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The following
content is provided

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by MIT OpenCourseWare under
a Creative Commons license.

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Additional information
about our license,

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and MIT OpenCourseWare
in general

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is available at ocw.mit.edu.

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PROFESSOR: Vasily.

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Vasily Strela who works
now for Morgan Stanley,

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did his PhD here in the math
department, and kindly said he

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would tell us about
financial mathematics.

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So, it's all yours.

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GUEST SPEAKER: Let me
thank Professor Strang

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for giving this
opportunity to talk here,

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and it feels very good to
be back, be back to 18.086.

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So, a few more
words about myself.

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I've been Professor Strang's
student in mathematics

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about ten years ago.

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So after receiving my
PhD, I taught mathematics

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for a few years.

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Then ended up working for
a financial institution,

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for investment bank, Morgan
Stanley in particular.

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I'm part of an analytic modeling
group in fixed income division.

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What we are doing, we are
doing math applications

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in finance and modeling
derivatives, fixed income

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derivatives.

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That's actually what I'm
going to talk about today.

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I want to show how 18.086,
it's a wonderful class

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which I admire a lot,
which applications

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it has in the real world,
and in particular in finance

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and derivatives pricing.

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Let's start with
a simple example,

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which actually comes not
from finance, but rather

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from gambling.

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Well, let's look at
horse racing or cockroach

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racing, if you prefer.

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Suppose there are two
horses, and sure enough,

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people bet on them and
bookie is a clever one, very

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scientific-minded
guy and he made

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a very good research of previous
history of these two horses.

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He found out that the first
horse has 20% chance to win.

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The second horse has
80% chance to win.

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He is actually right about his
knowledge about chances to win.

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On the other hand, general
public, people who bet,

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they don't have access
to all information,

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and the bets are split
slightly differently.

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So 10,000 is placed
on the first horse,

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and 50,000 is placed
on the second horse.

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Bookie, sticking to his
scientific knowledge,

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splits the odds 4 to 1, meaning
that if the first horse wins,

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then whoever put on the
horse gets his money back

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and four times his
money back on top of it.

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Or if the second horse wins then
whoever put money on this horse

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will get the money back
and 1/4 on top of it.

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So, let's see.

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What are chances for bookie to
win or lose in this situation?

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Well, if the first
horse wins then he

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has to give back 10,000
plus 40,000, 50,000,

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and he got 60,000,
so he gains 10,000.

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Well, good, good for him.

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On the other hand, if
the second horse wins,

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then he has to give back
50,000 plus 1/4 of 50,

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which is 12,500, so
62.50 altogether,

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and he loses $2,500.

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After many runs, the expected
win or loss of the bookmaker

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is the probability
of the first horse

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to win times the expected
win, plus the probability

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of second horse to win times the
expected loss, which turns out

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to be exactly zero.

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So in each particular run,
bookie may lose or win,

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but in the long run he
expects to break even.

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On the other hand, if he
would put the chances,

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he would set the odds according
to the money bet, 5 to 1.

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what would be the outcome?

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Well, if the first horse wins he
gives back 10,000 plus 50,000,

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60,000, exactly the
amount he collected.

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Or if the second
horse wins, again, he

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gives back 50 plus 1/5 of
that, 60, he breaks even.

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So no matter which horse
wins in this scenario,

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the bookie breaks even.

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How bookie operates,
well he actually

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charges a fee for
each bet, right.

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So the second situation is
much more preferable for him.

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When he doesn't care which horse
wins, he just collects the fee.

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Well here, he may
lose or gain money.

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This is quite
beautiful observation,

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which we will see how
it works in derivatives.

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So now back to finance,
back to derivatives.

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So we are actually
interested in pricing

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a few financial derivatives, and
what is a financial derivative?

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Well, a financial
derivative is a contract,

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payoff of which at
maturity, at some time T,

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depends on underlying
security -- in our case,

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we always we will be talking
about a stock as underlying

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security, and probably
interest rates.

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What are the examples of
financial derivatives?

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Well, the most simple example
is probably a forward contract.

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Forward contracts is a contract
when you agree to purchase

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the security for a price
which is set today --

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you've agreed to purchase
the security in the future

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for the price agreed today.

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Well, for example, if you needed
1,000 barrel of oil to heat

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your house, but not today, but
rather for the next winter,

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on the other hand,
you don't want

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to take the risks of waiting
until the next winter

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and buying oil then, you would
rather agree on the price

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now and pay it in the
future and get the oil.

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What the price should be?

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What is the fair price
for this contract?

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Well, we will see
how to price it.

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Well, the few observation here
is that this line represents

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the payout -- it's always
useful to represent the payout

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graphically.

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This is just a straight
line because the payout

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of our contract is
S minus K at time T.

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This actually gives the current
price of the contract for all

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different values
of the underlying.

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Usually, the price of
the forward contract

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is set such that for the
current value of the underlying,

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the price of the
contract is zero.

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It costs nothing to
enter a forward contract,

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so that's why it
intersects zero here.

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What are other
common derivatives?

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Another common derivative
are calls and puts.

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And I put European
call and put here.

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Don't be confused by
European or American.

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It has nothing to do
with Europe or America,

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it has to do with the
structure of the contract.

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European basically
means that the contract

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expires at certain
time T. American means

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that's you can exercise
this contract at any time

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between now and future.

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We'll be talking only
about European contracts.

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So European call
option is a contract

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which gives you the
right, but not obligation,

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to purchase the underlying
security at set price K, which

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is called strike
price, at a future time

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T, which is expiration time.

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So if your security at
time T ends up below K,

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below the strike,
then sure enough

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there is no point of
buying the security

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for a more expensive price.

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So the contract
expires worthless.

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On the other hand, if your stock
ends up being greater than K

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at expiration time
C, then you would

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make money but my
purchasing this stock for K

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dollars and your payout
will be S minus K,

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and this is a graph
of your payout.

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This line here, as we will
see, is the current price

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of the contract, and
we'll see how to obtain

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this line in a few minutes.

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Another common
contract is a put.

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While call was basically a
bet that your stock will grow,

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right, the put is the bet
that your stock will not grow.

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So, in this case, the put is the
right, but not the obligation

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to sell the stock for
a certain price K.

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Here is the payout, which
is similar to the put,

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but just flipped.

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This is the current
price of a put option.

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Calls and puts, being
very common contracts,

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are traded on exchanges --
Chicago exchange is probably

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the most common place for
the calls and puts on stocks

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to trade.

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I just printed out
a Bloomberg screen,

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which gives the information
about a few calls

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and puts on IBM stock.

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So I did it on March
8, and the IBM stock

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was trading at this
time at $81.14,

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and here are descriptions
of the contract,

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they expire on 22nd
of April, so it's

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pretty short-dated contract.

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They can go as far as two
years from now, usually.

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Here is a set of strikes,
and here are a set of prices.

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As you can see, there
is no single price,

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there is always a bid and
ask, and that's how dealers

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and brokers make their
money -- like a bookie,

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they basically charge you
a fee for selling or buying

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the contract.

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That's how the money are made
-- they are made on this spread,

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but not on the price
of the contract itself,

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because as we will
see in a second,

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we actually can price
the contract exactly,

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and there is no uncertainty once
the price of the stock is set.

