?
You can invest money so that
you--you can buy the right interest rate derivatives so
you're able to make all the payments of the future bonds.
So this bond,
the actual payments the bond is going to make,
you can buy all those payments for 98.84 dollars.
I guess that's the simplest way
of saying it. I'm going a little bit in
circles. What is the mortgage security
down here if you know how the mortgage is going to pay?
All this is,
is a bunch of payments at different parts in the tree.
It pays 8 dollars here it pays
8 dollars here. Somewhere low in the tree,
it's going to just prepay for sure and pay the whole remaining
balance. Down here, it's just paid off
for sure, so it's prepaid. Sorry, it's prepaid at the
top--at the bottom, it's prepaid.
The fact that all these
payments together are worth 98.84 means that by taking 98.84
dollars and trading on the derivatives market,
you can buy all those payments for exactly 98.84.
So therefore as you go farther
into the tree, you always will have the
payments to make. You've sold the bond short.
What does that mean?
Any time this bond makes a
dividend payment, you have to make the payment.
But you can always buy those
payments for 98.84. How do you know you can buy
those payments for 98.84? Well, by backward induction.
Here, the payments are 8
dollars here, 92 dollars of remaining
dollars. So there's a way in the
interest rate derivatives market of buying 8 92.6,
that's 100.6 dollars here, and buying 107.11 dollars here.
So by spending this 98.84,
you can get, buy this many interest
derivatives, so you can get 100 dollars of
payments here, and 107 dollars of payments
here. Half of 100 half of 107,
discounted at 6 percent interest, gives you 98.84.
So it's possible on the market
to buy 100 dollars up here and 107 dollars down here with this
98.84. Now out of those 100 dollars up
here you've bought, give 8 of them to the guy you
sold the bond short to. That's the coupon payment.
Down here, give 8 of these
dollars to the same guy you sold the coupon short to.
Now that leaves you with 92
dollars up there and 99 dollars there.
But this 92 dollars is the
present value of what's going to happen next,
and what's going to happen next up here is that you're going to
owe 8 dollars to the guy here and you're going to owe 8
dollars to the guy here, but what's left will be worth
83 dollars and 97 dollars. So therefore with this 92
dollars, you in fact are able to buy this 92 105 here,
because the present value of 92, almost 92,105 here,
is exactly 92. That's how we got this number.
It's 1 half times this number
the coupon, 1 half times this number the coupon,
discounted at the interest rate corresponding to here.
That is 92 dollars.
So therefore,
by investing here in the interest rate market,
you can buy that cash flow here, the 84 8,92 dollars.
Pay the guy the 8,
you'll still have 83 dollars left and with the cash you have
over here, remember you bought exactly this much cash.
You've got this much cash left.
You can buy all the payments
you need to make here, the 8-dollar coupon,
plus you'll still have 97 left. So all the way forward through
the tree, you could always afford to buy all the cash
payments of this bond. So you can pay off the guy you
sold it short to all the coupon payments that this bond is
making, by having invested 98.84 at the beginning.
And similarly,
you got 120 by selling the bond short, so you've made 21.7
dollars and there's never any more cash coming out of your
pocket. You pocketed the 21.7 and you
invested the 98.84 in the derivatives market over and
over, changing your investment,
and therefore reproducing all the cash flows that you have to
make to the guy for having sold it short.
I didn't say it very well.
How followable was that?
Not very followable.
Sorry, I didn't do a good job.
Let's try again.
What is the essence of what's
going on? The essence of what's going on
is that gain-by-gain or month-by-month,
you can find someone else who'll always trade at fair odds
with you. Fair odds means--in the
interest market. They don't know,
these people trading the interest rate market,
they don't understand prepayments.
It's not like the bookie here.
They don't understand
prepayments. They're not willing to be with
you on what prepayments are going to be.
They're just going to bet with
you on what interest rate is going to turn up.
But you see,
the cash flows from the bond depended on prepayments and
depended on the interest rate. So you calculated it was worth
98.84 because you know what prepayments to put in the
future. So if you're absolutely
confident on your prepayments, you're going to know what this
bond is going to do in the future.
So remember the prepayments.
What are these prepayments?
It's this stuff.
So down here,
where the 1s are, the guy's prepaid for sure,
so you'd have to deliver to the market the entire remaining
balance at that point. But see, when you calculated
this bond, you were anticipating what all
the payments were going to be, whether they were going to be
the whole remaining balance or just the coupon.
Taking the present value,
what calculations were you doing?
In the present value,
you're doing calculations that this guy can do.
You're just taking 1 half times
this 1 half times that discounted.
The interest rate derivative
guy is willing to do that. He doesn't know what the
prepayments are going to be, but you're never betting on
prepayments with him. With the interest rate
derivatives guy, you're just saying,
"Let me pay you some money now.
