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Additional information
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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
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
00:12:32.710 --> 00:12:40.600
of pricing, just a few
observations and statements.
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First of all, it
turns out that thanks
00:12:43.550 --> 00:12:47.760
to developed mathematics,
mathematical theory,
00:12:47.760 --> 00:12:51.480
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.
00:12:56.390 --> 00:13:01.360
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.
00:13:08.020 --> 00:13:13.670
So dealers quote these contracts
and there is a great agreement
00:13:13.670 --> 00:13:17.380
on the prices.
00:13:17.380 --> 00:13:22.250
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.
00:13:29.820 --> 00:13:35.060
So it doesn't matter what
are your views on the growth
00:13:35.060 --> 00:13:38.080
prospects of the stock.
00:13:38.080 --> 00:13:44.150
It will not affect the price
of the derivative contract.
00:13:44.150 --> 00:13:48.210
As I said, so the
mathematical part of it
00:13:48.210 --> 00:13:52.190
comes into giving the exact
price without any uncertainty.
00:13:55.080 --> 00:13:59.880
So let's consider a
simple example now.
00:13:59.880 --> 00:14:02.690
Let's assume that we are
in a very simple world.
00:14:02.690 --> 00:14:07.180
Well, first of all, in our world
there are only three objects --
00:14:07.180 --> 00:14:11.690
the stock itself, the
riskless money market account,
00:14:11.690 --> 00:14:15.900
meaning that it is an account
where we can either borrow
00:14:15.900 --> 00:14:20.560
money or invest money
at the riskless rate r,
00:14:20.560 --> 00:14:22.800
and finally our
derivative contract.
00:14:22.800 --> 00:14:25.620
Here we are not making any
assumptions of what kind
00:14:25.620 --> 00:14:29.200
of derivative contract it
is -- it could be forward,
00:14:29.200 --> 00:14:33.280
it could be call, it could
be put, it can be anything.
00:14:33.280 --> 00:14:35.830
Moreover, our world is so
simple, that first of all,
00:14:35.830 --> 00:14:38.620
it's discrete,
and second of all,
00:14:38.620 --> 00:14:42.396
there is only one time step
to the expiration of power
00:14:42.396 --> 00:14:44.360
of contract, dt.
00:14:44.360 --> 00:14:46.570
Not only there is
only one step left,
00:14:46.570 --> 00:14:51.690
we actually know exactly what
our transition probabilities.
00:14:51.690 --> 00:14:53.430
There are only two
states at the end,
00:14:53.430 --> 00:14:55.170
and we know the
transition probability.
00:14:55.170 --> 00:14:58.420
So with probability p, we
move from the state zero
00:14:58.420 --> 00:15:01.890
to the state one, and with
probability of one minus p,
00:15:01.890 --> 00:15:04.750
we move to the state two.
00:15:04.750 --> 00:15:08.100
And just notice, because this is
riskless money market account,
00:15:08.100 --> 00:15:10.670
it's the same in both cases.
00:15:10.670 --> 00:15:15.892
You just invest money and it
grows with risk-free interest
00:15:15.892 --> 00:15:18.420
rate.
00:15:18.420 --> 00:15:23.830
So, what can we say about the
price of our derivative f?
00:15:23.830 --> 00:15:28.110
Well a simple-minded -- well,
let's start with the forward
00:15:28.110 --> 00:15:29.310
contract.
00:15:29.310 --> 00:15:32.490
We know what the payout in
delta t of our forward contract
00:15:32.490 --> 00:15:35.200
will be, it will be just the
difference between the stock
00:15:35.200 --> 00:15:39.390
price and our strike.
00:15:39.390 --> 00:15:42.940
Well, a simple-minded
approach would be -- well,
00:15:42.940 --> 00:15:44.990
we know the transition
probabilities,
00:15:44.990 --> 00:15:48.980
let's just compute the
expected value of our contract,
00:15:48.980 --> 00:15:54.510
and that's what we would expect
to get if there were many such
00:15:54.510 --> 00:15:55.820
experiments.
00:15:55.820 --> 00:15:58.470
Well, you take the probability
of going to state one,
00:15:58.470 --> 00:16:01.254
you multiply by the
payoff at stage one.
00:16:01.254 --> 00:16:03.420
Take, minus p for probability
of going to state two,
00:16:03.420 --> 00:16:08.590
multiply by the payout
in that state two.
00:16:08.590 --> 00:16:12.600
Sum them up and you
get the expression.
00:16:12.600 --> 00:16:16.720
As I said, the common
thing to choose
00:16:16.720 --> 00:16:19.280
the strike such
that the contract
00:16:19.280 --> 00:16:22.796
has zero value now, so
you get your strike.
00:16:22.796 --> 00:16:24.170
Well, in particular
you could say
00:16:24.170 --> 00:16:26.440
that if you research
the market well
00:16:26.440 --> 00:16:29.350
and you know that the stock
has equal probability of going
00:16:29.350 --> 00:16:33.552
up and down, then actually
you expect your strike
00:16:33.552 --> 00:16:40.400
to be an average of end
values of the stock.
00:16:40.400 --> 00:16:44.060
But as we can imagine,
following our bookie example,
00:16:44.060 --> 00:16:46.270
this is not the right price.
00:16:46.270 --> 00:16:48.820
There is actually
a definite price
00:16:48.820 --> 00:16:52.580
which doesn't depend on
transition probability.
00:16:52.580 --> 00:16:57.220
Here is the reason why
there is a definite price.
00:16:57.220 --> 00:17:02.400
Well let's just consider
a very simple strategy.
00:17:02.400 --> 00:17:05.960
Let's borrow just enough
to purchase a stock.
00:17:05.960 --> 00:17:09.520
So let's borrow S_0
dollars right now and buy
00:17:09.520 --> 00:17:12.110
the stock for this money.
00:17:12.110 --> 00:17:13.920
And let's enter the
forward contract.
