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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..