You give me money if things go
up here, or I'll pay you some money now
and you give me money if things go down here,
if interest rates go up here or interest rates go down
here." So because the present value of
the mortgage cash flows evaluated according to the
interest rate probabilities and discounting is 98.84,
that means that you can successively trade on the
interest rate market and reproduce all the mortgage
payments, assuming you're right about
what they're going to be. And therefore with 98.84
dollars, you can replicate what the mortgage is going to pay.
And therefore,
you make 21. 7 for free, because you sold it
for 120. You only need 98.84 of that to
reproduce all the payments of the mortgage bond.
And how do you know that you
can do that? Because it's just one step
repeated over and over again. At every step,
there's going to be a payment of say 8 dollars and a present
value of what's left. And there'll be a payment down
here of 8 dollars and a present value of what's left.
Or maybe this thing is just a
prepayment, in which case it's a single payment of 99.5.
That's the remaining balance
after 1 period. But when you figure out this
number, it's always taken by averaging this number discounted
with this number discounted. So therefore,
averaging at the odds the interest rate guy will give you.
So therefore,
using this money, we're now doing the opposite of
the World Series, using this money,
you can buy this total payment, and you can buy this total
payment, because that's exactly what the
interest rate guy will give you. This is equal to 1 half of all
that divided by 1 r_0, plus 1 half of all this divided
by 1 r_0. That is this number,
so therefore in the interest rate market, by using this cash,
you can buy that and you can buy that.
And that's enough to make the
payment A8 that the mortgage is paying, you keep your promise.
And over here,
if the mortgage prepaid, the remaining balance,
that's that number, 99.5, you can pay that too.
And then the mortgage is done.
Here the mortgage is going to
go on, but see, it's going on at present value
you've calculated at 92.6. You bought 92.6 8.
You use the 8 to pay off the
coupon. You still have 92.6 dollars in
your pocket here. You're going to use that to buy
the future payments of the mortgage,
some of which will be a coupon, and maybe this will be another
coupon, plus the present value of
what's left. If it goes down here again,
the guy might prepay, but you're going to have enough
money to make that prepayment. So with this 98.84 dollars,
you're just doing fair odds over and over again.
You're buying the future cash
flows of the mortgage and therefore with 98.84 dollars,
you can keep all your promises, and yet you've gotten 120 at
the beginning, so you've locked it in.
That's it.
Any questions?
I don't know if that was too
clear, so somebody ask a question.
You're great at asking
questions. Is this followable?
Student:
> Prof: Okay,
that's too good. Too great.
Okay, let's see if I can do
another example. Let's suppose that we've got
this bond. Let's say you can't trade in
the interest rate derivatives market directly,
but you can trade in the bond market.
So let's get more realistic now.
Let's suppose that there's a
bond market. Now here we had interest
rates--so it was a 30-year bond, the starting interest rate was
6, and the volatility was 20. So we had that sheet that was
bond market trading, so let's just go to that.
File, where's open.
So callable bond.
Here we are in the bond market.
Let's say there's a 9 percent
coupon bond, doesn't matter what it is.
We start at the interest rate
of .06. Was that where we were starting?
And the volatility was 20.
So here we have the same
exact--exactly the same interest rate process as before,
so those are the interest rates, just as we had before.
Now let's talk about a bond.
So it's a non-callable bond.
I forgot already what I said.
How long was this bond?
It was a 30-year bond.
Didn't have to be 30 years,
but anyway, it's a 30-year, 9 percent bond,
and its value was 140.93. So the bond is worth 140.93.
If the interest rates go up,
same interest rate process, it can go to 121.4 or to 159.4.
So now let's suppose you can't
trade in the interest rate market any more.
What should you do now?
Student:
Don't you need another bond? Prof: Well,
you've got the interest rate. You've got the 1-month
[correction: year] bond that you can trade at an
interest rate of 6 percent and you've got this bond market.
So anyone will buy and sell
with you. Remember, this is the 30-year,
9 percent coupon bond. They're all willing to buy and
sell this with you. And you've got this mortgage
that everybody else thinks is the price is worth 120,
but you're sure it's worth 98.84, because you know the
prepayments. The homeowners are smart and
they're going to prepay and nobody else realizes they're
going to do it. So what should you do now?
You can't trade interest rate
derivatives any more. All you can do is trade in the
bond market, so do you have any idea what you could do?
What would you do?
This is even slightly more
realistic example. You've got a mortgage that you
know is overvalued. You've got a bond that you can
trade, and of course you can trade a very short bond,
the 1-year bond, which is trading at 6 percent
interest. You can trade a 1 year
treasury, and here's the 30 year, 9 percent treasury,
which happens to be worth 140, so what should you do?
Well, you know that you could
sell this mortgage short, get 120 dollars for it,
and you think the value of the payments is only 98,
so you'd love to do this. The trouble is,
the future values depend on a lot of uncertainty.