00:17:13.920 --> 00:17:17.410
Well, by definition forward
contract has price zero now,
00:17:17.410 --> 00:17:20.440
so we enter the
forward contract.
00:17:20.440 --> 00:17:25.530
Now, at the time dt when our
contract expires, what happens?
00:17:25.530 --> 00:17:27.950
Well, we deliver our
stock, which we already
00:17:27.950 --> 00:17:32.970
have in our hand in
exchange of K dollars.
00:17:32.970 --> 00:17:35.220
That's our forward contract.
00:17:35.220 --> 00:17:38.160
On the other hand, we
have to repay our loan,
00:17:38.160 --> 00:17:40.580
and because it was
a loan, it grew.
00:17:40.580 --> 00:17:46.520
It grew to S_0
times e to the r*dt.
00:17:46.520 --> 00:17:47.870
Now, let's see.
00:17:47.870 --> 00:17:53.460
What would happen if K
was greater than S times e
00:17:53.460 --> 00:17:54.460
to the r*dt?
00:17:54.460 --> 00:17:58.210
Then we know for sure,
we know now for sure,
00:17:58.210 --> 00:18:02.460
that we would make money.
00:18:02.460 --> 00:18:06.060
There is no uncertainty
about it now.
00:18:06.060 --> 00:18:09.190
Similarly, if K is
less than this value,
00:18:09.190 --> 00:18:12.550
then we know that
we will lose money.
00:18:12.550 --> 00:18:14.670
That's not how the
rational market works.
00:18:14.670 --> 00:18:18.170
If everybody knew that
by setting this price
00:18:18.170 --> 00:18:21.430
you would make money, people
would do it all day long
00:18:21.430 --> 00:18:23.760
and make infinite money.
00:18:23.760 --> 00:18:26.720
So there will be no
other side of the market.
00:18:26.720 --> 00:18:28.280
So the price has to go down.
00:18:28.280 --> 00:18:35.280
So the only choice for K, the
only market-implied choice,
00:18:35.280 --> 00:18:42.622
is that K has to be equal
to S times e to the r*dt.
00:18:42.622 --> 00:18:44.580
As you can see, it doesn't
depend on transition
00:18:44.580 --> 00:18:46.200
probabilities at all.
00:18:46.200 --> 00:18:49.560
That's what market implies us.
00:18:49.560 --> 00:18:51.490
That's the price of
forward contract,
00:18:51.490 --> 00:18:56.190
and that actually
explains why, when
00:18:56.190 --> 00:18:59.130
I was plotting the
forward contract,
00:18:59.130 --> 00:19:02.010
current price was just
the straight line,
00:19:02.010 --> 00:19:06.890
it's just discounted payoff.
00:19:06.890 --> 00:19:15.630
The payout is linear, so just
the parallel to the payoff.
00:19:15.630 --> 00:19:17.690
That's the idea, basically.
00:19:17.690 --> 00:19:25.280
The idea is to try to find
such a portfolio of stock
00:19:25.280 --> 00:19:30.290
and the money market
account with such a payout ,
00:19:30.290 --> 00:19:35.620
which will exactly replicate
the payoff of our derivative.
00:19:35.620 --> 00:19:37.896
If we found such of a
portfolio, than we know for sure
00:19:37.896 --> 00:19:39.270
that the value of
this portfolio,
00:19:39.270 --> 00:19:41.520
the replicating
portfolio today is
00:19:41.520 --> 00:19:44.590
equal to the value of the
derivative, because otherwise,
00:19:44.590 --> 00:19:48.110
you would make or
lose money risklessly.
00:19:48.110 --> 00:19:51.550
That's no-arbitrage condition.
00:19:51.550 --> 00:19:56.300
So, can we apply it to our
general one-step world?
00:19:56.300 --> 00:20:01.910
Well, if we have a general
payout f, what we want to do,
00:20:01.910 --> 00:20:05.140
we want to form a
replicating portfolio such
00:20:05.140 --> 00:20:09.970
that at expiration time, it
will replicate our payouts.
00:20:09.970 --> 00:20:15.540
So we want to choose such
constants a and b that such
00:20:15.540 --> 00:20:17.550
that the combination of
stock and money market
00:20:17.550 --> 00:20:21.430
account in both
states will replicate
00:20:21.430 --> 00:20:23.570
the payout of our option.
00:20:23.570 --> 00:20:27.940
Then, if we are able to
find such constants a and b,
00:20:27.940 --> 00:20:33.900
then we just look at the
current price of the contract
00:20:33.900 --> 00:20:36.620
and it has to be equal
to the current price
00:20:36.620 --> 00:20:38.930
of our derivative.
00:20:38.930 --> 00:20:41.360
Well, but in our particular
case, this is easy.
00:20:41.360 --> 00:20:45.770
It's just two linear equations
with two unknowns, easily
00:20:45.770 --> 00:20:52.020
solved, and here is current
price of our derivative.
00:20:52.020 --> 00:20:54.500
No matter what payout is --
I mean you just substitute
00:20:54.500 --> 00:21:02.980
the payout here, and if you
know S_1 and S_2, that's it.
00:21:02.980 --> 00:21:06.320
A useful way to look at this,
just to re-write this equation,
00:21:06.320 --> 00:21:13.350
is in this form, and then notice
that actually the current price
00:21:13.350 --> 00:21:20.510
of our derivative can be
viewed as a discounted expected
00:21:20.510 --> 00:21:25.350
payout of the derivative, but
with very certain probability.
00:21:25.350 --> 00:21:28.460
This probability,
it doesn't come
00:21:28.460 --> 00:21:32.020
from statistical properties of
the stock or from any research,
00:21:32.020 --> 00:21:34.720
it actually is
defined by the market.
00:21:34.720 --> 00:21:37.080
So it's called a
risk-neutral probability.
00:21:37.080 --> 00:21:42.050
So this probability
doesn't depend
00:21:42.050 --> 00:21:48.660
on the views on the market
by the market participants.
