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

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example.

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00:00:27,520 --> 00:00:30,010
So suppose you're a bookie.

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00:00:30,010 --> 00:00:36,550
And what a bookie does--
he sets bets on the horses,

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00:00:36,550 --> 00:00:40,330
sets the odds, and
then pays money back.

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00:00:40,330 --> 00:00:46,050
Probably collects a fee
somewhere in between.

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00:00:46,050 --> 00:00:50,400
So suppose he is a
good bookie and he

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00:00:50,400 --> 00:00:54,370
knows quite well the horses,
and there are two horses.

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00:00:54,370 --> 00:00:57,545
He knows that for sure one
horse has 20% chance of winning

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00:00:57,545 --> 00:01:01,329
and another horse has
80% chance of winning.

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00:01:01,329 --> 00:01:02,870
Obviously, the
general public doesn't

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00:01:02,870 --> 00:01:05,019
have all of this information.

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00:01:05,019 --> 00:01:07,940
So they place a bet
slightly differently.

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00:01:07,940 --> 00:01:12,960
And then there is $10,000
bet on one horse and $50,000

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00:01:12,960 --> 00:01:16,250
bet on another horse.

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00:01:16,250 --> 00:01:20,040
Well, bookie is sure that he
possesses good information.

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00:01:20,040 --> 00:01:23,860
So he-- suppose he
sets the odds according

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00:01:23,860 --> 00:01:26,450
to real-life probability.

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00:01:26,450 --> 00:01:29,570
So he sets it four to one.

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00:01:29,570 --> 00:01:34,660
What would be possible
outcomes of the race for him?

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00:01:34,660 --> 00:01:35,500
Monetary.

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00:01:35,500 --> 00:01:38,820
So suppose the first horse wins.

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00:01:38,820 --> 00:01:40,420
Then what happens?

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00:01:40,420 --> 00:01:45,280
He has to pay back $10,000
and four times more.

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00:01:45,280 --> 00:01:48,130
So he pays out $50,000.

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00:01:48,130 --> 00:01:50,510
And he receives $60,000, right?

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00:01:50,510 --> 00:01:55,640
So he can keep
$10,000 out of it.

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00:01:55,640 --> 00:01:56,780
OK.

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00:01:56,780 --> 00:02:01,360
So what happens is the other
more probable horse wins.

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00:02:01,360 --> 00:02:06,240
Well, he'll have to pay back the
$50,000 and one quarter of it,

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00:02:06,240 --> 00:02:10,139
which is $12.25.

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00:02:10,139 --> 00:02:14,366
So at the end, he'll
pay 62 1/2 thousand,

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00:02:14,366 --> 00:02:16,690
while he collected
$60,000, out right?

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00:02:16,690 --> 00:02:23,630
So he will-- in this
situation, he will lose $2,500.

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00:02:23,630 --> 00:02:26,720
Well, all in all, he
expects to make nothing.

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00:02:26,720 --> 00:02:29,690
So he probably could
collect enough fees

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00:02:29,690 --> 00:02:32,560
to cover his potential loss.

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00:02:32,560 --> 00:02:35,380
But there is certainly a
variability in outcomes.

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00:02:35,380 --> 00:02:36,460
He can win a lot.

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00:02:36,460 --> 00:02:39,090
He can lose some.

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00:02:39,090 --> 00:02:42,980
Now, what if he forgets
about his knowledge

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00:02:42,980 --> 00:02:47,330
about the real-life
probabilities of horses winning

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00:02:47,330 --> 00:02:52,520
or losing and instead sets bets
according to the amount which

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00:02:52,520 --> 00:02:53,870
we are already bet.

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00:02:53,870 --> 00:02:56,190
According to the
market, effectively.

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00:02:56,190 --> 00:03:00,970
So what if he sets
the odds five to one,

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00:03:00,970 --> 00:03:03,910
according to the bets placed?

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00:03:03,910 --> 00:03:06,210
Well, in this situation,
if the first horse wins,

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00:03:06,210 --> 00:03:12,790
he pays back 10 plus
5 times 10, so 60.

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00:03:12,790 --> 00:03:13,950
He is 0.

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00:03:13,950 --> 00:03:19,710
And if the second horse wins,
he pays back 50 plus 1/5 of 50,

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00:03:19,710 --> 00:03:20,950
plus another 10.

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00:03:20,950 --> 00:03:22,460
Again 60.

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00:03:22,460 --> 00:03:26,850
So no matter which horse
wins, he will get 0.

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00:03:26,850 --> 00:03:29,250
We're 100% sure.

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00:03:29,250 --> 00:03:33,030
And if he collects
some fee on top of it,

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00:03:33,030 --> 00:03:35,940
he will make a riskless profit.

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00:03:35,940 --> 00:03:40,300
And that's how, actually,
bookies are operating.

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00:03:40,300 --> 00:03:42,270
So it's a simple example.

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

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and risk-neutral pricing works.

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So we are, here,
not in the business

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of making bets on horses.

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We are actually in the business
of pricing derivatives.

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So we will talk about the
simplest possible derivatives--

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mostly derivatives on stocks.

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But there are more
complicated derivatives,

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00:04:10,470 --> 00:04:14,720
underlying for which could be
interest rates, bonds, swaps,

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00:04:14,720 --> 00:04:17,950
commodities, whatever.

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So a derivative
contract is some--

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00:04:22,540 --> 00:04:26,840
in general speaking, a
formal pay-out connected

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to underlying.

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00:04:27,810 --> 00:04:29,900
Usually, the underlying
is a liquid instrument

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which is traded on exchanges.

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00:04:32,610 --> 00:04:35,600
And derivative may be
traded on exchanges.

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Actually, quite a
few equity options

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are traded on exchanges.

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00:04:39,250 --> 00:04:43,170
But in general, they are
over-the-counter contracts

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

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00:04:48,420 --> 00:04:53,540
One of the simpler derivatives
is a forward contract.

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00:04:53,540 --> 00:04:54,960
So what is a forward contract?

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00:04:54,960 --> 00:05:01,300
A forward contract is a
contract where one party agrees

90
00:05:01,300 --> 00:05:04,850
to buy an asset from another
party for a price which

91
00:05:04,850 --> 00:05:07,030
is agreed today.

92
00:05:07,030 --> 00:05:13,890
Usually, this forward
price is set in such a way

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00:05:13,890 --> 00:05:17,187
that right now, no
money changes hands.

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00:05:17,187 --> 00:05:20,040
Right?

95
00:05:20,040 --> 00:05:22,600
And here is an example.

96
00:05:22,600 --> 00:05:24,450
Well, suppose there
is a stock which,

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00:05:24,450 --> 00:05:27,020
right now, is priced at $80.

98
00:05:27,020 --> 00:05:29,560
And this is the
forward for two years.

99
00:05:29,560 --> 00:05:32,580
So somebody agrees
to buy the stock

100
00:05:32,580 --> 00:05:35,790
in two years for this price.

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00:05:35,790 --> 00:05:39,380
And not surprisingly, I
somehow set this price such

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00:05:39,380 --> 00:05:44,660
that currently the value
of the contract is 0.

103
00:05:44,660 --> 00:05:46,910
And we'll see how I'll
come up with the price.

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00:05:46,910 --> 00:05:49,340
So this blue line is
actually the pay-out,

105
00:05:49,340 --> 00:05:51,055
what will happen at the end.

106
00:05:51,055 --> 00:05:52,450
Right?

107
00:05:52,450 --> 00:05:55,990
The pay-out,
depending-- the graph

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00:05:55,990 --> 00:06:02,330
of F at time T, the
determination time or expiry--

109
00:06:02,330 --> 00:06:04,131
how it depends on
the stock price.

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00:06:04,131 --> 00:06:04,630
Right?

111
00:06:04,630 --> 00:06:07,170
So obviously, the
pay-out is S minus K,

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00:06:07,170 --> 00:06:12,030
where S is the stock price,
so it's a linear function.

113
00:06:12,030 --> 00:06:14,710
It turns out that the counter
price is also a linear function

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00:06:14,710 --> 00:06:15,840
but slightly shifted.

115
00:06:15,840 --> 00:06:18,830
And we'll see how come
it's slightly shifted

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00:06:18,830 --> 00:06:22,410
and how much it
should be shifted.

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00:06:22,410 --> 00:06:28,090
And K is usually referred
to as a strike price.

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00:06:28,090 --> 00:06:30,400
Another slightly more
complicated contract

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00:06:30,400 --> 00:06:33,200
is called a call option.

