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PROFESSOR: So let's start with
a simple but quite illustrative

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example.

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So suppose you're a bookie.

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And what a bookie does--
he sets bets on the horses,

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sets the odds, and
then pays money back.

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Probably collects a fee
somewhere in between.

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So suppose he is a
good bookie and he

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knows quite well the horses,
and there are two horses.

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He knows that for sure one
horse has 20% chance of winning

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and another horse has
80% chance of winning.

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Obviously, the
general public doesn't

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have all of this information.

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So they place a bet
slightly differently.

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And then there is $10,000
bet on one horse and $50,000

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bet on another horse.

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Well, bookie is sure that he
possesses good information.

00:01:20.040 --> 00:01:23.860
So he-- suppose he
sets the odds according

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to real-life probability.

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So he sets it four to one.

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What would be possible
outcomes of the race for him?

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Monetary.

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So suppose the first horse wins.

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Then what happens?

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He has to pay back $10,000
and four times more.

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So he pays out $50,000.

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And he receives $60,000, right?

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So he can keep
$10,000 out of it.

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OK.

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So what happens is the other
more probable horse wins.

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Well, he'll have to pay back the
$50,000 and one quarter of it,

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which is $12.25.

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So at the end, he'll
pay 62 1/2 thousand,

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while he collected
$60,000, out right?

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So he will-- in this
situation, he will lose $2,500.

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Well, all in all, he
expects to make nothing.

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So he probably could
collect enough fees

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to cover his potential loss.

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But there is certainly a
variability in outcomes.

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He can win a lot.

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He can lose some.

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Now, what if he forgets
about his knowledge

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about the real-life
probabilities of horses winning

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or losing and instead sets bets
according to the amount which

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we are already bet.

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According to the
market, effectively.

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So what if he sets
the odds five to one,

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according to the bets placed?

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Well, in this situation,
if the first horse wins,

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he pays back 10 plus
5 times 10, so 60.

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He is 0.

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And if the second horse wins,
he pays back 50 plus 1/5 of 50,

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plus another 10.

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Again 60.

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So no matter which horse
wins, he will get 0.

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We're 100% sure.

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And if he collects
some fee on top of it,

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he will make a riskless profit.

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And that's how, actually,
bookies are operating.

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So it's a simple example.

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But it gives us a first idea
of how a risk-neutral framework

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and risk-neutral pricing works.

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So we are, here,
not in the business

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of making bets on horses.

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We are actually in the business
of pricing derivatives.

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So we will talk about the
simplest possible derivatives--

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mostly derivatives on stocks.

00:04:06.160 --> 00:04:10.470
But there are more
complicated derivatives,

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underlying for which could be
interest rates, bonds, swaps,

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commodities, whatever.

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So a derivative
contract is some--

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in general speaking, a
formal pay-out connected

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to underlying.

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Usually, the underlying
is a liquid instrument

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which is traded on exchanges.

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And derivative may be
traded on exchanges.

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Actually, quite a
few equity options

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are traded on exchanges.

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But in general, they are
over-the-counter contracts

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where two counterparties just
agree on some kind of pay-out.

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One of the simpler derivatives
is a forward contract.

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So what is a forward contract?

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A forward contract is a
contract where one party agrees

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to buy an asset from another
party for a price which

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is agreed today.

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Usually, this forward
price is set in such a way

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that right now, no
money changes hands.

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Right?

00:05:20.040 --> 00:05:22.600
And here is an example.

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Well, suppose there
is a stock which,

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right now, is priced at $80.

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And this is the
forward for two years.

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So somebody agrees
to buy the stock

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in two years for this price.

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And not surprisingly, I
somehow set this price such

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that currently the value
of the contract is 0.

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And we'll see how I'll
come up with the price.

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So this blue line is
actually the pay-out,

00:05:49.340 --> 00:05:51.055
what will happen at the end.

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Right?

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The pay-out,
depending-- the graph

00:05:55.990 --> 00:06:02.330
of F at time T, the
determination time or expiry--

00:06:02.330 --> 00:06:04.131
how it depends on
the stock price.

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Right?

00:06:04.630 --> 00:06:07.170
So obviously, the
pay-out is S minus K,

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where S is the stock price,
so it's a linear function.

00:06:12.030 --> 00:06:14.710
It turns out that the counter
price is also a linear function

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but slightly shifted.

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And we'll see how come
it's slightly shifted

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and how much it
should be shifted.

00:06:22.410 --> 00:06:28.090
And K is usually referred
to as a strike price.

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Another slightly more
complicated contract

00:06:30.400 --> 00:06:33.200
is called a call option.

00:06:33.200 --> 00:06:37.570
So if previously the
forward is an obligation

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to buy the asset
for an agreed price,

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call option is
actually an option

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to buy an asset at the
agreed price today.

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You can view it--
a call option can

00:06:51.130 --> 00:06:54.760
be viewed as kind of
insurance that the--

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against the asset going down.

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Basically the pay-out
is always positive.

00:07:01.059 --> 00:07:02.100
You can never lose money.

00:07:02.100 --> 00:07:03.660
On the forward,
you can lose money.

00:07:03.660 --> 00:07:05.240
You agree on the price.

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The asset ends up being
lower than this price,

00:07:07.540 --> 00:07:09.271
but you still have to buy it.

00:07:09.271 --> 00:07:09.770
Right?

00:07:09.770 --> 00:07:16.470
Here, if the asset ends up
at expiry below strike price

00:07:16.470 --> 00:07:21.720
or out of the money, then
the pay-out will be 0.

00:07:21.720 --> 00:07:26.880
If, on the other hand, it ends
up being above the strike price

00:07:26.880 --> 00:07:30.580
or, it's called, the
option is in the money.

00:07:30.580 --> 00:07:37.530
Then the pay-out will
be S minus K as before.

00:07:37.530 --> 00:07:41.590
So in mathematical
terms, the pay-out

00:07:41.590 --> 00:07:44.430
is maximum of S minus K and 0.

00:07:44.430 --> 00:07:44.990
Right?

00:07:44.990 --> 00:07:49.730
And that's what happens at
expiry time-- this blue line.

00:07:49.730 --> 00:07:53.350
So what is the price
of this option now?

00:07:53.350 --> 00:07:57.250
Well, obviously it
should be slightly above

00:07:57.250 --> 00:08:01.810
because even if now
the asset is slightly

00:08:01.810 --> 00:08:04.050
out of the money--
below strike price--

00:08:04.050 --> 00:08:07.030
there is some volatility to
it, and there is a probability

00:08:07.030 --> 00:08:10.270
that we will still end up
in the money at expiry.

00:08:10.270 --> 00:08:13.170
So you would be
willing-- you should

00:08:13.170 --> 00:08:16.190
be willing to pay
something for this.

00:08:16.190 --> 00:08:19.250
Obviously, if it's way out
of the money, it should be 0.

00:08:19.250 --> 00:08:20.020
Right?

00:08:20.020 --> 00:08:23.900
On the other hand, if it's
way in the money, in fact,

00:08:23.900 --> 00:08:26.240
it should be just as forward.

00:08:26.240 --> 00:08:27.120
And in fact, it is.

00:08:27.120 --> 00:08:30.170
We'll see because
the probability

00:08:30.170 --> 00:08:34.850
for the asset going back to
the strike price and below

00:08:34.850 --> 00:08:36.940
will be low.

00:08:36.940 --> 00:08:39.880
And the Black-Scholes equation
and Black-Scholes formula

00:08:39.880 --> 00:08:43.240
is exactly the solution
for this curved line,

00:08:43.240 --> 00:08:46.690
which we'll see in a second.

