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PROFESSOR: All right,
OK, let's get started.
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00:00:26,390 --> 00:00:28,450
So before I make
introduction, let me just
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00:00:28,450 --> 00:00:30,670
make a few announcements.
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00:00:30,670 --> 00:00:37,000
A few of you came to us asking
about the grading for the term.
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00:00:37,000 --> 00:00:41,520
And some feel the problem sets
may be on the difficult side,
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00:00:41,520 --> 00:00:45,670
and some of you haven't done
all of them, and some of you
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00:00:45,670 --> 00:00:46,970
have done more.
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00:00:46,970 --> 00:00:50,720
So we just want to let you know
that the most important thing
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00:00:50,720 --> 00:00:56,280
to us in grading is really
you show your effort in terms
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00:00:56,280 --> 00:00:57,220
of learning.
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00:00:57,220 --> 00:00:59,280
And we purposely
made the problem
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00:00:59,280 --> 00:01:01,740
sets more difficult
than the lecture,
20
00:01:01,740 --> 00:01:05,080
so you can-- if you want to
dig in deeper so you have
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00:01:05,080 --> 00:01:06,820
the opportunity to learn more.
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00:01:06,820 --> 00:01:11,310
But by no means we
expect you to finish
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00:01:11,310 --> 00:01:17,980
or feel easy in solving
all the problem sets.
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00:01:17,980 --> 00:01:20,390
So I just want to
put you at ease
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00:01:20,390 --> 00:01:24,080
that if that's your
concern, that's definitely--
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00:01:24,080 --> 00:01:25,610
you don't need to
worry about it.
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00:01:25,610 --> 00:01:31,800
And we will be really just
evaluating your effort.
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00:01:31,800 --> 00:01:35,260
And based on what do
we observed so far,
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00:01:35,260 --> 00:01:38,770
we actually believe every single
one of you is doing quite well.
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00:01:38,770 --> 00:01:44,060
So you shouldn't worry about
your performance at the class.
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00:01:44,060 --> 00:01:49,120
So continue to do a good job
on your class participation,
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00:01:49,120 --> 00:01:51,250
and do some of the problem sets.
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00:01:51,250 --> 00:01:53,970
And then you will be in
fine shape for your grade.
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00:01:53,970 --> 00:01:56,130
So that's all of that.
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00:01:56,130 --> 00:01:58,340
So without any
further delay, let
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00:01:58,340 --> 00:02:02,480
me introduce my colleague,
Doctor Stephen Blythe.
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00:02:02,480 --> 00:02:03,720
I'll be very brief.
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00:02:03,720 --> 00:02:09,160
And he's-- Stephen is doing
two jobs at the same time.
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00:02:09,160 --> 00:02:14,430
He's responsible for the all
the public markets at Harvard
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00:02:14,430 --> 00:02:19,100
Management, as well as being
a professor of practice
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00:02:19,100 --> 00:02:20,070
at Harvard.
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00:02:20,070 --> 00:02:21,962
So with that, I turn to Stephen.
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00:02:21,962 --> 00:02:24,300
STEPHEN BLYTHE: OK, well,
thank you, and thank you
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for having me to
speak this afternoon.
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00:02:26,720 --> 00:02:33,640
Before I begin, I wanted
to ask you a question.
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00:02:33,640 --> 00:02:36,830
So I'm speaking, actually,
at almost exactly
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00:02:36,830 --> 00:02:39,640
the 20th anniversary of
something very important.
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So on the 19th of October,
1993, which I guess
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00:02:44,660 --> 00:02:46,680
might be the birthday
of some of you,
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but almost exactly 20 years
ago, Congress voted 264 to 159--
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I actually remember the count
of the vote-- to do something.
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00:02:58,340 --> 00:03:01,820
So anybody like
to guess what they
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00:03:01,820 --> 00:03:05,190
voted to do on the 19th
of October, 1993 that
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might be tangentially
relevant to finance
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and quantitative finance?
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Anyone here from HMC is
not allowed to answer.
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Anybody-- any guesses?
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Any ideas at all coming
to people's minds?
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AUDIENCE: Was it
Gramm-Leach-Bliley?
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STEPHEN BLYTHE: No.
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AUDIENCE: Commodity
Futures Modernization Act?
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STEPHEN BLYTHE:
No, but good guess.
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But actually, that is actually
too related to finance,
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actually.
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This is actually--
wasn't actually
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directly financially
related, so that
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was related to [INAUDIBLE].
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Anybody else think about?
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What does Congress usually do?
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AUDIENCE: [INAUDIBLE].
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[LAUGHTER]
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STEPHEN BLYTHE: No,
no ideas whatsoever?
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What do you think
Congress did 20 years ago?
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They voted to do something.
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OK, well, what
Congress do usually
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is they cut money for something.
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So they voted to cut
financing to something.
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So what did they
cut financing to?
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Anybody guess?
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I know this isn't
business school.
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In business school, it would
be, like, right, you're failed.
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No class participation--
you failed.
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00:04:20,100 --> 00:04:22,190
You've got to say something
in business school.
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So I know it's not
business school.
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But anyway-- and I don't teach
in business school, either.
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But this is
actually-- these round
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desks make me think
of business school
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and striding into the
middle of the room,
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and saying OK, come on.
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Fortunately, I don't have names,
otherwise I'd pick on you.
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00:04:36,710 --> 00:04:39,784
No, no guesses-- no
guesses whatsoever?
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00:04:39,784 --> 00:04:41,950
I've got to take this up
the road to Harvard Square,
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and say I've taught at MIT.
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00:04:43,800 --> 00:04:46,140
No one had any guesses with
this question-- one guess,
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00:04:46,140 --> 00:04:47,910
actually, the gentleman here.
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What did they cancel the
financing for in 1993?
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I'll say it was the
Superconducting Super
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Collider underneath Texas
just south of Dallas.
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So $2 billion had been
spent on the Super Collider.
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And the budget had
expanded from, I think,
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$6 to $11 billion.
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So they, by canceling, had
a $9 billion dollar savings.
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This is 20 years
ago-- almost exactly.
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00:05:14,990 --> 00:05:17,440
And as a result of that--
one result of that-- was,
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of course, the academic
job market for physicists
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00:05:20,390 --> 00:05:22,700
collapsed overnight.
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00:05:22,700 --> 00:05:26,360
And two of my roommates
were theoretical physicists
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at Harvard.
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00:05:27,470 --> 00:05:30,990
And they basically realized
their job prospects in academia
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had vaporized
instantaneously that day.
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And both of them,
within six months,
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00:05:36,530 --> 00:05:40,820
had found jobs with
Goldman Sachs in New York.
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And they catalyzed they--
they and the cohort--
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they're called the
Superconducting Super Collider
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00:05:46,740 --> 00:05:47,940
generation.
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If you ever wondered why people
like myself and like Jake got
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PhDs in quantitative subjects
around the turn of-- around
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00:05:54,680 --> 00:05:59,070
1990 to 1993-- all ended
up in a financial path,
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part of it is due to Congress
cancelling the Superconducting
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00:06:01,630 --> 00:06:02,254
Super Collider.
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00:06:02,254 --> 00:06:04,720
Because this cohort
catalyzed this growth
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in quantitative finance.
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00:06:06,310 --> 00:06:09,680
Actually, they created a
field-- financial engineering--
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which you are all somewhat
interested in by taking
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this class.
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And they also created a career
path-- quantitative analyst,
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or quant, which really
did not exist before 1993.
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00:06:21,790 --> 00:06:26,485
And that growth of
mathematical finance,
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00:06:26,485 --> 00:06:28,360
financial engineering,
quantitative finance--
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00:06:28,360 --> 00:06:29,900
however you want
to look at it-- was
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basically exponential
from 1993 up
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00:06:32,460 --> 00:06:37,610
until 2008 and the financial
crisis exactly five years ago,
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00:06:37,610 --> 00:06:38,840
funnily enough.
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And since then, it's been
a little bit rockier.
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So if you're actually
interested in this aftermath
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of the physics funding-- what's
interesting is the Large Hadron
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Collider, which you might know
is up and running in Geneva
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00:06:48,980 --> 00:06:51,720
and just found the
Higgs Boson, has sort of
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reversed the trend somewhat.
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00:06:53,280 --> 00:06:54,990
So there used to
be a whole cohort
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of people going into
finance instead of physics.
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Now, because finance has this
somewhat pejorative nature
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00:07:00,360 --> 00:07:02,780
to it-- people don't
like bankers generally,
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00:07:02,780 --> 00:07:05,170
and they kind of like physicists
who find the Higgs Boson
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00:07:05,170 --> 00:07:07,690
and get a Nobel Prize--
maybe we're getting reversal.
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00:07:07,690 --> 00:07:10,190
But anyway, we're
still in finance.
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00:07:10,190 --> 00:07:14,640
I've, as Jake mentioned,
well, I did mention,
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I was originally in academics.
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I was actually a mathematics
faculty member in London
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when I got my PhD-- I
got my PhD from Harvard.
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And in 1993, I was an academic.
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And all my friends-- I
saw them go to finance.
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So I followed them, spent
a career in New York,
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00:07:27,940 --> 00:07:31,120
and then came back
to Harvard in 2006
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to run a part of the endowment.
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00:07:32,790 --> 00:07:34,580
And I started teaching.
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So just as a plug--
for those of you
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interested in mathematical
finance and applications
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00:07:39,870 --> 00:07:42,320
of mathematics finance, I
teach a course at Harvard.
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It's an upper level
undergraduate course
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called Applied Quantitative
Finance, which of course you
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00:07:47,310 --> 00:07:49,770
can cross-register for.
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00:07:49,770 --> 00:07:53,130
And today is also the
one-week anniversary
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of the publication of my book.
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So if you're interested in
what my course is about,
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you can just buy my book.
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00:08:00,070 --> 00:08:01,612
It's only $30.
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00:08:01,612 --> 00:08:02,320
And I'll sign it.
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00:08:02,320 --> 00:08:04,500
It's first edition,
first printing,
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00:08:04,500 --> 00:08:06,410
first impression
book, Introduction
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00:08:06,410 --> 00:08:09,270
to Quantitative Finance.
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00:08:09,270 --> 00:08:10,890
And that is what the course is.
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It's quite distilled.
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00:08:12,240 --> 00:08:15,470
When this book came out, I
thought, that's really thin.
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00:08:15,470 --> 00:08:17,639
This is three years
of my life's work.
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00:08:17,639 --> 00:08:18,930
It's come out-- it's very thin.
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00:08:18,930 --> 00:08:20,513
But I like to think
it's like whiskey.
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00:08:20,513 --> 00:08:23,140
It's well distilled,
and highly potent,
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00:08:23,140 --> 00:08:25,400
and you have to sip it,
and take it bit by bit.
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00:08:25,400 --> 00:08:27,910
Anyway, that is the
book of my class.
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00:08:27,910 --> 00:08:35,090
And the genesis of
the class was really
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00:08:35,090 --> 00:08:37,162
that, when I've
been on Wall Street,
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00:08:37,162 --> 00:08:38,870
and I was a colleague
of Jake's at Morgan
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00:08:38,870 --> 00:08:46,460
Stanley in this rapidly growing
quantitative finance field,
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00:08:46,460 --> 00:08:48,870
we encountered on the trading
desk in the late 1990s
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00:08:48,870 --> 00:08:53,820
and the early 2000s problems
from the real economy-- things
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00:08:53,820 --> 00:08:55,050
that we had to trade.
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00:08:55,050 --> 00:08:57,880
We were-- things that were
coming to us on the trading
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00:08:57,880 --> 00:09:03,200
desk that required subtle
understanding of the underlying
190
00:09:03,200 --> 00:09:03,980
theory.
191
00:09:03,980 --> 00:09:09,929
So that we, in essence, we
built theoretical framework
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00:09:09,929 --> 00:09:11,470
to solve the problems
that were given
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00:09:11,470 --> 00:09:13,480
to us by the financial markets.
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00:09:13,480 --> 00:09:17,150
So that period, especially
around the turn of the century,
195
00:09:17,150 --> 00:09:20,750
there's a big growth in
derivatives markets, which--
196
00:09:20,750 --> 00:09:25,460
options, futures,
forwards, et cetera, swaps.
197
00:09:25,460 --> 00:09:27,426
And we needed to build
theoretical tools
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00:09:27,426 --> 00:09:28,050
to tackle them.
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00:09:28,050 --> 00:09:31,640
And that's really what the
course was evolved out of,
200
00:09:31,640 --> 00:09:34,090
to build the appropriate
theoretical framework,
201
00:09:34,090 --> 00:09:37,040
motivated by the
problems we encountered.
202
00:09:37,040 --> 00:09:40,694
Why I enthuse
about the subject--
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and I really like
teaching the subject--
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00:09:42,360 --> 00:09:46,052
is that there is an impression
that qualitative finance is
205
00:09:46,052 --> 00:09:49,720
a very arcane and contrived
subject-- just a whole
206
00:09:49,720 --> 00:09:52,799
bunch of PhDs on Wall Street
coming up with crazy ideas.
207
00:09:52,799 --> 00:09:54,340
And they need
complicated mathematics
208
00:09:54,340 --> 00:09:56,620
that's just complicated
for the sake of complexity.
209
00:09:56,620 --> 00:09:59,570
And the theory is just
sort of a contrived theory.
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00:09:59,570 --> 00:10:02,400
But in fact, at the
heart of Wall Street
211
00:10:02,400 --> 00:10:05,275
is that the real economy
demands some of these products
212
00:10:05,275 --> 00:10:06,150
by supply and demand.
213
00:10:06,150 --> 00:10:07,960
There are actual,
real participants
214
00:10:07,960 --> 00:10:09,790
in the financial markets who
want to trade derivatives.
215
00:10:09,790 --> 00:10:11,500
And therefore, in order
to understand them,
216
00:10:11,500 --> 00:10:12,708
you need to develop a theory.
217
00:10:12,708 --> 00:10:15,420
So it's actually driven
by real examples.
218
00:10:15,420 --> 00:10:16,860
That's one part.
219
00:10:16,860 --> 00:10:19,320
The other part is that
the theory that comes out
220
00:10:19,320 --> 00:10:21,810
of it, and in particular the
approach I take here, I think
221
00:10:21,810 --> 00:10:23,234
is just very elegant.
222
00:10:23,234 --> 00:10:25,400
OK, so there's some subtlety
and elegance and beauty
223
00:10:25,400 --> 00:10:28,410
to the underlying
theory that comes out
224
00:10:28,410 --> 00:10:30,870
of addressing real problems.
225
00:10:30,870 --> 00:10:34,830
This course, and the way
that I teach finance,
226
00:10:34,830 --> 00:10:37,340
is very probability centric.
227
00:10:37,340 --> 00:10:40,220
You probably realize from the
lectures you've seen already
228
00:10:40,220 --> 00:10:43,350
in this class, there are
many different approaches,
229
00:10:43,350 --> 00:10:46,350
many different methods that are
used in finance-- stochastic
230
00:10:46,350 --> 00:10:48,640
calculus, partial differential
equations, simulation,
231
00:10:48,640 --> 00:10:50,170
and so on.
232
00:10:50,170 --> 00:10:53,140
The classical derivation
of Black-Scholes
233
00:10:53,140 --> 00:10:56,700
is, well, it's the
solution of the PDE.