120
00:06:33,200 --> 00:06:37,570
So if previously the
forward is an obligation

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00:06:37,570 --> 00:06:40,990
to buy the asset
for an agreed price,

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00:06:40,990 --> 00:06:44,460
call option is
actually an option

123
00:06:44,460 --> 00:06:49,110
to buy an asset at the
agreed price today.

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00:06:49,110 --> 00:06:51,130
You can view it--
a call option can

125
00:06:51,130 --> 00:06:54,760
be viewed as kind of
insurance that the--

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00:06:54,760 --> 00:06:58,170
against the asset going down.

127
00:06:58,170 --> 00:07:01,059
Basically the pay-out
is always positive.

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00:07:01,059 --> 00:07:02,100
You can never lose money.

129
00:07:02,100 --> 00:07:03,660
On the forward,
you can lose money.

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00:07:03,660 --> 00:07:05,240
You agree on the price.

131
00:07:05,240 --> 00:07:07,540
The asset ends up being
lower than this price,

132
00:07:07,540 --> 00:07:09,271
but you still have to buy it.

133
00:07:09,271 --> 00:07:09,770
Right?

134
00:07:09,770 --> 00:07:16,470
Here, if the asset ends up
at expiry below strike price

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00:07:16,470 --> 00:07:21,720
or out of the money, then
the pay-out will be 0.

136
00:07:21,720 --> 00:07:26,880
If, on the other hand, it ends
up being above the strike price

137
00:07:26,880 --> 00:07:30,580
or, it's called, the
option is in the money.

138
00:07:30,580 --> 00:07:37,530
Then the pay-out will
be S minus K as before.

139
00:07:37,530 --> 00:07:41,590
So in mathematical
terms, the pay-out

140
00:07:41,590 --> 00:07:44,430
is maximum of S minus K and 0.

141
00:07:44,430 --> 00:07:44,990
Right?

142
00:07:44,990 --> 00:07:49,730
And that's what happens at
expiry time-- this blue line.

143
00:07:49,730 --> 00:07:53,350
So what is the price
of this option now?

144
00:07:53,350 --> 00:07:57,250
Well, obviously it
should be slightly above

145
00:07:57,250 --> 00:08:01,810
because even if now
the asset is slightly

146
00:08:01,810 --> 00:08:04,050
out of the money--
below strike price--

147
00:08:04,050 --> 00:08:07,030
there is some volatility to
it, and there is a probability

148
00:08:07,030 --> 00:08:10,270
that we will still end up
in the money at expiry.

149
00:08:10,270 --> 00:08:13,170
So you would be
willing-- you should

150
00:08:13,170 --> 00:08:16,190
be willing to pay
something for this.

151
00:08:16,190 --> 00:08:19,250
Obviously, if it's way out
of the money, it should be 0.

152
00:08:19,250 --> 00:08:20,020
Right?

153
00:08:20,020 --> 00:08:23,900
On the other hand, if it's
way in the money, in fact,

154
00:08:23,900 --> 00:08:26,240
it should be just as forward.

155
00:08:26,240 --> 00:08:27,120
And in fact, it is.

156
00:08:27,120 --> 00:08:30,170
We'll see because
the probability

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00:08:30,170 --> 00:08:34,850
for the asset going back to
the strike price and below

158
00:08:34,850 --> 00:08:36,940
will be low.

159
00:08:36,940 --> 00:08:39,880
And the Black-Scholes equation
and Black-Scholes formula

160
00:08:39,880 --> 00:08:43,240
is exactly the solution
for this curved line,

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00:08:43,240 --> 00:08:46,690
which we'll see in a second.

162
00:08:46,690 --> 00:08:51,710
Another simple contract, which
is kind of dual to call option,

163
00:08:51,710 --> 00:08:54,110
is a put option.

164
00:08:54,110 --> 00:08:57,230
So put option, on
the contrary, is

165
00:08:57,230 --> 00:09:01,390
a bet on the asset going
down, rather than up.

166
00:09:01,390 --> 00:09:01,890
Right?

167
00:09:01,890 --> 00:09:07,290
So the pay-out is maximum
of K minus S and 0.

168
00:09:07,290 --> 00:09:10,490
So it's kind of reversed.

169
00:09:10,490 --> 00:09:14,730
Also a ramp function,
at maturity.

170
00:09:14,730 --> 00:09:19,590
And here is the current price.

171
00:09:19,590 --> 00:09:22,220
Again, even if
it's in the money--

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00:09:22,220 --> 00:09:26,190
if it's way in the money,
we expect it to be 0.

173
00:09:26,190 --> 00:09:30,410
If it's way in the
money, we expect

174
00:09:30,410 --> 00:09:34,580
it to be slightly below forward,
just because of this counting.

175
00:09:37,270 --> 00:09:38,500
OK.

176
00:09:38,500 --> 00:09:44,780
So and here are a few--
three main points,

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00:09:44,780 --> 00:09:49,140
which we'll try to
follow, through the class.

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00:09:49,140 --> 00:09:52,220
So first of all,
what we'll see--

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00:09:52,220 --> 00:09:56,570
that if we have current
price of the underlying

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00:09:56,570 --> 00:10:03,310
and some assumptions on how
the market or the underlying

181
00:10:03,310 --> 00:10:08,040
behaves, there is
actually no uncertainty

182
00:10:08,040 --> 00:10:11,040
in the price of the
option, obviously,

183
00:10:11,040 --> 00:10:12,140
if we fix the pay-out.

184
00:10:12,140 --> 00:10:13,380
Right?

185
00:10:13,380 --> 00:10:14,980
So somehow there
is no uncertainty.

186
00:10:14,980 --> 00:10:18,082
It's completely
deterministic, once we

187
00:10:18,082 --> 00:10:19,290
know the price of underlying.

188
00:10:22,960 --> 00:10:27,660
The other interesting
fact, which we'll find out,

189
00:10:27,660 --> 00:10:29,450
is actually
risk-neutrality, meaning

190
00:10:29,450 --> 00:10:32,930
that in fact, the
price of the option

191
00:10:32,930 --> 00:10:36,480
has nothing to do with the
risk preferences of market

192
00:10:36,480 --> 00:10:40,840
participants or counter-parties.

193
00:10:40,840 --> 00:10:44,610
It actually only depends on
the dynamics of the stock,

194
00:10:44,610 --> 00:10:48,470
only depends on the
volatility of the stock.

195
00:10:48,470 --> 00:10:50,922
And finally, the
most important idea

196
00:10:50,922 --> 00:10:53,740
of this class-- that
mathematical apparatus

197
00:10:53,740 --> 00:10:57,750
allows you to figure out how
much this deterministic option

198
00:10:57,750 --> 00:10:59,002
price is now.

199
00:11:02,770 --> 00:11:09,830
So let's consider a very simple
example, a very simple market,

200
00:11:09,830 --> 00:11:10,880
two-period.

201
00:11:10,880 --> 00:11:14,710
So suppose our time is
discrete, and we are

202
00:11:14,710 --> 00:11:17,990
one step before the maturity.

203
00:11:17,990 --> 00:11:22,800
So right now, our
stock has price at 0.

204
00:11:22,800 --> 00:11:27,990
And there is some derivative
f_0 with some pay-out.

205
00:11:27,990 --> 00:11:29,440
We'll consider a few of those.

206
00:11:29,440 --> 00:11:30,370
Right?

207
00:11:30,370 --> 00:11:33,950
Also, we'll add to
the mix a bit of cash.

208
00:11:33,950 --> 00:11:34,450
Right?

209
00:11:37,380 --> 00:11:40,000
Some amount of
riskless cash B_0.

210
00:11:40,000 --> 00:11:45,700
And riskless meaning that
it grows exponentially

211
00:11:45,700 --> 00:11:47,800
with some interest rate r.

212
00:11:47,800 --> 00:11:49,360
And there is no uncertainty.

213
00:11:49,360 --> 00:11:53,820
It's completely-- if you
have now B_0, we know then,

214
00:11:53,820 --> 00:11:59,950
in time dt, our B_0
will grow exponentially.

215
00:11:59,950 --> 00:12:02,080
It will become B e to the rt.

216
00:12:02,080 --> 00:12:05,172
So a bond, basically,
zero-coupon bond.

217
00:12:05,172 --> 00:12:07,835
Or money market account, rather.

218
00:12:07,835 --> 00:12:14,980
If you go to Cambridge
Savings Bank, put $1 in today,

219
00:12:14,980 --> 00:12:20,270
then in a year, you'll get
$1 and basically nothing

220
00:12:20,270 --> 00:12:22,900
because interest rates are 0.

221
00:12:22,900 --> 00:12:27,115
So in time dt, we will assume
with some probability p,

222
00:12:27,115 --> 00:12:31,370
our market can go to the state
where stock becomes S_1--

223
00:12:31,370 --> 00:12:33,370
the price of stock becomes S_1.