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There are plenty
of other options.

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Slightly more exotics
contracts, either digital

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which pays either
zero or one depending

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on where your stock ends up.

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It probably is not
exchange-traded,

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also I'm not sure.

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There are hundreds,
if not thousands,

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of exotic options where
you can say that, well,

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how much would be
the right to purchase

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00:12:06,980 --> 00:12:12,040
a stock for the maximum price
between today and two years

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from now.

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So it will be past-dependent.

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Depending on how
the stock will go,

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the payout will be
defined by this path.

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There are American
options where you

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can exercise your option any
time between now and maturity,

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and so on and so forth.

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So, just before we
go into mathematics

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of pricing, just a few
observations and statements.

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First of all, it
turns out that thanks

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to developed mathematics,
mathematical theory,

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if you make certain assumptions
on the dynamics of the stock,

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then there is no uncertainty
in the price of the option.

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You can say exactly how
much the option costs now,

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and that's what
provides, and this

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is a big driver for the market.

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So dealers quote these contracts
and there is a great agreement

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on the prices.

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The price of the
derivative contract

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is defined completely
by the stock price

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and not by risk preferences
of the market participant.

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So it doesn't matter what
are your views on the growth

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prospects of the stock.

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00:13:38,080 --> 00:13:44,150
It will not affect the price
of the derivative contract.

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00:13:44,150 --> 00:13:48,210
As I said, so the
mathematical part of it

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comes into giving the exact
price without any uncertainty.

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So let's consider a
simple example now.

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Let's assume that we are
in a very simple world.

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Well, first of all, in our world
there are only three objects --

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the stock itself, the
riskless money market account,

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meaning that it is an account
where we can either borrow

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money or invest money
at the riskless rate r,

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and finally our
derivative contract.

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00:14:22,800 --> 00:14:25,620
Here we are not making any
assumptions of what kind

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of derivative contract it
is -- it could be forward,

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it could be call, it could
be put, it can be anything.

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Moreover, our world is so
simple, that first of all,

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it's discrete,
and second of all,

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there is only one time step
to the expiration of power

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of contract, dt.

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00:14:44,360 --> 00:14:46,570
Not only there is
only one step left,

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we actually know exactly what
our transition probabilities.

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00:14:51,690 --> 00:14:53,430
There are only two
states at the end,

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00:14:53,430 --> 00:14:55,170
and we know the
transition probability.

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00:14:55,170 --> 00:14:58,420
So with probability p, we
move from the state zero

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to the state one, and with
probability of one minus p,

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00:15:01,890 --> 00:15:04,750
we move to the state two.

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00:15:04,750 --> 00:15:08,100
And just notice, because this is
riskless money market account,

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00:15:08,100 --> 00:15:10,670
it's the same in both cases.

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00:15:10,670 --> 00:15:15,892
You just invest money and it
grows with risk-free interest

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rate.

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00:15:18,420 --> 00:15:23,830
So, what can we say about the
price of our derivative f?

244
00:15:23,830 --> 00:15:28,110
Well a simple-minded -- well,
let's start with the forward

245
00:15:28,110 --> 00:15:29,310
contract.

246
00:15:29,310 --> 00:15:32,490
We know what the payout in
delta t of our forward contract

247
00:15:32,490 --> 00:15:35,200
will be, it will be just the
difference between the stock

248
00:15:35,200 --> 00:15:39,390
price and our strike.

249
00:15:39,390 --> 00:15:42,940
Well, a simple-minded
approach would be -- well,

250
00:15:42,940 --> 00:15:44,990
we know the transition
probabilities,

251
00:15:44,990 --> 00:15:48,980
let's just compute the
expected value of our contract,

252
00:15:48,980 --> 00:15:54,510
and that's what we would expect
to get if there were many such

253
00:15:54,510 --> 00:15:55,820
experiments.

254
00:15:55,820 --> 00:15:58,470
Well, you take the probability
of going to state one,

255
00:15:58,470 --> 00:16:01,254
you multiply by the
payoff at stage one.

256
00:16:01,254 --> 00:16:03,420
Take, minus p for probability
of going to state two,

257
00:16:03,420 --> 00:16:08,590
multiply by the payout
in that state two.

258
00:16:08,590 --> 00:16:12,600
Sum them up and you
get the expression.

259
00:16:12,600 --> 00:16:16,720
As I said, the common
thing to choose

260
00:16:16,720 --> 00:16:19,280
the strike such
that the contract

261
00:16:19,280 --> 00:16:22,796
has zero value now, so
you get your strike.

262
00:16:22,796 --> 00:16:24,170
Well, in particular
you could say

263
00:16:24,170 --> 00:16:26,440
that if you research
the market well

264
00:16:26,440 --> 00:16:29,350
and you know that the stock
has equal probability of going

265
00:16:29,350 --> 00:16:33,552
up and down, then actually
you expect your strike

266
00:16:33,552 --> 00:16:40,400
to be an average of end
values of the stock.

267
00:16:40,400 --> 00:16:44,060
But as we can imagine,
following our bookie example,

268
00:16:44,060 --> 00:16:46,270
this is not the right price.

269
00:16:46,270 --> 00:16:48,820
There is actually
a definite price

270
00:16:48,820 --> 00:16:52,580
which doesn't depend on
transition probability.

271
00:16:52,580 --> 00:16:57,220
Here is the reason why
there is a definite price.

272
00:16:57,220 --> 00:17:02,400
Well let's just consider
a very simple strategy.

273
00:17:02,400 --> 00:17:05,960
Let's borrow just enough
to purchase a stock.

274
00:17:05,960 --> 00:17:09,520
So let's borrow S_0
dollars right now and buy

275
00:17:09,520 --> 00:17:12,110
the stock for this money.

276
00:17:12,110 --> 00:17:13,920
And let's enter the
forward contract.

277
00:17:13,920 --> 00:17:17,410
Well, by definition forward
contract has price zero now,

278
00:17:17,410 --> 00:17:20,440
so we enter the
forward contract.

279
00:17:20,440 --> 00:17:25,530
Now, at the time dt when our
contract expires, what happens?

280
00:17:25,530 --> 00:17:27,950
Well, we deliver our
stock, which we already

281
00:17:27,950 --> 00:17:32,970
have in our hand in
exchange of K dollars.

282
00:17:32,970 --> 00:17:35,220
That's our forward contract.

283
00:17:35,220 --> 00:17:38,160
On the other hand, we
have to repay our loan,

284
00:17:38,160 --> 00:17:40,580
and because it was
a loan, it grew.

285
00:17:40,580 --> 00:17:46,520
It grew to S_0
times e to the r*dt.

286
00:17:46,520 --> 00:17:47,870
Now, let's see.

287
00:17:47,870 --> 00:17:53,460
What would happen if K
was greater than S times e

288
00:17:53,460 --> 00:17:54,460
to the r*dt?

289
00:17:54,460 --> 00:17:58,210
Then we know for sure,
we know now for sure,

290
00:17:58,210 --> 00:18:02,460
that we would make money.

291
00:18:02,460 --> 00:18:06,060
There is no uncertainty
about it now.

