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PROFESSOR: So what I would
like to discuss today

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00:00:29,180 --> 00:00:32,970
is just touch upon
what kind of problems

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00:00:32,970 --> 00:00:37,460
the quantitative analysts
are solving in the commodity

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00:00:37,460 --> 00:00:40,550
world, the problems
that are somewhat

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00:00:40,550 --> 00:00:43,800
different from
the other markets.

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00:00:43,800 --> 00:00:47,590
And I'm sure you have the
whole year of lectures here

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00:00:47,590 --> 00:00:50,420
from people in
different markets.

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00:00:50,420 --> 00:00:55,160
And you will judge for yourself
that the models we're building

16
00:00:55,160 --> 00:00:59,200
are somewhat, if not
completely, different.

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00:00:59,200 --> 00:01:01,970
So this is my goal
today is just for you

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00:01:01,970 --> 00:01:06,590
to have some taste of what kind
of models we're looking at.

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00:01:10,910 --> 00:01:12,330
So let's start.

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00:01:25,320 --> 00:01:37,350
Let's start with the following
abstract from the Dow Jones

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00:01:37,350 --> 00:01:42,440
publication, the
dispatch which announced

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00:01:42,440 --> 00:01:49,730
in 2009 that the Trafigura--
which is the biggest

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00:01:49,730 --> 00:01:53,920
trader, one of the biggest
commodity and energy trader--

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00:01:53,920 --> 00:01:56,720
is potentially on track to
post its best results ever

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00:01:56,720 --> 00:02:01,490
in fiscal 2009 on lower oil
prices and contango markets.

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00:02:01,490 --> 00:02:07,650
Remember, 2008, just the year
before that, the oil prices

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00:02:07,650 --> 00:02:14,720
shot to the highest possible
level, to $159 per barrel.

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00:02:14,720 --> 00:02:17,680
And a lot of people
blamed the traders

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00:02:17,680 --> 00:02:20,730
for the high oil prices.

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00:02:20,730 --> 00:02:25,540
And yet, in 2008, when the
prices dropped-- in 2009,

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00:02:25,540 --> 00:02:27,620
I'm sorry, next
year-- when the prices

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00:02:27,620 --> 00:02:31,590
dropped to $30 per barrel, as
you remember, to the lowest

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00:02:31,590 --> 00:02:37,100
possible level that
we remember, Trafigura

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00:02:37,100 --> 00:02:40,330
is about to make the
biggest profit ever.

35
00:02:40,330 --> 00:02:42,280
So it sounds like
a contradiction.

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00:02:42,280 --> 00:02:46,150
So they're making money on the
low oil prices and contango.

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00:02:46,150 --> 00:02:50,850
And contango, which I have to
explain to you what it is-- I

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00:02:50,850 --> 00:02:58,520
assume that everybody here knows
what the futures contract is.

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00:02:58,520 --> 00:03:02,420
If not, just let me
know, and I will explain.

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00:03:02,420 --> 00:03:09,440
It's very simple concept, so
don't be shy and just tell me.

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00:03:09,440 --> 00:03:13,320
So I would like to
show you how Trafigura

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00:03:13,320 --> 00:03:16,350
will make the record
profit in the year

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00:03:16,350 --> 00:03:20,590
when the prices are at
the record low level,

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00:03:20,590 --> 00:03:21,950
but they are in contango.

45
00:03:27,510 --> 00:03:35,160
So this is the graph
of futures contracts,

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00:03:35,160 --> 00:03:41,040
futures prices, oil futures
prices on January 15, 2009.

47
00:03:41,040 --> 00:03:43,330
Futures contract is
simply the contract

48
00:03:43,330 --> 00:03:45,270
that allows you to
buy today-- let's

49
00:03:45,270 --> 00:03:49,390
say, in January--
the barrel of oil

50
00:03:49,390 --> 00:03:51,050
for delivery in
some future time.

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00:03:51,050 --> 00:03:52,720
That's all.

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00:03:52,720 --> 00:03:54,430
You always know
what the price will

53
00:03:54,430 --> 00:03:58,570
be by looking into Wall Street
Journal, Section C. Open,

54
00:03:58,570 --> 00:03:59,640
you will see.

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00:03:59,640 --> 00:04:04,980
If you want to have
delivery in August,

56
00:04:04,980 --> 00:04:08,350
the price will be around, what?

57
00:04:08,350 --> 00:04:10,040
$55?

58
00:04:10,040 --> 00:04:16,200
If you want delivery on February
of next year-- so February,

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00:04:16,200 --> 00:04:19,510
2010-- the price will be $60.

60
00:04:19,510 --> 00:04:23,555
If you want delivery now,
the price will be $35.

61
00:04:23,555 --> 00:04:24,930
So if you want to
buy, basically,

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00:04:24,930 --> 00:04:29,180
the spot price--
for your knowledge,

63
00:04:29,180 --> 00:04:31,510
there is no spot price, as such.

64
00:04:31,510 --> 00:04:36,580
And whatever you see on
CNBC, for example, on TV,

65
00:04:36,580 --> 00:04:38,700
when they give you
the spot price of oil,

66
00:04:38,700 --> 00:04:42,280
it's the price of the first
future contract-- so this one--

67
00:04:42,280 --> 00:04:45,560
of the nearby, the most
nearby futures contract.

68
00:04:45,560 --> 00:04:47,960
So as you can see,
this is the curve.

69
00:04:47,960 --> 00:04:53,130
And recall that the curve is
in contango if the prices are

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00:04:53,130 --> 00:04:54,940
monotone increasing.

71
00:04:54,940 --> 00:04:57,160
If they are decreasing,
it's called backwardation.

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00:04:57,160 --> 00:04:58,710
That's all.

73
00:04:58,710 --> 00:05:01,340
So just a useful
term for you to know.

74
00:05:01,340 --> 00:05:04,940
So at that time, January
15th, the prices indeed

75
00:05:04,940 --> 00:05:07,140
were monotone increasing.

76
00:05:07,140 --> 00:05:09,430
They're going from
$35 to February

77
00:05:09,430 --> 00:05:11,450
and to $60 in February.

78
00:05:11,450 --> 00:05:12,780
So this is February of 2009.

79
00:05:12,780 --> 00:05:15,110
This is February of 2010.

80
00:05:15,110 --> 00:05:19,300
And we can see the prices
were going from $35 to $60.

81
00:05:19,300 --> 00:05:23,970
So let's now see how
Trafigura made money.

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00:05:23,970 --> 00:05:25,980
Can you guess, by the way?

83
00:05:25,980 --> 00:05:26,812
Yes.

84
00:05:26,812 --> 00:05:28,728
AUDIENCE: They borrowed
money, bought the spot

85
00:05:28,728 --> 00:05:30,579
and then sold the
futures contract?

86
00:05:30,579 --> 00:05:31,370
PROFESSOR: Exactly.

87
00:05:31,370 --> 00:05:34,830
So that's precisely
what they've done.

88
00:05:34,830 --> 00:05:37,202
What is needed for that, though?

89
00:05:37,202 --> 00:05:39,474
There's one little
thing that's required.

90
00:05:39,474 --> 00:05:43,360
AUDIENCE: Low interest rates
and ability to buy the spot?

91
00:05:43,360 --> 00:05:46,200
PROFESSOR: This price
is the ability to buy.

92
00:05:46,200 --> 00:05:49,591
But you need a little bit more.

93
00:05:49,591 --> 00:05:50,090
You need--

94
00:05:50,090 --> 00:05:50,980
AUDIENCE: [INAUDIBLE]

95
00:05:50,980 --> 00:05:52,317
PROFESSOR: I'm sorry?

96
00:05:52,317 --> 00:05:53,750
AUDIENCE: [INAUDIBLE]

97
00:05:53,750 --> 00:05:55,170
PROFESSOR: No, no, you lock in.

98
00:05:55,170 --> 00:05:56,950
You bought at $35.

99
00:05:56,950 --> 00:05:58,220
AUDIENCE: Storage.

100
00:05:58,220 --> 00:05:59,970
PROFESSOR: Storage--
you need the storage.

101
00:05:59,970 --> 00:06:03,800
You need to be able to wait one
year, because you already sold

102
00:06:03,800 --> 00:06:07,110
at $60 next year, in February.

103
00:06:07,110 --> 00:06:10,040
So you locked in
a massive profit

104
00:06:10,040 --> 00:06:14,450
of-- so this is your strategy.

105
00:06:14,450 --> 00:06:16,640
Just borrow 35.

106
00:06:16,640 --> 00:06:17,480
Buy one barrel.

107
00:06:17,480 --> 00:06:19,120
Store it.

108
00:06:19,120 --> 00:06:21,720
Then immediately sell,
so short means just go

109
00:06:21,720 --> 00:06:25,710
and sell the one barrel
for next year at $60.

110
00:06:25,710 --> 00:06:29,340
So you have-- you
made $25-- I'm sorry,

111
00:06:29,340 --> 00:06:32,680
you made $25 just on commodity.

112
00:06:32,680 --> 00:06:35,940
You have to-- on February,
you will get this-- February

113
00:06:35,940 --> 00:06:38,140
next year you will get this $25.

114
00:06:38,140 --> 00:06:41,600
You'll pay the interest, which
will be maybe, let's say 10%.

115
00:06:41,600 --> 00:06:45,590
So it will be $3.5.

116
00:06:45,590 --> 00:06:48,410
So you made total profit $21.

117
00:06:48,410 --> 00:06:53,120
If, like Trafigura, you have 50,
60 million barrels of storage,

118
00:06:53,120 --> 00:06:55,590
you can easily
calculate how much money

119
00:06:55,590 --> 00:07:00,570
they have without any risk they
made in this particular year.

120
00:07:00,570 --> 00:07:04,350
So whenever you hear that
traders are benefiting

121
00:07:04,350 --> 00:07:05,990
from the high prices, it's not.

122
00:07:05,990 --> 00:07:09,370
Actually, if the same situations
existed in the high prices,

123
00:07:09,370 --> 00:07:12,100
then the interest would
be substantially lower.

124
00:07:12,100 --> 00:07:15,480
They could drop
their profit by 50%,

125
00:07:15,480 --> 00:07:21,087
if, let's say, the same
where the prices $125

126
00:07:21,087 --> 00:07:23,231
to $135 to $160.

127
00:07:23,231 --> 00:07:23,730
Right?

128
00:07:23,730 --> 00:07:26,750
You would have the
enormous interest payment.

129
00:07:26,750 --> 00:07:28,740
So in reality, you
need the low prices,

130
00:07:28,740 --> 00:07:30,636
but you would like
to have a contango.

131
00:07:30,636 --> 00:07:33,670
You really don't care,
price is low or high.

132
00:07:33,670 --> 00:07:42,380
So to summarize,
we need to have--

133
00:07:42,380 --> 00:07:46,840
I mean, just a little thing
here-- that the strategy works

134
00:07:46,840 --> 00:07:47,640
like a charm.

135
00:07:47,640 --> 00:07:49,330
The only thing you
need is storage.

136
00:07:49,330 --> 00:07:57,010
OK, let me now
invert the question.

137
00:07:57,010 --> 00:07:58,510
Let me ask you this
question-- let's

138
00:07:58,510 --> 00:08:01,690
say we have the same curve.

139
00:08:01,690 --> 00:08:07,870
We are on January
1st, 2009, and we

140
00:08:07,870 --> 00:08:12,680
are asked to get the storage.

141
00:08:12,680 --> 00:08:16,650
You want to have your bosses
calling you and saying,

142
00:08:16,650 --> 00:08:19,000
I mean, I need
storage, oil storage

143
00:08:19,000 --> 00:08:21,940
from August to December.

144
00:08:21,940 --> 00:08:28,450
So in August, we have the price
is, what, $55, in December $58.

145
00:08:28,450 --> 00:08:32,260
You go and get me--
here's your credit card.

146
00:08:32,260 --> 00:08:34,990
Just go and get it.

147
00:08:34,990 --> 00:08:39,600
Well, storage is usually--
you get it at the auctions.

148
00:08:39,600 --> 00:08:43,740
So a lot of people come to
the auction, and they'll bid.

149
00:08:43,740 --> 00:08:47,370
So now you're going with the
boss's credit card, right?

150
00:08:47,370 --> 00:08:50,310
And you have the
following dilemma--

151
00:08:50,310 --> 00:08:56,280
if you borrow too little,
you will never get it.

152
00:08:56,280 --> 00:09:01,940
If you bid too much, you'll have
the winner's curse-- namely,

153
00:09:01,940 --> 00:09:04,640
you will never be able
to recover the money

154
00:09:04,640 --> 00:09:08,700
that you paid for that storage.

155
00:09:08,700 --> 00:09:14,270
So you have to have-- before you
bid, you have to have a plan.

156
00:09:14,270 --> 00:09:18,580
How do you-- or strategy--
how do you recover the money

157
00:09:18,580 --> 00:09:23,390
that you pay for the storage
through some foolproof,

158
00:09:23,390 --> 00:09:28,560
riskless activity-- some
strategy like Trafigura.

159
00:09:28,560 --> 00:09:31,710
Remember, they have
the riskless strategy.

160
00:09:31,710 --> 00:09:33,240
They lock in the profit.

161
00:09:33,240 --> 00:09:36,970
They can go to sleep for
a year, and then vacation,

162
00:09:36,970 --> 00:09:39,140
and then come back and
get this profit back.

163
00:09:39,140 --> 00:09:44,010
How do you-- in this
particular case, you have,

164
00:09:44,010 --> 00:09:46,070
whenever you bid
something, you also

165
00:09:46,070 --> 00:09:48,270
have to have a plan
in mind how to recover

166
00:09:48,270 --> 00:09:52,090
this money and even more, to
get a little bit of profit.

167
00:09:52,090 --> 00:09:57,350
So my question to you-- you
need storage from August

168
00:09:57,350 --> 00:09:59,190
to December.

169
00:09:59,190 --> 00:10:00,240
How much would you bid?

170
00:10:05,440 --> 00:10:07,310
These are the prices.

171
00:10:07,310 --> 00:10:10,930
If you need more prices,
they're all here.

172
00:10:10,930 --> 00:10:12,930
So you need the storage
from August to December.

173
00:10:18,654 --> 00:10:21,195
AUDIENCE: Do the same thing?

174
00:10:21,195 --> 00:10:23,010
Do the same thing
they did before?

175
00:10:23,010 --> 00:10:25,218
PROFESSOR: So how much would
you bid for the storage?

176
00:10:25,218 --> 00:10:28,400
You don't have the storage, so
you have to-- when you get it,

177
00:10:28,400 --> 00:10:33,050
then you do what we
did before, because

178
00:10:33,050 --> 00:10:36,340
before what we did was assume
that we already have it.

179
00:10:36,340 --> 00:10:41,670
Right now, you don't, but if
you win it, you will have it.

180
00:10:41,670 --> 00:10:43,670
But if you win it, you
have to have the strategy

181
00:10:43,670 --> 00:10:44,815
how to recover this money.

182
00:10:54,550 --> 00:10:55,220
Yes?

183
00:10:55,220 --> 00:10:57,470
AUDIENCE: Depends on how
much profit you want to make.

184
00:10:57,470 --> 00:11:00,060
PROFESSOR: Well, it's
not, because remember,

185
00:11:00,060 --> 00:11:03,081
you are competing
against other people.

186
00:11:03,081 --> 00:11:03,580
Right?

187
00:11:03,580 --> 00:11:08,730
So if you become too greedy,
they will outbid you.

188
00:11:08,730 --> 00:11:11,790
So let's say bid
one penny, right?

