WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.830 at ocw.mit.edu. 00:00:21.630 --> 00:00:25.120 ANDREW LO: I hope you all had a good Columbus Day weekend. 00:00:25.120 --> 00:00:26.450 The stock market certainly did. 00:00:30.090 --> 00:00:31.790 Any questions from last time? 00:00:35.774 --> 00:00:36.280 No? 00:00:36.280 --> 00:00:37.540 OK. 00:00:37.540 --> 00:00:40.420 So what I want to do today is to continue 00:00:40.420 --> 00:00:44.260 talking about futures and forward contracts. 00:00:44.260 --> 00:00:46.990 Today we're going to finish up on these interesting 00:00:46.990 --> 00:00:49.480 instruments, with a couple of examples, 00:00:49.480 --> 00:00:54.970 and then with a specific method for pricing forward and futures 00:00:54.970 --> 00:00:56.720 contracts. 00:00:56.720 --> 00:01:00.400 So let me refresh your memory, it's been a week, so I know. 00:01:00.400 --> 00:01:06.140 So we're going to go back and look at a specific futures 00:01:06.140 --> 00:01:07.640 contract. 00:01:07.640 --> 00:01:09.710 And I'm going to take this contract 00:01:09.710 --> 00:01:14.150 and then try to talk a bit about how you might use contracts 00:01:14.150 --> 00:01:18.050 like this in hedging your risks, as well 00:01:18.050 --> 00:01:23.150 as in making certain kinds of bets, if you will. 00:01:23.150 --> 00:01:28.550 So remember that this contract is a contract that 00:01:28.550 --> 00:01:33.710 was issued on July 27, 2007-- 00:01:33.710 --> 00:01:35.940 so it was the middle of the summer-- 00:01:35.940 --> 00:01:40.890 for oil to be delivered in December. 00:01:40.890 --> 00:01:44.610 And there's a specific date in December 00:01:44.610 --> 00:01:50.190 where all oil futures contracts of this type will settle-- 00:01:50.190 --> 00:01:52.410 that is will come to maturity-- where 00:01:52.410 --> 00:01:55.290 the date is going to be specified in advance 00:01:55.290 --> 00:01:56.970 and everybody knows it. 00:01:56.970 --> 00:02:00.150 And so in July, when you buy this contract 00:02:00.150 --> 00:02:05.520 at a price of $75.06 per barrel, and each contract 00:02:05.520 --> 00:02:07.650 is for 1,000 barrels. 00:02:07.650 --> 00:02:10.680 When you quote "buy the contract," 00:02:10.680 --> 00:02:15.840 what that means is that you are agreeing today, July 27-- 00:02:15.840 --> 00:02:19.050 you are agreeing today, that come 00:02:19.050 --> 00:02:24.540 December you are going to buy 1,000 barrels at a price 00:02:24.540 --> 00:02:29.010 of $75.06 per barrel. 00:02:29.010 --> 00:02:31.380 So that's the agreement. 00:02:31.380 --> 00:02:36.330 And the party who is selling you the contract, the counterparty, 00:02:36.330 --> 00:02:39.570 is agreeing to provide you with that oil 00:02:39.570 --> 00:02:42.730 at that price in December. 00:02:42.730 --> 00:02:46.050 So the futures price is $75.06. 00:02:46.050 --> 00:02:49.620 And as we said last time, the current market 00:02:49.620 --> 00:02:55.540 price on July 27, 2007, that's called the spot price. 00:02:55.540 --> 00:02:59.850 The spot price may be higher or lower than the futures price, 00:02:59.850 --> 00:03:03.720 depending on what expectations are 00:03:03.720 --> 00:03:07.650 for what's going to happen with oil over the subsequent six 00:03:07.650 --> 00:03:10.870 month period. 00:03:10.870 --> 00:03:14.410 Now the initial margin, as I mentioned last time, 00:03:14.410 --> 00:03:16.870 was $4,050. 00:03:16.870 --> 00:03:19.270 The maintenance margin, the margin 00:03:19.270 --> 00:03:21.790 that you need to maintain. 00:03:21.790 --> 00:03:26.020 So that if the initial margin goes down in value, 00:03:26.020 --> 00:03:28.630 you have to actually put money back into your brokerage 00:03:28.630 --> 00:03:31.120 account, your margin account. 00:03:31.120 --> 00:03:35.600 And if you fall below that $3,000 threshold, 00:03:35.600 --> 00:03:40.980 you'll get a phone call, which is known as a margin call. 00:03:40.980 --> 00:03:43.890 Several weeks ago, somebody told me 00:03:43.890 --> 00:03:47.180 that they've been getting lots of phone calls 00:03:47.180 --> 00:03:50.710 all from the same person, a person called margin. 00:03:50.710 --> 00:03:56.070 And you know that can happen when markets go awry. 00:03:56.070 --> 00:04:01.290 Now again no cash changes hands today, 00:04:01.290 --> 00:04:05.220 because the value of the contract when it's struck 00:04:05.220 --> 00:04:10.150 is a zero NPV transaction. 00:04:10.150 --> 00:04:12.310 And how do you know it's zero NPV? 00:04:12.310 --> 00:04:16.360 Again, because if it's positive for one party, 00:04:16.360 --> 00:04:18.470 it's coming from the other party. 00:04:18.470 --> 00:04:20.320 So which party would you like to be? 00:04:20.320 --> 00:04:22.960 You'd like to be the party receiving that positive NPV, 00:04:22.960 --> 00:04:27.290 nobody wants to be the party that is losing the NPV. 00:04:27.290 --> 00:04:30.310 So the futures price will adjust, in order 00:04:30.310 --> 00:04:32.260 to make it zero NPV. 00:04:32.260 --> 00:04:36.250 In fact, that's what we mean when we say that it's zero NPV. 00:04:36.250 --> 00:04:41.781 It is the futures price that makes it so. 00:04:41.781 --> 00:04:44.240 So I'll give me an example. 00:04:44.240 --> 00:04:48.860 If it turns out that somebody suggests a futures price 00:04:48.860 --> 00:04:53.560 of $60 a barrel on that day. 00:04:53.560 --> 00:04:55.450 Lots and lots of people are going 00:04:55.450 --> 00:04:58.060 to want to buy that contract, because that's 00:04:58.060 --> 00:05:01.960 a good deal relative to where oil really should be. 00:05:01.960 --> 00:05:05.560 And that means lots of people are going to want to buy, 00:05:05.560 --> 00:05:11.231 but nobody's going to want to sell at that price of $60. 00:05:11.231 --> 00:05:13.480 So if everybody wants to buy and nobody wants to sell, 00:05:13.480 --> 00:05:16.080 what has to happen to the price? 00:05:16.080 --> 00:05:17.290 Exactly, it goes up. 00:05:17.290 --> 00:05:20.650 And it keeps going up until the number of buyers 00:05:20.650 --> 00:05:22.000 equals the number of sellers. 00:05:22.000 --> 00:05:28.180 That's the point at which it's a zero NPV transaction. 00:05:28.180 --> 00:05:31.450 Now let's take a look at what the payoff is 00:05:31.450 --> 00:05:38.320 of such a contract on day zero, in this case July 27, 2007, 00:05:38.320 --> 00:05:40.660 the contract is worth nothing. 00:05:40.660 --> 00:05:46.730 But if the futures price moves tomorrow, 00:05:46.730 --> 00:05:49.390 then the contract could actually have value. 00:05:49.390 --> 00:05:55.150 And a diagram of how that works is something like this. 00:05:55.150 --> 00:05:58.000 If the futures price-- 00:05:58.000 --> 00:06:01.434 not the spot-- the spot obviously will move also. 00:06:01.434 --> 00:06:03.100 But I'm talking about the futures price, 00:06:03.100 --> 00:06:07.150 because the futures contract is specified 00:06:07.150 --> 00:06:13.630 so that every day's worth of gains or losses in the futures 00:06:13.630 --> 00:06:17.740 contract relative to its price, you 00:06:17.740 --> 00:06:22.030 will have to either get paid, or you will have to pay. 00:06:22.030 --> 00:06:25.900 So this blue line shows you the payoff 00:06:25.900 --> 00:06:29.590 if you're holding a long position in one 00:06:29.590 --> 00:06:33.460 of these contracts-- if you bought a contract. 00:06:33.460 --> 00:06:36.490 If you sold the contract, then your payoff diagram 00:06:36.490 --> 00:06:38.500 is the dotted line. 00:06:38.500 --> 00:06:40.870 Now the blue line basically says that if the futures 00:06:40.870 --> 00:06:46.480 price goes above $75.06, then you make money. 00:06:46.480 --> 00:06:51.830 If the futures price goes below $75.06, you lose money. 00:06:51.830 --> 00:06:56.200 So when you buy a contract like this, 00:06:56.200 --> 00:07:01.680 it is as if you actually bought the oil. 00:07:01.680 --> 00:07:03.330 But you haven't really bought the oil, 00:07:03.330 --> 00:07:07.410 all you've done is to buy the right 00:07:07.410 --> 00:07:10.635 and obligation to purchase the oil in the future. 00:07:14.040 --> 00:07:16.120 Let me let me give you another example that will 00:07:16.120 --> 00:07:18.310 make this even more concrete. 00:07:18.310 --> 00:07:20.020 The only way to understand this-- 00:07:20.020 --> 00:07:24.580 because this is not a natural security for most of us-- 00:07:24.580 --> 00:07:28.030 stocks and bonds you might find natural, 00:07:28.030 --> 00:07:31.960 futures contracts are weird in that they 00:07:31.960 --> 00:07:36.500 have zero investment today, and so they're worthless today. 00:07:36.500 --> 00:07:39.850 But they're not worthless after the initial date when 00:07:39.850 --> 00:07:41.150 you enter into that agreement. 00:07:41.150 --> 00:07:45.550 So let's do an example Yesterday, 00:07:45.550 --> 00:07:50.920 you bought 10 December live cattle contracts on the Chicago 00:07:50.920 --> 00:07:56.260 Mercantile Exchange at a price of $0.7455 per pound. 00:07:56.260 --> 00:07:59.950 OK, so you basically bought some cows. 00:07:59.950 --> 00:08:05.074 And the contract size is 40,000 pounds of cow. 00:08:05.074 --> 00:08:06.490 I don't know how much cow that is, 00:08:06.490 --> 00:08:09.270 but even if you're on the Atkins diet that's plenty of cow. 00:08:09.270 --> 00:08:14.560 [LAUGHTER] And so what you have though in this contract 00:08:14.560 --> 00:08:19.240 is not the cows, but rather you have the obligation 00:08:19.240 --> 00:08:25.970 to buy the cows in December for a price of $0.7455 per pound, 00:08:25.970 --> 00:08:28.270 and there is 40,000 pounds of it. 00:08:28.270 --> 00:08:37.270 So the value of your position is the size 00:08:37.270 --> 00:08:40.120 of the contract, multiplied by the futures price, 00:08:40.120 --> 00:08:41.850 multiplied by the number of contracts. 00:08:41.850 --> 00:08:46.400 So it's $298,200. 00:08:46.400 --> 00:08:48.990 That's the size of your position, 00:08:48.990 --> 00:08:52.440 or sometimes that's also called the notional size. 00:08:52.440 --> 00:08:55.410 You've heard that term over the last few weeks-- notional. 00:08:55.410 --> 00:08:58.080 Well, this is an example of a notional. 00:08:58.080 --> 00:09:03.510 So you don't actually own $298,200 of anything, 00:09:03.510 --> 00:09:07.110 because of course, we've said that the contract is zero 00:09:07.110 --> 00:09:09.120 NPV when you enter into it. 00:09:09.120 --> 00:09:15.750 All you've done is to agree to buy 40,000 pounds of cow 00:09:15.750 --> 00:09:18.420 in December at a particular price. 00:09:18.420 --> 00:09:24.630 So the idea is that you control the notional amount 00:09:24.630 --> 00:09:29.790 of $298,200, and what you do specifically 00:09:29.790 --> 00:09:36.010 get is the profits and losses from that notional. 00:09:36.010 --> 00:09:37.600 So let's do an example. 00:09:37.600 --> 00:09:39.280 That was the position yesterday. 00:09:39.280 --> 00:09:41.110 No money changed hands. 00:09:41.110 --> 00:09:43.590 You got some initial margin that you had to put down, 00:09:43.590 --> 00:09:46.720 but that's really your money in a brokerage account. 00:09:46.720 --> 00:09:48.430 You're not giving it to anybody. 00:09:48.430 --> 00:09:51.670 It's safety money, it's collateral. 00:09:51.670 --> 00:09:53.310 Now today what happens? 00:09:53.310 --> 00:09:55.060 Let's suppose that today the futures price 00:09:55.060 --> 00:09:58.360 closes at $0.7435. 00:09:58.360 --> 00:10:03.750 All right, it's just gone down by 2/10 of a cent. 00:10:06.880 --> 00:10:10.060 The value of cattle has gone down. 00:10:10.060 --> 00:10:14.320 Your holding long this cattle contract, 00:10:14.320 --> 00:10:20.350 maybe you're a restaurateur, you have a chain of steakhouses, 00:10:20.350 --> 00:10:23.390 and so you need to buy cattle on a regular basis. 00:10:23.390 --> 00:10:26.770 So the price of cattle just went down. 00:10:26.770 --> 00:10:29.051 Did you make money or lose money? 00:10:29.051 --> 00:10:31.540 Yeah, you lost, if you're long. 00:10:31.540 --> 00:10:33.880 On the other hand, if you're a cattle farmer 00:10:33.880 --> 00:10:37.180 and you sold the contract, you did a good thing, 00:10:37.180 --> 00:10:41.170 because you locked in the price of $0.7455, 00:10:41.170 --> 00:10:42.610 and now the price went down. 00:10:42.610 --> 00:10:46.480 So let's calculate what the value of the notional size 00:10:46.480 --> 00:10:47.410 of the position is. 00:10:47.410 --> 00:10:51.130 It's $297,400. 00:10:51.130 --> 00:10:55.280 That yields a loss of $800. 00:10:55.280 --> 00:10:56.990 So you know what happens today? 00:10:56.990 --> 00:11:01.880 Today, your broker will deduct $800 from your account, 00:11:01.880 --> 00:11:06.380 from your margin account, and take that $800 00:11:06.380 --> 00:11:10.250 and put it into the cattle farmer's account. 00:11:10.250 --> 00:11:14.400 So now he has the $800. 00:11:14.400 --> 00:11:18.600 Now, if the day after, if tomorrow, it 00:11:18.600 --> 00:11:20.100 turns out that the price of cattle 00:11:20.100 --> 00:11:24.360 goes up by 2/10 of a cent, it goes back to $0.7455. 00:11:24.360 --> 00:11:26.600 You know what happens? 00:11:26.600 --> 00:11:28.280 You get $800 back. 00:11:28.280 --> 00:11:34.600 Now the cattle farmer loses that $800 and gives it back to you. 00:11:34.600 --> 00:11:41.520 You see this way you always settle up every day. 00:11:41.520 --> 00:11:46.500 So if for some reason the cattle farmer ends up going bankrupt, 00:11:46.500 --> 00:11:50.070 and isn't able to deliver any cattle to you, 00:11:50.070 --> 00:11:56.940 then you're at out at most one day's worth of movements. 