20. Option Price and Probability Duality

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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right,
OK, let's get started. So before I make
introduction, let me just make a few announcements. A few of you came to us asking
about the grading for the term. And some feel the problem sets
may be on the difficult side, and some of you haven't done
all of them, and some of you have done more. So we just want to let you know
that the most important thing to us in grading is really
you show your effort in terms of learning. And we purposely
made the problem sets more difficult
than the lecture, so you can– if you want to
dig in deeper so you have the opportunity to learn more.

But by no means we
expect you to finish or feel easy in solving
all the problem sets. So I just want to
put you at ease that if that's your
concern, that's definitely– you don't need to
worry about it. And we will be really just
evaluating your effort. And based on what do
we observed so far, we actually believe every single
one of you is doing quite well. So you shouldn't worry about
your performance at the class. So continue to do a good job
on your class participation, and do some of the problem sets. And then you will be in
fine shape for your grade. So that's all of that. So without any
further delay, let me introduce my colleague,
Doctor Stephen Blythe. I'll be very brief. And he's– Stephen is doing
two jobs at the same time. He's responsible for the all
the public markets at Harvard Management, as well as being
a professor of practice at Harvard. So with that, I turn to Stephen. STEPHEN BLYTHE: OK, well,
thank you, and thank you for having me to
speak this afternoon. Before I begin, I wanted
to ask you a question.

So I'm speaking, actually,
at almost exactly the 20th anniversary of
something very important. So on the 19th of October,
1993, which I guess might be the birthday
of some of you, but almost exactly 20 years
ago, Congress voted 264 to 159– I actually remember the count
of the vote– to do something. So anybody like
to guess what they voted to do on the 19th
of October, 1993 that might be tangentially
relevant to finance and quantitative finance? Anyone here from HMC is
not allowed to answer. Anybody– any guesses? Any ideas at all coming
to people's minds? AUDIENCE: Was it
Gramm-Leach-Bliley? STEPHEN BLYTHE: No. AUDIENCE: Commodity
Futures Modernization Act? STEPHEN BLYTHE:
No, but good guess. But actually, that is actually
too related to finance, actually. This is actually–
wasn't actually directly financially
related, so that was related to [INAUDIBLE]. Anybody else think about? What does Congress usually do? AUDIENCE: [INAUDIBLE]. [LAUGHTER] STEPHEN BLYTHE: No,
no ideas whatsoever? What do you think
Congress did 20 years ago? They voted to do something. OK, well, what
Congress do usually is they cut money for something.

So they voted to cut
financing to something. So what did they
cut financing to? Anybody guess? I know this isn't
business school. In business school, it would
be, like, right, you're failed. No class participation–
you failed. You've got to say something
in business school. So I know it's not
business school. But anyway– and I don't teach
in business school, either. But this is
actually– these round desks make me think
of business school and striding into the
middle of the room, and saying OK, come on. Fortunately, I don't have names,
otherwise I'd pick on you. No, no guesses– no
guesses whatsoever? I've got to take this up
the road to Harvard Square, and say I've taught at MIT. No one had any guesses with
this question– one guess, actually, the gentleman here. What did they cancel the
financing for in 1993? I'll say it was the
Superconducting Super Collider underneath Texas
just south of Dallas.

So $2 billion had been
spent on the Super Collider. And the budget had
expanded from, I think, $6 to $11 billion. So they, by canceling, had
a $9 billion dollar savings. This is 20 years
ago– almost exactly. And as a result of that–
one result of that– was, of course, the academic
job market for physicists collapsed overnight. And two of my roommates
were theoretical physicists at Harvard. And they basically realized
their job prospects in academia had vaporized
instantaneously that day.

And both of them,
within six months, had found jobs with
Goldman Sachs in New York. And they catalyzed they–
they and the cohort– they're called the
Superconducting Super Collider generation. If you ever wondered why people
like myself and like Jake got PhDs in quantitative subjects
around the turn of– around 1990 to 1993– all ended
up in a financial path, part of it is due to Congress
cancelling the Superconducting Super Collider. Because this cohort
catalyzed this growth in quantitative finance. Actually, they created a
field– financial engineering– which you are all somewhat
interested in by taking this class. And they also created a career
path– quantitative analyst, or quant, which really
did not exist before 1993.

And that growth of
mathematical finance, financial engineering,
quantitative finance– however you want
to look at it– was basically exponential
from 1993 up until 2008 and the financial
crisis exactly five years ago, funnily enough. And since then, it's been
a little bit rockier. So if you're actually
interested in this aftermath of the physics funding– what's
interesting is the Large Hadron Collider, which you might know
is up and running in Geneva and just found the
Higgs Boson, has sort of reversed the trend somewhat.

So there used to
be a whole cohort of people going into
finance instead of physics. Now, because finance has this
somewhat pejorative nature to it– people don't
like bankers generally, and they kind of like physicists
who find the Higgs Boson and get a Nobel Prize–
maybe we're getting reversal. But anyway, we're
still in finance. I've, as Jake mentioned,
well, I did mention, I was originally in academics. I was actually a mathematics
faculty member in London when I got my PhD– I
got my PhD from Harvard. And in 1993, I was an academic. And all my friends– I
saw them go to finance. So I followed them, spent
a career in New York, and then came back
to Harvard in 2006 to run a part of the endowment. And I started teaching.

So just as a plug–
for those of you interested in mathematical
finance and applications of mathematics finance, I
teach a course at Harvard. It's an upper level
undergraduate course called Applied Quantitative
Finance, which of course you can cross-register for. And today is also the
one-week anniversary of the publication of my book. So if you're interested in
what my course is about, you can just buy my book.

It's only $30. And I'll sign it. It's first edition,
first printing, first impression
book, Introduction to Quantitative Finance. And that is what the course is. It's quite distilled. When this book came out, I
thought, that's really thin. This is three years
of my life's work. It's come out– it's very thin. But I like to think
it's like whiskey. It's well distilled,
and highly potent, and you have to sip it,
and take it bit by bit. Anyway, that is the
book of my class. And the genesis of
the class was really that, when I've
been on Wall Street, and I was a colleague
of Jake's at Morgan Stanley in this rapidly growing
quantitative finance field, we encountered on the trading
desk in the late 1990s and the early 2000s problems
from the real economy– things that we had to trade. We were– things that were
coming to us on the trading desk that required subtle
understanding of the underlying theory.