189
00:11:11,790 --> 00:11:17,020
Assuming that you'll make--
so it should be-- what

190
00:11:17,020 --> 00:11:19,940
is the highest you can bid?

191
00:11:19,940 --> 00:11:21,335
AUDIENCE: $3?

192
00:11:21,335 --> 00:11:23,200
PROFESSOR: Uh-huh.

193
00:11:23,200 --> 00:11:23,924
You say $3.

194
00:11:26,610 --> 00:11:31,544
All right, give me the
strategy that recovers that $3.

195
00:11:31,544 --> 00:11:35,030
AUDIENCE: But we do
the same process.

196
00:11:35,030 --> 00:11:37,130
We borrow money, buy--

197
00:11:37,130 --> 00:11:38,630
PROFESSOR: You don't
have to borrow,

198
00:11:38,630 --> 00:11:39,880
because it's futures contract.

199
00:11:39,880 --> 00:11:45,940
You don't need-- you
just go long the August.

200
00:11:45,940 --> 00:11:48,740
So buy August, using
the futures contract,

201
00:11:48,740 --> 00:11:51,160
and sell December using
the futures contract.

202
00:11:51,160 --> 00:11:52,770
You don't even need
to borrow money.

203
00:11:52,770 --> 00:11:57,250
You will need to borrow, maybe,
when you get to August, right?

204
00:11:57,250 --> 00:12:01,280
Then when you have to pay $55,
then you'll borrow, right?

205
00:12:01,280 --> 00:12:03,570
So you bid $3, you recovered $3.

206
00:12:03,570 --> 00:12:04,220
No profits.

207
00:12:04,220 --> 00:12:07,690
So most likely, you
will bid $2.99, right?

208
00:12:07,690 --> 00:12:09,200
That's the highest
probably, can go.

209
00:12:09,200 --> 00:12:12,601
You have to give yourself at
least a penny of a profit,

210
00:12:12,601 --> 00:12:13,100
right?

211
00:12:13,100 --> 00:12:13,860
AUDIENCE: The interest.

212
00:12:13,860 --> 00:12:14,706
PROFESSOR: I'm sorry?

213
00:12:14,706 --> 00:12:15,872
AUDIENCE: Also the interest.

214
00:12:15,872 --> 00:12:18,320
PROFESSOR: Let's forget
about interest for a moment,

215
00:12:18,320 --> 00:12:20,237
just for simplicity.

216
00:12:20,237 --> 00:12:22,570
In reality, of course, you
never should forget about it,

217
00:12:22,570 --> 00:12:27,610
but in order not to make our
discussions too complicated,

218
00:12:27,610 --> 00:12:28,300
no interest.

219
00:12:28,300 --> 00:12:35,030
So you're basically will be at
$2.99, it seems to me, right?

220
00:12:35,030 --> 00:12:38,130
You get a penny of profit,
and you have a strategy

221
00:12:38,130 --> 00:12:39,227
if you get the storage.

222
00:12:39,227 --> 00:12:40,060
You know what to do.

223
00:12:40,060 --> 00:12:42,480
You immediately go buy August.

224
00:12:42,480 --> 00:12:45,430
Using futures contract, buy
August, and sell December,

225
00:12:45,430 --> 00:12:48,170
locking in $3.

226
00:12:48,170 --> 00:12:51,670
Pay $2.99, if you manage
to win, if everybody is not

227
00:12:51,670 --> 00:12:54,140
as smart as you
are, and just bid--

228
00:12:54,140 --> 00:12:56,330
or maybe they want
bigger profit.

229
00:12:56,330 --> 00:12:57,570
You will win the storage.

230
00:12:57,570 --> 00:12:59,670
You will lock in.

231
00:12:59,670 --> 00:13:03,870
OK, and everybody agrees, right?

232
00:13:03,870 --> 00:13:07,010
This is actually a
standard strategy.

233
00:13:07,010 --> 00:13:09,280
People use it all the time.

234
00:13:09,280 --> 00:13:11,900
That's what I would
call-- that's what

235
00:13:11,900 --> 00:13:12,820
the trader would do.

236
00:13:12,820 --> 00:13:18,220
That's kind of the trader, so
a business guy would do that.

237
00:13:18,220 --> 00:13:21,520
This is a common strategy--
was very, very common,

238
00:13:21,520 --> 00:13:22,540
let's say in the '90s.

239
00:13:25,350 --> 00:13:27,910
What the quant will
do-- and that's

240
00:13:27,910 --> 00:13:31,290
where the added value of the
quant is to the organization--

241
00:13:31,290 --> 00:13:34,030
they will do something
completely different.

242
00:13:34,030 --> 00:13:38,770
What they will do-- they
will, on January 1st,

243
00:13:38,770 --> 00:13:42,430
they will sell something that
is called August-December spread

244
00:13:42,430 --> 00:13:43,220
option.

245
00:13:43,220 --> 00:13:45,100
You heard the word
option, right?

246
00:13:45,100 --> 00:13:50,390
Option is characterized by
the payout at expiration.

247
00:13:50,390 --> 00:13:52,020
So we have expiration.

248
00:13:52,020 --> 00:13:54,140
We have the payout.

249
00:13:54,140 --> 00:13:57,410
This is not your typical
options, not like IBM option.

250
00:13:57,410 --> 00:13:58,750
This is something different.

251
00:13:58,750 --> 00:14:04,730
The payout is determined at
the expiration, which is,

252
00:14:04,730 --> 00:14:08,261
let's say, July 31st, right
before the beginning of August,

253
00:14:08,261 --> 00:14:08,760
right?

254
00:14:08,760 --> 00:14:13,130
You will look again at
the Wall Street Journal

255
00:14:13,130 --> 00:14:20,140
and look at the December and
August prices on July 31st.

256
00:14:20,140 --> 00:14:22,450
And if the difference
is positive--

257
00:14:22,450 --> 00:14:25,700
this little plus sign means
that if it's positive--

258
00:14:25,700 --> 00:14:29,560
than you pay to the owner of
the option this difference.

259
00:14:29,560 --> 00:14:33,910
If it's negative, you pay zero.

260
00:14:33,910 --> 00:14:37,710
You're more familiar with the
options where one of this thing

261
00:14:37,710 --> 00:14:39,842
is just a fixed strike.

262
00:14:39,842 --> 00:14:41,050
It's called the fixed strike.

263
00:14:41,050 --> 00:14:42,350
Here, there's no strike.

264
00:14:42,350 --> 00:14:46,860
It simply the difference between
December and August contract.

265
00:14:46,860 --> 00:14:49,527
But it's a
two-dimensional object,

266
00:14:49,527 --> 00:14:51,610
so it's a little bit more
complicated to value it.

267
00:14:51,610 --> 00:14:54,780
But there is a whole methodology
developed, the same way

268
00:14:54,780 --> 00:14:55,780
for the regular options.

269
00:14:55,780 --> 00:14:57,890
This is for the spread options.

270
00:14:57,890 --> 00:15:03,990
So the quant will sell this
spread option on August 1st.

271
00:15:03,990 --> 00:15:05,000
Why is it better?

272
00:15:07,660 --> 00:15:11,670
Well, first of all, we have
to discuss how much will I

273
00:15:11,670 --> 00:15:13,920
get for this option?

274
00:15:17,420 --> 00:15:25,600
When I sold it, I give it to
you just for you to be confident

275
00:15:25,600 --> 00:15:28,760
that I'm not trying to deceive
you or anything like that.

276
00:15:28,760 --> 00:15:31,760
You will get-- this
is the formula that

277
00:15:31,760 --> 00:15:34,630
is used to compute the
value of that option.

278
00:15:34,630 --> 00:15:39,500
And when I substituted all the
parameters that are necessary,

279
00:15:39,500 --> 00:15:44,240
I get that the value
of the option is 447.

280
00:15:44,240 --> 00:15:50,890
So I immediately, on
January 1st, got 447.

281
00:15:50,890 --> 00:15:55,250
Everybody else is
going ready to bid $3.

282
00:15:55,250 --> 00:15:58,050
I have in my pocket 447.

283
00:15:58,050 --> 00:16:00,880
I can bid, let's say, 420.

284
00:16:00,880 --> 00:16:02,350
Right?

285
00:16:02,350 --> 00:16:06,272
Guaranteed if I know that
everybody else is bidding $30,

286
00:16:06,272 --> 00:16:07,230
I will win the storage.

287
00:16:07,230 --> 00:16:11,230
Plus my profit margin
is not a penny anymore.

288
00:16:11,230 --> 00:16:14,520
My profit margin
is $0.27, right?

289
00:16:14,520 --> 00:16:19,195
If I can do even 410, even
increase my profit margin

290
00:16:19,195 --> 00:16:22,090
if I really want to be greedy
in this particular case.

291
00:16:22,090 --> 00:16:24,690
But clearly, I can
have a bigger margin.

292
00:16:24,690 --> 00:16:27,660
I can have a bigger-- you
can guess, by the way,

293
00:16:27,660 --> 00:16:29,785
without even looking
at the formula, why

294
00:16:29,785 --> 00:16:37,750
the value of this option
was the payout like this--

295
00:16:37,750 --> 00:16:38,800
was this payout.

296
00:16:38,800 --> 00:16:40,845
Why is it bigger than $3?

297
00:16:40,845 --> 00:16:41,970
It's always bigger than $3.

298
00:16:44,850 --> 00:16:46,220
AUDIENCE: Discount?

299
00:16:46,220 --> 00:16:48,970
PROFESSOR: No, forget about,
again, interest rates.

300
00:16:48,970 --> 00:16:50,360
It's not the discount.

301
00:16:50,360 --> 00:16:54,370
What is the-- on January 1st,
what is the intrinsic value?

302
00:16:54,370 --> 00:16:56,860
If there's no
volatility at all, then

303
00:16:56,860 --> 00:16:58,980
the intrinsic value
of the options-- so

304
00:16:58,980 --> 00:17:01,560
at zero volatility,
the value of the option

305
00:17:01,560 --> 00:17:04,284
is exactly the difference
between December and August

306
00:17:04,284 --> 00:17:06,650
on January 1st, which is $3.

307
00:17:06,650 --> 00:17:10,369
So $3 is the intrinsic
value of that option.

308
00:17:10,369 --> 00:17:12,520
And we all know that
the value of the option

309
00:17:12,520 --> 00:17:15,569
is greater than the intrinsic
value if there is a volatility.

310
00:17:15,569 --> 00:17:19,490
Because there is a volatility,
the value is greater than $3.

311
00:17:19,490 --> 00:17:21,910
And actually, it's
substantially greater than $3,

312
00:17:21,910 --> 00:17:25,750
because the volatility in the
energy markets is very high.

313
00:17:25,750 --> 00:17:27,250
It's much, much
higher than what you

314
00:17:27,250 --> 00:17:33,150
see in the interest rates, or FX
or equities-- indices, I mean.

315
00:17:33,150 --> 00:17:33,650
Yes?

316
00:17:33,650 --> 00:17:35,066
AUDIENCE: So you're
getting money

317
00:17:35,066 --> 00:17:37,110
from taking on more
risk, basically?

318
00:17:37,110 --> 00:17:39,790
PROFESSOR: Let's--
we'll get to that.

319
00:17:39,790 --> 00:17:44,140
You are asking exactly
the correct question,

320
00:17:44,140 --> 00:17:47,620
because yes, I am proud.

321
00:17:47,620 --> 00:17:48,330
I got 447.

322
00:17:48,330 --> 00:17:51,250
I paid 420.

323
00:17:51,250 --> 00:17:52,690
I mean, I was bidding 420.

324
00:17:52,690 --> 00:17:54,670
Of course I won.

325
00:17:54,670 --> 00:17:55,700
I got my storage.

326
00:17:55,700 --> 00:17:57,960
I bring it back home.

327
00:17:57,960 --> 00:18:01,310
Now let's see what kind
of risk I brought home.

328
00:18:01,310 --> 00:18:03,560
I have $0.27 in my pocket left.

329
00:18:03,560 --> 00:18:05,710
That's my profit.

330
00:18:05,710 --> 00:18:10,130
But now let's
assume on July 31st,

331
00:18:10,130 --> 00:18:13,710
let's say December
price goes to $80

332
00:18:13,710 --> 00:18:20,350
and the August
price goes to $55.

333
00:18:20,350 --> 00:18:21,930
I sold this option.

334
00:18:21,930 --> 00:18:27,630
How much do I owe on July
31st-- option is exercised.

335
00:18:27,630 --> 00:18:31,050
How much do I owe to
the owner of the option

336
00:18:31,050 --> 00:18:35,995
if December, on July
31st, 80 and August is 55?

337
00:18:41,190 --> 00:18:43,420
Yes, $25.

338
00:18:43,420 --> 00:18:48,610
I have only $0.27
in my pocket, right?

339
00:18:48,610 --> 00:18:50,910
So what do I do?

340
00:18:50,910 --> 00:18:52,950
That's my risk, as
you're telling me.

341
00:18:52,950 --> 00:18:56,470
This is a risk that
all of a sudden,

342
00:18:56,470 --> 00:19:00,180
I owe astronomical
amount of money

343
00:19:00,180 --> 00:19:04,550
to the person to whom
I sold this option.

344
00:19:04,550 --> 00:19:06,620
So what do I do?

345
00:19:06,620 --> 00:19:10,680
Do I run to Venezuela or
what, I'm interested to know.

346
00:19:10,680 --> 00:19:12,712
Where?

347
00:19:12,712 --> 00:19:13,295
Can you guess?

348
00:19:19,810 --> 00:19:21,930
Remember, I have storage.

349
00:19:21,930 --> 00:19:26,350
And on July 31st, the
August price is $55,

350
00:19:26,350 --> 00:19:29,400
and the December is $80.

351
00:19:29,400 --> 00:19:32,910
So you already told me what
to do in this situation.

352
00:19:32,910 --> 00:19:35,290
That's what traders would do.

353
00:19:35,290 --> 00:19:38,180
They will buy immediately
the August at $55,

354
00:19:38,180 --> 00:19:41,570
immediately sell December,
because I have the storage.

355
00:19:41,570 --> 00:19:46,540
Now I can extract this $25
using my physical asset.

356
00:19:46,540 --> 00:19:51,040
So this is the beauty of the
physical or real options--

357
00:19:51,040 --> 00:19:54,220
that I can, by doing
certain things,

358
00:19:54,220 --> 00:19:56,410
I can extract the
payout of the option.

359
00:19:56,410 --> 00:19:59,770
So I'm completely protected.

360
00:19:59,770 --> 00:20:03,500
So it seems to me that my
storage and that spread option

361
00:20:03,500 --> 00:20:06,240
is the same thing.

362
00:20:06,240 --> 00:20:09,790
Because I'm fully
hedged with my storage.

363
00:20:09,790 --> 00:20:12,080
Right, is that clear?

364
00:20:12,080 --> 00:20:18,090
Do you understand how I managed
to escape terrible predicament?

365
00:20:18,090 --> 00:20:18,590
Yes?

366
00:20:20,765 --> 00:20:22,640
AUDIENCE: What happens
when the value of what

367
00:20:22,640 --> 00:20:25,090
you have in storage
falls greater than what

368
00:20:25,090 --> 00:20:28,530
you received from the auction?

369
00:20:28,530 --> 00:20:31,730
PROFESSOR: But it's
an auction, right?

370
00:20:31,730 --> 00:20:34,260
So I received 447.

371
00:20:34,260 --> 00:20:40,270
Everybody bids $3 or
$2.99 and then 420.

372
00:20:40,270 --> 00:20:42,600
There's no value of the
storage, except for the value

373
00:20:42,600 --> 00:20:45,120
what people are bidding for.