00:21:48.660 --> 00:21:51.860
An interesting observation
is that actually,
00:21:51.860 --> 00:21:57.435
the value of this stock, the
discounted value of the stock
00:21:57.435 --> 00:22:02.860
is actually also is expected
value of our outcomes
00:22:02.860 --> 00:22:06.850
under this risk-neutral
probability.
00:22:06.850 --> 00:22:09.450
That's basically general idea.
00:22:09.450 --> 00:22:14.330
Now let's move one notch
up and try to apply
00:22:14.330 --> 00:22:17.760
these idea to continuous case.
00:22:17.760 --> 00:22:20.600
Well, if you live in
continuous world now,
00:22:20.600 --> 00:22:27.270
we need to make some assumptions
on the behavior of the stock.
00:22:27.270 --> 00:22:31.225
The very common assumption is
that the dynamics of the stock
00:22:31.225 --> 00:22:32.770
is log-normal.
00:22:32.770 --> 00:22:35.870
Log-normal meaning that
the logarithm of the stock
00:22:35.870 --> 00:22:38.300
is actually normally
distributed.
00:22:38.300 --> 00:22:43.520
So, here mu is some drift, sigma
is the volatility of our stock,
00:22:43.520 --> 00:22:48.530
and dW is a Wiener process,
W is a Wiener process
00:22:48.530 --> 00:22:50.970
such that dW is
normally distributed
00:22:50.970 --> 00:22:56.320
with mean zero and
variance square root dt.
00:22:56.320 --> 00:23:00.530
Our approach would be to find
the replicating portfolio.
00:23:00.530 --> 00:23:01.550
And what does it mean?
00:23:01.550 --> 00:23:07.020
It means that we want to find
such constants, over time dt --
00:23:07.020 --> 00:23:10.900
so we assume that a and b are
constant over the next step,
00:23:10.900 --> 00:23:15.730
dt -- such that the change
in our derivative is a linear
00:23:15.730 --> 00:23:20.600
combination with this constant
of the change of our underlying
00:23:20.600 --> 00:23:25.740
security and the change
of money market account.
00:23:25.740 --> 00:23:31.800
Now we just need to look more
closely at this equation.
00:23:31.800 --> 00:23:34.520
First of all, let's
concentrate on df.
00:23:34.520 --> 00:23:44.910
So, f, our derivative,
is a function
00:23:44.910 --> 00:23:48.660
of stock value and time.
00:23:48.660 --> 00:23:52.060
But unfortunately, our
stock value is stochastic,
00:23:52.060 --> 00:24:00.110
so df is not that simple, and
to write df out we have to use
00:24:00.110 --> 00:24:03.180
a famous -- Ito's formula
from stochastic calculus,
00:24:03.180 --> 00:24:10.910
which actually is analogous of
Taylor's formula for stochastic
00:24:10.910 --> 00:24:12.080
variables.
00:24:12.080 --> 00:24:12.710
Let's see.
00:24:12.710 --> 00:24:15.240
If our S will not
be stochastic, if it
00:24:15.240 --> 00:24:17.110
would be completely
deterministic
00:24:17.110 --> 00:24:19.640
and depend only
on dt, then there
00:24:19.640 --> 00:24:24.670
would be no term and
differential f is just
00:24:24.670 --> 00:24:26.800
the standard expression.
00:24:26.800 --> 00:24:31.070
On the other hand,
if we have dependence
00:24:31.070 --> 00:24:35.040
on stochastic
variables, then we have
00:24:35.040 --> 00:24:38.130
to have more terms,
and why this happens?
00:24:38.130 --> 00:24:41.330
Well, in very rough words
is that because the order
00:24:41.330 --> 00:24:45.510
of magnitude of dW is
higher than dt's --
00:24:45.510 --> 00:24:47.930
it's square root of dt.
00:24:47.930 --> 00:24:51.340
So we have to make into
account more terms,
00:24:51.340 --> 00:24:56.970
and in particular, we have to
take into account next order
00:24:56.970 --> 00:24:59.070
of dS squared.
00:24:59.070 --> 00:25:01.970
Formally, dS square can
be written this way,
00:25:01.970 --> 00:25:04.900
and again, very rough
explanation is as follows.
00:25:04.900 --> 00:25:07.970
If we would square
this equation there
00:25:07.970 --> 00:25:09.490
will be three terms there.
00:25:09.490 --> 00:25:11.900
One would come from the
square of this term,
00:25:11.900 --> 00:25:16.530
and this would be of
the order of dt squared,
00:25:16.530 --> 00:25:20.240
next order of magnitude
-- much smaller than dt.
00:25:20.240 --> 00:25:23.060
The second term will be
cross-product of dW*dt.
00:25:23.060 --> 00:25:26.200
What order of magnitude
we are talking about,
00:25:26.200 --> 00:25:32.990
it is dt to the power 3/2,
again, much smaller than dt.
00:25:32.990 --> 00:25:37.090
On the other hand, the third
term will be the square of dW,
00:25:37.090 --> 00:25:40.520
this is of order
of magnitude of dt,
00:25:40.520 --> 00:25:43.500
so that's what we have to keep.
00:25:43.500 --> 00:25:48.990
And that's what Ito's
formula is about.
00:25:48.990 --> 00:25:56.210
Now, we are basically,
we know all terms here,
00:25:56.210 --> 00:26:01.560
and let me stress out that this
term, dB, it is not stochastic,
00:26:01.560 --> 00:26:04.860
it's completely
deterministic because we
00:26:04.860 --> 00:26:09.350
know that B grows with the
rate r, that's what it is.
00:26:09.350 --> 00:26:14.510
So we substitute all those terms
into our replicating equation.
00:26:14.510 --> 00:26:16.000
We collect the terms.
00:26:16.000 --> 00:26:17.120
We get this equation.
00:26:17.120 --> 00:26:19.580
And again, there is
the deterministic part,
00:26:19.580 --> 00:26:20.850
there is stochastic part.