234
00:10:56,700 --> 00:10:59,374
OK, that has appealed to people.
235
00:10:59,374 --> 00:11:00,790
In fact, this is
why in some ways,
236
00:11:00,790 --> 00:11:02,414
quantitative finance
is a broad church,
237
00:11:02,414 --> 00:11:05,280
because whether you're a
physicist, or probabilist,
238
00:11:05,280 --> 00:11:08,910
or a chemical engineer, all
the techniques you learn
239
00:11:08,910 --> 00:11:09,652
can be applied.
240
00:11:09,652 --> 00:11:10,860
You know stochastic calculus.
241
00:11:10,860 --> 00:11:12,610
You know differential equations.
242
00:11:12,610 --> 00:11:13,900
They can be applied.
243
00:11:13,900 --> 00:11:15,940
But the path that I
take in this class
244
00:11:15,940 --> 00:11:19,770
is very much through the
probabilistic route, which
245
00:11:19,770 --> 00:11:23,580
is my background as
a probabilist, as
246
00:11:23,580 --> 00:11:25,800
an academic, or a
statistician as an academic.
247
00:11:25,800 --> 00:11:28,905
And this is, in particular,
I think, a very elegant path
248
00:11:28,905 --> 00:11:30,770
to understand finance,
and the linkage
249
00:11:30,770 --> 00:11:32,480
between derivative
products-- which
250
00:11:32,480 --> 00:11:34,910
might seem contrived-- and
probability distributions,
251
00:11:34,910 --> 00:11:37,250
which is sort of natural
things for probabilists.
252
00:11:37,250 --> 00:11:39,810
So this, what we're going
to talk about today,
253
00:11:39,810 --> 00:11:44,340
is really this
link, which I call
254
00:11:44,340 --> 00:11:52,870
option-probability duality.
255
00:11:52,870 --> 00:11:58,660
Which, in essence, in the
simplest form, is just saying,
256
00:11:58,660 --> 00:12:00,970
option prices-- they're just
probability distributions.
257
00:12:00,970 --> 00:12:03,199
Therefore, these
complicated derivatives
258
00:12:03,199 --> 00:12:04,990
that people talk about--
all these options,
259
00:12:04,990 --> 00:12:06,906
these financial engineers,
these quants, these
260
00:12:06,906 --> 00:12:09,130
exotics-- we're really just
talking about probability
261
00:12:09,130 --> 00:12:09,713
distributions.
262
00:12:09,713 --> 00:12:12,680
We can go between them--
option prices, probabilities,
263
00:12:12,680 --> 00:12:15,600
and distributions-- back and
forth in a very elegant way.
264
00:12:15,600 --> 00:12:18,180
What I love about this
subject in particular
265
00:12:18,180 --> 00:12:22,070
is that to get to that point
where we see this duality does
266
00:12:22,070 --> 00:12:27,310
not need a whole framework
and infrastructure
267
00:12:27,310 --> 00:12:32,679
of complicated definitions,
or formulae, or option pricing
268
00:12:32,679 --> 00:12:33,470
formulae, or so on.
269
00:12:33,470 --> 00:12:36,080
So that's what I'm going to
try and do in this hour or so,
270
00:12:36,080 --> 00:12:39,750
is introduce this concept
of option price, probability
271
00:12:39,750 --> 00:12:40,380
duality.
272
00:12:40,380 --> 00:12:42,974
And show how the
natural-- so there's
273
00:12:42,974 --> 00:12:44,390
a natural duality
that can be seen
274
00:12:44,390 --> 00:12:45,860
in a number of different ways.
275
00:12:45,860 --> 00:12:49,870
OK, so we're going to need
a few definitions that
276
00:12:49,870 --> 00:12:52,137
should be familiar to you.
277
00:12:52,137 --> 00:12:53,595
We're going to
define three assets.
278
00:12:56,300 --> 00:13:00,306
We have a call option,
which we know about,
279
00:13:00,306 --> 00:13:06,659
a zero-coupon bond--
called a zed cee bee.
280
00:13:06,659 --> 00:13:08,950
This is the one thing I
haven't become Americanized on.
281
00:13:08,950 --> 00:13:10,400
I still call this zed.
282
00:13:10,400 --> 00:13:13,410
It's a-- other
things I've become--
283
00:13:13,410 --> 00:13:15,480
and then a digital option.
284
00:13:19,190 --> 00:13:21,830
OK, all right, so what are they?
285
00:13:21,830 --> 00:13:24,880
Well, they're all going to
be defined by their payouts
286
00:13:24,880 --> 00:13:26,260
at maturity.
287
00:13:26,260 --> 00:13:29,180
OK, so we're going to have
some maturity capital T,
288
00:13:29,180 --> 00:13:33,700
and some underlying asset, S,
the stock, with some price S_T.
289
00:13:33,700 --> 00:13:36,770
OK, so we know that the call
option has payout at T--
290
00:13:36,770 --> 00:13:41,260
So that's called
payout at T. So T is
291
00:13:41,260 --> 00:13:43,090
some fixed time in the future.
292
00:13:43,090 --> 00:13:45,930
We will change in the
future to some fixed time.
293
00:13:45,930 --> 00:13:53,940
This is simply the max
of S_T minus K and 0.
294
00:13:53,940 --> 00:13:56,830
That's a call option.
295
00:13:56,830 --> 00:14:00,021
You can go through the right
to buy, et cetera, et cetera.
296
00:14:00,021 --> 00:14:01,770
But it's clear it's
just value at maturity
297
00:14:01,770 --> 00:14:05,530
is just the max of
S_T minus K and 0.
298
00:14:05,530 --> 00:14:09,450
The zero-coupon
bond with maturity T
299
00:14:09,450 --> 00:14:11,840
is just something that's
worth 1 at time T.
300
00:14:11,840 --> 00:14:15,181
So that's just payout one.
301
00:14:15,181 --> 00:14:16,680
That's definition--
so you can think
302
00:14:16,680 --> 00:14:18,760
of these all as definitions.
303
00:14:18,760 --> 00:14:21,220
And then the digital
option is just
304
00:14:21,220 --> 00:14:31,760
the indicator function of S_T
being greater than K. So here,
305
00:14:31,760 --> 00:14:33,170
T is the maturity.
306
00:14:33,170 --> 00:14:36,055
K is the strike.
307
00:14:36,055 --> 00:14:41,510
So T maturity, K is strike.
308
00:14:44,540 --> 00:14:46,920
And these are three assets.
309
00:14:46,920 --> 00:14:49,540
So this is, in some sense,
the payout function.
310
00:14:49,540 --> 00:14:51,650
All derivative
products can be defined
311
00:14:51,650 --> 00:14:54,150
in terms of a function--
not all of them.
312
00:14:54,150 --> 00:14:56,170
Many derivative products
can be defined just
313
00:14:56,170 --> 00:14:57,620
as a function of S_T.
314
00:14:57,620 --> 00:15:00,870
And here are three functions
of S_T. [INAUDIBLE]
315
00:15:00,870 --> 00:15:06,320
And then I'm just going to get
notation for the price at t
316
00:15:06,320 --> 00:15:08,760
less than or equal to T.
We can think about little t
317
00:15:08,760 --> 00:15:10,600
as current time
today, or we can think
318
00:15:10,600 --> 00:15:13,460
of some future time
between now and capital
319
00:15:13,460 --> 00:15:18,350
T. I'm just going to
introduce notation.
320
00:15:18,350 --> 00:15:20,760
Every different finance book
uses different notation, so
321
00:15:20,760 --> 00:15:27,990
just C for call price,
with strike K, at little t
322
00:15:27,990 --> 00:15:32,100
with maturity big T.
OK, just that notation.
323
00:15:32,100 --> 00:15:37,160
The zero-coupon bond--
the price at little t--
324
00:15:37,160 --> 00:15:39,810
let's call that Z. That's
the price of little t.
325
00:15:39,810 --> 00:15:44,110
And the digital--
we'll just call that D.
326
00:15:44,110 --> 00:15:48,042
So this is what we're
going to set this up.
327
00:15:48,042 --> 00:15:50,000
Actually, you could have
a whole lecture on why
328
00:15:50,000 --> 00:15:51,930
notation-- different notation.
329
00:15:51,930 --> 00:15:54,690
K and capital T are
actually embedded
330
00:15:54,690 --> 00:15:56,410
in the terms of the contract.
331
00:15:56,410 --> 00:15:58,050
Little t is in my calendar time.
332
00:15:58,050 --> 00:16:00,350
So you might think why
don't you put K and capital
333
00:16:00,350 --> 00:16:01,714
T somewhere else?
334
00:16:01,714 --> 00:16:03,880
Well, when you get actually
to modeling derivatives,
335
00:16:03,880 --> 00:16:06,920
you like to be moving both
maturity and a forward time
336
00:16:06,920 --> 00:16:07,740
and calendar time.
337
00:16:07,740 --> 00:16:09,281
That's why I just
write it like that.
338
00:16:09,281 --> 00:16:12,440
But there's no-- so
C sub K, little t,
339
00:16:12,440 --> 00:16:14,380
big T is the price
at time of little t
340
00:16:14,380 --> 00:16:19,410
of a call with maturity
capital T and strike K.
341
00:16:19,410 --> 00:16:22,100
Black-Scholes and other
option pricing formula
342
00:16:22,100 --> 00:16:28,380
are all about determining
this-- for t less than T.
343
00:16:28,380 --> 00:16:33,190
Because clearly we know
that the price at maturity
344
00:16:33,190 --> 00:16:34,250
is simply the payout.
345
00:16:34,250 --> 00:16:38,200
I mean, that's, again,
just the definition.
346
00:16:38,200 --> 00:16:39,130
So that's trivial.
347
00:16:39,130 --> 00:16:43,280
But we want to find out what
the price is at little t.
348
00:16:43,280 --> 00:16:45,990
So that's the whole path
of finance-- Black-Scholes
349
00:16:45,990 --> 00:16:48,130
and other option
pricing methodology
350
00:16:48,130 --> 00:16:49,631
is working out this.
351
00:16:49,631 --> 00:16:51,880
But we're actually going to
go down a different route.
352
00:16:54,105 --> 00:16:55,480
So what we're
going to do-- we're
353
00:16:55,480 --> 00:16:59,520
going to construct a portfolio.
354
00:16:59,520 --> 00:17:07,020
So consider as a
portfolio of what?
355
00:17:07,020 --> 00:17:09,660
We're going to
consist of two calls.
356
00:17:09,660 --> 00:17:17,770
OK, we're going to have lambda
calls with strike K. OK,
357
00:17:17,770 --> 00:17:19,020
so this is the amount holding.
358
00:17:19,020 --> 00:17:22,119
And everything is going to
be with maturity capital T.
359
00:17:22,119 --> 00:17:24,810
So lambda calls with
strike K, and maturity T,
360
00:17:24,810 --> 00:17:32,740
and minus lambda calls with
strike K plus 1 over lambda.
361
00:17:35,852 --> 00:17:37,310
We'll just consider
that portfolio.
362
00:17:37,310 --> 00:17:40,960
It consists of two options.
363
00:17:40,960 --> 00:17:48,620
All right, well, this
price at T-- that's easy.
364
00:17:48,620 --> 00:17:54,200
We just write it
in terms of lambda
365
00:17:54,200 --> 00:17:57,660
times the price of the
call with strike K,
366
00:17:57,660 --> 00:18:06,390
minus lambda call with strike
K plus 1 over lambda-- just
367
00:18:06,390 --> 00:18:07,230
by definition.
368
00:18:07,230 --> 00:18:09,920
This is price at T.
369
00:18:09,920 --> 00:18:16,670
OK, well, let's look at
its payout at time capital
370
00:18:16,670 --> 00:18:17,495
T graphically.
371
00:18:20,090 --> 00:18:23,250
So we know about call options.
372
00:18:23,250 --> 00:18:27,130
The payout function is just the
hockey stick shape, clearly.
373
00:18:27,130 --> 00:18:29,840
That's confusing to people
from the UK, because in the UK,
374
00:18:29,840 --> 00:18:32,322
hockey means field
hockey, not ice hockey.
375
00:18:32,322 --> 00:18:34,530
And of course, the hockey
stick shape in field hockey
376
00:18:34,530 --> 00:18:35,405
looks very different.
377
00:18:35,405 --> 00:18:38,160
Anyway, that's-- you understand
what the payout of a call is.
378
00:18:38,160 --> 00:18:40,860
Clearly, this payout function
of a call looks like this.
379
00:18:44,910 --> 00:18:51,280
Well, putting this
payout of lambda calls
380
00:18:51,280 --> 00:18:53,030
of strike K minus
lambda calls of strike K
381
00:18:53,030 --> 00:18:55,105
plus 1 over lambda--
let's assume lambda
382
00:18:55,105 --> 00:18:57,230
is positive for the time being.
383
00:18:57,230 --> 00:18:58,750
What's it look like?
384
00:18:58,750 --> 00:19:06,000
Well, 0 below K, is flat
above K plus 1 over lambda.
385
00:19:06,000 --> 00:19:10,820
It has slope lambda,
and has value 1 here.
386
00:19:13,730 --> 00:19:18,230
You should be able
to see that easily.
387
00:19:18,230 --> 00:19:19,690
So that's the payout.
388
00:19:19,690 --> 00:19:23,670
This is called call spread--
just the spread between two
389
00:19:23,670 --> 00:19:27,390
calls, and has this
payout function.
390
00:19:27,390 --> 00:19:30,590
OK, so a natural
thing to do here,
391
00:19:30,590 --> 00:19:33,020
it being a mathematics
class, let's take limits.
392
00:19:33,020 --> 00:19:34,635
Just let lambda
tend to infinity.
393
00:19:41,310 --> 00:19:44,260
Well, then, this becomes
the partial derivative
394
00:19:44,260 --> 00:19:48,010
of the call price with respect
to K, or the negative of it.
395
00:19:48,010 --> 00:19:49,535
So this tends to minus.
396
00:19:52,700 --> 00:19:59,950
OK, let's just-- so that's that.
397
00:19:59,950 --> 00:20:03,350
And then this, of course
as lambda goes to infinity,
398
00:20:03,350 --> 00:20:04,730
this stays at 1.
399
00:20:04,730 --> 00:20:13,510
So this tends to payout
function that looks like that.
400
00:20:13,510 --> 00:20:14,010
OK?
401
00:20:16,830 --> 00:20:19,420
This is easy calculus.
402
00:20:19,420 --> 00:20:23,740
This is just by inspection.
403
00:20:23,740 --> 00:20:28,140
OK, so this, clearly, is
the payout of the digital.
404
00:20:39,170 --> 00:20:42,120
Of the-- strictly
a digital call,
405
00:20:42,120 --> 00:20:44,010
but that's called
the digital option.
406
00:20:46,560 --> 00:20:49,190
Just as a note, here
it's, just greater than.
407
00:20:49,190 --> 00:20:51,570
You might think, OK, it
doesn't matter if it's
408
00:20:51,570 --> 00:20:53,330
greater than or equal to.
409
00:20:53,330 --> 00:20:57,490
Well, in practice, the chance
of something equalling a number
410
00:20:57,490 --> 00:21:01,960
exactly is 0-- I mean, if it's
a continuous distribution.