374
00:20:45,120 --> 00:20:48,790
So if my bid is the
highest, that's it.

375
00:20:48,790 --> 00:20:49,947
I received it.

376
00:20:49,947 --> 00:20:52,530
So is that your question, or are
you asking me something else?

377
00:20:55,868 --> 00:20:57,696
AUDIENCE: I want to
visualize the payoff.

378
00:20:57,696 --> 00:20:59,070
It's when the--

379
00:20:59,070 --> 00:21:00,736
PROFESSOR: Oh, you're
talking about what

380
00:21:00,736 --> 00:21:03,730
happens if vice versa, December
not goes to $80, but to $20,

381
00:21:03,730 --> 00:21:04,710
right?

382
00:21:04,710 --> 00:21:07,150
And this one still
remains at $55.

383
00:21:07,150 --> 00:21:12,890
I do nothing, because the payout
of the option is equal to zero,

384
00:21:12,890 --> 00:21:15,370
because the difference
is negative.

385
00:21:15,370 --> 00:21:19,470
So I owe nothing to the
owner of the option,

386
00:21:19,470 --> 00:21:21,180
and I do nothing
with my storage.

387
00:21:21,180 --> 00:21:24,280
It's a fully hedged proposition.

388
00:21:24,280 --> 00:21:30,400
So the storage and this
option is one and the same.

389
00:21:30,400 --> 00:21:32,430
OK?

390
00:21:32,430 --> 00:21:33,920
Any questions?

391
00:21:33,920 --> 00:21:40,150
So conceptually, we're
on the same page, right?

392
00:21:40,150 --> 00:21:45,070
OK, so that was of
course of the caricature

393
00:21:45,070 --> 00:21:46,365
of the real situation.

394
00:21:49,980 --> 00:21:58,010
So reality is somewhat more
complicated, as always.

395
00:21:58,010 --> 00:22:02,300
In reality, let's say I
go and bid for two years.

396
00:22:02,300 --> 00:22:03,830
I need the storage
for two years.

397
00:22:07,800 --> 00:22:11,520
There are many spread
options I can sell, right?

398
00:22:11,520 --> 00:22:15,040
In this, my example-- again,
in the caricature example,

399
00:22:15,040 --> 00:22:16,890
I sold August-December.

400
00:22:16,890 --> 00:22:18,610
I could have sold
August-November,

401
00:22:18,610 --> 00:22:20,950
August-October, right?

402
00:22:20,950 --> 00:22:26,660
Or I can sell-- if I
have for two years,

403
00:22:26,660 --> 00:22:32,480
I can say May, November,
June, September, and so on.

404
00:22:32,480 --> 00:22:34,500
If I have two years, it's 24.

405
00:22:34,500 --> 00:22:37,820
You can understand how
many options I can sell.

406
00:22:37,820 --> 00:22:42,010
I mean, just options
where-- options

407
00:22:42,010 --> 00:22:47,690
to put the oil into the
tank today, and extract it

408
00:22:47,690 --> 00:22:49,910
six months later
with some profit,

409
00:22:49,910 --> 00:22:51,150
or maybe three months later.

410
00:22:51,150 --> 00:22:52,830
And so there are a
lot of these options.

411
00:22:52,830 --> 00:22:55,430
So first of all I
have to determine--

412
00:22:55,430 --> 00:23:00,860
so I can sell a lot of this
option against the storage.

413
00:23:00,860 --> 00:23:07,810
I have to determine A: what
is the most valuable portfolio

414
00:23:07,810 --> 00:23:10,210
of options I can sell
against the storage?

415
00:23:10,210 --> 00:23:15,180
This is-- our strategy
starts shaping up.

416
00:23:15,180 --> 00:23:17,876
So we want to
optimize something.

417
00:23:17,876 --> 00:23:19,000
And what are we optimizing?

418
00:23:19,000 --> 00:23:22,170
We're optimizing the
value of the portfolio

419
00:23:22,170 --> 00:23:25,670
that we can sell
against the storage.

420
00:23:25,670 --> 00:23:27,720
Value is-- remember
this cash that I

421
00:23:27,720 --> 00:23:29,560
get from selling the option?

422
00:23:29,560 --> 00:23:32,680
I want to get not just 447.

423
00:23:32,680 --> 00:23:39,130
I want to get 447,000--
just whatever.

424
00:23:39,130 --> 00:23:43,020
just I want to
maximize, and bring it

425
00:23:43,020 --> 00:23:47,000
to the highest possible level
based on the information--

426
00:23:47,000 --> 00:23:49,850
price information, volatility
information-- that exists

427
00:23:49,850 --> 00:23:51,520
at this particular moment.

428
00:23:51,520 --> 00:23:57,660
I want to determine what
kind of portfolio options.

429
00:23:57,660 --> 00:24:01,030
Now, this is optimization.

430
00:24:01,030 --> 00:24:02,960
What are the constraints?

431
00:24:02,960 --> 00:24:08,560
And now the constraints
are requiring

432
00:24:08,560 --> 00:24:12,560
that you have some technical,
contractual, legal,

433
00:24:12,560 --> 00:24:16,130
and environmental understanding
of what's going on.

434
00:24:16,130 --> 00:24:21,560
The simplest constraints I can
give you is that you cannot put

435
00:24:21,560 --> 00:24:25,870
the gas in the ground or oil
in the tank as quickly as you

436
00:24:25,870 --> 00:24:26,680
want.

437
00:24:26,680 --> 00:24:29,522
There are certain constraints
on the injection rate.

438
00:24:29,522 --> 00:24:31,730
There are certain constraints
on the withdrawal rate.

439
00:24:31,730 --> 00:24:35,310
You cannot instantaneously
extract gas from the ground,

440
00:24:35,310 --> 00:24:37,160
or oil from the tank.

441
00:24:37,160 --> 00:24:39,460
You can never have a
situation-- remember

442
00:24:39,460 --> 00:24:42,400
that whenever the
option expires,

443
00:24:42,400 --> 00:24:45,030
you have to do
something to extract.

444
00:24:45,030 --> 00:24:49,610
Remember, to extract the value,
whenever you owe somebody--

445
00:24:49,610 --> 00:24:53,250
the option holder--
$15 or $25, you

446
00:24:53,250 --> 00:24:58,690
have to put the oil in the tank,
wait six months, extract it.

447
00:24:58,690 --> 00:25:01,970
Under no circumstances--
so your option portfolio

448
00:25:01,970 --> 00:25:05,010
should be sold in such a
way against the storage

449
00:25:05,010 --> 00:25:08,930
that under no circumstances,
you are in the situation when

450
00:25:08,930 --> 00:25:10,270
the option is in the money.

451
00:25:10,270 --> 00:25:14,230
So option is positive.

452
00:25:14,230 --> 00:25:16,350
You have to do something
to extract the value.

453
00:25:16,350 --> 00:25:18,200
So you have to inject,
and there's no space,

454
00:25:18,200 --> 00:25:20,750
because you've done
something before,

455
00:25:20,750 --> 00:25:23,180
because some other
options expired before,

456
00:25:23,180 --> 00:25:25,220
and you put already
oil in the tank.

457
00:25:25,220 --> 00:25:26,690
And the tank is full.

458
00:25:26,690 --> 00:25:28,990
And now a new option
expires, and tells

459
00:25:28,990 --> 00:25:32,950
you put more into the
tank, and there's no space.

460
00:25:32,950 --> 00:25:35,080
In the opposite
situation, when you

461
00:25:35,080 --> 00:25:37,140
need to sell from
the tank and the tank

462
00:25:37,140 --> 00:25:40,810
is empty, because some other
options from that portfolio

463
00:25:40,810 --> 00:25:47,056
managed to completely deplete
the oil from the tank.

464
00:25:47,056 --> 00:25:47,555
Questions?

465
00:25:50,250 --> 00:25:56,020
So let's try to make an attempt
to write this optimization

466
00:25:56,020 --> 00:25:57,580
problem.

467
00:25:57,580 --> 00:26:02,150
Just for fun, so that you
have an idea what's going on.

468
00:26:06,390 --> 00:26:11,085
So first of all, let's see
what we're trying to optimize.

469
00:26:15,610 --> 00:26:17,230
Let's start from the end.

470
00:26:17,230 --> 00:26:21,040
This contract simply
tells me that I

471
00:26:21,040 --> 00:26:25,170
wanted some particular-- F
is the price of the futures

472
00:26:25,170 --> 00:26:29,860
contract with expiration,
let's say, the month of June.

473
00:26:29,860 --> 00:26:34,090
So this particular
term of the sum

474
00:26:34,090 --> 00:26:39,800
tells me that how
much I will have

475
00:26:39,800 --> 00:26:45,430
to pay in June if I
buy-- this is the volume.

476
00:26:45,430 --> 00:26:46,950
Let's say right now is January.

477
00:26:46,950 --> 00:26:52,590
If I buy in June this amount of
oil using my futures contract,

478
00:26:52,590 --> 00:26:53,850
how much I have to pay?

479
00:26:53,850 --> 00:26:57,510
This is kind of
cash from my pocket.

480
00:26:57,510 --> 00:26:58,010
Right?

481
00:26:58,010 --> 00:26:59,787
I mean, it's clear.

482
00:26:59,787 --> 00:27:00,620
The same thing here.

483
00:27:00,620 --> 00:27:01,495
There is option here.

484
00:27:01,495 --> 00:27:04,970
This is just straightforward
buying and selling,

485
00:27:04,970 --> 00:27:08,500
buying into the storage and
selling from the storage.

486
00:27:08,500 --> 00:27:09,000
Agreed?

487
00:27:11,860 --> 00:27:14,750
A little bit more
complicated is this list.

488
00:27:14,750 --> 00:27:22,220
This is the option as I told
you before-- the option that

489
00:27:22,220 --> 00:27:26,980
injects oil into the
tank at one month,

490
00:27:26,980 --> 00:27:29,840
and extract it in
month i, and extract it

491
00:27:29,840 --> 00:27:31,850
in the month j, which is later.

492
00:27:31,850 --> 00:27:36,160
So you extract in June,
extract in November.

493
00:27:36,160 --> 00:27:37,690
So this is the
value of the option.

494
00:27:37,690 --> 00:27:40,550
This is the volume
associated with this option.

495
00:27:40,550 --> 00:27:42,230
I sold this option.

496
00:27:42,230 --> 00:27:45,300
Therefore, I have the
positive cash flow.

497
00:27:45,300 --> 00:27:50,740
So this is 447 in my pocket.

498
00:27:50,740 --> 00:27:52,710
This is another
option that I sold.

499
00:27:52,710 --> 00:27:56,420
This is an option which
is kind of opposite, which

500
00:27:56,420 --> 00:28:01,710
is first you extract,
then you inject,

501
00:28:01,710 --> 00:28:04,860
which is typical situation
when you receive your storage

502
00:28:04,860 --> 00:28:10,140
with already oil in the tank.

503
00:28:10,140 --> 00:28:13,330
The curve is not contango,
but backwardation.

504
00:28:13,330 --> 00:28:16,940
So it's more profitable
for you to sell it now

505
00:28:16,940 --> 00:28:20,540
because the prices are high,
and then replace it later.

506
00:28:20,540 --> 00:28:23,490
So it's kind of symmetric
situation to this one,

507
00:28:23,490 --> 00:28:26,470
to the options that
we already discussed.

508
00:28:26,470 --> 00:28:28,920
It's just it works
when the curve is

509
00:28:28,920 --> 00:28:31,020
in the opposite direction.

510
00:28:31,020 --> 00:28:34,450
So you try to make
money no matter how.

511
00:28:34,450 --> 00:28:37,415
Just the prices look.

512
00:28:37,415 --> 00:28:38,790
You're trying to
make this money.

513
00:28:38,790 --> 00:28:43,530
So you try to optimize
this portfolio.

514
00:28:43,530 --> 00:28:44,030
OK?

515
00:28:48,730 --> 00:28:50,520
Now, let's talk
about constraints.

516
00:28:55,010 --> 00:28:59,190
To determine the constraints,
I introduce right now

517
00:28:59,190 --> 00:29:05,160
the Boolean variable, which is
1 or 0, which basically tells

518
00:29:05,160 --> 00:29:09,040
me if, when I come to the
exercise of this option--

519
00:29:09,040 --> 00:29:12,950
like in our case on July
31st, in our first example--

520
00:29:12,950 --> 00:29:15,020
and I see the option
is in the money or not.

521
00:29:15,020 --> 00:29:16,899
So do I have to do
something or not?

522
00:29:16,899 --> 00:29:18,440
Do I have to inject
into the storage?

523
00:29:18,440 --> 00:29:21,670
And so is it $15
or it's minus $15?

524
00:29:21,670 --> 00:29:24,930
If it's minus $15, the
difference between the prices,

525
00:29:24,930 --> 00:29:26,320
I don't do anything.

526
00:29:26,320 --> 00:29:27,490
So then it's 0.

527
00:29:27,490 --> 00:29:30,800
If it's plus $15, I
have to do something,

528
00:29:30,800 --> 00:29:35,040
because the owner of the
option asks me $15, right?

529
00:29:35,040 --> 00:29:38,920
And so I have to put
oil into the tank,

530
00:29:38,920 --> 00:29:41,560
and extract it sometime later.

531
00:29:41,560 --> 00:29:43,240
So that's one.

532
00:29:43,240 --> 00:29:45,170
The same thing with
the option which

533
00:29:45,170 --> 00:29:48,150
is opposite-- first
withdrawal, then injection.

534
00:29:50,990 --> 00:29:58,910
So then injection constraints
will be quite simple.

535
00:29:58,910 --> 00:30:02,510
So at the expiration, I look.

536
00:30:02,510 --> 00:30:08,280
If this one is 1, then it means
that I have to do something.

537
00:30:08,280 --> 00:30:13,630
And what I have to do, I have
to inject-- at the month i,

538
00:30:13,630 --> 00:30:17,110
I have to inject
this volume x_(i,j),

539
00:30:17,110 --> 00:30:19,510
which was withdrawn later.

540
00:30:19,510 --> 00:30:22,300
So that is injection.

541
00:30:22,300 --> 00:30:26,090
If some option before
that was in the minus--

542
00:30:26,090 --> 00:30:29,990
so I have to inject
before and withdraw now,

543
00:30:29,990 --> 00:30:31,340
that will cancel my injection.

544
00:30:31,340 --> 00:30:33,990
So if simultaneously,
I have injection,

545
00:30:33,990 --> 00:30:36,280
because one option is in
the money, and withdrawal,

546
00:30:36,280 --> 00:30:39,070
because the option that
was exercised before

547
00:30:39,070 --> 00:30:43,110
is in the money, they
kind of cancel each other.

548
00:30:43,110 --> 00:30:45,270
So that's why this
sign is plus and minus.

549
00:30:45,270 --> 00:30:47,820
The same thing with others
options, and the same thing

550
00:30:47,820 --> 00:30:49,300
with pure futures contract.

551
00:30:49,300 --> 00:30:52,790
And that should be less
than the injection rate.

552
00:30:52,790 --> 00:30:55,900
The same thing with
withdrawal rate, right?

553
00:30:55,900 --> 00:31:02,160
I mean, it's a
similar consideration.

554
00:31:02,160 --> 00:31:04,030
Now, maximum
capacity constraints,

555
00:31:04,030 --> 00:31:07,920
it tells me that if I start
with this inventory in my tank

556
00:31:07,920 --> 00:31:11,380
at time 0, and this
is basically how much

557
00:31:11,380 --> 00:31:14,770
I will get at any
time i, for any month

558
00:31:14,770 --> 00:31:20,540
i, that's how much I will have
in my tank by that time i.

