1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,850 Commons license. 3 00:00:03,850 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high-quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,860 at ocw.mit.edu. 8 00:00:21,780 --> 00:00:23,670 PROFESSOR: So today, what I want to do 9 00:00:23,670 --> 00:00:27,330 is to continue where we were last time with option pricing. 10 00:00:27,330 --> 00:00:30,090 As I promised you last time, having 11 00:00:30,090 --> 00:00:31,950 gone through the history of option pricing 12 00:00:31,950 --> 00:00:34,170 and the special role that MIT played, 13 00:00:34,170 --> 00:00:37,470 today I actually want to do some option pricing. 14 00:00:37,470 --> 00:00:40,410 I want to show you a simple but extraordinarily powerful 15 00:00:40,410 --> 00:00:44,910 model for actually coming up with a theoretical pricing 16 00:00:44,910 --> 00:00:48,810 formula for options, and frankly, 17 00:00:48,810 --> 00:00:50,120 all derivative securities. 18 00:00:50,120 --> 00:00:52,620 So we're gonna actually do that in the space of about a half 19 00:00:52,620 --> 00:00:54,600 an hour, and then we're going to conclude. 20 00:00:54,600 --> 00:00:57,270 And I want to turn, then, to the next lecture, which 21 00:00:57,270 --> 00:00:59,010 is on risk and return. 22 00:00:59,010 --> 00:01:01,770 I want to now, after we finish option pricing, 23 00:01:01,770 --> 00:01:06,360 take on the challenge of trying to understand risk 24 00:01:06,360 --> 00:01:10,890 in a much more concrete way than we've done up until now. 25 00:01:10,890 --> 00:01:19,170 OK, so let's turn to lecture 10 and 11. 26 00:01:19,170 --> 00:01:25,345 And I'd like you to take a look at slide 16. 27 00:01:28,680 --> 00:01:37,880 OK, this will be the first model of option pricing 28 00:01:37,880 --> 00:01:39,374 that any of you have ever seen. 29 00:01:39,374 --> 00:01:40,790 You've all heard of Black-Scholes. 30 00:01:40,790 --> 00:01:42,710 We talked a bit about it last time. 31 00:01:42,710 --> 00:01:46,730 Frankly, this is a simpler version of option pricing 32 00:01:46,730 --> 00:01:49,880 that ultimately can actually be used 33 00:01:49,880 --> 00:01:52,400 to derive the Black-Scholes formula as well. 34 00:01:53,460 --> 00:01:55,550 But the reason I love this model is 35 00:01:55,550 --> 00:01:59,360 because it is so simple that with only basic high school 36 00:01:59,360 --> 00:02:03,500 algebra, you can actually work out all of the analytics. 37 00:02:03,500 --> 00:02:05,720 So all of you already have the math 38 00:02:05,720 --> 00:02:09,139 that it takes to implement this formula, 39 00:02:09,139 --> 00:02:11,690 and even to derive the formula. 40 00:02:11,690 --> 00:02:15,110 But the underlying economics is extraordinarily deep, 41 00:02:15,110 --> 00:02:17,390 and so it's a wonderful way of sort 42 00:02:17,390 --> 00:02:22,430 of getting a handle on how these very complex formulas work. 43 00:02:22,430 --> 00:02:23,880 So here's what we're going to do. 44 00:02:23,880 --> 00:02:27,690 We're going to simplify the problem in the following way. 45 00:02:27,690 --> 00:02:29,000 We're going to use-- 46 00:02:29,000 --> 00:02:30,530 the framework, by the way, is called 47 00:02:30,530 --> 00:02:32,630 the binomial option-pricing model 48 00:02:32,630 --> 00:02:37,820 that was derived by our very own John Cox, Steve Ross, and Mark 49 00:02:37,820 --> 00:02:40,100 Rubenstein of UC Berkeley. 50 00:02:40,100 --> 00:02:43,310 And although this is a simpler version 51 00:02:43,310 --> 00:02:45,710 of option pricing than Black and Scholes, 52 00:02:45,710 --> 00:02:51,080 it turns out that on the street, this is used much more commonly 53 00:02:51,080 --> 00:02:53,480 than the Black-Scholes formula. 