224
00:12:33,370 --> 00:12:36,580
Our bond grows exponentially--
no uncertainty.

225
00:12:36,580 --> 00:12:40,210
And our derivative becomes f_1.

226
00:12:40,210 --> 00:12:44,550
Or with probability 1 minus p--
only two states, so-- our stock

227
00:12:44,550 --> 00:12:46,200
becomes S_2.

228
00:12:46,200 --> 00:12:47,200
Bond stays the same.

229
00:12:47,200 --> 00:12:51,870
And the derivative is some f_2.

230
00:12:51,870 --> 00:12:58,870
So let's start with our simple
contract, the forward contract.

231
00:12:58,870 --> 00:13:04,670
So one can naively
approach a problem, trying

232
00:13:04,670 --> 00:13:08,500
to get the price
of the derivative,

233
00:13:08,500 --> 00:13:12,740
using the real-world
probabilities, p and 1 minus p.

234
00:13:12,740 --> 00:13:13,240
Right?

235
00:13:13,240 --> 00:13:18,030
So we know that the pay-out
is S minus K. That's given.

236
00:13:18,030 --> 00:13:21,452
So one would say
that if we know we

237
00:13:21,452 --> 00:13:24,290
are one step before the
pay-out, so let's just

238
00:13:24,290 --> 00:13:26,590
compute expected
value of the pay-out,

239
00:13:26,590 --> 00:13:31,797
using real-world
probabilities, get this value.

240
00:13:31,797 --> 00:13:33,380
And actually, what
we are looking here

241
00:13:33,380 --> 00:13:38,340
is to set K such that the
price now at time t is 0.

242
00:13:38,340 --> 00:13:39,840
That's usual convention.

243
00:13:39,840 --> 00:13:48,500
So we'll then set K
to this probability,

244
00:13:48,500 --> 00:13:51,760
to this number, which depends
on real-world probability

245
00:13:51,760 --> 00:13:54,680
and obviously depends on
the stock price at expiry.

246
00:13:57,450 --> 00:14:00,130
But obviously, we don't know
real-world probabilities.

247
00:14:00,130 --> 00:14:00,830
We can guess.

248
00:14:00,830 --> 00:14:06,860
We can say, oh, this stock is
as likely to go up then down.

249
00:14:06,860 --> 00:14:12,560
Then it's just an average of end
stock prices or something else.

250
00:14:12,560 --> 00:14:14,280
But it's all hand-wavy.

251
00:14:14,280 --> 00:14:17,630
And actually, we
never will be right.

252
00:14:17,630 --> 00:14:21,320
Instead of doing
this-- we're kind

253
00:14:21,320 --> 00:14:26,270
of following bookie example--
let's try to do something else.

254
00:14:26,270 --> 00:14:27,920
Let's think a little bit.

255
00:14:27,920 --> 00:14:33,120
So we have a stock which
is trading at market now

256
00:14:33,120 --> 00:14:36,070
for the price S_0.

257
00:14:36,070 --> 00:14:41,740
How about we go to the bank and
borrow S_0 dollars right now

258
00:14:41,740 --> 00:14:44,810
and immediately go to the
market and buy the stock.

259
00:14:44,810 --> 00:14:47,910
So right now we are net 0.

260
00:14:47,910 --> 00:14:49,330
We borrowed S_0.

261
00:14:49,330 --> 00:14:51,910
We paid it immediately
to buy the stock.

262
00:14:51,910 --> 00:14:53,990
So we have stock at hand.

263
00:14:53,990 --> 00:14:57,830
Then we'll wait for one period.

264
00:14:57,830 --> 00:15:00,910
And at the same
time-- sorry-- we

265
00:15:00,910 --> 00:15:03,970
enter on the short side
of the forward contract.

266
00:15:03,970 --> 00:15:10,830
So we agree to sell the
stock for some price K_0.

267
00:15:10,830 --> 00:15:16,290
So in dt, in one period of
time, the contract expires.

268
00:15:16,290 --> 00:15:17,640
We already have stock.

269
00:15:17,640 --> 00:15:21,540
So we just go and exchange
it for K_0 dollars.

270
00:15:21,540 --> 00:15:23,870
Right?

271
00:15:23,870 --> 00:15:29,310
But at the same time, we need
to repay our loan which now have

272
00:15:29,310 --> 00:15:34,122
become S_0 times e to the r*dt.

273
00:15:34,122 --> 00:15:35,330
This is deterministic, right?

274
00:15:35,330 --> 00:15:36,640
We borrowed S_0.

275
00:15:36,640 --> 00:15:41,360
In time dt, it became
S times e to the r*dt.

276
00:15:41,360 --> 00:15:44,320
So what's our net?

277
00:15:44,320 --> 00:15:51,560
The net is K_0 minus
S times e r*dt.

278
00:15:51,560 --> 00:15:56,490
So suppose K_0 is
greater than this value.

279
00:15:56,490 --> 00:15:58,750
Then we made riskless profit.

280
00:15:58,750 --> 00:16:06,200
There is no risk in the
strategy which we proposed.

281
00:16:06,200 --> 00:16:07,740
So this is good.

282
00:16:07,740 --> 00:16:11,410
But why wouldn't everybody
do it all day long?

283
00:16:11,410 --> 00:16:14,910
On the other hand, if
K_0 is less than S_0,

284
00:16:14,910 --> 00:16:18,200
that's a loss for sure.

285
00:16:18,200 --> 00:16:21,540
And if anybody thinks,
as we did-- and we assume

286
00:16:21,540 --> 00:16:24,390
that everybody can
do it-- then nobody

287
00:16:24,390 --> 00:16:27,220
would want to enter
it, which means

288
00:16:27,220 --> 00:16:34,130
that in order for our
forward to be price 0 now,

289
00:16:34,130 --> 00:16:38,580
the strike price has to
be equal to this amount.

290
00:16:38,580 --> 00:16:42,550
And there is no
uncertainty about it.

291
00:16:42,550 --> 00:16:48,690
So let's stop and
think a little bit.

292
00:16:48,690 --> 00:16:58,450
Well, actually, just
to see how it works.

293
00:16:58,450 --> 00:17:04,530
And that's exactly why I
set this K to this number.

294
00:17:04,530 --> 00:17:07,140
So by the way, who
can tell me which

295
00:17:07,140 --> 00:17:08,790
interest rate does it imply?

296
00:17:15,599 --> 00:17:24,020
If our strike-- our stock price
is $80, our strike is 88.41.

297
00:17:24,020 --> 00:17:29,604
And the expiry is in two
years, approximately.

298
00:17:29,604 --> 00:17:31,460
AUDIENCE: 2.5?

299
00:17:31,460 --> 00:17:33,120
PROFESSOR: 2.5.

300
00:17:33,120 --> 00:17:37,530
So in two years, it will be 5%.

301
00:17:37,530 --> 00:17:39,920
So roughly speaking,
without compounding, it

302
00:17:39,920 --> 00:17:43,110
should be 5% of-- 80 plus 5%.

303
00:17:43,110 --> 00:17:43,920
It would be 84.

304
00:17:47,800 --> 00:17:48,940
So 10% for two years.

305
00:17:48,940 --> 00:17:51,245
So the interest rate is 5%.

306
00:17:51,245 --> 00:17:51,745
Yeah.

307
00:17:51,745 --> 00:17:52,245
So yeah.

308
00:17:52,245 --> 00:17:58,270
That's actually exactly 5
exponentially compounded.

309
00:17:58,270 --> 00:17:58,850
Yeah.

310
00:17:58,850 --> 00:18:01,690
Well, in a good world--
probably five years ago,

311
00:18:01,690 --> 00:18:03,190
that's how it would work.

312
00:18:03,190 --> 00:18:06,840
The two-years interest rates
now, the last time I checked,

313
00:18:06,840 --> 00:18:08,520
was, I think, 30 pips.

314
00:18:08,520 --> 00:18:13,855
We can check where the
bond is trading now.

315
00:18:13,855 --> 00:18:14,790
All right.

316
00:18:14,790 --> 00:18:16,240
Give me a sec.

317
00:18:16,240 --> 00:18:16,740
Now.

318
00:18:22,208 --> 00:18:23,080
Yep.

319
00:18:23,080 --> 00:18:26,920
32 1/2 basis points.

320
00:18:26,920 --> 00:18:30,220
1.6 basis points up,
since the morning.

321
00:18:30,220 --> 00:18:31,370
Quite a bit, by the way.

322
00:18:31,370 --> 00:18:32,300
So yeah.