292
00:18:06,060 --> 00:18:09,190
Similarly, if K is
less than this value,

293
00:18:09,190 --> 00:18:12,550
then we know that
we will lose money.

294
00:18:12,550 --> 00:18:14,670
That's not how the
rational market works.

295
00:18:14,670 --> 00:18:18,170
If everybody knew that
by setting this price

296
00:18:18,170 --> 00:18:21,430
you would make money, people
would do it all day long

297
00:18:21,430 --> 00:18:23,760
and make infinite money.

298
00:18:23,760 --> 00:18:26,720
So there will be no
other side of the market.

299
00:18:26,720 --> 00:18:28,280
So the price has to go down.

300
00:18:28,280 --> 00:18:35,280
So the only choice for K, the
only market-implied choice,

301
00:18:35,280 --> 00:18:42,622
is that K has to be equal
to S times e to the r*dt.

302
00:18:42,622 --> 00:18:44,580
As you can see, it doesn't
depend on transition

303
00:18:44,580 --> 00:18:46,200
probabilities at all.

304
00:18:46,200 --> 00:18:49,560
That's what market implies us.

305
00:18:49,560 --> 00:18:51,490
That's the price of
forward contract,

306
00:18:51,490 --> 00:18:56,190
and that actually
explains why, when

307
00:18:56,190 --> 00:18:59,130
I was plotting the
forward contract,

308
00:18:59,130 --> 00:19:02,010
current price was just
the straight line,

309
00:19:02,010 --> 00:19:06,890
it's just discounted payoff.

310
00:19:06,890 --> 00:19:15,630
The payout is linear, so just
the parallel to the payoff.

311
00:19:15,630 --> 00:19:17,690
That's the idea, basically.

312
00:19:17,690 --> 00:19:25,280
The idea is to try to find
such a portfolio of stock

313
00:19:25,280 --> 00:19:30,290
and the money market
account with such a payout ,

314
00:19:30,290 --> 00:19:35,620
which will exactly replicate
the payoff of our derivative.

315
00:19:35,620 --> 00:19:37,896
If we found such of a
portfolio, than we know for sure

316
00:19:37,896 --> 00:19:39,270
that the value of
this portfolio,

317
00:19:39,270 --> 00:19:41,520
the replicating
portfolio today is

318
00:19:41,520 --> 00:19:44,590
equal to the value of the
derivative, because otherwise,

319
00:19:44,590 --> 00:19:48,110
you would make or
lose money risklessly.

320
00:19:48,110 --> 00:19:51,550
That's no-arbitrage condition.

321
00:19:51,550 --> 00:19:56,300
So, can we apply it to our
general one-step world?

322
00:19:56,300 --> 00:20:01,910
Well, if we have a general
payout f, what we want to do,

323
00:20:01,910 --> 00:20:05,140
we want to form a
replicating portfolio such

324
00:20:05,140 --> 00:20:09,970
that at expiration time, it
will replicate our payouts.

325
00:20:09,970 --> 00:20:15,540
So we want to choose such
constants a and b that such

326
00:20:15,540 --> 00:20:17,550
that the combination of
stock and money market

327
00:20:17,550 --> 00:20:21,430
account in both
states will replicate

328
00:20:21,430 --> 00:20:23,570
the payout of our option.

329
00:20:23,570 --> 00:20:27,940
Then, if we are able to
find such constants a and b,

330
00:20:27,940 --> 00:20:33,900
then we just look at the
current price of the contract

331
00:20:33,900 --> 00:20:36,620
and it has to be equal
to the current price

332
00:20:36,620 --> 00:20:38,930
of our derivative.

333
00:20:38,930 --> 00:20:41,360
Well, but in our particular
case, this is easy.

334
00:20:41,360 --> 00:20:45,770
It's just two linear equations
with two unknowns, easily

335
00:20:45,770 --> 00:20:52,020
solved, and here is current
price of our derivative.

336
00:20:52,020 --> 00:20:54,500
No matter what payout is --
I mean you just substitute

337
00:20:54,500 --> 00:21:02,980
the payout here, and if you
know S_1 and S_2, that's it.

338
00:21:02,980 --> 00:21:06,320
A useful way to look at this,
just to re-write this equation,

339
00:21:06,320 --> 00:21:13,350
is in this form, and then notice
that actually the current price

340
00:21:13,350 --> 00:21:20,510
of our derivative can be
viewed as a discounted expected

341
00:21:20,510 --> 00:21:25,350
payout of the derivative, but
with very certain probability.

342
00:21:25,350 --> 00:21:28,460
This probability,
it doesn't come

343
00:21:28,460 --> 00:21:32,020
from statistical properties of
the stock or from any research,

344
00:21:32,020 --> 00:21:34,720
it actually is
defined by the market.

345
00:21:34,720 --> 00:21:37,080
So it's called a
risk-neutral probability.

346
00:21:37,080 --> 00:21:42,050
So this probability
doesn't depend

347
00:21:42,050 --> 00:21:48,660
on the views on the market
by the market participants.

348
00:21:48,660 --> 00:21:51,860
An interesting observation
is that actually,

349
00:21:51,860 --> 00:21:57,435
the value of this stock, the
discounted value of the stock

350
00:21:57,435 --> 00:22:02,860
is actually also is expected
value of our outcomes

351
00:22:02,860 --> 00:22:06,850
under this risk-neutral
probability.

352
00:22:06,850 --> 00:22:09,450
That's basically general idea.

353
00:22:09,450 --> 00:22:14,330
Now let's move one notch
up and try to apply

354
00:22:14,330 --> 00:22:17,760
these idea to continuous case.

355
00:22:17,760 --> 00:22:20,600
Well, if you live in
continuous world now,

356
00:22:20,600 --> 00:22:27,270
we need to make some assumptions
on the behavior of the stock.

357
00:22:27,270 --> 00:22:31,225
The very common assumption is
that the dynamics of the stock

358
00:22:31,225 --> 00:22:32,770
is log-normal.

359
00:22:32,770 --> 00:22:35,870
Log-normal meaning that
the logarithm of the stock

360
00:22:35,870 --> 00:22:38,300
is actually normally
distributed.

361
00:22:38,300 --> 00:22:43,520
So, here mu is some drift, sigma
is the volatility of our stock,

362
00:22:43,520 --> 00:22:48,530
and dW is a Wiener process,
W is a Wiener process

363
00:22:48,530 --> 00:22:50,970
such that dW is
normally distributed

364
00:22:50,970 --> 00:22:56,320
with mean zero and
variance square root dt.

365
00:22:56,320 --> 00:23:00,530
Our approach would be to find
the replicating portfolio.

366
00:23:00,530 --> 00:23:01,550
And what does it mean?

367
00:23:01,550 --> 00:23:07,020
It means that we want to find
such constants, over time dt --

368
00:23:07,020 --> 00:23:10,900
so we assume that a and b are
constant over the next step,

369
00:23:10,900 --> 00:23:15,730
dt -- such that the change
in our derivative is a linear

370
00:23:15,730 --> 00:23:20,600
combination with this constant
of the change of our underlying

371
00:23:20,600 --> 00:23:25,740
security and the change
of money market account.