559
00:31:20,540 --> 00:31:23,390
You have to believe me, because
it's not a trivial thing.

560
00:31:23,390 --> 00:31:25,590
But the same thing with
minimum constraints.

561
00:31:25,590 --> 00:31:27,630
What you really
have to understand,

562
00:31:27,630 --> 00:31:36,890
that unknowns are x, y,
z, v, and unfortunately,

563
00:31:36,890 --> 00:31:39,540
omega is also unknown.

564
00:31:39,540 --> 00:31:43,220
Omega is just-- it's basically
these are the volumes.

565
00:31:43,220 --> 00:31:47,240
This is the kind of control
variable, the price.

566
00:31:47,240 --> 00:31:51,020
And the price can be-- the
spread can be positive,

567
00:31:51,020 --> 00:31:53,620
can be negative, can be 0.

568
00:31:53,620 --> 00:31:58,610
And this is becoming a very
ugly, non-linear problem,

569
00:31:58,610 --> 00:31:59,610
very quickly.

570
00:31:59,610 --> 00:32:04,690
Very big, so you have
a lot of variables.

571
00:32:04,690 --> 00:32:16,180
For two years, you have, what,
12 times 11, which is 132.

572
00:32:16,180 --> 00:32:19,310
Let me just-- no, 24.

573
00:32:19,310 --> 00:32:20,170
I'm sorry.

574
00:32:20,170 --> 00:32:24,080
It's 24 times 23, so
it's 12 times-- so it's

575
00:32:24,080 --> 00:32:28,807
a big number of
variables, plus an equally

576
00:32:28,807 --> 00:32:31,390
big number of constraints, and
the constraints are non-linear.

577
00:32:35,760 --> 00:32:38,280
So the problem is pretty hard.

578
00:32:43,480 --> 00:32:47,730
I leave it up to you to
decide how to solve it.

579
00:32:47,730 --> 00:32:51,250
That's why they take
optimization courses here.

580
00:32:51,250 --> 00:32:56,150
I can suggest that there are
several approaches people

581
00:32:56,150 --> 00:32:58,310
would take.

582
00:32:58,310 --> 00:33:01,100
There's approximation, where
this problem is approximated,

583
00:33:01,100 --> 00:33:03,940
let's say by linear programming,
or quadratic programming,

584
00:33:03,940 --> 00:33:06,170
or whatever you want.

585
00:33:06,170 --> 00:33:09,320
You can do it through
Monte-Carlo simulations.

586
00:33:09,320 --> 00:33:11,800
Or there is an
interesting approach

587
00:33:11,800 --> 00:33:13,240
through the stochastic control.

588
00:33:13,240 --> 00:33:18,050
I recommend you a paper
by Carmona and Ludkovski,

589
00:33:18,050 --> 00:33:23,485
exactly how to do this-- how
to make a decision of injecting

590
00:33:23,485 --> 00:33:27,830
oil or withdrawing oil based
on the stochastic control,

591
00:33:27,830 --> 00:33:31,450
stochastic optimization
methodology.

592
00:33:31,450 --> 00:33:37,750
But it is quite an
interesting paper.

593
00:33:37,750 --> 00:33:40,540
So either of these
approaches can be used.

594
00:33:40,540 --> 00:33:47,170
And they are used, and give you
sometimes different results,

595
00:33:47,170 --> 00:33:50,985
but you can look.

596
00:33:50,985 --> 00:33:51,860
Any questions so far?

597
00:33:57,632 --> 00:34:01,910
AUDIENCE: Is the
stochastic control solution

598
00:34:01,910 --> 00:34:05,830
an optimal solution
or exact solution?

599
00:34:10,240 --> 00:34:13,670
Is it giving the solution that
the Monte-Carlo simulation's

600
00:34:13,670 --> 00:34:14,650
approximating?

601
00:34:14,650 --> 00:34:17,739
PROFESSOR: Let me
put it this way.

602
00:34:17,739 --> 00:34:26,620
None of these solutions
is better, or prettier,

603
00:34:26,620 --> 00:34:29,500
or whatever, because
of the parameters

604
00:34:29,500 --> 00:34:31,230
that go into the problem.

605
00:34:31,230 --> 00:34:35,440
And you can have the
most precise methodology,

606
00:34:35,440 --> 00:34:38,010
but if your parameters
are only 50%

607
00:34:38,010 --> 00:34:41,726
accurate-- in reality,
all those parameters

608
00:34:41,726 --> 00:34:43,350
necessarily for
Monte-Carlo simulations

609
00:34:43,350 --> 00:34:44,899
with stochastic
control, we really

610
00:34:44,899 --> 00:34:45,940
don't know what they are.

611
00:34:45,940 --> 00:34:50,800
We can only can guess through
some implied market parameter

612
00:34:50,800 --> 00:34:53,227
volatility, and so on.

613
00:34:53,227 --> 00:34:54,889
And if you're
wrong, you're wrong.

614
00:34:54,889 --> 00:34:59,250
Even your method can
be absolutely precise.

615
00:34:59,250 --> 00:35:03,340
I like it, because it's really
very nice mathematics there.

616
00:35:06,690 --> 00:35:09,100
We personally don't do that.

617
00:35:09,100 --> 00:35:11,560
I'm not going to tell
you what we're using,

618
00:35:11,560 --> 00:35:14,260
but we do something different.

619
00:35:14,260 --> 00:35:19,090
And our methodology is
chosen to be robust.

620
00:35:19,090 --> 00:35:21,930
We chose it because we
want the methodology

621
00:35:21,930 --> 00:35:23,750
to be extremely robust.

622
00:35:23,750 --> 00:35:26,797
So we don't want
small changes-- we

623
00:35:26,797 --> 00:35:28,380
don't want to have
the situation where

624
00:35:28,380 --> 00:35:32,210
small changes, which usually
comes when you overparametrize.

625
00:35:32,210 --> 00:35:35,980
I mean, people like to go and
later-- in the second half--

626
00:35:35,980 --> 00:35:38,430
I will discuss some of
the models people use.

627
00:35:38,430 --> 00:35:41,970
For example, they
see the richness

628
00:35:41,970 --> 00:35:44,520
of the behavior of
the prices, and they

629
00:35:44,520 --> 00:35:46,140
want to introduce
a lot of parameters

630
00:35:46,140 --> 00:35:47,570
to capture this richness.

631
00:35:47,570 --> 00:35:50,270
By that, they sacrifice the
stability and robustness.

632
00:35:50,270 --> 00:35:53,380
So little changes in the
parameters and the value

633
00:35:53,380 --> 00:35:55,060
can change substantially.

634
00:35:55,060 --> 00:35:57,330
So we prefer a
different approach,

635
00:35:57,330 --> 00:36:00,880
where we maybe a sacrifice
some of the value,

636
00:36:00,880 --> 00:36:04,500
but we will gain the
robustness, stability.

637
00:36:04,500 --> 00:36:07,750
And this is the most
important thing.

638
00:36:07,750 --> 00:36:10,800
And most importantly, everything
that we use in the model

639
00:36:10,800 --> 00:36:14,850
can be verified by the outside
regulators and controllers,

640
00:36:14,850 --> 00:36:18,550
which in this day and age
is extremely important.

641
00:36:18,550 --> 00:36:20,770
Because these days,
to have a model that

642
00:36:20,770 --> 00:36:23,990
is calibrated-- even
the word calibrated

643
00:36:23,990 --> 00:36:29,360
causes a lot of
antennas just going up.

644
00:36:29,360 --> 00:36:30,860
So we try not to
calibrate anything.

645
00:36:30,860 --> 00:36:33,370
Everything that we
do can be verified--

646
00:36:33,370 --> 00:36:35,269
every little brick that
goes into the model

647
00:36:35,269 --> 00:36:37,310
can be traded in the
market, and can be verified.

648
00:36:40,650 --> 00:36:45,010
This is very, very important
in this day and age.

649
00:36:45,010 --> 00:36:45,678
Yes?

650
00:36:45,678 --> 00:36:46,594
AUDIENCE: [INAUDIBLE].

651
00:36:46,594 --> 00:36:49,660
What do you mean by
calibration [INAUDIBLE]?

652
00:36:49,660 --> 00:36:52,940
PROFESSOR: We'll get to
that, but very quick answer

653
00:36:52,940 --> 00:36:57,300
is this-- just you have a lot
of parameters in the model.

654
00:36:57,300 --> 00:36:59,640
I'll show some of the
models people use.

655
00:36:59,640 --> 00:37:02,560
These parameters are not-- you
don't see them in the market.

656
00:37:02,560 --> 00:37:05,080
It's the result of your model.

657
00:37:05,080 --> 00:37:09,460
It's because you chose
this particular model.

658
00:37:09,460 --> 00:37:11,700
Black-Scholes uses
one parameter.

659
00:37:11,700 --> 00:37:14,850
If you use Black-Scholes
with jump-diffusion term,

660
00:37:14,850 --> 00:37:17,770
and some other, you introduce
another ten parameters.

661
00:37:17,770 --> 00:37:21,370
These parameters
are not observable,

662
00:37:21,370 --> 00:37:25,257
so you have to somehow calibrate
it to some market data.

663
00:37:25,257 --> 00:37:27,340
And the calibration becoming
a pretty interesting,

664
00:37:27,340 --> 00:37:30,040
because calibration usually is
done through some kind of least

665
00:37:30,040 --> 00:37:31,000
squares approach.

666
00:37:31,000 --> 00:37:35,250
So you try to look at
the observable data,

667
00:37:35,250 --> 00:37:39,030
and then adjust your parameters
using some least squares

668
00:37:39,030 --> 00:37:42,000
approach to match that data.

669
00:37:42,000 --> 00:37:46,100
Least squares is
notoriously bad-- problem,

670
00:37:46,100 --> 00:37:48,010
so you may have many solutions.

671
00:37:48,010 --> 00:37:49,930
The solution set
can be very flat,

672
00:37:49,930 --> 00:37:51,470
so you can stop very early.

673
00:37:51,470 --> 00:37:54,290
So there are problems
with least squares.

674
00:37:54,290 --> 00:37:58,780
It's non-linear optimization, so
by itself, it's very difficult.

675
00:37:58,780 --> 00:38:00,760
So calibration can be
very, very unstable.

676
00:38:04,870 --> 00:38:05,641
Any more?

677
00:38:05,641 --> 00:38:06,140
OK.

678
00:38:13,590 --> 00:38:15,280
Now I have to tell
you the secret--

679
00:38:15,280 --> 00:38:16,740
there is no Santa Claus.

680
00:38:16,740 --> 00:38:23,680
Just remember, I was telling you
that I go and I sold the option

681
00:38:23,680 --> 00:38:26,670
to get 447.

682
00:38:26,670 --> 00:38:27,616
I've got the option.

683
00:38:27,616 --> 00:38:30,760
In reality there is
no option market.

684
00:38:30,760 --> 00:38:32,401
Nobody will buy this
option from you.

685
00:38:32,401 --> 00:38:34,150
The market for this
option-- non-existent.

686
00:38:37,120 --> 00:38:43,130
Now what do we do in this
particular situation?

687
00:38:43,130 --> 00:38:49,260
That's where the whole beauty of
Black-Scholes comes into play.

688
00:38:49,260 --> 00:38:51,990
Black-Scholes didn't
get the Nobel Prize

689
00:38:51,990 --> 00:38:56,290
for integrating some very
simple payoff function that you

690
00:38:56,290 --> 00:39:00,045
can do in first year of school.

691
00:39:00,045 --> 00:39:05,470
They got the Nobel Prize-- or
they got kind of appreciation

692
00:39:05,470 --> 00:39:07,220
for what they've done,
because they showed

693
00:39:07,220 --> 00:39:11,710
that, with this value, they also
have the strategy that allows

694
00:39:11,710 --> 00:39:14,630
you by doing something every
day or every half a day,

695
00:39:14,630 --> 00:39:15,880
you can replicate this value.

696
00:39:15,880 --> 00:39:20,770
So if you paid for
the option $5, then

697
00:39:20,770 --> 00:39:25,430
you can show what
to do every day.

698
00:39:25,430 --> 00:39:27,660
They tell you what to
do every day in order

699
00:39:27,660 --> 00:39:30,690
to get back this $5 at the end.

700
00:39:30,690 --> 00:39:33,850
So that's what their
main achievement.

701
00:39:33,850 --> 00:39:38,560
Or if I sold an option,
and received $5,

702
00:39:38,560 --> 00:39:42,910
then I will do
something that-- I

703
00:39:42,910 --> 00:39:47,620
will use this $5 to replicate
any payout I owe at the end.

704
00:39:47,620 --> 00:39:49,680
So I sold it to
you, for example.

705
00:39:49,680 --> 00:39:52,300
So at the end, I owe you $50.

706
00:39:52,300 --> 00:39:56,000
But they showed how,
through the dynamic hedging,

707
00:39:56,000 --> 00:39:59,540
and using this $5 that
I receive up front,

708
00:39:59,540 --> 00:40:02,250
I can be able to meet
my obligations to you.

709
00:40:02,250 --> 00:40:04,770
So they showed how to replicate
the payout of the option.

710
00:40:07,720 --> 00:40:12,310
So in reality, I don't
have to have the market.

711
00:40:12,310 --> 00:40:15,020
If I sold somebody
an option for 47,

712
00:40:15,020 --> 00:40:16,790
I don't have to have
anybody who buys it.

713
00:40:16,790 --> 00:40:19,800
What I will do, I will simply
use this dynamic hedging

714
00:40:19,800 --> 00:40:22,410
strategy that
Black-Scholes advised me,

715
00:40:22,410 --> 00:40:25,680
adopt it to the spread
options, which at the end

716
00:40:25,680 --> 00:40:30,710
will produce for me 447.

717
00:40:30,710 --> 00:40:32,450
That's it.

718
00:40:32,450 --> 00:40:35,440
But here again,
this is another task

719
00:40:35,440 --> 00:40:37,170
that quants will have to do.

720
00:40:37,170 --> 00:40:39,020
Not only they will
find the value.

721
00:40:39,020 --> 00:40:42,690
They will produce every
day the strategy that

722
00:40:42,690 --> 00:40:46,510
allows you to extract this 447.

723
00:40:46,510 --> 00:40:49,070
So we already showed what
to do with the storage.

724
00:40:49,070 --> 00:40:53,610
But now you have to do on
the next level of complexity.

725
00:40:53,610 --> 00:40:57,420
You have to tell to the trader
what to do every day in order

726
00:40:57,420 --> 00:41:01,340
to, in reality, get this 447
from selling this option.

727
00:41:01,340 --> 00:41:05,505
So you told this trader,
I'm selling this portfolio

728
00:41:05,505 --> 00:41:09,550
for $100,000 for the storage.

729
00:41:09,550 --> 00:41:12,500
OK, so, but there's
nobody to sell it.

730
00:41:12,500 --> 00:41:13,910
And you'll say, don't worry.

731
00:41:13,910 --> 00:41:16,320
I will do it through
dynamic hedging.

732
00:41:16,320 --> 00:41:19,520
At the end, you will
get your $100,000.

733
00:41:19,520 --> 00:41:21,210
So he will believe you.

734
00:41:21,210 --> 00:41:22,220
And then he will do it.

735
00:41:22,220 --> 00:41:23,300
He will get $100,000.

736
00:41:23,300 --> 00:41:25,850
Then, on the next level you
have to work with the guy who

737
00:41:25,850 --> 00:41:27,260
operates the storage.

