WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.850 Commons license. 00:00:03.850 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.150 continue to offer high-quality educational resources for free. 00:00:10.150 --> 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 --> 00:00:17.860 at ocw.mit.edu. 00:00:21.780 --> 00:00:23.670 PROFESSOR: So today, what I want to do 00:00:23.670 --> 00:00:27.330 is to continue where we were last time with option pricing. 00:00:27.330 --> 00:00:30.090 As I promised you last time, having 00:00:30.090 --> 00:00:31.950 gone through the history of option pricing 00:00:31.950 --> 00:00:34.170 and the special role that MIT played, 00:00:34.170 --> 00:00:37.470 today I actually want to do some option pricing. 00:00:37.470 --> 00:00:40.410 I want to show you a simple but extraordinarily powerful 00:00:40.410 --> 00:00:44.910 model for actually coming up with a theoretical pricing 00:00:44.910 --> 00:00:48.810 formula for options, and frankly, 00:00:48.810 --> 00:00:50.120 all derivative securities. 00:00:50.120 --> 00:00:52.620 So we're gonna actually do that in the space of about a half 00:00:52.620 --> 00:00:54.600 an hour, and then we're going to conclude. 00:00:54.600 --> 00:00:57.270 And I want to turn, then, to the next lecture, which 00:00:57.270 --> 00:00:59.010 is on risk and return. 00:00:59.010 --> 00:01:01.770 I want to now, after we finish option pricing, 00:01:01.770 --> 00:01:06.360 take on the challenge of trying to understand risk 00:01:06.360 --> 00:01:10.890 in a much more concrete way than we've done up until now. 00:01:10.890 --> 00:01:19.170 OK, so let's turn to lecture 10 and 11. 00:01:19.170 --> 00:01:25.345 And I'd like you to take a look at slide 16. 00:01:28.680 --> 00:01:37.880 OK, this will be the first model of option pricing 00:01:37.880 --> 00:01:39.374 that any of you have ever seen. 00:01:39.374 --> 00:01:40.790 You've all heard of Black-Scholes. 00:01:40.790 --> 00:01:42.710 We talked a bit about it last time. 00:01:42.710 --> 00:01:46.730 Frankly, this is a simpler version of option pricing 00:01:46.730 --> 00:01:49.880 that ultimately can actually be used 00:01:49.880 --> 00:01:52.400 to derive the Black-Scholes formula as well. 00:01:53.460 --> 00:01:55.550 But the reason I love this model is 00:01:55.550 --> 00:01:59.360 because it is so simple that with only basic high school 00:01:59.360 --> 00:02:03.500 algebra, you can actually work out all of the analytics. 00:02:03.500 --> 00:02:05.720 So all of you already have the math 00:02:05.720 --> 00:02:09.139 that it takes to implement this formula, 00:02:09.139 --> 00:02:11.690 and even to derive the formula. 00:02:11.690 --> 00:02:15.110 But the underlying economics is extraordinarily deep, 00:02:15.110 --> 00:02:17.390 and so it's a wonderful way of sort 00:02:17.390 --> 00:02:22.430 of getting a handle on how these very complex formulas work. 00:02:22.430 --> 00:02:23.880 So here's what we're going to do. 00:02:23.880 --> 00:02:27.690 We're going to simplify the problem in the following way. 00:02:27.690 --> 00:02:29.000 We're going to use-- 00:02:29.000 --> 00:02:30.530 the framework, by the way, is called 00:02:30.530 --> 00:02:32.630 the binomial option-pricing model 00:02:32.630 --> 00:02:37.820 that was derived by our very own John Cox, Steve Ross, and Mark 00:02:37.820 --> 00:02:40.100 Rubenstein of UC Berkeley. 00:02:40.100 --> 00:02:43.310 And although this is a simpler version 00:02:43.310 --> 00:02:45.710 of option pricing than Black and Scholes, 00:02:45.710 --> 00:02:51.080 it turns out that on the street, this is used much more commonly 00:02:51.080 --> 00:02:53.480 than the Black-Scholes formula. 00:02:53.480 --> 00:02:55.110 So let me show you how it works. 00:02:55.110 --> 00:02:59.360 We're going to start with a very simple framework of one period 00:02:59.360 --> 00:03:01.520 option pricing, meaning we're going 00:03:01.520 --> 00:03:07.400 to focus on a stock that survives for two periods-- 00:03:07.400 --> 00:03:09.840 this period and the next period. 00:03:09.840 --> 00:03:12.230 And then we're going to consider the pricing of an option 00:03:12.230 --> 00:03:16.070 on that stock that expires next period. 00:03:16.070 --> 00:03:20.050 We're going to figure out what the price is this period. 00:03:20.050 --> 00:03:23.150 So we've got a stock XYZ, and let's suppose 00:03:23.150 --> 00:03:25.540 the current stock price is S0. 00:03:25.540 --> 00:03:28.300 And let's suppose that we have a call option 00:03:28.300 --> 00:03:31.030 on this stock with a strike price of K, 00:03:31.030 --> 00:03:35.710 and where the option expires tomorrow. 00:03:35.710 --> 00:03:39.950 And so tomorrow's value of the option 00:03:39.950 --> 00:03:44.780 is simply equal to C1, which is the maximum of tomorrow's stock 00:03:44.780 --> 00:03:50.180 price minus the strike, or 0, the bigger of those two. 00:03:50.180 --> 00:03:52.940 That's the payoff for the call option. 00:03:52.940 --> 00:03:55.040 And the question that we want to attack 00:03:55.040 --> 00:03:58.260 is, what is the option's price today? 00:03:58.260 --> 00:04:00.110 In other words, what is C0? 00:04:00.110 --> 00:04:02.930 So we draw a timeline, as I've told you, 00:04:02.930 --> 00:04:04.460 for every one of these problems. 00:04:04.460 --> 00:04:07.530 Draw a timeline just so that there's no confusion. 00:04:07.530 --> 00:04:11.330 So tomorrow, the stock price is going to be worth S1, 00:04:11.330 --> 00:04:14.540 and the option price is just equal to C1, 00:04:14.540 --> 00:04:17.660 which is the payoff, since it expires tomorrow. 00:04:17.660 --> 00:04:21.019 And the payoff is just the maximum of S1 minus K and 0. 00:04:21.019 --> 00:04:25.220 And the object of our focus is to try 00:04:25.220 --> 00:04:28.495 to figure out what the value of the option is today. 00:04:28.495 --> 00:04:30.620 And so I'm going to argue that if we can figure out 00:04:30.620 --> 00:04:33.230 what it is today, based upon this, 00:04:33.230 --> 00:04:35.690 then we can actually generalize it in a very natural way 00:04:35.690 --> 00:04:38.557 to figure out what the price is for any number of periods 00:04:38.557 --> 00:04:39.140 in the future. 00:04:41.660 --> 00:04:42.960 So how do we do that? 00:04:42.960 --> 00:04:48.470 Well, we first have to make an assumption about how 00:04:48.470 --> 00:04:50.960 the stock price behaves. 00:04:50.960 --> 00:04:52.700 As I mentioned last time, we need 00:04:52.700 --> 00:04:55.880 to say something about the dynamics of stock prices, 00:04:55.880 --> 00:04:59.360 and remember that Bachelier, that French mathematician that 00:04:59.360 --> 00:05:02.720 came up with a rudimentary version of an option pricing 00:05:02.720 --> 00:05:06.170 formula in 1900, he developed the mathematics 00:05:06.170 --> 00:05:08.630 for Brownian motion, or a random walk, 00:05:08.630 --> 00:05:10.040 for the particular stock price. 00:05:10.040 --> 00:05:12.680 So you have to assume something about how 00:05:12.680 --> 00:05:14.510 the stock price moves. 00:05:14.510 --> 00:05:17.660 So what we're going to do in a simplified version 00:05:17.660 --> 00:05:23.920 is to assume that the stock price tomorrow is a coin flip. 00:05:23.920 --> 00:05:25.910 It's a Bernoulli trial. 00:05:25.910 --> 00:05:27.380 That's the technical term. 00:05:27.380 --> 00:05:29.870 So it either goes up or down. 00:05:29.870 --> 00:05:34.880 And if it goes up, it goes up by a gross amount u. 00:05:34.880 --> 00:05:37.400 So the value of the stock tomorrow, 00:05:37.400 --> 00:05:42.530 S1, is going to be equal to u multiplied by S0. 00:05:42.530 --> 00:05:45.200 So if it goes up by 10% tomorrow, 00:05:45.200 --> 00:05:49.490 then u is equal to 1.1. 00:05:49.490 --> 00:05:56.920 Or it can go down by a factor of d tomorrow, 00:05:56.920 --> 00:06:03.820 and so if it goes down by 10%, then d is 0.9. 00:06:03.820 --> 00:06:06.280 So we're going to simply assert that this 00:06:06.280 --> 00:06:09.680 is the statistical behavior of stock prices. 00:06:09.680 --> 00:06:13.360 Now, granted, this is a very, very strong simplification, 00:06:13.360 --> 00:06:15.340 but bear with me. 00:06:15.340 --> 00:06:17.985 After I derive the simple version of the pricing formula, 00:06:17.985 --> 00:06:19.360 I'm going to show you how to make 00:06:19.360 --> 00:06:21.490 it much, much more complex. 00:06:21.490 --> 00:06:23.950 And the additional complexity will 00:06:23.950 --> 00:06:27.340 be really simple to achieve once we understand 00:06:27.340 --> 00:06:30.310 this very basic version. 00:06:30.310 --> 00:06:32.920 Now, the probability of going up or down 00:06:32.920 --> 00:06:35.860 is not 50/50-- doesn't have to be 50/50. 00:06:35.860 --> 00:06:39.110 So I'm going to assert that it's equal to some probability of p 00:06:39.110 --> 00:06:40.210 and 1 minus p. 00:06:40.210 --> 00:06:43.210 So it either goes up by p or goes down with probability 1 00:06:43.210 --> 00:06:45.520 minus p, and the amount that it goes up or down 00:06:45.520 --> 00:06:47.410 is given by u and d. 00:06:47.410 --> 00:06:50.110 And I'm going to assert that u is greater than d. 00:06:50.110 --> 00:06:51.952 Question? 00:06:51.952 --> 00:06:53.660 AUDIENCE: So u and d are not the changes, 00:06:53.660 --> 00:06:57.540 but you're just assuming that, in your case, [INAUDIBLE].. 00:06:57.540 --> 00:06:59.142 PROFESSOR: Sorry, that they're-- 00:06:59.142 --> 00:07:01.517 AUDIENCE: It's not necessary to always use [INAUDIBLE] d, 00:07:01.517 --> 00:07:02.334 or can you assume-- 00:07:02.334 --> 00:07:04.750 PROFESSOR: Yeah, I'm assuming as a matter of normalization 00:07:04.750 --> 00:07:06.310 that u is greater than d. 00:07:06.310 --> 00:07:07.060 It doesn't matter. 00:07:07.060 --> 00:07:09.531 I mean, one thing has to be bigger than the other, 00:07:09.531 --> 00:07:11.780 so I just may as well assume that u is greater than d. 00:07:11.780 --> 00:07:12.791 Yeah, question? 00:07:12.791 --> 00:07:15.041 AUDIENCE: Is it necessary to add up to 1 [INAUDIBLE]?? 00:07:15.041 --> 00:07:17.249 PROFESSOR: u and do don't have to add up to anything. 00:07:17.249 --> 00:07:18.090 That's right. 00:07:18.090 --> 00:07:21.440 p and 1 minus p always add up to 1, right. 00:07:21.440 --> 00:07:24.170 So for example, if this is a growth stock that's 00:07:24.170 --> 00:07:35.340 really doing well, then u may be 1.1, 10%, and d may be 0.99. 00:07:35.340 --> 00:07:38.270 So in other words, when it goes down, it goes down by 1%. 00:07:38.270 --> 00:07:40.320 When it goes up, it goes up by 10%. 00:07:40.320 --> 00:07:42.250 So on average, when you multiply by p, 00:07:42.250 --> 00:07:45.140 1 minus p, depending on what they are, 00:07:45.140 --> 00:07:48.145 you can get a stock that's got a positive drift. 00:07:48.145 --> 00:07:49.520 If, on the other hand, you've got 00:07:49.520 --> 00:07:51.720 a stock that's declining in value, 00:07:51.720 --> 00:07:58.490 then it may end up that d much smaller than u, and 1 minus p 00:07:58.490 --> 00:08:00.680 is bigger than p, which means that you're 00:08:00.680 --> 00:08:03.770 more likely to be going down than you are going up. 00:08:03.770 --> 00:08:05.207 So it's pretty general. 00:08:05.207 --> 00:08:06.640 Yeah? 00:08:06.640 --> 00:08:10.245 AUDIENCE: u is bigger than 1 and d is lower than 1, or you could 00:08:10.245 --> 00:08:10.856 [INAUDIBLE]? 00:08:10.856 --> 00:08:12.230 PROFESSOR: It doesn't have to be. 00:08:12.230 --> 00:08:15.560 But typically you would think that in the up state, 00:08:15.560 --> 00:08:17.762 it's going to be bigger than 1, and the down state, 00:08:17.762 --> 00:08:18.720 it will be less than 1. 