372
00:23:25,740 --> 00:23:31,800
Now we just need to look more
closely at this equation.

373
00:23:31,800 --> 00:23:34,520
First of all, let's
concentrate on df.

374
00:23:34,520 --> 00:23:44,910
So, f, our derivative,
is a function

375
00:23:44,910 --> 00:23:48,660
of stock value and time.

376
00:23:48,660 --> 00:23:52,060
But unfortunately, our
stock value is stochastic,

377
00:23:52,060 --> 00:24:00,110
so df is not that simple, and
to write df out we have to use

378
00:24:00,110 --> 00:24:03,180
a famous -- Ito's formula
from stochastic calculus,

379
00:24:03,180 --> 00:24:10,910
which actually is analogous of
Taylor's formula for stochastic

380
00:24:10,910 --> 00:24:12,080
variables.

381
00:24:12,080 --> 00:24:12,710
Let's see.

382
00:24:12,710 --> 00:24:15,240
If our S will not
be stochastic, if it

383
00:24:15,240 --> 00:24:17,110
would be completely
deterministic

384
00:24:17,110 --> 00:24:19,640
and depend only
on dt, then there

385
00:24:19,640 --> 00:24:24,670
would be no term and
differential f is just

386
00:24:24,670 --> 00:24:26,800
the standard expression.

387
00:24:26,800 --> 00:24:31,070
On the other hand,
if we have dependence

388
00:24:31,070 --> 00:24:35,040
on stochastic
variables, then we have

389
00:24:35,040 --> 00:24:38,130
to have more terms,
and why this happens?

390
00:24:38,130 --> 00:24:41,330
Well, in very rough words
is that because the order

391
00:24:41,330 --> 00:24:45,510
of magnitude of dW is
higher than dt's --

392
00:24:45,510 --> 00:24:47,930
it's square root of dt.

393
00:24:47,930 --> 00:24:51,340
So we have to make into
account more terms,

394
00:24:51,340 --> 00:24:56,970
and in particular, we have to
take into account next order

395
00:24:56,970 --> 00:24:59,070
of dS squared.

396
00:24:59,070 --> 00:25:01,970
Formally, dS square can
be written this way,

397
00:25:01,970 --> 00:25:04,900
and again, very rough
explanation is as follows.

398
00:25:04,900 --> 00:25:07,970
If we would square
this equation there

399
00:25:07,970 --> 00:25:09,490
will be three terms there.

400
00:25:09,490 --> 00:25:11,900
One would come from the
square of this term,

401
00:25:11,900 --> 00:25:16,530
and this would be of
the order of dt squared,

402
00:25:16,530 --> 00:25:20,240
next order of magnitude
-- much smaller than dt.

403
00:25:20,240 --> 00:25:23,060
The second term will be
cross-product of dW*dt.

404
00:25:23,060 --> 00:25:26,200
What order of magnitude
we are talking about,

405
00:25:26,200 --> 00:25:32,990
it is dt to the power 3/2,
again, much smaller than dt.

406
00:25:32,990 --> 00:25:37,090
On the other hand, the third
term will be the square of dW,

407
00:25:37,090 --> 00:25:40,520
this is of order
of magnitude of dt,

408
00:25:40,520 --> 00:25:43,500
so that's what we have to keep.

409
00:25:43,500 --> 00:25:48,990
And that's what Ito's
formula is about.

410
00:25:48,990 --> 00:25:56,210
Now, we are basically,
we know all terms here,

411
00:25:56,210 --> 00:26:01,560
and let me stress out that this
term, dB, it is not stochastic,

412
00:26:01,560 --> 00:26:04,860
it's completely
deterministic because we

413
00:26:04,860 --> 00:26:09,350
know that B grows with the
rate r, that's what it is.

414
00:26:09,350 --> 00:26:14,510
So we substitute all those terms
into our replicating equation.

415
00:26:14,510 --> 00:26:16,000
We collect the terms.

416
00:26:16,000 --> 00:26:17,120
We get this equation.

417
00:26:17,120 --> 00:26:19,580
And again, there is
the deterministic part,

418
00:26:19,580 --> 00:26:20,850
there is stochastic part.

419
00:26:20,850 --> 00:26:23,710
So the only way for
this equation to hold

420
00:26:23,710 --> 00:26:27,543
is this term to be equal
to this term, and this term

421
00:26:27,543 --> 00:26:32,170
to be equal to this term, and
that's what's written out here.

422
00:26:32,170 --> 00:26:34,780
So again, two equations,
these two unknowns, and here

423
00:26:34,780 --> 00:26:35,660
is answer.

424
00:26:38,210 --> 00:26:47,800
Finally, let's take
a*S to another part.

425
00:26:47,800 --> 00:26:50,720
Notice that this
part of our equation

426
00:26:50,720 --> 00:26:53,880
is completely deterministic.

427
00:26:53,880 --> 00:26:55,810
So we know how it will grow.

428
00:26:55,810 --> 00:27:01,020
So basically, d of f minus
a*S, which is b times dB,

429
00:27:01,020 --> 00:27:05,890
is r times b times dt.

430
00:27:05,890 --> 00:27:08,740
And we know all other
terms, we substitute them

431
00:27:08,740 --> 00:27:12,670
here, take something
to the left-hand side

432
00:27:12,670 --> 00:27:14,220
and get this equation.

433
00:27:14,220 --> 00:27:21,680
So this is partial differential
equation for our derivative f,

434
00:27:21,680 --> 00:27:26,670
as a function of S and
t, of second order,

435
00:27:26,670 --> 00:27:30,310
and this equation is the
famous Black-Scholes equation.

436
00:27:30,310 --> 00:27:34,200
It was derived by Fischer
Black and Myron Scholes

437
00:27:34,200 --> 00:27:38,940
in their famous paper
published in 1973.

438
00:27:38,940 --> 00:27:41,240
Myron Scholes and
Robert Merton actually

439
00:27:41,240 --> 00:27:44,640
received Nobel Prize
for deriving and solving

440
00:27:44,640 --> 00:27:47,560
this equation in '97.

441
00:27:47,560 --> 00:27:51,250
Black was already
dead by the time.

442
00:27:51,250 --> 00:27:55,320
This is really the
cornerstone of math finance.

443
00:27:59,290 --> 00:28:05,780
The cornerstone is because
using the replicating portfolio,

444
00:28:05,780 --> 00:28:12,080
using this reasoning, we were
able to find an exact equation

445
00:28:12,080 --> 00:28:14,040
for our derivative.

446
00:28:14,040 --> 00:28:16,980
So a few remarks
on Black-Scholes.

447
00:28:16,980 --> 00:28:20,930
So first of all, we
made some assumptions

448
00:28:20,930 --> 00:28:24,600
on the dynamic of the stock, but
we never made any assumptions

449
00:28:24,600 --> 00:28:27,150
on our derivative.

450
00:28:27,150 --> 00:28:32,610
Which means that any derivative
has to satisfy this equation,

451
00:28:32,610 --> 00:28:35,040
and that's very strong
result. So if you

452
00:28:35,040 --> 00:28:36,740
assume that our
stock is lognormal,

453
00:28:36,740 --> 00:28:39,780
which is not a bad assumption
and agrees quite well

454
00:28:39,780 --> 00:28:42,810
with the market, then we
basically, in principle,

455
00:28:42,810 --> 00:28:44,470
can price any derivative.

