Okay. Good morning. Welcome to 2020. I guess this
is the first lecture, certainly some of you. So we're here today
for mathematical models of financial derivatives. I'm
Sam Cohen. I'm one of the lecturers here in my finance. And what we're gonna do
is in this course, we are going to try and understand some
of the contracts that are commonly traded in finance. We're going to try and ask ourselves how we use
mathematics to understand the better. We understand the risk. How do we manage it? How
can we price it, et cetera. So before we begin, we really should
be asking ourselves, why are we wanting to do this?
And it's a it's a pertinent question because mathematical models of financial
derivatives have been more than a little bit controversial. Now,
you guys may not remember it as well as I do, but
12, 13 years ago, there was a very large financial crisis, the GFC.
And one of the things that are being blamed for this was an overreliance
on mathematical models of financial derivatives. You've got the article, the formula
that killed Wall Street. So is this a sensible
thing for us to be studying at all? It's an open question. It's a serious one.
I would argue it is for a variety of reasons. As
we'll see, financial derivatives have always existed pretty
much since agriculture. We have had financial derivatives
and because of that, we we can't close our eyes to the fact that these things exist,
that they are traded, that they are real and building models of
them trying to understand their risk is a sensible thing to do.
We also need to understand where those models are going to fail. We need to understand the restrictions
of what we're doing. And so that's something I'm hoping that we'll touch on quite a lot in this
course. Where is the model going to go wrong? Where are the restrictions? Where have we made
abstractions or simplifications that are a little too brave, a little
too far from reality? And I'm hoping that we'll cover some of those things as
we go along. So what are we trying to actually do in this course?
We are trying. As I say, we've got three broad goals.
The first is to build
models of contracts
traded on financial markets.
So that's the basic idea of what we're going to be doing. We're going to try and build models of some of
these contracts. We're going to try and understand their payoffs. And then from that, we want to
understand where they fail. We want to understand how to
use them,
where they fail.
And that's where we need to take particular care.
Well, we need to be careful.
And is the third point, we want to develop
the technical proficiency
to build, understand
and use better models.
So this course is not the end point. We're not going to get to the advanced things
that one can do in the world of mathematical finance, but we're going to try and get enough technical
proficiency that we could at least move on to that step at the end of this course.
Okay. So when we're building a model in finance, what
are the key concerns? Now, the usual thing people think is that you're trying to build
a model so that you make money. Sounds good. The truth
of the matter is that usually making money is not our first concern.
Usually our key concern in much
financial modelling, a key concern.
Is don't they exploited?
We are more worried about if we are trying to give a price for something, that we will give a price
that allows someone else to come and rip us off. That is actually our first concern.
And this is going to come up again and again to the concept of arbitrage that if we are giving prices
that allow us to be exploited, then we're not going to do very well. And so this is our first concern.
Our second concern is we need to understand
and manage our risk.
It's not enough to have something which pays off. Well, if it's so risky that we're going
to lose before we win, and so we need to understand to manage our
risk. And then a third goal is.
Make a profit in what we do in this course. This is actually not
going to be a key aim. We're going to be trying to understand these steps because only
once we've done these, that this is even possible to even talk about making profit, unless
you can understand your risk and you're sure you're not going to be exploited. There's no way you can turn a profit.
Okay. So what are the economic assumptions that underlie
most our modelling? Let's try and start moving towards actually building
a model of something. So will begin with
a fairly simple set of assumptions about how financial markets work.
And these are deliberately simplified, but they allow us to
start making some conclusions. So the first assumption that we're going to use in
a lot of the models is that there is a riskless
investment.
Now, what do I mean by riskless? I mean that if I put money into it now, I know how much money
I'm going to have in a year's time. So this is a fairly common
assumption. You can think of this as being cash or more usually as
being a bond or a bank account with a fixed interest rate. I can invest in it today and I know
how much money I'm going to have in the future. So we'll often refer to this either as
a bank account or
a bond. If you're
not sure a bond is just a an agreement to pay in the future. I thought of it.