323
00:18:32,300 --> 00:18:34,480
So right now interest
rates are basically 0.

324
00:18:34,480 --> 00:18:38,260
So these two lines would
be very close right now

325
00:18:38,260 --> 00:18:42,460
if we were for two
years, in that case.

326
00:18:42,460 --> 00:18:49,920
So coming back to our example.

327
00:18:49,920 --> 00:18:52,710
So what's important here?

328
00:18:52,710 --> 00:18:55,220
How did we arrive to
this strike price,

329
00:18:55,220 --> 00:18:58,430
or to this price of
the forward contract?

330
00:18:58,430 --> 00:19:03,810
We, in fact, tried-- we
took some amount of stock.

331
00:19:03,810 --> 00:19:06,650
In this particular case, it
was the whole price of stock.

332
00:19:06,650 --> 00:19:11,690
We took some amount of cash, and
by combining these two pieces,

333
00:19:11,690 --> 00:19:14,664
we somehow replicated
the final pay-off.

334
00:19:14,664 --> 00:19:15,510
Right?

335
00:19:15,510 --> 00:19:20,490
And that's the general idea
of risk-neutral pricing

336
00:19:20,490 --> 00:19:22,560
and replicating portfolio.

337
00:19:22,560 --> 00:19:25,410
What we will try to do,
in the rest of the class,

338
00:19:25,410 --> 00:19:31,640
is take a pay-off and try to
find a replicating portfolio,

339
00:19:31,640 --> 00:19:35,910
maybe more complicated, maybe
a dynamic such that at the end,

340
00:19:35,910 --> 00:19:38,510
this replicating portfolio
will be exactly our pay-off.

341
00:19:38,510 --> 00:19:39,420
Right?

342
00:19:39,420 --> 00:19:40,890
And what would it mean?

343
00:19:40,890 --> 00:19:46,380
Well, obviously it would
mean that the current price

344
00:19:46,380 --> 00:19:48,020
of the derivative
should be the price

345
00:19:48,020 --> 00:19:50,472
of our replicating
portfolio right now.

346
00:19:50,472 --> 00:19:51,950
Right?

347
00:19:51,950 --> 00:19:58,060
And that's how the
risk-neutral pricing works.

348
00:19:58,060 --> 00:20:01,180
So we are still in
this simple situation.

349
00:20:01,180 --> 00:20:05,760
But we will try to price
a general pay-off f_1--

350
00:20:05,760 --> 00:20:07,850
a general pay-off f.

351
00:20:07,850 --> 00:20:08,820
Right?

352
00:20:08,820 --> 00:20:10,650
And here's how it goes.

353
00:20:10,650 --> 00:20:16,760
So we still will try to form
our replicating portfolio out

354
00:20:16,760 --> 00:20:21,930
of the bond, of some amount of
bond, and some amount of stock.

355
00:20:21,930 --> 00:20:28,690
And we'll say that we will
need a S_1 and b of the bond.

356
00:20:28,690 --> 00:20:29,730
Right?

357
00:20:29,730 --> 00:20:35,860
And we'll try to find a and
b such that no matter what

358
00:20:35,860 --> 00:20:41,940
the real-world probability
is, at one step maturity,

359
00:20:41,940 --> 00:20:44,202
we'll replicate our
pay-off exactly.

360
00:20:44,202 --> 00:20:45,910
And fortunately, in
this particular case,

361
00:20:45,910 --> 00:20:46,810
it's very doable.

362
00:20:46,810 --> 00:20:48,310
It's just two equations.

363
00:20:48,310 --> 00:20:49,620
We use two variables.

364
00:20:49,620 --> 00:20:51,510
We should be able to do it.

365
00:20:51,510 --> 00:20:55,290
And we can solve it
and find this a and b.

366
00:20:55,290 --> 00:20:58,780
Then we'll substitute
them in the formula.

367
00:20:58,780 --> 00:20:59,350
Right?

368
00:20:59,350 --> 00:21:02,860
Take the current price of
the stock, which we know,

369
00:21:02,860 --> 00:21:11,010
and some cash, and
find the current price

370
00:21:11,010 --> 00:21:13,000
of the derivative.

371
00:21:13,000 --> 00:21:13,500
Right?

372
00:21:13,500 --> 00:21:15,730
And this works-- it should
work for any derivative.

373
00:21:15,730 --> 00:21:17,105
It doesn't matter,
is it forward,

374
00:21:17,105 --> 00:21:20,820
call, put, or some
complicated option,

375
00:21:20,820 --> 00:21:23,000
as long as it is
deterministic at expiry.

376
00:21:26,040 --> 00:21:29,040
An interesting way,
though, to look at it

377
00:21:29,040 --> 00:21:32,750
is to rewrite this
formula slightly,

378
00:21:32,750 --> 00:21:40,760
in such a way, which does remind
us, taking an expected value,

379
00:21:40,760 --> 00:21:44,050
maybe discounting it because
this is expected value

380
00:21:44,050 --> 00:21:46,090
at some time in the future.

381
00:21:46,090 --> 00:21:48,770
But this probability--
and it is a probability

382
00:21:48,770 --> 00:21:53,620
because this number q,
here, is between 0 and 1.

383
00:21:53,620 --> 00:21:59,470
But this probability has
little to do with real world.

384
00:21:59,470 --> 00:22:00,000
Right?

385
00:22:00,000 --> 00:22:03,960
In fact, it's
something different.

386
00:22:03,960 --> 00:22:06,230
But such probability exists.

387
00:22:06,230 --> 00:22:10,680
And it's called-- the measure
where our stock behaves

388
00:22:10,680 --> 00:22:12,590
like this is called a
risk-neutral measure

389
00:22:12,590 --> 00:22:16,080
or martingale measure.

390
00:22:16,080 --> 00:22:18,690
And in this measure,
as we will see,

391
00:22:18,690 --> 00:22:20,890
the value of the derivative
will be just expected

392
00:22:20,890 --> 00:22:24,820
value of our pay-out.

393
00:22:24,820 --> 00:22:25,740
And that's-- yeah.

394
00:22:25,740 --> 00:22:28,800
That's what I'm
trying to say, here.

395
00:22:33,200 --> 00:22:37,850
So now let's get into
continuous world.

396
00:22:37,850 --> 00:22:40,260
Right?

397
00:22:40,260 --> 00:22:43,000
In continuous world, we'll
need some assumptions

398
00:22:43,000 --> 00:22:46,930
on the dynamics of
our stock underlying.

399
00:22:46,930 --> 00:22:54,660
And let's make an assumption
that it is log-normal.

400
00:22:54,660 --> 00:22:56,630
What does it mean
that it's log-normal?

401
00:22:56,630 --> 00:23:04,250
It means that the proportional
change of the stock,

402
00:23:04,250 --> 00:23:08,970
over infinitely small
amount of time dt,

403
00:23:08,970 --> 00:23:13,780
has some drift mu, and
some stochastic component,

404
00:23:13,780 --> 00:23:15,910
which is just Brownian Motion.

405
00:23:15,910 --> 00:23:16,950
Right?

406
00:23:16,950 --> 00:23:21,130
So this dW is
distributed normally

407
00:23:21,130 --> 00:23:26,920
with mean 0 and standard
deviation, which

408
00:23:26,920 --> 00:23:28,480
is actually square root of dt.

409
00:23:28,480 --> 00:23:31,140
That's how Brownian
Motion works.

410
00:23:31,140 --> 00:23:33,720
And that's extremely important,
that the standard deviation

411
00:23:33,720 --> 00:23:39,180
of Brownian Motion is
square root of delta t.

412
00:23:39,180 --> 00:23:40,195
And that's how it works.

413
00:23:42,990 --> 00:23:49,520
And again, we will use this
idea of replicating portfolio.

414
00:23:49,520 --> 00:23:52,650
What would it mean in this case?

415
00:23:52,650 --> 00:23:58,880
Well, we would like to find
such coefficients a and b,

416
00:23:58,880 --> 00:24:02,580
on this infinitely
small period of time dt,

417
00:24:02,580 --> 00:24:06,670
such that by combining
small changes in stock,

418
00:24:06,670 --> 00:24:10,380
with coefficient a, and
small changes in bond,

419
00:24:10,380 --> 00:24:14,120
with coefficient b,
will exactly replicate

420
00:24:14,120 --> 00:24:18,030
the change in the derivative--
in the pay-out of derivative--

421
00:24:18,030 --> 00:24:18,630
not pay-out.

422
00:24:18,630 --> 00:24:19,590
In the derivative.

423
00:24:19,590 --> 00:24:23,529
In the change of the derivative,
over this infinitely small time

424
00:24:23,529 --> 00:24:24,028
t.