456
00:28:44,470 --> 00:28:48,400
We know the equation
for any derivative.

457
00:28:48,400 --> 00:28:51,800
The other thing is that
our Black-Scholes equation

458
00:28:51,800 --> 00:28:55,530
doesn't depend on
the actual drift mu

459
00:28:55,530 --> 00:28:57,180
in the dynamics of our stock.

460
00:28:57,180 --> 00:29:06,320
So again, it is the manifest
of risk-neutral dynamic.

461
00:29:06,320 --> 00:29:12,150
Not only we wrote down the
equation for our derivative,

462
00:29:12,150 --> 00:29:15,030
we also found a
replicating portfolio.

463
00:29:15,030 --> 00:29:18,280
So in other words, we
found a hedging strategy,

464
00:29:18,280 --> 00:29:21,795
meaning that at
any given time we

465
00:29:21,795 --> 00:29:28,360
can form this portfolio
with rates a and b.

466
00:29:28,360 --> 00:29:33,460
If we hold both the derivative
and both replicating portfolio,

467
00:29:33,460 --> 00:29:38,490
altogether, this
is zero sum gain.

468
00:29:38,490 --> 00:29:41,950
We know that no matter
where stock moves,

469
00:29:41,950 --> 00:29:44,710
we will not lose
money or gain money.

470
00:29:44,710 --> 00:29:51,110
So if we just charge
bid-offer on the derivative,

471
00:29:51,110 --> 00:29:53,330
if we charge a fee
on the contract,

472
00:29:53,330 --> 00:29:56,420
we can hedge ourself
perfectly, buy the contract

473
00:29:56,420 --> 00:29:58,970
or sell the contract,
hedge perfectly ourself

474
00:29:58,970 --> 00:30:01,460
and just make money
on the fee, that's it.

475
00:30:06,180 --> 00:30:09,060
Finally, more
mathematical remark

476
00:30:09,060 --> 00:30:13,250
is that actually after a few
manipulations, a few change

477
00:30:13,250 --> 00:30:16,710
of variables, the Black-Scholes
equation comes out

478
00:30:16,710 --> 00:30:19,980
to be just a heat
question, which you already

479
00:30:19,980 --> 00:30:22,190
saw in this class.

480
00:30:22,190 --> 00:30:23,650
This is very good news.

481
00:30:23,650 --> 00:30:25,000
Why is this good news?

482
00:30:25,000 --> 00:30:27,250
Well, because heat equation
is very well studied.

483
00:30:27,250 --> 00:30:32,986
So the solutions are well-known,
and numerical methods, the ways

484
00:30:32,986 --> 00:30:34,860
to solve it, in particular
the numerical ways

485
00:30:34,860 --> 00:30:36,760
to solve it are well-known.

486
00:30:36,760 --> 00:30:38,220
So we are in business.

487
00:30:38,220 --> 00:30:45,130
But as any partial
differential equation,

488
00:30:45,130 --> 00:30:49,450
the equation itself doesn't
make much sense because to find

489
00:30:49,450 --> 00:30:55,610
a particular solution we need
boundary and initial condition.

490
00:30:58,520 --> 00:31:01,940
And although any derivative
satisfies Black-Scholes

491
00:31:01,940 --> 00:31:08,310
equation, the final
and boundary conditions

492
00:31:08,310 --> 00:31:10,720
will vary from
contract to contract.

493
00:31:10,720 --> 00:31:15,510
Here are a few examples of the
final and boundary conditions.

494
00:31:15,510 --> 00:31:18,680
Here, an interesting
remark that if, usually,

495
00:31:18,680 --> 00:31:22,270
we would talk about
initial condition, here

496
00:31:22,270 --> 00:31:24,650
we are talking about
final condition.

497
00:31:24,650 --> 00:31:26,340
The time goes in
reverse; we know

498
00:31:26,340 --> 00:31:30,150
the state of the world at the
end, at expiration, not today.

499
00:31:30,150 --> 00:31:36,810
So, here are final and boundary
conditions for call and put,

500
00:31:36,810 --> 00:31:40,640
and let's look a little bit
at the pictures for our call

501
00:31:40,640 --> 00:31:44,250
and put to see where
they come from.

502
00:31:44,250 --> 00:31:48,950
So for example, for calls, well,
this is our final condition,

503
00:31:48,950 --> 00:31:50,930
right, this is
defined by the payout.

504
00:31:50,930 --> 00:31:54,690
On the other hand, the boundary
condition, well, what happens,

505
00:31:54,690 --> 00:31:57,646
we put them at zero at an
infinity, we [INAUDIBLE PHRASE]

506
00:31:57,646 --> 00:31:59,020
to put them at
zero and infinity.

507
00:31:59,020 --> 00:31:59,519
And why?

508
00:31:59,519 --> 00:32:03,340
Well, because if stock hits
zero then it stays at zero.

509
00:32:03,340 --> 00:32:06,330
That's what our dynamics show.

510
00:32:06,330 --> 00:32:11,400
So the value of our contract
at maturity will become zero.

511
00:32:11,400 --> 00:32:14,990
On the other hand, if the
stock grows, grows to infinity,

512
00:32:14,990 --> 00:32:17,360
a good assumption to
make is that actually it

513
00:32:17,360 --> 00:32:19,830
becomes similar to stock
itself, so it would just

514
00:32:19,830 --> 00:32:26,160
become parallel to the stock,
and that's the conditions

515
00:32:26,160 --> 00:32:27,150
which we impose here.

516
00:32:30,270 --> 00:32:36,290
Similarly for the put, you
can derive these conditions.

517
00:32:36,290 --> 00:32:41,740
And again, just because
it is a heat equation,

518
00:32:41,740 --> 00:32:44,910
it turns out that for a simple
derivative such as the calls

519
00:32:44,910 --> 00:32:50,070
and puts, it is possible to
find an exact analytic solution.

520
00:32:50,070 --> 00:32:54,350
Here are exact analytic set
of solutions for a call, put

521
00:32:54,350 --> 00:32:57,090
and the digital contracts.

522
00:32:57,090 --> 00:33:01,310
Well, not surprising
again, I mean

523
00:33:01,310 --> 00:33:04,310
they're all connected
to the error function,

524
00:33:04,310 --> 00:33:07,760
so to the normal
distribution, basically,

525
00:33:07,760 --> 00:33:12,400
as the solutions of heat
equation ought to be.

526
00:33:12,400 --> 00:33:14,580
Why do they look
exactly the same?

527
00:33:14,580 --> 00:33:17,020
If we have five
minutes at the end,

528
00:33:17,020 --> 00:33:21,690
we'll probably shed some
light on the specific form

529
00:33:21,690 --> 00:33:22,750
of equations.

530
00:33:22,750 --> 00:33:28,090
But let me just stress that we
can see that it's discounted,

531
00:33:28,090 --> 00:33:32,790
and what I'm claiming, it's
expected value of our payout

532
00:33:32,790 --> 00:33:36,810
under risk-neutral measure.