00:11:56.940 --> 00:12:00.480 And that's one of the reasons why futures markets and futures 00:12:00.480 --> 00:12:04.140 brokers are so careful about closing down 00:12:04.140 --> 00:12:07.430 accounts that don't meet their margin requirements. 00:12:07.430 --> 00:12:11.130 It's because they don't want to have credit risk lingering, 00:12:11.130 --> 00:12:14.670 growing, and unknown. 00:12:14.670 --> 00:12:18.450 The first moment that you do not make a margin call-- 00:12:18.450 --> 00:12:21.600 you do not deposit the requisite margin-- the first time 00:12:21.600 --> 00:12:24.930 that happens, they have the right, 00:12:24.930 --> 00:12:29.400 which they exercise always, to close down your position. 00:12:29.400 --> 00:12:32.040 You're out of the game, and that's the end. 00:12:32.040 --> 00:12:35.760 So it reduces dramatically, the amount of credit 00:12:35.760 --> 00:12:37.950 risk that either counterparty has. 00:12:37.950 --> 00:12:40.620 I don't have to trust you that three months from now 00:12:40.620 --> 00:12:44.230 you're going to actually have 40,000 pounds of cow for me. 00:12:44.230 --> 00:12:48.160 All I need to do is to make sure that this contract settles 00:12:48.160 --> 00:12:49.660 every day. 00:12:49.660 --> 00:12:54.370 And the uncertainty then gets resolved day by day, 00:12:54.370 --> 00:13:00.130 but your credit risk is very well managed, and mine is too. 00:13:00.130 --> 00:13:03.310 So this is a very important innovation. 00:13:03.310 --> 00:13:06.280 It's very different from a forward contract, in the sense 00:13:06.280 --> 00:13:10.251 that forward contracts contain enormous amounts of credit risk 00:13:10.251 --> 00:13:10.750 right. 00:13:10.750 --> 00:13:13.030 Because once we enter into a forward, 00:13:13.030 --> 00:13:15.260 that's just like a futures contract, 00:13:15.260 --> 00:13:19.810 but the difference is that we don't exchange any money ever 00:13:19.810 --> 00:13:22.120 until the settlement date. 00:13:22.120 --> 00:13:26.350 And by that point you could be so far out of the money, 00:13:26.350 --> 00:13:30.670 you could be so far in debt to me, 00:13:30.670 --> 00:13:33.100 as well as to other creditors, that you just 00:13:33.100 --> 00:13:34.510 can't afford to pay. 00:13:34.510 --> 00:13:36.760 And so I'm stuck with this piece of paper that 00:13:36.760 --> 00:13:39.520 says you're going to give me 40,000 pounds of cattle, 00:13:39.520 --> 00:13:42.820 and you can't even afford to buy me a steak dinner. 00:13:42.820 --> 00:13:45.400 That's a problem. 00:13:45.400 --> 00:13:50.090 So this futures exchange is a beautiful thing. 00:13:50.090 --> 00:13:52.280 It reduces credit risk. 00:13:52.280 --> 00:13:56.390 It also encourages liquidity, it encourages trading. 00:13:56.390 --> 00:13:57.010 Why? 00:13:57.010 --> 00:14:01.360 Because at any point in time, on any given day between now 00:14:01.360 --> 00:14:03.291 and December, if you decide that you 00:14:03.291 --> 00:14:05.040 want to get out of the restaurant business 00:14:05.040 --> 00:14:07.804 and you don't want this contract any more, 00:14:07.804 --> 00:14:08.720 you can get out of it. 00:14:08.720 --> 00:14:09.560 Poof. 00:14:09.560 --> 00:14:13.500 You just get out of it by doing an opposite transaction. 00:14:13.500 --> 00:14:16.154 So if you bought a contract for December, 00:14:16.154 --> 00:14:18.320 you know what you do when you want to get out of it? 00:14:18.320 --> 00:14:20.390 You sell a contract for December. 00:14:20.390 --> 00:14:21.750 You literally sell. 00:14:21.750 --> 00:14:25.160 So it's actually duplicated transaction, 00:14:25.160 --> 00:14:27.080 but it's of the opposite sign. 00:14:27.080 --> 00:14:30.620 And so they cancel out, because you're going to get delivery, 00:14:30.620 --> 00:14:34.460 and you will provide delivery, and those will cancel out. 00:14:34.460 --> 00:14:37.300 Yeah, Justin. 00:14:37.300 --> 00:14:40.180 AUDIENCE: The price of oil has been going down lately. 00:14:40.180 --> 00:14:42.420 So let's say I had a long position in oil, 00:14:42.420 --> 00:14:45.320 and then I found out that I was going to really lose 00:14:45.320 --> 00:14:47.650 half of that money, and I decided just 00:14:47.650 --> 00:14:50.532 to forego my margin. 00:14:50.532 --> 00:14:52.460 What else would I have to pay? 00:14:52.460 --> 00:14:56.660 ANDREW LO: Well, first of all, you 00:14:56.660 --> 00:15:00.860 are liable for all of the losses, not just the margin. 00:15:00.860 --> 00:15:03.440 So the margin account is not meant 00:15:03.440 --> 00:15:05.420 to be a non-recourse loan. 00:15:05.420 --> 00:15:07.340 They will go after your assets. 00:15:07.340 --> 00:15:09.230 Now you could declare bankruptcy, 00:15:09.230 --> 00:15:13.670 personal bankruptcy, and get protection under Chapter 7. 00:15:13.670 --> 00:15:16.520 But that will hurt your credit ratings 00:15:16.520 --> 00:15:19.550 and all other nasty things will happen to you if that occurs. 00:15:19.550 --> 00:15:21.710 AUDIENCE: So when you're saying that they close out 00:15:21.710 --> 00:15:26.930 your account, when your margins down. 00:15:26.930 --> 00:15:29.450 So they close it out, but if your losses 00:15:29.450 --> 00:15:31.860 are higher than your margin was anyway, 00:15:31.860 --> 00:15:35.450 so you're still liable for those in addition to [INAUDIBLE].. 00:15:35.450 --> 00:15:37.820 ANDREW LO: So you make a good point. 00:15:37.820 --> 00:15:40.670 Is it generally possible that your losses are greater 00:15:40.670 --> 00:15:41.970 than the amount of margin? 00:15:41.970 --> 00:15:44.270 So in that case, who gets left holding the bag? 00:15:44.270 --> 00:15:45.920 You know who gets left holding the bag? 00:15:45.920 --> 00:15:48.650 That blue box in the middle, the Futures Clearing corporation. 00:15:48.650 --> 00:15:52.160 But the reason that they establish a particular level 00:15:52.160 --> 00:15:56.750 of margin is to be able to ensure that that's 00:15:56.750 --> 00:15:58.400 a very unlikely event. 00:15:58.400 --> 00:16:03.080 And it goes back to what are the likely daily swings 00:16:03.080 --> 00:16:04.610 in the futures price. 00:16:04.610 --> 00:16:06.980 If you put enough margin in your account, 00:16:06.980 --> 00:16:10.910 so that I can be sure that 99% of the time you 00:16:10.910 --> 00:16:15.030 can cover the daily swing, then I don't have to worry. 00:16:15.030 --> 00:16:18.800 Now, of course, if we had a day like last Friday, or on Monday, 00:16:18.800 --> 00:16:21.660 you know that's pretty outrageous. 00:16:21.660 --> 00:16:24.560 That's one of the reasons why a number of futures exchanges 00:16:24.560 --> 00:16:26.080 have increased their margin levels. 00:16:26.080 --> 00:16:28.880 It's because the daily swings have gotten much bigger. 00:16:28.880 --> 00:16:31.130 But as long as they can cover the one day movement, 00:16:31.130 --> 00:16:34.689 they don't have to go after your home or your other assets. 00:16:34.689 --> 00:16:35.230 Yeah, Dennis. 00:16:35.230 --> 00:16:37.146 AUDIENCE: You said if I bought a contract now, 00:16:37.146 --> 00:16:39.230 then I just have to sit on the same contract. 00:16:39.230 --> 00:16:41.730 What happens if I bought at a $1.00, it's at $0.50 now, 00:16:41.730 --> 00:16:43.641 I can't sell at a $1.00. 00:16:43.641 --> 00:16:44.390 ANDREW LO: Oh, no. 00:16:44.390 --> 00:16:46.017 You certainly cannot sell at the $1.00, 00:16:46.017 --> 00:16:47.100 you have to sell at $0.50. 00:16:47.100 --> 00:16:47.943 Says 00:16:47.943 --> 00:16:49.980 AUDIENCE: So I'm not really out of the position. 00:16:49.980 --> 00:16:51.605 ANDREW LO: You are out of the position, 00:16:51.605 --> 00:16:55.100 because you don't have a claim, or you 00:16:55.100 --> 00:17:00.105 don't have a commitment to enter into that trade in December. 00:17:00.105 --> 00:17:02.480 That's what I mean when I say you're out of the position. 00:17:02.480 --> 00:17:04.909 You also happen to be out of money in your example. 00:17:04.909 --> 00:17:07.460 [LAUGHTER] In other words, you lost $0.50. 00:17:07.460 --> 00:17:09.980 That's not coming back. 00:17:09.980 --> 00:17:12.680 But what it means to sell is that you 00:17:12.680 --> 00:17:15.440 bought a contract that says in December you're 00:17:15.440 --> 00:17:19.109 going to buy 40,000 of cattle-- 00:17:19.109 --> 00:17:20.910 you're committed to doing that. 00:17:20.910 --> 00:17:23.980 Now if on the other hand, the next day 00:17:23.980 --> 00:17:26.357 you decide you want to get out of that commitment-- 00:17:26.357 --> 00:17:27.940 the way to get out of it is not to try 00:17:27.940 --> 00:17:29.440 to contact the counterparty and say, 00:17:29.440 --> 00:17:31.300 would you mind canceling my trade. 00:17:31.300 --> 00:17:35.150 The way to do it is to simply sell a commitment at 40,000 00:17:35.150 --> 00:17:37.520 pounds of cattle for December. 00:17:37.520 --> 00:17:39.490 So your commitment to buy and your commitment 00:17:39.490 --> 00:17:41.680 to sell, basically cancel each other out. 00:17:41.680 --> 00:17:44.710 So on settlement date, the Futures Clearing corporation 00:17:44.710 --> 00:17:48.550 will net out all of these buys and sells, 00:17:48.550 --> 00:17:51.070 and the net amount will be transacted 00:17:51.070 --> 00:17:53.612 between the providers of the cattle 00:17:53.612 --> 00:17:54.820 and the buyers of the cattle. 00:17:54.820 --> 00:17:56.320 AUDIENCE: So this means that there's 00:17:56.320 --> 00:17:57.430 no physical delivery then. 00:17:57.430 --> 00:17:58.680 ANDREW LO: That's right. 00:17:58.680 --> 00:18:01.120 So that's an example where if you bought and sold, 00:18:01.120 --> 00:18:03.040 then you would be netted out and you would not 00:18:03.040 --> 00:18:05.270 have a physical delivery. 00:18:05.270 --> 00:18:06.376 Yeah, [INAUDIBLE]. 00:18:06.376 --> 00:18:10.120 AUDIENCE: So if the margin is just 00:18:10.120 --> 00:18:12.470 a fund for exchanging commodities, 00:18:12.470 --> 00:18:15.175 what does the Futures Clearing corp-- what do they make? 00:18:15.175 --> 00:18:17.147 Is it a percentage of-- 00:18:17.147 --> 00:18:19.120 how do they make money? 00:18:19.120 --> 00:18:21.610 ANDREW LO: Well, their job is really not to make money, 00:18:21.610 --> 00:18:25.090 but to create an exchange for its members. 00:18:25.090 --> 00:18:28.070 So many exchanges are not for profit. 00:18:28.070 --> 00:18:30.640 Some of them are for profit, but the objective 00:18:30.640 --> 00:18:32.110 of the Futures Clearing corporation 00:18:32.110 --> 00:18:33.970 is really not to make a lot of money. 00:18:33.970 --> 00:18:36.310 What they're trying to do is just create a market 00:18:36.310 --> 00:18:39.220 and let people who want to trade with each other, 00:18:39.220 --> 00:18:42.340 trade freely and efficiently. 00:18:42.340 --> 00:18:44.980 They will charge perhaps a small transaction 00:18:44.980 --> 00:18:46.750 fee, that you have to pay in order 00:18:46.750 --> 00:18:48.550 to support the operations. 00:18:48.550 --> 00:18:51.040 But they're not trying to make tons of profits off of that 00:18:51.040 --> 00:18:52.090 necessarily. 00:18:52.090 --> 00:18:54.070 Now they may be trying to make profits off 00:18:54.070 --> 00:18:57.640 of other activities, but the objective of the Clearing 00:18:57.640 --> 00:19:00.380 Corporation itself is not to make tons of profit, 00:19:00.380 --> 00:19:02.380 It's really just to provide a stable environment 00:19:02.380 --> 00:19:03.940 where people can transact. 00:19:03.940 --> 00:19:07.870 And in some cases, the members of the exchange 00:19:07.870 --> 00:19:09.850 own the Clearing Corporation. 00:19:09.850 --> 00:19:12.790 So it's their own dollars that support 00:19:12.790 --> 00:19:15.865 the actual physical operations of the organization. 00:19:15.865 --> 00:19:17.656 AUDIENCE: And just going back to that point 00:19:17.656 --> 00:19:20.400 you had about no physical delivery. 00:19:20.400 --> 00:19:22.105 Two or three weeks ago, the price 00:19:22.105 --> 00:19:25.340 of oil spiked, [INAUDIBLE] I think 00:19:25.340 --> 00:19:27.152 the way I read in the papers, was people 00:19:27.152 --> 00:19:30.225 were trying to sell it, not buy it, because otherwise they 00:19:30.225 --> 00:19:31.420 would get physical delivery. 00:19:31.420 --> 00:19:35.250 So do you recall that? 00:19:35.250 --> 00:19:37.220 ANDREW LO: I recall the spike. 00:19:37.220 --> 00:19:40.620 I certainly don't recall the logic about physical delivery. 00:19:40.620 --> 00:19:43.710 I mean it could be that there are a number of people 00:19:43.710 --> 00:19:47.180 who are long the oil contracts that basically 00:19:47.180 --> 00:19:48.710 want it to be cash settled. 00:19:48.710 --> 00:19:50.460 And the way that they have it cash settled 00:19:50.460 --> 00:19:53.400 at a particular point in time, before settlement date, 00:19:53.400 --> 00:19:55.357 is they close out their positions. 00:19:55.357 --> 00:19:56.940 And so by closing out their positions, 00:19:56.940 --> 00:19:58.680 they basically reverse the trade, 00:19:58.680 --> 00:20:03.402 and that would actually push down the oil price. 00:20:03.402 --> 00:20:05.610 So maybe the reverse argument, a lot of short sellers 00:20:05.610 --> 00:20:08.670 were trying to argue that oil was going to go down, 00:20:08.670 --> 00:20:11.810 and they wanted to cover their position, so they bought. 00:20:11.810 --> 00:20:14.820 In any case, you don't have to take physical delivery 00:20:14.820 --> 00:20:17.730 if you specify to your broker that you want all of this 00:20:17.730 --> 00:20:19.482 to be cash settled. 00:20:19.482 --> 00:20:20.364 Yeah? 00:20:20.364 --> 00:20:21.700 AUDIENCE: Two part question. 