54 00:02:53,480 --> 00:02:55,110 So let me show you how it works. 55 00:02:55,110 --> 00:02:59,360 We're going to start with a very simple framework of one period 56 00:02:59,360 --> 00:03:01,520 option pricing, meaning we're going 57 00:03:01,520 --> 00:03:07,400 to focus on a stock that survives for two periods-- 58 00:03:07,400 --> 00:03:09,840 this period and the next period. 59 00:03:09,840 --> 00:03:12,230 And then we're going to consider the pricing of an option 60 00:03:12,230 --> 00:03:16,070 on that stock that expires next period. 61 00:03:16,070 --> 00:03:20,050 We're going to figure out what the price is this period. 62 00:03:20,050 --> 00:03:23,150 So we've got a stock XYZ, and let's suppose 63 00:03:23,150 --> 00:03:25,540 the current stock price is S0. 64 00:03:25,540 --> 00:03:28,300 And let's suppose that we have a call option 65 00:03:28,300 --> 00:03:31,030 on this stock with a strike price of K, 66 00:03:31,030 --> 00:03:35,710 and where the option expires tomorrow. 67 00:03:35,710 --> 00:03:39,950 And so tomorrow's value of the option 68 00:03:39,950 --> 00:03:44,780 is simply equal to C1, which is the maximum of tomorrow's stock 69 00:03:44,780 --> 00:03:50,180 price minus the strike, or 0, the bigger of those two. 70 00:03:50,180 --> 00:03:52,940 That's the payoff for the call option. 71 00:03:52,940 --> 00:03:55,040 And the question that we want to attack 72 00:03:55,040 --> 00:03:58,260 is, what is the option's price today? 73 00:03:58,260 --> 00:04:00,110 In other words, what is C0? 74 00:04:00,110 --> 00:04:02,930 So we draw a timeline, as I've told you, 75 00:04:02,930 --> 00:04:04,460 for every one of these problems. 76 00:04:04,460 --> 00:04:07,530 Draw a timeline just so that there's no confusion. 77 00:04:07,530 --> 00:04:11,330 So tomorrow, the stock price is going to be worth S1, 78 00:04:11,330 --> 00:04:14,540 and the option price is just equal to C1, 79 00:04:14,540 --> 00:04:17,660 which is the payoff, since it expires tomorrow. 80 00:04:17,660 --> 00:04:21,019 And the payoff is just the maximum of S1 minus K and 0. 81 00:04:21,019 --> 00:04:25,220 And the object of our focus is to try 82 00:04:25,220 --> 00:04:28,495 to figure out what the value of the option is today. 83 00:04:28,495 --> 00:04:30,620 And so I'm going to argue that if we can figure out 84 00:04:30,620 --> 00:04:33,230 what it is today, based upon this, 85 00:04:33,230 --> 00:04:35,690 then we can actually generalize it in a very natural way 86 00:04:35,690 --> 00:04:38,557 to figure out what the price is for any number of periods 87 00:04:38,557 --> 00:04:39,140 in the future. 88 00:04:41,660 --> 00:04:42,960 So how do we do that? 89 00:04:42,960 --> 00:04:48,470 Well, we first have to make an assumption about how 90 00:04:48,470 --> 00:04:50,960 the stock price behaves. 91 00:04:50,960 --> 00:04:52,700 As I mentioned last time, we need 92 00:04:52,700 --> 00:04:55,880 to say something about the dynamics of stock prices, 93 00:04:55,880 --> 00:04:59,360 and remember that Bachelier, that French mathematician that 94 00:04:59,360 --> 00:05:02,720 came up with a rudimentary version of an option pricing 95 00:05:02,720 --> 00:05:06,170 formula in 1900, he developed the mathematics 96 00:05:06,170 --> 00:05:08,630 for Brownian motion, or a random walk, 97 00:05:08,630 --> 00:05:10,040 for the particular stock price. 98 00:05:10,040 --> 00:05:12,680 So you have to assume something about how 99 00:05:12,680 --> 00:05:14,510 the stock price moves. 100 00:05:14,510 --> 00:05:17,660 So what we're going to do in a simplified version 101 00:05:17,660 --> 00:05:23,920 is to assume that the stock price tomorrow is a coin flip. 102 00:05:23,920 --> 00:05:25,910 It's a Bernoulli trial. 