00:26:20.850 --> 00:26:23.710
So the only way for
this equation to hold
00:26:23.710 --> 00:26:27.543
is this term to be equal
to this term, and this term
00:26:27.543 --> 00:26:32.170
to be equal to this term, and
that's what's written out here.
00:26:32.170 --> 00:26:34.780
So again, two equations,
these two unknowns, and here
00:26:34.780 --> 00:26:35.660
is answer.
00:26:38.210 --> 00:26:47.800
Finally, let's take
a*S to another part.
00:26:47.800 --> 00:26:50.720
Notice that this
part of our equation
00:26:50.720 --> 00:26:53.880
is completely deterministic.
00:26:53.880 --> 00:26:55.810
So we know how it will grow.
00:26:55.810 --> 00:27:01.020
So basically, d of f minus
a*S, which is b times dB,
00:27:01.020 --> 00:27:05.890
is r times b times dt.
00:27:05.890 --> 00:27:08.740
And we know all other
terms, we substitute them
00:27:08.740 --> 00:27:12.670
here, take something
to the left-hand side
00:27:12.670 --> 00:27:14.220
and get this equation.
00:27:14.220 --> 00:27:21.680
So this is partial differential
equation for our derivative f,
00:27:21.680 --> 00:27:26.670
as a function of S and
t, of second order,
00:27:26.670 --> 00:27:30.310
and this equation is the
famous Black-Scholes equation.
00:27:30.310 --> 00:27:34.200
It was derived by Fischer
Black and Myron Scholes
00:27:34.200 --> 00:27:38.940
in their famous paper
published in 1973.
00:27:38.940 --> 00:27:41.240
Myron Scholes and
Robert Merton actually
00:27:41.240 --> 00:27:44.640
received Nobel Prize
for deriving and solving
00:27:44.640 --> 00:27:47.560
this equation in '97.
00:27:47.560 --> 00:27:51.250
Black was already
dead by the time.
00:27:51.250 --> 00:27:55.320
This is really the
cornerstone of math finance.
00:27:59.290 --> 00:28:05.780
The cornerstone is because
using the replicating portfolio,
00:28:05.780 --> 00:28:12.080
using this reasoning, we were
able to find an exact equation
00:28:12.080 --> 00:28:14.040
for our derivative.
00:28:14.040 --> 00:28:16.980
So a few remarks
on Black-Scholes.
00:28:16.980 --> 00:28:20.930
So first of all, we
made some assumptions
00:28:20.930 --> 00:28:24.600
on the dynamic of the stock, but
we never made any assumptions
00:28:24.600 --> 00:28:27.150
on our derivative.
00:28:27.150 --> 00:28:32.610
Which means that any derivative
has to satisfy this equation,
00:28:32.610 --> 00:28:35.040
and that's very strong
result. So if you
00:28:35.040 --> 00:28:36.740
assume that our
stock is lognormal,
00:28:36.740 --> 00:28:39.780
which is not a bad assumption
and agrees quite well
00:28:39.780 --> 00:28:42.810
with the market, then we
basically, in principle,
00:28:42.810 --> 00:28:44.470
can price any derivative.
00:28:44.470 --> 00:28:48.400
We know the equation
for any derivative.
00:28:48.400 --> 00:28:51.800
The other thing is that
our Black-Scholes equation
00:28:51.800 --> 00:28:55.530
doesn't depend on
the actual drift mu
00:28:55.530 --> 00:28:57.180
in the dynamics of our stock.
00:28:57.180 --> 00:29:06.320
So again, it is the manifest
of risk-neutral dynamic.
00:29:06.320 --> 00:29:12.150
Not only we wrote down the
equation for our derivative,
00:29:12.150 --> 00:29:15.030
we also found a
replicating portfolio.
00:29:15.030 --> 00:29:18.280
So in other words, we
found a hedging strategy,
00:29:18.280 --> 00:29:21.795
meaning that at
any given time we
00:29:21.795 --> 00:29:28.360
can form this portfolio
with rates a and b.
00:29:28.360 --> 00:29:33.460
If we hold both the derivative
and both replicating portfolio,
00:29:33.460 --> 00:29:38.490
altogether, this
is zero sum gain.
00:29:38.490 --> 00:29:41.950
We know that no matter
where stock moves,
00:29:41.950 --> 00:29:44.710
we will not lose
money or gain money.
00:29:44.710 --> 00:29:51.110
So if we just charge
bid-offer on the derivative,
00:29:51.110 --> 00:29:53.330
if we charge a fee
on the contract,
00:29:53.330 --> 00:29:56.420
we can hedge ourself
perfectly, buy the contract
00:29:56.420 --> 00:29:58.970
or sell the contract,
hedge perfectly ourself
00:29:58.970 --> 00:30:01.460
and just make money
on the fee, that's it.
00:30:06.180 --> 00:30:09.060
Finally, more
mathematical remark
00:30:09.060 --> 00:30:13.250
is that actually after a few
manipulations, a few change
00:30:13.250 --> 00:30:16.710
of variables, the Black-Scholes
equation comes out
00:30:16.710 --> 00:30:19.980
to be just a heat
question, which you already
00:30:19.980 --> 00:30:22.190
saw in this class.
00:30:22.190 --> 00:30:23.650
This is very good news.
00:30:23.650 --> 00:30:25.000
Why is this good news?
00:30:25.000 --> 00:30:27.250
Well, because heat equation
is very well studied.
00:30:27.250 --> 00:30:32.986
So the solutions are well-known,
and numerical methods, the ways
00:30:32.986 --> 00:30:34.860
to solve it, in particular
the numerical ways
00:30:34.860 --> 00:30:36.760
to solve it are well-known.
00:30:36.760 --> 00:30:38.220
So we are in business.
00:30:38.220 --> 00:30:45.130
But as any partial
differential equation,
00:30:45.130 --> 00:30:49.450
the equation itself doesn't
make much sense because to find
00:30:49.450 --> 00:30:55.610
a particular solution we need
boundary and initial condition.