411
00:21:01,960 --> 00:21:03,532
In theory, I should
say, the chance
412
00:21:03,532 --> 00:21:05,240
of something actually
nailing the strike,
413
00:21:05,240 --> 00:21:07,490
actually being equal to K,
is 0, so it doesn't really
414
00:21:07,490 --> 00:21:09,750
matter whether you define
this as greater than,
415
00:21:09,750 --> 00:21:11,530
or greater than or equal to.
416
00:21:11,530 --> 00:21:14,970
But in practice, of course,
finance is in discrete time,
417
00:21:14,970 --> 00:21:17,884
because you don't quote things
to a million decimal places.
418
00:21:17,884 --> 00:21:19,300
So certain assets,
actually, which
419
00:21:19,300 --> 00:21:23,430
are quoted only in eighths
or 16ths or 32nds or 64ths,
420
00:21:23,430 --> 00:21:24,931
this matters,
actually, whether it's
421
00:21:24,931 --> 00:21:27,096
defined as greater than or
greater than or equal to.
422
00:21:27,096 --> 00:21:29,320
But theoretically, it
doesn't make any difference.
423
00:21:29,320 --> 00:21:38,120
OK, so we've got the call
spread tending to the digital.
424
00:21:38,120 --> 00:21:46,450
All right, so this tends to--
so the limit of this call
425
00:21:46,450 --> 00:21:52,730
spread-- of this price of the
call spread-- is the digital.
426
00:21:52,730 --> 00:22:04,880
And so we know that because
this is the price at t.
427
00:22:04,880 --> 00:22:08,670
This is the payout at capital
T. The price of the digital
428
00:22:08,670 --> 00:22:11,590
must equal just the partial
derivative with respect
429
00:22:11,590 --> 00:22:13,150
to strike of the call price.
430
00:22:16,330 --> 00:22:21,490
So that's just a nice, little
result. Where does this
431
00:22:21,490 --> 00:22:24,280
bring in probability?
432
00:22:24,280 --> 00:22:27,520
So this is the next.
433
00:22:27,520 --> 00:22:33,830
OK, so this is where
we'll make one assumption.
434
00:22:33,830 --> 00:22:36,520
And it's actually a very
important and fundamental
435
00:22:36,520 --> 00:22:37,855
assumption.
436
00:22:37,855 --> 00:22:39,230
And it's fundamental
because it's
437
00:22:39,230 --> 00:22:42,080
called The Fundamental
Theorem of Finance,
438
00:22:42,080 --> 00:22:46,690
or the Fundamental
Theorem of Asset Prices.
439
00:22:46,690 --> 00:22:51,795
So I call this FTAP--
Fundamental Theorem
440
00:22:51,795 --> 00:22:52,503
of Asset Pricing.
441
00:22:56,780 --> 00:22:59,500
By this theorem, which we
are going to assume here,
442
00:22:59,500 --> 00:23:06,530
the intuitive answer is correct,
meaning that prices today
443
00:23:06,530 --> 00:23:08,290
are expected values.
444
00:23:08,290 --> 00:23:11,580
It's the expectation
of a future payout.
445
00:23:11,580 --> 00:23:26,399
So by FTAP, the price
at t is expected payout
446
00:23:26,399 --> 00:23:31,365
at time capital T,
suitably discounted.
447
00:23:35,250 --> 00:23:40,050
So there's both something
very straightforward here,
448
00:23:40,050 --> 00:23:42,130
and something very deep.
449
00:23:42,130 --> 00:23:44,530
If you think about
how much would
450
00:23:44,530 --> 00:23:48,700
you pay for a contract
that gives you
451
00:23:48,700 --> 00:23:52,940
$1 if an event happens-- in
this case, the event being
452
00:23:52,940 --> 00:23:55,850
stock being greater
than K at maturity.
453
00:23:55,850 --> 00:23:57,620
You would intuitively
think that's related
454
00:23:57,620 --> 00:24:01,270
to the probability of
the event happening.
455
00:24:01,270 --> 00:24:03,610
How much will you
pay to see the dollar
456
00:24:03,610 --> 00:24:04,660
if a coin comes up heads?
457
00:24:04,660 --> 00:24:09,820
You'd pay a half, probably.
458
00:24:09,820 --> 00:24:11,480
It's very, very intuitive.
459
00:24:11,480 --> 00:24:15,550
But the deepness
is, this actually
460
00:24:15,550 --> 00:24:20,370
holds under a particular
probability distribution.
461
00:24:20,370 --> 00:24:23,050
I'm not going to
go into that here,
462
00:24:23,050 --> 00:24:27,640
but by the fundamental
theorem, this is true.
463
00:24:27,640 --> 00:24:30,602
So I can write, in the
case of the digital,
464
00:24:30,602 --> 00:24:39,680
the digital price
equals the discounted--
465
00:24:39,680 --> 00:24:43,430
and we'll explain why we want
to put the zero-coupon bond
466
00:24:43,430 --> 00:24:48,394
price here-- that's the present
value of a dollar at time t.
467
00:24:48,394 --> 00:24:49,560
It's just a discount factor.
468
00:24:49,560 --> 00:24:51,420
It's very trivial,
but it's written
469
00:24:51,420 --> 00:24:55,260
in terms of an asset price--
times the expected value
470
00:24:55,260 --> 00:24:57,400
of the payout.
471
00:25:03,700 --> 00:25:11,200
So either you take this as
this makes a lot of sense--
472
00:25:11,200 --> 00:25:14,030
the discounted expected
payout-- or you can say,
473
00:25:14,030 --> 00:25:15,030
I don't understand this.
474
00:25:15,030 --> 00:25:17,280
I want to find out about the
Fundamental Theorem Asset
475
00:25:17,280 --> 00:25:20,170
Pricing, which we will
prove in my class.
476
00:25:20,170 --> 00:25:22,440
But this intuitively
makes sense.
477
00:25:22,440 --> 00:25:26,630
The key here is that the
expected value actually
478
00:25:26,630 --> 00:25:30,480
has to be taken out under
the appropriate distribution,
479
00:25:30,480 --> 00:25:32,320
called the risk-neutral
distribution.
480
00:25:32,320 --> 00:25:38,660
But this formula holds--
in fact, strictly.
481
00:25:38,660 --> 00:25:41,730
I'll write this
is just for-- what
482
00:25:41,730 --> 00:25:48,400
holds is the price at
time little t divided
483
00:25:48,400 --> 00:25:55,130
by zero-coupon bond is a
martingale-- for those of you
484
00:25:55,130 --> 00:26:00,112
into probability theory.
485
00:26:00,112 --> 00:26:02,070
This gets probabilists
very excited, of course,
486
00:26:02,070 --> 00:26:03,230
because they love martingales.
487
00:26:03,230 --> 00:26:05,490
Everyone in probability
theory loves martingales-- lot
488
00:26:05,490 --> 00:26:07,770
of theorems about martingales.
489
00:26:07,770 --> 00:26:12,740
And you'll see, of course that
this is actually a restatement
490
00:26:12,740 --> 00:26:18,000
of this assertion.
491
00:26:18,000 --> 00:26:22,080
Because Z, capital
T, capital T is 1.
492
00:26:22,080 --> 00:26:26,280
So this statement here
is simply a re-expression
493
00:26:26,280 --> 00:26:28,640
of this martingale condition.
494
00:26:28,640 --> 00:26:29,917
So I'll just pause here.
495
00:26:29,917 --> 00:26:31,500
Just from a probability
point of view,
496
00:26:31,500 --> 00:26:33,833
when I learned probability,
it was under David Williams,
497
00:26:33,833 --> 00:26:36,840
who wrote the book
Probability With Martingales,
498
00:26:36,840 --> 00:26:37,930
which is a wonderful book.
499
00:26:37,930 --> 00:26:39,679
And I thought martingale
is a great thing.
500
00:26:39,679 --> 00:26:40,810
So I was sort of happy.
501
00:26:40,810 --> 00:26:42,470
It took me about
seven or eight years
502
00:26:42,470 --> 00:26:43,982
of being in finance
to realize there
503
00:26:43,982 --> 00:26:45,940
are a whole lot of
martingales floating around.
504
00:26:45,940 --> 00:26:49,210
Because this actual
approach-- this formalization
505
00:26:49,210 --> 00:26:52,930
of asset pricing
really only became
506
00:26:52,930 --> 00:26:56,770
embraced on the trade floor
around the early 2000's, even
507
00:26:56,770 --> 00:26:59,650
though the underlying
theory was always there--
508
00:26:59,650 --> 00:27:01,450
this idea of these martingales.
509
00:27:01,450 --> 00:27:06,730
Anyway, so this is--
and this, of course,
510
00:27:06,730 --> 00:27:14,225
is simply-- the expected value
of the indicator function
511
00:27:14,225 --> 00:27:16,430
is just the probability
of the event.
512
00:27:16,430 --> 00:27:17,870
OK.
513
00:27:17,870 --> 00:27:22,810
All right, so now
I've won by intuition.
514
00:27:22,810 --> 00:27:26,590
Just here's the probability
of the payout occurring.
515
00:27:26,590 --> 00:27:27,937
I've priced the digital.
516
00:27:27,937 --> 00:27:29,520
I've also priced the
digital by taking
517
00:27:29,520 --> 00:27:31,389
the limit of call spreads.
518
00:27:31,389 --> 00:27:32,930
So now I'm just
going to equate them.
519
00:27:37,200 --> 00:27:41,760
So by equating these two
prices for the digital,
520
00:27:41,760 --> 00:27:46,790
I simply get that the derivative
of the call price with respect
521
00:27:46,790 --> 00:27:57,410
to strike equals the discounted
probability of the stock being
522
00:27:57,410 --> 00:28:02,620
above K. I've just reorganized
a little bit, take 1 minus.
523
00:28:02,620 --> 00:28:08,520
So I get the probability
that-- well, I can clearly
524
00:28:08,520 --> 00:28:12,320
reorganize again
and get-- all right,
525
00:28:12,320 --> 00:28:15,420
so if I want to simply get
the cumulative distribution
526
00:28:15,420 --> 00:28:16,840
function, it's
just 1 minus this.
527
00:28:16,840 --> 00:28:19,930
So divide here, take 1 minus.
528
00:28:19,930 --> 00:28:22,880
OK, so I get the cumulative
distribution function
529
00:28:22,880 --> 00:28:31,000
for the stock price at T is
equal to 1 plus dC by dK times
530
00:28:31,000 --> 00:28:35,580
1 over Z. I'm just rearranging.
531
00:28:35,580 --> 00:28:38,200
So here now is the cumulative
distribution function.
532
00:28:38,200 --> 00:28:40,440
Clearly, I just need
to differentiate again
533
00:28:40,440 --> 00:28:43,670
to get the probability
density function.
534
00:28:43,670 --> 00:28:49,687
So here's where the
notation gets kind of messy,
535
00:28:49,687 --> 00:28:51,520
but clearly the probability
density function
536
00:28:51,520 --> 00:28:59,500
of-- f for my random
variable S sub T--
537
00:28:59,500 --> 00:29:03,452
so the density of-- express that
as-- I always-- probabilists,
538
00:29:03,452 --> 00:29:04,910
whenever they talk
about densities,
539
00:29:04,910 --> 00:29:06,290
they always want to say f of x.
540
00:29:06,290 --> 00:29:07,010
And it's the same with me.
541
00:29:07,010 --> 00:29:07,600
That's f of x.
542
00:29:07,600 --> 00:29:10,214
Here's the density is
simply just the next,
543
00:29:10,214 --> 00:29:11,130
the second derivative.
544
00:29:11,130 --> 00:29:12,700
We'll take the
derivative of this.
545
00:29:12,700 --> 00:29:16,130
It's the second derivative of
the call price with respect
546
00:29:16,130 --> 00:29:22,570
to strike, evaluated
at little x.
547
00:29:22,570 --> 00:29:25,680
All right, so what
we've done here
548
00:29:25,680 --> 00:29:29,000
is start off with
simple definition
549
00:29:29,000 --> 00:29:32,930
of three assets, price to
digital in two different ways.
550
00:29:32,930 --> 00:29:36,720
And now we have a
rather elegant linkage
551
00:29:36,720 --> 00:29:43,940
between call prices--
C-- and the density
552
00:29:43,940 --> 00:29:48,790
of the random variable that
is the underlying stock
553
00:29:48,790 --> 00:29:54,960
price at capital T. OK,
so we've established
554
00:29:54,960 --> 00:30:02,050
one side of the duality, which
is given the set of call prices
555
00:30:02,050 --> 00:30:10,720
for all K, I can then
uniquely determine the density
556
00:30:10,720 --> 00:30:15,303
of the underlying asset.
557
00:30:20,460 --> 00:30:25,490
So you might think, OK,
this is kind of nice.
558
00:30:25,490 --> 00:30:29,430
How does this actually
work in practice?
559
00:30:29,430 --> 00:30:33,450
Do we actually think in
terms of probability trading?
560
00:30:33,450 --> 00:30:37,600
We just said that call options
are equivalent to probability
561
00:30:37,600 --> 00:30:39,140
density functions.
562
00:30:39,140 --> 00:30:41,310
Well, actually,
there's a very neat way
563
00:30:41,310 --> 00:30:44,810
of accessing this
density function
564
00:30:44,810 --> 00:30:47,380
through another
portfolio of options.
565
00:30:47,380 --> 00:30:50,650
OK, so this is actually
where we get-- to me
566
00:30:50,650 --> 00:30:53,220
it's the practical relevance
of some of this theory.
567
00:30:53,220 --> 00:30:56,306
So let me just show you that.
568
00:30:56,306 --> 00:30:58,180
So we're going to consider
another portfolio.
569
00:30:58,180 --> 00:31:06,230
So here we consider
portfolio as follows--
570
00:31:06,230 --> 00:31:09,120
it's actually going to be the
difference between two call
571
00:31:09,120 --> 00:31:10,070
spreads.
572
00:31:10,070 --> 00:31:20,990
So lambda calls with strike
K minus 1 over lambda.
573
00:31:24,180 --> 00:31:31,340
Minus 2 lambda
calls with strike K,
574
00:31:31,340 --> 00:31:38,714
and lambda calls with strike
K plus 1 over lambda-- again,
575
00:31:38,714 --> 00:31:39,380
lambda positive.
576
00:31:43,750 --> 00:31:45,670
OK, why are we doing this?
577
00:31:45,670 --> 00:31:47,910
Let's just stop for
a bit of intuition.
578
00:31:47,910 --> 00:31:53,180
Here we see in the call spread
the discrete approximation
579
00:31:53,180 --> 00:31:58,032
to the first derivative of call
price with respect to strike.
580
00:31:58,032 --> 00:31:59,990
So clearly, if I want to
approximate the second
581
00:31:59,990 --> 00:32:04,230
derivative, I'm going to take
the difference between two call
582
00:32:04,230 --> 00:32:05,835
spreads appropriately scaled.
583
00:32:05,835 --> 00:32:07,960
You're now going to have
to have a little-- there's
584
00:32:07,960 --> 00:32:11,590
got to be another lambda
coming in here at some point.