I would have. I have one in my office and old physical bond. It's a contract
where what you do is a company or government will issue
bonds. They sell them in the market and they give the holder of the bonds the right to certain cash
flows. So if you buy a bond, you then can go to the company. And every month or however often the
payment is they will give you money. Now, to make
things simple. This grows
at a constant, continuously
compounded
interest rate,
which will tonight buy a lower case. Now,
the fact it's constant just simplifies things a lot. We'll come back to whether this is a good assumption at various
points. The continuous compounding, that's more of a just a technical
simplification. The fact that something is compounding continuously means that I can figure out the value
of my riskless investment at any point in time without worrying about when exactly
the interest is being determined. Now, in many cases, interest is calculated on a day
to day basis. So at the end of the day. And what that means is that this is
only an approximation. It's the first point where we've already started making bad assumptions,
assumptions that are not technically true, but they are good enough. If I'm pricing something which is a long
way in the future, say a few months, the fact that interest is being computed daily rather
than continuously makes very, very little difference. In most cases.
Okay. So what does this practically mean mathematically? Well,
it means that if a quantity MDT is invested
at time t
then it grows
to well how much a big T, which is E to
the R. Big T minus little T. Empty
time. And so this is this continuous
compounding, you get an exponential coming in. Okay.
A guaranteed
amount. B t to be
paid at time t.
Is therefore worth
little to which is E to the minus R t
tape. So if this is true, if I can invest
a time little T so today I invest into the future and my amount that I've invested
grows by multiplying by this call coefficient eight of the R big T minus subtlety,
the time duration. Well that means if I want to get a contract which will pay
me in the future a fixed amount B big T. How much do I have to invest today.
Will you rearrange this formula and you get it. But with a negative exponent,
we assume borrowing
and lending rates
are the same. Well, there's an assumption that's clearly
not true. But if you're a big financial institution. It gets
close to being true that the the amount that you're paying to borrow
and to lend is very similar. And so we're just going to assume that they're the same.
But we are going to remember that's an assumption which we know is sketchy.
OK. So this is the first thing that we've got. We've got this riskless investment
in the world where we can invest now for the future and
it's going to grow at a fixed rate that we know beforehand.
Okay. What are the assumptions about the market? Because so far we've said nothing about
a market. We've just said there is an investment. So we're going to make two assumptions,
which are sort of our basic modelling assumptions about financial markets.
So there are no
transaction costs.
So what does that mean? It means that if you want to buy something or if you want
to sell it, you're going to pay the same price to make either side of the trade.
So you pay the same price. You pay the same price
to buy or sell.
And for any quantity.
So the price that you would pay for buying one unit of an asset is the same as what you'd get for selling
one unit of an asset, which is one hundredth of what you'd get for buying or selling
100 units of the asset. Now, again. Is this assumption
reasonable? Well, it's a reasonable first approximation, but it's not
technically true. We all know that if I came to you and said I'd like to sell you a TV, maybe
you'd pay me one price if I said I'd like to buy a TV. You'd give me a different price. And if I said I'd like to sell
you a thousand TV shows, you'd look at me strangely and ask, where have I gotten a thousand TV's
from and why am I trying to sell them? So the quantity effects
are important, but we're just going to ignore them as a first approximation.
Okay. A related assumption is that assets
are infinitely divisible.
OK. You can own
zero point two shares. This is more technical
assumption than anything else. It means that we don't have to worry about discrete numbers of shares, that you've got to
buy one share or two shares. You can't buy one and a half shares
practically. It doesn't matter because once you're working on a large scale, if you're working
on a deal for ten thousand shares, the difference between buying.
Eight thousand and eight thousand and one shares is pretty small. It's a ization area,
but it's small enough that we don't need to worry about that as a first approximation.
Okay. Something a bit more controversial than those is that we're usually
going to build models where short selling
is allowed. OK, a little bit of terminology.
Whenever I say short, I mean, selling something. Whenever I say long. I mean owning something.
So what's a short sale? That's where I own a negative quantity
of an asset. Now, is this possible?
Well, it depends on what you mean. It's certainly true that in many markets,
what you can do, say, in the equity market for big stocks, if I want
to hold a negative amount of some big stock, what can I do? I can go
and I can borrow that stock from someone else. I pay a fee. We'll ignore the fee,
but I pay a fee to borrow the stock. I then hold the stock. I can sell the stock.
Now, what that means is I have an obligation in the future to buy the stock again and return it to the person
I borrowed it from. But in the meantime, I hold a negative quantity
of stock. Why is that? If the stock goes up in price? I lose. The stock goes down.
I win because I'm gonna have to buy it in the future.