738
00:41:27,260 --> 00:41:30,220
And you will tell the
person when to inject oil,

739
00:41:30,220 --> 00:41:33,542
and when to withdraw, and to
get all of this magic which

740
00:41:33,542 --> 00:41:34,500
we've discussed before.

741
00:41:38,630 --> 00:41:46,470
So I think it'll be
a very logical point

742
00:41:46,470 --> 00:41:47,980
for me right now to stop.

743
00:41:47,980 --> 00:41:50,710
And in the next
half, we'll discuss

744
00:41:50,710 --> 00:41:53,620
how to model the power plant.

745
00:41:53,620 --> 00:41:56,730
So we're going to
the next topic.

746
00:42:02,080 --> 00:42:06,850
So far, we have
covered how to model

747
00:42:06,850 --> 00:42:11,300
a physical asset named storage.

748
00:42:11,300 --> 00:42:17,970
The same approach can be used
to model practically everything

749
00:42:17,970 --> 00:42:19,240
that we're interested in.

750
00:42:19,240 --> 00:42:22,630
For example, we can
use the same thing

751
00:42:22,630 --> 00:42:27,440
to model the tankers,
power plants, these storage

752
00:42:27,440 --> 00:42:31,670
refineries, anything,
power lines, pipelines,

753
00:42:31,670 --> 00:42:36,890
everything can be modeled
using this methodology.

754
00:42:36,890 --> 00:42:41,670
Of course, the nature of
the beast will be different.

755
00:42:41,670 --> 00:42:45,150
And you have to understand the
nuances and the-- as I said,

756
00:42:45,150 --> 00:42:48,050
the constraints.

757
00:42:48,050 --> 00:42:50,380
This modeling--
for example, tanker

758
00:42:50,380 --> 00:42:53,440
requires understanding
of all the routes,

759
00:42:53,440 --> 00:42:59,780
and the constraints in the
ports, and all these things.

760
00:42:59,780 --> 00:43:03,350
But the conceptual philosophical
approach is the same.

761
00:43:03,350 --> 00:43:04,950
You have to find
some optionality

762
00:43:04,950 --> 00:43:07,530
and that's some
additional value.

763
00:43:07,530 --> 00:43:12,530
So I obviously don't have time
to go over the whole thing.

764
00:43:12,530 --> 00:43:16,770
But let's, again, for you
to have a taste of this,

765
00:43:16,770 --> 00:43:20,546
let me just-- let's decide
how to model the power plant.

766
00:43:24,410 --> 00:43:27,600
Assume that you are the manager
of the merchant power plant.

767
00:43:27,600 --> 00:43:31,780
Merchant means that you decide
when to run it, or not run it.

768
00:43:31,780 --> 00:43:36,661
You run it to maximize the
profit of this power plant.

769
00:43:40,220 --> 00:43:42,410
So how do you decide?

770
00:43:42,410 --> 00:43:45,310
Let's say you decide once
a day, in the morning,

771
00:43:45,310 --> 00:43:49,920
whether to turn it on or turn
it off, or not to turn it on.

772
00:43:49,920 --> 00:43:52,580
So how do you decide?

773
00:43:52,580 --> 00:43:55,090
Well, sounds complicated.

774
00:43:55,090 --> 00:43:56,794
In reality, it's very simple.

775
00:43:56,794 --> 00:44:00,660
You wake up in the morning.

776
00:44:00,660 --> 00:44:03,790
You look, first of
all, at the newspaper

777
00:44:03,790 --> 00:44:07,550
and find out what's the
price of electricity today.

778
00:44:07,550 --> 00:44:09,690
And price of electricity
you know for each hour--

779
00:44:09,690 --> 00:44:12,070
sometimes for each 15 minutes.

780
00:44:12,070 --> 00:44:13,430
It's determined.

781
00:44:13,430 --> 00:44:16,620
So you know for each hour,
or maybe the daily price,

782
00:44:16,620 --> 00:44:19,930
enough for the whole day.

783
00:44:19,930 --> 00:44:21,260
So that's the price.

784
00:44:21,260 --> 00:44:25,240
If you sell it, that's how
much money you will get.

785
00:44:25,240 --> 00:44:28,220
On the other hand, to produce
one megawatt-hour of power,

786
00:44:28,220 --> 00:44:30,080
you have to do something.

787
00:44:30,080 --> 00:44:32,910
You have to turn
on your turbines.

788
00:44:32,910 --> 00:44:34,795
So you have to bring some fuel.

789
00:44:34,795 --> 00:44:40,300
You have to put some fuel into
the turbines, just to make them

790
00:44:40,300 --> 00:44:45,210
move, and produce electricity.

791
00:44:45,210 --> 00:44:46,120
Fuel can be anything.

792
00:44:46,120 --> 00:44:50,770
Let's say it's natural
gas, but some fuel.

793
00:44:50,770 --> 00:44:53,850
Now you have to
determine-- so you

794
00:44:53,850 --> 00:44:58,120
know how much money you'll get
for one megawatt-hour of power,

795
00:44:58,120 --> 00:44:59,850
right?

796
00:44:59,850 --> 00:45:03,270
But how much it will
cost you to produce it?

797
00:45:03,270 --> 00:45:05,730
So how much fuel you
need to determine

798
00:45:05,730 --> 00:45:07,680
how much-- first of all,
how much it costs you

799
00:45:07,680 --> 00:45:10,940
in terms of fuel to produce
this one megawatt-hour,

800
00:45:10,940 --> 00:45:13,650
you have to know the
efficiency of the plant.

801
00:45:13,650 --> 00:45:17,090
Because efficiency of
the plant is telling you

802
00:45:17,090 --> 00:45:24,270
how much units of fuel--
let's say how much MmBTUs,

803
00:45:24,270 --> 00:45:27,610
so millions British thermal
units of natural gas, that's

804
00:45:27,610 --> 00:45:33,150
how it's measured-- how
much MmBTUs of natural gas

805
00:45:33,150 --> 00:45:36,590
you have to burn to
produce one megawatt-hour.

806
00:45:36,590 --> 00:45:41,570
Well, this measure of
efficiency for the power plant

807
00:45:41,570 --> 00:45:43,440
is something called heat rate.

808
00:45:43,440 --> 00:45:46,830
So heat rate is exactly
this coefficient

809
00:45:46,830 --> 00:45:54,485
by which you multiply the-- so
if I say that heat rate is 7,

810
00:45:54,485 --> 00:46:00,760
it means that I need 7
MmBTUs of natural gas

811
00:46:00,760 --> 00:46:05,720
to produce one megawatt-hour
of power-- that's all.

812
00:46:05,720 --> 00:46:09,790
So in our case, there's
nothing to be concerned about.

813
00:46:09,790 --> 00:46:15,190
It's just simply some
constant that is given.

814
00:46:15,190 --> 00:46:19,640
It's some constant
between 7 to 20.

815
00:46:19,640 --> 00:46:23,130
20 is a very inefficient
plant, very rarely run.

816
00:46:23,130 --> 00:46:26,690
Seven is right now more
kind of a standard constant.

817
00:46:26,690 --> 00:46:30,340
That's the constant
corresponding

818
00:46:30,340 --> 00:46:32,790
to the natural gas
power plant, which

819
00:46:32,790 --> 00:46:36,120
is-- right now the majority
of the new plants is

820
00:46:36,120 --> 00:46:38,600
the natural gas plants.

821
00:46:38,600 --> 00:46:40,410
And there are some
other costs associated

822
00:46:40,410 --> 00:46:42,735
with producing one
megawatt-hour-- just

823
00:46:42,735 --> 00:46:48,520
air-conditioning, labor
costs, and so on and so forth.

824
00:46:48,520 --> 00:46:50,950
Typically, they are not
the biggest component.

825
00:46:50,950 --> 00:46:54,102
So if, let's say,
this is seven, what

826
00:46:54,102 --> 00:46:55,685
is the price of
natural gas right now?

827
00:46:55,685 --> 00:46:58,010
Do you remember?

828
00:46:58,010 --> 00:46:59,410
AUDIENCE: $3.20.

829
00:46:59,410 --> 00:47:02,480
PROFESSOR: That's--
I wish, but it's not.

830
00:47:02,480 --> 00:47:08,060
It's around $4 right now,
let's say, per MmBTU.

831
00:47:08,060 --> 00:47:09,870
So you need $28.

832
00:47:09,870 --> 00:47:13,260
This is probably around
$3, so you need $31

833
00:47:13,260 --> 00:47:14,770
to produce one megawatt-hour.

834
00:47:14,770 --> 00:47:16,660
AUDIENCE: I was
thinking of gasoline.

835
00:47:16,660 --> 00:47:21,360
PROFESSOR: Oh, no, not
that gas-- natural gas.

836
00:47:21,360 --> 00:47:24,590
Sorry about that, but
just even gasoline

837
00:47:24,590 --> 00:47:26,082
is right now around $5.

838
00:47:26,082 --> 00:47:26,790
AUDIENCE: Really?

839
00:47:26,790 --> 00:47:27,415
PROFESSOR: Yes.

840
00:47:30,480 --> 00:47:34,510
Or maybe you are filling
your tank someplace

841
00:47:34,510 --> 00:47:36,348
which you'll share with us.

842
00:47:36,348 --> 00:47:43,420
[LAUGHS] So is that clear?

843
00:47:43,420 --> 00:47:45,810
That's basically--
so you wake up.

844
00:47:45,810 --> 00:47:48,060
You look at the
price of the power.

845
00:47:48,060 --> 00:47:49,980
Let's say it's $50.

846
00:47:49,980 --> 00:47:53,970
You know your plant is,
let's say, heat rate 8.

847
00:47:53,970 --> 00:47:55,610
You look at the
price of natural gas.

848
00:47:55,610 --> 00:47:58,700
It's $4, variable cost $3.

849
00:47:58,700 --> 00:47:59,700
Will we run the plant?

850
00:48:06,700 --> 00:48:10,460
What will be your profit?

851
00:48:10,460 --> 00:48:18,785
So $4, 8, 3, so it's
$35 it will cost you

852
00:48:18,785 --> 00:48:20,690
to produce one megawatt-hour.

853
00:48:20,690 --> 00:48:21,690
This is $50.

854
00:48:21,690 --> 00:48:24,600
You made $15 profit.

855
00:48:24,600 --> 00:48:26,390
You turn it on.

856
00:48:26,390 --> 00:48:27,400
It runs.

857
00:48:27,400 --> 00:48:29,480
If, on the other hand,
the price of power

858
00:48:29,480 --> 00:48:34,650
will be $30-- you look
in the newspaper--

859
00:48:34,650 --> 00:48:35,900
you go back to sleep.

860
00:48:35,900 --> 00:48:36,970
And nothing happens.

861
00:48:36,970 --> 00:48:38,690
You get zero for that day.

862
00:48:38,690 --> 00:48:41,020
So this is the maximum
between this price and zero.

863
00:48:41,020 --> 00:48:41,520
Agreed?

864
00:48:50,300 --> 00:48:55,980
Well now, before I
go to the next slide,

865
00:48:55,980 --> 00:49:02,150
I have to-- together, we'll
have to determine what

866
00:49:02,150 --> 00:49:07,520
is-- if I want to buy a plant--
if I want to buy a power plant,

867
00:49:07,520 --> 00:49:10,740
how much I'm willing
to pay for that.

868
00:49:10,740 --> 00:49:15,340
Well what I'm willing to pay--
I know that every day I'm

869
00:49:15,340 --> 00:49:17,840
getting this thing, right?

870
00:49:17,840 --> 00:49:19,030
But I don't know.

871
00:49:19,030 --> 00:49:21,340
Each of these things
is a random number.

872
00:49:21,340 --> 00:49:23,860
So I have to kind
of buy-- I have

873
00:49:23,860 --> 00:49:26,040
to construct the
distribution of this price,

874
00:49:26,040 --> 00:49:29,610
of this price, maybe
correlation between them.

875
00:49:29,610 --> 00:49:33,310
It's a two-dimensional
distribution, so correlation,

876
00:49:33,310 --> 00:49:34,740
find some expected value.

877
00:49:34,740 --> 00:49:39,650
And that's how much I will want
to pay for the power plant.

878
00:49:39,650 --> 00:49:42,060
So now we come to an
interesting question.

879
00:49:42,060 --> 00:49:44,060
So now, real work starts.

880
00:49:44,060 --> 00:49:47,260
So I know that this is the
value that I have to integrate,

881
00:49:47,260 --> 00:49:50,620
but I don't know with respect
to which distribution.

882
00:49:50,620 --> 00:49:54,490
So I have to now construct the
model of the power prices, fuel

883
00:49:54,490 --> 00:49:55,700
prices.

884
00:49:55,700 --> 00:49:59,190
And then take this
two-dimensional distribution

885
00:49:59,190 --> 00:50:02,000
and try to find the expectation.

886
00:50:02,000 --> 00:50:03,601
And that's how much
I'm willing to pay

887
00:50:03,601 --> 00:50:04,850
for this power plant-- agreed?

888
00:50:07,940 --> 00:50:13,660
Well, so let's start with kind
of building these components.

889
00:50:13,660 --> 00:50:15,660
That's what, again,
quants will do

890
00:50:15,660 --> 00:50:18,540
using already the experience
from other markets, and so on.

891
00:50:18,540 --> 00:50:21,200
You already heard
the other lectures.

892
00:50:21,200 --> 00:50:25,500
So what we'll do, we'll first
look at the distribution

893
00:50:25,500 --> 00:50:27,400
of power prices to see.

894
00:50:27,400 --> 00:50:30,060
Because I don't
know how to model

895
00:50:30,060 --> 00:50:32,590
the evolution of power prices.

896
00:50:32,590 --> 00:50:35,150
I know if I use simply
Brownian motion-- let's

897
00:50:35,150 --> 00:50:36,610
say it's my first idea.

898
00:50:36,610 --> 00:50:39,180
Well, let's see.

899
00:50:39,180 --> 00:50:42,560
Brownian motion gives
us the distribution

900
00:50:42,560 --> 00:50:45,810
which is, as you remember,
at any point of time is

901
00:50:45,810 --> 00:50:48,160
kind of normally distributed.

902
00:50:48,160 --> 00:50:52,940
If I look at the terminal
distribution of price, which

903
00:50:52,940 --> 00:50:56,810
is driven by Brownian motion--
if I assume that the price is

904
00:50:56,810 --> 00:50:58,800
driven by Brownian
motion, then at the end,

905
00:50:58,800 --> 00:51:01,360
I'll get the normal
distribution.

906
00:51:01,360 --> 00:51:04,900
So if I look at the, say
equity-- this is S&P 500--

907
00:51:04,900 --> 00:51:09,210
indeed it's-- we all know
that S&P has fat tails, right?

908
00:51:09,210 --> 00:51:13,690
But as we can see in a moment,
this is not a fat tail.

909
00:51:13,690 --> 00:51:16,840
This is very close to
the normal distribution.

910
00:51:16,840 --> 00:51:18,950
For the equity guys,
of course, for the guys

911
00:51:18,950 --> 00:51:22,830
who trade the stocks, it's
an enormously fat tail.

912
00:51:22,830 --> 00:51:25,260
But from the commodity
point of view,

913
00:51:25,260 --> 00:51:28,090
this is just perfect,
normal distribution.

914
00:51:28,090 --> 00:51:31,850
This is the distribution
that we deal with.

915
00:51:31,850 --> 00:51:36,920
This is-- the tails
here are just-- so

916
00:51:36,920 --> 00:51:40,790
the normal distribution just
out of the window immediately.