533
00:33:36,810 --> 00:33:42,290
Here is an example -- it's
of a particular call option

534
00:33:42,290 --> 00:33:44,160
on the same IBM stock.

535
00:33:44,160 --> 00:33:46,580
So I chose the
short-dated contract,

536
00:33:46,580 --> 00:33:50,300
just to avoid the
dividend payment.

537
00:33:50,300 --> 00:33:53,820
So it's a contract
expiring on March 18.

538
00:33:53,820 --> 00:33:56,750
So there is 10
days to expiration.

539
00:33:56,750 --> 00:34:02,170
The stock, as we saw,
the expirations of stock,

540
00:34:02,170 --> 00:34:06,750
as we saw, is what's
trading at 81.14.

541
00:34:06,750 --> 00:34:11,800
The volatility is
somewhere around 14%,

542
00:34:11,800 --> 00:34:17,110
estimated either from other
options or historically.

543
00:34:17,110 --> 00:34:20,210
Here is the price
of our contract.

544
00:34:20,210 --> 00:34:23,750
I also have a simple
Black-Scholes calculator here,

545
00:34:23,750 --> 00:34:28,800
and let's see if we
can match this price.

546
00:34:28,800 --> 00:34:39,280
So let's see, I believe the
volatility was 13%, right,

547
00:34:39,280 --> 00:34:42,240
13.47.

548
00:34:42,240 --> 00:34:45,220
The interest rate,
it's already here.

549
00:34:45,220 --> 00:34:48,540
As we all know, Fed just
bumped the interest rate,

550
00:34:48,540 --> 00:34:51,980
so they are at 4.75% right now.

551
00:34:51,980 --> 00:34:56,360
The strike of our option was 80.

552
00:34:56,360 --> 00:34:58,840
Time to expiration
was actually 10 days,

553
00:34:58,840 --> 00:35:02,390
and this should be measured
at a fraction of year.

554
00:35:02,390 --> 00:35:08,440
So we divide 10 by 365.

555
00:35:08,440 --> 00:35:15,350
This stock was trading at
81.14, if I'm not mistaken.

556
00:35:15,350 --> 00:35:21,210
Here is the price of our call
options contract, which is 150.

557
00:35:21,210 --> 00:35:26,630
Well, it's within the offer.

558
00:35:26,630 --> 00:35:29,070
So maybe our
volatility's slightly off

559
00:35:29,070 --> 00:35:35,140
and if we increase
[UNINTELLIGIBLE] to say,

560
00:35:35,140 --> 00:35:42,330
increase it to 14%, it
will go slightly up.

561
00:35:42,330 --> 00:35:44,800
152.

562
00:35:44,800 --> 00:35:48,580
Well, in general, let's
play a little bit with it.

563
00:35:48,580 --> 00:35:50,480
Well, it is very
short-dated option,

564
00:35:50,480 --> 00:35:55,990
so the value of our option
is very close to the payout.

565
00:35:55,990 --> 00:35:58,870
So if we increase
the time to maturity,

566
00:35:58,870 --> 00:36:02,280
let's make it two years
just to see where --

567
00:36:02,280 --> 00:36:08,980
so now value of our option is
-- well, that's what it is.

568
00:36:08,980 --> 00:36:15,040
If increase volatility, sure
enough, let's make it 30%.

569
00:36:15,040 --> 00:36:15,970
So what do we expect?

570
00:36:15,970 --> 00:36:19,340
We expect if volatility is
higher, the uncertainty higher,

571
00:36:19,340 --> 00:36:25,300
so the value of our contract
should go up, and it sure does.

572
00:36:29,120 --> 00:36:31,520
So basically that's how
Black-Scholes works.

573
00:36:39,690 --> 00:36:41,520
And plenty of those
contracts trade

574
00:36:41,520 --> 00:36:47,670
on the market, but unfortunately
not all of these contracts

575
00:36:47,670 --> 00:36:50,600
are so simple as calls and puts.

576
00:36:50,600 --> 00:36:55,610
First of all, there are many
more complicated products

577
00:36:55,610 --> 00:37:03,390
with more difficult payout,
which will constitute different

578
00:37:03,390 --> 00:37:08,800
and probably discontinuous final
conditions on our Black-Scholes

579
00:37:08,800 --> 00:37:10,930
equation.

580
00:37:10,930 --> 00:37:14,590
Moreover, we made an
assumption that the volatility

581
00:37:14,590 --> 00:37:16,650
is constant with time,
and interest rate

582
00:37:16,650 --> 00:37:18,210
is constant with time.

583
00:37:18,210 --> 00:37:21,860
It is certainly not
true for the real world.

584
00:37:21,860 --> 00:37:24,790
Volatility probably
should be time-dependent,

585
00:37:24,790 --> 00:37:26,540
and this would make
the coefficients

586
00:37:26,540 --> 00:37:30,090
in our Black-Scholes
equation time-dependent.

587
00:37:30,090 --> 00:37:33,250
Unfortunately, these cannot
be solve analytically.

588
00:37:33,250 --> 00:37:36,420
So in most of the
cases in practice,

589
00:37:36,420 --> 00:37:42,530
we will have to use some
kind of numerical solution.

590
00:37:42,530 --> 00:37:45,870
Finite difference methods
is the typical approach

591
00:37:45,870 --> 00:37:47,200
for the heat equation.

592
00:37:47,200 --> 00:37:50,330
As you know, both explicit
and implicit schemes,

593
00:37:50,330 --> 00:37:57,910
and you will discuss
some of those in 18.086.

594
00:37:57,910 --> 00:37:58,630
Tree methods.

595
00:37:58,630 --> 00:38:05,090
Tree methods meaning that we
go back to our one-step tree,

596
00:38:05,090 --> 00:38:08,380
and basically assume that
our time to expiration

597
00:38:08,380 --> 00:38:12,000
is many time steps away and
we'll grow the tree further,

598
00:38:12,000 --> 00:38:14,270
so from this node we
have two more nodes,

599
00:38:14,270 --> 00:38:15,550
and so and so forth.

600
00:38:15,550 --> 00:38:19,670
That would imply the final
condition at the end,

601
00:38:19,670 --> 00:38:23,640
and discount back using our
risk-neutral probabilities,

602
00:38:23,640 --> 00:38:25,170
and get the price now.

603
00:38:25,170 --> 00:38:27,240
So those are called
tree methods.

604
00:38:27,240 --> 00:38:30,330
One can show that actually those
tree methods are equivalent

605
00:38:30,330 --> 00:38:36,100
to finite difference -- explicit
finite difference schemes.

606
00:38:36,100 --> 00:38:38,110
Those are very popular.

607
00:38:38,110 --> 00:38:41,830
But again, in tree methods,
what is very important

608
00:38:41,830 --> 00:38:44,746
is to set the probabilities
from your tree, the transition

609
00:38:44,746 --> 00:38:46,120
probabilities, to
the right ones,

610
00:38:46,120 --> 00:38:48,960
and the right ones are
risk-neutral probabilities.

611
00:38:48,960 --> 00:38:53,170
Probabilities implied
by the market, actually.