585
00:32:11,590 --> 00:32:13,970
This is just the difference
between two call spreads,
586
00:32:13,970 --> 00:32:17,590
so that's the difference
between two approximations
587
00:32:17,590 --> 00:32:18,590
of the first derivative.
588
00:32:18,590 --> 00:32:21,215
So I'm going to have to scale by
lambda in order to approximate
589
00:32:21,215 --> 00:32:22,660
the second derivative.
590
00:32:22,660 --> 00:32:25,070
So this is actually
called a call butterfly.
591
00:32:32,304 --> 00:32:34,220
And this is a beautiful
thing for two reasons.
592
00:32:34,220 --> 00:32:37,370
One is they actually trade
a lot-- surprisingly.
593
00:32:37,370 --> 00:32:39,965
This is not a contrived
thing I just made up.
594
00:32:39,965 --> 00:32:42,340
A, it trades a lot, so you
can actually trade this thing.
595
00:32:42,340 --> 00:32:46,320
The second is you
can kind of imagine
596
00:32:46,320 --> 00:32:49,240
the right scaling of
this call butterfly
597
00:32:49,240 --> 00:32:52,450
is going to approximate
the second derivative,
598
00:32:52,450 --> 00:32:55,960
and that's approximating
the density function.
599
00:32:55,960 --> 00:33:01,611
So this is a traded object that
will approximate the density
600
00:33:01,611 --> 00:33:02,110
function.
601
00:33:02,110 --> 00:33:03,194
Yeah, you have a question?
602
00:33:03,194 --> 00:33:04,610
AUDIENCE: Yeah, I
have a question.
603
00:33:04,610 --> 00:33:06,370
In the real world,
you cannot really--
604
00:33:06,370 --> 00:33:10,130
the strike distance cannot
really go to infinitely small,
605
00:33:10,130 --> 00:33:12,745
so they have some [INAUDIBLE]
way how to approximate that?
606
00:33:12,745 --> 00:33:14,680
STEPHEN BLYTHE: Yeah,
so that's a good point.
607
00:33:14,680 --> 00:33:16,860
Yeah, so the question
is how, in practice, we
608
00:33:16,860 --> 00:33:18,690
can't go infinitely
small, which is true.
609
00:33:18,690 --> 00:33:21,460
But we can go pretty small.
610
00:33:21,460 --> 00:33:24,860
So in interest
rates, we might be
611
00:33:24,860 --> 00:33:31,820
able to trade a
150, 160, 170 call
612
00:33:31,820 --> 00:33:34,640
butterfly or equivalent--
10 basis points wide.
613
00:33:34,640 --> 00:33:37,350
That's a-- it's a
reasonable approximation
614
00:33:37,350 --> 00:33:40,330
to the probability of
being in that interval.
615
00:33:40,330 --> 00:33:42,870
So these are all, I mean,
you make a good point.
616
00:33:42,870 --> 00:33:44,950
In fact, all of finance
is discrete, in my view.
617
00:33:44,950 --> 00:33:48,010
So continuous-time finance
is done in continuous time
618
00:33:48,010 --> 00:33:50,600
because the theory
is much more elegant.
619
00:33:50,600 --> 00:33:54,540
But in practice, it's
discrete in time and space.
620
00:33:54,540 --> 00:33:59,370
You can only trade finitely
often in a day, and so on.
621
00:33:59,370 --> 00:34:01,865
I won't going into the detail,
but you can see the price.
622
00:34:01,865 --> 00:34:03,240
Let me just write
down the first.
623
00:34:03,240 --> 00:34:07,090
The price of this I have just
expressed as the difference
624
00:34:07,090 --> 00:34:09,550
between two call spreads.
625
00:34:09,550 --> 00:34:15,710
So it's lambda times the call
spread from 1 minus lambda
626
00:34:15,710 --> 00:34:30,010
to K, so K, 1 minus lambda
to K, minus the call spread
627
00:34:30,010 --> 00:34:34,020
from K to K plus 1 over lambda.
628
00:34:39,380 --> 00:34:41,020
OK, so the difference
between two call
629
00:34:41,020 --> 00:34:46,638
spreads-- we'll call this--
this is the butterfly.
630
00:34:46,638 --> 00:34:48,429
We're just going to
use temporary notation,
631
00:34:48,429 --> 00:34:52,469
call that B, B for butterfly.
632
00:34:52,469 --> 00:34:54,795
So the price B, and
then you get confused.
633
00:34:54,795 --> 00:34:58,812
It's B centered at
K with width lambda.
634
00:34:58,812 --> 00:35:00,770
No one ever uses this
notation outside this one
635
00:35:00,770 --> 00:35:02,300
section of my class,
so that's why,
636
00:35:02,300 --> 00:35:04,120
but it's just handy for this.
637
00:35:04,120 --> 00:35:06,487
So that is-- the
butterfly price is
638
00:35:06,487 --> 00:35:08,570
equal to the difference
in these two call spreads.
639
00:35:08,570 --> 00:35:10,150
What I want to do
is, I want to take
640
00:35:10,150 --> 00:35:12,930
limits of this, suitably scaled,
to get the second derivative.
641
00:35:12,930 --> 00:35:22,220
And if you just take
lambda times B_K of lambda,
642
00:35:22,220 --> 00:35:25,810
t, T is indeed,
approximately-- if I
643
00:35:25,810 --> 00:35:31,725
take limits is exactly-- the
second derivative of call
644
00:35:31,725 --> 00:35:32,225
price.
645
00:35:35,440 --> 00:35:39,630
OK, so here's how I'm
accessing the second derivative
646
00:35:39,630 --> 00:35:41,911
through a portfolio
of traded options.
647
00:35:41,911 --> 00:35:42,410
All right?
648
00:35:42,410 --> 00:35:46,820
And so the price of
this butterfly, B,
649
00:35:46,820 --> 00:35:51,960
if I just reorganize and
substitute-- so I get
650
00:35:51,960 --> 00:35:58,560
B_K-- for large lambda, i.e.
651
00:35:58,560 --> 00:36:03,180
a small interval--
is approximately 1
652
00:36:03,180 --> 00:36:09,950
over lambda times the
density function-- actually,
653
00:36:09,950 --> 00:36:22,130
evaluated at K. So I have
obtained this density function
654
00:36:22,130 --> 00:36:23,410
by this traded portfolio.
655
00:36:23,410 --> 00:36:26,451
And to your point about we're
not getting infinitely small.
656
00:36:26,451 --> 00:36:27,450
That's absolutely right.
657
00:36:27,450 --> 00:36:29,366
But if you think about
what the density-- when
658
00:36:29,366 --> 00:36:33,450
you learn about density
functions for the first time,
659
00:36:33,450 --> 00:36:36,430
you say the density function
at x times a small interval
660
00:36:36,430 --> 00:36:38,650
is the probability of being
in that small interval.
661
00:36:38,650 --> 00:36:41,700
All right, so when we
think about the density
662
00:36:41,700 --> 00:36:50,370
function f of x, if you have
a small interval of delta x,
663
00:36:50,370 --> 00:36:54,030
then clearly the probability
of being in this interval
664
00:36:54,030 --> 00:36:59,020
is approximately
f of x, delta x.
665
00:36:59,020 --> 00:37:00,349
In the limit, that is true.
666
00:37:00,349 --> 00:37:02,140
So what we're showing
here, if you actually
667
00:37:02,140 --> 00:37:04,237
think about what interval
we're looking at,
668
00:37:04,237 --> 00:37:06,320
we're actually looking at
in this call butterfly--
669
00:37:06,320 --> 00:37:08,340
if you were actually
to draw it out,
670
00:37:08,340 --> 00:37:17,430
this call butterfly
looks like that around K.
671
00:37:17,430 --> 00:37:19,060
It actually-- it's
a little triangle.
672
00:37:19,060 --> 00:37:21,434
It's not actually a rectangle,
but it's a little triangle
673
00:37:21,434 --> 00:37:23,550
of width 2 over lambda.
674
00:37:23,550 --> 00:37:31,490
OK, so it is actually-- this
is the area of this triangle--
675
00:37:31,490 --> 00:37:33,870
2 over lambda times
1/2 times f of x.
676
00:37:33,870 --> 00:37:36,150
And that's actually this, right?
677
00:37:36,150 --> 00:37:41,700
So this has width 2 over lambda.
678
00:37:41,700 --> 00:37:43,710
OK, so in fact, we've
got here exactly
679
00:37:43,710 --> 00:37:45,600
an approximate--
exactly approximation,
680
00:37:45,600 --> 00:37:46,780
that doesn't sound right.
681
00:37:46,780 --> 00:37:50,020
But it's entirely analogous
to the approximation
682
00:37:50,020 --> 00:37:52,110
of the probability of
being a small interval.
683
00:37:52,110 --> 00:37:57,640
Here is the probability
of being in this interval
684
00:37:57,640 --> 00:38:04,900
here-- just the area
under that is exactly
685
00:38:04,900 --> 00:38:06,230
1 over lambda f of x.
686
00:38:06,230 --> 00:38:09,890
So here is actually something
that people do do, is they say,
687
00:38:09,890 --> 00:38:16,380
OK, I will look at the price of
this butterfly, which gives me
688
00:38:16,380 --> 00:38:19,010
the probability
of this underlying
689
00:38:19,010 --> 00:38:23,270
random variable
ending up around K.
690
00:38:23,270 --> 00:38:26,850
I'll make a judgment whether
I agree with that probability
691
00:38:26,850 --> 00:38:28,620
or not.
692
00:38:28,620 --> 00:38:32,300
And if I think that
probability is higher
693
00:38:32,300 --> 00:38:38,260
than this price implies,
then I'll do a trade.
694
00:38:38,260 --> 00:38:39,030
I'll buy it.
695
00:38:39,030 --> 00:38:41,990
I'll buy that butterfly.
696
00:38:41,990 --> 00:38:48,122
So there is actually an
active market in butterflies,
697
00:38:48,122 --> 00:38:49,580
and so I think an
active trading in
698
00:38:49,580 --> 00:38:54,340
probabilities-- probabilities
of the underlying variable being
699
00:38:54,340 --> 00:38:56,700
at K at maturity.
700
00:38:59,250 --> 00:39:03,770
So OK, so that's the
first linkage here.
701
00:39:03,770 --> 00:39:06,280
Both-- the density is
the second derivative,
702
00:39:06,280 --> 00:39:08,080
and the second
derivative is essentially
703
00:39:08,080 --> 00:39:09,790
a portfolio of traded options.
704
00:39:09,790 --> 00:39:15,822
And none of this is dependent
on the actual price of the call
705
00:39:15,822 --> 00:39:18,231
option, in the sense that
this holds regardless.
706
00:39:18,231 --> 00:39:20,730
Clearly, this is a function of
the price of the call option,
707
00:39:20,730 --> 00:39:24,660
but I don't need any
model for the option price
708
00:39:24,660 --> 00:39:27,850
to hold, in order for these
relationships to hold.
709
00:39:27,850 --> 00:39:31,350
So these are model-independent
relationships, clearly.
710
00:39:31,350 --> 00:39:39,130
If you were to put the
Black-Scholes formula into C--
711
00:39:39,130 --> 00:39:41,140
Black-Scholes formula
of the call price--
712
00:39:41,140 --> 00:39:44,420
and take the second derivative
with respect to K, which
713
00:39:44,420 --> 00:39:49,115
would be a mess, you'll end up
with a log-normal distribution.
714
00:39:49,115 --> 00:39:51,365
Because that's what actually
the Black-Scholes formula
715
00:39:51,365 --> 00:39:52,823
is, is expected
value of the payout
716
00:39:52,823 --> 00:39:54,320
under a log-normal distribution.
717
00:39:54,320 --> 00:39:55,660
And that will hold.
718
00:39:55,660 --> 00:39:57,226
So this will hold for that.
719
00:39:57,226 --> 00:39:59,100
AUDIENCE: [INAUDIBLE]?
720
00:39:59,100 --> 00:40:01,184
STEPHEN BLYTHE: Yes.
721
00:40:01,184 --> 00:40:03,184
AUDIENCE: The last
[INAUDIBLE] So left-hand side
722
00:40:03,184 --> 00:40:06,470
depends on the small t.
723
00:40:06,470 --> 00:40:07,964
STEPHEN BLYTHE: Yes, it does.
724
00:40:07,964 --> 00:40:08,880
AUDIENCE: But the
right-hand side does not.
725
00:40:08,880 --> 00:40:09,900
What's the role of that?
726
00:40:09,900 --> 00:40:11,941
STEPHEN BLYTHE: Yeah,
that's a really good point.
727
00:40:11,941 --> 00:40:13,250
I've been loose in my notation.
728
00:40:13,250 --> 00:40:15,050
So here what is it?
729
00:40:15,050 --> 00:40:17,540
It's actually the conditional
distribution of S capital
730
00:40:17,540 --> 00:40:19,780
T, given S little t.
731
00:40:19,780 --> 00:40:22,510
So this is the
conditional distribution,
732
00:40:22,510 --> 00:40:25,810
given that we're currently
at time little t with price S
733
00:40:25,810 --> 00:40:29,930
little t for the distribution
at time capital T.
734
00:40:29,930 --> 00:40:31,110
So that's where it comes in.
735
00:40:31,110 --> 00:40:32,110
That's absolutely right.
736
00:40:32,110 --> 00:40:37,540
So in fact, this
expected value strictly
737
00:40:37,540 --> 00:40:42,190
should be conditional on S_t.
738
00:40:42,190 --> 00:40:44,300
This probability
is a probability
739
00:40:44,300 --> 00:40:48,070
conditional on S_t-- absolutely.
740
00:40:48,070 --> 00:40:50,610
And in fact, this
martingale condition
741
00:40:50,610 --> 00:40:56,840
is-- the martingales
with respect to S_t.
742
00:40:56,840 --> 00:40:58,782
So that's where the
little t comes in.
743
00:40:58,782 --> 00:40:59,698
AUDIENCE: [INAUDIBLE]?
744
00:41:04,900 --> 00:41:06,730
STEPHEN BLYTHE:
Here, yes, sorry.
745
00:41:06,730 --> 00:41:11,740
That's 1 over Z. So it's
just a constant here.
746
00:41:11,740 --> 00:41:14,520
This number, especially
because interest rates
747
00:41:14,520 --> 00:41:18,474
are so low in US, so this
number is so close to 1
748
00:41:18,474 --> 00:41:19,890
that you always
forget about that.
749
00:41:19,890 --> 00:41:22,320
Not when we're trading,
but when you, oh well, this
750
00:41:22,320 --> 00:41:26,870
is just a-- if you just think
about which one is-- this
751
00:41:26,870 --> 00:41:29,570
is a quantity that's
in the future.
752
00:41:29,570 --> 00:41:34,460
It's call prices, so
that's how you kind of go.
753
00:41:34,460 --> 00:41:37,870
All right, so that's
the first bit.
754
00:41:41,320 --> 00:41:44,890
So when I was an undergraduate,
actually, learning probability,
755
00:41:44,890 --> 00:41:47,190
one thing I learned
about probability
756
00:41:47,190 --> 00:41:49,940
was from my probability
lecturer, who said,
757
00:41:49,940 --> 00:41:53,010
the attention span of
students is no more than about
758
00:41:53,010 --> 00:41:53,820
40 minutes.