So that we, in essence, we
built theoretical framework to solve the problems
that were given to us by the financial markets. So that period, especially
around the turn of the century, there's a big growth in
derivatives markets, which– options, futures,
forwards, et cetera, swaps. And we needed to build
theoretical tools to tackle them. And that's really what the
course was evolved out of, to build the appropriate
theoretical framework, motivated by the
problems we encountered. Why I enthuse
about the subject– and I really like
teaching the subject– is that there is an impression
that qualitative finance is a very arcane and contrived
subject– just a whole bunch of PhDs on Wall Street
coming up with crazy ideas. And they need
complicated mathematics that's just complicated
for the sake of complexity. And the theory is just
sort of a contrived theory. But in fact, at the
heart of Wall Street is that the real economy
demands some of these products by supply and demand. There are actual,
real participants in the financial markets who
want to trade derivatives. And therefore, in order
to understand them, you need to develop a theory.

So it's actually driven
by real examples. That's one part. The other part is that
the theory that comes out of it, and in particular the
approach I take here, I think is just very elegant. OK, so there's some subtlety
and elegance and beauty to the underlying
theory that comes out of addressing real problems. This course, and the way
that I teach finance, is very probability centric. You probably realize from the
lectures you've seen already in this class, there are
many different approaches, many different methods that are
used in finance– stochastic calculus, partial differential
equations, simulation, and so on.

The classical derivation
of Black-Scholes is, well, it's the
solution of the PDE. OK, that has appealed to people. In fact, this is
why in some ways, quantitative finance
is a broad church, because whether you're a
physicist, or probabilist, or a chemical engineer, all
the techniques you learn can be applied. You know stochastic calculus. You know differential equations. They can be applied. But the path that I
take in this class is very much through the
probabilistic route, which is my background as
a probabilist, as an academic, or a
statistician as an academic. And this is, in particular,
I think, a very elegant path to understand finance,
and the linkage between derivative
products– which might seem contrived– and
probability distributions, which is sort of natural
things for probabilists. So this, what we're going
to talk about today, is really this
link, which I call option-probability duality.

Which, in essence, in the
simplest form, is just saying, option prices– they're just
probability distributions. Therefore, these
complicated derivatives that people talk about–
all these options, these financial engineers,
these quants, these exotics– we're really just
talking about probability distributions. We can go between them–
option prices, probabilities, and distributions– back and
forth in a very elegant way. What I love about this
subject in particular is that to get to that point
where we see this duality does not need a whole framework
and infrastructure of complicated definitions,
or formulae, or option pricing formulae, or so on. So that's what I'm going to
try and do in this hour or so, is introduce this concept
of option price, probability duality. And show how the
natural– so there's a natural duality
that can be seen in a number of different ways. OK, so we're going to need
a few definitions that should be familiar to you. We're going to
define three assets. We have a call option,
which we know about, a zero-coupon bond–
called a zed cee bee.

This is the one thing I
haven't become Americanized on. I still call this zed. It's a– other
things I've become– and then a digital option. OK, all right, so what are they? Well, they're all going to
be defined by their payouts at maturity. OK, so we're going to have
some maturity capital T, and some underlying asset, S,
the stock, with some price S_T. OK, so we know that the call
option has payout at T– So that's called
payout at T. So T is some fixed time in the future. We will change in the
future to some fixed time. This is simply the max
of S_T minus K and 0. That's a call option. You can go through the right
to buy, et cetera, et cetera.

But it's clear it's
just value at maturity is just the max of
S_T minus K and 0. The zero-coupon
bond with maturity T is just something that's
worth 1 at time T. So that's just payout one. That's definition–
so you can think of these all as definitions. And then the digital
option is just the indicator function of S_T
being greater than K. So here, T is the maturity. K is the strike. So T maturity, K is strike. And these are three assets. So this is, in some sense,
the payout function. All derivative
products can be defined in terms of a function–
not all of them. Many derivative products
can be defined just as a function of S_T.

And here are three functions
of S_T. [INAUDIBLE] And then I'm just going to get
notation for the price at t less than or equal to T.
We can think about little t as current time
today, or we can think of some future time
between now and capital T. I'm just going to
introduce notation. Every different finance book
uses different notation, so just C for call price,
with strike K, at little t with maturity big T.
OK, just that notation. The zero-coupon bond–
the price at little t– let's call that Z. That's
the price of little t. And the digital–
we'll just call that D. So this is what we're
going to set this up. Actually, you could have
a whole lecture on why notation– different notation. K and capital T are
actually embedded in the terms of the contract. Little t is in my calendar time. So you might think why
don't you put K and capital T somewhere else? Well, when you get actually
to modeling derivatives, you like to be moving both
maturity and a forward time and calendar time. That's why I just
write it like that.

But there's no– so
C sub K, little t, big T is the price
at time of little t of a call with maturity
capital T and strike K. Black-Scholes and other
option pricing formula are all about determining
this– for t less than T. Because clearly we know
that the price at maturity is simply the payout. I mean, that's, again,
just the definition. So that's trivial. But we want to find out what
the price is at little t. So that's the whole path
of finance– Black-Scholes and other option
pricing methodology is working out this. But we're actually going to
go down a different route. So what we're
going to do– we're going to construct a portfolio. So consider as a
portfolio of what? We're going to
consist of two calls. OK, we're going to have lambda
calls with strike K. OK, so this is the amount holding. And everything is going to
be with maturity capital T. So lambda calls with
strike K, and maturity T, and minus lambda calls with
strike K plus 1 over lambda.

We'll just consider
that portfolio. It consists of two options. All right, well, this
price at T– that's easy. We just write it
in terms of lambda times the price of the
call with strike K, minus lambda call with strike
K plus 1 over lambda– just by definition. This is price at T. OK, well, let's look at
its payout at time capital T graphically. So we know about call options. The payout function is just the
hockey stick shape, clearly. That's confusing to people
from the UK, because in the UK, hockey means field
hockey, not ice hockey.