00:08:46.690 --> 00:08:51.710
Another simple contract, which
is kind of dual to call option,

00:08:51.710 --> 00:08:54.110
is a put option.

00:08:54.110 --> 00:08:57.230
So put option, on
the contrary, is

00:08:57.230 --> 00:09:01.390
a bet on the asset going
down, rather than up.

00:09:01.390 --> 00:09:01.890
Right?

00:09:01.890 --> 00:09:07.290
So the pay-out is maximum
of K minus S and 0.

00:09:07.290 --> 00:09:10.490
So it's kind of reversed.

00:09:10.490 --> 00:09:14.730
Also a ramp function,
at maturity.

00:09:14.730 --> 00:09:19.590
And here is the current price.

00:09:19.590 --> 00:09:22.220
Again, even if
it's in the money--

00:09:22.220 --> 00:09:26.190
if it's way in the money,
we expect it to be 0.

00:09:26.190 --> 00:09:30.410
If it's way in the
money, we expect

00:09:30.410 --> 00:09:34.580
it to be slightly below forward,
just because of this counting.

00:09:37.270 --> 00:09:38.500
OK.

00:09:38.500 --> 00:09:44.780
So and here are a few--
three main points,

00:09:44.780 --> 00:09:49.140
which we'll try to
follow, through the class.

00:09:49.140 --> 00:09:52.220
So first of all,
what we'll see--

00:09:52.220 --> 00:09:56.570
that if we have current
price of the underlying

00:09:56.570 --> 00:10:03.310
and some assumptions on how
the market or the underlying

00:10:03.310 --> 00:10:08.040
behaves, there is
actually no uncertainty

00:10:08.040 --> 00:10:11.040
in the price of the
option, obviously,

00:10:11.040 --> 00:10:12.140
if we fix the pay-out.

00:10:12.140 --> 00:10:13.380
Right?

00:10:13.380 --> 00:10:14.980
So somehow there
is no uncertainty.

00:10:14.980 --> 00:10:18.082
It's completely
deterministic, once we

00:10:18.082 --> 00:10:19.290
know the price of underlying.

00:10:22.960 --> 00:10:27.660
The other interesting
fact, which we'll find out,

00:10:27.660 --> 00:10:29.450
is actually
risk-neutrality, meaning

00:10:29.450 --> 00:10:32.930
that in fact, the
price of the option

00:10:32.930 --> 00:10:36.480
has nothing to do with the
risk preferences of market

00:10:36.480 --> 00:10:40.840
participants or counter-parties.

00:10:40.840 --> 00:10:44.610
It actually only depends on
the dynamics of the stock,

00:10:44.610 --> 00:10:48.470
only depends on the
volatility of the stock.

00:10:48.470 --> 00:10:50.922
And finally, the
most important idea

00:10:50.922 --> 00:10:53.740
of this class-- that
mathematical apparatus

00:10:53.740 --> 00:10:57.750
allows you to figure out how
much this deterministic option

00:10:57.750 --> 00:10:59.002
price is now.

00:11:02.770 --> 00:11:09.830
So let's consider a very simple
example, a very simple market,

00:11:09.830 --> 00:11:10.880
two-period.

00:11:10.880 --> 00:11:14.710
So suppose our time is
discrete, and we are

00:11:14.710 --> 00:11:17.990
one step before the maturity.

00:11:17.990 --> 00:11:22.800
So right now, our
stock has price at 0.

00:11:22.800 --> 00:11:27.990
And there is some derivative
f_0 with some pay-out.

00:11:27.990 --> 00:11:29.440
We'll consider a few of those.

00:11:29.440 --> 00:11:30.370
Right?

00:11:30.370 --> 00:11:33.950
Also, we'll add to
the mix a bit of cash.

00:11:33.950 --> 00:11:34.450
Right?

00:11:37.380 --> 00:11:40.000
Some amount of
riskless cash B_0.

00:11:40.000 --> 00:11:45.700
And riskless meaning that
it grows exponentially

00:11:45.700 --> 00:11:47.800
with some interest rate r.

00:11:47.800 --> 00:11:49.360
And there is no uncertainty.

00:11:49.360 --> 00:11:53.820
It's completely-- if you
have now B_0, we know then,

00:11:53.820 --> 00:11:59.950
in time dt, our B_0
will grow exponentially.

00:11:59.950 --> 00:12:02.080
It will become B e to the rt.

00:12:02.080 --> 00:12:05.172
So a bond, basically,
zero-coupon bond.

00:12:05.172 --> 00:12:07.835
Or money market account, rather.

00:12:07.835 --> 00:12:14.980
If you go to Cambridge
Savings Bank, put $1 in today,

00:12:14.980 --> 00:12:20.270
then in a year, you'll get
$1 and basically nothing

00:12:20.270 --> 00:12:22.900
because interest rates are 0.

00:12:22.900 --> 00:12:27.115
So in time dt, we will assume
with some probability p,

00:12:27.115 --> 00:12:31.370
our market can go to the state
where stock becomes S_1--

00:12:31.370 --> 00:12:33.370
the price of stock becomes S_1.

00:12:33.370 --> 00:12:36.580
Our bond grows exponentially--
no uncertainty.

00:12:36.580 --> 00:12:40.210
And our derivative becomes f_1.

00:12:40.210 --> 00:12:44.550
Or with probability 1 minus p--
only two states, so-- our stock

00:12:44.550 --> 00:12:46.200
becomes S_2.

00:12:46.200 --> 00:12:47.200
Bond stays the same.

00:12:47.200 --> 00:12:51.870
And the derivative is some f_2.

00:12:51.870 --> 00:12:58.870
So let's start with our simple
contract, the forward contract.

00:12:58.870 --> 00:13:04.670
So one can naively
approach a problem, trying

00:13:04.670 --> 00:13:08.500
to get the price
of the derivative,

00:13:08.500 --> 00:13:12.740
using the real-world
probabilities, p and 1 minus p.

00:13:12.740 --> 00:13:13.240
Right?

00:13:13.240 --> 00:13:18.030
So we know that the pay-out
is S minus K. That's given.

00:13:18.030 --> 00:13:21.452
So one would say
that if we know we

00:13:21.452 --> 00:13:24.290
are one step before the
pay-out, so let's just

00:13:24.290 --> 00:13:26.590
compute expected
value of the pay-out,

00:13:26.590 --> 00:13:31.797
using real-world
probabilities, get this value.

00:13:31.797 --> 00:13:33.380
And actually, what
we are looking here

00:13:33.380 --> 00:13:38.340
is to set K such that the
price now at time t is 0.

00:13:38.340 --> 00:13:39.840
That's usual convention.

00:13:39.840 --> 00:13:48.500
So we'll then set K
to this probability,

00:13:48.500 --> 00:13:51.760
to this number, which depends
on real-world probability

00:13:51.760 --> 00:13:54.680
and obviously depends on
the stock price at expiry.

00:13:57.450 --> 00:14:00.130
But obviously, we don't know
real-world probabilities.

00:14:00.130 --> 00:14:00.830
We can guess.

00:14:00.830 --> 00:14:06.860
We can say, oh, this stock is
as likely to go up then down.

00:14:06.860 --> 00:14:12.560
Then it's just an average of end
stock prices or something else.

00:14:12.560 --> 00:14:14.280
But it's all hand-wavy.

00:14:14.280 --> 00:14:17.630
And actually, we
never will be right.

00:14:17.630 --> 00:14:21.320
Instead of doing
this-- we're kind

00:14:21.320 --> 00:14:26.270
of following bookie example--
let's try to do something else.

00:14:26.270 --> 00:14:27.920
Let's think a little bit.

00:14:27.920 --> 00:14:33.120
So we have a stock which
is trading at market now

00:14:33.120 --> 00:14:36.070
for the price S_0.