759
00:41:53,820 --> 00:41:56,780
So there's no point lecturing
continuously for 40 minutes,
760
00:41:56,780 --> 00:42:00,110
because people will just start
switching off after 40 minutes.
761
00:42:00,110 --> 00:42:03,125
So rather than wait, just have
a break and waste the time,
762
00:42:03,125 --> 00:42:04,500
the lecturer said,
I'm just going
763
00:42:04,500 --> 00:42:06,510
to give you some random
information in the break,
764
00:42:06,510 --> 00:42:08,120
and then we'll go
back to probability.
765
00:42:08,120 --> 00:42:10,014
So I learned that
from 25 years ago.
766
00:42:10,014 --> 00:42:12,180
I can't remember-- I actually
remember the material.
767
00:42:12,180 --> 00:42:13,450
I can't remember any
of the random material.
768
00:42:13,450 --> 00:42:14,908
So that's what I
do in my lectures,
769
00:42:14,908 --> 00:42:17,290
is I break them up, and
talk about something random.
770
00:42:17,290 --> 00:42:19,680
So I thought I'd do that
here as well, with some--
771
00:42:19,680 --> 00:42:21,090
not completely random.
772
00:42:21,090 --> 00:42:23,740
So this is somewhat applicable,
this being a math class.
773
00:42:23,740 --> 00:42:28,547
So how many of you are math
concentrators or applied math
774
00:42:28,547 --> 00:42:29,130
concentrators?
775
00:42:29,130 --> 00:42:31,930
One, two-- a lot, applied
math concentrators--
776
00:42:31,930 --> 00:42:36,424
especially for the applied
math concentrators,
777
00:42:36,424 --> 00:42:37,840
going straight to
the conclusion--
778
00:42:37,840 --> 00:42:39,270
your entire syllabus
was generated
779
00:42:39,270 --> 00:42:40,270
at Cambridge University.
780
00:42:40,270 --> 00:42:41,520
That's the conclusion.
781
00:42:41,520 --> 00:42:43,450
So anyway, here's the story.
782
00:42:43,450 --> 00:42:49,670
So back in the 19th century, the
Cambridge Mathematics degree--
783
00:42:49,670 --> 00:42:51,810
the undergraduate
Mathematics degree--
784
00:42:51,810 --> 00:42:55,430
was the most prestigious
degree in the world.
785
00:42:55,430 --> 00:42:59,130
In fact, it was actually the
first undergraduate degree
786
00:42:59,130 --> 00:43:02,350
with a written examination
was Cambridge Mathematics.
787
00:43:02,350 --> 00:43:06,190
So they have a lot to
be responsible for.
788
00:43:06,190 --> 00:43:09,770
And each year,
people took the exam.
789
00:43:09,770 --> 00:43:11,640
And they were ranked.
790
00:43:11,640 --> 00:43:14,850
And that ranking was published
in the Times of London--
791
00:43:14,850 --> 00:43:16,207
so the national newspaper.
792
00:43:16,207 --> 00:43:18,040
And the people who got
first-class degrees--
793
00:43:18,040 --> 00:43:20,950
so summa degrees-- in
Cambridge Mathematics
794
00:43:20,950 --> 00:43:23,920
were called wranglers, and still
are called wranglers, actually.
795
00:43:23,920 --> 00:43:25,545
And the reason they're
called wranglers
796
00:43:25,545 --> 00:43:27,770
was from way back
in the 17th century
797
00:43:27,770 --> 00:43:29,937
where, before they had
exams-- or 18th century,
798
00:43:29,937 --> 00:43:32,620
I should say, before they had
exams-- instead of writing down
799
00:43:32,620 --> 00:43:34,350
exam, you have to
argue, or dispute,
800
00:43:34,350 --> 00:43:38,020
or wrangle with your professor
to get to pass the class.
801
00:43:38,020 --> 00:43:39,800
So that's where
wrangler comes from.
802
00:43:39,800 --> 00:43:42,822
So these people who got
the first-class degree
803
00:43:42,822 --> 00:43:44,530
are called wranglers,
and they're ranked.
804
00:43:44,530 --> 00:43:46,390
And basically, the
senior wrangler
805
00:43:46,390 --> 00:43:50,420
was a very famous person in
the UK in the 19th century.
806
00:43:50,420 --> 00:43:53,410
And a lot of them turned
out to be quite successful.
807
00:43:53,410 --> 00:43:55,900
So here are a few wranglers.
808
00:43:55,900 --> 00:44:01,180
I've just got this one-- I can't
reach that, but [INAUDIBLE].
809
00:44:01,180 --> 00:44:04,400
So some of you might
recognize-- and I just
810
00:44:04,400 --> 00:44:07,680
want to tell you a quick
story about one of them.
811
00:44:12,290 --> 00:44:18,210
OK, so let's start
1841, senior wrangler
812
00:44:18,210 --> 00:44:22,690
was George Stokes-- so
basic fluid dynamics--
813
00:44:22,690 --> 00:44:25,094
the whole of fluid dynamics--
that's George Stokes.
814
00:44:31,216 --> 00:44:32,986
1854, second
wrangler-- this is--
815
00:44:32,986 --> 00:44:34,110
who was the first wrangler?
816
00:44:34,110 --> 00:44:37,670
The second wrangler was James
Maxwell, so electrodynamics,
817
00:44:37,670 --> 00:44:39,195
Maxwell equations.
818
00:44:39,195 --> 00:44:39,945
He was the second.
819
00:44:39,945 --> 00:44:44,090
And I can't quite work
out who was the first.
820
00:44:44,090 --> 00:44:50,450
1880, the second wrangler was
J.J. Thompson, so electrons,
821
00:44:50,450 --> 00:44:55,190
atomic physics, that comes
out of-- he was only second.
822
00:44:55,190 --> 00:44:57,890
1865, senior wrangler
Lord Rayleigh.
823
00:44:57,890 --> 00:45:00,350
So he was the sky is blue.
824
00:45:03,322 --> 00:45:05,090
He was first.
825
00:45:05,090 --> 00:45:08,570
So they're a pretty good bunch.
826
00:45:08,570 --> 00:45:12,290
So the story--
the best of 1845--
827
00:45:12,290 --> 00:45:14,920
I'm going back--
the second wrangler
828
00:45:14,920 --> 00:45:18,971
was Lord Kelvin, so absolute
zero, amongst other things,
829
00:45:18,971 --> 00:45:19,470
of course.
830
00:45:19,470 --> 00:45:21,900
But absolute zero-- he
was second wrangler.
831
00:45:21,900 --> 00:45:23,930
And the great
story about him, he
832
00:45:23,930 --> 00:45:26,020
was the most talented
mathematician of his--
833
00:45:26,020 --> 00:45:28,040
of the decade.
834
00:45:28,040 --> 00:45:30,030
And he was such a lock
for senior wrangler
835
00:45:30,030 --> 00:45:31,755
that-- and I actually
read the biography,
836
00:45:31,755 --> 00:45:34,390
so this is a sort
of true statement--
837
00:45:34,390 --> 00:45:37,800
that he sent his servant
to the Senate House where
838
00:45:37,800 --> 00:45:40,680
these things are being read out,
and with a request, "Tell me
839
00:45:40,680 --> 00:45:42,120
who is second wrangler."
840
00:45:42,120 --> 00:45:44,720
And the servant came
back, and said, you, sir.
841
00:45:44,720 --> 00:45:49,150
And because he was such a
lock to be first wrangler.
842
00:45:49,150 --> 00:45:51,370
And in fact, what
happened was a question
843
00:45:51,370 --> 00:45:55,880
on the mathematical
exam was a theorem
844
00:45:55,880 --> 00:45:58,440
that he had proved two years
before in the Cambridge
845
00:45:58,440 --> 00:46:00,730
Mathematical Journal.
846
00:46:00,730 --> 00:46:02,570
So his theorem was
set on the exam.
847
00:46:02,570 --> 00:46:05,040
Because he had not
memorized it, so he
848
00:46:05,040 --> 00:46:07,890
had to reprove it,
whereas the person who
849
00:46:07,890 --> 00:46:10,610
became senior wrangler
had memorized the proof,
850
00:46:10,610 --> 00:46:11,860
and was able to regenerate it.
851
00:46:11,860 --> 00:46:13,609
In those days, there
was a lot of cramming
852
00:46:13,609 --> 00:46:14,850
to be done in these exams.
853
00:46:14,850 --> 00:46:18,250
So the guy who-- Stephen
Parkinson was senior wrangler.
854
00:46:18,250 --> 00:46:19,870
He went on to be
FRS, and eminent.
855
00:46:19,870 --> 00:46:24,350
But he wasn't-- so anyway,
so here's the applied math
856
00:46:24,350 --> 00:46:25,477
syllabus.
857
00:46:25,477 --> 00:46:27,560
Here's a couple of other
ones which I really like.
858
00:46:27,560 --> 00:46:35,090
In 1904, John Maynard
Keynes was at 12th wrangler.
859
00:46:35,090 --> 00:46:37,181
Now, I can tell the
story either way,
860
00:46:37,181 --> 00:46:39,430
depending on whether I'm in
an audience of economists,
861
00:46:39,430 --> 00:46:40,804
or an audience of
mathematicians.
862
00:46:40,804 --> 00:46:43,120
Since I'm in an audience
of mathematicians,
863
00:46:43,120 --> 00:46:45,500
I like to say the
greatest economist was
864
00:46:45,500 --> 00:46:48,190
so poor at mathematics, he only
managed to be 12th wrangler.
865
00:46:48,190 --> 00:46:49,770
There are 11 better
mathematicians
866
00:46:49,770 --> 00:46:52,380
in the UK in that year.
867
00:46:52,380 --> 00:46:54,587
So he was obviously
not that great.
868
00:46:54,587 --> 00:46:56,420
If I was talking to
economists, I would say,
869
00:46:56,420 --> 00:46:59,050
this guy is so brilliant that
his main field was economics,
870
00:46:59,050 --> 00:47:02,540
and yet as part time, he's
able to be the 12th best
871
00:47:02,540 --> 00:47:04,050
mathematician in the UK.
872
00:47:04,050 --> 00:47:11,860
So last one I wanted to talk
about-- 1879-- here's a quiz.
873
00:47:11,860 --> 00:47:13,900
This one you have to
have some answers for.
874
00:47:13,900 --> 00:47:21,180
OK, so 1980 something-- I can't
remember what it is-- so here's
875
00:47:21,180 --> 00:47:24,880
one, here's two, here's three.
876
00:47:24,880 --> 00:47:26,420
I'm going to give
you one and two.
877
00:47:26,420 --> 00:47:27,615
You've got to fill in three.
878
00:47:27,615 --> 00:47:29,990
You probably aren't going to
be able to get this one yet,
879
00:47:29,990 --> 00:47:37,100
but this is-- Andrew Alan,
senior wrangler, George Walker,
880
00:47:37,100 --> 00:47:43,090
second wrangler, and number
three is the question.
881
00:47:43,090 --> 00:47:56,050
That's the question-- 1980,
Hakeem Olajuwon, Sam Bowie,
882
00:47:56,050 --> 00:47:57,620
question mark--
who's question mark?
883
00:48:00,780 --> 00:48:03,350
Do we know which sport
these people play?
884
00:48:03,350 --> 00:48:04,850
AUDIENCE: That one's
Michael Jordan.
885
00:48:04,850 --> 00:48:05,974
STEPHEN BLYTHE: Yes, right.
886
00:48:05,974 --> 00:48:09,490
There we go, that's
Michael Jordan-- exactly.
887
00:48:09,490 --> 00:48:12,000
This question could go on
forever in the UK because they
888
00:48:12,000 --> 00:48:13,900
don't-- so Michael
Jordan, famously,
889
00:48:13,900 --> 00:48:17,370
was only picked third in the
NBA draft in 1984, was that--
890
00:48:17,370 --> 00:48:19,497
four or five,
something like that.
891
00:48:19,497 --> 00:48:21,330
So Hakeem Olajuwon was
actually pretty good,
892
00:48:21,330 --> 00:48:23,780
but Sam Bowie was a total bust.
893
00:48:23,780 --> 00:48:25,290
But he was third.
894
00:48:25,290 --> 00:48:28,700
So in 1879, in the Cambridge
Mathematics Tripos,
895
00:48:28,700 --> 00:48:32,250
these two people you never heard
of, who were first and second.
896
00:48:32,250 --> 00:48:35,550
And the person who came third,
you've probably heard of him.
897
00:48:35,550 --> 00:48:39,940
This is more of a
statistics thing.
898
00:48:39,940 --> 00:48:41,974
People know about correlation?
899
00:48:41,974 --> 00:48:43,890
What's the correlation--
who's the correlation
900
00:48:43,890 --> 00:48:46,065
coefficient named after?
901
00:48:46,065 --> 00:48:46,815
AUDIENCE: Pearson.
902
00:48:46,815 --> 00:48:48,856
STEPHEN BLYTHE: Pearson,
you've got Karl Pearson.
903
00:48:48,856 --> 00:48:53,584
So Karl Pearson was the
third wrangler in 1879.
904
00:48:53,584 --> 00:48:55,000
And the founder
of statistics-- he
905
00:48:55,000 --> 00:48:56,874
founded the first ever
statistics department,
906
00:48:56,874 --> 00:48:59,140
and obviously
invented correlation
907
00:48:59,140 --> 00:49:00,786
with Gould-- Gould and Pearson.
908
00:49:00,786 --> 00:49:02,410
Anyway, he was only
the third wrangler.
909
00:49:02,410 --> 00:49:05,600
And unfortunately, these
people have very common names,
910
00:49:05,600 --> 00:49:07,920
so I have no idea what
they went on to do.
911
00:49:07,920 --> 00:49:11,170
To Google these people
is not very effective.
912
00:49:11,170 --> 00:49:16,090
Anyway, so that's the story
of Cambridge Mathematics--
913
00:49:16,090 --> 00:49:18,860
lots of good stuff.
914
00:49:28,400 --> 00:49:31,310
All right, so in
the last half hour,
915
00:49:31,310 --> 00:49:41,160
I just want to go the other
way from-- so the other way--
916
00:49:41,160 --> 00:49:44,020
we went from option
prices to probability.
917
00:49:44,020 --> 00:49:45,970
Let's go from probability
to option price.
918
00:49:45,970 --> 00:49:47,510
We sort of already
have, actually.
919
00:49:47,510 --> 00:49:49,500
This is what the
Fundamental Theorem does.
920
00:49:49,500 --> 00:49:51,770
If we're thinking--
if we take on trust
921
00:49:51,770 --> 00:49:53,230
that the Fundamental
Theorem holds,
922
00:49:53,230 --> 00:49:58,402
namely option prices today are
the discounted expected payout
923
00:49:58,402 --> 00:49:59,940
at maturity.
924
00:49:59,940 --> 00:50:01,450
OK, let's take that on trust.
925
00:50:01,450 --> 00:50:03,690
Then we're going from
probability distribution
926
00:50:03,690 --> 00:50:05,231
to option price in
the following way.
927
00:50:05,231 --> 00:50:08,730
So let's actually state the
Fundamental Theorem, FTAP.
928
00:50:11,630 --> 00:50:17,800
OK, so I'm going to go
general derivative D
929
00:50:17,800 --> 00:50:19,820
is-- D, digital D, derivative.