It may be that the payments you
might have to make are more than 98, so what are you going to do?
How would you lock in your
profit for sure? Can anybody figure out what to
do, just intuitively? It seems pretty complicated to
figure out what to do. So you sell this bond for 120.
You put the 21.7 dollars in
your pocket. You have 98.4 dollars now that
you have to spend to somehow acquire payments that are going
to just allow you to make the bond payments you've sold short,
the mortgage payments you've sold short at every node,
so you never have to give up any more money.
So what should you do?
You want to acquire some assets
with this 98.84 dollars that will put you in a position where
you have 100.6 dollars up here and 107.1 dollars down there.
So how can you do that?
Well, you have a combination of
the short--you have this 30-year bond you could invest in.
If you spent 98 dollars on the
30 year bond, that's 70 percent of that,
so that would give you 84 up here and 105 down there.
So it wouldn't match what you
really need. On the other hand,
you could put your 98 dollars in the 1-year bond,
which would give you 98.84 times 1.06 and 98.84 times 1.06.
So you've got this choice,
or you can put 100 dollars here and get 106 dollars there.
The interest rate we said was 6
percent, so that means there's a 1-year bond that pays 106 in
these two cases. There's the 30-year bond,
where 140 dollars could be worth this in the two cases.
So let's say you can't go trade
on the derivatives market, it doesn't exist.
Or at least you don't know how
to trade on it. But what you do know how to
trade is on the bond market, and--the long bond market and
the short bond market. So let's call this the long
bond market, and the short bond market.
So what would you do?
Any thoughts here?
Let me repeat the problem.
You find, like you typically do
when you're a trader, you know something other guys
don't do. That's what you're making your
whole livelihood on. You understand that prepayments
are going to be very fast. The people know how to prepay.
The rest of the market,
hasn't occurred to them yet that people can prepay.
They're grossly overpaying for
the mortgage, paying 120 dollars.
So you go and sell the mortgage
short. Now that means you have to
deliver what the mortgage is paying.
You have to deliver it to other
people, so that's a lot to take on.
But I say that you can sell
this bond for 120, put 21.7 in your pocket,
take the remaining 98.84 dollars and now buy the cash
flows that you're going to have to deliver in the future.
How do you know you can buy all
the cash flows? There are 30 years of cash
flows. There are coupon payments,
there are prepayments. There are all kinds of
complicated payments of the cash flows.
How, with this 98.84 dollars,
are you going to be able to buy them all exactly?
All you can do is trade,
buy the long bond and buy the short bond, or maybe sell the
long bond short or sell the short bond short.
So how, by trading these two
securities over and over again, can you replicate the cash
flows of the mortgage? That way you will have hedged
out your risk, and you'll have the 21.7 locked
in without ever having to worry about anything in the future,
provided your prepayment calculations are correct.
If you're wrong about
prepayments, then you're going to be in big trouble,
so you're betting on knowing what the prepayments are.
So what should you do?
Well, here this thing is going
to--the cash flows at this point are going to be worth 100.6.
The 8 that you have to deliver,
92.6 for the present value of the future cash flows.
Here they're worth--let's just
do the real thing. Never mind that stuff--here
they're worth 107.11. Now you know that 1 half times
1 over (1 r_0) of 106 1 half over (1 .06) times 107 is
98.84. So the average of these
discounted is that number. You know that the average of
these numbers--so this is the non-callable bond,
so it's not these numbers. It's this number 9 and this
number 9. It's paid a coupon,
so the average of these numbers, discounted by 6
percent, is that. So what does that tell you you
should do? I claim by combining this bond
and this bond, you can produce 100.6 and
107.11 here. So we know that this top bond
is going to pay you 130.4 up there and 168.4 down here.
If you bought X units of the
long bond, that's what your payoffs would be.
The short bond is going to pay
you 106 in both cases. So if you bought X units of the
short bond, that's what your payoff would be.
You want the payoff to be 100.6
and 107.11. So there must be an
X_L and an X_S that equals that.
How do you know there has to be?
Because it's 2 equations and 2
unknowns and these are independent equations.
In fact, you can tell what
X_L has to be. This gap between 100.6 and 107,
that gap is about 7, and this gap is about 38.
So 7 is like 1 fifth of 38,
so X_L is going to have to be 1 fifth.
I know in advance that this is
going to turn out to be about 7 over 38.
Just by general principles,
you know there's a solution for X_L and X_S.
So I know there's a solution to
that because it's 2 linear equations and 2 unknowns and
they're not degenerate, so I know there has to be a
solution to that. And I also can tell you what
X_L has to be, because the gap,
that's the difference between the top number and the bottom
number, is 7. That gap over there is 38.