00:14:36.070 --> 00:14:41.740
How about we go to the bank and
borrow S_0 dollars right now

00:14:41.740 --> 00:14:44.810
and immediately go to the
market and buy the stock.

00:14:44.810 --> 00:14:47.910
So right now we are net 0.

00:14:47.910 --> 00:14:49.330
We borrowed S_0.

00:14:49.330 --> 00:14:51.910
We paid it immediately
to buy the stock.

00:14:51.910 --> 00:14:53.990
So we have stock at hand.

00:14:53.990 --> 00:14:57.830
Then we'll wait for one period.

00:14:57.830 --> 00:15:00.910
And at the same
time-- sorry-- we

00:15:00.910 --> 00:15:03.970
enter on the short side
of the forward contract.

00:15:03.970 --> 00:15:10.830
So we agree to sell the
stock for some price K_0.

00:15:10.830 --> 00:15:16.290
So in dt, in one period of
time, the contract expires.

00:15:16.290 --> 00:15:17.640
We already have stock.

00:15:17.640 --> 00:15:21.540
So we just go and exchange
it for K_0 dollars.

00:15:21.540 --> 00:15:23.870
Right?

00:15:23.870 --> 00:15:29.310
But at the same time, we need
to repay our loan which now have

00:15:29.310 --> 00:15:34.122
become S_0 times e to the r*dt.

00:15:34.122 --> 00:15:35.330
This is deterministic, right?

00:15:35.330 --> 00:15:36.640
We borrowed S_0.

00:15:36.640 --> 00:15:41.360
In time dt, it became
S times e to the r*dt.

00:15:41.360 --> 00:15:44.320
So what's our net?

00:15:44.320 --> 00:15:51.560
The net is K_0 minus
S times e r*dt.

00:15:51.560 --> 00:15:56.490
So suppose K_0 is
greater than this value.

00:15:56.490 --> 00:15:58.750
Then we made riskless profit.

00:15:58.750 --> 00:16:06.200
There is no risk in the
strategy which we proposed.

00:16:06.200 --> 00:16:07.740
So this is good.

00:16:07.740 --> 00:16:11.410
But why wouldn't everybody
do it all day long?

00:16:11.410 --> 00:16:14.910
On the other hand, if
K_0 is less than S_0,

00:16:14.910 --> 00:16:18.200
that's a loss for sure.

00:16:18.200 --> 00:16:21.540
And if anybody thinks,
as we did-- and we assume

00:16:21.540 --> 00:16:24.390
that everybody can
do it-- then nobody

00:16:24.390 --> 00:16:27.220
would want to enter
it, which means

00:16:27.220 --> 00:16:34.130
that in order for our
forward to be price 0 now,

00:16:34.130 --> 00:16:38.580
the strike price has to
be equal to this amount.

00:16:38.580 --> 00:16:42.550
And there is no
uncertainty about it.

00:16:42.550 --> 00:16:48.690
So let's stop and
think a little bit.

00:16:48.690 --> 00:16:58.450
Well, actually, just
to see how it works.

00:16:58.450 --> 00:17:04.530
And that's exactly why I
set this K to this number.

00:17:04.530 --> 00:17:07.140
So by the way, who
can tell me which

00:17:07.140 --> 00:17:08.790
interest rate does it imply?

00:17:15.599 --> 00:17:24.020
If our strike-- our stock price
is $80, our strike is 88.41.

00:17:24.020 --> 00:17:29.604
And the expiry is in two
years, approximately.

00:17:29.604 --> 00:17:31.460
AUDIENCE: 2.5?

00:17:31.460 --> 00:17:33.120
PROFESSOR: 2.5.

00:17:33.120 --> 00:17:37.530
So in two years, it will be 5%.

00:17:37.530 --> 00:17:39.920
So roughly speaking,
without compounding, it

00:17:39.920 --> 00:17:43.110
should be 5% of-- 80 plus 5%.

00:17:43.110 --> 00:17:43.920
It would be 84.

00:17:47.800 --> 00:17:48.940
So 10% for two years.

00:17:48.940 --> 00:17:51.245
So the interest rate is 5%.

00:17:51.245 --> 00:17:51.745
Yeah.

00:17:51.745 --> 00:17:52.245
So yeah.

00:17:52.245 --> 00:17:58.270
That's actually exactly 5
exponentially compounded.

00:17:58.270 --> 00:17:58.850
Yeah.

00:17:58.850 --> 00:18:01.690
Well, in a good world--
probably five years ago,

00:18:01.690 --> 00:18:03.190
that's how it would work.

00:18:03.190 --> 00:18:06.840
The two-years interest rates
now, the last time I checked,

00:18:06.840 --> 00:18:08.520
was, I think, 30 pips.

00:18:08.520 --> 00:18:13.855
We can check where the
bond is trading now.

00:18:13.855 --> 00:18:14.790
All right.

00:18:14.790 --> 00:18:16.240
Give me a sec.

00:18:16.240 --> 00:18:16.740
Now.

00:18:22.208 --> 00:18:23.080
Yep.

00:18:23.080 --> 00:18:26.920
32 1/2 basis points.

00:18:26.920 --> 00:18:30.220
1.6 basis points up,
since the morning.

00:18:30.220 --> 00:18:31.370
Quite a bit, by the way.

00:18:31.370 --> 00:18:32.300
So yeah.

00:18:32.300 --> 00:18:34.480
So right now interest
rates are basically 0.

00:18:34.480 --> 00:18:38.260
So these two lines would
be very close right now

00:18:38.260 --> 00:18:42.460
if we were for two
years, in that case.

00:18:42.460 --> 00:18:49.920
So coming back to our example.

00:18:49.920 --> 00:18:52.710
So what's important here?

00:18:52.710 --> 00:18:55.220
How did we arrive to
this strike price,

00:18:55.220 --> 00:18:58.430
or to this price of
the forward contract?

00:18:58.430 --> 00:19:03.810
We, in fact, tried-- we
took some amount of stock.

00:19:03.810 --> 00:19:06.650
In this particular case, it
was the whole price of stock.

00:19:06.650 --> 00:19:11.690
We took some amount of cash, and
by combining these two pieces,

00:19:11.690 --> 00:19:14.664
we somehow replicated
the final pay-off.

00:19:14.664 --> 00:19:15.510
Right?

00:19:15.510 --> 00:19:20.490
And that's the general idea
of risk-neutral pricing

00:19:20.490 --> 00:19:22.560
and replicating portfolio.

00:19:22.560 --> 00:19:25.410
What we will try to do,
in the rest of the class,

00:19:25.410 --> 00:19:31.640
is take a pay-off and try to
find a replicating portfolio,

00:19:31.640 --> 00:19:35.910
maybe more complicated, maybe
a dynamic such that at the end,

00:19:35.910 --> 00:19:38.510
this replicating portfolio
will be exactly our pay-off.

00:19:38.510 --> 00:19:39.420
Right?

00:19:39.420 --> 00:19:40.890
And what would it mean?

00:19:40.890 --> 00:19:46.380
Well, obviously it would
mean that the current price

00:19:46.380 --> 00:19:48.020
of the derivative
should be the price

00:19:48.020 --> 00:19:50.472
of our replicating
portfolio right now.

00:19:50.472 --> 00:19:51.950
Right?

00:19:51.950 --> 00:19:58.060
And that's how the
risk-neutral pricing works.

00:19:58.060 --> 00:20:01.180
So we are still in
this simple situation.

00:20:01.180 --> 00:20:05.760
But we will try to price
a general pay-off f_1--

00:20:05.760 --> 00:20:07.850
a general pay-off f.

00:20:07.850 --> 00:20:08.820
Right?