612
00:38:53,170 --> 00:38:58,280
Another important numerical
method is Monte Carlo

613
00:38:58,280 --> 00:39:02,910
simulation where you would
simulate many different

614
00:39:02,910 --> 00:39:05,650
scenarios of the development of
your stock up to the maturity,

615
00:39:05,650 --> 00:39:09,840
and then, basically
find -- using this path,

616
00:39:09,840 --> 00:39:15,990
you will find the expected
value of your payout.

617
00:39:15,990 --> 00:39:20,210
But again, in order
for this expected value

618
00:39:20,210 --> 00:39:24,810
to be the same as the
risk-neutral value,

619
00:39:24,810 --> 00:39:27,050
as the arbitrage-free
value, you have

620
00:39:27,050 --> 00:39:29,910
to develop your Monte
Carlo simulations

621
00:39:29,910 --> 00:39:31,800
with risk-neutral probabilities.

622
00:39:31,800 --> 00:39:37,360
So, risk-neutral valuation
is extremely important.

623
00:39:40,780 --> 00:39:45,810
Here is actually the general
risk-neutral statement, which

624
00:39:45,810 --> 00:39:51,580
one can prove, is that actually,
the value of any derivative

625
00:39:51,580 --> 00:39:56,140
is just discounted expected
value of the payout

626
00:39:56,140 --> 00:39:59,280
of this derivative
at maturity, but you

627
00:39:59,280 --> 00:40:02,140
have to take this expectation
at the right measure.

628
00:40:02,140 --> 00:40:05,230
Using the right measure, meaning
that you have to set correctly

629
00:40:05,230 --> 00:40:08,705
the transition probability
-- you have to make them

630
00:40:08,705 --> 00:40:09,330
market-neutral.

631
00:40:12,710 --> 00:40:19,950
Under this measure, actually
the dynamics of our stocks

632
00:40:19,950 --> 00:40:22,660
looks slightly different,
and as you can see,

633
00:40:22,660 --> 00:40:25,240
our drift becomes
the interest rate.

634
00:40:25,240 --> 00:40:27,810
So under risk-neutral
measure, everything

635
00:40:27,810 --> 00:40:32,430
grows with our
risk-free interest rate.

636
00:40:32,430 --> 00:40:38,720
Just to shed a little
bit of light on how we go

637
00:40:38,720 --> 00:40:42,630
at the solutions
for calls and puts,

638
00:40:42,630 --> 00:40:44,920
Black-Scholes solutions
for call and put,

639
00:40:44,920 --> 00:40:47,920
well this is the
distribution of our stock,

640
00:40:47,920 --> 00:40:51,790
log-normal distribution
of our stock at time T,

641
00:40:51,790 --> 00:40:57,660
and if we take this distribution
and integrate our payout

642
00:40:57,660 --> 00:41:01,950
of our call option against this
distribution -- in other words,

643
00:41:01,950 --> 00:41:10,260
find the expected value of
payout of our call option under

644
00:41:10,260 --> 00:41:17,810
risk-neutral measure,
then, sure enough,

645
00:41:17,810 --> 00:41:21,200
you will get
[UNINTELLIGIBLE PHRASE].

646
00:41:21,200 --> 00:41:24,000
This illustrates the best
-- because what is digital?

647
00:41:24,000 --> 00:41:26,750
Digital is just the
probability to end up

648
00:41:26,750 --> 00:41:29,590
at above the strike
at time T, right?

649
00:41:29,590 --> 00:41:34,110
So if you integrate
this log-normal pdf

650
00:41:34,110 --> 00:41:42,310
from the strike K to infinity,
that will be your answer.

651
00:41:42,310 --> 00:41:44,930
This is a good exercise
in integration,

652
00:41:44,930 --> 00:41:47,760
to make sure that it's correct.

653
00:41:47,760 --> 00:41:49,270
So let's see.

654
00:41:49,270 --> 00:41:52,170
To conclude, what we've seen.

655
00:41:52,170 --> 00:42:02,180
So, we have seen that modern
derivatives business makes

656
00:42:02,180 --> 00:42:04,600
use of quite
advanced mathematics,

657
00:42:04,600 --> 00:42:08,050
and what kinds of
mathematics is used there?

658
00:42:08,050 --> 00:42:12,630
Well, partial differential
equations are used heavily.

659
00:42:12,630 --> 00:42:14,530
Numerical methods
for the solution

660
00:42:14,530 --> 00:42:20,470
of this partial differential
equations are naturally used.

661
00:42:20,470 --> 00:42:23,890
In order to get these
equations, we actually

662
00:42:23,890 --> 00:42:26,870
need to operate in terms
of stochastic calculus,

663
00:42:26,870 --> 00:42:31,180
meaning that we need to know
how to deal with Ito calculus,

664
00:42:31,180 --> 00:42:36,580
Ito formula, Girsanov theorem,
and so on and so forth.

665
00:42:36,580 --> 00:42:41,330
The other thing is to be
able to build simulations

666
00:42:41,330 --> 00:42:45,380
to solve the heat equation
and all other equations

667
00:42:45,380 --> 00:42:47,780
that you might encounter.

668
00:42:47,780 --> 00:42:51,230
The topic which we
didn't touch upon

669
00:42:51,230 --> 00:42:54,660
is statistics because,
of course, very advanced

670
00:42:54,660 --> 00:42:58,690
statistics is used
for many, many things,

671
00:42:58,690 --> 00:43:02,320
for analyzing
historical data, which

672
00:43:02,320 --> 00:43:04,040
can be quite
beautiful for trading

673
00:43:04,040 --> 00:43:07,660
strategies and many others.

674
00:43:07,660 --> 00:43:12,450
Besides these five topics,
there is much, much more

675
00:43:12,450 --> 00:43:17,400
to mathematical finance,
which makes it a very, very

676
00:43:17,400 --> 00:43:19,760
exciting field to work in.

677
00:43:19,760 --> 00:43:22,126
That's what I wanted
to talk about.

678
00:43:22,126 --> 00:43:23,750
Thank you very much
for your attention.

679
00:43:30,890 --> 00:43:35,430
PROFESSOR: Maybe I'll
ask a firm question

680
00:43:35,430 --> 00:43:38,100
about boundary
conditions, because you

681
00:43:38,100 --> 00:43:40,060
had said that
those are different

682
00:43:40,060 --> 00:43:42,860
for different contracts.

683
00:43:42,860 --> 00:43:47,290
And how do you deal with them
in the finite differences

684
00:43:47,290 --> 00:43:52,140
or the tree model or whatever?

685
00:43:52,140 --> 00:43:54,750
What would be a typical one?

686
00:43:54,750 --> 00:43:57,560
GUEST SPEAKER: Well, typical
one -- two very typical ones.

687
00:43:57,560 --> 00:44:05,500
So those, you basically
make a grid of your problem,

688
00:44:05,500 --> 00:44:08,100
in particular, you
build a tree, which

689
00:44:08,100 --> 00:44:11,650
is actually a grid of
all possible outcomes.