425
00:24:27,240 --> 00:24:31,170
Well, to do this, we'll
need to use Ito's formula.

426
00:24:31,170 --> 00:24:33,040
Did you talk about Ito already?

427
00:24:33,040 --> 00:24:33,540
OK.

428
00:24:33,540 --> 00:24:34,040
Cool.

429
00:24:34,040 --> 00:24:35,320
That's great.

430
00:24:35,320 --> 00:24:38,480
So just to remind you
that Ito's formula

431
00:24:38,480 --> 00:24:41,580
is nothing more than the
Taylor rule, actually--

432
00:24:41,580 --> 00:24:45,560
the first
approximation up to dt.

433
00:24:45,560 --> 00:24:50,080
But because of the standard
deviation of the Brownian

434
00:24:50,080 --> 00:24:53,680
Motion being on the scale
of square root of t,

435
00:24:53,680 --> 00:24:55,410
we will need one
more term there.

436
00:24:55,410 --> 00:24:55,910
Right?

437
00:24:55,910 --> 00:24:59,130
So one term is df/dt by dt.

438
00:24:59,130 --> 00:25:01,480
Another is df by dS by dS.

439
00:25:01,480 --> 00:25:05,560
And the square of
dS now is actually

440
00:25:05,560 --> 00:25:07,740
of order of magnitude of dt.

441
00:25:07,740 --> 00:25:12,000
So we'll need a
quadratic term there.

442
00:25:12,000 --> 00:25:12,500
All right.

443
00:25:12,500 --> 00:25:21,290
So if this is our df, so what
we'll do-- we'll differentiate.

444
00:25:21,290 --> 00:25:24,000
We'll just substitute it here.

445
00:25:24,000 --> 00:25:25,960
Right?

446
00:25:25,960 --> 00:25:27,110
We'll substitute it here.

447
00:25:27,110 --> 00:25:33,630
We'll substitute df taken from
our dS, which is like this,

448
00:25:33,630 --> 00:25:34,130
and dB.

449
00:25:40,380 --> 00:25:45,030
Let's not forget that dB--
that B is deterministic.

450
00:25:45,030 --> 00:25:45,660
Right?

451
00:25:45,660 --> 00:25:47,390
There is nothing
uncertain about it.

452
00:25:47,390 --> 00:25:51,464
So dB is actually r*B*dt.

453
00:25:51,464 --> 00:25:52,330
All right?

454
00:25:52,330 --> 00:25:57,900
Because our B
grows exponentially

455
00:25:57,900 --> 00:26:00,120
with interest rate r.

456
00:26:00,120 --> 00:26:04,850
So we substitute everything
into the formula above.

457
00:26:04,850 --> 00:26:10,630
This is just our df with
dS expanded and everything.

458
00:26:10,630 --> 00:26:13,940
And then when we start
comparing the terms.

459
00:26:13,940 --> 00:26:20,970
One immediate thing
to notice-- that a

460
00:26:20,970 --> 00:26:25,990
has to be equal to df
over dS, for this to hold.

461
00:26:25,990 --> 00:26:27,360
Right?

462
00:26:27,360 --> 00:26:30,510
And if you compare
the terms near dt,

463
00:26:30,510 --> 00:26:32,560
we'll get this expression here.

464
00:26:32,560 --> 00:26:37,660
But that's actually even
more the most important part.

465
00:26:37,660 --> 00:26:45,340
Then we'll go and use our
knowledge that some part

466
00:26:45,340 --> 00:26:52,555
of our equation is deterministic
and basically take f and a*S

467
00:26:52,555 --> 00:26:55,330
on one side and leave
the deterministic part,

468
00:26:55,330 --> 00:27:00,000
on the other side,
differentiated once again.

469
00:27:00,000 --> 00:27:04,940
And left side will
be just r*B*dt.

470
00:27:04,940 --> 00:27:08,870
And if we substitute
once again df--

471
00:27:08,870 --> 00:27:11,430
and don't forget
that what we learned

472
00:27:11,430 --> 00:27:15,530
is that a is equal to df by dS.

473
00:27:15,530 --> 00:27:19,780
Then we collect all
the terms and arrive

474
00:27:19,780 --> 00:27:24,750
to this partial
differential equation which

475
00:27:24,750 --> 00:27:27,540
connects-- which basically
is a partial differential

476
00:27:27,540 --> 00:27:31,920
equation for the current
price of a derivative--

477
00:27:31,920 --> 00:27:34,480
of any derivative.

478
00:27:34,480 --> 00:27:38,480
And how if we solve it,
then we should actually

479
00:27:38,480 --> 00:27:41,980
be able to know the
price of the derivative.

480
00:27:41,980 --> 00:27:45,850
So now how do we solve this
partial differential equation?

481
00:27:45,850 --> 00:27:47,000
Well, for-- yeah.

482
00:27:47,000 --> 00:27:51,330
So a few observations
about this equation.

483
00:27:51,330 --> 00:27:59,930
Well, the first observation
is that any tradable

484
00:27:59,930 --> 00:28:04,330
derivative-- we made no
assumptions about the pay-off.

485
00:28:04,330 --> 00:28:07,080
So any tradable
derivative as any pay-off

486
00:28:07,080 --> 00:28:10,780
should satisfy this equation.

487
00:28:10,780 --> 00:28:14,890
The other observation
is as we expected,

488
00:28:14,890 --> 00:28:19,050
there is no dependency
on real-world drift

489
00:28:19,050 --> 00:28:23,445
or any probability of
it going up or down.

490
00:28:23,445 --> 00:28:27,530
The only dependence is on
the volatility of the stock.

491
00:28:27,530 --> 00:28:28,030
Right?

492
00:28:31,110 --> 00:28:34,400
Not only we found the
value of the derivative--

493
00:28:34,400 --> 00:28:38,620
most importantly,
we actually were

494
00:28:38,620 --> 00:28:42,760
able to come up with
a hedging strategy.

495
00:28:42,760 --> 00:28:45,810
And what does it mean, we came
up with a hedging strategy?

496
00:28:45,810 --> 00:28:49,490
Well, we found
coefficients-- for any time,

497
00:28:49,490 --> 00:28:51,960
we found the
coefficients, a and b,

498
00:28:51,960 --> 00:28:55,460
such that we have a
replicating portfolio.

499
00:28:55,460 --> 00:28:58,090
So what we could do,
at any point of time,

500
00:28:58,090 --> 00:29:02,560
we can hold the derivative--
short derivative and long

501
00:29:02,560 --> 00:29:06,010
the portfolio of stock
itself, and some cash,

502
00:29:06,010 --> 00:29:07,704
and then know how
much it should be.

503
00:29:07,704 --> 00:29:08,870
Here, it's more complicated.

504
00:29:08,870 --> 00:29:11,510
We have to dynamically
change these numbers,

505
00:29:11,510 --> 00:29:12,760
as time develops.

506
00:29:12,760 --> 00:29:15,890
Every time dt we will
have to rebalance.

507
00:29:15,890 --> 00:29:20,480
But both parts will replicate
each other perfectly.

508
00:29:20,480 --> 00:29:22,590
It's like in a bookie's example.

509
00:29:22,590 --> 00:29:29,470
We can go to a
counterparty, agree

510
00:29:29,470 --> 00:29:32,030
for some derivative contract.

511
00:29:32,030 --> 00:29:33,810
Probably there will be some fee.

512
00:29:33,810 --> 00:29:37,040
And then we'll go to
exchange and buy the stock,

513
00:29:37,040 --> 00:29:39,475
and we will get just
cash from the bank.

514
00:29:39,475 --> 00:29:41,990
And we'll maintain this
at some amount of stock

515
00:29:41,990 --> 00:29:43,800
and some amount of cash.

516
00:29:43,800 --> 00:29:46,610
And we'll be sure
that we are hedged.

517
00:29:46,610 --> 00:29:50,570
There is no risk in this
combination of the derivative

518
00:29:50,570 --> 00:29:52,100
and our hedge.

519
00:29:52,100 --> 00:29:56,040
So we will just collect
a fee on the transaction.

520
00:29:56,040 --> 00:30:01,280
So that's what actually--
how the business is working.

521
00:30:01,280 --> 00:30:05,180
Traders are trading and hedging
their positions immediately.

522
00:30:05,180 --> 00:30:07,590
I mean, they do take
some market risks.

523
00:30:07,590 --> 00:30:11,410
But you want to take very
little and very directional,

524
00:30:11,410 --> 00:30:16,890
very specific market
risks and not everything.

525
00:30:16,890 --> 00:30:20,370
So our strategy
allows us to have

526
00:30:20,370 --> 00:30:24,910
a hedging portfolio at the
same time-- hedging strategy.