00:20:08.820 --> 00:20:10.650
And here's how it goes.

00:20:10.650 --> 00:20:16.760
So we still will try to form
our replicating portfolio out

00:20:16.760 --> 00:20:21.930
of the bond, of some amount of
bond, and some amount of stock.

00:20:21.930 --> 00:20:28.690
And we'll say that we will
need a S_1 and b of the bond.

00:20:28.690 --> 00:20:29.730
Right?

00:20:29.730 --> 00:20:35.860
And we'll try to find a and
b such that no matter what

00:20:35.860 --> 00:20:41.940
the real-world probability
is, at one step maturity,

00:20:41.940 --> 00:20:44.202
we'll replicate our
pay-off exactly.

00:20:44.202 --> 00:20:45.910
And fortunately, in
this particular case,

00:20:45.910 --> 00:20:46.810
it's very doable.

00:20:46.810 --> 00:20:48.310
It's just two equations.

00:20:48.310 --> 00:20:49.620
We use two variables.

00:20:49.620 --> 00:20:51.510
We should be able to do it.

00:20:51.510 --> 00:20:55.290
And we can solve it
and find this a and b.

00:20:55.290 --> 00:20:58.780
Then we'll substitute
them in the formula.

00:20:58.780 --> 00:20:59.350
Right?

00:20:59.350 --> 00:21:02.860
Take the current price of
the stock, which we know,

00:21:02.860 --> 00:21:11.010
and some cash, and
find the current price

00:21:11.010 --> 00:21:13.000
of the derivative.

00:21:13.000 --> 00:21:13.500
Right?

00:21:13.500 --> 00:21:15.730
And this works-- it should
work for any derivative.

00:21:15.730 --> 00:21:17.105
It doesn't matter,
is it forward,

00:21:17.105 --> 00:21:20.820
call, put, or some
complicated option,

00:21:20.820 --> 00:21:23.000
as long as it is
deterministic at expiry.

00:21:26.040 --> 00:21:29.040
An interesting way,
though, to look at it

00:21:29.040 --> 00:21:32.750
is to rewrite this
formula slightly,

00:21:32.750 --> 00:21:40.760
in such a way, which does remind
us, taking an expected value,

00:21:40.760 --> 00:21:44.050
maybe discounting it because
this is expected value

00:21:44.050 --> 00:21:46.090
at some time in the future.

00:21:46.090 --> 00:21:48.770
But this probability--
and it is a probability

00:21:48.770 --> 00:21:53.620
because this number q,
here, is between 0 and 1.

00:21:53.620 --> 00:21:59.470
But this probability has
little to do with real world.

00:21:59.470 --> 00:22:00.000
Right?

00:22:00.000 --> 00:22:03.960
In fact, it's
something different.

00:22:03.960 --> 00:22:06.230
But such probability exists.

00:22:06.230 --> 00:22:10.680
And it's called-- the measure
where our stock behaves

00:22:10.680 --> 00:22:12.590
like this is called a
risk-neutral measure

00:22:12.590 --> 00:22:16.080
or martingale measure.

00:22:16.080 --> 00:22:18.690
And in this measure,
as we will see,

00:22:18.690 --> 00:22:20.890
the value of the derivative
will be just expected

00:22:20.890 --> 00:22:24.820
value of our pay-out.

00:22:24.820 --> 00:22:25.740
And that's-- yeah.

00:22:25.740 --> 00:22:28.800
That's what I'm
trying to say, here.

00:22:33.200 --> 00:22:37.850
So now let's get into
continuous world.

00:22:37.850 --> 00:22:40.260
Right?

00:22:40.260 --> 00:22:43.000
In continuous world, we'll
need some assumptions

00:22:43.000 --> 00:22:46.930
on the dynamics of
our stock underlying.

00:22:46.930 --> 00:22:54.660
And let's make an assumption
that it is log-normal.

00:22:54.660 --> 00:22:56.630
What does it mean
that it's log-normal?

00:22:56.630 --> 00:23:04.250
It means that the proportional
change of the stock,

00:23:04.250 --> 00:23:08.970
over infinitely small
amount of time dt,

00:23:08.970 --> 00:23:13.780
has some drift mu, and
some stochastic component,

00:23:13.780 --> 00:23:15.910
which is just Brownian Motion.

00:23:15.910 --> 00:23:16.950
Right?

00:23:16.950 --> 00:23:21.130
So this dW is
distributed normally

00:23:21.130 --> 00:23:26.920
with mean 0 and standard
deviation, which

00:23:26.920 --> 00:23:28.480
is actually square root of dt.

00:23:28.480 --> 00:23:31.140
That's how Brownian
Motion works.

00:23:31.140 --> 00:23:33.720
And that's extremely important,
that the standard deviation

00:23:33.720 --> 00:23:39.180
of Brownian Motion is
square root of delta t.

00:23:39.180 --> 00:23:40.195
And that's how it works.

00:23:42.990 --> 00:23:49.520
And again, we will use this
idea of replicating portfolio.

00:23:49.520 --> 00:23:52.650
What would it mean in this case?

00:23:52.650 --> 00:23:58.880
Well, we would like to find
such coefficients a and b,

00:23:58.880 --> 00:24:02.580
on this infinitely
small period of time dt,

00:24:02.580 --> 00:24:06.670
such that by combining
small changes in stock,

00:24:06.670 --> 00:24:10.380
with coefficient a, and
small changes in bond,

00:24:10.380 --> 00:24:14.120
with coefficient b,
will exactly replicate

00:24:14.120 --> 00:24:18.030
the change in the derivative--
in the pay-out of derivative--

00:24:18.030 --> 00:24:18.630
not pay-out.

00:24:18.630 --> 00:24:19.590
In the derivative.

00:24:19.590 --> 00:24:23.529
In the change of the derivative,
over this infinitely small time

00:24:23.529 --> 00:24:24.028
t.

00:24:27.240 --> 00:24:31.170
Well, to do this, we'll
need to use Ito's formula.

00:24:31.170 --> 00:24:33.040
Did you talk about Ito already?

00:24:33.040 --> 00:24:33.540
OK.

00:24:33.540 --> 00:24:34.040
Cool.

00:24:34.040 --> 00:24:35.320
That's great.

00:24:35.320 --> 00:24:38.480
So just to remind you
that Ito's formula

00:24:38.480 --> 00:24:41.580
is nothing more than the
Taylor rule, actually--

00:24:41.580 --> 00:24:45.560
the first
approximation up to dt.

00:24:45.560 --> 00:24:50.080
But because of the standard
deviation of the Brownian

00:24:50.080 --> 00:24:53.680
Motion being on the scale
of square root of t,

00:24:53.680 --> 00:24:55.410
we will need one
more term there.

00:24:55.410 --> 00:24:55.910
Right?

00:24:55.910 --> 00:24:59.130
So one term is df/dt by dt.

00:24:59.130 --> 00:25:01.480
Another is df by dS by dS.

00:25:01.480 --> 00:25:05.560
And the square of
dS now is actually

00:25:05.560 --> 00:25:07.740
of order of magnitude of dt.

00:25:07.740 --> 00:25:12.000
So we'll need a
quadratic term there.

00:25:12.000 --> 00:25:12.500
All right.

00:25:12.500 --> 00:25:21.290
So if this is our df, so what
we'll do-- we'll differentiate.

00:25:21.290 --> 00:25:24.000
We'll just substitute it here.

00:25:24.000 --> 00:25:25.960
Right?

00:25:25.960 --> 00:25:27.110
We'll substitute it here.

00:25:27.110 --> 00:25:33.630
We'll substitute df taken from
our dS, which is like this,

00:25:33.630 --> 00:25:34.130
and dB.