00:30:58.520 --> 00:31:01.940
And although any derivative
satisfies Black-Scholes
00:31:01.940 --> 00:31:08.310
equation, the final
and boundary conditions
00:31:08.310 --> 00:31:10.720
will vary from
contract to contract.
00:31:10.720 --> 00:31:15.510
Here are a few examples of the
final and boundary conditions.
00:31:15.510 --> 00:31:18.680
Here, an interesting
remark that if, usually,
00:31:18.680 --> 00:31:22.270
we would talk about
initial condition, here
00:31:22.270 --> 00:31:24.650
we are talking about
final condition.
00:31:24.650 --> 00:31:26.340
The time goes in
reverse; we know
00:31:26.340 --> 00:31:30.150
the state of the world at the
end, at expiration, not today.
00:31:30.150 --> 00:31:36.810
So, here are final and boundary
conditions for call and put,
00:31:36.810 --> 00:31:40.640
and let's look a little bit
at the pictures for our call
00:31:40.640 --> 00:31:44.250
and put to see where
they come from.
00:31:44.250 --> 00:31:48.950
So for example, for calls, well,
this is our final condition,
00:31:48.950 --> 00:31:50.930
right, this is
defined by the payout.
00:31:50.930 --> 00:31:54.690
On the other hand, the boundary
condition, well, what happens,
00:31:54.690 --> 00:31:57.646
we put them at zero at an
infinity, we [INAUDIBLE PHRASE]
00:31:57.646 --> 00:31:59.020
to put them at
zero and infinity.
00:31:59.020 --> 00:31:59.519
And why?
00:31:59.519 --> 00:32:03.340
Well, because if stock hits
zero then it stays at zero.
00:32:03.340 --> 00:32:06.330
That's what our dynamics show.
00:32:06.330 --> 00:32:11.400
So the value of our contract
at maturity will become zero.
00:32:11.400 --> 00:32:14.990
On the other hand, if the
stock grows, grows to infinity,
00:32:14.990 --> 00:32:17.360
a good assumption to
make is that actually it
00:32:17.360 --> 00:32:19.830
becomes similar to stock
itself, so it would just
00:32:19.830 --> 00:32:26.160
become parallel to the stock,
and that's the conditions
00:32:26.160 --> 00:32:27.150
which we impose here.
00:32:30.270 --> 00:32:36.290
Similarly for the put, you
can derive these conditions.
00:32:36.290 --> 00:32:41.740
And again, just because
it is a heat equation,
00:32:41.740 --> 00:32:44.910
it turns out that for a simple
derivative such as the calls
00:32:44.910 --> 00:32:50.070
and puts, it is possible to
find an exact analytic solution.
00:32:50.070 --> 00:32:54.350
Here are exact analytic set
of solutions for a call, put
00:32:54.350 --> 00:32:57.090
and the digital contracts.
00:32:57.090 --> 00:33:01.310
Well, not surprising
again, I mean
00:33:01.310 --> 00:33:04.310
they're all connected
to the error function,
00:33:04.310 --> 00:33:07.760
so to the normal
distribution, basically,
00:33:07.760 --> 00:33:12.400
as the solutions of heat
equation ought to be.
00:33:12.400 --> 00:33:14.580
Why do they look
exactly the same?
00:33:14.580 --> 00:33:17.020
If we have five
minutes at the end,
00:33:17.020 --> 00:33:21.690
we'll probably shed some
light on the specific form
00:33:21.690 --> 00:33:22.750
of equations.
00:33:22.750 --> 00:33:28.090
But let me just stress that we
can see that it's discounted,
00:33:28.090 --> 00:33:32.790
and what I'm claiming, it's
expected value of our payout
00:33:32.790 --> 00:33:36.810
under risk-neutral measure.
00:33:36.810 --> 00:33:42.290
Here is an example -- it's
of a particular call option
00:33:42.290 --> 00:33:44.160
on the same IBM stock.
00:33:44.160 --> 00:33:46.580
So I chose the
short-dated contract,
00:33:46.580 --> 00:33:50.300
just to avoid the
dividend payment.
00:33:50.300 --> 00:33:53.820
So it's a contract
expiring on March 18.
00:33:53.820 --> 00:33:56.750
So there is 10
days to expiration.
00:33:56.750 --> 00:34:02.170
The stock, as we saw,
the expirations of stock,
00:34:02.170 --> 00:34:06.750
as we saw, is what's
trading at 81.14.
00:34:06.750 --> 00:34:11.800
The volatility is
somewhere around 14%,
00:34:11.800 --> 00:34:17.110
estimated either from other
options or historically.
00:34:17.110 --> 00:34:20.210
Here is the price
of our contract.
00:34:20.210 --> 00:34:23.750
I also have a simple
Black-Scholes calculator here,
00:34:23.750 --> 00:34:28.800
and let's see if we
can match this price.
00:34:28.800 --> 00:34:39.280
So let's see, I believe the
volatility was 13%, right,
00:34:39.280 --> 00:34:42.240
13.47.
00:34:42.240 --> 00:34:45.220
The interest rate,
it's already here.
00:34:45.220 --> 00:34:48.540
As we all know, Fed just
bumped the interest rate,
00:34:48.540 --> 00:34:51.980
so they are at 4.75% right now.
00:34:51.980 --> 00:34:56.360
The strike of our option was 80.
00:34:56.360 --> 00:34:58.840
Time to expiration
was actually 10 days,
00:34:58.840 --> 00:35:02.390
and this should be measured
at a fraction of year.
00:35:02.390 --> 00:35:08.440
So we divide 10 by 365.
00:35:08.440 --> 00:35:15.350
This stock was trading at
81.14, if I'm not mistaken.
00:35:15.350 --> 00:35:21.210
Here is the price of our call
options contract, which is 150.
00:35:21.210 --> 00:35:26.630
Well, it's within the offer.
00:35:26.630 --> 00:35:29.070
So maybe our
volatility's slightly off
00:35:29.070 --> 00:35:35.140
and if we increase
[UNINTELLIGIBLE] to say,
00:35:35.140 --> 00:35:42.330
increase it to 14%, it
will go slightly up.