527
00:30:24,910 --> 00:30:29,400
And now there are more
mathematical but practical

528
00:30:29,400 --> 00:30:34,790
consequences that actually,
by certain-- not very easy--

529
00:30:34,790 --> 00:30:38,080
change of variables, we can
take the Black-Scholes equation

530
00:30:38,080 --> 00:30:40,350
and put it back
to heat equation.

531
00:30:40,350 --> 00:30:45,710
Actually, I suggest it as one of
the topics for the final paper,

532
00:30:45,710 --> 00:30:50,320
for you to do it or check
it out in the books.

533
00:30:50,320 --> 00:30:52,990
Go and understand it.

534
00:30:52,990 --> 00:30:55,630
But the good part of it--
that heat equation is

535
00:30:55,630 --> 00:30:57,400
well known and well understood.

536
00:30:57,400 --> 00:31:01,790
There are many, many ways
to solve it numerically.

537
00:31:01,790 --> 00:31:06,012
For simple pay-outs,
for calls and puts,

538
00:31:06,012 --> 00:31:07,470
we don't have to
do it numerically,

539
00:31:07,470 --> 00:31:10,320
but if the pay-outs
are more complicated

540
00:31:10,320 --> 00:31:18,730
or the dynamics is different,
then numerical methods

541
00:31:18,730 --> 00:31:22,180
will be needed, for sure.

542
00:31:22,180 --> 00:31:24,770
So again, to solve
this equation,

543
00:31:24,770 --> 00:31:29,090
we'll need, as for any
partial differential equation,

544
00:31:29,090 --> 00:31:32,290
we'll need some boundary
and initial conditions.

545
00:31:32,290 --> 00:31:36,665
And these come from
our final pay-out

546
00:31:36,665 --> 00:31:39,600
of the option, which we know.

547
00:31:39,600 --> 00:31:41,750
We will know what
happens at expiry.

548
00:31:41,750 --> 00:31:44,110
And some boundary conditions.

549
00:31:44,110 --> 00:31:49,670
For call and put, the
final pay-out we know.

550
00:31:49,670 --> 00:31:50,510
Right?

551
00:31:50,510 --> 00:31:56,090
So at time T. And the
boundary conditions

552
00:31:56,090 --> 00:32:01,830
we discussed, we can
observe them graphically.

553
00:32:01,830 --> 00:32:09,850
So basically for call, as
we said, at current time t,

554
00:32:09,850 --> 00:32:12,930
and boundary 0, it should be 0.

555
00:32:12,930 --> 00:32:14,400
The price should be 0.

556
00:32:14,400 --> 00:32:20,700
And at infinity, it should be
actually the forward price.

557
00:32:20,700 --> 00:32:28,940
So it should be just discounted
S minus K. Discounted pay-out.

558
00:32:28,940 --> 00:32:29,440
Right?

559
00:32:32,910 --> 00:32:34,490
And similarly for put.

560
00:32:40,190 --> 00:32:47,500
So given these conditions,
we can solve the equation.

561
00:32:47,500 --> 00:32:52,670
And as I said, for call and
put and for simple dynamics--

562
00:32:52,670 --> 00:32:58,090
Black-Scholes dynamical or
log-normal dynamics-- actually,

563
00:32:58,090 --> 00:33:02,050
these equations can be
solved exactly-- exactly

564
00:33:02,050 --> 00:33:07,170
meaning up to this term, the
normal distribution, which

565
00:33:07,170 --> 00:33:10,610
still has to be computed
numerically, obviously.

566
00:33:10,610 --> 00:33:13,200
But here are the formulas.

567
00:33:13,200 --> 00:33:15,210
They do kind of
look a little bit--

568
00:33:15,210 --> 00:33:19,300
and we'll see about it--
there is some kind of expected

569
00:33:19,300 --> 00:33:20,270
volume going on.

570
00:33:20,270 --> 00:33:21,030
Right?

571
00:33:21,030 --> 00:33:25,130
One probability times another.

572
00:33:25,130 --> 00:33:26,600
But these are the formulas.

573
00:33:26,600 --> 00:33:29,590
And that's how I drew
the lines on the graphs.

574
00:33:34,440 --> 00:33:45,420
And as I said, in
fact, the whole world,

575
00:33:45,420 --> 00:33:49,900
instead of solving the whole
partial differential equation,

576
00:33:49,900 --> 00:33:55,450
we can approach it from
a risk-neutral position

577
00:33:55,450 --> 00:33:58,970
and say that, in fact, the
price of our derivative

578
00:33:58,970 --> 00:34:06,310
now is just expected value of
pay-out, discounted, probably,

579
00:34:06,310 --> 00:34:10,429
from the maturity.

580
00:34:10,429 --> 00:34:13,679
But not in real time
or real-world measure,

581
00:34:13,679 --> 00:34:16,179
but in some specific
risk-neutral measure.

582
00:34:16,179 --> 00:34:18,790
And how do we find this
risk-neutral measure?

583
00:34:18,790 --> 00:34:21,980
Well, the risk-neutral
measure is such

584
00:34:21,980 --> 00:34:24,909
that the drift of our stock
is actually interest rate.

585
00:34:24,909 --> 00:34:26,500
It's riskless.

586
00:34:26,500 --> 00:34:32,700
That's exactly how we saw
it in our binary example.

587
00:34:36,239 --> 00:34:36,739
All right?

588
00:34:36,739 --> 00:34:42,610
So even in our binary
example, our expected value

589
00:34:42,610 --> 00:34:44,730
of our stock, under
risk-neutral measure,

590
00:34:44,730 --> 00:34:50,159
meaning using the
risk-neutral probability,

591
00:34:50,159 --> 00:34:54,429
was drifting with
interest rate r.

592
00:34:54,429 --> 00:34:59,540
So that the same happens
in continuous case.

593
00:34:59,540 --> 00:35:01,950
And that's another
good exercise--

594
00:35:01,950 --> 00:35:10,900
and I would accept it as a
final paper-- is deriving

595
00:35:10,900 --> 00:35:16,600
the Black-Scholes formula
just by the expected value

596
00:35:16,600 --> 00:35:23,019
of the call and put pay-out with
the log-normal distribution--

597
00:35:23,019 --> 00:35:23,935
terminal distribution.

598
00:35:28,180 --> 00:35:28,950
All right.

599
00:35:28,950 --> 00:35:36,835
So for more
complicated pay-offs,

600
00:35:36,835 --> 00:35:39,960
the life becomes
more complicated.

601
00:35:39,960 --> 00:35:43,180
And some finite
differences should

602
00:35:43,180 --> 00:35:49,070
be used for more complicated
pay-offs or American pay-offs

603
00:35:49,070 --> 00:35:53,960
or path-dependent pay-offs,
tree methods or Monte Carlo

604
00:35:53,960 --> 00:35:54,510
simulations.

605
00:35:54,510 --> 00:35:57,150
And that's what was
happening in real life.

606
00:36:03,755 --> 00:36:04,255
Yeah.

607
00:36:08,200 --> 00:36:13,640
Now, since we have,
actually, plenty of time,

608
00:36:13,640 --> 00:36:21,620
I would like to give an example
of how replicating-- idea

609
00:36:21,620 --> 00:36:24,960
of replicating portfolio works.

610
00:36:24,960 --> 00:36:26,685
I give a couple more examples.

611
00:36:34,670 --> 00:36:35,430
So OK.

612
00:36:35,430 --> 00:36:38,600
Here is a Bloomberg screen
for foreign options--

613
00:36:38,600 --> 00:36:40,910
call options on IBM stock.

614
00:36:40,910 --> 00:36:45,280
It actually was taken a
while ago-- a few years ago.

615
00:36:45,280 --> 00:36:49,820
And so here are different
strikes for a call option.

616
00:36:49,820 --> 00:36:55,350
The current price of
the stock is $81.14.

617
00:36:55,350 --> 00:36:58,650
And here are the
strikes of the call.

618
00:36:58,650 --> 00:37:04,470
So obviously, if the option
is way out of the money,

619
00:37:04,470 --> 00:37:11,490
meaning the strike is very high
compared to the stock price,

620
00:37:11,490 --> 00:37:13,470
the value of the option is 0.

621
00:37:13,470 --> 00:37:20,900
If it's way in the money, in
fact, it is just S minus K.

622
00:37:20,900 --> 00:37:24,240
So S being $81.

623
00:37:24,240 --> 00:37:27,040
And say, the strike being $55.

624
00:37:27,040 --> 00:37:28,740
So it's $26.

625
00:37:28,740 --> 00:37:29,360
Right?