00:25:40.380 --> 00:25:45.030
Let's not forget that dB--
that B is deterministic.

00:25:45.030 --> 00:25:45.660
Right?

00:25:45.660 --> 00:25:47.390
There is nothing
uncertain about it.

00:25:47.390 --> 00:25:51.464
So dB is actually r*B*dt.

00:25:51.464 --> 00:25:52.330
All right?

00:25:52.330 --> 00:25:57.900
Because our B
grows exponentially

00:25:57.900 --> 00:26:00.120
with interest rate r.

00:26:00.120 --> 00:26:04.850
So we substitute everything
into the formula above.

00:26:04.850 --> 00:26:10.630
This is just our df with
dS expanded and everything.

00:26:10.630 --> 00:26:13.940
And then when we start
comparing the terms.

00:26:13.940 --> 00:26:20.970
One immediate thing
to notice-- that a

00:26:20.970 --> 00:26:25.990
has to be equal to df
over dS, for this to hold.

00:26:25.990 --> 00:26:27.360
Right?

00:26:27.360 --> 00:26:30.510
And if you compare
the terms near dt,

00:26:30.510 --> 00:26:32.560
we'll get this expression here.

00:26:32.560 --> 00:26:37.660
But that's actually even
more the most important part.

00:26:37.660 --> 00:26:45.340
Then we'll go and use our
knowledge that some part

00:26:45.340 --> 00:26:52.555
of our equation is deterministic
and basically take f and a*S

00:26:52.555 --> 00:26:55.330
on one side and leave
the deterministic part,

00:26:55.330 --> 00:27:00.000
on the other side,
differentiated once again.

00:27:00.000 --> 00:27:04.940
And left side will
be just r*B*dt.

00:27:04.940 --> 00:27:08.870
And if we substitute
once again df--

00:27:08.870 --> 00:27:11.430
and don't forget
that what we learned

00:27:11.430 --> 00:27:15.530
is that a is equal to df by dS.

00:27:15.530 --> 00:27:19.780
Then we collect all
the terms and arrive

00:27:19.780 --> 00:27:24.750
to this partial
differential equation which

00:27:24.750 --> 00:27:27.540
connects-- which basically
is a partial differential

00:27:27.540 --> 00:27:31.920
equation for the current
price of a derivative--

00:27:31.920 --> 00:27:34.480
of any derivative.

00:27:34.480 --> 00:27:38.480
And how if we solve it,
then we should actually

00:27:38.480 --> 00:27:41.980
be able to know the
price of the derivative.

00:27:41.980 --> 00:27:45.850
So now how do we solve this
partial differential equation?

00:27:45.850 --> 00:27:47.000
Well, for-- yeah.

00:27:47.000 --> 00:27:51.330
So a few observations
about this equation.

00:27:51.330 --> 00:27:59.930
Well, the first observation
is that any tradable

00:27:59.930 --> 00:28:04.330
derivative-- we made no
assumptions about the pay-off.

00:28:04.330 --> 00:28:07.080
So any tradable
derivative as any pay-off

00:28:07.080 --> 00:28:10.780
should satisfy this equation.

00:28:10.780 --> 00:28:14.890
The other observation
is as we expected,

00:28:14.890 --> 00:28:19.050
there is no dependency
on real-world drift

00:28:19.050 --> 00:28:23.445
or any probability of
it going up or down.

00:28:23.445 --> 00:28:27.530
The only dependence is on
the volatility of the stock.

00:28:27.530 --> 00:28:28.030
Right?

00:28:31.110 --> 00:28:34.400
Not only we found the
value of the derivative--

00:28:34.400 --> 00:28:38.620
most importantly,
we actually were

00:28:38.620 --> 00:28:42.760
able to come up with
a hedging strategy.

00:28:42.760 --> 00:28:45.810
And what does it mean, we came
up with a hedging strategy?

00:28:45.810 --> 00:28:49.490
Well, we found
coefficients-- for any time,

00:28:49.490 --> 00:28:51.960
we found the
coefficients, a and b,

00:28:51.960 --> 00:28:55.460
such that we have a
replicating portfolio.

00:28:55.460 --> 00:28:58.090
So what we could do,
at any point of time,

00:28:58.090 --> 00:29:02.560
we can hold the derivative--
short derivative and long

00:29:02.560 --> 00:29:06.010
the portfolio of stock
itself, and some cash,

00:29:06.010 --> 00:29:07.704
and then know how
much it should be.

00:29:07.704 --> 00:29:08.870
Here, it's more complicated.

00:29:08.870 --> 00:29:11.510
We have to dynamically
change these numbers,

00:29:11.510 --> 00:29:12.760
as time develops.

00:29:12.760 --> 00:29:15.890
Every time dt we will
have to rebalance.

00:29:15.890 --> 00:29:20.480
But both parts will replicate
each other perfectly.

00:29:20.480 --> 00:29:22.590
It's like in a bookie's example.

00:29:22.590 --> 00:29:29.470
We can go to a
counterparty, agree

00:29:29.470 --> 00:29:32.030
for some derivative contract.

00:29:32.030 --> 00:29:33.810
Probably there will be some fee.

00:29:33.810 --> 00:29:37.040
And then we'll go to
exchange and buy the stock,

00:29:37.040 --> 00:29:39.475
and we will get just
cash from the bank.

00:29:39.475 --> 00:29:41.990
And we'll maintain this
at some amount of stock

00:29:41.990 --> 00:29:43.800
and some amount of cash.

00:29:43.800 --> 00:29:46.610
And we'll be sure
that we are hedged.

00:29:46.610 --> 00:29:50.570
There is no risk in this
combination of the derivative

00:29:50.570 --> 00:29:52.100
and our hedge.

00:29:52.100 --> 00:29:56.040
So we will just collect
a fee on the transaction.

00:29:56.040 --> 00:30:01.280
So that's what actually--
how the business is working.

00:30:01.280 --> 00:30:05.180
Traders are trading and hedging
their positions immediately.

00:30:05.180 --> 00:30:07.590
I mean, they do take
some market risks.

00:30:07.590 --> 00:30:11.410
But you want to take very
little and very directional,

00:30:11.410 --> 00:30:16.890
very specific market
risks and not everything.

00:30:16.890 --> 00:30:20.370
So our strategy
allows us to have

00:30:20.370 --> 00:30:24.910
a hedging portfolio at the
same time-- hedging strategy.

00:30:24.910 --> 00:30:29.400
And now there are more
mathematical but practical

00:30:29.400 --> 00:30:34.790
consequences that actually,
by certain-- not very easy--

00:30:34.790 --> 00:30:38.080
change of variables, we can
take the Black-Scholes equation

00:30:38.080 --> 00:30:40.350
and put it back
to heat equation.

00:30:40.350 --> 00:30:45.710
Actually, I suggest it as one of
the topics for the final paper,

00:30:45.710 --> 00:30:50.320
for you to do it or check
it out in the books.

00:30:50.320 --> 00:30:52.990
Go and understand it.

00:30:52.990 --> 00:30:55.630
But the good part of it--
that heat equation is

00:30:55.630 --> 00:30:57.400
well known and well understood.

00:30:57.400 --> 00:31:01.790
There are many, many ways
to solve it numerically.

00:31:01.790 --> 00:31:06.012
For simple pay-outs,
for calls and puts,

00:31:06.012 --> 00:31:07.470
we don't have to
do it numerically,

00:31:07.470 --> 00:31:10.320
but if the pay-outs
are more complicated

00:31:10.320 --> 00:31:18.730
or the dynamics is different,
then numerical methods

00:31:18.730 --> 00:31:22.180
will be needed, for sure.