917
00:51:40,790 --> 00:51:42,470
So Brownian motion
out of the window--

918
00:51:42,470 --> 00:51:45,260
we cannot construct this
distribution of the terminal

919
00:51:45,260 --> 00:51:50,280
prices of power using that.

920
00:51:50,280 --> 00:51:51,060
That's one thing.

921
00:51:51,060 --> 00:51:54,080
So we have very fat tails.

922
00:51:54,080 --> 00:51:55,960
And this is the kind
of the parameters

923
00:51:55,960 --> 00:51:58,110
that specify the distribution.

924
00:51:58,110 --> 00:52:01,450
If I look at the, let's
say the equity index,

925
00:52:01,450 --> 00:52:03,790
the kurtosis is very
close to the kurtosis

926
00:52:03,790 --> 00:52:10,380
of the normal distribution,
which is 3 exactly.

927
00:52:10,380 --> 00:52:11,930
But if I look at
the Nord Pool, which

928
00:52:11,930 --> 00:52:15,970
is the power prices in
Scandinavian countries,

929
00:52:15,970 --> 00:52:17,200
kurtosis is 26.

930
00:52:17,200 --> 00:52:19,940
If I look at the
one-hour price, it's 76.

931
00:52:19,940 --> 00:52:27,130
So it is, as you can see, as
far from normal as possible.

932
00:52:27,130 --> 00:52:28,505
And we've seen it
in the picture.

933
00:52:28,505 --> 00:52:31,230
And these are the numbers
corresponding to that.

934
00:52:31,230 --> 00:52:35,730
Moreover, look at the
behavior of the prices.

935
00:52:35,730 --> 00:52:39,750
This is the price in Texas, for
example, power prices in Texas.

936
00:52:39,750 --> 00:52:42,890
What immediately jumps-- I
mean, look at the prices,

937
00:52:42,890 --> 00:52:45,660
and say, wow, how is
it different from what

938
00:52:45,660 --> 00:52:49,200
you see, let's say, in the
equity world, the stock market,

939
00:52:49,200 --> 00:52:50,270
for example?

940
00:52:50,270 --> 00:52:53,220
What is it that
immediately jumps at you?

941
00:52:56,330 --> 00:52:57,661
What is different here?

942
00:53:01,430 --> 00:53:02,260
Go ahead.

943
00:53:02,260 --> 00:53:03,703
There's no right
or wrong answer.

944
00:53:03,703 --> 00:53:04,430
AUDIENCE: Spikes.

945
00:53:04,430 --> 00:53:07,460
PROFESSOR: Spikes--
that's the key word.

946
00:53:07,460 --> 00:53:10,570
Like no other market,
we have spikes here.

947
00:53:10,570 --> 00:53:14,460
That's a major,
major issue for us

948
00:53:14,460 --> 00:53:16,660
from the modeling point of view.

949
00:53:16,660 --> 00:53:18,530
You take any standard,
say Brownian motion,

950
00:53:18,530 --> 00:53:20,580
it will never
exhibit the spikes.

951
00:53:20,580 --> 00:53:22,310
Not only that, the
volatility, of course,

952
00:53:22,310 --> 00:53:26,090
of the prices, as we
already expect, is huge.

953
00:53:26,090 --> 00:53:27,710
It's hundreds.

954
00:53:27,710 --> 00:53:30,290
S&P volatility right
now is 10% percent.

955
00:53:30,290 --> 00:53:35,460
This is 100, 200,
sometimes 1,000, right?

956
00:53:35,460 --> 00:53:38,340
All the intuition you
have about the behavior

957
00:53:38,340 --> 00:53:43,370
of the prices, the behavior
of the random variables,

958
00:53:43,370 --> 00:53:47,570
is mostly evolved,
all this intuition,

959
00:53:47,570 --> 00:53:51,000
under this 10, 15, 20%
volatility assumption.

960
00:53:54,140 --> 00:53:56,990
When the variables behave in
a completely different way

961
00:53:56,990 --> 00:54:03,510
when they have volatility
of 100% or 200% or 300%,

962
00:54:03,510 --> 00:54:08,890
so that's another thing that
will be challenging us, spikes

963
00:54:08,890 --> 00:54:11,080
and high volatility.

964
00:54:11,080 --> 00:54:12,710
It's just the same thing.

965
00:54:12,710 --> 00:54:14,470
It's everywhere,
so I just wanted

966
00:54:14,470 --> 00:54:16,750
to bring-- a different
region of the country,

967
00:54:16,750 --> 00:54:18,640
it will be exactly
the same thing.

968
00:54:18,640 --> 00:54:22,010
So it's common thing.

969
00:54:22,010 --> 00:54:24,900
So if you look, and start
to summarize what we're

970
00:54:24,900 --> 00:54:28,090
trying to capture
is-- mean reversion

971
00:54:28,090 --> 00:54:30,350
and spikes are more or
less the same thing,

972
00:54:30,350 --> 00:54:34,890
so we know if we go far
away, it will go back--

973
00:54:34,890 --> 00:54:37,390
high kurtosis,
regime switching, I

974
00:54:37,390 --> 00:54:42,510
will talk about that in a
moment-- and non-stationarity.

975
00:54:42,510 --> 00:54:45,930
That goes without saying.

976
00:54:45,930 --> 00:54:50,380
That's true for
most of the markets.

977
00:54:50,380 --> 00:54:52,360
And now the problem that
we have to capture--

978
00:54:52,360 --> 00:54:55,250
or not the problem
but another phenomenon

979
00:54:55,250 --> 00:54:59,880
that we have to capture--
that the power and the fuel,

980
00:54:59,880 --> 00:55:04,870
natural gas, exhibit a
very particular structure

981
00:55:04,870 --> 00:55:05,750
of the correlation.

982
00:55:05,750 --> 00:55:07,560
So correlation is not a number.

983
00:55:07,560 --> 00:55:10,540
It's dependent the heat
rate in the market.

984
00:55:10,540 --> 00:55:13,239
Remember heat rate is the
efficiency of the power plant

985
00:55:13,239 --> 00:55:14,530
that are running in the market.

986
00:55:14,530 --> 00:55:18,120
Depending on the efficiency, you
may have very high correlation

987
00:55:18,120 --> 00:55:19,330
or low correlation.

988
00:55:19,330 --> 00:55:21,460
We'll discuss that at the
end of this talk, which

989
00:55:21,460 --> 00:55:23,370
is why this here.

990
00:55:23,370 --> 00:55:25,800
Right now, I'm giving you
something that we observe,

991
00:55:25,800 --> 00:55:31,040
and our models, preferably,
should capture that as well.

992
00:55:31,040 --> 00:55:35,990
So requirements for our
model are pretty intense,

993
00:55:35,990 --> 00:55:40,880
and pretty difficult. I
don't think that you ever

994
00:55:40,880 --> 00:55:43,390
assume the requirement
of the correlation

995
00:55:43,390 --> 00:55:46,240
should have some particular
structure, just depending

996
00:55:46,240 --> 00:55:47,590
on some parameters, and so on.

997
00:55:54,320 --> 00:55:56,815
OK, any questions?

998
00:56:01,490 --> 00:56:04,840
So the models typically--
the first thing people do,

999
00:56:04,840 --> 00:56:08,370
they take the models that they
know, and they try to apply.

1000
00:56:08,370 --> 00:56:10,834
Let's just very quickly
go through the models.

1001
00:56:10,834 --> 00:56:12,500
I mean, you've seen
these models already

1002
00:56:12,500 --> 00:56:14,520
in the previous lectures, right?

1003
00:56:14,520 --> 00:56:16,900
Let's start with the
straightforward geometric

1004
00:56:16,900 --> 00:56:20,830
Brownian motion, which out of
the window right away, right?

1005
00:56:20,830 --> 00:56:25,810
I mean you agree, because
no spikes, no high--

1006
00:56:25,810 --> 00:56:28,620
no correlation
structure, nothing.

1007
00:56:28,620 --> 00:56:34,111
So it's clear that in order
to-- we want to kind of start

1008
00:56:34,111 --> 00:56:36,110
with Brownian motion,
geometric Brownian motion,

1009
00:56:36,110 --> 00:56:40,340
try to maybe modify
it a little bit.

1010
00:56:40,340 --> 00:56:42,060
So what is spike?

1011
00:56:42,060 --> 00:56:43,972
Spike means that
just things go up,

1012
00:56:43,972 --> 00:56:45,180
and then they're pulled back.

1013
00:56:45,180 --> 00:56:47,290
So there should be some
mean reversion, right?

1014
00:56:47,290 --> 00:56:50,204
So mean reversion
is good, right?

1015
00:56:50,204 --> 00:56:51,620
But unfortunately,
mean reversion,

1016
00:56:51,620 --> 00:56:56,490
if you-- first of all, it's
pretty strong, so it goes,

1017
00:56:56,490 --> 00:56:59,330
remember, if the price
goes from $30 to $1,000,

1018
00:56:59,330 --> 00:57:03,840
and go back to $30,
mean reversion should

1019
00:57:03,840 --> 00:57:05,670
be extremely strong, right?

1020
00:57:05,670 --> 00:57:08,870
Second of all, what pushes
it to the $1,000 level?

1021
00:57:08,870 --> 00:57:13,450
You need to have a jump.

1022
00:57:13,450 --> 00:57:16,910
So that's why people
introduce the jumps.

1023
00:57:16,910 --> 00:57:21,100
So we have mean
reversion, jumps; jumps

1024
00:57:21,100 --> 00:57:23,250
and mean reversion.

1025
00:57:23,250 --> 00:57:30,430
Still it is becoming
pretty clear

1026
00:57:30,430 --> 00:57:32,970
that this will not work
for the following reason:

1027
00:57:32,970 --> 00:57:35,590
it goes from $30
to $1,000, and then

1028
00:57:35,590 --> 00:57:37,380
it goes back,
because it's a spike.

1029
00:57:37,380 --> 00:57:39,760
It goes back within
three or four days.

1030
00:57:39,760 --> 00:57:40,505
It comes back.

1031
00:57:43,170 --> 00:57:45,620
So the mean reversion
should be extremely strong.

1032
00:57:45,620 --> 00:57:48,100
The force of pulling back
should be extremely strong.

1033
00:57:48,100 --> 00:57:52,500
But if it's so strong
that it moves you back

1034
00:57:52,500 --> 00:57:56,200
from $1,000 to $30,
imagine what it does when

1035
00:57:56,200 --> 00:57:57,980
you're on the level of $30.

1036
00:57:57,980 --> 00:58:01,120
Because it will be completely
flat-- just nothing,

1037
00:58:01,120 --> 00:58:02,490
you could not move there, right?

1038
00:58:02,490 --> 00:58:06,860
So people observed that,
and introduced, well, they

1039
00:58:06,860 --> 00:58:08,360
figured out maybe
the mean reversion

1040
00:58:08,360 --> 00:58:10,470
will be different at
the level of $1,000

1041
00:58:10,470 --> 00:58:13,450
than when you are at
the normal level of $30,

1042
00:58:13,450 --> 00:58:14,740
so the difference.

1043
00:58:14,740 --> 00:58:17,600
So you introduce right now
so-called regime switching.

1044
00:58:17,600 --> 00:58:19,410
So it means that
all the parameters

1045
00:58:19,410 --> 00:58:22,510
change, so you introduce
the kind of the high price

1046
00:58:22,510 --> 00:58:25,060
level, low price level,
parameters change.

1047
00:58:25,060 --> 00:58:29,220
And now we're getting
to the example

1048
00:58:29,220 --> 00:58:32,410
that we discussed
in the first half.

1049
00:58:32,410 --> 00:58:35,560
You end up with the
model with 10 parameters,

1050
00:58:35,560 --> 00:58:38,890
12 parameters, which is
absolutely impossible

1051
00:58:38,890 --> 00:58:41,010
to manage.

1052
00:58:41,010 --> 00:58:44,630
I mean, I can give
you another, probably,

1053
00:58:44,630 --> 00:58:48,220
hour discussing why
this approach in general

1054
00:58:48,220 --> 00:58:50,480
is not good.

1055
00:58:50,480 --> 00:58:53,290
And not just in commodities,
but everywhere else.

1056
00:58:53,290 --> 00:58:56,270
I mean, I'm not a big fan of
making the models extremely

1057
00:58:56,270 --> 00:59:01,410
complicated, because
introduction of one complexity

1058
00:59:01,410 --> 00:59:03,827
leads to introduction
of another complexity.

1059
00:59:03,827 --> 00:59:05,410
And you just cannot
manage this thing.

1060
00:59:05,410 --> 00:59:06,990
It's impossible to manage.

1061
00:59:06,990 --> 00:59:10,820
Theoretically, it looks
fantastic-- unmanageable.

1062
00:59:10,820 --> 00:59:15,070
So we have to figure
out something else.

1063
00:59:15,070 --> 00:59:17,080
So all these
methodologies-- and I

1064
00:59:17,080 --> 00:59:19,590
put them there-- the
methodologies people actually

1065
00:59:19,590 --> 00:59:25,230
use, the stochastic volatility,
the regime switching,

1066
00:59:25,230 --> 00:59:28,150
multiple jumps, and so on.

1067
00:59:28,150 --> 00:59:35,080
This-- it's all used, but it's
not what I want to suggest.

1068
00:59:35,080 --> 00:59:37,650
I want to suggest something
completely different.

1069
00:59:37,650 --> 00:59:39,620
These are the methodologies
that typically

1070
00:59:39,620 --> 00:59:43,845
are coming from fixed-income
world or foreign exchange

1071
00:59:43,845 --> 00:59:45,040
or equities and so on.

1072
00:59:45,040 --> 00:59:49,050
I want to introduce something
completely different.

1073
00:59:49,050 --> 00:59:52,180
Any questions so far?

1074
00:59:52,180 --> 00:59:54,970
So I want to introduce
the methodology which

1075
00:59:54,970 --> 00:59:58,070
is more suitable and
more understandable from

1076
00:59:58,070 --> 00:59:59,580
the commodity point of view.

1077
00:59:59,580 --> 01:00:02,110
Because actually, what's
the price of commodity?

1078
01:00:02,110 --> 01:00:06,380
Price of commodity is
driven by only two things.

1079
01:00:06,380 --> 01:00:08,394
Can you guess which?

1080
01:00:08,394 --> 01:00:09,560
AUDIENCE: Supply and demand.

1081
01:00:09,560 --> 01:00:13,080
PROFESSOR: Exactly, supply
and demand-- that's all it is.

1082
01:00:13,080 --> 01:00:14,400
That's what we'll try to do.

1083
01:00:14,400 --> 01:00:19,100
Maybe we can-- we have a hard
time modeling the commodity

1084
01:00:19,100 --> 01:00:22,190
prices using just these
standard methodologies

1085
01:00:22,190 --> 01:00:27,430
from different markets,
which rely only on the prices

1086
01:00:27,430 --> 01:00:28,760
themselves.

1087
01:00:28,760 --> 01:00:32,900
Maybe if I introduce some
fundamental modeling as well,

1088
01:00:32,900 --> 01:00:36,820
I can do it without losing
the most important part

1089
01:00:36,820 --> 01:00:39,930
of this model, is it
matches the market data.

1090
01:00:39,930 --> 01:00:44,460
So I have to-- A: I want
to model supply and demand.

1091
01:00:44,460 --> 01:00:50,210
But I also would like to match
every market data that I have.

1092
01:00:50,210 --> 01:00:52,750
So maybe I introduce
a completely different

1093
01:00:52,750 --> 01:00:54,890
complexity, maybe not.

1094
01:00:54,890 --> 01:01:01,710
So let's try to see if I
can succeed here or not.