Now, that is what's called a covered short sale. I have gone. I have
borrowed the stock from someone. So the stock is physically there and I'm able to sell
it on. When I say physically, I don't mean physically. Pretty much everything's
done electronically these days, but don't let that worry you. There is also
what's called a naked short sale. And naked short sale is basically
where I go into the market and say, I'll send you some stocks. And you say,
all right, I'll buy them. And I go, great. And I know that the contract
gives me two days to make good on the deal. So if you've bought stock from me,
I know that I have two days in which I. In two days, time will buy
two days time. I have to deliver the stock. So I am now currently
short the stock, because the stock goes up in value, I'm in trouble. But
there's the idea that I could go and if I do this, if I make a naked short sale, I say I'm
gonna sell it. I could then go and buy it in the market or borrow it from someone after I've made the deal.
This is a naked short sale. This has been very controversial in the past few years.
Why is it? Because if I haven't already identified the stock that I'm
going to be borrowing or if I haven't already borrowed it, it's very easy for me to start
selling things I don't own. Very easy if I'm allowed to do that for me to go and say
I'm going to sell 10000, 10 million, however many stocks I want.
And that looks like lots of people are trying to sell a lot of stock. Even though I
don't own it. And so that's been linked to the idea that it's going to push prices down.
And so it's been quite controversial. We're going to assume that it's allowed.
We're not going to go into the details of how short selling is actually manipulated.
But we're just going to say you can hold
a negative quantity.
Of an asset.
Okay, so these are the basic assumptions of how we are going to
model markets, unless I say otherwise, we're going to take these is given, even though we know that most
of them sort of. Not true. But this is true in most mathematical modelling. We have to
abstract. We approximate. And we hope that we've captured the key features of what we're
trying to understand. OK. So how do we proceed?
Well, we haven't yet given any principles which will allow us to find a price
of anything. But we have said one thing earlier here.
That we're going to avoid being exploited. And the most basic form of exploitation
is arbitration.
So what's an arbitrage, an arbitrage is a deal that is too
good to be true. So an arbitrage.
Let's get the definition right. So is an investment.
Which costs? Nothing.
To set up.
At time t.
I. So if I think of the
cost or value of this investment, it starts off being negative or most Zira.
But. At a later time,
later time. Big teeth. We have.
A zero probability
of it being negative.
So the probability that X a big T is less than
zero zero. Because if it was worth a negative
amount. That would be costly for me in the future. I'd have to pay it.
So this is a deal which costs me nothing to set up in the future. I can't
lose the probability of me losing is zero and
it has a strictly positive
probability of a
strictly positive
payoff. So, I mean that the
probability that X T is greater than zero should
be positive. So an arbitrage, if we think of simple
games, is something like the game where we flip a coin. If it's heads, I'll give you
a pound. If it's a tails, nothing happens. Now, if that doesn't
cost you anything to enter into, it's a very good game to play because, well, maybe
you win something, maybe you don't. But you're never going to lose. So that is an arbitrage.
And we are going to basically assume that arbitragers shouldn't exist.
So, Al, the way we're gonna build models is based on the principle of no arbitrage.
Now, does that mean arbitrages actually don't exist? Well, no, they do exist, but
on well functioning markets with lots of players. Whenever
an arbitrage exists, someone will exploit it and some will exploit it quickly.
And if someone exploits it, they start changing prices. And when
they change prices, they do so in a way which gets rid of the arbitrage, sort of
eat up all of the money that can be made out of the arbitrage. By changing the price.
So we're going to assume there are no arbitragers when we build models. What does that mean?
It means we're not building a model to try and detect and exploit arbitrage.
We're going to be building models which say, let's assume we're not fast enough. We're not aiming
to exploit this. What can we then say about the market?
OK, so we
assume. No.
Arbitragers, no arbitragers
exist.
So we've got some basic assumptions about markets and how they work,
but we haven't actually considered any contracts yet. So where we'll begin
is with probably the oldest form of financial contract. This is what's called a forward
and what it is. I'll give you a simple example. Imagine for a moment that you are
an exporter. So you are based here
in the UK. You are selling things in the US,
for example. So in one year's time. You will make
a sale. You've already signed the deal today to make a sale in the US
and you will receive some quantity of US dollars. You know
that you're going to receive them. On that day, you will have to pay
all your costs in the UK. All those costs have to be paid
in pound sterling. OK. You have calculated
that if the exchange rate is as it is today, then this is a good deal.
You will make money, but you're worried. You've looked at the
last couple of years and the price of pound sterling versus the US dollar has fluctuated
quite a lot. And your worries that in one year's time when
you receive a whole lot of U.S. dollars but have a whole lot of expenses in pounds, that you
won't have the money to cover it and you might lose. Even though you think today this is a good deal.