So obviously the middle gap,
106 and 106 is 0. So if I take X_L to
be 7 thirty-eighths of that number, the gap is going to go
down from 38 to 7. Then I'll just have to find the
right X_S to make that equal.
So that combination of
X_L and X_S will produce this payoff here
and this payoff there. So I know what combination of
the long bond and the short bond to hold so I produce exactly
these things. So I'm going to get a cash flow
of 100.6 and 107.11 and I'll be able to use that to make the
coupon payment of 8 in both cases and on top of that,
have enough money to continue buying future cash flows.
Now what will the cost of the
X_L and X short be? It will be X_L--I'll
have to pay 100 dollars for the short bond,
if I have to do X_S, X_short,
and I'll have to take 7 thirty-eighths of 140,
that's how much it'll cost me to buy the long bond.
So the right combination of
short bond and long bond will give me the right payoffs.
And it's clear that there is an
X_S here and an X_L here,
such that I get the payoffs I want of 100 and 107.
But the last thing to notice is
the cost of this X_L and this X_S,
which is going to be 140 times X_L 100 times
X_S. That cost will be exactly 98.84.
How do I know that--therefore
I'll just be able to do it. I'll ask you that question in a
second. Here's what this hedging
amounts to. The hedging amounts to again,
you know because of your superior knowledge of
prepayments that the mortgage cash flows,
not the mortgage itself, the cash flows that are going
to come from the mortgage are only worth 98.8.
Someone's handing you 120
dollars in exchange for making those mortgage cash flows.
Okay, so what do you do for
that? You take the 98.84 dollars out
of the 120. You take 21.7.
That goes into your pocket.
The 98.4 you have to use to buy
the future cash flows. The first cash flows are going
to be 8 dollars, but you're going to need more
money to buy the cash flows that come after.
How much more money do you need?
92.6 here, 99.1 there.
So how can you put yourself in
a position to have 100.6 dollars here and 107.11 dollars here?
Can you use this amount of
money to buy this value at the next step?
We saw in the derivatives
market, yes you can. But even if you can't trade in
the derivatives market, you don't need to if you could
trade a short bond and a long bond.
You would just find how much of
the long bond do I hold and how much of the short bond do I
hold, so I can get these cash flows of 100.6 and 107.11?
The answer is,
you have to solve that equation up there.
What X_L and what
X_S gives you that number?
You know that there's some
X_L and some X_S that will solve
that, because it's 2 equations and 2 unknowns.
I'm repeating myself.
You even know what
X_L is without writing down the equations,
because the gap has to be--one thing,
there's no risk, and the other thing,
there's a big risk. So the only risk in the payoff
has to come from the long bond. So that gap of 38,
if you take 7 thirty-eighths of that, you're going to get the
gap to go down to 7. So the X_L's going to
turn out to be 7 thirty-eighths. X_S you have to solve
by algebra. So you know how much that
X_L is and what the X_S is.
That's what you hold,
that combination of X_L and X_S,
you hold here and you get exactly the payoffs.
And then the final step is to
notice that you can exactly afford it with 98.84 dollars.
How do you know that you can
afford it? Because what you want is 100.6
and 107.11 and that costs 98.84. You're buying two things
together whose payoffs together are worth 98.84.
So therefore the cost of the
two things separately have to be worth 98.84.
So it will turn out when you
solve for X_L and X_S,
you'll exactly be able to afford it with this amount of
cash. So you can just do that going
forward all the time, constantly re-hedging your
portfolio. So that's the essence of
dynamic hedging. It's a very beautiful idea
which I probably haven't explained in the optimal way,
but the point is that, to summarize the whole thing
again, you know something about a
bond, but you're subject to more risk besides what you know.
You hedge out the extra risk,
still relying on yourself to be right about what you know.
You relied on your being right
about the prepayments. You don't know anything about
the future interest rates. But there's another guy like
the trader in the World Series, the broker in the World Series,
who's willing to make bets on the interest rate with you.
Or to say the same thing,
who's been calculating the values of the bonds,
the short and the long bond, as if he was making bets on the
interest rate. He's calculated at the same
50/50 odds, discounting by the interest rate.
So because there are all these
guys on the market who are willing to make these game by
game bets, either directly--I'm almost
done--they either make the game by game bets and the interest
rate directly in the derivatives market,
or what's equivalent to that, they've used that calculus to
figure out the value of these long and short bonds.
So by trading through the long
and short bonds, you're effectively doing the
same interest rate bet. So either way,
by trading through the two bonds,
or trading directly in the interest rate derivative market,
you're able to buy, by going game at a time.
Year by year,
you're able to buy all the cash flows of your mortgage,
the cash flows that you're predicting it's going to have.
And therefore,
you can make the profit for sure.
It's predicting something,
being confident in your prediction,
then being able to buy what you've predicted the cash flows
are for a smaller price than you can sell the security for.
That's how you locked in your
profit. Okay.