1095
01:01:01,710 --> 01:01:04,800
Before we're going to go
into the depths of this,

1096
01:01:04,800 --> 01:01:07,466
let's discuss how the
power prices are formed.

1097
01:01:19,130 --> 01:01:22,550
Power prices are formed
in the following way--

1098
01:01:22,550 --> 01:01:29,520
let's say that the market
consists of two power plants.

1099
01:01:29,520 --> 01:01:32,930
And there's Generator
One and Generator Two.

1100
01:01:32,930 --> 01:01:40,830
Every day, they get prices of
power formed at the auction.

1101
01:01:40,830 --> 01:01:42,960
So every time the
generators will

1102
01:01:42,960 --> 01:01:48,100
submit the bids, which says
the following-- I can generate

1103
01:01:48,100 --> 01:01:52,570
50 megawatt for a given hour.

1104
01:01:52,570 --> 01:01:55,190
I will generate 50 megawatt.

1105
01:01:55,190 --> 01:01:57,210
If you ask me only
to generate 50,

1106
01:01:57,210 --> 01:02:01,180
I will generate them at $20.

1107
01:02:01,180 --> 01:02:01,680
Why?

1108
01:02:01,680 --> 01:02:05,435
Because I will run my most
efficient power plant,

1109
01:02:05,435 --> 01:02:06,810
which is the
cheapest one to run.

1110
01:02:06,810 --> 01:02:10,260
I will generate 50 megawatts.

1111
01:02:10,260 --> 01:02:12,570
If you want more, if
you want me to-- I'm

1112
01:02:12,570 --> 01:02:15,250
bidding-- if you
want me to do 100,

1113
01:02:15,250 --> 01:02:18,660
then you'll have to
pay me $25, because I

1114
01:02:18,660 --> 01:02:21,450
will have to introduce less
efficient power plants,

1115
01:02:21,450 --> 01:02:22,410
and so on.

1116
01:02:22,410 --> 01:02:25,930
200 to $30, and if you
want me to me generate 600,

1117
01:02:25,930 --> 01:02:29,090
you will have to pay me $50.

1118
01:02:29,090 --> 01:02:32,470
So that's my bid,
so-called bid stack.

1119
01:02:32,470 --> 01:02:35,230
That's what I'm
sending to the auction.

1120
01:02:35,230 --> 01:02:38,220
The other guy is
sending similar things.

1121
01:02:38,220 --> 01:02:41,030
The auctioneer-- there
is an auctioneer,

1122
01:02:41,030 --> 01:02:46,210
organization called independent
system operator, ISO.

1123
01:02:46,210 --> 01:02:48,680
They collect all these bids.

1124
01:02:48,680 --> 01:02:52,050
They know what demand
will be tomorrow.

1125
01:02:52,050 --> 01:02:55,440
They collect, and then
knowing this demand,

1126
01:02:55,440 --> 01:02:58,260
they will-- first of all, they
will sort out all the bids

1127
01:02:58,260 --> 01:03:00,170
in the most optimal way.

1128
01:03:00,170 --> 01:03:06,120
So they'll put it all
together by sorting.

1129
01:03:06,120 --> 01:03:10,555
So this is the auctioneer
combined all these two

1130
01:03:10,555 --> 01:03:14,430
bids, created this
graph-- so basically based

1131
01:03:14,430 --> 01:03:17,970
on how much they
need to generate.

1132
01:03:17,970 --> 01:03:21,524
So based on the demand,
this will be the price.

1133
01:03:21,524 --> 01:03:23,440
That will be the clearing
price of the market.

1134
01:03:23,440 --> 01:03:26,650
And then, because the
auctioneer knows all the bids,

1135
01:03:26,650 --> 01:03:29,367
the auctioneer will
send the demand

1136
01:03:29,367 --> 01:03:30,950
for this generator
and that generator.

1137
01:03:30,950 --> 01:03:35,840
So you will generate 60, you
will generate 600, or whatever.

1138
01:03:35,840 --> 01:03:40,590
The final price is the highest,
basically highest price

1139
01:03:40,590 --> 01:03:43,220
that is necessary
to meet the demand.

1140
01:03:43,220 --> 01:03:46,310
So if the demand is 600,
so this will be the price.

1141
01:03:46,310 --> 01:03:48,600
If it's 800, this
will be the price.

1142
01:03:48,600 --> 01:03:56,500
So the price is
clearly the function

1143
01:03:56,500 --> 01:03:57,840
of demand for any given day.

1144
01:04:01,160 --> 01:04:05,650
Well, even if this
is the case, then

1145
01:04:05,650 --> 01:04:09,050
if I look at the-- if I plot,
do the scatter-plot demand

1146
01:04:09,050 --> 01:04:13,740
versus price, I'll have
to see something similar.

1147
01:04:13,740 --> 01:04:15,080
Well, let's see.

1148
01:04:15,080 --> 01:04:17,670
When we take a
particular market,

1149
01:04:17,670 --> 01:04:22,680
and do the scattered graph,
so demand versus the price,

1150
01:04:22,680 --> 01:04:28,020
you see this is the graph
that we expect to see.

1151
01:04:28,020 --> 01:04:29,390
It's a little bit fat.

1152
01:04:29,390 --> 01:04:31,530
Why is it fat?

1153
01:04:31,530 --> 01:04:32,820
Why is it not a straight line?

1154
01:04:32,820 --> 01:04:36,304
Why is it not the line
as before, but it's fat?

1155
01:04:39,060 --> 01:04:42,740
What is random here?

1156
01:04:42,740 --> 01:04:46,500
Remember I told you
that each generator will

1157
01:04:46,500 --> 01:04:49,900
bid approximately how much
it will cost them to generate

1158
01:04:49,900 --> 01:04:52,850
at the particular day.

1159
01:04:52,850 --> 01:04:55,330
That cost depends
on the fuel price,

1160
01:04:55,330 --> 01:04:56,610
and that is a random number.

1161
01:04:56,610 --> 01:05:00,380
So what you can see here
is a lot of these curves,

1162
01:05:00,380 --> 01:05:04,370
but they are kind of randomly
moving because of the fuel

1163
01:05:04,370 --> 01:05:07,420
price affects the
cost of running.

1164
01:05:07,420 --> 01:05:11,030
But conceptually,
philosophically, this

1165
01:05:11,030 --> 01:05:14,660
is exactly what we discussed--
that these guys will

1166
01:05:14,660 --> 01:05:18,380
bid this curve.

1167
01:05:18,380 --> 01:05:21,730
And depending on the
demand, the price

1168
01:05:21,730 --> 01:05:25,370
will be simply the
value on that curve.

1169
01:05:28,340 --> 01:05:31,320
Are you with me?

1170
01:05:31,320 --> 01:05:33,090
So far so good?

1171
01:05:33,090 --> 01:05:37,500
So this is-- that's what
we're trying to model.

1172
01:05:37,500 --> 01:05:42,040
So if we can model that, if
I can model for every day

1173
01:05:42,040 --> 01:05:46,060
the bid curve that
the auctioneer,

1174
01:05:46,060 --> 01:05:48,350
the independent
system operator, sees,

1175
01:05:48,350 --> 01:05:50,170
I know the price at
that particular day.

1176
01:05:52,820 --> 01:05:56,450
Moreover, I don't want
to get too complicated,

1177
01:05:56,450 --> 01:05:57,950
but in reality,
I don't even have

1178
01:05:57,950 --> 01:05:59,280
to know precisely the curve.

1179
01:05:59,280 --> 01:06:02,590
I kind of have to know
precisely the distribution,

1180
01:06:02,590 --> 01:06:04,650
because remember, for
the value of the option,

1181
01:06:04,650 --> 01:06:07,790
you don't need to know where
the price will be at expiration.

1182
01:06:07,790 --> 01:06:11,970
You just need to know how the
price will be distributed.

1183
01:06:11,970 --> 01:06:14,780
If you catch the
distribution correctly

1184
01:06:14,780 --> 01:06:16,565
through, again,
dynamic hedging and so

1185
01:06:16,565 --> 01:06:18,445
on-- that's what
Black-Scholes tells you--

1186
01:06:18,445 --> 01:06:20,960
you can actually value
the option correctly.

1187
01:06:20,960 --> 01:06:23,330
So you can value the
power plant correctly.

1188
01:06:23,330 --> 01:06:25,240
But that's besides the point.

1189
01:06:25,240 --> 01:06:27,820
The point is right now
I'm trying to model,

1190
01:06:27,820 --> 01:06:33,220
somehow, the randomness
of this bid curve.

1191
01:06:33,220 --> 01:06:41,110
So now, to summarize, the power
price is the function of what?

1192
01:06:41,110 --> 01:06:43,720
Of demand, clearly-- we
already know it is that.

1193
01:06:43,720 --> 01:06:46,560
But also the fuel
prices, because the fuel

1194
01:06:46,560 --> 01:06:49,350
prices determine the
cost of generation

1195
01:06:49,350 --> 01:06:51,950
and therefore, how much
each generator will

1196
01:06:51,950 --> 01:06:54,560
bid into the market.

1197
01:06:54,560 --> 01:06:56,520
That's dependent on the cost.

1198
01:06:56,520 --> 01:06:59,300
If the natural gas
goes through the roof,

1199
01:06:59,300 --> 01:07:02,180
the price of generating one
megawatt will be very high.

1200
01:07:02,180 --> 01:07:03,770
So the person will
be-- the generator

1201
01:07:03,770 --> 01:07:06,520
will be bidding
very high prices.

1202
01:07:06,520 --> 01:07:08,520
And that's what we see here.

1203
01:07:08,520 --> 01:07:10,600
And also outages-- we
have to model outages,

1204
01:07:10,600 --> 01:07:14,220
because the market
is only finite, kind

1205
01:07:14,220 --> 01:07:15,710
of fleet of the power plants.

1206
01:07:15,710 --> 01:07:20,645
And if a couple of them will
go down, the price of power

1207
01:07:20,645 --> 01:07:22,440
can be affected dramatically.

1208
01:07:22,440 --> 01:07:33,330
The 1997, the Indiana-- the
price went from $40 to $7,000

1209
01:07:33,330 --> 01:07:36,960
because tornado hit
the nuclear plant.

1210
01:07:36,960 --> 01:07:41,580
You take a big chunk of the
generation from the stack,

1211
01:07:41,580 --> 01:07:44,740
and all of a sudden,
you have to run

1212
01:07:44,740 --> 01:07:49,190
absolutely anything, including
some very expensive diesels,

1213
01:07:49,190 --> 01:07:49,760
and so on.

1214
01:07:49,760 --> 01:07:53,100
So the price of power,
obviously, goes very high.

1215
01:07:55,980 --> 01:08:00,670
OK, so these are three
things that we try to model.

1216
01:08:00,670 --> 01:08:04,070
So how will we model that?

1217
01:08:04,070 --> 01:08:14,340
Well, before even I get
to the modeling of them,

1218
01:08:14,340 --> 01:08:19,040
let me just again outline
our modeling approach.

1219
01:08:19,040 --> 01:08:23,819
If I know the fuel-- let's say
there are no outages-- if I

1220
01:08:23,819 --> 01:08:29,850
know the fuel price,
I know then how much

1221
01:08:29,850 --> 01:08:31,430
each of the
generators-- how much

1222
01:08:31,430 --> 01:08:34,410
it will cost for each of the
generators in the market.

1223
01:08:34,410 --> 01:08:37,069
Because I know
everybody in the market.

1224
01:08:37,069 --> 01:08:42,069
I know how much for each of
them it will cost to generate.

1225
01:08:42,069 --> 01:08:46,020
So fuel, if I know fuel
price, I can generate

1226
01:08:46,020 --> 01:08:47,620
so-called generation stack.

1227
01:08:47,620 --> 01:08:51,810
It means the cost of generation
for each of the generators,

1228
01:08:51,810 --> 01:08:54,340
for each of the
participants in the market.

1229
01:08:54,340 --> 01:08:57,229
The outages will simply
allow me to take out

1230
01:08:57,229 --> 01:09:00,180
some of these participants
out of the picture, basically.

1231
01:09:00,180 --> 01:09:03,830
So I need to know that, but
that's-- at any given day,

1232
01:09:03,830 --> 01:09:06,010
if I know the outages,
I know the fuel prices,

1233
01:09:06,010 --> 01:09:10,569
I can construct the cost of
generating for everybody.

1234
01:09:10,569 --> 01:09:14,529
So it's close to the bid stack.

1235
01:09:14,529 --> 01:09:18,180
It's something that they
will bid-- their cost,

1236
01:09:18,180 --> 01:09:19,260
maybe plus some profit.

1237
01:09:19,260 --> 01:09:21,710
I don't know what their
profit margins are.

1238
01:09:21,710 --> 01:09:25,779
But what I know, I
know the market prices.

1239
01:09:25,779 --> 01:09:28,100
I know the options, how
they trade, and so on.

1240
01:09:28,100 --> 01:09:31,430
So the bid stack
will, in general,

1241
01:09:31,430 --> 01:09:33,120
follow the generation stack.

1242
01:09:33,120 --> 01:09:35,779
It will be, more or less,
the same thing, maybe some

1243
01:09:35,779 --> 01:09:38,000
added profit requirements.

1244
01:09:38,000 --> 01:09:41,340
But I can back these
profit requirements

1245
01:09:41,340 --> 01:09:43,290
from the market prices.

1246
01:09:43,290 --> 01:09:47,840
That's where my supply-demand
approach and the market

1247
01:09:47,840 --> 01:09:49,630
will get together.

1248
01:09:49,630 --> 01:09:52,750
I will adjust the generation
stack in such a way,

1249
01:09:52,750 --> 01:09:54,920
maybe moving it up and
down in such a way,

1250
01:09:54,920 --> 01:09:57,740
that I matched the
prices, and I matched

1251
01:09:57,740 --> 01:10:02,490
the option prices, so measured
the volatility of the market.

1252
01:10:02,490 --> 01:10:05,830
So now the circle is
completely closed,

1253
01:10:05,830 --> 01:10:09,870
if I can succeed in doing that.

1254
01:10:09,870 --> 01:10:11,060
Questions?

1255
01:10:11,060 --> 01:10:12,390
This is the key.

1256
01:10:17,200 --> 01:10:20,530
Well, let's see
if I can do that.

1257
01:10:20,530 --> 01:10:27,560
So first I want to-- now if
I manage to do that, then

1258
01:10:27,560 --> 01:10:31,725
if I know how to model the
evolution of fuel prices,

1259
01:10:31,725 --> 01:10:35,210
if I know how the
outages are modeled,

1260
01:10:35,210 --> 01:10:39,520
and I model the demand, then
I can determine how the power

1261
01:10:39,520 --> 01:10:43,080
prices will be moving in time.

1262
01:10:43,080 --> 01:10:46,370
So power price is the function
of the bid stack and demand,

1263
01:10:46,370 --> 01:10:47,300
as you remember.

1264
01:10:47,300 --> 01:10:50,510
So if I know the evolution of
the bid stack, the evolution

1265
01:10:50,510 --> 01:10:55,210
of the demand, I can determine
the evolution of the power

1266
01:10:55,210 --> 01:10:56,862
price-- completely
different approach.

1267
01:11:00,114 --> 01:11:02,030
So let's start with the
evolution of the fuel,

1268
01:11:02,030 --> 01:11:04,940
with building the fuel model.

1269
01:11:04,940 --> 01:11:07,250
Well, I told you that
there is natural gas.

1270
01:11:07,250 --> 01:11:10,510
In reality, we have to
model all that stuff.