So what could you do? Well, what you do is you can go to a bank
and you say to them, look, this is my situation. I would like an agreement with
you to sell us dollars into pound
sterling in one year's time. Now, there's nothing strange about this contract. All
it is, is in one year I will be giving you say one million U.S.
dollars. And I would like you to give me however many it is pound sterling.
And what you try and do is you go to the bank and say, could we please agree today
on the price at which we will do this deal? OK. So we're going to agree today
on the exchange rate at which we will trade in one year's time. And that's the basic idea
of a forward. It's about taking a price today
at which we will trade in the future. The question is,
what price should we trade at? So how do we determine the price today?
Okay, so let's try and write this out. Okay.
So a forward is an
agreement to
trade.
On a fixed date in the future.
For a price determined.
Tonight. Now, one thing to notice, this is an agreement to trade.
All parties must trade on that date. There's no way
I can get out of this once I've signed the contract with the bank. I am legally bound to deliver
US dollars in one year's time. And they are legally bound to deliver. Pound sterling
in one year's time. No one is allowed to back out of the deal.
So both parties
to the contract have
the obligation
to try and. So
how are we going to find a price for this? Okay, let's
think about this. How could I? What other options
could I do? Well, I know that I'm going to be.
I'm going to have to sell us dollars in one year to get pound sterling.
So what I could do. This is called the short
position from the U.K. perspective, because I am selling the asset,
the asset is US dollars. OK. So, for
example, suppose
we wish to sell
one U. S. T in one
year. For simplicity,
we assume no interest
is paid for holding US dollars,
although we'll come back to that assumption. But let's begin. As soon as you hold any US dollars,
there's no interest being paid on that. So what could I do?
Well, one thing I could say is I could.
Let's suppose
the price of one U.S.
dollar today is
s little take. So that's the price at which I can buy one U.S. dollar
in pounds right now. And we've assumed no transaction costs so I could buy or sell
at the same price. So what could I do? Well,
I could. So consider
borrowing S-T.
Hey. Buying
one new installer and holding it.
For a year.
The payoff for doing this.
Is well. What's the value of my US dollar? That I've bought in one years time?
S. S t. That's the future value of one U.S. dollar.
I don't know its value now. I'll only know that at time, big T. But I've also had to borrow
money. And we remember by borrowing S-T today, I've had to
pay interest on that. So I've got E to the R team
honesty. This is my interest rate on the money that I've borrowed in Pownce times. The amount that I
borrowed, which is as little T.
OK, if I hold
this and move forward,
which has pay off, let's think about
this, what's the forward contract worth? Five agreed a price today.
De. Then I've got. Oh.
My US dollar in one year's time, I can sell it for some amount.
Which is the forward price.
De. So I
agreed. Agreed price
F.T. So if I've agreed that price. What's the value of this contract?
Well, in one year I have agreed to traded s little T. The asset,
the U.S. dollar is worth. S big T.
So I have a pay off, which is evidently
which is what I'm actually getting minus
S Big T, which is what the price would be if I didn't have the contract.
I didn't have the forward. OK. So if I
hold this and the forward my total
cash flow is
o f t minus eight to the R.
Hey, let's take. So it's my cash
flow story titled Profit.
Everyone see where I've got that from. So I've got my terminal pay off. This is what I paid
for, the amount that I borrowed. I've got my forward contract, which I agree a price on today,
but I'm effectively losing out because I'm not getting the value of the stock in
the future. But if I add these two together, I just get this payoff here.
The interesting thing is, while the problem of figuring out the value of this was that I don't
know the value of a yet, that I'll only know in one year's time at the terminal point
of my contracts. But this depends on things that I know today.
There's nothing risky in here. I've got. This is the amount that I borrowed. I know what I'm going to pay for that. This is
the price I'm agreeing on today. So there's my total profit, which I know today
by no. Of a trash.
This cannot. Uh, to have
a positive value.
That has to be negative because if that was positive, this would be free money.
I'd go and do this deal, I'd enter into the forward contract. I would borrow money out by U.S.
dollars. And in one year's time, I would have money. It costs me
nothing today. Forward contracts. I'm not paying you anything today, we're just agreeing a price.
No cash exchange tonight. And so if this were a positive amount.
Then I'll be able to make money for nothing. So that's an arbitrage which we've excluded from our models.
So we certainly know that this quantity must be negative.