00:08:18.720 --> 00:08:21.110 But it doesn't have to be. 00:08:21.110 --> 00:08:25.610 What I'm going to normalize it to be is u is greater than d. 00:08:25.610 --> 00:08:28.281 And later on, we may make some other economic assumptions 00:08:28.281 --> 00:08:30.530 that I'll come to that will tell you a little bit more 00:08:30.530 --> 00:08:32.000 about what u and d are. 00:08:36.059 --> 00:08:40.770 Now, if it's true that the stock price can only 00:08:40.770 --> 00:08:45.050 take on two values tomorrow, then it 00:08:45.050 --> 00:08:49.700 stands to reason that the option can only take on two values 00:08:49.700 --> 00:08:51.470 tomorrow. 00:08:51.470 --> 00:08:52.700 And those are the two values. 00:08:52.700 --> 00:08:56.060 It's going to be Cu and Cd. 00:08:56.060 --> 00:09:00.320 Cu is where the stock price goes up to u times S0. 00:09:00.320 --> 00:09:04.040 Therefore, the option's going to be worth u S0 minus K, 0, 00:09:04.040 --> 00:09:06.200 maximum of those two. 00:09:06.200 --> 00:09:10.340 And similarly, if it turns out that the stock price goes down 00:09:10.340 --> 00:09:13.510 tomorrow, then the option is worth this tomorrow. 00:09:16.700 --> 00:09:18.530 Two values for the stock tomorrow 00:09:18.530 --> 00:09:23.030 implies two values for the option tomorrow. 00:09:23.030 --> 00:09:26.430 Any questions about that? 00:09:26.430 --> 00:09:31.680 OK, so having said that, we can now 00:09:31.680 --> 00:09:38.100 proceed to ask the question, given this simple framework, 00:09:38.100 --> 00:09:43.270 what should the option price today depend on? 00:09:43.270 --> 00:09:46.720 It's going to be a function of a bunch of parameters. 00:09:46.720 --> 00:09:50.040 So what should it depend upon? 00:09:50.040 --> 00:09:55.210 Well, the parameters that are given are these-- 00:09:55.210 --> 00:10:00.630 the stock price today, the strike price, u and d, 00:10:00.630 --> 00:10:05.610 p, and the interest rate between today and tomorrow. 00:10:05.610 --> 00:10:08.099 Those are the only parameters that we have. 00:10:08.099 --> 00:10:08.640 These are it. 00:10:08.640 --> 00:10:11.310 This is everything. 00:10:11.310 --> 00:10:15.650 It's going to turn out that with the simple framework that 00:10:15.650 --> 00:10:21.500 I've put down, we will be able to derive a closed-form 00:10:21.500 --> 00:10:25.970 analytical expression for what the option price has to be 00:10:25.970 --> 00:10:28.010 today-- 00:10:28.010 --> 00:10:28.910 C0. 00:10:28.910 --> 00:10:31.845 I'm going to do that for you in just a minute. 00:10:31.845 --> 00:10:33.970 But it's going to turn out that that option pricing 00:10:33.970 --> 00:10:37.960 formula, that f of stuff, is going 00:10:37.960 --> 00:10:42.800 to depend on all of these parameters except for one. 00:10:42.800 --> 00:10:46.190 One of these parameters is going to drop out. 00:10:46.190 --> 00:10:50.760 In other words, one of these parameters is redundant. 00:10:50.760 --> 00:10:53.070 And anybody want to take a guess as to what 00:10:53.070 --> 00:10:54.450 that parameter might be? 00:10:54.450 --> 00:10:57.570 What parameter do you think might not 00:10:57.570 --> 00:11:00.260 matter for pricing an option? 00:11:00.260 --> 00:11:01.035 Yeah, Terry. 00:11:01.035 --> 00:11:02.762 AUDIENCE: The interest rate, the r? 00:11:02.762 --> 00:11:03.970 PROFESSOR: The interest rate. 00:11:03.970 --> 00:11:10.060 Well, that's a good guess, but that's not the case. 00:11:10.060 --> 00:11:11.880 That's what I would have guessed, 00:11:11.880 --> 00:11:14.280 because that seems to be the thing that 00:11:14.280 --> 00:11:17.430 should matter the least, given how important all 00:11:17.430 --> 00:11:19.122 of these other parameters are. 00:11:19.122 --> 00:11:20.580 Anybody want to take another guess? 00:11:20.580 --> 00:11:21.200 Yeah, Ken. 00:11:21.200 --> 00:11:22.449 AUDIENCE: Today's stock price. 00:11:22.449 --> 00:11:23.850 PROFESSOR: Today's stock price. 00:11:23.850 --> 00:11:24.940 That's another good guess. 00:11:24.940 --> 00:11:26.520 [LAUGHTER] 00:11:26.520 --> 00:11:30.660 Although that's not correct, because in both the case 00:11:30.660 --> 00:11:33.060 of the interest rate and today's stock price, 00:11:33.060 --> 00:11:35.400 you could ask the question, suppose the stock price 00:11:35.400 --> 00:11:38.370 were at $1,000 versus $10. 00:11:38.370 --> 00:11:40.400 That would matter, wouldn't it? 00:11:40.400 --> 00:11:43.800 Or if the interest rate were at 20% versus 1%, 00:11:43.800 --> 00:11:46.200 that should matter, shouldn't it? 00:11:46.200 --> 00:11:47.850 And it does. 00:11:47.850 --> 00:11:51.690 In fact, if you look at every single one of these parameters, 00:11:51.690 --> 00:11:57.030 none of them looks like they're unnecessary. 00:11:57.030 --> 00:11:59.080 It looks like all of them are required. 00:11:59.080 --> 00:12:01.468 Yeah, John. 00:12:01.468 --> 00:12:03.910 AUDIENCE: [INAUDIBLE] 00:12:03.910 --> 00:12:06.330 PROFESSOR: Well, the strike price remains the same, 00:12:06.330 --> 00:12:08.790 but the thing is that the question is whether or not 00:12:08.790 --> 00:12:11.680 the value of the option depends on the strike price. 00:12:11.680 --> 00:12:13.950 And if the option is, for example, in the money 00:12:13.950 --> 00:12:15.908 or out of the money, you would expect that that 00:12:15.908 --> 00:12:17.346 would make a big difference. 00:12:17.346 --> 00:12:19.130 AUDIENCE: [INAUDIBLE] 00:12:19.130 --> 00:12:20.420 PROFESSOR: Right. 00:12:20.420 --> 00:12:22.910 In fact, let me tell you that there 00:12:22.910 --> 00:12:26.390 is no good answer to this, because all of these parameters 00:12:26.390 --> 00:12:28.130 look like they belong. 00:12:28.130 --> 00:12:31.250 But I want to tell you that one of them will not. 00:12:31.250 --> 00:12:32.960 One of them will not be in here. 00:12:32.960 --> 00:12:38.300 And this is going to be a major source of both confusion 00:12:38.300 --> 00:12:42.770 and illumination for what really depends on-- 00:12:42.770 --> 00:12:44.627 what option pricing really depends on. 00:12:44.627 --> 00:12:47.210 All right, so let me just show you how we're going to do this. 00:12:47.210 --> 00:12:50.030 Let me illustrate to you the method. 00:12:50.030 --> 00:12:53.720 And we're going to do this in the exact same way 00:12:53.720 --> 00:12:57.830 that we've priced virtually everything under the sun. 00:12:57.830 --> 00:13:01.310 We're going to use an arbitrage argument. 00:13:01.310 --> 00:13:03.650 I'm going to construct a portfolio 00:13:03.650 --> 00:13:08.940 that will have the identical payoff to the option, 00:13:08.940 --> 00:13:12.090 and therefore if the portfolio has the exact same cash 00:13:12.090 --> 00:13:15.180 flows as the option, then the cost 00:13:15.180 --> 00:13:17.070 of constructing that portfolio has 00:13:17.070 --> 00:13:20.860 to be the price of the option. 00:13:20.860 --> 00:13:23.850 Moreover, if it's not, you're going 00:13:23.850 --> 00:13:27.250 to be very happy, because that will mean that there 00:13:27.250 --> 00:13:28.810 is an arbitrage opportunity. 00:13:28.810 --> 00:13:30.820 That is, there's money to be made. 00:13:30.820 --> 00:13:33.970 If this theory fails, then you're 00:13:33.970 --> 00:13:37.780 going to be able to get rich beyond your wildest dreams. 00:13:37.780 --> 00:13:42.254 So we're hoping for a violation of this. 00:13:42.254 --> 00:13:43.420 So let's see how we do that. 00:13:45.990 --> 00:13:49.250 I want you to now forget about the option for a moment, 00:13:49.250 --> 00:13:51.980 and I want you to imagine that at time 0, 00:13:51.980 --> 00:13:56.890 we construct a portfolio consisting of stocks 00:13:56.890 --> 00:14:00.740 and riskless bonds, in particular delta 00:14:00.740 --> 00:14:06.610 shares of stocks and B dollars of riskless bonds-- 00:14:06.610 --> 00:14:10.920 riskless in terms of default. 00:14:10.920 --> 00:14:20.530 And the total cost of this portfolio today, time 0, 00:14:20.530 --> 00:14:23.140 is simply equal to the price per share times 00:14:23.140 --> 00:14:27.400 the number of shares of stock, so that's S0 times delta, 00:14:27.400 --> 00:14:30.910 plus the value of the bonds that I'm 00:14:30.910 --> 00:14:33.190 buying-- the market value of the bonds 00:14:33.190 --> 00:14:35.650 that I'm buying today, or selling. 00:14:35.650 --> 00:14:38.050 So B could be a positive or negative number. 00:14:38.050 --> 00:14:41.860 Delta could be a positive or a negative number. 00:14:41.860 --> 00:14:48.110 And that's my cost today, time 0. 00:14:48.110 --> 00:14:50.070 Now, I want to look at the payoff 00:14:50.070 --> 00:14:52.050 tomorrow for this portfolio. 00:14:52.050 --> 00:14:54.040 So V1 is the payoff for the portfolio. 00:14:54.040 --> 00:14:56.010 That's what it's worth tomorrow. 00:14:56.010 --> 00:15:02.450 And V1 is going to be given by the value of the stocks 00:15:02.450 --> 00:15:04.440 and the value of the bonds. 00:15:04.440 --> 00:15:06.410 Now, the stocks are going to be-- 00:15:06.410 --> 00:15:07.590 there are two possibilities. 00:15:07.590 --> 00:15:10.820 Either the stock goes up or the stock goes down. 00:15:10.820 --> 00:15:14.340 And if it goes up, it'll be worth u S0 times delta, 00:15:14.340 --> 00:15:17.510 and if it goes down it'll be worth d S0 times delta. 00:15:17.510 --> 00:15:21.410 I don't know whether it'll go up or down, but whatever it does, 00:15:21.410 --> 00:15:24.170 this is the value tomorrow. 00:15:24.170 --> 00:15:26.090 Now, what about my bond portfolio? 00:15:26.090 --> 00:15:29.750 Well, I bought B bonds, and r now 00:15:29.750 --> 00:15:32.830 is the gross rate of return, the gross interest rate, 00:15:32.830 --> 00:15:37.040 so it's a number like 1.03 or 1.05. 00:15:37.040 --> 00:15:39.170 And the reason that I'm switching notation 00:15:39.170 --> 00:15:42.610 is I'm following the notation used originally by Cox, Ross, 00:15:42.610 --> 00:15:46.370 and Rubenstein, so I apologize for the kind 00:15:46.370 --> 00:15:49.760 of cognitive dissonance that this may generate. 00:15:49.760 --> 00:15:53.720 But this r, the way that Cox, Ross, and Rubenstein wrote it, 00:15:53.720 --> 00:15:56.160 was meant to be a gross rate of return. 00:15:56.160 --> 00:15:58.910 So you'll never see a 1 plus r, because this-- 00:15:58.910 --> 00:16:02.859 in their framework, because this already contains the 1. 00:16:02.859 --> 00:16:04.650 So just keep that in the back of your mind, 00:16:04.650 --> 00:16:07.730 and make a note of that. r is the gross interest rate. 00:16:07.730 --> 00:16:10.680 It's 1 plus the net interest rate, 00:16:10.680 --> 00:16:16.491 so it's a number like 1.03 for a 3% rate of return. 00:16:16.491 --> 00:16:17.990 Actually nowadays, it should be more 00:16:17.990 --> 00:16:23.667 like 1.01 for short-term interest rates, or less. 00:16:23.667 --> 00:16:25.250 Now, you'll notice that whether or not 00:16:25.250 --> 00:16:27.230 stocks go up or down has no impact 00:16:27.230 --> 00:16:28.640 on your riskless borrowing. 00:16:28.640 --> 00:16:31.070 You're going to get r times B no matter what, 00:16:31.070 --> 00:16:34.320 or you're going to owe r times B if B 00:16:34.320 --> 00:16:38.360 was a negative number, in both cases, because it's riskless. 00:16:38.360 --> 00:16:40.340 It has nothing to do with whether or not 00:16:40.340 --> 00:16:43.700 stocks go up or down. 00:16:43.700 --> 00:16:45.860 OK, now here's what I want you to do. 00:16:45.860 --> 00:16:51.530 I want you to select a specific amount of stocks and bonds 00:16:51.530 --> 00:16:58.350 at date zero in order to make two things true. 00:16:58.350 --> 00:17:05.369 I want you to select delta and B so as to satisfy these two 00:17:05.369 --> 00:17:07.700 equations. 00:17:07.700 --> 00:17:10.990 I want you to pick delta and B so that in the up state, 00:17:10.990 --> 00:17:16.089 you get Cu and in the down state you get Cd. 00:17:16.089 --> 00:17:17.261 Now, what are Cu and Cd? 00:17:17.261 --> 00:17:18.219 Remember what they are? 00:17:18.219 --> 00:17:20.950 They're the value of the call option in the up 00:17:20.950 --> 00:17:22.990 state and the down state. 00:17:22.990 --> 00:17:24.329 And you know that in advance. 