1271
01:11:10,510 --> 01:11:15,330
Because each of them has a
curve because of volatility.

1272
01:11:15,330 --> 01:11:18,150
It looks a lot,
but in reality it's

1273
01:11:18,150 --> 01:11:21,490
not, because we have
a pretty good idea.

1274
01:11:21,490 --> 01:11:23,810
Unlike the power prices
with the spikes and all

1275
01:11:23,810 --> 01:11:26,330
this crazy behavior, we
have a pretty good idea

1276
01:11:26,330 --> 01:11:29,360
of the behavior of these things,
because they're all storable.

1277
01:11:29,360 --> 01:11:30,940
By the way, power
prices are jumping

1278
01:11:30,940 --> 01:11:33,760
because of non-storability
of power prices,

1279
01:11:33,760 --> 01:11:35,800
because they're not storable.

1280
01:11:35,800 --> 01:11:42,050
Because they're non-storable,
you cannot smooth out

1281
01:11:42,050 --> 01:11:43,960
the changes in the demand.

1282
01:11:43,960 --> 01:11:46,610
Just so you have-- your
action's immediate.

1283
01:11:46,610 --> 01:11:48,440
These are all
storable commodities,

1284
01:11:48,440 --> 01:11:51,020
so the behavior is
much more regular.

1285
01:11:51,020 --> 01:11:54,320
They're much closer, let's
say, to the equity prices.

1286
01:11:54,320 --> 01:11:56,150
And so we can use
some standard models.

1287
01:11:56,150 --> 01:11:59,020
So from the modeling
point of view,

1288
01:11:59,020 --> 01:12:03,320
they are not particularly
difficult to model.

1289
01:12:03,320 --> 01:12:07,550
So particularly just natural
gas, heating oil and fuel oil,

1290
01:12:07,550 --> 01:12:10,440
just the coal-- just we have
a pretty good understanding

1291
01:12:10,440 --> 01:12:12,552
of how these things are modeled.

1292
01:12:18,270 --> 01:12:22,120
Outage, well, we take
the standard model from--

1293
01:12:22,120 --> 01:12:24,950
and reliability
theory provides us

1294
01:12:24,950 --> 01:12:29,520
with a very well-developed
mechanism and apparatus

1295
01:12:29,520 --> 01:12:30,690
how to model these things.

1296
01:12:30,690 --> 01:12:32,940
I mean, usually we do it
through some kind of Poisson,

1297
01:12:32,940 --> 01:12:34,890
or version of Poisson process.

1298
01:12:34,890 --> 01:12:36,891
It is very well understood.

1299
01:12:36,891 --> 01:12:38,390
There's a lot of
literature on that.

1300
01:12:38,390 --> 01:12:40,890
We can model that very easily.

1301
01:12:40,890 --> 01:12:44,520
Where do we have the
parameters for these Poissons?

1302
01:12:44,520 --> 01:12:46,570
The government
provides us with these.

1303
01:12:46,570 --> 01:12:47,700
There's government data.

1304
01:12:47,700 --> 01:12:50,790
Sometimes we get it directly
from the power plants.

1305
01:12:50,790 --> 01:12:55,130
Everybody keeps track of
the frequency of the outages

1306
01:12:55,130 --> 01:12:59,860
and so on and so forth.

1307
01:12:59,860 --> 01:13:02,470
Demand-- how will
we model demand?

1308
01:13:02,470 --> 01:13:07,250
Well, we'll model demand through
the temperature, typically.

1309
01:13:07,250 --> 01:13:07,940
Why temperature?

1310
01:13:07,940 --> 01:13:10,340
Because for temperature,
we have a lot of data.

1311
01:13:10,340 --> 01:13:13,310
It's statistically
a very stable thing.

1312
01:13:13,310 --> 01:13:16,750
So there are many
different approaches

1313
01:13:16,750 --> 01:13:17,750
to modeling temperature.

1314
01:13:22,190 --> 01:13:22,966
It's up to you.

1315
01:13:22,966 --> 01:13:24,590
There's a lot of
literature, so I'm not

1316
01:13:24,590 --> 01:13:26,090
going to go into detail.

1317
01:13:26,090 --> 01:13:28,200
So we choose something.

1318
01:13:28,200 --> 01:13:29,970
We model temperature.

1319
01:13:29,970 --> 01:13:31,460
From that, we model the demand.

1320
01:13:31,460 --> 01:13:33,718
And it works pretty well.

1321
01:13:36,970 --> 01:13:40,480
So now we have-- we modeled
the evolution of temperature,

1322
01:13:40,480 --> 01:13:42,950
evolution of demand,
evolution of fuel,

1323
01:13:42,950 --> 01:13:45,420
evolution of the outages.

1324
01:13:45,420 --> 01:13:50,950
Now we can construct
the generation stack.

1325
01:13:50,950 --> 01:13:57,450
So remember, the generation
was this curve, this curve that

1326
01:13:57,450 --> 01:14:01,820
was a function of demand,
but also the fraction

1327
01:14:01,820 --> 01:14:05,490
of the outages, variable
costs, and so on,

1328
01:14:05,490 --> 01:14:10,800
and the fuel, the vector fuels.

1329
01:14:10,800 --> 01:14:13,240
And then there's
these parameters alpha

1330
01:14:13,240 --> 01:14:15,050
and these alpha
parameters we choose

1331
01:14:15,050 --> 01:14:19,330
to match the futures, the
forward curve for the power

1332
01:14:19,330 --> 01:14:22,730
prices and all other market
parameters as we need.

1333
01:14:22,730 --> 01:14:25,090
So that's how we get-- so
we're matching the market,

1334
01:14:25,090 --> 01:14:34,650
and we're matching the kind of
supply and demand formation.

1335
01:14:34,650 --> 01:14:39,330
Now, it is very clear
why with this approach,

1336
01:14:39,330 --> 01:14:44,540
I can capture spikes without
effort, without any effort.

1337
01:14:44,540 --> 01:14:46,537
And the reason is very simple.

1338
01:14:51,427 --> 01:15:00,260
This is my-- remember the stack,
my bid stack or generation

1339
01:15:00,260 --> 01:15:01,250
stack, whatever.

1340
01:15:01,250 --> 01:15:06,530
So for high generation
volumes, it's

1341
01:15:06,530 --> 01:15:09,190
becoming more and more
expensive to generate.

1342
01:15:09,190 --> 01:15:11,570
And after a certain
time, you've exhausted

1343
01:15:11,570 --> 01:15:13,270
all your cheap plants.

1344
01:15:13,270 --> 01:15:15,410
You have to go to very,
very expensive plants,

1345
01:15:15,410 --> 01:15:17,740
like diesel plants
and so on, which

1346
01:15:17,740 --> 01:15:20,180
runs maybe once in a
lifetime, maybe once a year,

1347
01:15:20,180 --> 01:15:21,050
twice a year.

1348
01:15:21,050 --> 01:15:22,062
And very expensive.

1349
01:15:22,062 --> 01:15:23,270
But they determine the price.

1350
01:15:23,270 --> 01:15:25,960
They are the ones that
determined the clearing price

1351
01:15:25,960 --> 01:15:27,790
at the market.

1352
01:15:27,790 --> 01:15:29,160
Now, let's see.

1353
01:15:29,160 --> 01:15:31,346
If we are somewhere
here-- this is demand.

1354
01:15:31,346 --> 01:15:32,990
Demand is driven by temperature.

1355
01:15:32,990 --> 01:15:37,070
Temperature is
typically a normal thing

1356
01:15:37,070 --> 01:15:38,460
with maybe mean reversion.

1357
01:15:38,460 --> 01:15:42,330
If it goes up, it typically
reverses some mean.

1358
01:15:42,330 --> 01:15:45,400
So this is the distribution
of demand around here.

1359
01:15:45,400 --> 01:15:48,420
Well, if demand
moves left or right,

1360
01:15:48,420 --> 01:15:53,720
up, down, the prices
don't change much.

1361
01:15:53,720 --> 01:15:55,540
So that's your normal regime.

1362
01:15:55,540 --> 01:15:59,840
But let's assume demand
is somewhere here.

1363
01:15:59,840 --> 01:16:03,050
If you are to the left,
the prices are very small.

1364
01:16:03,050 --> 01:16:07,130
But the moment you move a
little bit to the right,

1365
01:16:07,130 --> 01:16:11,240
your temperature moves a
little bit to the high region.

1366
01:16:11,240 --> 01:16:15,810
You immediately spike
into the $1,000 territory.

1367
01:16:15,810 --> 01:16:17,640
But remember, temperature's
mean reverting,

1368
01:16:17,640 --> 01:16:22,100
so within five, six
days, it comes back.

1369
01:16:22,100 --> 01:16:23,100
You go back.

1370
01:16:23,100 --> 01:16:28,520
Your price goes from high
levels back to the normal level.

1371
01:16:28,520 --> 01:16:31,130
That's your spikes.

1372
01:16:31,130 --> 01:16:33,260
You get it completely for free.

1373
01:16:33,260 --> 01:16:36,380
And I'll show you right
now a couple of graphs.

1374
01:16:36,380 --> 01:16:37,810
One is the actual graph.

1375
01:16:37,810 --> 01:16:41,460
The other the simulated
using this approach

1376
01:16:41,460 --> 01:16:43,470
for the same market.

1377
01:16:43,470 --> 01:16:47,930
So as you can see, this
is the actual price

1378
01:16:47,930 --> 01:16:50,360
for that particular period.

1379
01:16:50,360 --> 01:16:53,430
These are the prices
that we generate-- just

1380
01:16:53,430 --> 01:16:55,802
plain Monte-Carlo simulation.

1381
01:16:55,802 --> 01:17:00,610
As you can see, this
can be easily this.

1382
01:17:00,610 --> 01:17:03,750
Just from the point of view--
distribution of the spikes

1383
01:17:03,750 --> 01:17:06,610
and so on, it's
exactly the same thing.

1384
01:17:06,610 --> 01:17:08,270
This is what we were after.

1385
01:17:08,270 --> 01:17:11,590
Moreover, I'll
tell you even more.

1386
01:17:11,590 --> 01:17:16,270
If I knew the past of
the fuel, then obviously

1387
01:17:16,270 --> 01:17:19,800
my approach-- there
are two graphs here.

1388
01:17:19,800 --> 01:17:21,700
You cannot distinguish
between them,

1389
01:17:21,700 --> 01:17:26,720
because I substituted exactly
the right price of the fuel,

1390
01:17:26,720 --> 01:17:28,660
just actual price,
which was historically.

1391
01:17:28,660 --> 01:17:32,410
And therefore my price and the
actual price were the same.

1392
01:17:32,410 --> 01:17:36,690
So, but the reality, of
course, I don't know the past.

1393
01:17:36,690 --> 01:17:38,740
But as I explained to
you, I don't need to know.

1394
01:17:38,740 --> 01:17:41,570
I just need to know the
distribution of fuel prices

1395
01:17:41,570 --> 01:17:42,890
is correctly captured.

1396
01:17:42,890 --> 01:17:45,370
Once I have the correctly
captured distribution

1397
01:17:45,370 --> 01:17:48,390
of the fuel prices, I
have, according to this,

1398
01:17:48,390 --> 01:17:51,580
correctly captured distribution
of the power prices.

1399
01:17:51,580 --> 01:17:57,500
Moreover, so as you can see,
we have a very nice behavior.

1400
01:17:57,500 --> 01:18:01,470
Now from the parameter's point
of view, this is the model.

1401
01:18:01,470 --> 01:18:04,330
Look at the kurtosis.

1402
01:18:04,330 --> 01:18:05,770
This is summers.

1403
01:18:05,770 --> 01:18:08,570
Kurtosis of the model and
kurtosis of the empirical

1404
01:18:08,570 --> 01:18:09,870
is very, very close.

1405
01:18:09,870 --> 01:18:12,580
So the distributions
are very close.

1406
01:18:12,580 --> 01:18:15,230
I mean, the skewness
and so on-- it's

1407
01:18:15,230 --> 01:18:21,930
a very, very good capturing
of the distribution.

1408
01:18:21,930 --> 01:18:26,554
So this approach
works very nicely.

1409
01:18:26,554 --> 01:18:28,720
It's completely different
from what you get used to,

1410
01:18:28,720 --> 01:18:33,530
but it's really
the one that works.

1411
01:18:33,530 --> 01:18:39,601
Moreover, final
benefit-- bonus point--

1412
01:18:39,601 --> 01:18:41,684
this is simulated correlation
structure, remember?

1413
01:18:44,920 --> 01:18:52,540
And this is the actual
one-- very close.

1414
01:18:52,540 --> 01:18:56,650
And the beauty of it is you
don't have-- it's not an input.

1415
01:18:56,650 --> 01:18:57,960
It's an output.

1416
01:18:57,960 --> 01:19:01,470
I don't ever use the correlation
as an input in my model.

1417
01:19:01,470 --> 01:19:02,410
I got it.

1418
01:19:02,410 --> 01:19:04,870
I got the distribution of
power, got distribution of fuel,

1419
01:19:04,870 --> 01:19:07,780
natural gas, put it all
together, compute correlation

1420
01:19:07,780 --> 01:19:09,450
and that's what I've got.

1421
01:19:09,450 --> 01:19:15,000
And this means that this is
really a correct approach.

1422
01:19:15,000 --> 01:19:16,900
And I don't need
to put it-- I don't

1423
01:19:16,900 --> 01:19:18,930
need to look for the
distribution that

1424
01:19:18,930 --> 01:19:20,030
has this property.

1425
01:19:20,030 --> 01:19:23,140
This property comes
for free, just

1426
01:19:23,140 --> 01:19:25,985
as a result of the
completely alternative way

1427
01:19:25,985 --> 01:19:27,780
of modeling this thing.

1428
01:19:27,780 --> 01:19:30,930
Now, what is the
negative side of this?

1429
01:19:30,930 --> 01:19:32,840
The negative side is
it's extremely difficult

1430
01:19:32,840 --> 01:19:34,960
to do it and maintain
it, because you

1431
01:19:34,960 --> 01:19:37,430
have to maintain the
information of every power plant

1432
01:19:37,430 --> 01:19:41,247
and was built, was retired,
and will be built, and so on.

1433
01:19:41,247 --> 01:19:43,080
Because you have to
look at the power prices

1434
01:19:43,080 --> 01:19:45,300
10 years, 20 years from now.

1435
01:19:45,300 --> 01:19:48,080
You have to know what's going
to be there, what kind of stack

1436
01:19:48,080 --> 01:19:49,490
you have to forecast.

1437
01:19:49,490 --> 01:19:51,260
That's a lot of
information to keep.

1438
01:19:51,260 --> 01:19:54,400
I mean, you have to have a big
organization to work on that,

1439
01:19:54,400 --> 01:19:58,560
to maintain it, to
build the model.

1440
01:19:58,560 --> 01:20:02,070
Because each region in this
country has a different market.

1441
01:20:02,070 --> 01:20:06,790
So you have to-- I mean,
it's a massive undertaking.

1442
01:20:06,790 --> 01:20:08,040
It takes years.

1443
01:20:08,040 --> 01:20:12,474
So it's not like you
can get it from the can.

1444
01:20:12,474 --> 01:20:14,100
It's expensive.

1445
01:20:14,100 --> 01:20:19,825
So I think that will be a
good point for me to stop.

1446
01:20:23,380 --> 01:20:25,971
If you have questions,
please let me know.

1447
01:20:29,944 --> 01:20:30,444
Questions?

1448
01:20:37,402 --> 01:20:40,510
I think it's music to my ears.

1449
01:20:40,510 --> 01:20:42,360
Thank you.