Well, what about the other side? So this was if I was
if I was thinking as myself. But what I could do is I could
instead think, what if I was my bank? So what's my bank's position? They're doing
everything opposite to me. So they're selling me. They're agreeing the price to do the trade the other way.
And they could borrow money. They could effectively
take money invested and sell us dollars. OK, so instead of buying
a US dollar, they short selling US dollar, instead of borrowing money, they invest money.
So they do everything with the opposite sign to me. So considering
the long forward
with the direction
of trades. First.
We get that. Because if it was possible for my bank to just go and make
money. Then that's an albatross for them. In fact, if
they're offering me a deal, which OK. There's money on the table, I can make
money for doing nothing. Well, we do this. That's an arbitrage. So we're going to assume this can't
happen either. But now we put those two together and we find
that the forward price is equal to the R
t take t times. Let's take.
That is the current price of one U.S. dollar. So the price at
which we agree to trade in the future. Depends on the current exchange
rate. Well, the current exchange rate and the interest rate payments.
Okay, so just before we continue, we should just think
a little bit about this this pay off here.
OK, so the
pay off of a forward
is. So if this is the
stock price. OK.
This is the payoff. Well, if I go for the
short forward, which is what I've described here. So that's an agreement to sell the asset.
Then the payoff looks like this.
On the other hand, if I look at the other side of the deal, an agreement to buy the asset
in one year's time, then I get the opposite.
This is the long forward.
OK. So what are some peculiar things that we notice
when we're doing this?
Observations of
the forward value forward price
is based on current.
Pryce's. And interest payments.
It does not
depend on whether.
S t is a fair price
for the asset.
I have made no assumption anywhere that the current exchange
rate is a sensible exchange rate. And there's something
a little bit weird going on here that I haven't even assumed anything about the future
evolution of the exchange rate. We're making an agreement for what's
going to happen in one year's time. And I have told you nothing about what I think the exchange
rate will be in one year's time. That has not entered
into our calculations at all. So even though we've got a contract, the payoff of
which depends on the exchange rate in one year's time.
Whether it's profitable or whether I make a profit or loss depends
on what the exchange rate one year time is. We have not used any statements
about what I think this is what I think the exchange rate will be.
Now, if you've got an efficient market, then this should be a fair price.
And there is a secret assumption. It's not so secret. We made
it up here. There are no transaction costs. And I can buy and sell any
amount for the same price. So because of that, if there was
money on the table, if this was a bad price. This would be a very strange assumption
to be making, because what it's saying is I can start just buying as much as
I like without changing the price
anyway. But our calculation for the forward price actually
doesn't depend on whether we have a fair price for the asset at all.
We did not need to model.
Didn't need a model for what the future exchange rate will be. So if
interest rates are
positive, she's usually a good assumption.
OK? And
there is no cost or benefit.
Holding the asset.
Then f t you look at this and you say, well, this is a positive.
This is a number. Well, if are is
positive. This quantity is going to be bigger than one.
If I made a silly mistake, someone.
And F.T. is.
Above the current.
Prior asset price.
OK. But we've made an assumption in there, so
question.
What if. Interest
is earned on holding.
That's right.
So let's quickly. In the last minute. So what happens if interest isn't on holding you
dollar solicitor, right? Aha. Well.
Okay. The thing is here, we made our contract
about one U.S. dollar. But if I'm gonna earn interest on holding U.S. dollars
in this deal, I'd buy one U.S. dollar and I hold it to the end.
So how about I change this to holding E to the minus hat?
Do you want to see yourselves? Why is that amount? Well,
this is the amount of U.S. dollars I need to buy today to have one U.S. dollar in
a year's time or in the future. So if I hope if I do this,
I get the same payoff. OK,
but what's it going to cost me? Well, I have to correct this because I no longer.
How much do I borrow? Well, I get eight to the minus our hat.
Got to change the amount I buy. These combine.
OK, so you get now this is E to the minus,
I had to take T think.
So this changes and you end up.
With each of the R minus R hat to take
T. S t. And this is the sort
of formula you get whenever there is a cost or benefit for holding the asset. Now,
forwards are traded on many things, including commodities. That was the
classic example, is buying agricultural products. They've been traded since
antiquity. So there's evidence for them in the Code of Hammurabi, which is 18th century
B.C. There's examples of them in Aristotle, in the politics.
On the other hand, trading on financial products like interest rates, exchange
rates is much more recent. That's really since the 1970s. So they were
out of time. We'll continue talking about this tomorrow morning where we
will also discuss what are the criticisms of what we've done, whereas what we've
done gone wrong.