00:35:42.330 --> 00:35:44.800
152.
00:35:44.800 --> 00:35:48.580
Well, in general, let's
play a little bit with it.
00:35:48.580 --> 00:35:50.480
Well, it is very
short-dated option,
00:35:50.480 --> 00:35:55.990
so the value of our option
is very close to the payout.
00:35:55.990 --> 00:35:58.870
So if we increase
the time to maturity,
00:35:58.870 --> 00:36:02.280
let's make it two years
just to see where --
00:36:02.280 --> 00:36:08.980
so now value of our option is
-- well, that's what it is.
00:36:08.980 --> 00:36:15.040
If increase volatility, sure
enough, let's make it 30%.
00:36:15.040 --> 00:36:15.970
So what do we expect?
00:36:15.970 --> 00:36:19.340
We expect if volatility is
higher, the uncertainty higher,
00:36:19.340 --> 00:36:25.300
so the value of our contract
should go up, and it sure does.
00:36:29.120 --> 00:36:31.520
So basically that's how
Black-Scholes works.
00:36:39.690 --> 00:36:41.520
And plenty of those
contracts trade
00:36:41.520 --> 00:36:47.670
on the market, but unfortunately
not all of these contracts
00:36:47.670 --> 00:36:50.600
are so simple as calls and puts.
00:36:50.600 --> 00:36:55.610
First of all, there are many
more complicated products
00:36:55.610 --> 00:37:03.390
with more difficult payout,
which will constitute different
00:37:03.390 --> 00:37:08.800
and probably discontinuous final
conditions on our Black-Scholes
00:37:08.800 --> 00:37:10.930
equation.
00:37:10.930 --> 00:37:14.590
Moreover, we made an
assumption that the volatility
00:37:14.590 --> 00:37:16.650
is constant with time,
and interest rate
00:37:16.650 --> 00:37:18.210
is constant with time.
00:37:18.210 --> 00:37:21.860
It is certainly not
true for the real world.
00:37:21.860 --> 00:37:24.790
Volatility probably
should be time-dependent,
00:37:24.790 --> 00:37:26.540
and this would make
the coefficients
00:37:26.540 --> 00:37:30.090
in our Black-Scholes
equation time-dependent.
00:37:30.090 --> 00:37:33.250
Unfortunately, these cannot
be solve analytically.
00:37:33.250 --> 00:37:36.420
So in most of the
cases in practice,
00:37:36.420 --> 00:37:42.530
we will have to use some
kind of numerical solution.
00:37:42.530 --> 00:37:45.870
Finite difference methods
is the typical approach
00:37:45.870 --> 00:37:47.200
for the heat equation.
00:37:47.200 --> 00:37:50.330
As you know, both explicit
and implicit schemes,
00:37:50.330 --> 00:37:57.910
and you will discuss
some of those in 18.086.
00:37:57.910 --> 00:37:58.630
Tree methods.
00:37:58.630 --> 00:38:05.090
Tree methods meaning that we
go back to our one-step tree,
00:38:05.090 --> 00:38:08.380
and basically assume that
our time to expiration
00:38:08.380 --> 00:38:12.000
is many time steps away and
we'll grow the tree further,
00:38:12.000 --> 00:38:14.270
so from this node we
have two more nodes,
00:38:14.270 --> 00:38:15.550
and so and so forth.
00:38:15.550 --> 00:38:19.670
That would imply the final
condition at the end,
00:38:19.670 --> 00:38:23.640
and discount back using our
risk-neutral probabilities,
00:38:23.640 --> 00:38:25.170
and get the price now.
00:38:25.170 --> 00:38:27.240
So those are called
tree methods.
00:38:27.240 --> 00:38:30.330
One can show that actually those
tree methods are equivalent
00:38:30.330 --> 00:38:36.100
to finite difference -- explicit
finite difference schemes.
00:38:36.100 --> 00:38:38.110
Those are very popular.
00:38:38.110 --> 00:38:41.830
But again, in tree methods,
what is very important
00:38:41.830 --> 00:38:44.746
is to set the probabilities
from your tree, the transition
00:38:44.746 --> 00:38:46.120
probabilities, to
the right ones,
00:38:46.120 --> 00:38:48.960
and the right ones are
risk-neutral probabilities.
00:38:48.960 --> 00:38:53.170
Probabilities implied
by the market, actually.
00:38:53.170 --> 00:38:58.280
Another important numerical
method is Monte Carlo
00:38:58.280 --> 00:39:02.910
simulation where you would
simulate many different
00:39:02.910 --> 00:39:05.650
scenarios of the development of
your stock up to the maturity,
00:39:05.650 --> 00:39:09.840
and then, basically
find -- using this path,
00:39:09.840 --> 00:39:15.990
you will find the expected
value of your payout.
00:39:15.990 --> 00:39:20.210
But again, in order
for this expected value
00:39:20.210 --> 00:39:24.810
to be the same as the
risk-neutral value,
00:39:24.810 --> 00:39:27.050
as the arbitrage-free
value, you have
00:39:27.050 --> 00:39:29.910
to develop your Monte
Carlo simulations
00:39:29.910 --> 00:39:31.800
with risk-neutral probabilities.
00:39:31.800 --> 00:39:37.360
So, risk-neutral valuation
is extremely important.
00:39:40.780 --> 00:39:45.810
Here is actually the general
risk-neutral statement, which
00:39:45.810 --> 00:39:51.580
one can prove, is that actually,
the value of any derivative
00:39:51.580 --> 00:39:56.140
is just discounted expected
value of the payout
00:39:56.140 --> 00:39:59.280
of this derivative
at maturity, but you
00:39:59.280 --> 00:40:02.140
have to take this expectation
at the right measure.
00:40:02.140 --> 00:40:05.230
Using the right measure, meaning
that you have to set correctly
00:40:05.230 --> 00:40:08.705
the transition probability
-- you have to make them
00:40:08.705 --> 00:40:09.330
market-neutral.
