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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.