00:31:22.180 --> 00:31:24.770
So again, to solve
this equation,

00:31:24.770 --> 00:31:29.090
we'll need, as for any
partial differential equation,

00:31:29.090 --> 00:31:32.290
we'll need some boundary
and initial conditions.

00:31:32.290 --> 00:31:36.665
And these come from
our final pay-out

00:31:36.665 --> 00:31:39.600
of the option, which we know.

00:31:39.600 --> 00:31:41.750
We will know what
happens at expiry.

00:31:41.750 --> 00:31:44.110
And some boundary conditions.

00:31:44.110 --> 00:31:49.670
For call and put, the
final pay-out we know.

00:31:49.670 --> 00:31:50.510
Right?

00:31:50.510 --> 00:31:56.090
So at time T. And the
boundary conditions

00:31:56.090 --> 00:32:01.830
we discussed, we can
observe them graphically.

00:32:01.830 --> 00:32:09.850
So basically for call, as
we said, at current time t,

00:32:09.850 --> 00:32:12.930
and boundary 0, it should be 0.

00:32:12.930 --> 00:32:14.400
The price should be 0.

00:32:14.400 --> 00:32:20.700
And at infinity, it should be
actually the forward price.

00:32:20.700 --> 00:32:28.940
So it should be just discounted
S minus K. Discounted pay-out.

00:32:28.940 --> 00:32:29.440
Right?

00:32:32.910 --> 00:32:34.490
And similarly for put.

00:32:40.190 --> 00:32:47.500
So given these conditions,
we can solve the equation.

00:32:47.500 --> 00:32:52.670
And as I said, for call and
put and for simple dynamics--

00:32:52.670 --> 00:32:58.090
Black-Scholes dynamical or
log-normal dynamics-- actually,

00:32:58.090 --> 00:33:02.050
these equations can be
solved exactly-- exactly

00:33:02.050 --> 00:33:07.170
meaning up to this term, the
normal distribution, which

00:33:07.170 --> 00:33:10.610
still has to be computed
numerically, obviously.

00:33:10.610 --> 00:33:13.200
But here are the formulas.

00:33:13.200 --> 00:33:15.210
They do kind of
look a little bit--

00:33:15.210 --> 00:33:19.300
and we'll see about it--
there is some kind of expected

00:33:19.300 --> 00:33:20.270
volume going on.

00:33:20.270 --> 00:33:21.030
Right?

00:33:21.030 --> 00:33:25.130
One probability times another.

00:33:25.130 --> 00:33:26.600
But these are the formulas.

00:33:26.600 --> 00:33:29.590
And that's how I drew
the lines on the graphs.

00:33:34.440 --> 00:33:45.420
And as I said, in
fact, the whole world,

00:33:45.420 --> 00:33:49.900
instead of solving the whole
partial differential equation,

00:33:49.900 --> 00:33:55.450
we can approach it from
a risk-neutral position

00:33:55.450 --> 00:33:58.970
and say that, in fact, the
price of our derivative

00:33:58.970 --> 00:34:06.310
now is just expected value of
pay-out, discounted, probably,

00:34:06.310 --> 00:34:10.429
from the maturity.

00:34:10.429 --> 00:34:13.679
But not in real time
or real-world measure,

00:34:13.679 --> 00:34:16.179
but in some specific
risk-neutral measure.

00:34:16.179 --> 00:34:18.790
And how do we find this
risk-neutral measure?

00:34:18.790 --> 00:34:21.980
Well, the risk-neutral
measure is such

00:34:21.980 --> 00:34:24.909
that the drift of our stock
is actually interest rate.

00:34:24.909 --> 00:34:26.500
It's riskless.

00:34:26.500 --> 00:34:32.700
That's exactly how we saw
it in our binary example.

00:34:36.239 --> 00:34:36.739
All right?

00:34:36.739 --> 00:34:42.610
So even in our binary
example, our expected value

00:34:42.610 --> 00:34:44.730
of our stock, under
risk-neutral measure,

00:34:44.730 --> 00:34:50.159
meaning using the
risk-neutral probability,

00:34:50.159 --> 00:34:54.429
was drifting with
interest rate r.

00:34:54.429 --> 00:34:59.540
So that the same happens
in continuous case.

00:34:59.540 --> 00:35:01.950
And that's another
good exercise--

00:35:01.950 --> 00:35:10.900
and I would accept it as a
final paper-- is deriving

00:35:10.900 --> 00:35:16.600
the Black-Scholes formula
just by the expected value

00:35:16.600 --> 00:35:23.019
of the call and put pay-out with
the log-normal distribution--

00:35:23.019 --> 00:35:23.935
terminal distribution.

00:35:28.180 --> 00:35:28.950
All right.

00:35:28.950 --> 00:35:36.835
So for more
complicated pay-offs,

00:35:36.835 --> 00:35:39.960
the life becomes
more complicated.

00:35:39.960 --> 00:35:43.180
And some finite
differences should

00:35:43.180 --> 00:35:49.070
be used for more complicated
pay-offs or American pay-offs

00:35:49.070 --> 00:35:53.960
or path-dependent pay-offs,
tree methods or Monte Carlo

00:35:53.960 --> 00:35:54.510
simulations.

00:35:54.510 --> 00:35:57.150
And that's what was
happening in real life.

00:36:03.755 --> 00:36:04.255
Yeah.

00:36:08.200 --> 00:36:13.640
Now, since we have,
actually, plenty of time,

00:36:13.640 --> 00:36:21.620
I would like to give an example
of how replicating-- idea

00:36:21.620 --> 00:36:24.960
of replicating portfolio works.

00:36:24.960 --> 00:36:26.685
I give a couple more examples.

00:36:34.670 --> 00:36:35.430
So OK.

00:36:35.430 --> 00:36:38.600
Here is a Bloomberg screen
for foreign options--

00:36:38.600 --> 00:36:40.910
call options on IBM stock.

00:36:40.910 --> 00:36:45.280
It actually was taken a
while ago-- a few years ago.

00:36:45.280 --> 00:36:49.820
And so here are different
strikes for a call option.

00:36:49.820 --> 00:36:55.350
The current price of
the stock is $81.14.

00:36:55.350 --> 00:36:58.650
And here are the
strikes of the call.

00:36:58.650 --> 00:37:04.470
So obviously, if the option
is way out of the money,

00:37:04.470 --> 00:37:11.490
meaning the strike is very high
compared to the stock price,

00:37:11.490 --> 00:37:13.470
the value of the option is 0.

00:37:13.470 --> 00:37:20.900
If it's way in the money, in
fact, it is just S minus K.

00:37:20.900 --> 00:37:24.240
So S being $81.

00:37:24.240 --> 00:37:27.040
And say, the strike being $55.

00:37:27.040 --> 00:37:28.740
So it's $26.

00:37:28.740 --> 00:37:29.360
Right?

00:37:29.360 --> 00:37:30.900
So there is some difference.

00:37:30.900 --> 00:37:32.780
But actually, here
it's a bit small

00:37:32.780 --> 00:37:35.410
because the difference should
be just discounting, as we know.

00:37:35.410 --> 00:37:35.909
Right?

00:37:35.909 --> 00:37:38.460
But it's pretty
short-dated options.

00:37:38.460 --> 00:37:42.460
They are probably a month
long, so there is not

00:37:42.460 --> 00:37:44.990
much discounting.

00:37:44.990 --> 00:37:47.400
So it becomes pretty parallel.

00:37:47.400 --> 00:37:49.040
It's similar here, right?

00:37:49.040 --> 00:37:53.530
So I mean, this changes by 5.

00:37:53.530 --> 00:37:56.239
This changes by 5.

00:37:56.239 --> 00:37:57.030
It's pretty linear.