And of course, the hockey
stick shape in field hockey looks very different. Anyway, that's– you understand
what the payout of a call is. Clearly, this payout function
of a call looks like this. Well, putting this
payout of lambda calls of strike K minus
lambda calls of strike K plus 1 over lambda–
let's assume lambda is positive for the time being. What's it look like? Well, 0 below K, is flat
above K plus 1 over lambda. It has slope lambda,
and has value 1 here. You should be able
to see that easily.

So that's the payout. This is called call spread–
just the spread between two calls, and has this
payout function. OK, so a natural
thing to do here, it being a mathematics
class, let's take limits. Just let lambda
tend to infinity. Well, then, this becomes
the partial derivative of the call price with respect
to K, or the negative of it. So this tends to minus. OK, let's just– so that's that. And then this, of course
as lambda goes to infinity, this stays at 1. So this tends to payout
function that looks like that. OK? This is easy calculus. This is just by inspection.

OK, so this, clearly, is
the payout of the digital. Of the– strictly
a digital call, but that's called
the digital option. Just as a note, here
it's, just greater than. You might think, OK, it
doesn't matter if it's greater than or equal to. Well, in practice, the chance
of something equalling a number exactly is 0– I mean, if it's
a continuous distribution. In theory, I should
say, the chance of something actually
nailing the strike, actually being equal to K,
is 0, so it doesn't really matter whether you define
this as greater than, or greater than or equal to. But in practice, of course,
finance is in discrete time, because you don't quote things
to a million decimal places.

So certain assets,
actually, which are quoted only in eighths
or 16ths or 32nds or 64ths, this matters,
actually, whether it's defined as greater than or
greater than or equal to. But theoretically, it
doesn't make any difference. OK, so we've got the call
spread tending to the digital. All right, so this tends to–
so the limit of this call spread– of this price of the
call spread– is the digital. And so we know that because
this is the price at t. This is the payout at capital
T. The price of the digital must equal just the partial
derivative with respect to strike of the call price. So that's just a nice, little
result. Where does this bring in probability? So this is the next. OK, so this is where
we'll make one assumption. And it's actually a very
important and fundamental assumption. And it's fundamental
because it's called The Fundamental
Theorem of Finance, or the Fundamental
Theorem of Asset Prices.

So I call this FTAP–
Fundamental Theorem of Asset Pricing. By this theorem, which we
are going to assume here, the intuitive answer is correct,
meaning that prices today are expected values. It's the expectation
of a future payout. So by FTAP, the price
at t is expected payout at time capital T,
suitably discounted. So there's both something
very straightforward here, and something very deep. If you think about
how much would you pay for a contract
that gives you $1 if an event happens– in
this case, the event being stock being greater
than K at maturity. You would intuitively
think that's related to the probability of
the event happening. How much will you
pay to see the dollar if a coin comes up heads? You'd pay a half, probably. It's very, very intuitive. But the deepness
is, this actually holds under a particular
probability distribution.

I'm not going to
go into that here, but by the fundamental
theorem, this is true. So I can write, in the
case of the digital, the digital price
equals the discounted– and we'll explain why we want
to put the zero-coupon bond price here– that's the present
value of a dollar at time t. It's just a discount factor. It's very trivial,
but it's written in terms of an asset price–
times the expected value of the payout. So either you take this as
this makes a lot of sense– the discounted expected
payout– or you can say, I don't understand this.

I want to find out about the
Fundamental Theorem Asset Pricing, which we will
prove in my class. But this intuitively
makes sense. The key here is that the
expected value actually has to be taken out under
the appropriate distribution, called the risk-neutral
distribution. But this formula holds–
in fact, strictly. I'll write this
is just for– what holds is the price at
time little t divided by zero-coupon bond is a
martingale– for those of you into probability theory. This gets probabilists
very excited, of course, because they love martingales. Everyone in probability
theory loves martingales– lot of theorems about martingales. And you'll see, of course that
this is actually a restatement of this assertion.

Because Z, capital
T, capital T is 1. So this statement here
is simply a re-expression of this martingale condition. So I'll just pause here. Just from a probability
point of view, when I learned probability,
it was under David Williams, who wrote the book
Probability With Martingales, which is a wonderful book. And I thought martingale
is a great thing. So I was sort of happy. It took me about
seven or eight years of being in finance
to realize there are a whole lot of
martingales floating around. Because this actual
approach– this formalization of asset pricing
really only became embraced on the trade floor
around the early 2000's, even though the underlying
theory was always there– this idea of these martingales.

Anyway, so this is–
and this, of course, is simply– the expected value
of the indicator function is just the probability
of the event. OK. All right, so now
I've won by intuition. Just here's the probability
of the payout occurring. I've priced the digital. I've also priced the
digital by taking the limit of call spreads. So now I'm just
going to equate them. So by equating these two
prices for the digital, I simply get that the derivative
of the call price with respect to strike equals the discounted
probability of the stock being above K. I've just reorganized
a little bit, take 1 minus. So I get the probability
that– well, I can clearly reorganize again
and get– all right, so if I want to simply get
the cumulative distribution function, it's
just 1 minus this.

So divide here, take 1 minus. OK, so I get the cumulative
distribution function for the stock price at T is
equal to 1 plus dC by dK times 1 over Z. I'm just rearranging. So here now is the cumulative
distribution function. Clearly, I just need
to differentiate again to get the probability
density function. So here's where the
notation gets kind of messy, but clearly the probability
density function of– f for my random
variable S sub T– so the density of– express that
as– I always– probabilists, whenever they talk
about densities, they always want to say f of x. And it's the same with me. That's f of x. Here's the density is
simply just the next, the second derivative. We'll take the
derivative of this.

It's the second derivative of
the call price with respect to strike, evaluated
at little x. All right, so what
we've done here is start off with
simple definition of three assets, price to
digital in two different ways. And now we have a
rather elegant linkage between call prices–
C– and the density of the random variable that
is the underlying stock price at capital T. OK,
so we've established one side of the duality, which
is given the set of call prices for all K, I can then
uniquely determine the density of the underlying asset.