626
00:37:29,360 --> 00:37:30,900
So there is some difference.

627
00:37:30,900 --> 00:37:32,780
But actually, here
it's a bit small

628
00:37:32,780 --> 00:37:35,410
because the difference should
be just discounting, as we know.

629
00:37:35,410 --> 00:37:35,909
Right?

630
00:37:35,909 --> 00:37:38,460
But it's pretty
short-dated options.

631
00:37:38,460 --> 00:37:42,460
They are probably a month
long, so there is not

632
00:37:42,460 --> 00:37:44,990
much discounting.

633
00:37:44,990 --> 00:37:47,400
So it becomes pretty parallel.

634
00:37:47,400 --> 00:37:49,040
It's similar here, right?

635
00:37:49,040 --> 00:37:53,530
So I mean, this changes by 5.

636
00:37:53,530 --> 00:37:56,239
This changes by 5.

637
00:37:56,239 --> 00:37:57,030
It's pretty linear.

638
00:37:57,030 --> 00:37:59,620
But it becomes non-linear
around the money,

639
00:37:59,620 --> 00:38:01,770
around current stock price.

640
00:38:01,770 --> 00:38:02,270
Right?

641
00:38:02,270 --> 00:38:07,640
So we do observe this behavior.

642
00:38:07,640 --> 00:38:12,580
But to tell you the
truth, if you were to-- I

643
00:38:12,580 --> 00:38:18,040
didn't put implied
volatilities here.

644
00:38:18,040 --> 00:38:22,150
But actually, you would
observe that the world is not

645
00:38:22,150 --> 00:38:25,225
Black-Scholes,
meaning that-- what's

646
00:38:25,225 --> 00:38:26,840
the assumption of Black-Scholes.

647
00:38:26,840 --> 00:38:28,256
The assumption of
Black-Scholes is

648
00:38:28,256 --> 00:38:31,807
that every option, for any
strike, on a given stock,

649
00:38:31,807 --> 00:38:33,890
on a given expiry, would
have the same volatility.

650
00:38:33,890 --> 00:38:35,150
Right?

651
00:38:35,150 --> 00:38:38,000
So if we went through exercise
of implying the volatility

652
00:38:38,000 --> 00:38:39,940
according to
Black-Scholes formula,

653
00:38:39,940 --> 00:38:42,260
from the option
price which is traded

654
00:38:42,260 --> 00:38:46,130
on the market and
the current price,

655
00:38:46,130 --> 00:38:52,110
we would find out that,
actually, the volatility is not

656
00:38:52,110 --> 00:38:52,985
constant with strike.

657
00:38:57,146 --> 00:39:00,430
Well, it's actually skewed.

658
00:39:00,430 --> 00:39:03,230
Well, actually it is smiled.

659
00:39:03,230 --> 00:39:08,140
They would find
something like this,

660
00:39:08,140 --> 00:39:11,980
which means that Black-Scholes
theory is not perfectly good.

661
00:39:11,980 --> 00:39:12,480
Right?

662
00:39:12,480 --> 00:39:16,340
So something more
complicated should be done.

663
00:39:16,340 --> 00:39:18,820
But in some cases,
we even don't need

664
00:39:18,820 --> 00:39:21,130
to do something
more complicated.

665
00:39:21,130 --> 00:39:27,230
One example, being
so-called put-call parity.

666
00:39:27,230 --> 00:39:28,690
Right?

667
00:39:28,690 --> 00:39:31,040
So let's see.

668
00:39:31,040 --> 00:39:32,490
Suppose we look at the screen.

669
00:39:32,490 --> 00:39:37,790
So we know all prices for all
call options for all strikes.

670
00:39:37,790 --> 00:39:40,700
Well, probably will be some
granularity, but we know those.

671
00:39:40,700 --> 00:39:42,950
But instead of pricing a
call, we need to price a put.

672
00:39:45,470 --> 00:39:48,470
Somehow, we don't know how
the dynamics of our stock

673
00:39:48,470 --> 00:39:49,040
looks like.

674
00:39:49,040 --> 00:39:54,540
So we have strong suspicion that
it's not exactly log-normal.

675
00:39:54,540 --> 00:39:56,710
So there is some
volatility smile.

676
00:39:56,710 --> 00:39:57,540
It's not constant.

677
00:39:57,540 --> 00:40:00,230
The world is slightly
not Black-Scholes.

678
00:40:00,230 --> 00:40:04,600
So how do we price put?

679
00:40:04,600 --> 00:40:05,400
Well, let's see.

680
00:40:08,910 --> 00:40:14,330
We'll stare long enough at the
pay-outs of the call and put.

681
00:40:14,330 --> 00:40:19,162
So what's the pay-out of
a call with some strike?

682
00:40:19,162 --> 00:40:20,250
It looks like this.

683
00:40:20,250 --> 00:40:21,740
Right?

684
00:40:21,740 --> 00:40:25,390
The pay-out of the put,
with the same strike,

685
00:40:25,390 --> 00:40:27,860
would look like this.

686
00:40:27,860 --> 00:40:35,550
So what if we take, we
buy a call and sell a put?

687
00:40:45,230 --> 00:40:47,480
So this would go like this.

688
00:40:47,480 --> 00:40:49,900
Right?

689
00:40:49,900 --> 00:40:51,680
Straight line.

690
00:40:51,680 --> 00:40:55,000
Looks very much
like forward, right?

691
00:40:55,000 --> 00:41:00,830
So if we actually subtract
the stock from here,

692
00:41:00,830 --> 00:41:12,570
move it from here,
then it should

693
00:41:12,570 --> 00:41:17,008
be-- yeah-- minus K. Yeah.

694
00:41:19,830 --> 00:41:22,100
I think I got the signs correct.

695
00:41:22,100 --> 00:41:22,800
Right?

696
00:41:22,800 --> 00:41:24,210
And this is just a number.

697
00:41:24,210 --> 00:41:25,152
Right?

698
00:41:25,152 --> 00:41:26,610
And that's what
happens at pay-out.

699
00:41:26,610 --> 00:41:30,480
So if we take this portfolio,
if we action now, buy a call,

700
00:41:30,480 --> 00:41:34,320
sell a put, and sell a stock,
we know that at the end,

701
00:41:34,320 --> 00:41:37,410
we'll for sure get K in money.

702
00:41:37,410 --> 00:41:39,890
Right?

703
00:41:39,890 --> 00:41:45,310
So which means that now--
so this is at time t.

704
00:41:50,340 --> 00:42:02,780
So right now, it looks, to
me, that if we do write this,

705
00:42:02,780 --> 00:42:05,470
and that's just the
current price of the stock,

706
00:42:05,470 --> 00:42:11,880
this should be-- right?

707
00:42:11,880 --> 00:42:15,395
We just need to discount
this price to now,

708
00:42:15,395 --> 00:42:23,060
in this amount of cash, which
means that our put, at any time

709
00:42:23,060 --> 00:42:36,130
t, is stock minus K. Right?

710
00:42:36,130 --> 00:42:39,810
So if we know all of the
prices for any strike K--

711
00:42:39,810 --> 00:42:44,170
if we know price of a call, we
don't need any Black-Scholes

712
00:42:44,170 --> 00:42:45,690
or anything.

713
00:42:45,690 --> 00:42:50,700
We can immediately tell
everybody how much is a put.

714
00:42:50,700 --> 00:42:51,200
Right?

715
00:42:51,200 --> 00:42:57,570
So then this relationship is
actually a call-put parity.

716
00:42:57,570 --> 00:43:00,360
And that's, again-- that's
a replicating portfolio.

717
00:43:00,360 --> 00:43:01,860
It's a simple
replicating portfolio.

718
00:43:01,860 --> 00:43:04,750
It's static, meaning
that we fixed it now

719
00:43:04,750 --> 00:43:07,360
and we don't change
it to expiry.

720
00:43:07,360 --> 00:43:10,130
So it's quite good this way.

721
00:43:10,130 --> 00:43:16,863
But that's how it works.

722
00:43:16,863 --> 00:43:19,070
Another example.

723
00:43:19,070 --> 00:43:24,520
So for this, I have,
actually, a picture.

724
00:43:24,520 --> 00:43:30,540
So again, we have
the same situation.

725
00:43:30,540 --> 00:43:33,030
We have prices of calls.

726
00:43:33,030 --> 00:43:36,490
But instead of pricing a call,
we want to price a digital.

727
00:43:36,490 --> 00:43:38,290
So what is digital?

728
00:43:38,290 --> 00:43:41,130
Digital is such
a weird contract,

729
00:43:41,130 --> 00:43:43,815
which pay-out is just
a function-- Basically,

730
00:43:43,815 --> 00:43:47,770
it's a bet on the
stock to finish

731
00:43:47,770 --> 00:43:50,380
above strike price, K. Right?