00:37:57.030 --> 00:37:59.620
But it becomes non-linear
around the money,

00:37:59.620 --> 00:38:01.770
around current stock price.

00:38:01.770 --> 00:38:02.270
Right?

00:38:02.270 --> 00:38:07.640
So we do observe this behavior.

00:38:07.640 --> 00:38:12.580
But to tell you the
truth, if you were to-- I

00:38:12.580 --> 00:38:18.040
didn't put implied
volatilities here.

00:38:18.040 --> 00:38:22.150
But actually, you would
observe that the world is not

00:38:22.150 --> 00:38:25.225
Black-Scholes,
meaning that-- what's

00:38:25.225 --> 00:38:26.840
the assumption of Black-Scholes.

00:38:26.840 --> 00:38:28.256
The assumption of
Black-Scholes is

00:38:28.256 --> 00:38:31.807
that every option, for any
strike, on a given stock,

00:38:31.807 --> 00:38:33.890
on a given expiry, would
have the same volatility.

00:38:33.890 --> 00:38:35.150
Right?

00:38:35.150 --> 00:38:38.000
So if we went through exercise
of implying the volatility

00:38:38.000 --> 00:38:39.940
according to
Black-Scholes formula,

00:38:39.940 --> 00:38:42.260
from the option
price which is traded

00:38:42.260 --> 00:38:46.130
on the market and
the current price,

00:38:46.130 --> 00:38:52.110
we would find out that,
actually, the volatility is not

00:38:52.110 --> 00:38:52.985
constant with strike.

00:38:57.146 --> 00:39:00.430
Well, it's actually skewed.

00:39:00.430 --> 00:39:03.230
Well, actually it is smiled.

00:39:03.230 --> 00:39:08.140
They would find
something like this,

00:39:08.140 --> 00:39:11.980
which means that Black-Scholes
theory is not perfectly good.

00:39:11.980 --> 00:39:12.480
Right?

00:39:12.480 --> 00:39:16.340
So something more
complicated should be done.

00:39:16.340 --> 00:39:18.820
But in some cases,
we even don't need

00:39:18.820 --> 00:39:21.130
to do something
more complicated.

00:39:21.130 --> 00:39:27.230
One example, being
so-called put-call parity.

00:39:27.230 --> 00:39:28.690
Right?

00:39:28.690 --> 00:39:31.040
So let's see.

00:39:31.040 --> 00:39:32.490
Suppose we look at the screen.

00:39:32.490 --> 00:39:37.790
So we know all prices for all
call options for all strikes.

00:39:37.790 --> 00:39:40.700
Well, probably will be some
granularity, but we know those.

00:39:40.700 --> 00:39:42.950
But instead of pricing a
call, we need to price a put.

00:39:45.470 --> 00:39:48.470
Somehow, we don't know how
the dynamics of our stock

00:39:48.470 --> 00:39:49.040
looks like.

00:39:49.040 --> 00:39:54.540
So we have strong suspicion that
it's not exactly log-normal.

00:39:54.540 --> 00:39:56.710
So there is some
volatility smile.

00:39:56.710 --> 00:39:57.540
It's not constant.

00:39:57.540 --> 00:40:00.230
The world is slightly
not Black-Scholes.

00:40:00.230 --> 00:40:04.600
So how do we price put?

00:40:04.600 --> 00:40:05.400
Well, let's see.

00:40:08.910 --> 00:40:14.330
We'll stare long enough at the
pay-outs of the call and put.

00:40:14.330 --> 00:40:19.162
So what's the pay-out of
a call with some strike?

00:40:19.162 --> 00:40:20.250
It looks like this.

00:40:20.250 --> 00:40:21.740
Right?

00:40:21.740 --> 00:40:25.390
The pay-out of the put,
with the same strike,

00:40:25.390 --> 00:40:27.860
would look like this.

00:40:27.860 --> 00:40:35.550
So what if we take, we
buy a call and sell a put?

00:40:45.230 --> 00:40:47.480
So this would go like this.

00:40:47.480 --> 00:40:49.900
Right?

00:40:49.900 --> 00:40:51.680
Straight line.

00:40:51.680 --> 00:40:55.000
Looks very much
like forward, right?

00:40:55.000 --> 00:41:00.830
So if we actually subtract
the stock from here,

00:41:00.830 --> 00:41:12.570
move it from here,
then it should

00:41:12.570 --> 00:41:17.008
be-- yeah-- minus K. Yeah.

00:41:19.830 --> 00:41:22.100
I think I got the signs correct.

00:41:22.100 --> 00:41:22.800
Right?

00:41:22.800 --> 00:41:24.210
And this is just a number.

00:41:24.210 --> 00:41:25.152
Right?

00:41:25.152 --> 00:41:26.610
And that's what
happens at pay-out.

00:41:26.610 --> 00:41:30.480
So if we take this portfolio,
if we action now, buy a call,

00:41:30.480 --> 00:41:34.320
sell a put, and sell a stock,
we know that at the end,

00:41:34.320 --> 00:41:37.410
we'll for sure get K in money.

00:41:37.410 --> 00:41:39.890
Right?

00:41:39.890 --> 00:41:45.310
So which means that now--
so this is at time t.

00:41:50.340 --> 00:42:02.780
So right now, it looks, to
me, that if we do write this,

00:42:02.780 --> 00:42:05.470
and that's just the
current price of the stock,

00:42:05.470 --> 00:42:11.880
this should be-- right?

00:42:11.880 --> 00:42:15.395
We just need to discount
this price to now,

00:42:15.395 --> 00:42:23.060
in this amount of cash, which
means that our put, at any time

00:42:23.060 --> 00:42:36.130
t, is stock minus K. Right?

00:42:36.130 --> 00:42:39.810
So if we know all of the
prices for any strike K--

00:42:39.810 --> 00:42:44.170
if we know price of a call, we
don't need any Black-Scholes

00:42:44.170 --> 00:42:45.690
or anything.

00:42:45.690 --> 00:42:50.700
We can immediately tell
everybody how much is a put.

00:42:50.700 --> 00:42:51.200
Right?

00:42:51.200 --> 00:42:57.570
So then this relationship is
actually a call-put parity.

00:42:57.570 --> 00:43:00.360
And that's, again-- that's
a replicating portfolio.

00:43:00.360 --> 00:43:01.860
It's a simple
replicating portfolio.

00:43:01.860 --> 00:43:04.750
It's static, meaning
that we fixed it now

00:43:04.750 --> 00:43:07.360
and we don't change
it to expiry.

00:43:07.360 --> 00:43:10.130
So it's quite good this way.

00:43:10.130 --> 00:43:16.863
But that's how it works.

00:43:16.863 --> 00:43:19.070
Another example.

00:43:19.070 --> 00:43:24.520
So for this, I have,
actually, a picture.

00:43:24.520 --> 00:43:30.540
So again, we have
the same situation.

00:43:30.540 --> 00:43:33.030
We have prices of calls.

00:43:33.030 --> 00:43:36.490
But instead of pricing a call,
we want to price a digital.

00:43:36.490 --> 00:43:38.290
So what is digital?

00:43:38.290 --> 00:43:41.130
Digital is such
a weird contract,

00:43:41.130 --> 00:43:43.815
which pay-out is just
a function-- Basically,

00:43:43.815 --> 00:43:47.770
it's a bet on the
stock to finish

00:43:47.770 --> 00:43:50.380
above strike price, K. Right?

00:43:50.380 --> 00:43:56.280
If at expiry, the stock
is above K, you get 1.

00:43:56.280 --> 00:43:58.030
You'd get $1.

00:43:58.030 --> 00:44:01.730
If it's below, you'd
get nothing, 0.

00:44:01.730 --> 00:44:02.230
Right?