So you might think, OK,
this is kind of nice. How does this actually
work in practice? Do we actually think in
terms of probability trading? We just said that call options
are equivalent to probability density functions. Well, actually,
there's a very neat way of accessing this
density function through another
portfolio of options. OK, so this is actually
where we get– to me it's the practical relevance
of some of this theory. So let me just show you that. So we're going to consider
another portfolio. So here we consider
portfolio as follows– it's actually going to be the
difference between two call spreads. So lambda calls with strike
K minus 1 over lambda. Minus 2 lambda
calls with strike K, and lambda calls with strike
K plus 1 over lambda– again, lambda positive. OK, why are we doing this? Let's just stop for
a bit of intuition. Here we see in the call spread
the discrete approximation to the first derivative of call
price with respect to strike. So clearly, if I want to
approximate the second derivative, I'm going to take
the difference between two call spreads appropriately scaled.

You're now going to have
to have a little– there's got to be another lambda
coming in here at some point. This is just the difference
between two call spreads, so that's the difference
between two approximations of the first derivative. So I'm going to have to scale by
lambda in order to approximate the second derivative. So this is actually
called a call butterfly. And this is a beautiful
thing for two reasons.

One is they actually trade
a lot– surprisingly. This is not a contrived
thing I just made up. A, it trades a lot, so you
can actually trade this thing. The second is you
can kind of imagine the right scaling of
this call butterfly is going to approximate
the second derivative, and that's approximating
the density function. So this is a traded object that
will approximate the density function. Yeah, you have a question? AUDIENCE: Yeah, I
have a question. In the real world,
you cannot really– the strike distance cannot
really go to infinitely small, so they have some [INAUDIBLE]
way how to approximate that? STEPHEN BLYTHE: Yeah,
so that's a good point. Yeah, so the question
is how, in practice, we can't go infinitely
small, which is true. But we can go pretty small. So in interest
rates, we might be able to trade a
150, 160, 170 call butterfly or equivalent–
10 basis points wide. That's a– it's a
reasonable approximation to the probability of
being in that interval.

So these are all, I mean,
you make a good point. In fact, all of finance
is discrete, in my view. So continuous-time finance
is done in continuous time because the theory
is much more elegant. But in practice, it's
discrete in time and space. You can only trade finitely
often in a day, and so on. I won't going into the detail,
but you can see the price. Let me just write
down the first. The price of this I have just
expressed as the difference between two call spreads. So it's lambda times the call
spread from 1 minus lambda to K, so K, 1 minus lambda
to K, minus the call spread from K to K plus 1 over lambda. OK, so the difference
between two call spreads– we'll call this–
this is the butterfly. We're just going to
use temporary notation, call that B, B for butterfly.

So the price B, and
then you get confused. It's B centered at
K with width lambda. No one ever uses this
notation outside this one section of my class,
so that's why, but it's just handy for this. So that is– the
butterfly price is equal to the difference
in these two call spreads. What I want to do
is, I want to take limits of this, suitably scaled,
to get the second derivative. And if you just take
lambda times B_K of lambda, t, T is indeed,
approximately– if I take limits is exactly– the
second derivative of call price. OK, so here's how I'm
accessing the second derivative through a portfolio
of traded options. All right? And so the price of
this butterfly, B, if I just reorganize and
substitute– so I get B_K– for large lambda, i.e. a small interval–
is approximately 1 over lambda times the
density function– actually, evaluated at K. So I have
obtained this density function by this traded portfolio.

And to your point about we're
not getting infinitely small. That's absolutely right. But if you think about
what the density– when you learn about density
functions for the first time, you say the density function
at x times a small interval is the probability of being
in that small interval. All right, so when we
think about the density function f of x, if you have
a small interval of delta x, then clearly the probability
of being in this interval is approximately
f of x, delta x. In the limit, that is true. So what we're showing
here, if you actually think about what interval
we're looking at, we're actually looking at
in this call butterfly– if you were actually
to draw it out, this call butterfly
looks like that around K.

It actually– it's
a little triangle. It's not actually a rectangle,
but it's a little triangle of width 2 over lambda. OK, so it is actually– this
is the area of this triangle– 2 over lambda times
1/2 times f of x. And that's actually this, right? So this has width 2 over lambda. OK, so in fact, we've
got here exactly an approximate–
exactly approximation, that doesn't sound right. But it's entirely analogous
to the approximation of the probability of
being a small interval. Here is the probability
of being in this interval here– just the area
under that is exactly 1 over lambda f of x. So here is actually something
that people do do, is they say, OK, I will look at the price of
this butterfly, which gives me the probability
of this underlying random variable
ending up around K.

I'll make a judgment whether
I agree with that probability or not. And if I think that
probability is higher than this price implies,
then I'll do a trade. I'll buy it. I'll buy that butterfly. So there is actually an
active market in butterflies, and so I think an
active trading in probabilities– probabilities
of the underlying variable being at K at maturity. So OK, so that's the
first linkage here. Both– the density is
the second derivative, and the second
derivative is essentially a portfolio of traded options. And none of this is dependent
on the actual price of the call option, in the sense that
this holds regardless.

Clearly, this is a function of
the price of the call option, but I don't need any
model for the option price to hold, in order for these
relationships to hold. So these are model-independent
relationships, clearly. If you were to put the
Black-Scholes formula into C– Black-Scholes formula
of the call price– and take the second derivative
with respect to K, which would be a mess, you'll end up
with a log-normal distribution. Because that's what actually
the Black-Scholes formula is, is expected
value of the payout under a log-normal distribution. And that will hold. So this will hold for that. AUDIENCE: [INAUDIBLE]? STEPHEN BLYTHE: Yes. AUDIENCE: The last
[INAUDIBLE] So left-hand side depends on the small t. STEPHEN BLYTHE: Yes, it does. AUDIENCE: But the
right-hand side does not. What's the role of that? STEPHEN BLYTHE: Yeah,
that's a really good point. I've been loose in my notation. So here what is it? It's actually the conditional
distribution of S capital T, given S little t.