732
00:43:50,380 --> 00:43:56,280
If at expiry, the stock
is above K, you get 1.

733
00:43:56,280 --> 00:43:58,030
You'd get $1.

734
00:43:58,030 --> 00:44:01,730
If it's below, you'd
get nothing, 0.

735
00:44:01,730 --> 00:44:02,230
Right?

736
00:44:02,230 --> 00:44:03,950
So

737
00:44:03,950 --> 00:44:07,110
So such an interesting contract.

738
00:44:07,110 --> 00:44:10,310
The question is,
can we price it,

739
00:44:10,310 --> 00:44:13,510
given that we know
the prices of Calls?

740
00:44:17,040 --> 00:44:20,400
And I suggest we use the idea
of replicating portfolio.

741
00:44:20,400 --> 00:44:25,305
Any ideas how to do it?

742
00:44:25,305 --> 00:44:31,030
It's my typical
interview question.

743
00:44:31,030 --> 00:44:32,780
So just pretend that
you are interviewing.

744
00:44:41,640 --> 00:44:42,330
Yep?

745
00:44:42,330 --> 00:44:45,430
AUDIENCE: You long the call,
and then you short the call,

746
00:44:45,430 --> 00:44:49,256
just like smaller
or a higher strike.

747
00:44:49,256 --> 00:44:49,880
PROFESSOR: Yep.

748
00:44:49,880 --> 00:44:50,546
The call strike.

749
00:44:50,546 --> 00:44:53,190
Yeah, you're absolutely right.

750
00:44:53,190 --> 00:44:54,390
Good.

751
00:44:54,390 --> 00:44:57,040
You've got an offer.

752
00:44:57,040 --> 00:44:57,820
Yeah.

753
00:44:57,820 --> 00:44:59,120
So here's how it goes.

754
00:44:59,120 --> 00:45:02,660
So this is a strike K. Right?

755
00:45:02,660 --> 00:45:16,070
So let's buy a call
with strike K minus 1/2

756
00:45:16,070 --> 00:45:21,890
and sell a Call with
strike K plus 1/2.

757
00:45:21,890 --> 00:45:23,070
Right?

758
00:45:23,070 --> 00:45:25,420
We just sold.

759
00:45:25,420 --> 00:45:31,842
So if we combine these two--
well, actually, if this is 1--

760
00:45:31,842 --> 00:45:32,342
yeah.

761
00:45:38,150 --> 00:45:41,054
If this is 1, it should
look something like this.

762
00:45:45,910 --> 00:45:48,080
Great.

763
00:45:48,080 --> 00:45:50,380
So how will it look like?

764
00:45:50,380 --> 00:45:51,675
So obviously, here, it's 0.

765
00:45:51,675 --> 00:45:53,030
Right?

766
00:45:53,030 --> 00:45:55,290
Then it will be like this.

767
00:45:55,290 --> 00:45:56,210
Right?

768
00:45:56,210 --> 00:45:58,390
And after that, it will be what?

769
00:45:58,390 --> 00:45:59,280
AUDIENCE: Constant.

770
00:45:59,280 --> 00:46:00,000
PROFESSOR: It will be constant.

771
00:46:00,000 --> 00:46:00,500
Right?

772
00:46:00,500 --> 00:46:04,170
And because this is K minus
1/2 and this is K plus 1/2,

773
00:46:04,170 --> 00:46:05,300
it will be exactly 1.

774
00:46:05,300 --> 00:46:06,820
Right?

775
00:46:06,820 --> 00:46:07,320
Good.

776
00:46:07,320 --> 00:46:11,520
So our pay-out, at the
end, will be like this.

777
00:46:16,230 --> 00:46:17,520
So that's good.

778
00:46:17,520 --> 00:46:21,270
But there is quite
a bit of slope here.

779
00:46:21,270 --> 00:46:25,330
So how can we do
better than this?

780
00:46:25,330 --> 00:46:37,770
Well, if we buy it at K minus
1/4, and sell it at K plus 1/4,

781
00:46:37,770 --> 00:46:41,380
and just combine those, it
will be exactly the same,

782
00:46:41,380 --> 00:46:43,940
but the level will be 1/2.

783
00:46:43,940 --> 00:46:48,200
So we need to buy two of those
and to sell two of those.

784
00:46:48,200 --> 00:46:50,510
Right?

785
00:46:50,510 --> 00:46:54,900
Well, we might as well go
K minus epsilon and K plus

786
00:46:54,900 --> 00:47:04,910
epsilon, so it'll be call price
at strike K minus epsilon,

787
00:47:04,910 --> 00:47:13,640
minus call price at K plus
epsilon, divided by 2*epsilon.

788
00:47:13,640 --> 00:47:15,310
Right?

789
00:47:15,310 --> 00:47:20,940
This 2*epsilon coefficient
needed rescale it back to 1.

790
00:47:20,940 --> 00:47:22,310
Right?

791
00:47:22,310 --> 00:47:27,960
So in fact, if we go small
epsilon, we need a lot of both.

792
00:47:27,960 --> 00:47:30,680
Right?

793
00:47:30,680 --> 00:47:37,320
And that's how-- that's
the approximation

794
00:47:37,320 --> 00:47:39,050
of our digital price.

795
00:47:39,050 --> 00:47:43,340
And that's actually how
people on the market

796
00:47:43,340 --> 00:47:46,750
do price and hedge,
most importantly,

797
00:47:46,750 --> 00:47:54,340
the digital contracts, because
call contracts are liquid,

798
00:47:54,340 --> 00:47:57,710
and they are traded on
exchanges while digitals

799
00:47:57,710 --> 00:47:59,420
are way less liquid.

800
00:47:59,420 --> 00:48:02,030
So somebody would call
again-- to counterparty,

801
00:48:02,030 --> 00:48:06,910
enter into digital, and
hedge it on the exchange.

802
00:48:06,910 --> 00:48:08,650
These two calls
with a call spread.

803
00:48:08,650 --> 00:48:12,200
But now tell me,
is it surprising

804
00:48:12,200 --> 00:48:18,480
that-- I mean, what
does it remind you?

805
00:48:18,480 --> 00:48:18,980
Yeah.

806
00:48:18,980 --> 00:48:26,600
So it's derivative
of the call price

807
00:48:26,600 --> 00:48:28,840
but with respect to strike.

808
00:48:28,840 --> 00:48:30,922
Right?

809
00:48:30,922 --> 00:48:31,630
Is it surprising?

810
00:48:38,280 --> 00:48:41,350
How did our call
price look like?

811
00:48:41,350 --> 00:48:41,850
It's a ramp.

812
00:48:41,850 --> 00:48:44,580
Right?

813
00:48:44,580 --> 00:48:48,926
If we take a derivative
of this, what will we get?

814
00:48:48,926 --> 00:48:49,425
Yeah.

815
00:48:49,425 --> 00:48:49,880
AUDIENCE: [INAUDIBLE].

816
00:48:49,880 --> 00:48:50,588
PROFESSOR: Right.

817
00:48:50,588 --> 00:48:54,790
So in fact, if we do something
even more weird with this,

818
00:48:54,790 --> 00:48:59,990
and then I'll take a
square or something else,

819
00:48:59,990 --> 00:49:01,240
the same will apply.

820
00:49:03,930 --> 00:49:05,520
So it's not surprising at all.

821
00:49:08,364 --> 00:49:09,790
All right.

822
00:49:09,790 --> 00:49:15,390
So that's basically how
the replicate-- this idea

823
00:49:15,390 --> 00:49:21,050
of replicating portfolios
is extremely powerful.

824
00:49:21,050 --> 00:49:24,790
And in fact, that's what
happens in real life.

825
00:49:24,790 --> 00:49:29,480
In real life, you have
some complicated derivative

826
00:49:29,480 --> 00:49:30,830
which you need to hedge.

827
00:49:30,830 --> 00:49:34,070
And how to hedge-- you'll
find something else which

828
00:49:34,070 --> 00:49:36,900
replicates-- to
a certain extent,

829
00:49:36,900 --> 00:49:38,140
replicates your pay-off.

830
00:49:38,140 --> 00:49:40,110
That's what you'll try to do.

831
00:49:40,110 --> 00:49:42,180
And this will be
your hedge portfolio.

832
00:49:42,180 --> 00:49:43,310
Usually, it's dynamic.

833
00:49:43,310 --> 00:49:45,720
So you'll have to rebalance.

834
00:49:45,720 --> 00:49:51,570
And that's how you
basically reduce the risks.