00:17:24.329 --> 00:17:26.079 You know what those two possibilities are. 00:17:26.079 --> 00:17:27.940 You don't know which one's going to occur, 00:17:27.940 --> 00:17:31.570 but you know that if the up state occurs, it'll be Cu, 00:17:31.570 --> 00:17:36.300 and if the down state occurs, it'll be Cd. 00:17:36.300 --> 00:17:42.540 So I want you to find two numbers, delta and B, 00:17:42.540 --> 00:17:45.370 that make those two equations true. 00:17:45.370 --> 00:17:47.935 Can you always do that? 00:17:47.935 --> 00:17:49.560 How do you know you can always do that? 00:17:53.420 --> 00:17:58.180 OK, can you always find a delta and a B 00:17:58.180 --> 00:18:03.980 to make those two relationships true? 00:18:03.980 --> 00:18:04.741 Yeah. 00:18:04.741 --> 00:18:07.200 AUDIENCE: You have two equations and two variables. 00:18:07.200 --> 00:18:11.060 PROFESSOR: Ah, you have two linear equations 00:18:11.060 --> 00:18:13.310 in two unknowns. 00:18:13.310 --> 00:18:15.960 And from basic high school algebra, 00:18:15.960 --> 00:18:19.550 you know that unless those two linear equations are 00:18:19.550 --> 00:18:24.820 multiples of each other, you can always find one-- 00:18:24.820 --> 00:18:29.430 exactly one solution that satisfies those two 00:18:29.430 --> 00:18:32.200 equations in two unknowns. 00:18:32.200 --> 00:18:37.910 Kind of a handy feature about linear equations. 00:18:37.910 --> 00:18:41.120 So as long as these two equations are said to be 00:18:41.120 --> 00:18:43.912 linearly independent-- that's a fancy way of saying that 00:18:43.912 --> 00:18:45.620 they're actually two different equations, 00:18:45.620 --> 00:18:48.690 they're not multiples of each other-- 00:18:48.690 --> 00:18:50.790 as long as these two equations are not 00:18:50.790 --> 00:18:53.070 multiples of each other, you can always 00:18:53.070 --> 00:18:57.900 find two numbers, delta and B, to make that true. 00:18:57.900 --> 00:18:58.890 And here they are. 00:18:58.890 --> 00:19:01.000 Those are the two numbers, delta star and B star. 00:19:01.000 --> 00:19:02.801 I solved the equation for you, not 00:19:02.801 --> 00:19:04.300 that you couldn't do it on your own, 00:19:04.300 --> 00:19:09.560 but for convenience, there it is. 00:19:09.560 --> 00:19:11.900 So let me tell you what we've done. 00:19:11.900 --> 00:19:16.990 We've put together a portfolio of stocks and bonds at date 0 00:19:16.990 --> 00:19:22.900 such that at time 1, the value of this portfolio 00:19:22.900 --> 00:19:26.950 is always equal to the value of the call 00:19:26.950 --> 00:19:32.960 option in no matter what state of the world actually occurs. 00:19:32.960 --> 00:19:36.870 Well, by the principle of arbitrage, 00:19:36.870 --> 00:19:41.610 what this tells us is that the cost of putting together 00:19:41.610 --> 00:19:47.490 this portfolio that replicates the call option's cash flows, 00:19:47.490 --> 00:19:49.860 the value of that portfolio at date 0 00:19:49.860 --> 00:19:54.680 must equal the price of a call option. 00:19:54.680 --> 00:19:58.350 So we're done. 00:19:58.350 --> 00:20:04.860 The solution of what is the call option price at date 0, 00:20:04.860 --> 00:20:06.900 it's given by this formula right here. 00:20:06.900 --> 00:20:11.530 There it is-- a closed-form solution. 00:20:11.530 --> 00:20:13.860 Now, before we beat up on it and say, 00:20:13.860 --> 00:20:16.590 gee, there's only two possibilities, 00:20:16.590 --> 00:20:18.210 life is more complicated than that, 00:20:18.210 --> 00:20:21.180 and also there's only one period, let's not-- 00:20:21.180 --> 00:20:23.050 let's not beat up on it just yet. 00:20:23.050 --> 00:20:26.340 Let's take a look to see whether or not this makes sense 00:20:26.340 --> 00:20:29.760 and whether we agree and understand 00:20:29.760 --> 00:20:32.070 that if, in fact, the assumptions are true, 00:20:32.070 --> 00:20:35.190 that this is indeed the price of an option. 00:20:35.190 --> 00:20:38.010 Because this is a pretty remarkable formula. 00:20:38.010 --> 00:20:41.190 It's a remarkable formula for its simplicity, 00:20:41.190 --> 00:20:45.060 and for the fact that we actually 00:20:45.060 --> 00:20:49.280 have been able to derive it explicitly. 00:20:49.280 --> 00:20:51.030 Now, the other amazing thing is that there 00:20:51.030 --> 00:20:53.340 is a missing parameter here. 00:20:53.340 --> 00:20:56.040 Now you see what the missing parameter is. 00:20:56.040 --> 00:21:01.410 What this formula doesn't depend on 00:21:01.410 --> 00:21:07.000 is the probability of the thing going up or down. 00:21:07.000 --> 00:21:08.820 Now, that's astonishing. 00:21:08.820 --> 00:21:13.320 It's astonishing because what it says is that you and I, 00:21:13.320 --> 00:21:17.370 we can disagree on whether General Electric is going 00:21:17.370 --> 00:21:21.250 to go up tomorrow or down tomorrow, 00:21:21.250 --> 00:21:24.460 and yet we still are going to agree 00:21:24.460 --> 00:21:27.580 on what the value of a General Electric call option 00:21:27.580 --> 00:21:29.710 is tomorrow. 00:21:29.710 --> 00:21:32.580 That's a remarkable fact, and it has 00:21:32.580 --> 00:21:36.810 to do with a very deep, deep phenomenon 00:21:36.810 --> 00:21:41.310 going on in option pricing, which is that option pricing is 00:21:41.310 --> 00:21:47.340 all about pricing the relative magnitude of the security 00:21:47.340 --> 00:21:50.190 relative to the stock price. 00:21:50.190 --> 00:21:55.020 And once we understand the basic features of the stock price, 00:21:55.020 --> 00:21:57.990 like whether or not it can go up or down 00:21:57.990 --> 00:22:04.800 by u or d, that's more important than the actual probabilities 00:22:04.800 --> 00:22:06.810 of u and d. 00:22:06.810 --> 00:22:11.670 So this expression-- and when you fill in for Cu and Cd, 00:22:11.670 --> 00:22:17.760 you can plug in for that maximum of u S0 minus k, 0, 00:22:17.760 --> 00:22:20.130 you'll see that there's no p in there as well. 00:22:23.260 --> 00:22:26.606 So any questions about this? 00:22:26.606 --> 00:22:27.105 Yeah. 00:22:27.800 --> 00:22:30.100 AUDIENCE: You just said that we could 00:22:30.100 --> 00:22:34.780 disagree on what we-- if the stock will go up or down, 00:22:34.780 --> 00:22:35.436 [INAUDIBLE]. 00:22:35.436 --> 00:22:38.060 PROFESSOR: Yeah, the probability of it going up or down, right. 00:22:38.060 --> 00:22:42.290 AUDIENCE: And what if we disagree on the actual number? 00:22:42.290 --> 00:22:44.520 PROFESSOR: Then we will disagree on the option price. 00:22:44.520 --> 00:22:47.494 So we have to agree on the u and the d. 00:22:47.494 --> 00:22:48.470 AUDIENCE: But if we-- 00:22:48.470 --> 00:22:50.236 PROFESSOR: Sorry-- yeah, the u and the d. 00:22:50.236 --> 00:22:52.569 AUDIENCE: [INAUDIBLE] we will disagree the option price, 00:22:52.569 --> 00:22:55.649 or that's why this market is possible, 00:22:55.649 --> 00:22:57.581 because someone will think it will go up-- 00:22:57.581 --> 00:22:59.490 PROFESSOR: No, no, it's not the reason 00:22:59.490 --> 00:23:01.220 that the market will be possible. 00:23:01.220 --> 00:23:03.260 The possibility of the market actually 00:23:03.260 --> 00:23:05.450 does depend on whether or not there's 00:23:05.450 --> 00:23:07.730 a demand for this particular kind of payoff. 00:23:07.730 --> 00:23:10.505 But that doesn't necessarily hinge on the u or the d. 00:23:10.505 --> 00:23:12.760 In other words, we can agree on the u and the d, 00:23:12.760 --> 00:23:15.440 but it turns out that you think that the price is going 00:23:15.440 --> 00:23:18.270 to go up, therefore, you want to have that kind of a call option 00:23:18.270 --> 00:23:18.770 bet. 00:23:18.770 --> 00:23:20.460 I think the price is going to go down, 00:23:20.460 --> 00:23:21.830 so I'm happy to sell it to you, because I think I'm 00:23:21.830 --> 00:23:23.450 going to get a good deal on it. 00:23:23.450 --> 00:23:25.760 So we disagree on the p. 00:23:25.760 --> 00:23:27.546 You think that there's a high p. 00:23:27.546 --> 00:23:29.600 I think it's a low p. 00:23:29.600 --> 00:23:31.160 That's what drives the market. 00:23:31.160 --> 00:23:34.880 And the beauty of this particular setup 00:23:34.880 --> 00:23:40.020 is that it tells you that you can actually agree on a price, 00:23:40.020 --> 00:23:41.790 but you have very different reasons 00:23:41.790 --> 00:23:43.920 for engaging in the transaction. 00:23:43.920 --> 00:23:49.330 And then you will have markets for this particular security. 00:23:49.330 --> 00:23:51.520 Now, I want to go through and look at this formula 00:23:51.520 --> 00:23:54.130 and try to understand it. 00:23:54.130 --> 00:23:58.300 First of all, we see that this formula is a weighted average 00:23:58.300 --> 00:24:00.730 of the value of the call option in the up 00:24:00.730 --> 00:24:02.980 state and the down state. 00:24:02.980 --> 00:24:05.270 It's a weighted average. 00:24:05.270 --> 00:24:08.300 And this part inside the bracket you 00:24:08.300 --> 00:24:11.870 can think of as a weighted average of the outcome. 00:24:11.870 --> 00:24:18.045 But then you discount it back to the 0th period using the one 00:24:18.045 --> 00:24:18.920 period interest rate. 00:24:18.920 --> 00:24:21.590 And again, remember this is not meant 00:24:21.590 --> 00:24:24.670 to be a perpetuity kind of expression. 00:24:24.670 --> 00:24:27.170 This r is a gross interest rate, so it 00:24:27.170 --> 00:24:30.440 is equivalent to our old 1 over 1 plus 00:24:30.440 --> 00:24:36.560 r, where the r that we used is the net interest rate. 00:24:36.560 --> 00:24:40.280 Here, because of the Cox, Ross, and Rubenstein notation, 00:24:40.280 --> 00:24:44.080 this is meant to be the gross interest rate. 00:24:44.080 --> 00:24:48.510 So this looks like a present value, 00:24:48.510 --> 00:24:51.120 because whatever is inside the bracket, 00:24:51.120 --> 00:24:54.090 you can think of as some weighted average 00:24:54.090 --> 00:25:02.280 of the value at date 1, and then this brings it back to date 0. 00:25:02.280 --> 00:25:04.200 But now let's look at the weighted average. 00:25:04.200 --> 00:25:09.720 The weights r minus d and u minus d, those-- 00:25:09.720 --> 00:25:15.550 it turns out that this plus this adds up to 1, 00:25:15.550 --> 00:25:18.520 so indeed, it is a weighted average. 00:25:18.520 --> 00:25:23.560 When you multiply by theta and 1 minus theta, 00:25:23.560 --> 00:25:25.030 the weights add up to 1. 00:25:25.030 --> 00:25:28.180 You're basically taking a weighted average. 00:25:28.180 --> 00:25:31.970 But I want to argue that it's more than just 00:25:31.970 --> 00:25:34.060 a simple weighted average. 00:25:34.060 --> 00:25:38.600 I'm gonna argue that these weights are always 00:25:38.600 --> 00:25:41.050 non-negative. 00:25:41.050 --> 00:25:44.890 So in fact, this looks like not just a weighted average, 00:25:44.890 --> 00:25:50.020 this looks an awful lot like a kind of an expected value, 00:25:50.020 --> 00:25:52.210 like a probability weighted average. 00:25:52.210 --> 00:25:54.170 This looks like a probability. 00:25:54.170 --> 00:25:56.980 It's not a probability, but I want 00:25:56.980 --> 00:25:59.740 to argue that this number is always non-negative, 00:25:59.740 --> 00:26:00.799 and they add up to 1. 00:26:00.799 --> 00:26:02.965 So when you've got two numbers that are not negative 00:26:02.965 --> 00:26:04.798 and they add up to 1, you can interpret them 00:26:04.798 --> 00:26:07.960 as a probability. 00:26:07.960 --> 00:26:11.440 Now, what's the argument for why this number is always 00:26:11.440 --> 00:26:13.480 going to be non-negative? 00:26:13.480 --> 00:26:17.380 The condition that's required for these numbers 00:26:17.380 --> 00:26:21.100 to be non-negative is that the interest rate 00:26:21.100 --> 00:26:24.910 r, the risk-free rate, is strictly 00:26:24.910 --> 00:26:28.420 contained in between u and d. 00:26:28.420 --> 00:26:32.340 So you've got u here, d here. 00:26:32.340 --> 00:26:34.070 r has to be in the middle. 