So this is the
conditional distribution, given that we're currently
at time little t with price S little t for the distribution
at time capital T. So that's where it comes in. That's absolutely right. So in fact, this
expected value strictly should be conditional on S_t. This probability
is a probability conditional on S_t– absolutely. And in fact, this
martingale condition is– the martingales
with respect to S_t.

So that's where the
little t comes in. AUDIENCE: [INAUDIBLE]? STEPHEN BLYTHE:
Here, yes, sorry. That's 1 over Z. So it's
just a constant here. This number, especially
because interest rates are so low in US, so this
number is so close to 1 that you always
forget about that. Not when we're trading,
but when you, oh well, this is just a– if you just think
about which one is– this is a quantity that's
in the future. It's call prices, so
that's how you kind of go. All right, so that's
the first bit. So when I was an undergraduate,
actually, learning probability, one thing I learned
about probability was from my probability
lecturer, who said, the attention span of
students is no more than about 40 minutes. So there's no point lecturing
continuously for 40 minutes, because people will just start
switching off after 40 minutes. So rather than wait, just have
a break and waste the time, the lecturer said,
I'm just going to give you some random
information in the break, and then we'll go
back to probability.

So I learned that
from 25 years ago. I can't remember– I actually
remember the material. I can't remember any
of the random material. So that's what I
do in my lectures, is I break them up, and
talk about something random. So I thought I'd do that
here as well, with some– not completely random. So this is somewhat applicable,
this being a math class. So how many of you are math
concentrators or applied math concentrators? One, two– a lot, applied
math concentrators– especially for the applied
math concentrators, going straight to
the conclusion– your entire syllabus
was generated at Cambridge University. That's the conclusion.

So anyway, here's the story. So back in the 19th century, the
Cambridge Mathematics degree– the undergraduate
Mathematics degree– was the most prestigious
degree in the world. In fact, it was actually the
first undergraduate degree with a written examination
was Cambridge Mathematics. So they have a lot to
be responsible for. And each year,
people took the exam. And they were ranked. And that ranking was published
in the Times of London– so the national newspaper. And the people who got
first-class degrees– so summa degrees– in
Cambridge Mathematics were called wranglers, and still
are called wranglers, actually. And the reason they're
called wranglers was from way back
in the 17th century where, before they had
exams– or 18th century, I should say, before they had
exams– instead of writing down exam, you have to
argue, or dispute, or wrangle with your professor
to get to pass the class.

So that's where
wrangler comes from. So these people who got
the first-class degree are called wranglers,
and they're ranked. And basically, the
senior wrangler was a very famous person in
the UK in the 19th century. And a lot of them turned
out to be quite successful. So here are a few wranglers. I've just got this one– I can't
reach that, but [INAUDIBLE]. So some of you might
recognize– and I just want to tell you a quick
story about one of them.

OK, so let's start
1841, senior wrangler was George Stokes– so
basic fluid dynamics– the whole of fluid dynamics–
that's George Stokes. 1854, second
wrangler– this is– who was the first wrangler? The second wrangler was James
Maxwell, so electrodynamics, Maxwell equations. He was the second. And I can't quite work
out who was the first. 1880, the second wrangler was
J.J. Thompson, so electrons, atomic physics, that comes
out of– he was only second. 1865, senior wrangler
Lord Rayleigh. So he was the sky is blue. He was first.

So they're a pretty good bunch. So the story–
the best of 1845– I'm going back–
the second wrangler was Lord Kelvin, so absolute
zero, amongst other things, of course. But absolute zero– he
was second wrangler. And the great
story about him, he was the most talented
mathematician of his– of the decade. And he was such a lock
for senior wrangler that– and I actually
read the biography, so this is a sort
of true statement– that he sent his servant
to the Senate House where these things are being read out,
and with a request, "Tell me who is second wrangler." And the servant came
back, and said, you, sir. And because he was such a
lock to be first wrangler. And in fact, what
happened was a question on the mathematical
exam was a theorem that he had proved two years
before in the Cambridge Mathematical Journal. So his theorem was
set on the exam. Because he had not
memorized it, so he had to reprove it,
whereas the person who became senior wrangler
had memorized the proof, and was able to regenerate it.

In those days, there
was a lot of cramming to be done in these exams. So the guy who– Stephen
Parkinson was senior wrangler. He went on to be
FRS, and eminent. But he wasn't– so anyway,
so here's the applied math syllabus. Here's a couple of other
ones which I really like. In 1904, John Maynard
Keynes was at 12th wrangler. Now, I can tell the
story either way, depending on whether I'm in
an audience of economists, or an audience of
mathematicians. Since I'm in an audience
of mathematicians, I like to say the
greatest economist was so poor at mathematics, he only
managed to be 12th wrangler. There are 11 better
mathematicians in the UK in that year. So he was obviously
not that great. If I was talking to
economists, I would say, this guy is so brilliant that
his main field was economics, and yet as part time, he's
able to be the 12th best mathematician in the UK. So last one I wanted to talk
about– 1879– here's a quiz.

This one you have to
have some answers for. OK, so 1980 something– I can't
remember what it is– so here's one, here's two, here's three. I'm going to give
you one and two. You've got to fill in three. You probably aren't going to
be able to get this one yet, but this is– Andrew Alan,
senior wrangler, George Walker, second wrangler, and number
three is the question. That's the question– 1980,
Hakeem Olajuwon, Sam Bowie, question mark–
who's question mark? Do we know which sport
these people play? AUDIENCE: That one's
Michael Jordan. STEPHEN BLYTHE: Yes, right. There we go, that's
Michael Jordan– exactly. This question could go on
forever in the UK because they don't– so Michael
Jordan, famously, was only picked third in the
NBA draft in 1984, was that– four or five,
something like that. So Hakeem Olajuwon was
actually pretty good, but Sam Bowie was a total bust.

But he was third. So in 1879, in the Cambridge
Mathematics Tripos, these two people you never heard
of, who were first and second. And the person who came third,
you've probably heard of him. This is more of a
statistics thing. People know about correlation? What's the correlation–
who's the correlation coefficient named after? AUDIENCE: Pearson. STEPHEN BLYTHE: Pearson,
you've got Karl Pearson. So Karl Pearson was the
third wrangler in 1879. And the founder
of statistics– he founded the first ever
statistics department, and obviously
invented correlation with Gould– Gould and Pearson. Anyway, he was only
the third wrangler. And unfortunately, these
people have very common names, so I have no idea what
they went on to do.