00:40:12.710 --> 00:40:19.950
Under this measure, actually
the dynamics of our stocks
00:40:19.950 --> 00:40:22.660
looks slightly different,
and as you can see,
00:40:22.660 --> 00:40:25.240
our drift becomes
the interest rate.
00:40:25.240 --> 00:40:27.810
So under risk-neutral
measure, everything
00:40:27.810 --> 00:40:32.430
grows with our
risk-free interest rate.
00:40:32.430 --> 00:40:38.720
Just to shed a little
bit of light on how we go
00:40:38.720 --> 00:40:42.630
at the solutions
for calls and puts,
00:40:42.630 --> 00:40:44.920
Black-Scholes solutions
for call and put,
00:40:44.920 --> 00:40:47.920
well this is the
distribution of our stock,
00:40:47.920 --> 00:40:51.790
log-normal distribution
of our stock at time T,
00:40:51.790 --> 00:40:57.660
and if we take this distribution
and integrate our payout
00:40:57.660 --> 00:41:01.950
of our call option against this
distribution -- in other words,
00:41:01.950 --> 00:41:10.260
find the expected value of
payout of our call option under
00:41:10.260 --> 00:41:17.810
risk-neutral measure,
then, sure enough,
00:41:17.810 --> 00:41:21.200
you will get
[UNINTELLIGIBLE PHRASE].
00:41:21.200 --> 00:41:24.000
This illustrates the best
-- because what is digital?
00:41:24.000 --> 00:41:26.750
Digital is just the
probability to end up
00:41:26.750 --> 00:41:29.590
at above the strike
at time T, right?
00:41:29.590 --> 00:41:34.110
So if you integrate
this log-normal pdf
00:41:34.110 --> 00:41:42.310
from the strike K to infinity,
that will be your answer.
00:41:42.310 --> 00:41:44.930
This is a good exercise
in integration,
00:41:44.930 --> 00:41:47.760
to make sure that it's correct.
00:41:47.760 --> 00:41:49.270
So let's see.
00:41:49.270 --> 00:41:52.170
To conclude, what we've seen.
00:41:52.170 --> 00:42:02.180
So, we have seen that modern
derivatives business makes
00:42:02.180 --> 00:42:04.600
use of quite
advanced mathematics,
00:42:04.600 --> 00:42:08.050
and what kinds of
mathematics is used there?
00:42:08.050 --> 00:42:12.630
Well, partial differential
equations are used heavily.
00:42:12.630 --> 00:42:14.530
Numerical methods
for the solution
00:42:14.530 --> 00:42:20.470
of this partial differential
equations are naturally used.
00:42:20.470 --> 00:42:23.890
In order to get these
equations, we actually
00:42:23.890 --> 00:42:26.870
need to operate in terms
of stochastic calculus,
00:42:26.870 --> 00:42:31.180
meaning that we need to know
how to deal with Ito calculus,
00:42:31.180 --> 00:42:36.580
Ito formula, Girsanov theorem,
and so on and so forth.
00:42:36.580 --> 00:42:41.330
The other thing is to be
able to build simulations
00:42:41.330 --> 00:42:45.380
to solve the heat equation
and all other equations
00:42:45.380 --> 00:42:47.780
that you might encounter.
00:42:47.780 --> 00:42:51.230
The topic which we
didn't touch upon
00:42:51.230 --> 00:42:54.660
is statistics because,
of course, very advanced
00:42:54.660 --> 00:42:58.690
statistics is used
for many, many things,
00:42:58.690 --> 00:43:02.320
for analyzing
historical data, which
00:43:02.320 --> 00:43:04.040
can be quite
beautiful for trading
00:43:04.040 --> 00:43:07.660
strategies and many others.
00:43:07.660 --> 00:43:12.450
Besides these five topics,
there is much, much more
00:43:12.450 --> 00:43:17.400
to mathematical finance,
which makes it a very, very
00:43:17.400 --> 00:43:19.760
exciting field to work in.
00:43:19.760 --> 00:43:22.126
That's what I wanted
to talk about.
00:43:22.126 --> 00:43:23.750
Thank you very much
for your attention.
00:43:30.890 --> 00:43:35.430
PROFESSOR: Maybe I'll
ask a firm question
00:43:35.430 --> 00:43:38.100
about boundary
conditions, because you
00:43:38.100 --> 00:43:40.060
had said that
those are different
00:43:40.060 --> 00:43:42.860
for different contracts.
00:43:42.860 --> 00:43:47.290
And how do you deal with them
in the finite differences
00:43:47.290 --> 00:43:52.140
or the tree model or whatever?
00:43:52.140 --> 00:43:54.750
What would be a typical one?
00:43:54.750 --> 00:43:57.560
GUEST SPEAKER: Well, typical
one -- two very typical ones.
00:43:57.560 --> 00:44:05.500
So those, you basically
make a grid of your problem,
00:44:05.500 --> 00:44:08.100
in particular, you
build a tree, which
00:44:08.100 --> 00:44:11.650
is actually a grid of
all possible outcomes.
00:44:11.650 --> 00:44:28.490
You set them up at the
end, so your tree grows --
00:44:28.490 --> 00:44:36.310
so you set your boundary here
at the end, and well, you set,
00:44:36.310 --> 00:44:45.480
probably, some initial --
this is final condition,
00:44:45.480 --> 00:44:48.690
so you set some boundary
conditions here.
00:44:48.690 --> 00:44:55.710
So this is your time T. This
is t = 0, time t -- this is 0,
00:44:55.710 --> 00:45:04.030
this is 1, this is 2, this is
T. So you set your payout here,
00:45:04.030 --> 00:45:11.340
so it will be maximum
of S minus K and 0.
00:45:11.340 --> 00:45:14.940
PROFESSOR: How many time
steps might you take in this?
00:45:14.940 --> 00:45:20.315
GUEST SPEAKER: Well, you would
do like daily for three months
00:45:20.315 --> 00:45:22.476
-- if you three-month options.