00:20:21.700 --> 00:20:23.500 Would you say that the credit risk involved 00:20:23.500 --> 00:20:26.440 in a forward contract is somewhat similar to the credit 00:20:26.440 --> 00:20:29.530 risk in credit default swaps? 00:20:29.530 --> 00:20:32.590 And if so, is there something analogous to a credit default 00:20:32.590 --> 00:20:35.702 swap that's similar to a futures contract? 00:20:35.702 --> 00:20:37.660 ANDREW LO: So the answer to your first question 00:20:37.660 --> 00:20:42.280 is yes, because a credit default swap contract is basically 00:20:42.280 --> 00:20:44.320 a kind of a forward contract. 00:20:44.320 --> 00:20:46.690 It does involve intermediate payments, 00:20:46.690 --> 00:20:51.064 but if it turns out that the credit changes dramatically, 00:20:51.064 --> 00:20:52.480 those intermediate payments either 00:20:52.480 --> 00:20:55.120 may be too much or not enough to cover the underlying 00:20:55.120 --> 00:20:56.390 value of the contract. 00:20:56.390 --> 00:20:58.420 And after you strike a credit default swap, 00:20:58.420 --> 00:21:01.330 it will take on value. 00:21:01.330 --> 00:21:05.230 As for an exchange, what a wonderful idea. 00:21:05.230 --> 00:21:07.660 That is exactly what's being proposed now. 00:21:07.660 --> 00:21:09.880 It hasn't been done yet, but there 00:21:09.880 --> 00:21:11.710 have been a number of proposals to set up 00:21:11.710 --> 00:21:15.580 exactly a structure like this for credit default swaps. 00:21:15.580 --> 00:21:20.380 In order to do that, you have to standardize those contracts, 00:21:20.380 --> 00:21:23.440 and you have to be able to do the paperwork in a relatively 00:21:23.440 --> 00:21:24.640 efficient manner. 00:21:24.640 --> 00:21:27.910 And so that's actually being discussed, debated, 00:21:27.910 --> 00:21:30.250 and I think that there's a proposal by the Chicago 00:21:30.250 --> 00:21:32.950 Mercantile Exchange to start doing that. 00:21:32.950 --> 00:21:36.070 If you do start doing that, you will see that market growing 00:21:36.070 --> 00:21:39.610 even bigger than it is today, and at the same time, 00:21:39.610 --> 00:21:42.280 the risks are actually going to decrease. 00:21:42.280 --> 00:21:45.400 Because with daily settlement of credit default swaps, 00:21:45.400 --> 00:21:47.860 just like with futures contracts, all you need 00:21:47.860 --> 00:21:55.280 is one day margin in order to eliminate 99% of the problems. 00:21:55.280 --> 00:22:05.534 AUDIENCE: [INAUDIBLE] 00:22:05.534 --> 00:22:07.450 ANDREW LO: So let me-- that's a good question. 00:22:07.450 --> 00:22:09.850 Let me now talk about how to price futures 00:22:09.850 --> 00:22:12.850 and we'll take in interest rates explicitly. 00:22:12.850 --> 00:22:17.490 So the question is what determines either a forward 00:22:17.490 --> 00:22:19.620 or a futures price? 00:22:19.620 --> 00:22:21.750 You now know what a futures price is, right? 00:22:21.750 --> 00:22:25.080 It's the price at which you're willing to do 00:22:25.080 --> 00:22:27.286 a future transaction. 00:22:27.286 --> 00:22:28.410 What determines that price? 00:22:28.410 --> 00:22:30.750 We say supply and demand and the market, 00:22:30.750 --> 00:22:34.750 but is there some logic that we can give to this process? 00:22:34.750 --> 00:22:36.240 And the answer is yes, we're going 00:22:36.240 --> 00:22:38.040 to use the exact same argument that we use 00:22:38.040 --> 00:22:39.900 for pricing everything else. 00:22:39.900 --> 00:22:44.160 We're going to come up with two identical cash flows. 00:22:44.160 --> 00:22:48.450 And two assets that have identical cash flows 00:22:48.450 --> 00:22:51.246 have to have the same what? 00:22:51.246 --> 00:22:52.070 AUDIENCE: Price. 00:22:52.070 --> 00:22:55.430 ANDREW LO: Price, value, right. 00:22:55.430 --> 00:23:00.110 So for now, I'm going to actually ignore the difference 00:23:00.110 --> 00:23:02.030 between futures and forwards. 00:23:02.030 --> 00:23:06.530 The only difference is the back and forth amount of money 00:23:06.530 --> 00:23:08.810 that we give to each other, and therefore, 00:23:08.810 --> 00:23:12.410 the accumulated interest or foregone interest 00:23:12.410 --> 00:23:16.040 that we pay when we put our money back and forth 00:23:16.040 --> 00:23:17.870 into each other's accounts. 00:23:17.870 --> 00:23:19.799 So let me abstract from that and-- you 00:23:19.799 --> 00:23:21.590 know if you're interested, you can actually 00:23:21.590 --> 00:23:25.250 see the derivation of that in recitation. 00:23:25.250 --> 00:23:27.740 I want to focus on the bigger question 00:23:27.740 --> 00:23:33.140 of how these things are priced with respect to other prices. 00:23:33.140 --> 00:23:36.350 So let me start with some notation. 00:23:36.350 --> 00:23:39.335 I've got a particular contract, let's say a futures 00:23:39.335 --> 00:23:41.240 or a forward contract. 00:23:41.240 --> 00:23:44.780 And I've also got the spot price of the asset 00:23:44.780 --> 00:23:46.170 at a point in time. 00:23:46.170 --> 00:23:49.220 So I'm going to let St denote my spot price. 00:23:49.220 --> 00:23:53.470 I'm going to let F of little t big T 00:23:53.470 --> 00:23:55.610 determine the forward price. 00:23:55.610 --> 00:23:59.619 And H of little t big T, the futures price. 00:23:59.619 --> 00:24:01.160 And for now, I'm going to just assume 00:24:01.160 --> 00:24:03.860 that H and F are pretty close. 00:24:03.860 --> 00:24:06.700 Now notice that when I write down 00:24:06.700 --> 00:24:08.960 a futures price or a forward price, 00:24:08.960 --> 00:24:11.360 I've got two sub-indexes. 00:24:11.360 --> 00:24:13.160 I've got little t and big T. The reason 00:24:13.160 --> 00:24:17.780 I need two is that for every forward or futures contract, 00:24:17.780 --> 00:24:20.600 there are two dates you need to worry about. 00:24:20.600 --> 00:24:24.410 The date at which you are pricing the contract, namely 00:24:24.410 --> 00:24:27.620 today, and the settlement date. 00:24:27.620 --> 00:24:29.460 So you need to have those two indexes. 00:24:29.460 --> 00:24:32.180 So right away we know that this contract 00:24:32.180 --> 00:24:35.450 is a little bit more complicated than say a stock, 00:24:35.450 --> 00:24:38.000 where there is no settlement date. 00:24:41.150 --> 00:24:42.914 So I want to go back to a comment that 00:24:42.914 --> 00:24:45.080 was made by one of you when we first started talking 00:24:45.080 --> 00:24:46.370 about futures and forwards. 00:24:46.370 --> 00:24:49.310 And the comment was why go to the trouble of using 00:24:49.310 --> 00:24:50.810 these contracts, when you could just 00:24:50.810 --> 00:24:53.810 buy the asset itself and hold onto it. 00:24:53.810 --> 00:24:56.600 If you need oil in December, in order 00:24:56.600 --> 00:24:58.550 to make sure you have oil in December, 00:24:58.550 --> 00:25:00.560 why don't you just buy it in October 00:25:00.560 --> 00:25:01.810 and hold it for two months. 00:25:01.810 --> 00:25:03.950 Then you have it in December. 00:25:03.950 --> 00:25:06.650 And in fact, that's exactly what we're 00:25:06.650 --> 00:25:10.820 going to do to figure out what the appropriate price is 00:25:10.820 --> 00:25:14.250 of the specific futures or forward contract. 00:25:14.250 --> 00:25:15.680 So here we go. 00:25:15.680 --> 00:25:18.170 I'm going to do my exact same analysis 00:25:18.170 --> 00:25:20.300 that I've done many times before, 00:25:20.300 --> 00:25:23.660 when we tried to price bonds, and stocks, 00:25:23.660 --> 00:25:27.380 and other basic securities. 00:25:27.380 --> 00:25:29.690 The left hand column here is going 00:25:29.690 --> 00:25:33.400 to be the cash flows associated with 00:25:33.400 --> 00:25:36.140 a typical forward contract. 00:25:36.140 --> 00:25:37.760 So a forward contract is one way. 00:25:37.760 --> 00:25:41.330 You enter into the contract, let's say at date zero. 00:25:41.330 --> 00:25:43.540 And you pay nothing for the contract right, 00:25:43.540 --> 00:25:46.070 this is a zero NPV transaction. 00:25:46.070 --> 00:25:49.820 And you are long the forward contract, 00:25:49.820 --> 00:25:55.220 with the forward price F of 0,T. 00:25:55.220 --> 00:25:58.280 The only cash flow that occurs with a forward contract 00:25:58.280 --> 00:26:00.350 is on settlement date. 00:26:00.350 --> 00:26:05.690 And on settlement date, you've agreed to pay F of 0,T 00:26:05.690 --> 00:26:08.840 for delivery of whatever it is that you bought the forward 00:26:08.840 --> 00:26:09.620 contract on. 00:26:12.830 --> 00:26:18.080 So the only cash flow that comes out of a forward contract 00:26:18.080 --> 00:26:21.730 is this F right here. 00:26:21.730 --> 00:26:23.800 Everybody see that? 00:26:23.800 --> 00:26:29.860 Nothing up my sleeve, it's very simple calculation. 00:26:29.860 --> 00:26:32.290 Now, I want you to look at the right hand column, which 00:26:32.290 --> 00:26:34.540 is going to be less simple. 00:26:34.540 --> 00:26:36.820 The right hand column, I want you 00:26:36.820 --> 00:26:40.420 to imagine doing the following. 00:26:40.420 --> 00:26:45.430 I want you to imagine buying the commodity at date zero. 00:26:45.430 --> 00:26:49.490 However, I don't want you to use any money. 00:26:49.490 --> 00:26:52.330 I want you to buy it with no money down. 00:26:52.330 --> 00:26:56.720 That's the start of a scam, it sounds like it, 00:26:56.720 --> 00:26:59.360 but I promise you it's not. 00:26:59.360 --> 00:27:02.135 So the way you're going to buy the commodity 00:27:02.135 --> 00:27:05.750 is you have to pay the price, the spot price. 00:27:05.750 --> 00:27:08.150 And the spot prices is S sub 0 You don't 00:27:08.150 --> 00:27:12.980 have S sub 0, so borrow it. 00:27:12.980 --> 00:27:14.900 Now, I'm going to abstract from credit 00:27:14.900 --> 00:27:17.862 risk, which I know is on everybody's minds today. 00:27:17.862 --> 00:27:19.820 But let's suppose that you're all good credits, 00:27:19.820 --> 00:27:23.420 so I'm not worried about loaning you the money at the risk 00:27:23.420 --> 00:27:25.560 free rate. 00:27:25.560 --> 00:27:28.820 So now you've borrowed S sub 0 dollars, 00:27:28.820 --> 00:27:33.800 and then you spent it right away buying the asset. 00:27:33.800 --> 00:27:40.730 So as of date zero, in the right hand column, you own the asset. 00:27:40.730 --> 00:27:49.810 Now you have to wait T periods, and while you wait 00:27:49.810 --> 00:27:51.370 you may have some costs. 00:27:51.370 --> 00:27:56.000 For example, if the asset that you bought is gasoline, 00:27:56.000 --> 00:27:59.206 well you've got to store it in just the right way. 00:27:59.206 --> 00:28:00.580 You probably don't want to put it 00:28:00.580 --> 00:28:02.980 next to your furnace in the basement. 00:28:02.980 --> 00:28:05.830 You probably want to put it in a cool place, isolated, 00:28:05.830 --> 00:28:07.550 and so on and so forth. 00:28:07.550 --> 00:28:11.440 On the other hand, if what you bought is pork bellies, 00:28:11.440 --> 00:28:14.890 you probably want to put that in a freezer compartment, as 00:28:14.890 --> 00:28:18.280 opposed to in your garage. 00:28:18.280 --> 00:28:22.900 So you might have to pay costs for storing. 00:28:22.900 --> 00:28:26.530 And at the end of that time T, you 00:28:26.530 --> 00:28:29.000 have to pay interest on your loan. 00:28:29.000 --> 00:28:31.149 So you borrowed S sub 0 dollars, you 00:28:31.149 --> 00:28:33.440 don't get that for free, you got to pay interest on it. 00:28:33.440 --> 00:28:34.856 This is a question about interest, 00:28:34.856 --> 00:28:37.910 so you've got to pay interest on that money. 00:28:37.910 --> 00:28:42.355 And so you have to pay back at this point T-- 00:28:42.355 --> 00:28:44.230 you have to pay back the money you borrowed-- 00:28:44.230 --> 00:28:47.830 S sub 0, 1 plus R to the capital T, 00:28:47.830 --> 00:28:50.150 plus whatever your storage costs are. 00:28:50.150 --> 00:28:54.220 But I'm going to allow that having the asset 00:28:54.220 --> 00:28:57.700 around might be kind of convenient. 00:28:57.700 --> 00:29:01.090 There might be a benefit to having the asset around-- 00:29:01.090 --> 00:29:02.530 a convenience yield. 00:29:02.530 --> 00:29:05.440 Maybe if you need to use it sooner, you have it there. 00:29:05.440 --> 00:29:11.290 And having it there saves you a little bit of trouble in order 00:29:11.290 --> 00:29:13.060 to be able to get whatever it is you 00:29:13.060 --> 00:29:15.820 need to get done with that underlying asset. 00:29:15.820 --> 00:29:20.740 So I'm going to deduct from my cumulative storage costs 00:29:20.740 --> 00:29:23.260 any convenience yield-- 00:29:23.260 --> 00:29:27.310 that's future speak for any kind of benefits 00:29:27.310 --> 00:29:31.150 that you get from holding onto the physical asset. 00:29:31.150 --> 00:29:35.020 So your net storage costs are given here-- 00:29:35.020 --> 00:29:37.705 that's what you pay at the end of T periods. 00:29:40.420 --> 00:29:45.610 I argue that these two cash flows give you 00:29:45.610 --> 00:29:50.120 the exact same value of the asset. 00:29:50.120 --> 00:29:52.900 In other words, in both cases you 00:29:52.900 --> 00:29:56.570 happen to have the asset at the time T. 00:29:56.570 --> 00:30:02.140 So these two contracts have to have the same value 00:30:02.140 --> 00:30:07.660 because they offer the same set of cash flows, 00:30:07.660 --> 00:30:10.960 in terms of the underlying commodity. 00:30:10.960 --> 00:30:14.550 You get the commodity in both cases. 00:30:14.550 --> 00:30:17.930 So another way of thinking about it is if your objective is 00:30:17.930 --> 00:30:23.450 to have 40,000 pounds of cattle in December, both of these 00:30:23.450 --> 00:30:26.270 will get you to the exact same point. 