00:44:02.230 --> 00:44:03.950
So

00:44:03.950 --> 00:44:07.110
So such an interesting contract.

00:44:07.110 --> 00:44:10.310
The question is,
can we price it,

00:44:10.310 --> 00:44:13.510
given that we know
the prices of Calls?

00:44:17.040 --> 00:44:20.400
And I suggest we use the idea
of replicating portfolio.

00:44:20.400 --> 00:44:25.305
Any ideas how to do it?

00:44:25.305 --> 00:44:31.030
It's my typical
interview question.

00:44:31.030 --> 00:44:32.780
So just pretend that
you are interviewing.

00:44:41.640 --> 00:44:42.330
Yep?

00:44:42.330 --> 00:44:45.430
AUDIENCE: You long the call,
and then you short the call,

00:44:45.430 --> 00:44:49.256
just like smaller
or a higher strike.

00:44:49.256 --> 00:44:49.880
PROFESSOR: Yep.

00:44:49.880 --> 00:44:50.546
The call strike.

00:44:50.546 --> 00:44:53.190
Yeah, you're absolutely right.

00:44:53.190 --> 00:44:54.390
Good.

00:44:54.390 --> 00:44:57.040
You've got an offer.

00:44:57.040 --> 00:44:57.820
Yeah.

00:44:57.820 --> 00:44:59.120
So here's how it goes.

00:44:59.120 --> 00:45:02.660
So this is a strike K. Right?

00:45:02.660 --> 00:45:16.070
So let's buy a call
with strike K minus 1/2

00:45:16.070 --> 00:45:21.890
and sell a Call with
strike K plus 1/2.

00:45:21.890 --> 00:45:23.070
Right?

00:45:23.070 --> 00:45:25.420
We just sold.

00:45:25.420 --> 00:45:31.842
So if we combine these two--
well, actually, if this is 1--

00:45:31.842 --> 00:45:32.342
yeah.

00:45:38.150 --> 00:45:41.054
If this is 1, it should
look something like this.

00:45:45.910 --> 00:45:48.080
Great.

00:45:48.080 --> 00:45:50.380
So how will it look like?

00:45:50.380 --> 00:45:51.675
So obviously, here, it's 0.

00:45:51.675 --> 00:45:53.030
Right?

00:45:53.030 --> 00:45:55.290
Then it will be like this.

00:45:55.290 --> 00:45:56.210
Right?

00:45:56.210 --> 00:45:58.390
And after that, it will be what?

00:45:58.390 --> 00:45:59.280
AUDIENCE: Constant.

00:45:59.280 --> 00:46:00.000
PROFESSOR: It will be constant.

00:46:00.000 --> 00:46:00.500
Right?

00:46:00.500 --> 00:46:04.170
And because this is K minus
1/2 and this is K plus 1/2,

00:46:04.170 --> 00:46:05.300
it will be exactly 1.

00:46:05.300 --> 00:46:06.820
Right?

00:46:06.820 --> 00:46:07.320
Good.

00:46:07.320 --> 00:46:11.520
So our pay-out, at the
end, will be like this.

00:46:16.230 --> 00:46:17.520
So that's good.

00:46:17.520 --> 00:46:21.270
But there is quite
a bit of slope here.

00:46:21.270 --> 00:46:25.330
So how can we do
better than this?

00:46:25.330 --> 00:46:37.770
Well, if we buy it at K minus
1/4, and sell it at K plus 1/4,

00:46:37.770 --> 00:46:41.380
and just combine those, it
will be exactly the same,

00:46:41.380 --> 00:46:43.940
but the level will be 1/2.

00:46:43.940 --> 00:46:48.200
So we need to buy two of those
and to sell two of those.

00:46:48.200 --> 00:46:50.510
Right?

00:46:50.510 --> 00:46:54.900
Well, we might as well go
K minus epsilon and K plus

00:46:54.900 --> 00:47:04.910
epsilon, so it'll be call price
at strike K minus epsilon,

00:47:04.910 --> 00:47:13.640
minus call price at K plus
epsilon, divided by 2*epsilon.

00:47:13.640 --> 00:47:15.310
Right?

00:47:15.310 --> 00:47:20.940
This 2*epsilon coefficient
needed rescale it back to 1.

00:47:20.940 --> 00:47:22.310
Right?

00:47:22.310 --> 00:47:27.960
So in fact, if we go small
epsilon, we need a lot of both.

00:47:27.960 --> 00:47:30.680
Right?

00:47:30.680 --> 00:47:37.320
And that's how-- that's
the approximation

00:47:37.320 --> 00:47:39.050
of our digital price.

00:47:39.050 --> 00:47:43.340
And that's actually how
people on the market

00:47:43.340 --> 00:47:46.750
do price and hedge,
most importantly,

00:47:46.750 --> 00:47:54.340
the digital contracts, because
call contracts are liquid,

00:47:54.340 --> 00:47:57.710
and they are traded on
exchanges while digitals

00:47:57.710 --> 00:47:59.420
are way less liquid.

00:47:59.420 --> 00:48:02.030
So somebody would call
again-- to counterparty,

00:48:02.030 --> 00:48:06.910
enter into digital, and
hedge it on the exchange.

00:48:06.910 --> 00:48:08.650
These two calls
with a call spread.

00:48:08.650 --> 00:48:12.200
But now tell me,
is it surprising

00:48:12.200 --> 00:48:18.480
that-- I mean, what
does it remind you?

00:48:18.480 --> 00:48:18.980
Yeah.

00:48:18.980 --> 00:48:26.600
So it's derivative
of the call price

00:48:26.600 --> 00:48:28.840
but with respect to strike.

00:48:28.840 --> 00:48:30.922
Right?

00:48:30.922 --> 00:48:31.630
Is it surprising?

00:48:38.280 --> 00:48:41.350
How did our call
price look like?

00:48:41.350 --> 00:48:41.850
It's a ramp.

00:48:41.850 --> 00:48:44.580
Right?

00:48:44.580 --> 00:48:48.926
If we take a derivative
of this, what will we get?

00:48:48.926 --> 00:48:49.425
Yeah.

00:48:49.425 --> 00:48:49.880
AUDIENCE: [INAUDIBLE].

00:48:49.880 --> 00:48:50.588
PROFESSOR: Right.

00:48:50.588 --> 00:48:54.790
So in fact, if we do something
even more weird with this,

00:48:54.790 --> 00:48:59.990
and then I'll take a
square or something else,

00:48:59.990 --> 00:49:01.240
the same will apply.

00:49:03.930 --> 00:49:05.520
So it's not surprising at all.

00:49:08.364 --> 00:49:09.790
All right.

00:49:09.790 --> 00:49:15.390
So that's basically how
the replicate-- this idea

00:49:15.390 --> 00:49:21.050
of replicating portfolios
is extremely powerful.

00:49:21.050 --> 00:49:24.790
And in fact, that's what
happens in real life.

00:49:24.790 --> 00:49:29.480
In real life, you have
some complicated derivative

00:49:29.480 --> 00:49:30.830
which you need to hedge.

00:49:30.830 --> 00:49:34.070
And how to hedge-- you'll
find something else which

00:49:34.070 --> 00:49:36.900
replicates-- to
a certain extent,

00:49:36.900 --> 00:49:38.140
replicates your pay-off.

00:49:38.140 --> 00:49:40.110
That's what you'll try to do.

00:49:40.110 --> 00:49:42.180
And this will be
your hedge portfolio.

00:49:42.180 --> 00:49:43.310
Usually, it's dynamic.

00:49:43.310 --> 00:49:45.720
So you'll have to rebalance.

00:49:45.720 --> 00:49:51.570
And that's how you
basically reduce the risks.