930
00:50:19,820 --> 00:50:35,190
It's-- so derivative
with payout at T.
931
00:50:35,190 --> 00:50:36,950
So this could be
the digital payout.
932
00:50:36,950 --> 00:50:38,241
It could be call option payout.
933
00:50:38,241 --> 00:50:40,880
It could be one.
934
00:50:40,880 --> 00:50:50,190
And price-- OK, so
often, we actually
935
00:50:50,190 --> 00:50:52,850
think about payout function
as just a simple function
936
00:50:52,850 --> 00:50:55,870
of the stock price.
937
00:50:55,870 --> 00:50:58,765
But this notation is useful
when we think about the price
938
00:50:58,765 --> 00:51:01,170
as being martingales.
939
00:51:01,170 --> 00:51:05,040
Then what is FTAP?
940
00:51:05,040 --> 00:51:13,270
D-- the ratio of the price
to the zero coupon bond
941
00:51:13,270 --> 00:51:15,000
is equal to-- is a martingale.
942
00:51:15,000 --> 00:51:17,210
In other words,
its expected value
943
00:51:17,210 --> 00:51:20,880
under the special distribution,
risk-neutral distribution,
944
00:51:20,880 --> 00:51:22,860
of the payout at maturity.
945
00:51:27,110 --> 00:51:29,680
And to you point, it's
conditional on S_t.
946
00:51:29,680 --> 00:51:32,120
So this is the proper statement.
947
00:51:32,120 --> 00:51:34,460
So this is the Fundamental
Theorem of Asset Prices.
948
00:51:34,460 --> 00:51:37,080
In words, it's
saying this ratio is
949
00:51:37,080 --> 00:51:39,510
a martingale with
respect to the stock
950
00:51:39,510 --> 00:51:41,980
price under the
risk-neutral distribution.
951
00:51:45,020 --> 00:51:47,670
That's the statement of
the Fundamental Theorem.
952
00:51:47,670 --> 00:51:52,600
This is actually rather neat
to prove in the binomial tree,
953
00:51:52,600 --> 00:51:54,460
two-state world.
954
00:51:54,460 --> 00:51:57,920
It's very, very difficult
to prove in continuous time.
955
00:51:57,920 --> 00:52:01,270
This is Harrison
and Kreps in 1979.
956
00:52:01,270 --> 00:52:04,250
It's the proof that, however
many times you look at it,
957
00:52:04,250 --> 00:52:06,010
you're only probably going to
get through two or three pages
958
00:52:06,010 --> 00:52:07,390
before thinking,
OK, that's hard.
959
00:52:07,390 --> 00:52:09,390
But it was done.
960
00:52:09,390 --> 00:52:12,160
So this is, you can
imagine continuous time,
961
00:52:12,160 --> 00:52:16,740
infinite amount of trading,
infinite states of the world.
962
00:52:16,740 --> 00:52:18,130
OK, so now this,
of course, is 1.
963
00:52:21,196 --> 00:52:22,070
And this can come up.
964
00:52:22,070 --> 00:52:23,620
These are known
at time little t.
965
00:52:23,620 --> 00:52:26,270
So if I'm thinking at-- if
I'm at current time little t,
966
00:52:26,270 --> 00:52:30,150
therefore, the
derivative price is
967
00:52:30,150 --> 00:52:33,490
what we had before--
the expected payout.
968
00:52:39,130 --> 00:52:41,430
OK, so this is rather
a nice expression.
969
00:52:41,430 --> 00:52:44,541
And now we can actually just
write down what this is.
970
00:52:44,541 --> 00:52:50,500
This is the expected value of a
function of a random variable.
971
00:52:50,500 --> 00:52:56,305
So this is just
the integral of g
972
00:52:56,305 --> 00:53:03,030
of x, f of x, dx, where
this is the density
973
00:53:03,030 --> 00:53:06,030
of the random variable
at time capital T,
974
00:53:06,030 --> 00:53:07,630
conditional on being at S_t.
975
00:53:07,630 --> 00:53:09,970
So this is conditional at S_t.
976
00:53:09,970 --> 00:53:14,452
So this is a restatement
of the Fundamental Theorem.
977
00:53:14,452 --> 00:53:16,410
So this is essentially
the Fundamental Theorem.
978
00:53:16,410 --> 00:53:19,080
And this is a
intuition made good,
979
00:53:19,080 --> 00:53:21,515
or intuition made
real-- expected payouts.
980
00:53:24,580 --> 00:53:26,497
This is sometimes
called LOTUS-- the lure
981
00:53:26,497 --> 00:53:27,830
of the unconscious statistician.
982
00:53:27,830 --> 00:53:33,010
Just the expected value of g of
x is integral g of x, f of x.
983
00:53:33,010 --> 00:53:35,890
That's not immediate from the
definition of expected value.
984
00:53:35,890 --> 00:53:40,350
You should really work
out the density of g of x.
985
00:53:40,350 --> 00:53:44,020
And then integral of x--
the density of g of x dx,
986
00:53:44,020 --> 00:53:45,780
but it turns out to be this.
987
00:53:45,780 --> 00:53:47,820
So that's a very
nice, nice result.
988
00:53:47,820 --> 00:53:54,240
OK, so here is now a way of
going from density to price.
989
00:53:54,240 --> 00:53:58,376
If I put in the call
option payout for g,
990
00:53:58,376 --> 00:54:03,350
and I have the density, I
can then derive the price C.
991
00:54:03,350 --> 00:54:09,170
So If you like, the way I go
from density to probability
992
00:54:09,170 --> 00:54:11,600
distribution to option
price is exactly
993
00:54:11,600 --> 00:54:12,990
the Fundamental Theorem.
994
00:54:12,990 --> 00:54:15,682
The route I take is the
Fundamental Theorem.
995
00:54:23,720 --> 00:54:27,880
OK, so FTAP, the Fundamental
Theorem of Asset Pricing,
996
00:54:27,880 --> 00:54:31,230
means I can going from
the probability density
997
00:54:31,230 --> 00:54:37,930
to the price of a derivative,
for any derivative.
998
00:54:37,930 --> 00:54:43,000
All right, OK, so now we
can go either way-- density
999
00:54:43,000 --> 00:54:48,712
to derivative or call
price to density.
1000
00:54:48,712 --> 00:54:49,920
You might say, hang on a sec.
1001
00:54:49,920 --> 00:54:51,560
We've only gone from--
we need the call
1002
00:54:51,560 --> 00:54:53,810
prices to get the density.
1003
00:54:53,810 --> 00:54:59,480
Well, of course, we can go
via an intermediate step.
1004
00:54:59,480 --> 00:55:02,920
So to get from the call price to
an arbitrary derivative price,
1005
00:55:02,920 --> 00:55:06,450
I just go via the density.
1006
00:55:06,450 --> 00:55:14,520
So in particular-- this
is restating-- knowledge
1007
00:55:14,520 --> 00:55:26,710
of all the call prices
for all K determines
1008
00:55:26,710 --> 00:55:29,580
this derivative
payout for any g.
1009
00:55:33,310 --> 00:55:35,726
So if I know all calls,
I know the density.
1010
00:55:35,726 --> 00:55:37,100
And then if I know
the density, I
1011
00:55:37,100 --> 00:55:39,090
know an arbitrary
derivative price.
1012
00:55:39,090 --> 00:55:41,040
It's obvious as we stated here.
1013
00:55:41,040 --> 00:55:44,095
But what this is saying
is, the call options often
1014
00:55:44,095 --> 00:55:46,890
are introduced as this--
why are they important--
1015
00:55:46,890 --> 00:55:50,620
are the spanning set of
all derivative prices.
1016
00:55:50,620 --> 00:56:02,479
So calls span-- call prices
span all derivative prices.
1017
00:56:02,479 --> 00:56:04,770
And they are a particular
type of derivative-- the ones
1018
00:56:04,770 --> 00:56:09,355
that are determined exactly
by their payout at maturity.
1019
00:56:09,355 --> 00:56:11,855
One can imagine other things
that are a function of the path
1020
00:56:11,855 --> 00:56:12,435
or whatever.
1021
00:56:12,435 --> 00:56:14,580
But this is a particular
derivative price.
1022
00:56:14,580 --> 00:56:19,860
European derivative prices
are determined by calls.
1023
00:56:19,860 --> 00:56:23,970
OK, so that's kind of nice--
sort of obvious, elegant.
1024
00:56:23,970 --> 00:56:28,890
There's two other ways of
looking at this, though.
1025
00:56:28,890 --> 00:56:42,190
If I think about my function
g-- so consider function g-- OK,
1026
00:56:42,190 --> 00:56:47,330
so that determines
my derivative.
1027
00:56:47,330 --> 00:56:49,450
So it determines,
defines the derivative
1028
00:56:49,450 --> 00:56:50,760
by its payout at maturity.
1029
00:56:50,760 --> 00:56:54,270
Let's just graph it.
1030
00:56:54,270 --> 00:56:57,190
OK, so it might
look-- let's just
1031
00:56:57,190 --> 00:56:59,290
assume first it's
piecewise linear,
1032
00:56:59,290 --> 00:57:07,352
so it looks like--
so suppose this is g.
1033
00:57:13,490 --> 00:57:15,750
Well you can kind of
see I can replicate
1034
00:57:15,750 --> 00:57:21,870
this portfolio, or this option,
by a portfolio of calls--
1035
00:57:21,870 --> 00:57:24,410
in fact, a linear
combination of calls.
1036
00:57:24,410 --> 00:57:28,040
Right, I have no calls,
but if this is say K_1,
1037
00:57:28,040 --> 00:57:37,260
this is K_2, K_3,
K_4, K_5, et cetera.
1038
00:57:37,260 --> 00:57:41,065
You can see what the
portfolio of calls
1039
00:57:41,065 --> 00:57:43,770
will be in order to replicate
this payout at maturity.
1040
00:57:43,770 --> 00:57:46,670
There'll be a certain amount
of calls with strike K_1,
1041
00:57:46,670 --> 00:57:49,750
so that the slope is right,
minus a certain amount of K_2
1042
00:57:49,750 --> 00:57:52,480
to get this slope, plus
a certain about of K_3,
1043
00:57:52,480 --> 00:57:55,370
minus K_4, minus K_5,
plus K_6, et cetera.
1044
00:57:55,370 --> 00:57:56,880
So, in this case,
if the piecewise
1045
00:57:56,880 --> 00:58:13,860
linear g, replicating portfolio
of calls, it's obvious.
1046
00:58:18,380 --> 00:58:22,610
So if I can replicate the
payout exactly at maturity,
1047
00:58:22,610 --> 00:58:26,830
the price at time little
t of this derivative
1048
00:58:26,830 --> 00:58:31,220
must be the price at little t
of the replicating portfolio.
1049
00:58:31,220 --> 00:58:35,955
That's actually a-- I'll do
that early on in my class,
1050
00:58:35,955 --> 00:58:39,250
and of the 100 people, everyone
says, OK, that makes sense.
1051
00:58:39,250 --> 00:58:42,879
And someone says, does that
always have to be the case?
1052
00:58:42,879 --> 00:58:44,920
And it's actually a really,
really good question.
1053
00:58:44,920 --> 00:58:48,680
Here, I was about to
just hand wave over it.
1054
00:58:48,680 --> 00:58:52,530
Is it the case that if I
have one derivative contract
1055
00:58:52,530 --> 00:58:57,770
with this payout at maturity,
and I have a linear combination
1056
00:58:57,770 --> 00:59:01,340
of calls with the identical
payout at maturity,
1057
00:59:01,340 --> 00:59:05,610
capital T, must these
two portfolios have
1058
00:59:05,610 --> 00:59:07,470
the same value at little t?
1059
00:59:11,810 --> 00:59:14,424
Well, one would think
so, because they're
1060
00:59:14,424 --> 00:59:15,840
both the same at
maturity, so they
1061
00:59:15,840 --> 00:59:17,006
must both be the same thing.
1062
00:59:17,006 --> 00:59:18,620
They're just
constructed differently.
1063
00:59:18,620 --> 00:59:20,240
And the assumption
of no arbitrage--
1064
00:59:20,240 --> 00:59:23,150
which underpins everything, in
some sense, what we're doing--
1065
00:59:23,150 --> 00:59:27,370
would allow you to say
yes, indeed, that is true.
1066
00:59:27,370 --> 00:59:30,860
And in fact, it's actually the
fundamental of finance, right?
1067
00:59:30,860 --> 00:59:33,030
If two things are worth a
dollar in a year's time,
1068
00:59:33,030 --> 00:59:36,409
they're going to be
worth the same today.
1069
00:59:36,409 --> 00:59:37,450
That's what we're saying.
1070
00:59:37,450 --> 00:59:40,030
If you can match the portfolio
at t, that is actually
1071
00:59:40,030 --> 00:59:44,970
the definition of-- it follows
immediately from no arbitrage.
1072
00:59:44,970 --> 00:59:51,230
What has been interesting in
finance, especially since 2008,
1073
00:59:51,230 --> 00:59:57,470
is that that-- this
assumption-- has broken down.
1074
00:59:57,470 --> 01:00:01,060
In other words, I can
hold a portfolio of things
1075
01:00:01,060 --> 01:00:05,220
when aggregated have
exactly this payout,
1076
01:00:05,220 --> 01:00:09,840
against an option with
exactly this payout,
1077
01:00:09,840 --> 01:00:13,171
and be paid for that.
1078
01:00:13,171 --> 01:00:15,670
And this is actually really--
it's a very fascinating thing,
1079
01:00:15,670 --> 01:00:18,630
to think about actually, the
dynamics of financial markets
1080
01:00:18,630 --> 01:00:20,030
when arbitrage can break down.
1081
01:00:20,030 --> 01:00:23,160
What is the main theme
here is that when
1082
01:00:23,160 --> 01:00:29,520
capital T is a long way
in the future-- 10 years,
1083
01:00:29,520 --> 01:00:33,120
20 years-- there's
nothing to stop
1084
01:00:33,120 --> 01:00:36,230
the price of the option and
the replicating portfolio
1085
01:00:36,230 --> 01:00:39,240
going arbitrarily wide,
other than people believing
1086
01:00:39,240 --> 01:00:40,560
that it has to be equal.
1087
01:00:43,430 --> 01:00:46,060
The only way you can guarantee
the two things to be equal
1088
01:00:46,060 --> 01:00:52,060
is by holding it until capital
T-- for 10 years, 20 years.
1089
01:00:52,060 --> 01:00:56,420
In the meantime,
those prices can move.
1090
01:00:56,420 --> 01:01:01,000
Empirically, they've been shown
to move away from each other.
1091
01:01:01,000 --> 01:01:04,670
So there's actually a deep
economic question here.
1092
01:01:04,670 --> 01:01:09,950
So if there is the presence
of arbitrage in the markets,
1093
01:01:09,950 --> 01:01:11,969
then arbitrage can
be arbitrarily big.
1094
01:01:11,969 --> 01:01:14,260
Because you're saying there
aren't enough-- there's not
1095
01:01:14,260 --> 01:01:17,230
enough capital, or that's
not enough risk capital,
1096
01:01:17,230 --> 01:01:21,200
for people to come in and
say, OK, these two things
1097
01:01:21,200 --> 01:01:23,720
have to be worth the
same in 10 year's time.