00:30:26.270 --> 00:30:32.110 Both of these costs you nothing on date zero. 00:30:32.110 --> 00:30:35.950 And therefore, if they cost you nothing, 00:30:35.950 --> 00:30:40.490 and they give you the same outcome at the end, 00:30:40.490 --> 00:30:43.390 they've got to sell for the same price. 00:30:43.390 --> 00:30:49.101 So this has to equal this. 00:30:49.101 --> 00:30:49.600 That's it. 00:30:49.600 --> 00:30:51.220 That's the simple argument. 00:30:51.220 --> 00:30:57.020 And the counter argument or proof that this has to be true 00:30:57.020 --> 00:30:59.030 is-- let's assume it's not. 00:30:59.030 --> 00:31:02.660 Let's assume that this is a lot bigger than this. 00:31:02.660 --> 00:31:07.512 Well, if this is bigger than this, then what should you do? 00:31:07.512 --> 00:31:08.012 What? 00:31:08.012 --> 00:31:12.112 AUDIENCE: [INAUDIBLE] 00:31:12.112 --> 00:31:12.820 ANDREW LO: Right. 00:31:12.820 --> 00:31:14.410 Which one? 00:31:14.410 --> 00:31:16.360 Which one? 00:31:16.360 --> 00:31:19.360 Sell the forward contract, and then 00:31:19.360 --> 00:31:22.892 buy this thing, whatever it is, do this. 00:31:22.892 --> 00:31:24.100 Now what if it's the reverse? 00:31:24.100 --> 00:31:26.260 What if this is bigger than this? 00:31:26.260 --> 00:31:32.320 Then buy the forward, and then do the opposite of this. 00:31:32.320 --> 00:31:33.460 Flip it around. 00:31:33.460 --> 00:31:38.050 Short sell the asset if you can, and then take the money 00:31:38.050 --> 00:31:42.040 and lend it out at interest rate r, and dot dot dot, 00:31:42.040 --> 00:31:44.980 you follow the logic. 00:31:44.980 --> 00:31:50.170 So that gives us a relationship between the forward price 00:31:50.170 --> 00:31:51.920 and other stuff. 00:31:51.920 --> 00:31:53.320 And what is the other stuff? 00:31:53.320 --> 00:31:56.680 The forward price has to be related 00:31:56.680 --> 00:32:02.560 to the spot price, the interest rate, the time to settlement, 00:32:02.560 --> 00:32:07.030 and any other weird things about the commodity that 00:32:07.030 --> 00:32:08.740 may affect the value of it. 00:32:08.740 --> 00:32:14.436 Like the storage costs or the convenience yield-- 00:32:14.436 --> 00:32:17.760 you've got to factor that in. 00:32:17.760 --> 00:32:19.980 So this is the relationship that tells you how 00:32:19.980 --> 00:32:22.140 to price a forward contract. 00:32:22.140 --> 00:32:25.470 Now a futures contract is almost like a forward. 00:32:25.470 --> 00:32:28.450 The only difference is the interest differential 00:32:28.450 --> 00:32:32.190 on a daily basis, where you actually are moving money back 00:32:32.190 --> 00:32:34.230 and forth into our accounts. 00:32:34.230 --> 00:32:37.470 But the cumulative sum is going to end up 00:32:37.470 --> 00:32:39.270 being approximately the same. 00:32:39.270 --> 00:32:41.500 So for the purposes of this class, 00:32:41.500 --> 00:32:44.760 I'm going to assert that this is approximately the same. 00:32:44.760 --> 00:32:46.350 In fact, you can show that there's 00:32:46.350 --> 00:32:48.630 another relationship that looks at the interest 00:32:48.630 --> 00:32:50.410 rate per period. 00:32:50.410 --> 00:32:53.454 And it's a little bit more complicated, but not 00:32:53.454 --> 00:32:54.370 much more complicated. 00:32:54.370 --> 00:32:57.280 You can see that in your textbook, if you're interested. 00:32:57.280 --> 00:33:00.240 But for now, I want to just focus on this relationship. 00:33:00.240 --> 00:33:06.070 This relationship tells us how to price futures and forwards. 00:33:06.070 --> 00:33:11.760 And now if I divide by 1 plus r, f to the T, then what I've got 00:33:11.760 --> 00:33:16.890 is that the forward price divided by the interest rate, 00:33:16.890 --> 00:33:22.140 that calculates the current value of that forward price, 00:33:22.140 --> 00:33:25.680 has got to equal the spot price plus the present value 00:33:25.680 --> 00:33:27.702 of the net storage costs. 00:33:27.702 --> 00:33:29.910 This is the relationship that we've been looking for, 00:33:29.910 --> 00:33:33.270 and you guys have been struggling for the last couple 00:33:33.270 --> 00:33:33.810 of lectures. 00:33:33.810 --> 00:33:35.520 You've been asking well, gee, doesn't the interest 00:33:35.520 --> 00:33:37.880 rate belong in there, and what about having the asset, 00:33:37.880 --> 00:33:40.420 wouldn't it be nice to have it, and so on and so forth. 00:33:40.420 --> 00:33:43.020 All of those considerations are summed up 00:33:43.020 --> 00:33:44.870 in this one expression. 00:33:44.870 --> 00:33:46.830 A very nice expression. 00:33:46.830 --> 00:33:48.930 Very intuitive. 00:33:48.930 --> 00:33:54.160 What you pay at date T, when you take the present value of it, 00:33:54.160 --> 00:33:59.500 that has to be equal to what the thing is worth today 00:33:59.500 --> 00:34:05.230 plus any benefits for having the thing, as opposed 00:34:05.230 --> 00:34:09.600 to not having the thing between now and settlement. 00:34:09.600 --> 00:34:11.090 That's it. 00:34:11.090 --> 00:34:13.300 Now this is for the very beginning 00:34:13.300 --> 00:34:15.949 when you strike the contract. 00:34:15.949 --> 00:34:20.659 What about at an arbitrary point in time between 0 and T? 00:34:20.659 --> 00:34:24.530 Well, all of these arguments work exactly the same way 00:34:24.530 --> 00:34:30.830 when you're looking at two dates t and T, as opposed to 0 and T. 00:34:30.830 --> 00:34:32.836 So the relationship that I showed you, 00:34:32.836 --> 00:34:34.460 it's a little bit more complicated now, 00:34:34.460 --> 00:34:37.550 because you've got to take into account the fact that the time 00:34:37.550 --> 00:34:39.889 to settlement is not capital T, it's 00:34:39.889 --> 00:34:43.310 capital T minus where you are today. 00:34:43.310 --> 00:34:45.050 But that's the only change. 00:34:45.050 --> 00:34:48.020 Other than that, everything is the same. 00:34:51.969 --> 00:34:55.330 And you have to make sure that you accumulate 00:34:55.330 --> 00:34:58.840 the future value of all the net storage costs, 00:34:58.840 --> 00:35:01.750 so that you actually move all of the costs to the end, 00:35:01.750 --> 00:35:06.340 and then you bring it back to time T. 00:35:06.340 --> 00:35:08.380 Now, let's take this out for a spin. 00:35:08.380 --> 00:35:10.070 Let's see how this works. 00:35:10.070 --> 00:35:12.250 Let's take a look at gold. 00:35:14.800 --> 00:35:19.170 Gold is easy to store. 00:35:19.170 --> 00:35:22.470 There's no storage costs really. 00:35:22.470 --> 00:35:25.010 I mean gold is relatively compact, a little heavy, 00:35:25.010 --> 00:35:27.510 so you're going to have to lift it and put it in your vault, 00:35:27.510 --> 00:35:29.385 as some of you, I'm sure, are doing nowadays. 00:35:29.385 --> 00:35:35.490 [LAUGHTER] But the bottom line is that the storage costs 00:35:35.490 --> 00:35:37.320 are negligible. 00:35:37.320 --> 00:35:38.400 There are no dividends. 00:35:38.400 --> 00:35:41.640 Gold does not pay out dividends. 00:35:41.640 --> 00:35:44.040 There are no real benefits either, 00:35:44.040 --> 00:35:45.765 there is no convenience yield. 00:35:45.765 --> 00:35:48.480 It's not like you need a little piece of gold 00:35:48.480 --> 00:35:50.500 every once in a while for your pleasure, 00:35:50.500 --> 00:35:52.500 and so you want to scrape that off and enjoy it. 00:35:52.500 --> 00:35:55.120 [LAUGHTER] It just sort of sits there. 00:35:55.120 --> 00:35:57.810 So if that's the case, you factor that 00:35:57.810 --> 00:35:59.640 into that relationship that I showed you, 00:35:59.640 --> 00:36:04.980 and that last term, the PV of net storage costs is nothing. 00:36:04.980 --> 00:36:08.720 And so the relationship is really simple. 00:36:08.720 --> 00:36:14.390 The forward slash futures price today 00:36:14.390 --> 00:36:17.930 is just equal to what the current spot 00:36:17.930 --> 00:36:22.730 price is multiplied by 1 plus the risk free rate of interest 00:36:22.730 --> 00:36:27.090 between today and a settlement date. 00:36:27.090 --> 00:36:29.340 If this relationship is violated-- 00:36:29.340 --> 00:36:33.210 when you look at gold futures, and gold spot, 00:36:33.210 --> 00:36:37.140 and you see that this relationship was violated, 00:36:37.140 --> 00:36:39.300 that's a sign that there's an arbitrage. 00:36:39.300 --> 00:36:41.380 You can make money off of that. 00:36:41.380 --> 00:36:43.890 So that really is the way to make a million dollars 00:36:43.890 --> 00:36:45.350 with no money down-- 00:36:45.350 --> 00:36:50.160 is to try to find violations of this arbitrage relationship. 00:36:50.160 --> 00:36:52.770 It's going to be hard, because there are a lot of people that 00:36:52.770 --> 00:36:55.080 are looking at it all the time. 00:36:55.080 --> 00:36:58.150 And so when there is an inequality of some sort, 00:36:58.150 --> 00:36:59.820 it's probably not going to be very big, 00:36:59.820 --> 00:37:01.770 and it probably won't last very long. 00:37:01.770 --> 00:37:04.590 But to the person who found it first, 00:37:04.590 --> 00:37:07.020 they might actually be able to make a little bit off 00:37:07.020 --> 00:37:10.920 of that discrepancy, by either buying or selling gold, 00:37:10.920 --> 00:37:13.860 and transacting these markets quickly. 00:37:13.860 --> 00:37:15.160 Let me do another example. 00:37:15.160 --> 00:37:16.140 So this is gold. 00:37:16.140 --> 00:37:18.370 What about gasoline? 00:37:18.370 --> 00:37:20.850 Gasoline it turns out, is very different from gold. 00:37:20.850 --> 00:37:24.910 First of all, it's a pain in the neck to store safely. 00:37:24.910 --> 00:37:27.300 So if you don't want to be blown up in the middle-- 00:37:27.300 --> 00:37:31.590 and this is what I really mean by blowing up right, gasoline-- 00:37:31.590 --> 00:37:33.610 you want to prevent that from happening, 00:37:33.610 --> 00:37:35.310 you've got to pay a storage cost. 00:37:35.310 --> 00:37:37.310 On the other hand, there is a convenience yield. 00:37:37.310 --> 00:37:39.810 If you've got the gasoline, you can actually 00:37:39.810 --> 00:37:41.244 use it along the way. 00:37:41.244 --> 00:37:42.660 You have to replenish it, in order 00:37:42.660 --> 00:37:45.930 to get the same level of stock at the end, 00:37:45.930 --> 00:37:48.420 but it's convenient that you have it, 00:37:48.420 --> 00:37:50.010 instead of having to go get it. 00:37:50.010 --> 00:37:54.150 Because getting it involves trouble and costs. 00:37:54.150 --> 00:37:56.850 So that's the convenience yield. 00:37:56.850 --> 00:38:02.360 So if you factor that in, then what you get is the futures 00:38:02.360 --> 00:38:13.470 or forward price is equal to 1 plus r,f plus a plus a storage 00:38:13.470 --> 00:38:17.370 cost per period, minus a convenience yield per period, 00:38:17.370 --> 00:38:20.790 and then raised to the T minus t power, 00:38:20.790 --> 00:38:24.720 multiplied by the current spot price. 00:38:24.720 --> 00:38:27.570 If this is violated, then you're going 00:38:27.570 --> 00:38:29.244 to want to do one of two things. 00:38:29.244 --> 00:38:31.410 Either you're going to want to buy your own gasoline 00:38:31.410 --> 00:38:35.370 and store it, or you're going to want to short it and do 00:38:35.370 --> 00:38:37.650 the opposite. 00:38:37.650 --> 00:38:40.440 After Hurricane Katrina hit, we had 00:38:40.440 --> 00:38:45.750 violations from this for a period of time, which suggested 00:38:45.750 --> 00:38:47.880 that it was actually worthwhile for you 00:38:47.880 --> 00:38:51.060 to go out and build your own storage facilities, 00:38:51.060 --> 00:38:53.760 because the storage facilities were destroyed. 00:38:53.760 --> 00:38:55.520 Now it's only if you had the technology 00:38:55.520 --> 00:38:57.270 to build those storage facilities that you 00:38:57.270 --> 00:38:59.660 could actually profit from it. 00:38:59.660 --> 00:39:04.160 But there are periods of time where market dislocation can 00:39:04.160 --> 00:39:10.190 occur, and the discrepancy between futures prices and spot 00:39:10.190 --> 00:39:11.240 prices-- 00:39:11.240 --> 00:39:13.160 that gives you valuable information 00:39:13.160 --> 00:39:16.920 about what's happening in markets, and in some cases, 00:39:16.920 --> 00:39:20.300 in non-financial contexts, like commodities. 00:39:20.300 --> 00:39:23.480 Whether there's a shortage or whether there's a glut. 00:39:23.480 --> 00:39:27.110 Weather impacts these commodities. 00:39:27.110 --> 00:39:28.910 And so by looking at this relationship 00:39:28.910 --> 00:39:32.990 you gather very valuable information. 00:39:32.990 --> 00:39:34.340 Here's another example. 00:39:34.340 --> 00:39:36.110 Another example is financials. 00:39:36.110 --> 00:39:38.684 I'm going to take this example as the last one 00:39:38.684 --> 00:39:40.100 that I want to focus on, because I 00:39:40.100 --> 00:39:46.940 want to now talk about how to use this for your own purposes. 00:39:46.940 --> 00:39:52.100 A financial future is a futures contract 00:39:52.100 --> 00:39:55.200 on an index like the S&P 500. 00:39:55.200 --> 00:39:59.810 So there, all of those contracts are cash settled, 00:39:59.810 --> 00:40:01.270 there is no physical delivery. 00:40:01.270 --> 00:40:03.890 Although, you can easily imagine a situation where 00:40:03.890 --> 00:40:05.310 you could have physical delivery. 00:40:05.310 --> 00:40:08.240 Somebody literally delivers 500 shares 00:40:08.240 --> 00:40:13.430 of stocks, 500 stocks with a certain number of shares 00:40:13.430 --> 00:40:17.140 for each, in order to get the S&P 500. 