To Google these people
is not very effective. Anyway, so that's the story
of Cambridge Mathematics– lots of good stuff. All right, so in
the last half hour, I just want to go the other
way from– so the other way– we went from option
prices to probability. Let's go from probability
to option price. We sort of already
have, actually. This is what the
Fundamental Theorem does. If we're thinking–
if we take on trust that the Fundamental
Theorem holds, namely option prices today are
the discounted expected payout at maturity. OK, let's take that on trust. Then we're going from
probability distribution to option price in
the following way. So let's actually state the
Fundamental Theorem, FTAP. OK, so I'm going to go
general derivative D is– D, digital D, derivative. It's– so derivative
with payout at T. So this could be
the digital payout. It could be call option payout. It could be one. And price– OK, so
often, we actually think about payout function
as just a simple function of the stock price.

But this notation is useful
when we think about the price as being martingales. Then what is FTAP? D– the ratio of the price
to the zero coupon bond is equal to– is a martingale. In other words,
its expected value under the special distribution,
risk-neutral distribution, of the payout at maturity. And to you point, it's
conditional on S_t. So this is the proper statement. So this is the Fundamental
Theorem of Asset Prices. In words, it's
saying this ratio is a martingale with
respect to the stock price under the
risk-neutral distribution. That's the statement of
the Fundamental Theorem. This is actually rather neat
to prove in the binomial tree, two-state world. It's very, very difficult
to prove in continuous time. This is Harrison
and Kreps in 1979. It's the proof that, however
many times you look at it, you're only probably going to
get through two or three pages before thinking,
OK, that's hard.

But it was done. So this is, you can
imagine continuous time, infinite amount of trading,
infinite states of the world. OK, so now this,
of course, is 1. And this can come up. These are known
at time little t. So if I'm thinking at– if
I'm at current time little t, therefore, the
derivative price is what we had before–
the expected payout. OK, so this is rather
a nice expression. And now we can actually just
write down what this is. This is the expected value of a
function of a random variable. So this is just
the integral of g of x, f of x, dx, where
this is the density of the random variable
at time capital T, conditional on being at S_t. So this is conditional at S_t.

So this is a restatement
of the Fundamental Theorem. So this is essentially
the Fundamental Theorem. And this is a
intuition made good, or intuition made
real– expected payouts. This is sometimes
called LOTUS– the lure of the unconscious statistician. Just the expected value of g of
x is integral g of x, f of x. That's not immediate from the
definition of expected value. You should really work
out the density of g of x.

And then integral of x–
the density of g of x dx, but it turns out to be this. So that's a very
nice, nice result. OK, so here is now a way of
going from density to price. If I put in the call
option payout for g, and I have the density, I
can then derive the price C. So If you like, the way I go
from density to probability distribution to option
price is exactly the Fundamental Theorem. The route I take is the
Fundamental Theorem. OK, so FTAP, the Fundamental
Theorem of Asset Pricing, means I can going from
the probability density to the price of a derivative,
for any derivative. All right, OK, so now we
can go either way– density to derivative or call
price to density. You might say, hang on a sec. We've only gone from–
we need the call prices to get the density. Well, of course, we can go
via an intermediate step. So to get from the call price to
an arbitrary derivative price, I just go via the density.

So in particular– this
is restating– knowledge of all the call prices
for all K determines this derivative
payout for any g. So if I know all calls,
I know the density. And then if I know
the density, I know an arbitrary
derivative price. It's obvious as we stated here. But what this is saying
is, the call options often are introduced as this–
why are they important– are the spanning set of
all derivative prices. So calls span– call prices
span all derivative prices. And they are a particular
type of derivative– the ones that are determined exactly
by their payout at maturity. One can imagine other things
that are a function of the path or whatever.

But this is a particular
derivative price. European derivative prices
are determined by calls. OK, so that's kind of nice–
sort of obvious, elegant. There's two other ways of
looking at this, though. If I think about my function
g– so consider function g– OK, so that determines
my derivative. So it determines,
defines the derivative by its payout at maturity. Let's just graph it. OK, so it might
look– let's just assume first it's
piecewise linear, so it looks like–
so suppose this is g. Well you can kind of
see I can replicate this portfolio, or this option,
by a portfolio of calls– in fact, a linear
combination of calls. Right, I have no calls,
but if this is say K_1, this is K_2, K_3,
K_4, K_5, et cetera.

You can see what the
portfolio of calls will be in order to replicate
this payout at maturity. There'll be a certain amount
of calls with strike K_1, so that the slope is right,
minus a certain amount of K_2 to get this slope, plus
a certain about of K_3, minus K_4, minus K_5,
plus K_6, et cetera. So, in this case,
if the piecewise linear g, replicating portfolio
of calls, it's obvious. So if I can replicate the
payout exactly at maturity, the price at time little
t of this derivative must be the price at little t
of the replicating portfolio. That's actually a– I'll do
that early on in my class, and of the 100 people, everyone
says, OK, that makes sense.

And someone says, does that
always have to be the case? And it's actually a really,
really good question. Here, I was about to
just hand wave over it. Is it the case that if I
have one derivative contract with this payout at maturity,
and I have a linear combination of calls with the identical
payout at maturity, capital T, must these
two portfolios have the same value at little t? Well, one would think
so, because they're both the same at
maturity, so they must both be the same thing.

They're just
constructed differently. And the assumption
of no arbitrage– which underpins everything, in
some sense, what we're doing– would allow you to say
yes, indeed, that is true. And in fact, it's actually the
fundamental of finance, right? If two things are worth a
dollar in a year's time, they're going to be
worth the same today. That's what we're saying. If you can match the portfolio
at t, that is actually the definition of– it follows
immediately from no arbitrage.