1098
01:01:23,720 --> 01:01:28,889
Therefore, I'm prepared to buy
one $1 cheaper than the other.
1099
01:01:28,889 --> 01:01:31,180
It's actually a question
really relevant to the Harvard
1100
01:01:31,180 --> 01:01:31,679
endowment.
1101
01:01:31,679 --> 01:01:32,910
We're a long-term investor.
1102
01:01:32,910 --> 01:01:35,310
You say, why doesn't
the Harvard endowment,
1103
01:01:35,310 --> 01:01:38,646
if these two things
are $1 apart,
1104
01:01:38,646 --> 01:01:40,020
buy the things
that's $1 cheaper,
1105
01:01:40,020 --> 01:01:41,976
and just hold them 10
years, make the dollar?
1106
01:01:41,976 --> 01:01:43,490
Well, we'd like
to, but if we think
1107
01:01:43,490 --> 01:01:45,260
they're going to be $1 apart,
and they're going to go to $10
1108
01:01:45,260 --> 01:01:47,790
apart, we don't want to
buy them at $1 apart.
1109
01:01:47,790 --> 01:01:49,316
We want to buy
them at $10 apart.
1110
01:01:49,316 --> 01:01:51,570
I mean, yes, we're a
long-term investor,
1111
01:01:51,570 --> 01:01:54,357
but we care about our annual
returns, or five-year returns.
1112
01:01:54,357 --> 01:01:55,690
Suppose this is a 20-year trade.
1113
01:01:55,690 --> 01:01:58,900
This is very prevalent when
these things are 20 years out.
1114
01:01:58,900 --> 01:02:01,630
Anyway, it's a whole-- this
is-- it's a little bit-- it's
1115
01:02:01,630 --> 01:02:03,100
a foundational issue.
1116
01:02:03,100 --> 01:02:04,600
It's this thing
where it could shake
1117
01:02:04,600 --> 01:02:06,849
the foundational underpinnings
of quantitative finance
1118
01:02:06,849 --> 01:02:09,400
if you don't allow this
replicating portfolio to have
1119
01:02:09,400 --> 01:02:11,182
the same price as
the actual option.
1120
01:02:11,182 --> 01:02:13,390
But mathematically, you can
see you can replicate it,
1121
01:02:13,390 --> 01:02:15,200
certainly at capital
T, and therefore
1122
01:02:15,200 --> 01:02:18,280
the price at time little t is
just the linear combination
1123
01:02:18,280 --> 01:02:19,690
of call prices.
1124
01:02:19,690 --> 01:02:20,830
OK, so let's assume that.
1125
01:02:20,830 --> 01:02:23,531
And then obviously,
continuous function
1126
01:02:23,531 --> 01:02:25,030
can be arbitrarily
well approximated
1127
01:02:25,030 --> 01:02:26,750
by piecewise linear function.
1128
01:02:26,750 --> 01:02:33,270
Therefore, any
function at time--
1129
01:02:33,270 --> 01:02:37,300
any function of this
form-- a derivative
1130
01:02:37,300 --> 01:02:40,809
when compared to that
form can be replicated
1131
01:02:40,809 --> 01:02:42,100
by a portfolio of call options.
1132
01:02:42,100 --> 01:02:45,270
So we can sort of hand
wave to kind of say,
1133
01:02:45,270 --> 01:02:50,710
this must be true-- the
calls are a spanning set.
1134
01:02:50,710 --> 01:02:52,690
There's another way to
look at it, which is-- I
1135
01:02:52,690 --> 01:02:56,960
just-- like from calculus, where
we can actually make explicit
1136
01:02:56,960 --> 01:03:01,740
what this spanning-- what
this portfolio of calls looks
1137
01:03:01,740 --> 01:03:03,550
like in the arbitrary case.
1138
01:03:03,550 --> 01:03:05,535
So let me just do that.
1139
01:03:08,770 --> 01:03:12,940
So you can sort of see, there
must be a linear combination
1140
01:03:12,940 --> 01:03:15,800
by this for a piecewise linear.
1141
01:03:15,800 --> 01:03:18,140
Therefore in the limit,
any continuous function
1142
01:03:18,140 --> 01:03:20,510
must be able to be
replicated by calls.
1143
01:03:20,510 --> 01:03:21,720
How many of each?
1144
01:03:21,720 --> 01:03:33,860
Well, there's actually a
very, very simple result.
1145
01:03:33,860 --> 01:03:37,060
That is as follows--
and, well, let's just
1146
01:03:37,060 --> 01:03:40,040
write down an exact Taylor
series to the second order.
1147
01:03:40,040 --> 01:03:47,130
So this is-- so for any
function with second derivative,
1148
01:03:47,130 --> 01:03:52,659
let's just write down a Taylor
series-- the first two terms.
1149
01:03:52,659 --> 01:03:54,450
And let's put the second
term-- we can just
1150
01:03:54,450 --> 01:03:58,740
do an exact
second-order term, so 0
1151
01:03:58,740 --> 01:04:08,310
to infinity x minus c plus
g double prime of c dc.
1152
01:04:08,310 --> 01:04:11,560
c is my dummy variable.
1153
01:04:11,560 --> 01:04:13,817
Actually, I've gone
to plus notation.
1154
01:04:13,817 --> 01:04:15,025
Here's the max of this and 0.
1155
01:04:18,530 --> 01:04:20,470
OK, that's an exact
Taylor series,
1156
01:04:20,470 --> 01:04:22,370
true for any-- it's
not an approximation.
1157
01:04:22,370 --> 01:04:22,950
That's exact.
1158
01:04:22,950 --> 01:04:24,600
You just integrate the
right-hand side by parts
1159
01:04:24,600 --> 01:04:25,950
if you want to verify it.
1160
01:04:25,950 --> 01:04:28,210
Maybe it's obvious
to you, but I'm
1161
01:04:28,210 --> 01:04:30,404
so used to just doing
non-exact Taylor series.
1162
01:04:30,404 --> 01:04:31,570
So this is the second order.
1163
01:04:31,570 --> 01:04:35,190
So this holds for any g exactly.
1164
01:04:35,190 --> 01:04:37,720
And now I'm just going to
make one little change, which
1165
01:04:37,720 --> 01:04:39,855
sort of might make obvious
what we're trying to do.
1166
01:04:39,855 --> 01:04:42,230
I'm just going to take this
dummy variable c, which we're
1167
01:04:42,230 --> 01:04:48,500
integrating over from 0 to
infinity, and just call it K.
1168
01:04:48,500 --> 01:04:50,694
We can certainly do that.
1169
01:04:50,694 --> 01:04:52,860
All right, this now looks
like the payout of a call.
1170
01:04:56,304 --> 01:04:57,720
It's the payout
of the call price.
1171
01:04:57,720 --> 01:04:59,620
Now, I don't want
to be integrating.
1172
01:04:59,620 --> 01:05:04,010
Remember, if I want to
actually get the call price,
1173
01:05:04,010 --> 01:05:06,870
I take the expected
value of this.
1174
01:05:06,870 --> 01:05:10,970
I integrate x over x with
respect to its density.
1175
01:05:10,970 --> 01:05:14,670
This is g of a
payout function of x.
1176
01:05:14,670 --> 01:05:17,607
Here I'm integrating
over K, so I'm
1177
01:05:17,607 --> 01:05:18,940
doing something a bit different.
1178
01:05:18,940 --> 01:05:22,470
But this is the
call option payout.
1179
01:05:22,470 --> 01:05:23,230
So this holds.
1180
01:05:23,230 --> 01:05:26,200
It's a linear
equation, obviously.
1181
01:05:26,200 --> 01:05:30,260
And of course, expectation
is a linear operator.
1182
01:05:30,260 --> 01:05:35,165
So I'm just going to take,
well, what are the two steps?
1183
01:05:35,165 --> 01:05:36,790
First of all, I'm
just going to replace
1184
01:05:36,790 --> 01:05:42,060
x with my random variable
S sub T. So that I can do.
1185
01:05:46,230 --> 01:05:48,850
This also holds.
1186
01:05:48,850 --> 01:05:51,100
And formally, of course, S
sub T is a random variable,
1187
01:05:51,100 --> 01:05:53,516
so it's a function from the
sample space of the real line.
1188
01:05:53,516 --> 01:05:56,170
But this holds for every
point on the sample space.
1189
01:05:56,170 --> 01:05:58,310
So I can write
down this equation
1190
01:05:58,310 --> 01:05:59,351
between random variables.
1191
01:06:06,900 --> 01:06:10,220
Here it's just the
integral over dK.
1192
01:06:10,220 --> 01:06:11,180
So that holds.
1193
01:06:11,180 --> 01:06:16,090
Now I'm going to take
the expectation operator.
1194
01:06:16,090 --> 01:06:27,470
So take discounted expected
value, of each side.
1195
01:06:32,546 --> 01:06:34,670
So in other words, what is
my operator [INAUDIBLE]?
1196
01:06:34,670 --> 01:06:41,696
It looks like Z(t, T),
expected value of, given S_t.
1197
01:06:41,696 --> 01:06:42,196
All right?
1198
01:06:45,690 --> 01:06:48,630
OK, so this one is a
discounted expected value.
1199
01:06:48,630 --> 01:06:50,094
That's the price.
1200
01:06:55,470 --> 01:07:01,410
So this becomes price of
the derivative with payout
1201
01:07:01,410 --> 01:07:03,567
at maturity g.
1202
01:07:03,567 --> 01:07:04,900
All right, what do we have here?
1203
01:07:04,900 --> 01:07:07,570
Well, first we've
got a constant.
1204
01:07:07,570 --> 01:07:17,768
So we've got a constant times--
OK, so that's a constant.
1205
01:07:22,450 --> 01:07:27,440
OK, now we've got the
discounted expected stock price.
1206
01:07:27,440 --> 01:07:29,020
A little bit of
thought on the terms
1207
01:07:29,020 --> 01:07:30,686
of the Fundamental
Theorem will show you
1208
01:07:30,686 --> 01:07:33,745
that the discounted expected
stock price under this operator
1209
01:07:33,745 --> 01:07:35,336
is the current stock price.
1210
01:07:35,336 --> 01:07:40,930
It's actually non-trivial, but
just think of the stock itself
1211
01:07:40,930 --> 01:07:43,520
as a derivative,
with the payout S,
1212
01:07:43,520 --> 01:07:45,950
and apply the
Fundamental Theorem.
1213
01:07:45,950 --> 01:07:49,220
This has to be the case,
because a replicating portfolio
1214
01:07:49,220 --> 01:07:52,800
of the stock is just a
holding of the stock.
1215
01:07:52,800 --> 01:07:55,580
Plus-- and then we
just take the integral.
1216
01:07:55,580 --> 01:08:05,910
So the expectation inside the
integral-- OK, so now I've got
1217
01:08:05,910 --> 01:08:08,455
discounted expected
payout of this.
1218
01:08:08,455 --> 01:08:10,490
And the discounted
expected payout of this
1219
01:08:10,490 --> 01:08:21,930
is just the call
price, with strike K.
1220
01:08:21,930 --> 01:08:23,689
OK, so I really
like this formula.
1221
01:08:23,689 --> 01:08:28,399
In some sense, there's nothing
too complicated about how
1222
01:08:28,399 --> 01:08:30,250
to derive it.
1223
01:08:30,250 --> 01:08:36,250
But it says explicitly now,
how do I replicate an arbitrary
1224
01:08:36,250 --> 01:08:40,990
derivative product with payout
g of x or g of S at maturity?
1225
01:08:40,990 --> 01:08:41,810
Well, it's clear.
1226
01:08:41,810 --> 01:08:48,399
I replicate it by g(0)
zero-coupon bonds.
1227
01:08:48,399 --> 01:08:52,650
So I have g(0) of
zero coupon bonds.
1228
01:08:52,650 --> 01:08:54,170
That's this.
1229
01:08:54,170 --> 01:08:59,450
I have g prime zero of
stock-- that's this.
1230
01:08:59,450 --> 01:09:02,270
And I have this linear
combination of calls.
1231
01:09:12,160 --> 01:09:15,779
So there-- this kind
of makes sense, right?
1232
01:09:15,779 --> 01:09:18,870
You want the zero-coupon
bond amount is just
1233
01:09:18,870 --> 01:09:21,229
the intercept of g.
1234
01:09:21,229 --> 01:09:24,960
The number of stocks is
just the slope of g at 0.
1235
01:09:24,960 --> 01:09:28,882
And then I have this linear
combination of call prices.
1236
01:09:28,882 --> 01:09:32,490
I've just proved that by
taking this, and taking
1237
01:09:32,490 --> 01:09:35,160
expected values.
1238
01:09:35,160 --> 01:09:44,399
So this is sort of looking at
the duality of option prices
1239
01:09:44,399 --> 01:09:45,899
and probabilities
in different ways.
1240
01:09:45,899 --> 01:09:49,160
But then, also how
calls span everything.
1241
01:09:49,160 --> 01:09:54,000
So the calls, in some sense,
are the primitive information.
1242
01:09:54,000 --> 01:09:56,240
Once I know all
call option prices,
1243
01:09:56,240 --> 01:09:59,890
I know the probability
distribution exactly.
1244
01:09:59,890 --> 01:10:03,300
So there are a couple of sort
of interesting further questions
1245
01:10:03,300 --> 01:10:06,600
you might want to pose.
1246
01:10:06,600 --> 01:10:09,880
We seem to have
done everything here
1247
01:10:09,880 --> 01:10:14,140
with regard to the
distribution at time capital T.
1248
01:10:14,140 --> 01:10:14,880
And that's true.
1249
01:10:14,880 --> 01:10:15,754
I know all the calls.
1250
01:10:15,754 --> 01:10:18,735
I know the distribution
at time capital T.
1251
01:10:18,735 --> 01:10:19,610
I know all the calls.
1252
01:10:19,610 --> 01:10:24,810
I know the price of any option
with a payout defined solely
1253
01:10:24,810 --> 01:10:27,470
by a function at capital T.
1254
01:10:27,470 --> 01:10:30,590
But I said nothing
about the path that
1255
01:10:30,590 --> 01:10:34,300
takes the stock from
today until capital T.
1256
01:10:34,300 --> 01:10:37,170
So I'm just going to leave you
with two things to think about.
1257
01:10:39,834 --> 01:10:41,500
Actually, it's one
thing to think about.
1258
01:10:44,570 --> 01:10:45,920
Two people thought about a lot.
1259
01:10:50,850 --> 01:10:54,170
And it's the following
question, which now we'll
1260
01:10:54,170 --> 01:10:57,860
start transitioning into
stochastic calculus,
1261
01:10:57,860 --> 01:10:59,900
and stochastic
processes a little bit.
1262
01:10:59,900 --> 01:11:03,982
So we know-- let's
just imagine two times.
1263
01:11:09,290 --> 01:11:22,190
So suppose we know-- so we
know the set of all call prices
1264
01:11:22,190 --> 01:11:32,190
with maturity T_1, for all
K, and the set of all call
1265
01:11:32,190 --> 01:11:41,560
prices with maturity
T_2 for all K.
1266
01:11:41,560 --> 01:11:43,810
OK, so then we know
the distribution.