00:26:34.070 --> 00:26:37.600 And when that's the case, then you've 00:26:37.600 --> 00:26:41.030 got these things looking like probabilities. 00:26:41.030 --> 00:26:43.480 Now, the question is, is that a reasonable assumption? 00:26:43.480 --> 00:26:49.150 Is it reasonable to assume that d is less 00:26:49.150 --> 00:26:52.270 than r is less than u? 00:26:52.270 --> 00:26:54.880 Can anybody give me some intuition 00:26:54.880 --> 00:26:58.400 for why that makes economic sense? 00:26:58.400 --> 00:27:00.199 It has nothing to do with mathematics. 00:27:00.199 --> 00:27:02.490 The mathematics couldn't care less as to whether or not 00:27:02.490 --> 00:27:03.600 that inequality held. 00:27:03.600 --> 00:27:04.227 Brian? 00:27:04.227 --> 00:27:08.560 AUDIENCE: If the downside was less than the rate, 00:27:08.560 --> 00:27:11.452 then you'd just automatically buy the security. 00:27:11.452 --> 00:27:12.160 PROFESSOR: Right. 00:27:12.160 --> 00:27:15.940 AUDIENCE: And if the upside was less than the risk-free rate, 00:27:15.940 --> 00:27:18.060 they'd you'd just go into the risk-free bills. 00:27:18.060 --> 00:27:18.830 PROFESSOR: Right. 00:27:18.830 --> 00:27:20.010 That's exactly right. 00:27:20.010 --> 00:27:22.290 That's a very important economic insight. 00:27:22.290 --> 00:27:23.740 Let me go through that slowly. 00:27:23.740 --> 00:27:28.830 So Brian, you said if r is less than the downside, then 00:27:28.830 --> 00:27:31.070 what happens in that case? 00:27:31.070 --> 00:27:34.020 AUDIENCE: Then you'd want to buy the stock, the security. 00:27:34.020 --> 00:27:38.190 PROFESSOR: If the stock in its worst possible state 00:27:38.190 --> 00:27:41.820 offers more than T-bills, why would 00:27:41.820 --> 00:27:43.290 you ever want to buy T-bills? 00:27:43.290 --> 00:27:44.730 In fact, you wouldn't. 00:27:44.730 --> 00:27:46.559 And if that were true, then what would 00:27:46.559 --> 00:27:47.850 happen to the price of T-bills? 00:27:51.670 --> 00:27:53.525 The price of T-bills-- 00:27:53.525 --> 00:27:54.650 AUDIENCE: It would go down. 00:27:54.650 --> 00:27:56.250 PROFESSOR: It would go to 0. 00:27:56.250 --> 00:27:59.240 Nobody would hold it, and therefore the value of it 00:27:59.240 --> 00:28:00.020 would go to zero. 00:28:00.020 --> 00:28:02.370 It would not exist any longer. 00:28:02.370 --> 00:28:04.010 So if we're going to assume that there 00:28:04.010 --> 00:28:06.560 exists riskless borrowing, that can't be true. 00:28:06.560 --> 00:28:08.847 We can't have r over here. 00:28:08.847 --> 00:28:10.430 Now, what about the other side, Brian? 00:28:10.430 --> 00:28:12.560 What happens if r is over here? 00:28:12.560 --> 00:28:13.895 What did you say? 00:28:13.895 --> 00:28:16.020 AUDIENCE: Then you'd want to go into the risk-free. 00:28:16.020 --> 00:28:16.250 PROFESSOR: Right. 00:28:16.250 --> 00:28:18.083 You would never hold the stock, because even 00:28:18.083 --> 00:28:21.020 in the best possible world for the stock, 00:28:21.020 --> 00:28:24.650 you would not be able to get as good a return as T-bills, 00:28:24.650 --> 00:28:27.650 in which case the value of the stock would go to zero, 00:28:27.650 --> 00:28:30.020 and therefore there'd be no more stocks in the economy. 00:28:30.020 --> 00:28:33.830 The only situation where you can have stocks and T-bills 00:28:33.830 --> 00:28:36.650 coexisting in this simple world-- 00:28:36.650 --> 00:28:42.210 the only case where that's true is if this inequality held. 00:28:42.210 --> 00:28:45.500 That's the economics of this pricing formula. 00:28:45.500 --> 00:28:47.830 It has nothing to do with math. 00:28:47.830 --> 00:28:49.080 It's the economics. 00:28:49.080 --> 00:28:51.780 And the economics tells you that these things 00:28:51.780 --> 00:28:52.980 have to be non-negative. 00:28:52.980 --> 00:28:54.730 That's good, because that suggests 00:28:54.730 --> 00:28:57.420 that the price of the call option at date 0 00:28:57.420 --> 00:29:01.680 can never be negative, because these guys, Cu and Cd, 00:29:01.680 --> 00:29:04.870 are non-negative, and 1 over r is non-negative. 00:29:04.870 --> 00:29:07.977 So if it turns out that the weights can never be negative, 00:29:07.977 --> 00:29:09.810 then you know that you've got something that 00:29:09.810 --> 00:29:11.199 really is a pricing formula. 00:29:11.199 --> 00:29:12.990 You're never going to punch in some numbers 00:29:12.990 --> 00:29:18.140 and get out a formula that says this thing is worth minus 2. 00:29:18.140 --> 00:29:20.510 But more importantly, it suggests 00:29:20.510 --> 00:29:23.480 that there is a probability interpretation. 00:29:23.480 --> 00:29:26.930 But the probability is not the mathematical probability 00:29:26.930 --> 00:29:27.780 that matters. 00:29:27.780 --> 00:29:31.160 It is the economic probability. 00:29:31.160 --> 00:29:34.970 And there is a term for this particular probability. 00:29:34.970 --> 00:29:39.950 This is known as the risk-neutral probabilities 00:29:39.950 --> 00:29:43.910 of the particular economy that we've created. 00:29:43.910 --> 00:29:46.400 And it turns out that these probabilities 00:29:46.400 --> 00:29:49.010 can be used to price not just options, 00:29:49.010 --> 00:29:50.430 but anything under the sun. 00:29:50.430 --> 00:29:52.880 So there's a very, very important 00:29:52.880 --> 00:29:56.030 property and very deep property that we can't go into here, 00:29:56.030 --> 00:29:59.930 but you'll cover in 15 437, about the so-called risk 00:29:59.930 --> 00:30:03.200 neutral probabilities. 00:30:03.200 --> 00:30:04.980 But now we've got a formula here. 00:30:04.980 --> 00:30:07.340 This is a bona fide pricing formula. 00:30:07.340 --> 00:30:09.680 And the beauty of it is that if it 00:30:09.680 --> 00:30:12.530 is violated-- if it is violated but the assumptions are 00:30:12.530 --> 00:30:15.860 correct, then there is a way to create a money 00:30:15.860 --> 00:30:20.180 machine, an arbitrage, either by buying the cheap stuff 00:30:20.180 --> 00:30:23.840 and shorting the expensive, or vice versa, 00:30:23.840 --> 00:30:27.890 in the case where the signs are flipped. 00:30:27.890 --> 00:30:29.480 So here's the argument. 00:30:29.480 --> 00:30:33.770 Suppose that C is greater than V. Then here's the arbitrage. 00:30:33.770 --> 00:30:36.650 Suppose it's less, and then you basically 00:30:36.650 --> 00:30:38.540 construct the opposite arbitrage. 00:30:38.540 --> 00:30:42.170 Therefore the cost of the option has to be equal to the value 00:30:42.170 --> 00:30:42.920 that we computed. 00:30:42.920 --> 00:30:45.135 Yeah, [INAUDIBLE]. 00:30:45.135 --> 00:30:51.100 AUDIENCE: Is this function sensitive [INAUDIBLE] 00:30:51.100 --> 00:30:53.150 PROFESSOR: Well, I mean, you tell me. 00:30:53.150 --> 00:30:56.250 It's a convex combination of these two things. 00:30:56.250 --> 00:30:57.410 So in that sense-- 00:30:57.410 --> 00:30:59.850 AUDIENCE: [INAUDIBLE] 00:30:59.850 --> 00:31:01.340 PROFESSOR: That's right, exactly. 00:31:01.340 --> 00:31:02.960 Yeah. 00:31:02.960 --> 00:31:07.410 AUDIENCE: And this is not going to [INAUDIBLE] 00:31:07.410 --> 00:31:08.900 PROFESSOR: Yeah. 00:31:08.900 --> 00:31:12.472 AUDIENCE: The u and d are determined by the market-- 00:31:12.472 --> 00:31:13.055 PROFESSOR: No. 00:31:13.055 --> 00:31:14.270 AUDIENCE: [INAUDIBLE] 00:31:14.270 --> 00:31:14.870 PROFESSOR: No. 00:31:14.870 --> 00:31:16.560 That's a modeling assumption. 00:31:16.560 --> 00:31:20.630 So in advance, we agree what u and d are. 00:31:20.630 --> 00:31:24.710 Now in a minute, I'm going to start relaxing 00:31:24.710 --> 00:31:26.865 all of these assumptions. 00:31:26.865 --> 00:31:28.490 But before we do that, I want make sure 00:31:28.490 --> 00:31:30.110 we all agree on what this says. 00:31:30.110 --> 00:31:30.900 Yeah. 00:31:30.900 --> 00:31:33.000 AUDIENCE: So the p is missing, as you said. 00:31:33.000 --> 00:31:33.625 PROFESSOR: Yes. 00:31:33.625 --> 00:31:35.460 AUDIENCE: But isn't that-- 00:31:35.460 --> 00:31:38.070 isn't that embedded in Cu and Cd, because you 00:31:38.070 --> 00:31:40.350 rely on a market price for Cu and Cd? 00:31:40.350 --> 00:31:44.640 PROFESSOR: No, there is no market price for Cu and Cd. 00:31:44.640 --> 00:31:48.307 Let's go back and take a look at what Cu and the Cd are. 00:31:51.290 --> 00:31:53.220 That's not a market price. 00:31:53.220 --> 00:31:55.310 This is not a market price. 00:31:55.310 --> 00:32:00.560 Cu is basically the outcome of u times S0 minus K, 00:32:00.560 --> 00:32:05.080 and Cd is the outcome of d S0 minus K, 0. 00:32:05.080 --> 00:32:07.650 That's not a market price. 00:32:07.650 --> 00:32:12.390 We have to agree in advance on what the possible outcomes are. 00:32:12.390 --> 00:32:14.640 But once we agree on those outcomes, 00:32:14.640 --> 00:32:17.280 everything follows from that. 00:32:17.280 --> 00:32:18.820 There's no market price here. 00:32:18.820 --> 00:32:20.880 The only market price is S0. 00:32:20.880 --> 00:32:22.890 That is the market price. 00:32:22.890 --> 00:32:24.000 That is determined today. 00:32:24.000 --> 00:32:28.290 But fortunately, that market price we observe. 00:32:28.290 --> 00:32:29.900 We can see it. 00:32:29.900 --> 00:32:37.150 Now, there is a link between the market price, u 00:32:37.150 --> 00:32:49.750 and d, p, because if the stock price today is worth $20, 00:32:49.750 --> 00:32:53.570 and tomorrow we say that there are two possibilities, 00:32:53.570 --> 00:33:00.530 either it's $30 or $10, then that tells you 00:33:00.530 --> 00:33:06.680 that p sort of has to be somewhere around 0.5. 00:33:06.680 --> 00:33:07.660 We may disagree. 00:33:07.660 --> 00:33:09.260 I may think it's 0.55. 00:33:09.260 --> 00:33:12.230 You may think it's 0.45, whatever. 00:33:12.230 --> 00:33:17.100 But when we aggregate all of our expectations, 00:33:17.100 --> 00:33:22.400 we come up with $20 for the stock today. 00:33:22.400 --> 00:33:24.770 So it's all related. 00:33:24.770 --> 00:33:26.670 It's in there. 00:33:26.670 --> 00:33:29.250 But we don't need to make an assumption explicitly 00:33:29.250 --> 00:33:30.360 for what p is. 00:33:30.360 --> 00:33:34.050 That is the power of this kind of option pricing approach. 00:33:34.050 --> 00:33:34.690 [INAUDIBLE] 00:33:34.690 --> 00:33:41.183 AUDIENCE: [INAUDIBLE] the reason you get the market here is we 00:33:41.183 --> 00:33:43.348 agree on everything except that I think that 00:33:43.348 --> 00:33:44.310 the higher [INAUDIBLE]. 00:33:44.310 --> 00:33:44.550 PROFESSOR: Yeah. 00:33:44.550 --> 00:33:46.032 AUDIENCE: --it will go up, and you think 00:33:46.032 --> 00:33:47.140 it will go down [INAUDIBLE]. 00:33:47.140 --> 00:33:47.540 PROFESSOR: Right. 00:33:47.540 --> 00:33:48.956 AUDIENCE: What if we fundamentally 00:33:48.956 --> 00:33:50.820 don't agree on u and d? 00:33:50.820 --> 00:33:52.530 PROFESSOR: Oh, then we have a problem. 00:33:52.530 --> 00:33:57.230 We need to assume a particular u and d that we can agree on. 00:33:57.230 --> 00:33:58.560 So let me turn to that now. 00:33:58.560 --> 00:34:00.570 Let me turn to the extension of this. 00:34:00.570 --> 00:34:04.320 So what I've derived is a one-period pricing model-- 00:34:04.320 --> 00:34:05.880 very, very simple. 00:34:05.880 --> 00:34:10.650 It turns out that you can do a multiperiod pricing model. 00:34:10.650 --> 00:34:13.409 And this multiperiod generalization 00:34:13.409 --> 00:34:15.480 is given by this. 00:34:15.480 --> 00:34:17.489 What is that multiperiod generalization? 00:34:17.489 --> 00:34:21.150 Basically you have-- let me see if I have the diagram here-- 00:34:24.040 --> 00:34:26.320 the multiperiod generalization is simply 00:34:26.320 --> 00:34:33.650 that you now have a bunch of possibilities, 00:34:33.650 --> 00:34:36.350 and you are figuring out what the price of the option 00:34:36.350 --> 00:34:42.159 is at date 0 when it pays off at date capital N, 00:34:42.159 --> 00:34:44.230 or lowercase n in this case-- 00:34:44.230 --> 00:34:46.310 n periods. 