00:40:17.140 --> 00:40:19.330 But that's a pain, and that defeats 00:40:19.330 --> 00:40:20.849 the purpose of the futures market, 00:40:20.849 --> 00:40:22.390 which is to try to make things simple 00:40:22.390 --> 00:40:25.340 and to make it more efficient. 00:40:25.340 --> 00:40:29.830 So a stock index future is really a pure bet 00:40:29.830 --> 00:40:32.770 on an underlying index. 00:40:32.770 --> 00:40:36.550 And it gives you, the investor or the hedger, 00:40:36.550 --> 00:40:40.570 a way to get exposure or get out of exposure of that underlying 00:40:40.570 --> 00:40:41.950 in a very direct way. 00:40:41.950 --> 00:40:47.395 Now in this case, there's no real convenience yield, 00:40:47.395 --> 00:40:50.170 but there is a dividend that gets 00:40:50.170 --> 00:40:54.670 paid by the particular set of securities. 00:40:54.670 --> 00:40:59.090 So if you're holding the S&P 500 portfolio, 00:40:59.090 --> 00:41:02.060 then you're going to be getting paid dividends 00:41:02.060 --> 00:41:05.970 for individual stocks in that portfolio. 00:41:05.970 --> 00:41:10.850 And so you'd want to factor in in your futures arbitrage 00:41:10.850 --> 00:41:15.560 relationship the fact that you're getting a benefit, 00:41:15.560 --> 00:41:19.850 like a convenience yield, that you have to subtract off 00:41:19.850 --> 00:41:21.830 of this relationship. 00:41:21.830 --> 00:41:24.470 So you don't have a cost of storage, 00:41:24.470 --> 00:41:26.330 because this is a financial futures, 00:41:26.330 --> 00:41:29.960 but you do have a convenience yield, in terms of a payment 00:41:29.960 --> 00:41:34.492 if you held the physical shares of the S&P 500. 00:41:34.492 --> 00:41:36.200 So that's the difference between futures. 00:41:36.200 --> 00:41:37.850 Futures, you don't get that dividend, 00:41:37.850 --> 00:41:42.190 so you got to take that out in order to do the calculation. 00:41:42.190 --> 00:41:47.180 That tells you what the futures price is relative to the spot. 00:41:47.180 --> 00:41:51.430 So now if I give you an exam question that says, 00:41:51.430 --> 00:41:54.220 today's spot price is such and such, 00:41:54.220 --> 00:41:57.340 and the risk free interest rate over the next three month 00:41:57.340 --> 00:42:00.080 period is such and such, you should 00:42:00.080 --> 00:42:04.670 be able to tell me what the no arbitrage futures price should 00:42:04.670 --> 00:42:06.560 be today. 00:42:06.560 --> 00:42:09.832 Or vice versa, if I told you what the futures price is, 00:42:09.832 --> 00:42:11.540 and I told you what the interest rate is, 00:42:11.540 --> 00:42:13.940 you should be able to infer from that what 00:42:13.940 --> 00:42:16.280 the spot price is going to be. 00:42:16.280 --> 00:42:22.550 On October 19, 1987, the morning before the New York Stock 00:42:22.550 --> 00:42:25.940 Exchange opened, there was a very big discrepancy 00:42:25.940 --> 00:42:28.880 between the spot price and the futures price. 00:42:31.440 --> 00:42:35.460 That discrepancy caused arbitrageurs to rub their hands 00:42:35.460 --> 00:42:38.580 and say, oh my god, this is Christmas coming early, 00:42:38.580 --> 00:42:40.450 I'm going to take advantage of this. 00:42:40.450 --> 00:42:41.990 And so what they ended up doing-- 00:42:41.990 --> 00:42:43.740 it turned out because the relationship was 00:42:43.740 --> 00:42:46.840 violated in one specific way-- 00:42:46.840 --> 00:42:51.880 they ended up buying the futures and shorting the stocks. 00:42:51.880 --> 00:42:56.170 That was the beginning of the October, 1987 crash, 00:42:56.170 --> 00:42:59.109 that within a day dropped the market by about 20%. 00:42:59.109 --> 00:43:01.150 Nowadays, that's no big deal, we're used to that. 00:43:01.150 --> 00:43:06.030 [LAUGHTER] But back then, it was really something. 00:43:06.030 --> 00:43:10.620 So now that you have examples of how these prices are 00:43:10.620 --> 00:43:15.115 determined, let me take this out for a different kind of a spin. 00:43:15.115 --> 00:43:17.240 I want to show you how you use one of these things. 00:43:19.910 --> 00:43:27.830 And the way I'm going to do that is with the S&P 500. 00:43:27.830 --> 00:43:30.140 What I skipped were more numerical examples 00:43:30.140 --> 00:43:34.350 that I would encourage you to go through on your own. 00:43:34.350 --> 00:43:36.330 But this is an example that's important, 00:43:36.330 --> 00:43:38.330 so I want to take you through it carefully, make 00:43:38.330 --> 00:43:40.430 sure everybody understands. 00:43:40.430 --> 00:43:42.560 Suppose that you've got $1 million 00:43:42.560 --> 00:43:45.770 to invest in the stock market, and you've 00:43:45.770 --> 00:43:48.360 decided that you want to invest it in the S&P 500. 00:43:48.360 --> 00:43:51.650 You don't want to invest it in any other individual stocks. 00:43:51.650 --> 00:43:54.770 You want a broadly diversified investment, 00:43:54.770 --> 00:43:58.520 and the S&P looks like a pretty good thing. 00:43:58.520 --> 00:44:01.380 So there are several ways of doing this, 00:44:01.380 --> 00:44:03.150 I'm going to focus just on two. 00:44:03.150 --> 00:44:06.380 One of them is you could put your money 00:44:06.380 --> 00:44:10.240 in 500 different stocks. 00:44:10.240 --> 00:44:12.534 And you have to spend a little bit of time figuring out 00:44:12.534 --> 00:44:14.200 what the proportions are, because if you 00:44:14.200 --> 00:44:17.761 want to replicate the S&P, the S&P is a value weighted index, 00:44:17.761 --> 00:44:18.760 it's not equal weighted. 00:44:18.760 --> 00:44:21.045 It's weighted by market capitalization. 00:44:21.045 --> 00:44:23.170 So you've got to actually go through and figure out 00:44:23.170 --> 00:44:25.870 how big each company the S&P is, and then 00:44:25.870 --> 00:44:27.910 calculate those weights. 00:44:27.910 --> 00:44:30.190 And then you've got to give this order to your broker, 00:44:30.190 --> 00:44:32.360 and $1 million dollars isn't what it used to be, 00:44:32.360 --> 00:44:34.060 so I suspect that that would generate 00:44:34.060 --> 00:44:35.380 some pretty tiny trades. 00:44:35.380 --> 00:44:37.600 You got 500 securities, and you've 00:44:37.600 --> 00:44:39.820 got a bunch of different odd lot trades. 00:44:39.820 --> 00:44:41.260 Good luck finding a broker that's 00:44:41.260 --> 00:44:42.940 willing to do it at a reasonable price. 00:44:42.940 --> 00:44:45.760 It's a pretty long list. 00:44:45.760 --> 00:44:50.630 Or you can buy a futures contract. 00:44:50.630 --> 00:44:53.330 In particular, you can buy a contract 00:44:53.330 --> 00:44:55.370 on the S&P 500 futures. 00:44:55.370 --> 00:44:59.090 So I want to go through and show you what that involves. 00:44:59.090 --> 00:45:01.855 Now let's take your $1 million and let's deposit 00:45:01.855 --> 00:45:06.050 it at the Futures Brokerage account. 00:45:06.050 --> 00:45:08.810 So the money is sitting there, earning whatever interest they 00:45:08.810 --> 00:45:10.350 pay you on that account. 00:45:10.350 --> 00:45:16.670 Which is not much, it's probably akin to a money market return. 00:45:16.670 --> 00:45:20.540 So what you do is you want to buy futures contracts 00:45:20.540 --> 00:45:22.460 and you want to have the equivalent 00:45:22.460 --> 00:45:29.270 exposure of $1 million invested in the S&P 500. 00:45:29.270 --> 00:45:32.900 Now the way that the S&P 500 futures contract works, 00:45:32.900 --> 00:45:37.580 is that the value of the contract, the notional amount 00:45:37.580 --> 00:45:43.020 of the contract, is 250 times the index. 00:45:43.020 --> 00:45:46.400 Whatever the index is worth, they just make up a number, 00:45:46.400 --> 00:45:49.670 like I don't know 250, and multiply that 00:45:49.670 --> 00:45:50.930 by the value of the index. 00:45:50.930 --> 00:45:57.014 And they say that is what your exposure is for one contract. 00:45:57.014 --> 00:45:57.680 So what is that? 00:45:57.680 --> 00:46:06.050 Let's suppose the the S&P index is now at a 1,000. 00:46:06.050 --> 00:46:09.140 So the value of the futures contract is 250 times 00:46:09.140 --> 00:46:16.910 that and that's going to be $250,000. 00:46:16.910 --> 00:46:19.520 In order for you to have the equivalent of $1 million 00:46:19.520 --> 00:46:24.990 in the S&P, you need four of those contracts. 00:46:24.990 --> 00:46:30.120 Four times a notional of 250 is equal to $1 million. 00:46:30.120 --> 00:46:31.230 Now what does this say? 00:46:31.230 --> 00:46:33.810 This says that you are essentially 00:46:33.810 --> 00:46:40.440 agreeing that you're going to buy the S&P 500 whenever 00:46:40.440 --> 00:46:42.060 it settles. 00:46:42.060 --> 00:46:43.820 But you're not really buying the S&P 500, 00:46:43.820 --> 00:46:50.130 you're buying a pure bet that is equivalent to 250 times 00:46:50.130 --> 00:46:53.700 the S&P 500. 00:46:53.700 --> 00:46:57.220 So let's take a look at what that means. 00:46:57.220 --> 00:47:02.230 Suppose that the S&P index fluctuates, bounces around, 00:47:02.230 --> 00:47:07.740 then it turns out that you'll see that your cash portfolio-- 00:47:07.740 --> 00:47:11.680 the portfolio fluctuations if you had put $1 million 00:47:11.680 --> 00:47:14.260 into the S&P directly-- 00:47:14.260 --> 00:47:17.440 it fluctuates in exactly the same way 00:47:17.440 --> 00:47:19.480 that your futures portfolio fluctuate. 00:47:19.480 --> 00:47:23.920 If the S&P goes down to 900, the notional value 00:47:23.920 --> 00:47:28.870 of your portfolio with four contracts is $900,000. 00:47:28.870 --> 00:47:31.720 So you've actually lost $100,000, 00:47:31.720 --> 00:47:35.580 and that's going to be deducted from your account. 00:47:35.580 --> 00:47:41.250 If on the other hand, the S&P goes up by 100, 00:47:41.250 --> 00:47:47.790 then your cash portfolio will be worth $1,100,000, 00:47:47.790 --> 00:47:50.640 and your futures portfolio will be worth the same. 00:47:50.640 --> 00:47:54.660 You will now get $100,000 deposited into your account. 00:47:54.660 --> 00:47:56.970 By holding this futures contract, 00:47:56.970 --> 00:48:01.860 it's as if you were actually invested in the S&P. 00:48:01.860 --> 00:48:05.830 What you're getting is the daily fluctuations. 00:48:05.830 --> 00:48:08.360 But you actually don't own the security, 00:48:08.360 --> 00:48:14.470 you simply agreed to buy this so-called index on the maturity 00:48:14.470 --> 00:48:15.220 date. 00:48:15.220 --> 00:48:19.210 And by doing so, and because that contract value 00:48:19.210 --> 00:48:22.930 is so closely tied to the S&P 500 index, 00:48:22.930 --> 00:48:27.400 it moves in lockstep with the cash portfolio. 00:48:30.260 --> 00:48:32.242 Any questions about this? 00:48:32.242 --> 00:48:34.572 Yeah. 00:48:34.572 --> 00:48:36.660 AUDIENCE: Does someone own those shares 00:48:36.660 --> 00:48:38.370 behind you or it's just-- 00:48:38.370 --> 00:48:39.010 ANDREW LO: No. 00:48:39.010 --> 00:48:41.150 AUDIENCE: --an agreement that we're going to wait on this-- 00:48:41.150 --> 00:48:41.960 ANDREW LO: Exactly. 00:48:41.960 --> 00:48:42.460 Right. 00:48:42.460 --> 00:48:45.860 So you and I, we're just going to agree, we're going to bet. 00:48:45.860 --> 00:48:47.660 We're going to bet and we're going 00:48:47.660 --> 00:48:52.130 to agree on a particular price for S&P 500 three months 00:48:52.130 --> 00:48:53.300 from now. 00:48:53.300 --> 00:48:57.170 And if it goes up, and I bought the contract, then I win. 00:48:57.170 --> 00:49:01.190 If it goes down, then you sold the contract, then you win. 00:49:01.190 --> 00:49:03.564 But it's a pure bet between you and me. 00:49:03.564 --> 00:49:05.589 AUDIENCE: In the middle is the Futures-- 00:49:05.589 --> 00:49:07.380 ANDREW LO: The Futures Clearing corporation 00:49:07.380 --> 00:49:09.963 sits in the middle to make sure that you and I don't run away. 00:49:09.963 --> 00:49:11.944 AUDIENCE: Why do they do this? 00:49:11.944 --> 00:49:13.235 ANDREW LO: Why do they do this? 00:49:13.235 --> 00:49:20.940 AUDIENCE: I mean why do they [INAUDIBLE] days of sunlight. 00:49:20.940 --> 00:49:24.650 ANDREW LO: Well, first of all, in some cases they do. 00:49:24.650 --> 00:49:27.710 So for example, you could buy a contract 00:49:27.710 --> 00:49:34.030 on the number of degree days of a certain amount in Florida. 00:49:34.030 --> 00:49:35.960 Now why would you want to do that? 00:49:35.960 --> 00:49:41.320 It turns out that one of the largest crops of oranges 00:49:41.320 --> 00:49:43.180 are grown in Florida. 00:49:43.180 --> 00:49:47.500 And it turns out that the output of oranges groves 00:49:47.500 --> 00:49:51.590 is very closely tied to temperature. 00:49:51.590 --> 00:49:56.595 So if it goes up to 39 degrees or below 32 degrees, 00:49:56.595 --> 00:49:58.720 you can actually have very different kind of crops. 00:49:58.720 --> 00:50:01.052 And so you can bet on that, and at some point 00:50:01.052 --> 00:50:02.260 you can actually trade on it. 00:50:02.260 --> 00:50:03.926 I don't know if you can trade on it now, 00:50:03.926 --> 00:50:08.290 but there are markets for some of the wildest things. 00:50:08.290 --> 00:50:10.040 And the reason that you have these markets 00:50:10.040 --> 00:50:13.670 is because when two mutually consenting adults have 00:50:13.670 --> 00:50:17.194 opposite views and they want to express them, then 00:50:17.194 --> 00:50:18.860 you want to be able to let them do that, 00:50:18.860 --> 00:50:22.700 and allow them to basically either hedge their risks, 00:50:22.700 --> 00:50:26.070 or take on risks that they're able to do. 00:50:26.070 --> 00:50:27.620 So this is an example of that. 00:50:27.620 --> 00:50:28.460 You're an investor. 