690
00:44:11,650 --> 00:44:28,490
You set them up at the
end, so your tree grows --

691
00:44:28,490 --> 00:44:36,310
so you set your boundary here
at the end, and well, you set,

692
00:44:36,310 --> 00:44:45,480
probably, some initial --
this is final condition,

693
00:44:45,480 --> 00:44:48,690
so you set some boundary
conditions here.

694
00:44:48,690 --> 00:44:55,710
So this is your time T. This
is t = 0, time t -- this is 0,

695
00:44:55,710 --> 00:45:04,030
this is 1, this is 2, this is
T. So you set your payout here,

696
00:45:04,030 --> 00:45:11,340
so it will be maximum
of S minus K and 0.

697
00:45:11,340 --> 00:45:14,940
PROFESSOR: How many time
steps might you take in this?

698
00:45:14,940 --> 00:45:20,315
GUEST SPEAKER: Well, you would
do like daily for three months

699
00:45:20,315 --> 00:45:22,476
-- if you three-month options.

700
00:45:22,476 --> 00:45:23,600
PROFESSOR: Maybe 100 steps.

701
00:45:23,600 --> 00:45:25,308
GUEST SPEAKER: Yeah,
something like that.

702
00:45:25,308 --> 00:45:28,470
Well, if it's two-year option,
that you probably would do it

703
00:45:28,470 --> 00:45:30,820
weekly or something like that.

704
00:45:30,820 --> 00:45:32,350
PROFESSOR: You don't
get into large,

705
00:45:32,350 --> 00:45:35,490
what would be scientifically,
large-scale calculating.

706
00:45:35,490 --> 00:45:37,490
PROFESSOR: No, in
finance we usually

707
00:45:37,490 --> 00:45:39,040
don't keep this problem--.

708
00:45:39,040 --> 00:45:40,810
PROFESSOR: In
finite differences,

709
00:45:40,810 --> 00:45:44,670
do you use like higher-order
-- suppose, well,

710
00:45:44,670 --> 00:45:47,230
you had second derivatives,
would you always use second

711
00:45:47,230 --> 00:45:51,449
differences or
second-order accuracy?

712
00:45:51,449 --> 00:45:52,740
GUEST SPEAKER: In general, yes.

713
00:45:52,740 --> 00:45:55,790
In general,
second-order accuracy.

714
00:45:55,790 --> 00:45:58,210
In general you don't go higher.

715
00:45:58,210 --> 00:46:01,920
I mean the precision -- well,
it's within cents, right.

716
00:46:01,920 --> 00:46:06,010
So you cannot do
better than that.

717
00:46:06,010 --> 00:46:11,650
So it depends -- well, it
depends what kind of amount you

718
00:46:11,650 --> 00:46:12,830
are dealing with.

719
00:46:12,830 --> 00:46:16,775
If you're actually selling
and buying units of stock,

720
00:46:16,775 --> 00:46:21,080
you might consider
something more precise.

721
00:46:21,080 --> 00:46:25,220
But it's very problem-defined.

722
00:46:30,280 --> 00:46:33,370
So that's how we deal with it.

723
00:46:33,370 --> 00:46:35,770
PROFESSOR: Any questions?

724
00:46:35,770 --> 00:46:38,440
You can put the mic on
if you have a question.

725
00:46:38,440 --> 00:46:45,080
STUDENT:
[UNINTELLIGIBLE PHRASE].

726
00:46:55,800 --> 00:46:58,000
GUEST SPEAKER: Well,
it is Markov process.

727
00:46:58,000 --> 00:46:58,500
Yes.

728
00:46:58,500 --> 00:47:01,480
I mean, this is just
a numerical solution.

729
00:47:01,480 --> 00:47:02,910
So yeah, it is Markov process.

730
00:47:02,910 --> 00:47:04,990
and basically all
stochastic calculus

731
00:47:04,990 --> 00:47:07,590
is about Markov process,
continuous Markov process.

732
00:47:15,030 --> 00:47:17,010
PROFESSOR: Is the
mathematics that you

733
00:47:17,010 --> 00:47:22,600
get involved with pretty
well set now or is there

734
00:47:22,600 --> 00:47:27,000
a need for more
mathematics, if I

735
00:47:27,000 --> 00:47:28,920
can ask the question that way?

736
00:47:28,920 --> 00:47:31,140
PROFESSOR: Yeah,
well, in this field

737
00:47:31,140 --> 00:47:33,610
it is probably quite well set.

738
00:47:33,610 --> 00:47:37,410
But if you get into
more complicated fields,

739
00:47:37,410 --> 00:47:44,690
especially into credit modeling,
the model for the credits

740
00:47:44,690 --> 00:47:49,810
of certain companies, then
mathematics is not quite set,

741
00:47:49,810 --> 00:47:53,730
because there, you start
talking about jump processes

742
00:47:53,730 --> 00:47:58,860
and not Wiener processes, not
just log-normal processes.

743
00:47:58,860 --> 00:48:03,070
This stochastic differential
equation become very hard,

744
00:48:03,070 --> 00:48:06,150
but maybe still
analytically tractable.

745
00:48:06,150 --> 00:48:09,330
So from this point of
view there is need --

746
00:48:09,330 --> 00:48:13,281
but it's not a
fundamental mathematics,

747
00:48:13,281 --> 00:48:15,030
it's not that you are
opening a new field,

748
00:48:15,030 --> 00:48:21,170
but definitely trying to solve
a stochastic differential

749
00:48:21,170 --> 00:48:23,840
equation -- which usually
boils down to solving a partial

750
00:48:23,840 --> 00:48:26,780
differential equation
analytically --

751
00:48:26,780 --> 00:48:30,280
can be pretty hard a
mathematical problem,

752
00:48:30,280 --> 00:48:32,280
viewed as a
mathematical problem.

753
00:48:32,280 --> 00:48:33,790
PROFESSOR: So you
showed the example

754
00:48:33,790 --> 00:48:37,700
of Black-Scholes solver.

755
00:48:37,700 --> 00:48:39,872
Everybody has that
available all the time?

756
00:48:39,872 --> 00:48:40,830
GUEST SPEAKER: Oh yeah.

757
00:48:40,830 --> 00:48:44,550
On Chicago trading
floor, the traders

758
00:48:44,550 --> 00:48:47,520
have calculators where
they just press a button

759
00:48:47,520 --> 00:48:49,104
and it's just hard-wired there.

760
00:48:49,104 --> 00:48:51,270
PROFESSOR: And they're
printing out error functions,

761
00:48:51,270 --> 00:48:55,060
basically -- a combination
of error function, yeah.

762
00:48:55,060 --> 00:49:01,110
GUEST SPEAKER: Well, sure
enough, nobody uses just --

763
00:49:01,110 --> 00:49:04,820
I mean this was very approximate
example and that's why I chose

764
00:49:04,820 --> 00:49:09,390
such short-dated stock, that
before it pays any dividends,

765
00:49:09,390 --> 00:49:12,920
and where we can assume
the volatility is constant,

766
00:49:12,920 --> 00:49:15,400
and so on and so forth,
to match the prices.

767
00:49:15,400 --> 00:49:28,430
Otherwise, the prices
wouldn't match.

768
00:49:28,430 --> 00:49:29,930
PROFESSOR: Thank you.