00:34:46.310 --> 00:34:50.000 And you can use exactly the same arbitrage argument that I just 00:34:50.000 --> 00:34:51.650 showed you, but it's a little bit more 00:34:51.650 --> 00:34:55.070 complicated now because you've got multiple branches. 00:34:55.070 --> 00:34:57.920 But it's still, at every step of the way, a binomial 00:34:57.920 --> 00:34:59.930 or a Bernoulli trial. 00:34:59.930 --> 00:35:04.950 And so in a multiperiod setting, you get a binomial tree. 00:35:04.950 --> 00:35:08.700 Now, the reason that this is such a powerful extension 00:35:08.700 --> 00:35:13.500 is that nowhere have I specified what a period is. 00:35:13.500 --> 00:35:15.990 I just said it's a period, today versus tomorrow. 00:35:15.990 --> 00:35:19.680 But it could be today versus three minutes from now, 00:35:19.680 --> 00:35:24.720 or three femtoseconds from now, or three years from now. 00:35:24.720 --> 00:35:26.630 I haven't specified. 00:35:26.630 --> 00:35:29.280 So if you say we can't agree on a u 00:35:29.280 --> 00:35:32.820 and a d, fine, let's not agree on a u and a d. 00:35:32.820 --> 00:35:37.410 Let's agree that between now and five minutes from now, 00:35:37.410 --> 00:35:41.320 there are 256 possible outcomes for the stock price. 00:35:41.320 --> 00:35:42.819 Do you agree on that? 00:35:42.819 --> 00:35:44.110 You think we can agree on that? 00:35:44.110 --> 00:35:45.860 Is that something that's easy to agree on? 00:35:45.860 --> 00:35:47.940 Well, if that's the case, then all 00:35:47.940 --> 00:35:51.450 I need to do is to have enough steps between now 00:35:51.450 --> 00:35:55.470 and five minutes from now to have 256 possibilities. 00:35:55.470 --> 00:35:57.810 And by the way, I chose that number specifically 00:35:57.810 --> 00:36:01.470 as a power of 2 because with these kinds of branches, 00:36:01.470 --> 00:36:04.320 it's actually very easy to be able to get 00:36:04.320 --> 00:36:07.500 that kind of a tree, with that many branches. 00:36:07.500 --> 00:36:10.860 So now you see that the u and the d, that's 00:36:10.860 --> 00:36:12.840 not relevant, because we can make 00:36:12.840 --> 00:36:14.340 it as small as you would like. 00:36:14.340 --> 00:36:16.860 If you would like to have it really, really fine, 00:36:16.860 --> 00:36:21.180 I can get it down to double precision, 32 decimal places, 00:36:21.180 --> 00:36:25.460 by basically taking one period to be a millisecond. 00:36:25.460 --> 00:36:28.610 And this binomial option pricing formula 00:36:28.610 --> 00:36:32.850 will apply exactly in the same way. 00:36:32.850 --> 00:36:35.780 It turns out that when you let the number of periods 00:36:35.780 --> 00:36:40.170 go to infinity, and at the same time, 00:36:40.170 --> 00:36:41.820 you control the u and the d and make 00:36:41.820 --> 00:36:44.140 them smaller and smaller and smaller and smaller 00:36:44.140 --> 00:36:49.350 so as to be able to get a tree that is reasonably realistic, 00:36:49.350 --> 00:36:51.452 you know what you get? 00:36:51.452 --> 00:36:54.370 You get the Black-Scholes formula. 00:36:54.370 --> 00:36:58.090 The pricing formula that you get is a solution 00:36:58.090 --> 00:36:59.980 to this parabolic partial differential 00:36:59.980 --> 00:37:03.010 equation with the following boundary conditions. 00:37:03.010 --> 00:37:09.260 And so using the simple binomial two-step kind of process, when 00:37:09.260 --> 00:37:12.020 you let it go to infinity and you 00:37:12.020 --> 00:37:13.970 shrink the probabilities and the u 00:37:13.970 --> 00:37:18.390 and the d to make it more and more refined, 00:37:18.390 --> 00:37:21.400 you get the Black-Scholes formula. 00:37:21.400 --> 00:37:23.440 This is something that Black and Scholes 00:37:23.440 --> 00:37:25.870 never, never contemplated. 00:37:25.870 --> 00:37:28.980 So this is a completely different approach 00:37:28.980 --> 00:37:33.600 that allows you to reach the exact same conclusion, which 00:37:33.600 --> 00:37:35.910 is a startling one. 00:37:35.910 --> 00:37:38.730 Now, as I told you at the beginning, 00:37:38.730 --> 00:37:42.960 when people apply option pricing formulas, most of the time 00:37:42.960 --> 00:37:44.400 they do not do this. 00:37:44.400 --> 00:37:47.520 They do not solve the heat equation. 00:37:47.520 --> 00:37:50.170 What they do is that. 00:37:50.170 --> 00:37:51.910 They do a binomial tree. 00:37:51.910 --> 00:37:55.030 The reason is because in order to solve these PDEs, 00:37:55.030 --> 00:37:59.650 except in a very, very small number of textbook example 00:37:59.650 --> 00:38:03.190 cases, you can't solve this analytically anyway. 00:38:03.190 --> 00:38:04.840 You can't get a formula. 00:38:04.840 --> 00:38:06.581 You have to solve it numerically. 00:38:06.581 --> 00:38:08.830 And so if you're going to go to the trouble of solving 00:38:08.830 --> 00:38:11.746 these differential equations numerically, 00:38:11.746 --> 00:38:13.870 you may as well just do the binomial option pricing 00:38:13.870 --> 00:38:17.390 formula, because that's numerical as well. 00:38:17.390 --> 00:38:20.800 And it's a lot simpler computationally to be 00:38:20.800 --> 00:38:22.420 able to do that binomial tree. 00:38:22.420 --> 00:38:25.030 By the way, for those of you computing 00:38:25.030 --> 00:38:29.680 fans who like to think about parallel processing, 00:38:29.680 --> 00:38:32.950 these kinds of binomials trees are extraordinarily 00:38:32.950 --> 00:38:34.810 easy to parallelize. 00:38:34.810 --> 00:38:36.700 So if you thought about the old days, where 00:38:36.700 --> 00:38:38.290 you had a connection machine that 00:38:38.290 --> 00:38:40.810 was developed by Danny Hillis, you 00:38:40.810 --> 00:38:44.020 had 64,000 processors in parallel. 00:38:44.020 --> 00:38:45.670 You can actually make use of that 00:38:45.670 --> 00:38:47.500 by implementing a binomial tree. 00:38:47.500 --> 00:38:49.060 Nowadays we've got grid computing. 00:38:49.060 --> 00:38:52.450 The most recent advance is to be able to use both hardware 00:38:52.450 --> 00:38:55.600 and software to do distributed computing. 00:38:55.600 --> 00:38:58.510 The binomial tree is ideally suited 00:38:58.510 --> 00:38:59.810 for being able to do that. 00:38:59.810 --> 00:39:02.230 So you can evaluate extraordinarily complex 00:39:02.230 --> 00:39:08.030 derivatives very, very quickly using this kind of a framework. 00:39:08.030 --> 00:39:11.165 So you're not giving up a lot by the u and the d, 00:39:11.165 --> 00:39:14.170 because we can make the u and the d as fine as possible 00:39:14.170 --> 00:39:16.420 so that ultimately we would all say, yeah, enough. 00:39:16.420 --> 00:39:18.070 I agree, all right, leave me alone. 00:39:18.070 --> 00:39:20.110 I don't want any more binomial trees. 00:39:20.110 --> 00:39:22.350 This is complicated enough. 00:39:22.350 --> 00:39:24.910 256 of them over a five-minute interval 00:39:24.910 --> 00:39:28.780 is enough for all practical purposes. 00:39:28.780 --> 00:39:30.740 Yeah. 00:39:30.740 --> 00:39:36.124 AUDIENCE: [INAUDIBLE] if that cannot be solved, 00:39:36.124 --> 00:39:38.245 why was it so important? 00:39:38.245 --> 00:39:40.310 PROFESSOR: Oh, no, this can be solved. 00:39:40.310 --> 00:39:43.834 The solution of this equation is the Black-Scholes formula. 00:39:43.834 --> 00:39:46.000 What I said cannot be solved is when you have a more 00:39:46.000 --> 00:39:47.530 complicated security. 00:39:47.530 --> 00:39:49.630 So for example, the option pricing 00:39:49.630 --> 00:39:53.050 formula that we looked at with the simple plain vanilla call 00:39:53.050 --> 00:39:56.420 and put option, that's relatively straightforward. 00:39:56.420 --> 00:39:59.920 But think about something like a mortgage-backed security 00:39:59.920 --> 00:40:04.000 that has all sorts of conversion features and knockout features, 00:40:04.000 --> 00:40:07.450 and other types of legal restrictions, 00:40:07.450 --> 00:40:10.600 as well as certain rights and requirements. 00:40:10.600 --> 00:40:13.120 Then it's not so easy. 00:40:13.120 --> 00:40:14.930 It looks much more complicated. 00:40:14.930 --> 00:40:17.980 For example, this particular coefficient 00:40:17.980 --> 00:40:20.020 that multiplies this second derivative 00:40:20.020 --> 00:40:22.510 ends up being a highly non-linear function, not just 00:40:22.510 --> 00:40:23.740 a quadratic. 00:40:23.740 --> 00:40:26.380 Or this piece here becomes a nonlinear function, 00:40:26.380 --> 00:40:29.890 or the boundary conditions are kind of weird. 00:40:29.890 --> 00:40:32.560 In that case, you can't solve it analytically. 00:40:32.560 --> 00:40:35.068 You have to use numerical methods to solve it. 00:40:35.068 --> 00:40:37.060 AUDIENCE: [INAUDIBLE] 00:40:37.060 --> 00:40:39.520 PROFESSOR: This is just the arbitrage condition 00:40:39.520 --> 00:40:44.750 that says that the solution C will give you a null arbitrage 00:40:44.750 --> 00:40:47.680 price for the call option. 00:40:47.680 --> 00:40:53.271 So the equivalent of this PDE, partial differential equation, 00:40:53.271 --> 00:40:53.770 is-- 00:40:56.280 --> 00:41:02.530 go back-- is this, the simultaneous equation 00:41:02.530 --> 00:41:07.520 up there and down here, and then this expression 00:41:07.520 --> 00:41:10.300 that says that the price of the option 00:41:10.300 --> 00:41:13.600 has to be given by this particular portfolio. 00:41:13.600 --> 00:41:18.550 That's what the PDE looks like in continuous time, 00:41:18.550 --> 00:41:20.635 or when you have an infinite number of time steps. 00:41:23.980 --> 00:41:25.970 So it is not-- 00:41:25.970 --> 00:41:27.790 that's absolutely a good question, 00:41:27.790 --> 00:41:30.400 because this is solvable. 00:41:30.400 --> 00:41:34.570 But very quickly, when you change the terms of a contract, 00:41:34.570 --> 00:41:36.920 it turns out that it's very hard to model. 00:41:36.920 --> 00:41:37.420 Yes. 00:41:39.780 --> 00:41:41.670 AUDIENCE: Question about the random walk. 00:41:41.670 --> 00:41:42.400 PROFESSOR: Yes. 00:41:42.400 --> 00:41:46.050 AUDIENCE: Can you just briefly mention how that feeds 00:41:46.050 --> 00:41:49.920 into the final answer, and how it will change things if it's-- 00:41:49.920 --> 00:41:53.200 PROFESSOR: Well, the random walk hypothesis is implicit in here, 00:41:53.200 --> 00:41:56.160 because I've got a coin toss. 00:41:56.160 --> 00:42:00.670 And the coin toss is independent period by period. 00:42:00.670 --> 00:42:03.620 If the coin toss is not independent, 00:42:03.620 --> 00:42:06.020 then that's the wrong formula. 00:42:06.020 --> 00:42:09.230 In other words, you don't have a simple binomial distribution 00:42:09.230 --> 00:42:11.540 if you don't have IID coin tosses. 00:42:11.540 --> 00:42:14.180 The random walk is basically the assumption 00:42:14.180 --> 00:42:17.510 of IID coin tosses-- independently and identically 00:42:17.510 --> 00:42:18.120 distributed. 00:42:18.120 --> 00:42:19.730 That's what IID stands for-- 00:42:19.730 --> 00:42:21.480 IID coin tosses. 00:42:21.480 --> 00:42:23.885 So that's where the Bachelier assumption came in. 00:42:23.885 --> 00:42:27.470 In order for Bachelier to derive the heat equation, 00:42:27.470 --> 00:42:30.410 or some variant of the heat equation, 00:42:30.410 --> 00:42:34.340 he was implicitly assuming that what happens in one period 00:42:34.340 --> 00:42:37.730 for the stock has no bearing on what happens in next period. 00:42:37.730 --> 00:42:42.080 If stock prices are correlated over time, 00:42:42.080 --> 00:42:44.240 then these formulas do not work. 00:42:44.240 --> 00:42:46.460 You need a different kind of formula. 00:42:46.460 --> 00:42:47.900 It's actually not that far off. 00:42:47.900 --> 00:42:52.190 You can derive an expression for an option pricing formula 00:42:52.190 --> 00:42:53.600 with correlated returns. 