00:50:28.460 --> 00:50:31.310 You want to buy stocks, but you don't 00:50:31.310 --> 00:50:34.790 want to buy 500 little stocks one by one. 00:50:34.790 --> 00:50:36.834 You want to get the exposure right away. 00:50:36.834 --> 00:50:38.750 Now of course, there's another way to do this, 00:50:38.750 --> 00:50:40.670 you can put it in a mutual fund. 00:50:40.670 --> 00:50:42.170 But the problem with the mutual fund 00:50:42.170 --> 00:50:45.980 is that it only gets priced once a day, whereas this thing gets 00:50:45.980 --> 00:50:49.460 priced every second of the day when the futures exchange is 00:50:49.460 --> 00:50:50.540 open. 00:50:50.540 --> 00:50:53.600 Of course, nowadays, you can buy an ETF, an Exchange Traded 00:50:53.600 --> 00:50:54.930 Fund. 00:50:54.930 --> 00:50:57.180 So that's another way of getting exposure. 00:50:57.180 --> 00:51:00.050 But the S&P futures was around long before ETFs 00:51:00.050 --> 00:51:04.479 and allowed people to do all sorts of hedging transactions. 00:51:04.479 --> 00:51:06.770 Now I'm going to give you a second example that I think 00:51:06.770 --> 00:51:08.311 will make it a little bit more clear, 00:51:08.311 --> 00:51:12.020 and actually will answer a question that was asked, 00:51:12.020 --> 00:51:14.240 I think two lectures ago. 00:51:14.240 --> 00:51:15.680 When I first started this lecture, 00:51:15.680 --> 00:51:17.450 I said that maybe a company would only 00:51:17.450 --> 00:51:19.250 want to hedge 25% of its risk. 00:51:19.250 --> 00:51:21.447 And somebody asked well, what does that mean 25%? 00:51:21.447 --> 00:51:23.030 And I said, I'll answer that question. 00:51:23.030 --> 00:51:25.850 Well, so I'm going to answer that question now. 00:51:25.850 --> 00:51:30.440 So suppose now as a different example, 00:51:30.440 --> 00:51:34.880 you have a diversified portfolio of large cap stocks 00:51:34.880 --> 00:51:36.800 worth $5 million. 00:51:36.800 --> 00:51:41.310 So you already own the stocks, and it's currently 00:51:41.310 --> 00:51:44.400 worth $5 million, but you don't have any confidence 00:51:44.400 --> 00:51:47.160 that the market is going to stay where it is. 00:51:47.160 --> 00:51:48.690 You think it's going to go down. 00:51:48.690 --> 00:51:51.360 And so you want to hedge some of that risk. 00:51:51.360 --> 00:51:52.980 You don't want to hedge all of it, 00:51:52.980 --> 00:51:57.540 because you do have faith that over time markets will do well, 00:51:57.540 --> 00:52:00.000 but you just want to be able to dampen 00:52:00.000 --> 00:52:04.780 a little bit of the downward spiral if it does occur. 00:52:04.780 --> 00:52:09.960 So you might consider selling 25% of your portfolio. 00:52:09.960 --> 00:52:13.050 Getting rid of 25% of it and putting that in cash. 00:52:13.050 --> 00:52:14.710 That's one way to do it. 00:52:14.710 --> 00:52:17.160 But the problem as you know is that it's not 00:52:17.160 --> 00:52:21.390 that easy to sell 25% of 500 stocks, 00:52:21.390 --> 00:52:24.960 because you have to again, slice the portfolio, stock by stock. 00:52:24.960 --> 00:52:29.280 You're going to have a trade list of 500 stocks, which 00:52:29.280 --> 00:52:32.920 comprise 25% of your portfolio. 00:52:32.920 --> 00:52:34.800 So it's a pain. 00:52:34.800 --> 00:52:37.650 But here's an easier way to do it. 00:52:37.650 --> 00:52:45.170 You can short sell five S&P contracts. 00:52:45.170 --> 00:52:48.230 And I'm arguing that that will do the exact same as 00:52:48.230 --> 00:52:53.530 if you just liquidated 25% of your portfolio. 00:52:53.530 --> 00:52:57.130 Now let's see if that's right. 00:52:57.130 --> 00:52:59.950 So let's go through the exact same table. 00:52:59.950 --> 00:53:01.420 The cash portfolio-- let's see what 00:53:01.420 --> 00:53:05.350 happens to the cash portfolio if the S&P goes up or down by 100 00:53:05.350 --> 00:53:06.730 points. 00:53:06.730 --> 00:53:09.610 If it goes up by 100 points, then you've made money. 00:53:09.610 --> 00:53:11.200 You've got $5.5 million. 00:53:11.200 --> 00:53:13.510 If it goes down by a 100 points, you've lost money. 00:53:13.510 --> 00:53:17.110 You've lost to $4.5 million. 00:53:17.110 --> 00:53:19.510 Now let's see what happens if you don't do anything 00:53:19.510 --> 00:53:22.630 with the cash portfolio, but you simply short sell 00:53:22.630 --> 00:53:26.170 five S&P futures contracts. 00:53:26.170 --> 00:53:30.850 If you do that then obviously if the S&P doesn't change, 00:53:30.850 --> 00:53:33.140 then nothing happens to your portfolio. 00:53:33.140 --> 00:53:35.830 But if the S&P goes up, then you're 00:53:35.830 --> 00:53:38.790 going to make some money. 00:53:38.790 --> 00:53:40.570 Sorry. 00:53:40.570 --> 00:53:42.670 So yeah, if the S&P goes up, you're 00:53:42.670 --> 00:53:47.080 going to lose money in the sense that what's going to happen 00:53:47.080 --> 00:53:53.500 is that your short positions are going to cost you $125,000. 00:53:53.500 --> 00:53:55.149 How did I get $125,000? 00:53:55.149 --> 00:53:56.690 Anybody work through the math for me? 00:54:03.030 --> 00:54:06.310 The S&P 500 goes up by a 100 points. 00:54:06.310 --> 00:54:13.640 The futures price goes up by 250 times 100 points. 00:54:13.640 --> 00:54:16.190 My position, I've got five of these contracts, 00:54:16.190 --> 00:54:19.700 I've just lost $25,000 per contract. 00:54:19.700 --> 00:54:25.460 I've got five of these contracts, I lost $125,000. 00:54:25.460 --> 00:54:27.120 Now what about the downside? 00:54:27.120 --> 00:54:32.090 The downside if the S&P goes down by 100, 00:54:32.090 --> 00:54:38.330 then the price goes down by $25,000. 00:54:38.330 --> 00:54:42.110 I'm short, so I make $25,000 per contract. 00:54:42.110 --> 00:54:46.190 I've got five contracts, I've made $125,000. 00:54:46.190 --> 00:54:48.320 So look what happens. 00:54:48.320 --> 00:54:51.980 In this case, when the S&P goes up, 00:54:51.980 --> 00:54:56.070 I don't make as much, because my hedge works against me. 00:54:56.070 --> 00:54:58.490 On the other hand, when the S&P goes down, 00:54:58.490 --> 00:55:02.030 I don't lose as much, because the hedge is working for me. 00:55:02.030 --> 00:55:06.440 Because I've only taken out 25% of my portfolio 00:55:06.440 --> 00:55:12.190 with this hedge, it's dampening, but not eliminating 00:55:12.190 --> 00:55:13.794 that kind of fluctuation. 00:55:13.794 --> 00:55:15.276 Yeah? 00:55:15.276 --> 00:55:17.808 AUDIENCE: I think that this an obvious question, but why 00:55:17.808 --> 00:55:21.520 do you do that, versus just putting it in cash. 00:55:21.520 --> 00:55:24.516 Because you can make the argument that if you had 25%, 00:55:24.516 --> 00:55:28.241 and had it earning interest, and so you'd still be up too. 00:55:28.241 --> 00:55:29.740 ANDREW LO: Well, that's the argument 00:55:29.740 --> 00:55:31.614 that I gave earlier, which is that you'd have 00:55:31.614 --> 00:55:34.350 to sell 25% of your portfolio. 00:55:34.350 --> 00:55:36.490 This is a way of doing it. 00:55:36.490 --> 00:55:39.450 And not only that, if you did it this way, 00:55:39.450 --> 00:55:42.000 it would be a lot cheaper to implement in the sense 00:55:42.000 --> 00:55:47.100 that you don't have to do 500 transactions, 00:55:47.100 --> 00:55:50.760 you do one transaction. 00:55:50.760 --> 00:55:53.700 So the transactions cost is a lot cheaper, 00:55:53.700 --> 00:55:55.427 and it's also easier to keep track of. 00:55:55.427 --> 00:55:57.510 You don't have to figure out what the price of 500 00:55:57.510 --> 00:55:59.400 securities are. 00:55:59.400 --> 00:56:01.780 You've got the price of just one security to worry about. 00:56:01.780 --> 00:56:02.280 Yeah. 00:56:02.280 --> 00:56:04.560 AUDIENCE: And I think also you're 00:56:04.560 --> 00:56:06.380 not losing out on what you could've 00:56:06.380 --> 00:56:08.120 had in cash in terms of interest, 00:56:08.120 --> 00:56:10.320 because that interest is factored in to the futures. 00:56:10.320 --> 00:56:10.710 ANDREW LO: That's right. 00:56:10.710 --> 00:56:12.540 Remember we used that interest equation 00:56:12.540 --> 00:56:16.850 so all the foregone interest is in there. 00:56:16.850 --> 00:56:21.840 OK, so the meaning of I want to hedge 25% 00:56:21.840 --> 00:56:25.710 means I'm going to use the futures contract, 00:56:25.710 --> 00:56:31.380 so that the notional exposure is 25% of the current value 00:56:31.380 --> 00:56:34.280 of my portfolio. 00:56:34.280 --> 00:56:38.260 So if you're Merck pharmaceutical company that 00:56:38.260 --> 00:56:41.650 has a certain percentage of their revenues 00:56:41.650 --> 00:56:43.870 in foreign denominated currencies, 00:56:43.870 --> 00:56:47.950 you can hedge half of the risk of those exchange rate 00:56:47.950 --> 00:56:51.130 fluctuations by taking half of the revenue stream-- 00:56:51.130 --> 00:56:54.096 let's say it's $10 billion-- 00:56:54.096 --> 00:56:59.080 and buying or selling, depending on which way you're going, 00:56:59.080 --> 00:57:03.190 the amount of futures or forwards to 00:57:03.190 --> 00:57:04.737 get rid of that exposure. 00:57:04.737 --> 00:57:05.551 Yeah. 00:57:05.551 --> 00:57:10.320 AUDIENCE: In this example, we put our million in the margin 00:57:10.320 --> 00:57:15.066 account, but we only should put as much as 00:57:15.066 --> 00:57:15.940 [INTERPOSING VOICES]. 00:57:15.940 --> 00:57:16.979 ANDREW LO: That's right. 00:57:16.979 --> 00:57:19.270 You don't have to put $1 million in the margin account, 00:57:19.270 --> 00:57:21.160 because typically the margin is going 00:57:21.160 --> 00:57:25.300 to be something like in this case 7% or 8% 00:57:25.300 --> 00:57:27.100 of the notional exposure. 00:57:27.100 --> 00:57:30.220 So you could take the rest of that money and go to Las Vegas 00:57:30.220 --> 00:57:31.379 if you like. 00:57:31.379 --> 00:57:33.670 Although, some would say this is better than Las Vegas. 00:57:33.670 --> 00:57:34.115 Yeah. 00:57:34.115 --> 00:57:35.656 AUDIENCE: This is the main reason why 00:57:35.656 --> 00:57:37.656 we buy futures and not ETFs. 00:57:37.656 --> 00:57:40.360 You can leverage your bet as much as you want. 00:57:40.360 --> 00:57:42.850 ANDREW LO: That's right with an ETF, if you want $1 million 00:57:42.850 --> 00:57:47.350 of exposure, you got to put $1 million into the ETF. 00:57:47.350 --> 00:57:50.170 With the futures contract, if you want to put $1 million 00:57:50.170 --> 00:57:53.140 of exposure on, you need 7%. 00:57:53.140 --> 00:57:55.030 And the reason is obvious, it's because 00:57:55.030 --> 00:57:58.220 of that daily mark to market. 00:57:58.220 --> 00:58:01.270 So ETFs have not killed the futures market, 00:58:01.270 --> 00:58:04.180 but it does provide another vehicle for retail investors 00:58:04.180 --> 00:58:07.900 who may not want the leverage, who may not need to leverage, 00:58:07.900 --> 00:58:11.140 to not have to worry about the leverage. 00:58:11.140 --> 00:58:15.650 This leverage-- leverage is a scary thing, as I said before. 00:58:15.650 --> 00:58:18.370 This is the chain saw that you don't want to be giving 00:58:18.370 --> 00:58:20.350 your eight-year-old as a toy. 00:58:20.350 --> 00:58:24.340 Because when prices move quickly, 00:58:24.340 --> 00:58:27.640 you're going to have very big swings in the underlying 00:58:27.640 --> 00:58:30.340 value of your margin account. 00:58:30.340 --> 00:58:35.170 So if you've got only 7% margin in an account, 00:58:35.170 --> 00:58:39.400 think about it, that means that if the prices go down by 7%, 00:58:39.400 --> 00:58:40.540 you are wiped out. 00:58:40.540 --> 00:58:42.790 Your entire margin account is gone. 00:58:42.790 --> 00:58:47.410 When futures brokers take your money, 00:58:47.410 --> 00:58:49.480 they assume that you know what you're doing. 00:58:49.480 --> 00:58:52.960 And so they assume that the margin that you're putting down 00:58:52.960 --> 00:58:56.620 is margin that you can afford to lose, 00:58:56.620 --> 00:58:59.440 and that you understand that what you're getting 00:58:59.440 --> 00:59:02.590 is much bigger exposure that presumably is either 00:59:02.590 --> 00:59:04.240 for speculative purposes, in which case 00:59:04.240 --> 00:59:07.750 you won't over leverage, or for hedging purposes, in which case 00:59:07.750 --> 00:59:10.690 you've got some other assets that are counterbalancing 00:59:10.690 --> 00:59:11.350 these swings. 00:59:11.350 --> 00:59:12.556 Like in this case. 00:59:12.556 --> 00:59:13.930 You know obviously, when you look 00:59:13.930 --> 00:59:17.440 at the fluctuations in your positions, 00:59:17.440 --> 00:59:20.995 they are extraordinarily big relative to the margin. 00:59:23.590 --> 00:59:26.180 Let's do a quick back of the envelope calculation. 00:59:26.180 --> 00:59:27.920 Let me tell you what I mean. 00:59:27.920 --> 00:59:31.640 Suppose that you put 5% margin down. 00:59:31.640 --> 00:59:34.600 You buy a contract, put 5% margin down, 00:59:34.600 --> 00:59:37.810 and let's suppose that the price of the futures contract 00:59:37.810 --> 00:59:40.140 drops by 2.5%. 00:59:43.080 --> 00:59:46.830 What is the rate of return on the amount of money 00:59:46.830 --> 00:59:51.944 you've put down as margin, if that's your initial investment? 00:59:51.944 --> 00:59:53.610 You can think about it as an investment, 00:59:53.610 --> 00:59:56.130 because that's the only way a futures broker will 00:59:56.130 --> 00:59:57.840 let you buy a contract. 00:59:57.840 --> 01:00:04.320 If you put down $100,000 and the futures price goes down 01:00:04.320 --> 01:00:08.218 by $50,000, what's the rate of return on your investment? 01:00:10.910 --> 01:00:14.960 Yeah, it's minus 50%, that's a big move. 01:00:14.960 --> 01:00:18.230 That's a huge move in a day. 01:00:18.230 --> 01:00:20.960 So when you deal with margin, you 01:00:20.960 --> 01:00:22.910 have to be extraordinarily careful. 