1267
01:11:43,810 --> 01:11:45,960
Well, there are
two distributions.
1268
01:11:45,960 --> 01:11:48,597
We know the distribution
of T_1 given S_t,
1269
01:11:48,597 --> 01:12:00,970
and-- but do we know
the distribution
1270
01:12:00,970 --> 01:12:03,930
of the stock at T_2 given T_1?
1271
01:12:10,930 --> 01:12:23,250
More of a general point--
suppose I know this for all T.
1272
01:12:23,250 --> 01:12:24,800
Let's put T_0 here.
1273
01:12:27,800 --> 01:12:32,365
OK, I know all option prices of
all maturities and all strikes.
1274
01:12:35,230 --> 01:12:52,990
Can I determine the stochastic
process for S_T over this time?
1275
01:12:52,990 --> 01:12:56,920
Is the underlying stochastic
process for the stock price
1276
01:12:56,920 --> 01:12:59,500
fully determined by
knowing all call option
1277
01:12:59,500 --> 01:13:01,250
prices for all strikes
and all maturities?
1278
01:13:04,460 --> 01:13:07,320
The marginal distributions or
the conditional distributions
1279
01:13:07,320 --> 01:13:08,980
for all maturities
are determined,
1280
01:13:08,980 --> 01:13:12,416
because we know that here.
1281
01:13:12,416 --> 01:13:13,790
Well, you'll
probably see this is
1282
01:13:13,790 --> 01:13:16,030
a rephrasing of a
finite-dimensional problem
1283
01:13:16,030 --> 01:13:17,802
from probability.
1284
01:13:17,802 --> 01:13:18,510
The answer is no.
1285
01:13:25,190 --> 01:13:26,890
And the reason to
think about is,
1286
01:13:26,890 --> 01:13:29,500
if I know all the--
my intuition for this
1287
01:13:29,500 --> 01:13:34,000
is if I know all the
distributions that--
1288
01:13:34,000 --> 01:13:36,900
think about just a denser
and denser grid of times
1289
01:13:36,900 --> 01:13:39,400
that I know the distribution
of-- getting closer and closer.
1290
01:13:39,400 --> 01:13:44,757
I can still allow the stock to
flip instantaneously quickly.
1291
01:13:44,757 --> 01:13:47,090
Imagine they're all essentially
symmetric distributions,
1292
01:13:47,090 --> 01:13:49,048
and they're all roughly
the same expanding out.
1293
01:13:49,048 --> 01:13:54,060
I can imagine the stock
flipping discontinuously
1294
01:13:54,060 --> 01:13:56,220
over an arbitrarily
small time interval.
1295
01:13:56,220 --> 01:14:00,930
So without a constraint on the
continuity of this process,
1296
01:14:00,930 --> 01:14:03,190
or mathematical constraints
on this process,
1297
01:14:03,190 --> 01:14:07,870
you can't determine the
actual process for the stock,
1298
01:14:07,870 --> 01:14:11,920
even given all the option
prices-- call option prices.
1299
01:14:11,920 --> 01:14:15,386
So there are two--
so Emanuel Derman,
1300
01:14:15,386 --> 01:14:20,390
who was at Goldman Sachs,
now at Columbia-- and Bruno
1301
01:14:20,390 --> 01:14:24,530
Dupire-- who's, I think,
still at Bloomberg--
1302
01:14:24,530 --> 01:14:27,360
this is the early '90s--
basically determined
1303
01:14:27,360 --> 01:14:28,665
the conditions that you need.
1304
01:14:28,665 --> 01:14:30,680
And the basic conditions
are that just the stock
1305
01:14:30,680 --> 01:14:32,412
has to be a diffusion process.
1306
01:14:32,412 --> 01:14:36,100
If it is a diffusion
process-- the random term is
1307
01:14:36,100 --> 01:14:39,130
Brownian motion-- then it is,
actually, fully determined.
1308
01:14:39,130 --> 01:14:41,690
And it's a really
nice, elegant result.
1309
01:14:41,690 --> 01:14:44,360
So this is what gets
mathematically quite
1310
01:14:44,360 --> 01:14:47,740
nice, and a little tricky.
1311
01:14:47,740 --> 01:14:51,530
But there's a practical
implication of this,
1312
01:14:51,530 --> 01:14:58,770
as well, which is in practice,
I will know a finite subset
1313
01:14:58,770 --> 01:15:01,260
of call options.
1314
01:15:01,260 --> 01:15:04,510
Those prices will be
available to me in the market.
1315
01:15:04,510 --> 01:15:06,750
So they will be given.
1316
01:15:06,750 --> 01:15:11,400
So one thing I know
for sure is that even
1317
01:15:11,400 --> 01:15:15,690
with a very densely set
of call option prices,
1318
01:15:15,690 --> 01:15:18,330
there will be some
other derivative prices
1319
01:15:18,330 --> 01:15:21,840
whose price is not
exactly determined
1320
01:15:21,840 --> 01:15:22,810
by that set of calls.
1321
01:15:22,810 --> 01:15:25,660
Because in particular, I
know that the set of calls
1322
01:15:25,660 --> 01:15:28,050
does not determine the
underlying stochastic process,
1323
01:15:28,050 --> 01:15:30,050
even if I knew all of them.
1324
01:15:30,050 --> 01:15:33,140
So that's a very important
thing for traders to understand,
1325
01:15:33,140 --> 01:15:36,690
is that even if I know a lot
of market information-- so I'm
1326
01:15:36,690 --> 01:15:39,690
given here are the
prices of a large number
1327
01:15:39,690 --> 01:15:43,060
of European options, European
call options I can trade--
1328
01:15:43,060 --> 01:15:45,940
there may be a complex or
nonstandard derivative product,
1329
01:15:45,940 --> 01:15:49,160
whose price is not
determined uniquely, simply
1330
01:15:49,160 --> 01:15:50,680
by knowing those options.
1331
01:15:50,680 --> 01:15:52,155
And that is one
of the challenges
1332
01:15:52,155 --> 01:15:55,730
for some of the quant groups.
1333
01:15:55,730 --> 01:16:00,190
So anyway, with that, that
is all I wanted to convey.
1334
01:16:00,190 --> 01:16:01,660
I'm happy to take
some questions.
1335
01:16:01,660 --> 01:16:04,474
And thank you for your time.
1336
01:16:04,474 --> 01:16:05,473
Thank you for having me.
1337
01:16:05,473 --> 01:16:06,952
I appreciate it.
1338
01:16:06,952 --> 01:16:13,854
[APPLAUSE]
1339
01:16:13,854 --> 01:16:16,040
AUDIENCE: Yeah, I
have-- I was just
1340
01:16:16,040 --> 01:16:19,950
wondering, so you the call,
or the set of all calls
1341
01:16:19,950 --> 01:16:23,797
basically spans the space of
all possible payouts, right?
1342
01:16:23,797 --> 01:16:24,630
STEPHEN BLYTHE: Yes.
1343
01:16:24,630 --> 01:16:26,005
AUDIENCE: So I
was just wondering
1344
01:16:26,005 --> 01:16:30,700
if maybe if we could change,
and select some other such basis
1345
01:16:30,700 --> 01:16:32,457
for spanning it?
1346
01:16:32,457 --> 01:16:34,040
Instead of call,
maybe some other kind
1347
01:16:34,040 --> 01:16:39,260
of basic payoff that could
still span the same thing,
1348
01:16:39,260 --> 01:16:42,190
and maybe it's more easily
tradable, or something?
1349
01:16:42,190 --> 01:16:44,820
STEPHEN BLYTHE: Yeah, that's
a good-- there must be many,
1350
01:16:44,820 --> 01:16:48,060
if I can-- but this,
given that this
1351
01:16:48,060 --> 01:16:54,140
is the simplest expansion
of the function g,
1352
01:16:54,140 --> 01:16:56,650
an arbitrary function g,
and the second term comes in
1353
01:16:56,650 --> 01:17:01,300
with this call payout,
gives us this elegance.
1354
01:17:01,300 --> 01:17:04,507
Of course, if I know
all the digitals,
1355
01:17:04,507 --> 01:17:06,340
I know the cumulative
distribution function,
1356
01:17:06,340 --> 01:17:07,550
and therefore, I
know the density.
1357
01:17:07,550 --> 01:17:09,050
So I mean, the
digitals do the same.
1358
01:17:09,050 --> 01:17:10,784
And in fact,
Arrow-Debreu securities,
1359
01:17:10,784 --> 01:17:13,200
which is building blocks, which
is something that pays off
1360
01:17:13,200 --> 01:17:16,200
one in a particular
state, sample state,
1361
01:17:16,200 --> 01:17:18,508
also are building blocks.
1362
01:17:18,508 --> 01:17:20,047
AUDIENCE: [INAUDIBLE].
1363
01:17:20,047 --> 01:17:21,630
STEPHEN BLYTHE: I
mean, sometimes, you
1364
01:17:21,630 --> 01:17:23,440
could think about an
arbitrary basis that
1365
01:17:23,440 --> 01:17:27,160
will span-- an arbitrary
basis of functions that will
1366
01:17:27,160 --> 01:17:28,560
span any continuous function.
1367
01:17:28,560 --> 01:17:31,709
And sometimes, you can do it
in any polynomial expansion.
1368
01:17:31,709 --> 01:17:33,500
If I have a price and
any of those payouts,
1369
01:17:33,500 --> 01:17:35,356
and I've got my spanning set.
1370
01:17:35,356 --> 01:17:36,730
But this is the
most elegant one.
1371
01:17:41,380 --> 01:17:42,464
Yeah, next question there.
1372
01:17:42,464 --> 01:17:44,713
AUDIENCE: I have a question
about the last [INAUDIBLE]
1373
01:17:44,713 --> 01:17:45,295
mentioned.
1374
01:17:45,295 --> 01:17:50,225
[INAUDIBLE] because
market's incomplete,
1375
01:17:50,225 --> 01:17:54,255
so you can not sort
of use call option
1376
01:17:54,255 --> 01:17:57,412
to replicate the stock itself.
1377
01:17:57,412 --> 01:17:59,120
STEPHEN BLYTHE: You
can use a call option
1378
01:17:59,120 --> 01:18:02,467
to replicate a stock.
1379
01:18:02,467 --> 01:18:03,800
As long as you have zero-coupon.
1380
01:18:03,800 --> 01:18:06,580
You can see from here, I can
just reorganize everything here
1381
01:18:06,580 --> 01:18:09,640
to zero-coupon bond
stock, and a set of calls
1382
01:18:09,640 --> 01:18:12,880
will span anything-- with
maturity T. What they're
1383
01:18:12,880 --> 01:18:17,080
sort of saying is, if I
have this strange process
1384
01:18:17,080 --> 01:18:21,360
with jumps, and flips,
and discontinuities,
1385
01:18:21,360 --> 01:18:24,320
then the market is incomplete,
I guess is what this is saying.
1386
01:18:24,320 --> 01:18:25,860
AUDIENCE: OK, yeah,
so [INAUDIBLE]
1387
01:18:25,860 --> 01:18:27,130
is due to the incompleteness.
1388
01:18:27,130 --> 01:18:28,546
STEPHEN BLYTHE:
Yeah, in the sense
1389
01:18:28,546 --> 01:18:31,700
of most finance-- in fact,
all continuous-time finance
1390
01:18:31,700 --> 01:18:33,660
will assume there's
some diffusion
1391
01:18:33,660 --> 01:18:38,540
process for-- some
process for stock,
1392
01:18:38,540 --> 01:18:41,530
which has some Brownian motion.
1393
01:18:41,530 --> 01:18:43,030
There's some function
here, and some
1394
01:18:43,030 --> 01:18:45,070
function for the drift term.
1395
01:18:45,070 --> 01:18:48,890
In that case, then all the
call prices do determine.
1396
01:18:48,890 --> 01:18:52,986
If you think there's some
exogenous flipping parameter--
1397
01:18:52,986 --> 01:18:54,110
that's my intuition for it.
1398
01:18:54,110 --> 01:18:56,110
So there's some-- that's
why this is incomplete.
1399
01:18:56,110 --> 01:18:57,235
So this will not determine.
1400
01:18:57,235 --> 01:18:59,443
So in particular, I could
know all these call prices.
1401
01:18:59,443 --> 01:19:01,900
Then I could determine a
particular derivative product.
1402
01:19:01,900 --> 01:19:03,680
It could be the
number of times that
1403
01:19:03,680 --> 01:19:05,170
in an arbitrarily
small interval,
1404
01:19:05,170 --> 01:19:06,904
the stock flips this many times.
1405
01:19:06,904 --> 01:19:08,820
I mean, there's some--
you can create whatever
1406
01:19:08,820 --> 01:19:10,319
you like for a
derivative that would
1407
01:19:10,319 --> 01:19:11,846
be incomplete for these calls.
1408
01:19:11,846 --> 01:19:14,220
AUDIENCE: So in this case, go
back to a previous question
1409
01:19:14,220 --> 01:19:17,200
as we just mentioned-- the
second-order derivative
1410
01:19:17,200 --> 01:19:20,295
of a call option with
respect to a strike
1411
01:19:20,295 --> 01:19:22,510
is [INAUDIBLE]
risk-neutral density.
1412
01:19:22,510 --> 01:19:26,555
So in this case, it was not--
that risk-neutral density,
1413
01:19:26,555 --> 01:19:28,630
or a particular
instance of that,
1414
01:19:28,630 --> 01:19:30,089
rather, is not
uniquely determined.
1415
01:19:30,089 --> 01:19:31,921
STEPHEN BLYTHE: No, the
risk-neutral density
1416
01:19:31,921 --> 01:19:32,930
is uniquely determined.
1417
01:19:32,930 --> 01:19:35,670
The stochastic process
for S_t over all time
1418
01:19:35,670 --> 01:19:38,220
is not uniquely determined.
1419
01:19:38,220 --> 01:19:42,530
So this is uniquely determined
by call option prices.
1420
01:19:45,490 --> 01:19:48,814
That is uniquely determined.
1421
01:19:48,814 --> 01:19:50,480
But knowing the
conditional distribution
1422
01:19:50,480 --> 01:19:55,760
of S capital T given S
little t doesn't determine
1423
01:19:55,760 --> 01:19:57,064
the process of the stock price.
1424
01:19:57,064 --> 01:19:59,355
To get there-- I can think
of infinitely many processes
1425
01:19:59,355 --> 01:20:03,580
of the stock price that can
give rise to this distribution.
1426
01:20:03,580 --> 01:20:04,824
That's what's not determined.
1427
01:20:09,280 --> 01:20:11,540
The terminal distribution
is uniquely determined
1428
01:20:11,540 --> 01:20:13,653
by the call option
prices-- nothing else.
1429
01:20:13,653 --> 01:20:15,920
AUDIENCE: So in this case,
if we take Z over theta,
1430
01:20:15,920 --> 01:20:19,401
so we'll get a particular
risk-neutral density
1431
01:20:19,401 --> 01:20:20,590
for each particular stock?
1432
01:20:20,590 --> 01:20:21,881
STEPHEN BLYTHE: That's correct.
1433
01:20:25,465 --> 01:20:26,590
Right, thank you very much.
1434
01:20:26,590 --> 01:20:28,140
Appreciate it.