00:42:53.600 --> 00:42:56.720 In fact, professor Wang and I published a paper, 00:42:56.720 --> 00:42:58.820 I think it's maybe close to 10 years ago, 00:42:58.820 --> 00:43:01.160 where we worked out that case. 00:43:01.160 --> 00:43:03.220 But up until then, most people assumed 00:43:03.220 --> 00:43:05.760 that stock prices are not correlated, 00:43:05.760 --> 00:43:09.890 so the Brownian motion or random walk idea fit in very nicely 00:43:09.890 --> 00:43:12.080 with this binomial. 00:43:12.080 --> 00:43:15.410 If they're correlated, then you no longer have IID Bernoulli 00:43:15.410 --> 00:43:17.690 trials, you have a Markov chain, and you 00:43:17.690 --> 00:43:19.190 have to use Markov pricing in order 00:43:19.190 --> 00:43:22.820 to be able to get this formula. 00:43:22.820 --> 00:43:25.730 If you're interested in this, I urge you to take 15 437, 00:43:25.730 --> 00:43:28.000 because that's where we go into it in much more depth. 00:43:28.000 --> 00:43:28.872 Yeah. 00:43:28.872 --> 00:43:34.420 AUDIENCE: [INAUDIBLE] use the binomial coin to value options, 00:43:34.420 --> 00:43:37.540 and we see a range of prices. 00:43:39.050 --> 00:43:39.830 PROFESSOR: Yeah. 00:43:39.830 --> 00:43:41.881 AUDIENCE: So how do we approach that? 00:43:41.881 --> 00:43:44.470 Do we take some kind of average? 00:43:44.470 --> 00:43:47.832 Is this common, or do we receive a specific u and a d each time? 00:43:47.832 --> 00:43:49.956 I mean, I imagine it could be a range [INAUDIBLE].. 00:43:49.956 --> 00:43:51.060 PROFESSOR: No, no. 00:43:51.060 --> 00:43:52.760 So the way that you would apply this 00:43:52.760 --> 00:43:55.100 is that you would, first of all, pick 00:43:55.100 --> 00:43:58.460 the number of periods that are appropriate to the problem 00:43:58.460 --> 00:43:59.240 at hand. 00:43:59.240 --> 00:44:01.070 So if you have got an option that's 00:44:01.070 --> 00:44:03.920 expiring in three months, then typically, 00:44:03.920 --> 00:44:07.130 if you did it on a daily basis or an hourly basis, 00:44:07.130 --> 00:44:08.840 that would be more than enough. 00:44:08.840 --> 00:44:11.780 And then you would assume that there would be a u and a d 00:44:11.780 --> 00:44:16.040 in order to match the approximate outcomes that you 00:44:16.040 --> 00:44:17.060 would expect. 00:44:17.060 --> 00:44:20.010 And then out of that, you would actually get a number. 00:44:20.010 --> 00:44:24.750 So this, this C0, when you plug in all of these parameters, 00:44:24.750 --> 00:44:30.810 you actually get a number, like $30.25. 00:44:30.810 --> 00:44:33.160 That's the price of the option. 00:44:33.160 --> 00:44:35.050 And of course if you change the parameters, 00:44:35.050 --> 00:44:38.020 you change the strike price, the interest rate changes, the u 00:44:38.020 --> 00:44:40.540 and the d changes, that will change the value of the option 00:44:40.540 --> 00:44:41.622 price as well. 00:44:41.622 --> 00:44:44.520 AUDIENCE: [INAUDIBLE] every now and then, 00:44:44.520 --> 00:44:46.900 [INAUDIBLE] to receive a range from you, and a range-- 00:44:46.900 --> 00:44:48.280 PROFESSOR: No, no, no. 00:44:48.280 --> 00:44:50.440 What you do is you start off with an assumption 00:44:50.440 --> 00:44:53.200 for what u and d exactly are. 00:44:53.200 --> 00:44:57.190 Not a range, but actually if it goes up, it goes up by 1.05. 00:44:57.190 --> 00:45:01.120 If it goes down it goes down by 0.92. 00:45:01.120 --> 00:45:02.673 Yeah. 00:45:02.673 --> 00:45:04.374 AUDIENCE: [INAUDIBLE] 00:45:04.374 --> 00:45:06.040 PROFESSOR: Oh, well, it varies depending 00:45:06.040 --> 00:45:09.620 on the particular instrument that you're trying to price. 00:45:09.620 --> 00:45:13.030 So-- well, no, what I mean is options on what stock? 00:45:13.030 --> 00:45:16.250 So in other words, with any kind of option pricing formula, 00:45:16.250 --> 00:45:18.250 you actually have to calibrate these parameters. 00:45:18.250 --> 00:45:20.416 So you have to figure out what the interest rate is, 00:45:20.416 --> 00:45:22.540 and then typically what is done is you assume 00:45:22.540 --> 00:45:26.440 a particular grid, and then use a u and a d that will capture 00:45:26.440 --> 00:45:27.830 all the elements of that grid. 00:45:27.830 --> 00:45:30.010 So for example, let's assume that u 00:45:30.010 --> 00:45:37.130 is 25 basis points plus 1, and d is 1 minus 25 basis points. 00:45:37.130 --> 00:45:40.150 So that means you can capture stock price movements that 00:45:40.150 --> 00:45:42.760 go up by 25 basis points or down, 00:45:42.760 --> 00:45:46.240 and you assume a number of n in order 00:45:46.240 --> 00:45:48.700 to get that tree to be as fine as you 00:45:48.700 --> 00:45:53.110 would like for the particular time that you're pricing it at. 00:45:53.110 --> 00:45:58.000 So in other words, if I use 25 basis points and n equal to 1, 00:45:58.000 --> 00:46:01.990 that means that I can capture a situation where, 00:46:01.990 --> 00:46:06.370 at maturity, the stock price goes up or down by 25 basis 00:46:06.370 --> 00:46:08.230 points. 00:46:08.230 --> 00:46:10.690 If I now go four periods, then I can 00:46:10.690 --> 00:46:14.200 capture a situation where the stock price goes up by 1% 00:46:14.200 --> 00:46:18.100 or down by 1% in 25-basis-point increments. 00:46:18.100 --> 00:46:21.725 And if I want more refinements, then I keep going, 00:46:21.725 --> 00:46:24.160 let n get bigger and bigger and bigger. 00:46:24.160 --> 00:46:28.120 And then whatever that is, that final number of nodes 00:46:28.120 --> 00:46:33.646 will be the possible stock price values. 00:46:33.646 --> 00:46:38.012 AUDIENCE: [INAUDIBLE] historical data on the specific stock to-- 00:46:38.012 --> 00:46:39.720 PROFESSOR: You would use historical data. 00:46:39.720 --> 00:46:42.053 You would use historical-- because the way you calibrate 00:46:42.053 --> 00:46:45.460 this is you can show that the expected value-- 00:46:45.460 --> 00:46:49.260 so the expected value of S1 is just 00:46:49.260 --> 00:46:54.450 equal to the probability of u S0 plus 1 minus 00:46:54.450 --> 00:46:57.150 probability of d S0. 00:46:57.150 --> 00:46:58.740 So you've got the expected value. 00:46:58.740 --> 00:47:02.550 Calculate the variance of S1, and you'll 00:47:02.550 --> 00:47:05.610 get another expression with u and d and p, 00:47:05.610 --> 00:47:07.320 and then you simply use historical data 00:47:07.320 --> 00:47:10.200 to match the parameters and pick them 00:47:10.200 --> 00:47:13.260 so that they give you a reasonable approximation 00:47:13.260 --> 00:47:14.830 to reality. 00:47:14.830 --> 00:47:16.830 AUDIENCE: [INAUDIBLE] doesn't continue to behave 00:47:16.830 --> 00:47:18.830 as the history-- 00:47:18.830 --> 00:47:19.484 PROFESSOR: Yes. 00:47:19.484 --> 00:47:20.650 AUDIENCE: --so the options-- 00:47:20.650 --> 00:47:21.190 PROFESSOR: Yeah. 00:47:21.190 --> 00:47:21.890 AUDIENCE: --don't match. 00:47:21.890 --> 00:47:22.500 PROFESSOR: Absolutely. 00:47:22.500 --> 00:47:23.950 That's always the case, isn't it? 00:47:23.950 --> 00:47:27.270 In other words, if you don't have IID, 00:47:27.270 --> 00:47:28.860 you're going to get a problem. 00:47:28.860 --> 00:47:31.830 But remember, it doesn't depend upon the p. 00:47:31.830 --> 00:47:35.250 And so in that sense, if there's a change in p, 00:47:35.250 --> 00:47:38.670 as long as the u and the d are appropriate, 00:47:38.670 --> 00:47:41.970 you'll still be able to capture the value of the option. 00:47:41.970 --> 00:47:42.749 Question. 00:47:42.749 --> 00:47:44.186 AUDIENCE: I'm trying to figure out 00:47:44.186 --> 00:47:47.060 the analogy with the [INAUDIBLE].. 00:47:47.060 --> 00:47:47.802 PROFESSOR: Yeah. 00:47:47.802 --> 00:47:52.540 AUDIENCE: So I understand how it works in temperature. 00:47:52.540 --> 00:47:56.260 What would be here that [INAUDIBLE].. 00:47:56.260 --> 00:47:58.590 PROFESSOR: Let me-- let me not talk about that now, 00:47:58.590 --> 00:48:00.690 because I suspect that while you may be interested 00:48:00.690 --> 00:48:02.670 and a couple of other people, we probably 00:48:02.670 --> 00:48:04.870 don't have everybody being physicists here. 00:48:04.870 --> 00:48:05.900 So we'll talk about that afterwards. 00:48:05.900 --> 00:48:07.525 And also, that's something that, again, 00:48:07.525 --> 00:48:09.270 in 437, they may touch upon. 00:48:09.270 --> 00:48:12.870 But I want to keep moving along, because this is already 00:48:12.870 --> 00:48:16.170 more complicated than the nature of what 00:48:16.170 --> 00:48:17.994 I want to cover in this course. 00:48:17.994 --> 00:48:19.410 So let me get back to you on that, 00:48:19.410 --> 00:48:21.159 but we can talk about it afterwards. 00:48:21.159 --> 00:48:22.450 Any other questions about this? 00:48:22.450 --> 00:48:23.250 Yeah. 00:48:23.250 --> 00:48:25.250 AUDIENCE: I have a question about volatility, 00:48:25.250 --> 00:48:28.250 and how it is going to play in the equation. 00:48:28.250 --> 00:48:30.070 Like for example, I have two scenarios. 00:48:30.070 --> 00:48:35.170 They all, in three months, could go up or down by u or d. 00:48:35.170 --> 00:48:38.922 But the volatility of those to scenarios vary dramatically. 00:48:38.922 --> 00:48:39.630 PROFESSOR: Right. 00:48:39.630 --> 00:48:40.912 AUDIENCE: So how does-- 00:48:40.912 --> 00:48:42.870 PROFESSOR: How does volatility enter into this. 00:48:42.870 --> 00:48:43.828 That's a good question. 00:48:43.828 --> 00:48:46.320 Well, what do you think volatility 00:48:46.320 --> 00:48:48.740 is captured by in this simple Bernoulli trial? 00:48:49.332 --> 00:48:51.040 AUDIENCE: The difference between u and d. 00:48:51.040 --> 00:48:52.290 PROFESSOR: Exactly, exactly. 00:48:52.290 --> 00:48:58.660 Volatility is a measure of the spread between u and d. 00:48:58.660 --> 00:49:01.030 Holding other things equal-- 00:49:01.030 --> 00:49:03.730 by that, I mean holding the current stock price equal, 00:49:03.730 --> 00:49:06.580 holding the probability p equal-- 00:49:06.580 --> 00:49:12.700 so fixing that, as I increase the spread between u and d, 00:49:12.700 --> 00:49:14.270 I'm increasing the volatility. 00:49:16.890 --> 00:49:21.840 And if there's one thing that we see that matters 00:49:21.840 --> 00:49:24.750 is the spread between u and d. 00:49:30.730 --> 00:49:35.170 So if the spread between you and d increases, 00:49:35.170 --> 00:49:39.400 that actually will have an impact on this formula, 00:49:39.400 --> 00:49:42.190 and you have to work out the effects, which 00:49:42.190 --> 00:49:43.790 is a very easy thing to do. 00:49:43.790 --> 00:49:45.580 You can even do this in a spreadsheet. 00:49:45.580 --> 00:49:49.330 But you can show that as the volatility increases, 00:49:49.330 --> 00:49:54.070 the value of the call option is actually increasing. 00:49:54.070 --> 00:49:56.770 So take a look at that, and you'll 00:49:56.770 --> 00:50:01.610 see that it behaves the way that we think it should. 00:50:01.610 --> 00:50:06.760 OK, other questions? 00:50:06.760 --> 00:50:11.740 OK, well, so I'll leave I'll leave it 00:50:11.740 --> 00:50:14.230 at this point, which is to say that the derivatives 00:50:14.230 --> 00:50:17.000 literature is huge. 00:50:17.000 --> 00:50:20.890 And it has really spawned a number of different not only 00:50:20.890 --> 00:50:24.610 securities, but also different methods 00:50:24.610 --> 00:50:28.780 for hedging and managing your portfolios, 00:50:28.780 --> 00:50:33.250 to the point where really, derivatives are everywhere. 00:50:33.250 --> 00:50:36.880 And there are some examples that I've given you here, 00:50:36.880 --> 00:50:39.910 but this is an area which is considered 00:50:39.910 --> 00:50:42.400 rocket science because of the analytics 00:50:42.400 --> 00:50:43.810 that are so demanding. 00:50:43.810 --> 00:50:46.720 So this is a natural area for students here at MIT 00:50:46.720 --> 00:50:49.510 to be involved in, but it's certainly not the only area. 00:50:49.510 --> 00:50:53.380 And ultimately, what's important about derivatives is not just 00:50:53.380 --> 00:50:56.660 the pricing and the hedging, but rather the application. 