00:45:22.476 --> 00:45:23.600
PROFESSOR: Maybe 100 steps.
00:45:23.600 --> 00:45:25.308
GUEST SPEAKER: Yeah,
something like that.
00:45:25.308 --> 00:45:28.470
Well, if it's two-year option,
that you probably would do it
00:45:28.470 --> 00:45:30.820
weekly or something like that.
00:45:30.820 --> 00:45:32.350
PROFESSOR: You don't
get into large,
00:45:32.350 --> 00:45:35.490
what would be scientifically,
large-scale calculating.
00:45:35.490 --> 00:45:37.490
PROFESSOR: No, in
finance we usually
00:45:37.490 --> 00:45:39.040
don't keep this problem--.
00:45:39.040 --> 00:45:40.810
PROFESSOR: In
finite differences,
00:45:40.810 --> 00:45:44.670
do you use like higher-order
-- suppose, well,
00:45:44.670 --> 00:45:47.230
you had second derivatives,
would you always use second
00:45:47.230 --> 00:45:51.449
differences or
second-order accuracy?
00:45:51.449 --> 00:45:52.740
GUEST SPEAKER: In general, yes.
00:45:52.740 --> 00:45:55.790
In general,
second-order accuracy.
00:45:55.790 --> 00:45:58.210
In general you don't go higher.
00:45:58.210 --> 00:46:01.920
I mean the precision -- well,
it's within cents, right.
00:46:01.920 --> 00:46:06.010
So you cannot do
better than that.
00:46:06.010 --> 00:46:11.650
So it depends -- well, it
depends what kind of amount you
00:46:11.650 --> 00:46:12.830
are dealing with.
00:46:12.830 --> 00:46:16.775
If you're actually selling
and buying units of stock,
00:46:16.775 --> 00:46:21.080
you might consider
something more precise.
00:46:21.080 --> 00:46:25.220
But it's very problem-defined.
00:46:30.280 --> 00:46:33.370
So that's how we deal with it.
00:46:33.370 --> 00:46:35.770
PROFESSOR: Any questions?
00:46:35.770 --> 00:46:38.440
You can put the mic on
if you have a question.
00:46:38.440 --> 00:46:45.080
STUDENT:
[UNINTELLIGIBLE PHRASE].
00:46:55.800 --> 00:46:58.000
GUEST SPEAKER: Well,
it is Markov process.
00:46:58.000 --> 00:46:58.500
Yes.
00:46:58.500 --> 00:47:01.480
I mean, this is just
a numerical solution.
00:47:01.480 --> 00:47:02.910
So yeah, it is Markov process.
00:47:02.910 --> 00:47:04.990
and basically all
stochastic calculus
00:47:04.990 --> 00:47:07.590
is about Markov process,
continuous Markov process.
00:47:15.030 --> 00:47:17.010
PROFESSOR: Is the
mathematics that you
00:47:17.010 --> 00:47:22.600
get involved with pretty
well set now or is there
00:47:22.600 --> 00:47:27.000
a need for more
mathematics, if I
00:47:27.000 --> 00:47:28.920
can ask the question that way?
00:47:28.920 --> 00:47:31.140
PROFESSOR: Yeah,
well, in this field
00:47:31.140 --> 00:47:33.610
it is probably quite well set.
00:47:33.610 --> 00:47:37.410
But if you get into
more complicated fields,
00:47:37.410 --> 00:47:44.690
especially into credit modeling,
the model for the credits
00:47:44.690 --> 00:47:49.810
of certain companies, then
mathematics is not quite set,
00:47:49.810 --> 00:47:53.730
because there, you start
talking about jump processes
00:47:53.730 --> 00:47:58.860
and not Wiener processes, not
just log-normal processes.
00:47:58.860 --> 00:48:03.070
This stochastic differential
equation become very hard,
00:48:03.070 --> 00:48:06.150
but maybe still
analytically tractable.
00:48:06.150 --> 00:48:09.330
So from this point of
view there is need --
00:48:09.330 --> 00:48:13.281
but it's not a
fundamental mathematics,
00:48:13.281 --> 00:48:15.030
it's not that you are
opening a new field,
00:48:15.030 --> 00:48:21.170
but definitely trying to solve
a stochastic differential
00:48:21.170 --> 00:48:23.840
equation -- which usually
boils down to solving a partial
00:48:23.840 --> 00:48:26.780
differential equation
analytically --
00:48:26.780 --> 00:48:30.280
can be pretty hard a
mathematical problem,
00:48:30.280 --> 00:48:32.280
viewed as a
mathematical problem.
00:48:32.280 --> 00:48:33.790
PROFESSOR: So you
showed the example
00:48:33.790 --> 00:48:37.700
of Black-Scholes solver.
00:48:37.700 --> 00:48:39.872
Everybody has that
available all the time?
00:48:39.872 --> 00:48:40.830
GUEST SPEAKER: Oh yeah.
00:48:40.830 --> 00:48:44.550
On Chicago trading
floor, the traders
00:48:44.550 --> 00:48:47.520
have calculators where
they just press a button
00:48:47.520 --> 00:48:49.104
and it's just hard-wired there.
00:48:49.104 --> 00:48:51.270
PROFESSOR: And they're
printing out error functions,
00:48:51.270 --> 00:48:55.060
basically -- a combination
of error function, yeah.
00:48:55.060 --> 00:49:01.110
GUEST SPEAKER: Well, sure
enough, nobody uses just --
00:49:01.110 --> 00:49:04.820
I mean this was very approximate
example and that's why I chose
00:49:04.820 --> 00:49:09.390
such short-dated stock, that
before it pays any dividends,
00:49:09.390 --> 00:49:12.920
and where we can assume
the volatility is constant,
00:49:12.920 --> 00:49:15.400
and so on and so forth,
to match the prices.
00:49:15.400 --> 00:49:28.430
Otherwise, the prices
wouldn't match.
00:49:28.430 --> 00:49:29.930
PROFESSOR: Thank you.