01:00:22.910 --> 01:00:25.730 You have to have very, very tight risk controls. 01:00:25.730 --> 01:00:29.270 You have to understand what the swings can be, 01:00:29.270 --> 01:00:32.900 and you have to manage that risk very, very carefully, 01:00:32.900 --> 01:00:35.150 on an intradaily basis in some cases, 01:00:35.150 --> 01:00:38.340 because these futures prices can swing a lot even within a day. 01:00:41.330 --> 01:00:44.130 Any other questions? 01:00:44.130 --> 01:00:46.950 Well, that's it for futures and forwards. 01:00:46.950 --> 01:00:49.890 You now know how to price them. 01:00:49.890 --> 01:00:53.290 You now know how to use them for hedging purposes. 01:00:53.290 --> 01:00:55.680 And there are all sorts of other kinds 01:00:55.680 --> 01:00:59.520 of futures and forwards-- interest rate, bond, currency, 01:00:59.520 --> 01:01:02.460 single stock futures now exist. 01:01:02.460 --> 01:01:06.405 In fact, there are even futures contracts on the VIX, 01:01:06.405 --> 01:01:11.805 there's futures contracts on electricity usage, 01:01:11.805 --> 01:01:15.180 there's futures contracts on the presidential election. 01:01:15.180 --> 01:01:18.840 If you go to the Iowa Electronic Markets, the University 01:01:18.840 --> 01:01:21.990 of Iowa, they created a futures exchange 01:01:21.990 --> 01:01:24.300 that has two contracts. 01:01:24.300 --> 01:01:26.700 One that pays $1 if McCain gets elected, 01:01:26.700 --> 01:01:29.460 and the other that pays $1 if Obama gets elected. 01:01:29.460 --> 01:01:31.530 And by looking at the prices, you 01:01:31.530 --> 01:01:34.830 can actually see what the folks that are trading these futures 01:01:34.830 --> 01:01:39.810 contracts are thinking, in terms of who's got the edge. 01:01:39.810 --> 01:01:44.230 So the futures prices contain an enormous amount of information. 01:01:44.230 --> 01:01:50.060 But keep in mind the information is only as good as you are. 01:01:50.060 --> 01:01:52.250 By you, I mean the market. 01:01:52.250 --> 01:01:54.300 If the market is comprised of knuckleheads, 01:01:54.300 --> 01:01:57.660 the prices you get will be knucklehead prices. 01:01:57.660 --> 01:02:00.210 If the market contains really smart 01:02:00.210 --> 01:02:03.060 sharp sophisticated individuals, you'll 01:02:03.060 --> 01:02:05.820 get extremely informative prices. 01:02:05.820 --> 01:02:09.360 So prices, while they are the best thing 01:02:09.360 --> 01:02:12.180 that we have as a guide for the future, 01:02:12.180 --> 01:02:13.890 they're clearly not perfect. 01:02:13.890 --> 01:02:16.740 And there are periods of time when the market prices 01:02:16.740 --> 01:02:18.690 are less perfect than others. 01:02:18.690 --> 01:02:21.540 And as I told you before, for the next three weeks, 01:02:21.540 --> 01:02:24.060 finance theory is going to be on vacation in the US stock 01:02:24.060 --> 01:02:27.120 market, because all the uncertainty that 01:02:27.120 --> 01:02:30.360 has been building up over the last several years 01:02:30.360 --> 01:02:33.060 are now focused on the next three weeks. 01:02:33.060 --> 01:02:36.390 Markets will be swinging back and forth pretty wildly, 01:02:36.390 --> 01:02:38.580 and it's because people are reacting emotionally, 01:02:38.580 --> 01:02:45.680 not necessarily with their full logical capabilities. 01:02:45.680 --> 01:02:48.680 That's it for futures and forwards, 01:02:48.680 --> 01:02:54.860 and now what I'm going to turn to is options. 01:02:54.860 --> 01:02:58.334 These are the last set of securities 01:02:58.334 --> 01:02:59.750 that I want to go through with you 01:02:59.750 --> 01:03:03.275 that are not like the securities that we've done before. 01:03:05.790 --> 01:03:11.605 And let me just pull up the lecture notes for options. 01:03:14.720 --> 01:03:18.170 I want to start with a little bit of an introduction for how 01:03:18.170 --> 01:03:19.370 to motivate options. 01:03:19.370 --> 01:03:22.970 I think most of you know what options are. 01:03:22.970 --> 01:03:25.970 Their name is quite apropos, because they 01:03:25.970 --> 01:03:27.840 do give you options. 01:03:27.840 --> 01:03:32.210 Futures and forwards require you to engage in a transaction, 01:03:32.210 --> 01:03:34.490 but options don't. 01:03:34.490 --> 01:03:38.370 They give you the right, but not the obligation. 01:03:38.370 --> 01:03:41.570 So you have the option of not entering 01:03:41.570 --> 01:03:45.230 into that final transaction at settlement date. 01:03:45.230 --> 01:03:47.210 I'm going to start with some motivation, 01:03:47.210 --> 01:03:49.820 then go through some payoff diagrams, 01:03:49.820 --> 01:03:52.190 go through options strategies, and then I'm 01:03:52.190 --> 01:03:55.100 going to talk very briefly about valuation of options. 01:03:55.100 --> 01:03:57.927 I have to talk to you guys about Black-Scholes. 01:03:57.927 --> 01:04:00.260 You can't leave MIT without hearing about Black-Scholes. 01:04:00.260 --> 01:04:02.360 [LAUGHTER] So I've got to do a little bit of that. 01:04:02.360 --> 01:04:06.080 But really the derivation is quite a bit more sophisticated, 01:04:06.080 --> 01:04:09.920 and that's why you might want to take 15.437 Options 01:04:09.920 --> 01:04:11.930 and Futures, where the entire course is 01:04:11.930 --> 01:04:13.310 devoted to these instruments. 01:04:13.310 --> 01:04:17.880 They are that complex and that important. 01:04:17.880 --> 01:04:21.530 So let me first describe exactly what an option is. 01:04:21.530 --> 01:04:24.530 An option actually is a specific example of something 01:04:24.530 --> 01:04:28.590 that you now know of more generally as a derivative. 01:04:28.590 --> 01:04:31.010 A derivative security gets its name 01:04:31.010 --> 01:04:33.710 because the value of the security 01:04:33.710 --> 01:04:38.750 is derived from yet another security. 01:04:38.750 --> 01:04:44.000 It's derivative, as opposed to I guess fundamental or primary. 01:04:44.000 --> 01:04:49.250 And examples of derivatives are warrants versus options. 01:04:49.250 --> 01:04:51.860 Options are securities that you can 01:04:51.860 --> 01:04:55.460 think of as pure bets between two parties. 01:04:55.460 --> 01:04:58.550 Warrants are options that are issued 01:04:58.550 --> 01:05:01.070 by a company on its own shares. 01:05:01.070 --> 01:05:05.900 So the net supply of options is zero, 01:05:05.900 --> 01:05:09.710 but the net supply of warrants is not zero, 01:05:09.710 --> 01:05:11.810 it's issued by companies. 01:05:11.810 --> 01:05:14.750 And there are two different kinds of options, 01:05:14.750 --> 01:05:15.950 calls and puts. 01:05:15.950 --> 01:05:19.550 A call option is a piece of paper 01:05:19.550 --> 01:05:21.830 that says the holder of this piece of paper 01:05:21.830 --> 01:05:26.930 is allowed to buy a security on or possibly 01:05:26.930 --> 01:05:31.340 before a particular date, usually called the exercise 01:05:31.340 --> 01:05:36.050 date or maturity date. 01:05:36.050 --> 01:05:39.920 And the difference between being able to exercise early 01:05:39.920 --> 01:05:42.890 versus exercising at the maturity only, 01:05:42.890 --> 01:05:46.640 is the difference between an American and a European option. 01:05:46.640 --> 01:05:50.990 An American option is one where you can exercise it early. 01:05:50.990 --> 01:05:54.890 And a European option is one where you can only exercise it 01:05:54.890 --> 01:06:00.440 on a specific date, the maturity date or the expiration date. 01:06:00.440 --> 01:06:03.740 And puts are the opposite of calls. 01:06:03.740 --> 01:06:05.930 Instead of giving you the right to buy, 01:06:05.930 --> 01:06:08.510 it gives you the right to sell or to put 01:06:08.510 --> 01:06:11.780 the stock to somebody else. 01:06:11.780 --> 01:06:14.150 And the prices at which you get to either 01:06:14.150 --> 01:06:18.140 buy in the case of calls, or sell in the case of puts, 01:06:18.140 --> 01:06:22.400 is called the strike price or the exercise price. 01:06:22.400 --> 01:06:23.200 All right. 01:06:23.200 --> 01:06:25.820 So I'm going to define a little bit of notation. 01:06:25.820 --> 01:06:27.890 Stock prices is S sub t. 01:06:27.890 --> 01:06:31.880 Strike price is K. Notice that K does not have a time subscript, 01:06:31.880 --> 01:06:34.880 because it's fixed at the time the options are issued 01:06:34.880 --> 01:06:37.580 and it doesn't change throughout the life of that option, 01:06:37.580 --> 01:06:40.580 it's part of the contract terms. 01:06:40.580 --> 01:06:42.440 And then the call price is C,t. 01:06:42.440 --> 01:06:44.360 Put price is P,t. 01:06:44.360 --> 01:06:47.240 And the value of these contracts at maturity 01:06:47.240 --> 01:06:49.350 is actually pretty simple. 01:06:49.350 --> 01:06:58.140 If today a particular stock is trading at $60 a share, 01:06:58.140 --> 01:07:02.610 and you purchase an option to buy that stock at $70 a share 01:07:02.610 --> 01:07:06.470 in three months, does that piece of paper have any value? 01:07:09.320 --> 01:07:12.110 The current price is $60, this piece of paper 01:07:12.110 --> 01:07:17.594 gives you the right to buy it at $70 in three months. 01:07:17.594 --> 01:07:21.140 Is that worthless? 01:07:21.140 --> 01:07:22.490 Why not? 01:07:22.490 --> 01:07:25.760 The price is at $60, you can get it at $60 today. 01:07:25.760 --> 01:07:28.510 So why would you want it at $70 three months from now? 01:07:31.020 --> 01:07:32.760 Exactly the price may go up. 01:07:32.760 --> 01:07:34.800 The reason the piece of paper is not worth 01:07:34.800 --> 01:07:38.040 zero today is that there is a chance, 01:07:38.040 --> 01:07:40.980 no matter how small you might think it is, 01:07:40.980 --> 01:07:44.250 there is still a chance that something wonderful might 01:07:44.250 --> 01:07:46.860 happen in the next three months, and then 01:07:46.860 --> 01:07:48.510 the price will go up to $80. 01:07:48.510 --> 01:07:51.300 And if it goes up to $80, you'll be very happy 01:07:51.300 --> 01:07:53.370 that you have the right to buy it at $70. 01:07:53.370 --> 01:07:54.930 How happy will you be? 01:07:54.930 --> 01:07:57.180 You'll be $10 per share happy. 01:07:57.180 --> 01:08:00.800 [LAUGHTER] That's what that expression says. 01:08:00.800 --> 01:08:07.020 On the expiration date, you get to buy the shares-- 01:08:07.020 --> 01:08:10.220 if you're holding a call option, you get to buy it for K 01:08:10.220 --> 01:08:14.760 dollars, but in fact the market has determined that the price 01:08:14.760 --> 01:08:19.010 at that time is really S,T dollars. 01:08:19.010 --> 01:08:22.870 So if you're holding this piece of paper, this is your profit-- 01:08:22.870 --> 01:08:25.560 S,T minus K per share. 01:08:25.560 --> 01:08:29.960 Now if it turns out that you get to buy it for $60, 01:08:29.960 --> 01:08:34.250 and it ends up trading at $40, well then 01:08:34.250 --> 01:08:36.649 you're not going to exercise that right. 01:08:36.649 --> 01:08:39.500 You're going to let the option expire, 01:08:39.500 --> 01:08:42.020 and when it expires it'll be worthless 01:08:42.020 --> 01:08:44.210 if this number is negative. 01:08:44.210 --> 01:08:46.520 It can be negative of course, but you're not 01:08:46.520 --> 01:08:49.130 obligated to buy it. 01:08:49.130 --> 01:08:52.100 On the other hand, if this were a futures contract, 01:08:52.100 --> 01:08:55.580 you certainly are obligated to buy it 01:08:55.580 --> 01:08:58.520 and then you'd get a negative return. 01:08:58.520 --> 01:09:03.410 But an option is a wonderful thing, in that 01:09:03.410 --> 01:09:07.069 the payoff is never negative. 01:09:07.069 --> 01:09:14.000 It's either zero or it's S,T minus K. That's for a call. 01:09:14.000 --> 01:09:16.340 Now a put option, it's exactly the reverse. 01:09:16.340 --> 01:09:21.140 If you get to sell the stock, then your profit 01:09:21.140 --> 01:09:23.840 is what you get to sell it at versus 01:09:23.840 --> 01:09:25.819 what it's really trading at. 01:09:25.819 --> 01:09:28.160 And so you actually hope that it's really 01:09:28.160 --> 01:09:30.380 trading at a very low price. 01:09:30.380 --> 01:09:32.420 Because if you get to sell it at a high price, 01:09:32.420 --> 01:09:36.920 but it's trading at a low price, you profit the difference. 01:09:36.920 --> 01:09:38.960 So the payoff for a put option is exactly 01:09:38.960 --> 01:09:45.120 the reverse, maximum of zero and K minus S. 01:09:45.120 --> 01:09:48.895 Now, it's very important that you understand this asymmetry, 01:09:48.895 --> 01:09:50.520 because that asymmetry is going to lead 01:09:50.520 --> 01:09:54.060 to all sorts of interesting things about these instruments. 01:09:54.060 --> 01:09:58.900 And before we go and talk about that kind of asymmetry, 01:09:58.900 --> 01:10:02.050 I want to give you another way of looking at options. 01:10:02.050 --> 01:10:07.110 Which is to look at options as a kind of insurance contract, 01:10:07.110 --> 01:10:11.130 because actually all insurance contracts are 01:10:11.130 --> 01:10:14.280 a form of an option. 01:10:14.280 --> 01:10:16.660 So let me give you an example. 01:10:16.660 --> 01:10:21.150 Suppose that you want to insure the value of a particular stock 01:10:21.150 --> 01:10:22.020 that you're holding. 01:10:22.020 --> 01:10:24.210 You're holding General Electric and it's 01:10:24.210 --> 01:10:26.910 trading at $20 a share, and you'd 01:10:26.910 --> 01:10:34.760 like to make sure that it never goes below $18 a share. 01:10:34.760 --> 01:10:39.720 You want to buy insurance that if it goes below $18 a share, 01:10:39.720 --> 01:10:44.210 you will get paid $18 a share. 