00:50:56.660 --> 00:50:59.440 So the fact that we spend a fair bit of time 00:50:59.440 --> 00:51:01.750 at the beginning of this lecture talking 00:51:01.750 --> 00:51:07.480 about payoff diagrams, that wasn't just for completeness. 00:51:07.480 --> 00:51:09.730 That really is one of the most important aspects, 00:51:09.730 --> 00:51:13.360 is how you use options in order to tailor 00:51:13.360 --> 00:51:16.900 the kinds of risks and return profiles 00:51:16.900 --> 00:51:18.190 that you'd like to have. 00:51:18.190 --> 00:51:19.930 And now that you know how to price them, 00:51:19.930 --> 00:51:22.000 you can have a very clear sense of 00:51:22.000 --> 00:51:24.130 whether or not they are appropriate 00:51:24.130 --> 00:51:25.732 from a risk-return tradeoff. 00:51:25.732 --> 00:51:27.190 But they are very different, as you 00:51:27.190 --> 00:51:30.070 can see, from the securities that we've done. 00:51:30.070 --> 00:51:32.260 However, having done it, having now 00:51:32.260 --> 00:51:35.750 priced options and other derivatives, 00:51:35.750 --> 00:51:39.780 which are really relatively straightforward extensions, 00:51:39.780 --> 00:51:42.360 we've now been able to price virtually 00:51:42.360 --> 00:51:46.470 99% of all the securities that you would ever run into. 00:51:46.470 --> 00:51:47.790 We've done stocks. 00:51:47.790 --> 00:51:48.720 We've done bonds. 00:51:48.720 --> 00:51:52.260 We've done futures, forwards, and now options, 00:51:52.260 --> 00:51:54.810 so there really isn't any other kind of financial security 00:51:54.810 --> 00:51:58.470 out there that you could possibly come across that you 00:51:58.470 --> 00:51:59.820 don't know how to price. 00:51:59.820 --> 00:52:03.120 You may not realize it yet, and the purpose 00:52:03.120 --> 00:52:05.250 of the second half of the course is 00:52:05.250 --> 00:52:08.730 to introduce risk and show you how to use all of these methods 00:52:08.730 --> 00:52:10.890 to price all of the other securities 00:52:10.890 --> 00:52:12.440 that you will come into contact with. 00:52:12.440 --> 00:52:15.390 And then, of course, in 402 and other finance courses, 00:52:15.390 --> 00:52:17.480 you'll see that much more closely. 00:52:17.480 --> 00:52:21.530 So for example, a revolving credit agreement, 00:52:21.530 --> 00:52:26.720 a sinking fund debt issue, a credit default swap, 00:52:26.720 --> 00:52:29.070 an interest rate swap-- 00:52:29.070 --> 00:52:33.840 all of these securities are mixtures of the securities 00:52:33.840 --> 00:52:35.730 that we've seen till now. 00:52:35.730 --> 00:52:38.490 And the pricing method in all of these cases 00:52:38.490 --> 00:52:43.080 is exactly the same, which is identify the cash flows, 00:52:43.080 --> 00:52:47.070 come up with another portfolio that has the same cash flows, 00:52:47.070 --> 00:52:49.860 but where you know how to construct it, 00:52:49.860 --> 00:52:51.780 therefore the price of that security 00:52:51.780 --> 00:52:53.760 has to be equal to the price of the thing 00:52:53.760 --> 00:52:56.160 that you're trying to value. 00:52:56.160 --> 00:53:00.030 That's the basic principle in virtually all financial pricing 00:53:00.030 --> 00:53:01.360 applications. 00:53:01.360 --> 00:53:04.020 So once you understand these concepts, 00:53:04.020 --> 00:53:06.810 you can literally price anything under the sun, 00:53:06.810 --> 00:53:10.420 and all you need between now and then is practice, practice, 00:53:10.420 --> 00:53:13.990 practice in doing that. 00:53:13.990 --> 00:53:17.560 All right, so that wraps up the lecture on derivatives. 00:53:17.560 --> 00:53:22.030 And now I want to turn to risk and reward, 00:53:22.030 --> 00:53:25.330 because up until now, we've really 00:53:25.330 --> 00:53:28.810 talked about risk in an indirect way, 00:53:28.810 --> 00:53:33.040 and I want to talk about it in a much more direct fashion 00:53:33.040 --> 00:53:36.640 by looking at measures of risk. 00:53:36.640 --> 00:53:38.530 So what I want to do now is to turn 00:53:38.530 --> 00:53:42.700 to a little bit of statistical background 00:53:42.700 --> 00:53:44.350 to talk about risk and return. 00:53:44.350 --> 00:53:46.270 I want to motivate it first, and then 00:53:46.270 --> 00:53:47.853 give you the measures that we're going 00:53:47.853 --> 00:53:49.990 to use for capturing risk and return, 00:53:49.990 --> 00:53:53.290 and then apply it to stocks, and get a sense of what 00:53:53.290 --> 00:53:56.730 kinds of anomalies are out there that we should be aware of. 00:53:56.730 --> 00:53:58.480 And then I'm going to take these measures, 00:53:58.480 --> 00:54:02.980 and then tell you how to come up with the one number 00:54:02.980 --> 00:54:08.240 that I've had to put off for the first half of the semester, 00:54:08.240 --> 00:54:10.900 which is the cost of capital-- 00:54:10.900 --> 00:54:14.140 the required rate of return, the risk-adjusted rate of return. 00:54:14.140 --> 00:54:16.120 We are now going to get to a point 00:54:16.120 --> 00:54:19.030 where we can actually identify what that number is, 00:54:19.030 --> 00:54:21.340 and how to make that risk adjustment. 00:54:21.340 --> 00:54:23.870 So that's where we're going. 00:54:23.870 --> 00:54:29.400 Now, to give you a quick summary of where we are, as I told you, 00:54:29.400 --> 00:54:32.010 we've priced all of these different securities. 00:54:32.010 --> 00:54:34.230 But underlying all of these prices 00:54:34.230 --> 00:54:37.290 is a kind of a net present value calculation where we're 00:54:37.290 --> 00:54:39.690 taking some kind of a payoff or expected payoff 00:54:39.690 --> 00:54:41.997 and discounting it at a particular rate, 00:54:41.997 --> 00:54:44.580 and we need to figure out what that appropriate rate of return 00:54:44.580 --> 00:54:45.360 is. 00:54:45.360 --> 00:54:47.220 I've said before that that rate of return 00:54:47.220 --> 00:54:49.380 is determined by the marketplace. 00:54:49.380 --> 00:54:52.900 But what we want to know is how. 00:54:52.900 --> 00:54:55.290 How does the market do that? 00:54:55.290 --> 00:54:58.440 Because unless we understand a little bit better what 00:54:58.440 --> 00:55:01.080 that mechanism is, we won't be in a position 00:55:01.080 --> 00:55:04.200 to be able to say that the particular market that we're 00:55:04.200 --> 00:55:08.250 using is either working very well or completely out 00:55:08.250 --> 00:55:11.130 to lunch and crazy. 00:55:11.130 --> 00:55:14.490 So we need to deconstruct the process by which 00:55:14.490 --> 00:55:17.280 the market gets to that. 00:55:17.280 --> 00:55:20.280 In order to do that, we have to go back even farther 00:55:20.280 --> 00:55:22.920 and peel back the onion and ask the question, 00:55:22.920 --> 00:55:29.010 how do people measure risk, and how do they engage in risk 00:55:29.010 --> 00:55:30.540 taking behavior? 00:55:30.540 --> 00:55:34.164 So we have to do a little bit more work in figuring out 00:55:34.164 --> 00:55:35.580 these different kinds of measures, 00:55:35.580 --> 00:55:39.810 and then talking explicitly about how individuals actually 00:55:39.810 --> 00:55:42.840 incorporate that into their world view. 00:55:42.840 --> 00:55:45.480 Along the way, we're going to ask questions like, 00:55:45.480 --> 00:55:47.850 is the market efficient, and how do 00:55:47.850 --> 00:55:52.470 we measure the performance of portfolio managers? 00:55:52.470 --> 00:55:56.790 This past year, the typical portfolio manager 00:55:56.790 --> 00:56:01.530 has lost about 30% to 40%. 00:56:01.530 --> 00:56:05.130 That's a pretty devastating kind of return. 00:56:05.130 --> 00:56:09.960 And in that environment, if you found a portfolio manager that 00:56:09.960 --> 00:56:14.160 ended up losing you 10%, you might think, gee, 00:56:14.160 --> 00:56:16.770 that's pretty good. 00:56:16.770 --> 00:56:18.030 Does that really make sense? 00:56:18.030 --> 00:56:21.180 Is it ever the case that we want to congratulate a portfolio 00:56:21.180 --> 00:56:23.490 manager for losing money for us? 00:56:23.490 --> 00:56:26.010 We have to answer that question in the context of how 00:56:26.010 --> 00:56:29.220 you figure out what an appropriate or fair rate 00:56:29.220 --> 00:56:30.150 of return is. 00:56:30.150 --> 00:56:32.660 So that's what we're going to be doing. 00:56:32.660 --> 00:56:35.390 Now, to do that, I need to develop 00:56:35.390 --> 00:56:36.680 a little bit of new notation. 00:56:36.680 --> 00:56:38.690 And so the notation that I'm going to develop 00:56:38.690 --> 00:56:42.200 is to talk about returns that are 00:56:42.200 --> 00:56:45.440 inclusive of any kind of distributions, like dividends. 00:56:45.440 --> 00:56:48.740 So when I talk about the returns of equities, 00:56:48.740 --> 00:56:53.060 I'm going to be talking explicitly about a return that 00:56:53.060 --> 00:56:55.940 includes the dividend. 00:56:55.940 --> 00:56:57.500 And so the concept that we're going 00:56:57.500 --> 00:56:59.690 to be working on, for the most part, 00:56:59.690 --> 00:57:05.130 for the next half of this course is the expected rate of return. 00:57:05.130 --> 00:57:08.420 We obviously will be talking about realized returns, 00:57:08.420 --> 00:57:10.700 but from a portfolio management perspective, 00:57:10.700 --> 00:57:13.660 we're going to be focusing not just on what happened this year 00:57:13.660 --> 00:57:15.530 or what happened last year, but we're 00:57:15.530 --> 00:57:18.710 going to be focusing on the average rate of return 00:57:18.710 --> 00:57:22.950 that we would expect over the course of the next five years. 00:57:22.950 --> 00:57:25.100 We're going to be looking at excess returns, which 00:57:25.100 --> 00:57:31.100 is in excess of the net risk-free rate, little rf. 00:57:31.100 --> 00:57:35.390 And what we refer to as a risk premium is simply 00:57:35.390 --> 00:57:38.810 the average rate of return of a risky security 00:57:38.810 --> 00:57:40.290 minus the risk-free rate. 00:57:40.290 --> 00:57:44.920 So the excess return you can think of as a realization 00:57:44.920 --> 00:57:46.480 of that risk premium. 00:57:46.480 --> 00:57:49.402 But on average over a long period of time, 00:57:49.402 --> 00:57:51.610 the number that we're going to be concerned with most 00:57:51.610 --> 00:57:55.240 is this risk premium number, the average rate of return 00:57:55.240 --> 00:57:57.100 minus the risk-free rate. 00:57:57.100 --> 00:57:59.860 Over the course of the last 100 years or so, 00:57:59.860 --> 00:58:04.000 US equity markets have provided an average rate 00:58:04.000 --> 00:58:10.120 of return minus the risk-free rate on the order of 7%. 00:58:10.120 --> 00:58:13.630 That's pretty good, but that's a long-run average. 00:58:13.630 --> 00:58:17.740 The realized excess rate of return this year 00:58:17.740 --> 00:58:19.890 is horrible, so I'm not even going 00:58:19.890 --> 00:58:23.050 to talk about what that number is, but it's bad. 00:58:23.050 --> 00:58:25.390 But do you see the difference between this year's rate 00:58:25.390 --> 00:58:28.420 of return versus the long-run average? 00:58:28.420 --> 00:58:30.910 And we can talk about both of them, 00:58:30.910 --> 00:58:34.240 but we're going to use different techniques for each. 00:58:34.240 --> 00:58:38.320 So the technique for talking about the statistical aspects 00:58:38.320 --> 00:58:41.560 of returns will be from the language of statistics. 00:58:41.560 --> 00:58:44.290 We're going to talk about the expected rate of return. 00:58:44.290 --> 00:58:47.890 I'm going to use the Greek letter mu to denote that. 00:58:47.890 --> 00:58:51.310 We're gonna also talk about the riskiness of returns, which 00:58:51.310 --> 00:58:56.080 I'm going to use the variance and the standard deviation 00:58:56.080 --> 00:58:57.880 to proxy for. 00:58:57.880 --> 00:59:01.870 So the variance is simply the expected value 00:59:01.870 --> 00:59:05.830 of the squared excess return. 