What has been interesting in
finance, especially since 2008, is that that– this
assumption– has broken down. In other words, I can
hold a portfolio of things when aggregated have
exactly this payout, against an option with
exactly this payout, and be paid for that. And this is actually really–
it's a very fascinating thing, to think about actually, the
dynamics of financial markets when arbitrage can break down. What is the main theme
here is that when capital T is a long way
in the future– 10 years, 20 years– there's
nothing to stop the price of the option and
the replicating portfolio going arbitrarily wide,
other than people believing that it has to be equal. The only way you can guarantee
the two things to be equal is by holding it until capital
T– for 10 years, 20 years.

In the meantime,
those prices can move. Empirically, they've been shown
to move away from each other. So there's actually a deep
economic question here. So if there is the presence
of arbitrage in the markets, then arbitrage can
be arbitrarily big. Because you're saying there
aren't enough– there's not enough capital, or that's
not enough risk capital, for people to come in and
say, OK, these two things have to be worth the
same in 10 year's time.

Therefore, I'm prepared to buy
one $1 cheaper than the other. It's actually a question
really relevant to the Harvard endowment. We're a long-term investor. You say, why doesn't
the Harvard endowment, if these two things
are $1 apart, buy the things
that's $1 cheaper, and just hold them 10
years, make the dollar? Well, we'd like
to, but if we think they're going to be $1 apart,
and they're going to go to $10 apart, we don't want to
buy them at $1 apart. We want to buy
them at $10 apart. I mean, yes, we're a
long-term investor, but we care about our annual
returns, or five-year returns. Suppose this is a 20-year trade. This is very prevalent when
these things are 20 years out. Anyway, it's a whole– this
is– it's a little bit– it's a foundational issue. It's this thing
where it could shake the foundational underpinnings
of quantitative finance if you don't allow this
replicating portfolio to have the same price as
the actual option. But mathematically, you can
see you can replicate it, certainly at capital
T, and therefore the price at time little t is
just the linear combination of call prices.

OK, so let's assume that. And then obviously,
continuous function can be arbitrarily
well approximated by piecewise linear function. Therefore, any
function at time– any function of this
form– a derivative when compared to that
form can be replicated by a portfolio of call options. So we can sort of hand
wave to kind of say, this must be true– the
calls are a spanning set. There's another way to
look at it, which is– I just– like from calculus, where
we can actually make explicit what this spanning– what
this portfolio of calls looks like in the arbitrary case. So let me just do that. So you can sort of see, there
must be a linear combination by this for a piecewise linear. Therefore in the limit,
any continuous function must be able to be
replicated by calls.

How many of each? Well, there's actually a
very, very simple result. That is as follows–
and, well, let's just write down an exact Taylor
series to the second order. So this is– so for any
function with second derivative, let's just write down a Taylor
series– the first two terms. And let's put the second
term– we can just do an exact
second-order term, so 0 to infinity x minus c plus
g double prime of c dc. c is my dummy variable. Actually, I've gone
to plus notation. Here's the max of this and 0. OK, that's an exact
Taylor series, true for any– it's
not an approximation. That's exact. You just integrate the
right-hand side by parts if you want to verify it. Maybe it's obvious
to you, but I'm so used to just doing
non-exact Taylor series.

So this is the second order. So this holds for any g exactly. And now I'm just going to
make one little change, which sort of might make obvious
what we're trying to do. I'm just going to take this
dummy variable c, which we're integrating over from 0 to
infinity, and just call it K. We can certainly do that. All right, this now looks
like the payout of a call. It's the payout
of the call price. Now, I don't want
to be integrating. Remember, if I want to
actually get the call price, I take the expected
value of this. I integrate x over x with
respect to its density. This is g of a
payout function of x. Here I'm integrating
over K, so I'm doing something a bit different. But this is the
call option payout. So this holds. It's a linear
equation, obviously. And of course, expectation
is a linear operator. So I'm just going to take,
well, what are the two steps? First of all, I'm
just going to replace x with my random variable
S sub T.

So that I can do. This also holds. And formally, of course, S
sub T is a random variable, so it's a function from the
sample space of the real line. But this holds for every
point on the sample space. So I can write
down this equation between random variables. Here it's just the
integral over dK. So that holds. Now I'm going to take
the expectation operator. So take discounted expected
value, of each side. So in other words, what is
my operator [INAUDIBLE]? It looks like Z(t, T),
expected value of, given S_t. All right? OK, so this one is a
discounted expected value. That's the price. So this becomes price of
the derivative with payout at maturity g. All right, what do we have here? Well, first we've
got a constant.

So we've got a constant times–
OK, so that's a constant. OK, now we've got the
discounted expected stock price. A little bit of
thought on the terms of the Fundamental
Theorem will show you that the discounted expected
stock price under this operator is the current stock price. It's actually non-trivial, but
just think of the stock itself as a derivative,
with the payout S, and apply the
Fundamental Theorem. This has to be the case,
because a replicating portfolio of the stock is just a
holding of the stock. Plus– and then we
just take the integral.

So the expectation inside the
integral– OK, so now I've got discounted expected
payout of this. And the discounted
expected payout of this is just the call
price, with strike K. OK, so I really
like this formula. In some sense, there's nothing
too complicated about how to derive it. But it says explicitly now,
how do I replicate an arbitrary derivative product with payout
g of x or g of S at maturity? Well, it's clear. I replicate it by g(0)
zero-coupon bonds.

So I have g(0) of
zero coupon bonds. That's this. I have g prime zero of
stock– that's this. And I have this linear
combination of calls. So there– this kind
of makes sense, right? You want the zero-coupon
bond amount is just the intercept of g. The number of stocks is
just the slope of g at 0. And then I have this linear
combination of call prices.

I've just proved that by
taking this, and taking expected values. So this is sort of looking at
the duality of option prices and probabilities
in different ways. But then, also how
calls span everything. So the calls, in some sense,
are the primitive information. Once I know all
call option prices, I know the probability
distribution exactly. So there are a couple of sort
of interesting further questions you might want to pose. We seem to have
done everything here with regard to the
distribution at time capital T. And that's true. I know all the calls. I know the distribution
at time capital T. I know all the calls. I know the price of any option
with a payout defined solely by a function at capital T.