01:10:44.210 --> 01:10:49.520 Well, the way you do that is you buy a put option. 01:10:49.520 --> 01:10:52.880 A put option on General Electric where the strike 01:10:52.880 --> 01:10:57.070 price is $18 a share. 01:10:57.070 --> 01:11:00.310 Because if it goes below $18 a share, 01:11:00.310 --> 01:11:03.370 you get to sell General Electric for that $18. 01:11:03.370 --> 01:11:05.650 So you'll get the $18, regardless 01:11:05.650 --> 01:11:10.330 of whether it goes to $10, or $5, or who knows what. 01:11:10.330 --> 01:11:13.700 It turns out that the put option is exactly like insurance, 01:11:13.700 --> 01:11:16.450 and let's take a look and see why. 01:11:16.450 --> 01:11:19.757 These are the typical terms of an insurance contract. 01:11:19.757 --> 01:11:21.340 What's the asset that you're insuring? 01:11:21.340 --> 01:11:23.170 General Electric. 01:11:23.170 --> 01:11:25.120 What's the current asset value? 01:11:25.120 --> 01:11:27.400 $20 a share. 01:11:27.400 --> 01:11:28.970 What's the term of the policy? 01:11:28.970 --> 01:11:30.880 How long do you have the policy for? 01:11:30.880 --> 01:11:33.490 It's the time to maturity. 01:11:33.490 --> 01:11:35.440 What's the maximum coverage? 01:11:35.440 --> 01:11:38.150 What are you covered for? 01:11:38.150 --> 01:11:39.274 $18 a share, that's right. 01:11:39.274 --> 01:11:40.940 That's what you bought the coverage for, 01:11:40.940 --> 01:11:43.330 that's what you're going to get if it goes below. 01:11:43.330 --> 01:11:45.290 What's the deductible? 01:11:45.290 --> 01:11:49.760 How much could you lose before the insurance kicks in? 01:11:49.760 --> 01:11:50.990 $2 a share, exactly. 01:11:50.990 --> 01:11:52.040 That's the deductible. 01:11:52.040 --> 01:11:55.450 And finally, what does it cost you 01:11:55.450 --> 01:11:57.850 to buy this insurance, what's the insurance premium? 01:12:00.590 --> 01:12:02.270 Exactly, the price of the put. 01:12:02.270 --> 01:12:03.800 That's it. 01:12:03.800 --> 01:12:05.270 Beautiful thing. 01:12:05.270 --> 01:12:08.510 A put option is nothing more than an insurance contract 01:12:08.510 --> 01:12:10.850 on the value of a stock. 01:12:10.850 --> 01:12:13.310 And it's going to turn out that a call 01:12:13.310 --> 01:12:18.740 option will be intimately tied to what a put option is. 01:12:18.740 --> 01:12:23.630 So every call option can be converted into a portfolio that 01:12:23.630 --> 01:12:25.340 includes a put. 01:12:25.340 --> 01:12:30.590 So all options you can think of as insurance contracts, 01:12:30.590 --> 01:12:33.860 but there are a few differences. 01:12:33.860 --> 01:12:35.450 The difference between an option is 01:12:35.450 --> 01:12:38.530 that you can exercise it early. 01:12:38.530 --> 01:12:41.720 So for example, for whatever reason, 01:12:41.720 --> 01:12:44.350 if you decide that you want to buy General Electric at $18 01:12:44.350 --> 01:12:48.230 a share, when it's trading at $17.50, 01:12:48.230 --> 01:12:49.960 and you still have one month to go. 01:12:49.960 --> 01:12:54.160 But you want to get paid that $18 now, you can do that. 01:12:54.160 --> 01:12:56.530 You can't do that with your car insurance, right? 01:12:56.530 --> 01:12:59.770 I guess you could, you could ram it into a post, 01:12:59.770 --> 01:13:01.480 and I want to get paid now, so let's-- 01:13:01.480 --> 01:13:04.600 [LAUGHTER] But that's not really considered a proper thing 01:13:04.600 --> 01:13:06.100 to do. 01:13:06.100 --> 01:13:07.600 So early exercise is one difference. 01:13:07.600 --> 01:13:10.300 Second difference is marketability. 01:13:10.300 --> 01:13:14.360 If at some point you don't want the insurance anymore, 01:13:14.360 --> 01:13:17.060 you can't get rid of it and give it to somebody else. 01:13:17.060 --> 01:13:21.350 You can't transfer your auto insurance to your friend, 01:13:21.350 --> 01:13:24.020 if you decide you don't need it anymore. 01:13:24.020 --> 01:13:27.650 But you can transfer the insurance policy here. 01:13:27.650 --> 01:13:31.650 You can sell the option, you can sell it. 01:13:31.650 --> 01:13:33.420 And also there are dividends that 01:13:33.420 --> 01:13:36.090 are being paid on the stock that you have to worry about 01:13:36.090 --> 01:13:39.840 with options, whereas with a typical insurance contract 01:13:39.840 --> 01:13:42.060 a car doesn't necessarily pay dividends. 01:13:42.060 --> 01:13:45.000 And the reason that's important is when it pays dividends, 01:13:45.000 --> 01:13:48.240 the value goes down, and so you have to make adjustments 01:13:48.240 --> 01:13:49.800 for that with an option. 01:13:49.800 --> 01:13:52.904 You have to protect an option for dividend payments. 01:13:52.904 --> 01:13:54.570 You don't need to do that for insurance, 01:13:54.570 --> 01:13:57.120 because typically you don't assume that the insurance 01:13:57.120 --> 01:14:03.311 value, the value of the asset goes down that much over time. 01:14:03.311 --> 01:14:03.810 Yep? 01:14:03.810 --> 01:14:05.460 AUDIENCE: When they buy the put option, 01:14:05.460 --> 01:14:10.530 they also eliminated the chance to enjoy it, 01:14:10.530 --> 01:14:14.960 from the prices are going to go up, 01:14:14.960 --> 01:14:18.410 with the futures we'd have a higher value. 01:14:18.410 --> 01:14:21.390 ANDREW LO: Well, no that's actually not true. 01:14:21.390 --> 01:14:23.070 With the put option, it gives you 01:14:23.070 --> 01:14:25.890 the right to sell the stock. 01:14:25.890 --> 01:14:28.680 If you buy the stock and you hold onto it, 01:14:28.680 --> 01:14:33.400 and you also buy a put, that protects the downside. 01:14:33.400 --> 01:14:35.610 But the upside, that's all yours. 01:14:38.600 --> 01:14:41.360 Because as the stock goes up, what 01:14:41.360 --> 01:14:43.010 happens to the value of the put? 01:14:43.010 --> 01:14:44.690 AUDIENCE: It's going to zero. 01:14:44.690 --> 01:14:46.610 ANDREW LO: Exactly, it stops at zero. 01:14:46.610 --> 01:14:52.430 So as the stock goes up, the put doesn't have any value anymore. 01:14:52.430 --> 01:14:54.560 It becomes worthless, worth less and less. 01:14:54.560 --> 01:14:57.170 And on the date of expiration, if the stock 01:14:57.170 --> 01:15:01.010 is way above the value of the strike, 01:15:01.010 --> 01:15:03.380 then it expires worthless. 01:15:03.380 --> 01:15:05.100 It doesn't go negative. 01:15:05.100 --> 01:15:08.210 If it went negative, if you had a futures position, 01:15:08.210 --> 01:15:12.070 then you'd be right, you've actually capped your gains. 01:15:12.070 --> 01:15:13.400 But this doesn't. 01:15:13.400 --> 01:15:17.920 See with this you get the best of both, or so it seems. 01:15:17.920 --> 01:15:22.360 You get the upside, but it protects the downside. 01:15:22.360 --> 01:15:27.050 And as you all probably know, insurance is not cheap. 01:15:27.050 --> 01:15:30.830 So it sounds good, but you've got to pay for this. 01:15:30.830 --> 01:15:36.910 And so you bet that the price of a call option or put option 01:15:36.910 --> 01:15:40.240 is not zero when you strike it. 01:15:40.240 --> 01:15:43.610 Unlike a futures contract that's worthless, 01:15:43.610 --> 01:15:46.970 an option is not worthless on day zero. 01:15:46.970 --> 01:15:48.200 It's worth a lot. 01:15:48.200 --> 01:15:52.550 For example, right now what's really expensive-- 01:15:52.550 --> 01:15:56.100 and if you want to check this, you could take a look for fun. 01:15:56.100 --> 01:15:59.130 If you want to buy insurance on the S&P 500-- 01:15:59.130 --> 01:16:01.230 now we've had a great rally on Monday, 01:16:01.230 --> 01:16:04.500 the S&P was up 1,000 points. 01:16:04.500 --> 01:16:08.010 If you want to buy insurance on the S&P 500 index, 01:16:08.010 --> 01:16:08.980 you can do that. 01:16:08.980 --> 01:16:12.010 There are options on the index. 01:16:12.010 --> 01:16:14.040 So you might say, OK let's say that the S&P is 01:16:14.040 --> 01:16:18.150 at 1,000 today, I would like to buy protection 01:16:18.150 --> 01:16:22.620 that over the next month it doesn't go down 01:16:22.620 --> 01:16:27.160 by more than 100 points, 10%. 01:16:27.160 --> 01:16:28.000 So what do you do? 01:16:28.000 --> 01:16:33.510 You buy a put option on the S&P with the strike price of what? 01:16:33.510 --> 01:16:34.920 900, right. 01:16:34.920 --> 01:16:37.954 OK, for a month. 01:16:37.954 --> 01:16:39.120 That's what you want to buy. 01:16:41.660 --> 01:16:43.640 Go out and calculate that price. 01:16:43.640 --> 01:16:46.460 You're going to be shocked at how expensive it is, 01:16:46.460 --> 01:16:48.860 to get that insurance for four weeks. 01:16:48.860 --> 01:16:50.570 Four weeks, that's all. 01:16:50.570 --> 01:16:55.220 It's really expensive today. 01:16:55.220 --> 01:16:58.400 I think it's approximately 10 times more expensive today, 01:16:58.400 --> 01:17:00.440 than it was a year ago. 01:17:00.440 --> 01:17:05.740 The implied volatility is up by at least an order of magnitude. 01:17:05.740 --> 01:17:10.710 So if you want that insurance, it's available, 01:17:10.710 --> 01:17:11.990 but you have to pay for it. 01:17:11.990 --> 01:17:15.800 So the question in all of these things is is it worth it? 01:17:15.800 --> 01:17:17.644 In order to decide whether it's worth it, 01:17:17.644 --> 01:17:18.810 you've got to do two things. 01:17:18.810 --> 01:17:22.100 First look into the inner most workings of your own soul 01:17:22.100 --> 01:17:25.335 and ask how frightened you truly are. 01:17:25.335 --> 01:17:27.710 And the second thing you got to do is look at the market. 01:17:27.710 --> 01:17:30.500 And is the market functioning reasonably well, 01:17:30.500 --> 01:17:36.634 or is the market reflecting all of these kinds of crazy fears. 01:17:36.634 --> 01:17:39.050 In order for us to be able to talk about it intelligently, 01:17:39.050 --> 01:17:41.510 we need a way to price it. 01:17:41.510 --> 01:17:43.702 We need the kind of logic that I showed you 01:17:43.702 --> 01:17:44.660 with futures contracts. 01:17:44.660 --> 01:17:45.990 And we're going to get that logic. 01:17:45.990 --> 01:17:48.198 I'm going to show you how to price these things using 01:17:48.198 --> 01:17:51.340 a very, very simple model that is incredibly powerful, 01:17:51.340 --> 01:17:52.340 but we're not there yet. 01:17:52.340 --> 01:17:53.310 Before we do that, I want to make sure 01:17:53.310 --> 01:17:54.768 you understand what these contracts 01:17:54.768 --> 01:17:58.220 can do for you in terms of changing your payoff 01:17:58.220 --> 01:17:59.911 profiles of your portfolio. 01:17:59.911 --> 01:18:00.411 Yeah? 01:18:00.411 --> 01:18:02.135 AUDIENCE: So wouldn't European option 01:18:02.135 --> 01:18:04.520 be similar to a futures, since you have 01:18:04.520 --> 01:18:06.650 that you can only exercise on maturity date? 01:18:06.650 --> 01:18:08.830 ANDREW LO: Well, no, that's not what 01:18:08.830 --> 01:18:10.240 makes it similar to a futures. 01:18:10.240 --> 01:18:13.360 Because while you cannot exercise it early, 01:18:13.360 --> 01:18:16.470 you never have to exercise it at all. 01:18:16.470 --> 01:18:21.380 So a European option gives you only one date 01:18:21.380 --> 01:18:24.920 where you are able to exercise, but even on that date 01:18:24.920 --> 01:18:27.220 you never have to exercise it. 01:18:27.220 --> 01:18:29.150 With the futures contract, you have 01:18:29.150 --> 01:18:31.010 to exercise it on that day. 01:18:31.010 --> 01:18:32.442 You've made a commitment. 01:18:32.442 --> 01:18:35.160 AUDIENCE: But it would have a net present value of zero. 01:18:35.160 --> 01:18:38.950 ANDREW LO: No, no, it won't, because still on that date 01:18:38.950 --> 01:18:41.830 you have a positive amount of protection. 01:18:41.830 --> 01:18:43.480 Like the example I gave you. 01:18:43.480 --> 01:18:47.020 Let's suppose that I bought a European S&P 01:18:47.020 --> 01:18:51.830 option for the day after election day, Wednesday, 01:18:51.830 --> 01:18:53.860 November 3rd is it. 01:18:53.860 --> 01:18:56.150 That will have positive value today. 01:18:56.150 --> 01:18:57.640 In other words, I'm going to have 01:18:57.640 --> 01:18:59.560 to pay money in order for you guys 01:18:59.560 --> 01:19:02.320 to sell it to me, because you're going to be providing me 01:19:02.320 --> 01:19:05.800 with some protection that if the wrong thing happens on Tuesday, 01:19:05.800 --> 01:19:08.154 the world is not going to blow up on Wednesday. 01:19:08.154 --> 01:19:09.820 I'm not telling you what the wrong thing 01:19:09.820 --> 01:19:12.100 is, I'm neutral in all of this. 01:19:12.100 --> 01:19:16.340 But that's an example where that insurance really has value. 01:19:16.340 --> 01:19:18.330 So you're not going to give it to me for free, 01:19:18.330 --> 01:19:19.579 and I'm willing to pay for it. 01:19:22.190 --> 01:19:23.780 All right, since we're out of time, 01:19:23.780 --> 01:19:26.570 I'm going to just leave you with this diagram that 01:19:26.570 --> 01:19:30.170 shows you the difference between a call option and a futures 01:19:30.170 --> 01:19:31.100 contract. 01:19:31.100 --> 01:19:33.380 Remember the futures contract what that looked like-- 01:19:33.380 --> 01:19:34.580 that was a straight line. 01:19:34.580 --> 01:19:35.750 Right Exactly. 01:19:35.750 --> 01:19:38.540 This is not a straight line, this is kinked-- 01:19:38.540 --> 01:19:40.610 very kinky security. 01:19:40.610 --> 01:19:42.770 And so we're going to talk next time 01:19:42.770 --> 01:19:45.290 about how to price kinky securities, 01:19:45.290 --> 01:19:47.570 and how to combine them, and engage in even more 01:19:47.570 --> 01:19:48.770 kinky kinds of payoffs. 01:19:48.770 --> 01:19:50.620 [LAUGHTER]