00:59:05.830 --> 00:59:09.910 That gives you a sense of the fluctuations around the mean. 00:59:09.910 --> 00:59:12.220 And the standard deviation is the square root 00:59:12.220 --> 00:59:13.030 of the variance. 00:59:13.030 --> 00:59:14.980 And we use the standard deviation simply 00:59:14.980 --> 00:59:16.960 because that's in the same units. 00:59:16.960 --> 00:59:20.380 It's in units of percent per year, 00:59:20.380 --> 00:59:24.496 whereas the variance is in units of percentage points squared 00:59:24.496 --> 00:59:25.870 per year, so it's a little easier 00:59:25.870 --> 00:59:29.380 to deal with the standard deviation. 00:59:29.380 --> 00:59:35.890 And those concepts are the theoretical or population 00:59:35.890 --> 00:59:39.310 values of the underlying securities 00:59:39.310 --> 00:59:40.670 that we're going to look at. 00:59:40.670 --> 00:59:43.510 We also want to look at the historical estimates, 00:59:43.510 --> 00:59:45.550 and the historical estimates are given 00:59:45.550 --> 00:59:47.900 by the sample counterparts. 00:59:47.900 --> 00:59:52.319 So this is the sample mean, the sample variance, and the sample 00:59:52.319 --> 00:59:53.110 standard deviation. 00:59:53.110 --> 00:59:56.740 You should all remember this from your DMD class. 00:59:56.740 --> 01:00:00.290 But if not, we'll have the TAs go over it during recitation. 01:00:00.290 --> 01:00:02.680 You can also look in the appendix of Brealey, Myers, 01:00:02.680 --> 01:00:07.330 and Allen, and they'll provide a little review about this. 01:00:07.330 --> 01:00:09.250 Now, there are lots of other statistics, 01:00:09.250 --> 01:00:12.280 and the only one that I'm gonna spend time on 01:00:12.280 --> 01:00:13.870 is the correlation. 01:00:13.870 --> 01:00:15.550 There's the median instead of the mean. 01:00:15.550 --> 01:00:18.790 You can look at skewness, which way the distribution leans. 01:00:18.790 --> 01:00:22.060 But what we're going to look at in just a little 01:00:22.060 --> 01:00:26.950 while is correlation, which is how closely do 01:00:26.950 --> 01:00:31.600 the returns of two investments move together. 01:00:31.600 --> 01:00:33.760 If they move together a lot, then we 01:00:33.760 --> 01:00:37.750 say that they're highly correlated, or co-related. 01:00:37.750 --> 01:00:40.540 And if they don't move together a lot, 01:00:40.540 --> 01:00:42.300 they're not very highly correlated. 01:00:42.300 --> 01:00:45.680 And in some cases, if they move in opposite directions, 01:00:45.680 --> 01:00:48.120 we say that they're negatively correlated. 01:00:48.120 --> 01:00:50.830 So correlation, as most of you already know, 01:00:50.830 --> 01:00:53.950 is a statistic that's a number between minus 1 and 1, 01:00:53.950 --> 01:00:57.040 or minus 100% and 100%, that measures 01:00:57.040 --> 01:01:02.860 the degree of association between these two securities. 01:01:02.860 --> 01:01:05.680 We're going to be making use of correlations a lot 01:01:05.680 --> 01:01:08.050 in the coming couple of lectures to try 01:01:08.050 --> 01:01:11.590 to get a sense of whether or not an investment is going to help 01:01:11.590 --> 01:01:14.920 you diversify your overall portfolio, 01:01:14.920 --> 01:01:17.140 or if an investment is only going to add 01:01:17.140 --> 01:01:19.660 to the risks of your portfolio. 01:01:19.660 --> 01:01:22.370 And you can guess as to how we're going to measure that. 01:01:22.370 --> 01:01:26.950 If the new investment is either zero correlated 01:01:26.950 --> 01:01:29.740 or negatively correlated with your current portfolio, 01:01:29.740 --> 01:01:33.460 that's going to help in terms of dampening your fluctuations. 01:01:33.460 --> 01:01:37.030 But if the two investments move at the same time, that's 01:01:37.030 --> 01:01:38.650 not only going to not help, that's 01:01:38.650 --> 01:01:41.080 going to actually add to your risks. 01:01:41.080 --> 01:01:42.520 And you don't want that, at least 01:01:42.520 --> 01:01:46.157 not without the proper reward. 01:01:46.157 --> 01:01:47.740 So that's a brief preview of how we're 01:01:47.740 --> 01:01:50.020 going to use these statistics. 01:01:50.020 --> 01:01:52.930 And you get some examples here about what 01:01:52.930 --> 01:01:54.570 correlation looks like. 01:01:54.570 --> 01:01:58.300 Here I've plotted four different scatter graphs 01:01:58.300 --> 01:02:01.960 of the return of one asset on the x-axis 01:02:01.960 --> 01:02:05.020 and the return of another asset on the y-axis. 01:02:05.020 --> 01:02:07.540 And the dots represent those pairs 01:02:07.540 --> 01:02:11.780 of returns for different assumptions about correlation. 01:02:11.780 --> 01:02:14.710 So the upper left-hand scatter graph 01:02:14.710 --> 01:02:17.770 is a graph where there's no correlation. 01:02:17.770 --> 01:02:19.540 The correlation is zero. 01:02:19.540 --> 01:02:21.880 The scatter graph on the lower left 01:02:21.880 --> 01:02:24.210 is where there's very high positive correlation-- 01:02:24.210 --> 01:02:27.490 80% correlation between the two. 01:02:27.490 --> 01:02:30.820 And the scatter graph on the lower right 01:02:30.820 --> 01:02:35.770 is where there's a negative 50% correlation. 01:02:35.770 --> 01:02:38.530 So we're going to use correlation, 01:02:38.530 --> 01:02:43.030 along with mean and variance, to try to put together 01:02:43.030 --> 01:02:47.460 good collections of securities-- in other words, good portfolios 01:02:47.460 --> 01:02:48.690 of securities. 01:02:48.690 --> 01:02:52.710 And by doing that, we're going to show that we can actually 01:02:52.710 --> 01:02:58.050 construct some very attractive kinds of investments using 01:02:58.050 --> 01:02:59.976 relatively simple information. 01:02:59.976 --> 01:03:01.350 But at the same time, we're going 01:03:01.350 --> 01:03:04.980 to use that insight to then deconstruct 01:03:04.980 --> 01:03:06.960 how to come up with the appropriate risk 01:03:06.960 --> 01:03:10.750 adjustment for cost of capital calculations. 01:03:10.750 --> 01:03:13.230 Now, there's a review here about normal distributions 01:03:13.230 --> 01:03:15.000 and confidence intervals, and I'd 01:03:15.000 --> 01:03:17.790 like you to go over that, either on your own or with the TAs 01:03:17.790 --> 01:03:19.390 during recitations. 01:03:19.390 --> 01:03:21.840 We're going to be using these kinds of concepts 01:03:21.840 --> 01:03:23.700 to try to measure the risk and return 01:03:23.700 --> 01:03:25.690 of various different investments. 01:03:25.690 --> 01:03:28.950 Here's an example of General Motors' monthly returns. 01:03:28.950 --> 01:03:34.170 That's a histogram in blue, and the line, the dark line, 01:03:34.170 --> 01:03:38.160 is the assumed normal distribution that 01:03:38.160 --> 01:03:40.830 has the same mean and variance. 01:03:40.830 --> 01:03:43.380 And you could see that it looks like it's 01:03:43.380 --> 01:03:47.580 sort of a good approximation, but there are actually 01:03:47.580 --> 01:03:52.650 little bits of extra probability stuck out here and stuck out 01:03:52.650 --> 01:03:56.140 here that don't exactly correspond to normal. 01:03:56.140 --> 01:03:58.740 In other words, the assumption of normality 01:03:58.740 --> 01:04:00.900 would say that the probability of getting 01:04:00.900 --> 01:04:05.280 a return of minus 15% is relatively low, 01:04:05.280 --> 01:04:10.230 then getting a return less than minus 20% is exceedingly low. 01:04:10.230 --> 01:04:12.630 But the reality is different. 01:04:12.630 --> 01:04:18.870 There are risks of having much lower returns in the data. 01:04:18.870 --> 01:04:20.855 And after this year, I can tell you 01:04:20.855 --> 01:04:23.760 that these tails are going to be fatter. 01:04:23.760 --> 01:04:28.939 So this is meant to be an approximation, not reality. 01:04:28.939 --> 01:04:30.480 The approximation is what we're going 01:04:30.480 --> 01:04:34.590 to go over in this course, and in the very last lecture, 01:04:34.590 --> 01:04:37.687 I want to tell you how good that approximation is. 01:04:37.687 --> 01:04:40.020 And then I'm going to tell you about a number of courses 01:04:40.020 --> 01:04:42.660 you might want to take that focus on getting 01:04:42.660 --> 01:04:45.240 that last 5% right. 01:04:45.240 --> 01:04:49.740 So 95% of the distribution is captured by what I'm going 01:04:49.740 --> 01:04:52.717 to teach you in this course, but if you want to get the other 5% 01:04:52.717 --> 01:04:55.050 right-- and by the way, if you're going into investments 01:04:55.050 --> 01:04:58.330 as a profession, it's all about that 5%-- 01:04:58.330 --> 01:05:02.850 then you'll want to take 15 433, investments. 01:05:02.850 --> 01:05:06.029 So with that as the basic preamble, 01:05:06.029 --> 01:05:08.320 let me tell you what I'm going to talk about next time, 01:05:08.320 --> 01:05:09.870 since we're almost out of time. 01:05:09.870 --> 01:05:11.820 What we're gonna do next time is I'm 01:05:11.820 --> 01:05:13.960 going to talk about the US stock market. 01:05:13.960 --> 01:05:17.160 I'm gonna talk about volatility, about predictability, 01:05:17.160 --> 01:05:19.290 and then I'm going to talk a bit about the notion 01:05:19.290 --> 01:05:21.360 of efficient markets, and try to describe 01:05:21.360 --> 01:05:24.720 to you what kinds of properties we expect 01:05:24.720 --> 01:05:26.334 from typical investments. 01:05:26.334 --> 01:05:28.500 And we're actually going to go through some numbers. 01:05:28.500 --> 01:05:32.670 I'm gonna show you some examples of basic statistics 01:05:32.670 --> 01:05:34.650 for the stock market that will give you 01:05:34.650 --> 01:05:38.880 a sense of how things have behaved over the last 50 years. 01:05:38.880 --> 01:05:41.400 And what you'll get a sense of is that in some cases, 01:05:41.400 --> 01:05:43.050 there is a lot of predictability. 01:05:43.050 --> 01:05:45.000 There are certain things that we can count on. 01:05:45.000 --> 01:05:48.360 For example, these are stock market returns 01:05:48.360 --> 01:05:50.640 from 1946 to 2001. 01:05:50.640 --> 01:05:54.180 This is monthly data, monthly returns 01:05:54.180 --> 01:05:58.990 of the S&P 500 over a fairly long period of time. 01:05:58.990 --> 01:06:02.760 And this might sort of look like a typical person's EKG 01:06:02.760 --> 01:06:04.260 over the last few weeks. 01:06:04.260 --> 01:06:07.020 Not surprisingly, there were periods where 01:06:07.020 --> 01:06:09.240 we had some pretty bad returns. 01:06:09.240 --> 01:06:11.340 We're going to see another one of these things 01:06:11.340 --> 01:06:14.610 as well over the more recent period. 01:06:14.610 --> 01:06:17.160 But when you look at this thing, you then 01:06:17.160 --> 01:06:19.800 begin to appreciate that what we're living through now, 01:06:19.800 --> 01:06:22.680 while it's bad and it's scary, it's 01:06:22.680 --> 01:06:26.640 not at all unusual or completely unheard of. 01:06:26.640 --> 01:06:28.620 There are periods in the stock market 01:06:28.620 --> 01:06:30.720 where we've seen really big swings. 01:06:30.720 --> 01:06:32.550 And by the way, this is just the US. 01:06:32.550 --> 01:06:36.270 If I had shown you some emerging market returns, 01:06:36.270 --> 01:06:38.020 it would go off the screen. 01:06:38.020 --> 01:06:40.120 So we're going to talk about that next time. 01:06:40.120 --> 01:06:42.270 And out of all of this chaos, we're 01:06:42.270 --> 01:06:45.396 going to distill a very important relationship. 01:06:45.396 --> 01:06:46.770 We're going to ultimately come up 01:06:46.770 --> 01:06:50.460 with a simple linear equation that shows you 01:06:50.460 --> 01:06:52.890 how to make that risk adjustment between the expected 01:06:52.890 --> 01:06:56.140 return and the underlying risk of a portfolio. 01:06:56.140 --> 01:06:58.890 So that's coming up, and we'll do that on Wednesday. 01:06:58.890 --> 01:07:01.180 All right, see you then.