But I said nothing
about the path that takes the stock from
today until capital T. So I'm just going to leave you
with two things to think about. Actually, it's one
thing to think about. Two people thought about a lot. And it's the following
question, which now we'll start transitioning into
stochastic calculus, and stochastic
processes a little bit. So we know– let's
just imagine two times. So suppose we know– so we
know the set of all call prices with maturity T_1, for all
K, and the set of all call prices with maturity
T_2 for all K. OK, so then we know
the distribution. Well, there are
two distributions. We know the distribution
of T_1 given S_t, and– but do we know
the distribution of the stock at T_2 given T_1? More of a general point–
suppose I know this for all T.

Let's put T_0 here. OK, I know all option prices of
all maturities and all strikes. Can I determine the stochastic
process for S_T over this time? Is the underlying stochastic
process for the stock price fully determined by
knowing all call option prices for all strikes
and all maturities? The marginal distributions or
the conditional distributions for all maturities
are determined, because we know that here. Well, you'll
probably see this is a rephrasing of a
finite-dimensional problem from probability. The answer is no. And the reason to
think about is, if I know all the–
my intuition for this is if I know all the
distributions that– think about just a denser
and denser grid of times that I know the distribution
of– getting closer and closer.

I can still allow the stock to
flip instantaneously quickly. Imagine they're all essentially
symmetric distributions, and they're all roughly
the same expanding out. I can imagine the stock
flipping discontinuously over an arbitrarily
small time interval. So without a constraint on the
continuity of this process, or mathematical constraints
on this process, you can't determine the
actual process for the stock, even given all the option
prices– call option prices. So there are two–
so Emanuel Derman, who was at Goldman Sachs,
now at Columbia– and Bruno Dupire– who's, I think,
still at Bloomberg– this is the early '90s–
basically determined the conditions that you need.

And the basic conditions
are that just the stock has to be a diffusion process. If it is a diffusion
process– the random term is Brownian motion– then it is,
actually, fully determined. And it's a really
nice, elegant result. So this is what gets
mathematically quite nice, and a little tricky. But there's a practical
implication of this, as well, which is in practice,
I will know a finite subset of call options. Those prices will be
available to me in the market. So they will be given. So one thing I know
for sure is that even with a very densely set
of call option prices, there will be some
other derivative prices whose price is not
exactly determined by that set of calls.

Because in particular, I
know that the set of calls does not determine the
underlying stochastic process, even if I knew all of them. So that's a very important
thing for traders to understand, is that even if I know a lot
of market information– so I'm given here are the
prices of a large number of European options, European
call options I can trade– there may be a complex or
nonstandard derivative product, whose price is not
determined uniquely, simply by knowing those options. And that is one
of the challenges for some of the quant groups. So anyway, with that, that
is all I wanted to convey. I'm happy to take
some questions. And thank you for your time. Thank you for having me. I appreciate it. [APPLAUSE] AUDIENCE: Yeah, I
have– I was just wondering, so you the call,
or the set of all calls basically spans the space of
all possible payouts, right? STEPHEN BLYTHE: Yes. AUDIENCE: So I
was just wondering if maybe if we could change,
and select some other such basis for spanning it? Instead of call,
maybe some other kind of basic payoff that could
still span the same thing, and maybe it's more easily
tradable, or something? STEPHEN BLYTHE: Yeah, that's
a good– there must be many, if I can– but this,
given that this is the simplest expansion
of the function g, an arbitrary function g,
and the second term comes in with this call payout,
gives us this elegance.

Of course, if I know
all the digitals, I know the cumulative
distribution function, and therefore, I
know the density. So I mean, the
digitals do the same. And in fact,
Arrow-Debreu securities, which is building blocks, which
is something that pays off one in a particular
state, sample state, also are building blocks. AUDIENCE: [INAUDIBLE]. STEPHEN BLYTHE: I
mean, sometimes, you could think about an
arbitrary basis that will span– an arbitrary
basis of functions that will span any continuous function. And sometimes, you can do it
in any polynomial expansion. If I have a price and
any of those payouts, and I've got my spanning set. But this is the
most elegant one. Yeah, next question there. AUDIENCE: I have a question
about the last [INAUDIBLE] mentioned. [INAUDIBLE] because
market's incomplete, so you can not sort
of use call option to replicate the stock itself. STEPHEN BLYTHE: You
can use a call option to replicate a stock.

As long as you have zero-coupon. You can see from here, I can
just reorganize everything here to zero-coupon bond
stock, and a set of calls will span anything– with
maturity T. What they're sort of saying is, if I
have this strange process with jumps, and flips,
and discontinuities, then the market is incomplete,
I guess is what this is saying. AUDIENCE: OK, yeah,
so [INAUDIBLE] is due to the incompleteness. STEPHEN BLYTHE:
Yeah, in the sense of most finance– in fact,
all continuous-time finance will assume there's
some diffusion process for– some
process for stock, which has some Brownian motion.

There's some function
here, and some function for the drift term. In that case, then all the
call prices do determine. If you think there's some
exogenous flipping parameter– that's my intuition for it. So there's some– that's
why this is incomplete. So this will not determine. So in particular, I could
know all these call prices. Then I could determine a
particular derivative product. It could be the
number of times that in an arbitrarily
small interval, the stock flips this many times. I mean, there's some–
you can create whatever you like for a
derivative that would be incomplete for these calls.

AUDIENCE: So in this case, go
back to a previous question as we just mentioned– the
second-order derivative of a call option with
respect to a strike is [INAUDIBLE]
risk-neutral density. So in this case, it was not–
that risk-neutral density, or a particular
instance of that, rather, is not
uniquely determined. STEPHEN BLYTHE: No, the
risk-neutral density is uniquely determined. The stochastic process
for S_t over all time is not uniquely determined. So this is uniquely determined
by call option prices. That is uniquely determined. But knowing the
conditional distribution of S capital T given S
little t doesn't determine the process of the stock price. To get there– I can think
of infinitely many processes of the stock price that can
give rise to this distribution. That's what's not determined. The terminal distribution
is uniquely determined by the call option
prices– nothing else. AUDIENCE: So in this case,
if we take Z over theta, so we'll get a particular
risk-neutral density for each particular stock? STEPHEN BLYTHE: That's correct.

Right, thank you very much. Appreciate it..

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