103 00:05:25,910 --> 00:05:27,380 That's the technical term. 104 00:05:27,380 --> 00:05:29,870 So it either goes up or down. 105 00:05:29,870 --> 00:05:34,880 And if it goes up, it goes up by a gross amount u. 106 00:05:34,880 --> 00:05:37,400 So the value of the stock tomorrow, 107 00:05:37,400 --> 00:05:42,530 S1, is going to be equal to u multiplied by S0. 108 00:05:42,530 --> 00:05:45,200 So if it goes up by 10% tomorrow, 109 00:05:45,200 --> 00:05:49,490 then u is equal to 1.1. 110 00:05:49,490 --> 00:05:56,920 Or it can go down by a factor of d tomorrow, 111 00:05:56,920 --> 00:06:03,820 and so if it goes down by 10%, then d is 0.9. 112 00:06:03,820 --> 00:06:06,280 So we're going to simply assert that this 113 00:06:06,280 --> 00:06:09,680 is the statistical behavior of stock prices. 114 00:06:09,680 --> 00:06:13,360 Now, granted, this is a very, very strong simplification, 115 00:06:13,360 --> 00:06:15,340 but bear with me. 116 00:06:15,340 --> 00:06:17,985 After I derive the simple version of the pricing formula, 117 00:06:17,985 --> 00:06:19,360 I'm going to show you how to make 118 00:06:19,360 --> 00:06:21,490 it much, much more complex. 119 00:06:21,490 --> 00:06:23,950 And the additional complexity will 120 00:06:23,950 --> 00:06:27,340 be really simple to achieve once we understand 121 00:06:27,340 --> 00:06:30,310 this very basic version. 122 00:06:30,310 --> 00:06:32,920 Now, the probability of going up or down 123 00:06:32,920 --> 00:06:35,860 is not 50/50-- doesn't have to be 50/50. 124 00:06:35,860 --> 00:06:39,110 So I'm going to assert that it's equal to some probability of p 125 00:06:39,110 --> 00:06:40,210 and 1 minus p. 126 00:06:40,210 --> 00:06:43,210 So it either goes up by p or goes down with probability 1 127 00:06:43,210 --> 00:06:45,520 minus p, and the amount that it goes up or down 128 00:06:45,520 --> 00:06:47,410 is given by u and d. 129 00:06:47,410 --> 00:06:50,110 And I'm going to assert that u is greater than d. 130 00:06:50,110 --> 00:06:51,952 Question? 131 00:06:51,952 --> 00:06:53,660 AUDIENCE: So u and d are not the changes, 132 00:06:53,660 --> 00:06:57,540 but you're just assuming that, in your case, [INAUDIBLE].. 133 00:06:57,540 --> 00:06:59,142 PROFESSOR: Sorry, that they're-- 134 00:06:59,142 --> 00:07:01,517 AUDIENCE: It's not necessary to always use [INAUDIBLE] d, 135 00:07:01,517 --> 00:07:02,334 or can you assume-- 136 00:07:02,334 --> 00:07:04,750 PROFESSOR: Yeah, I'm assuming as a matter of normalization 137 00:07:04,750 --> 00:07:06,310 that u is greater than d. 138 00:07:06,310 --> 00:07:07,060 It doesn't matter. 139 00:07:07,060 --> 00:07:09,531 I mean, one thing has to be bigger than the other, 140 00:07:09,531 --> 00:07:11,780 so I just may as well assume that u is greater than d. 141 00:07:11,780 --> 00:07:12,791 Yeah, question? 142 00:07:12,791 --> 00:07:15,041 AUDIENCE: Is it necessary to add up to 1 [INAUDIBLE]?? 143 00:07:15,041 --> 00:07:17,249 PROFESSOR: u and do don't have to add up to anything. 144 00:07:17,249 --> 00:07:18,090 That's right. 145 00:07:18,090 --> 00:07:21,440 p and 1 minus p always add up to 1, right. 146 00:07:21,440 --> 00:07:24,170 So for example, if this is a growth stock that's 147 00:07:24,170 --> 00:07:35,340 really doing well, then u may be 1.1, 10%, and d may be 0.99. 148 00:07:35,340 --> 00:07:38,270 So in other words, when it goes down, it goes down by 1%. 149 00:07:38,270 --> 00:07:40,320 When it goes up, it goes up by 10%. 150 00:07:40,320 --> 00:07:42,250 So on average, when you multiply by p, 151 00:07:42,250 --> 00:07:45,140 1 minus p, depending on what they are, 152 00:07:45,140 --> 00:07:48,145 you can get a stock that's got a positive drift. 153 00:07:48,145 --> 00:07:49,520 If, on the other hand, you've got 154 00:07:49,520 --> 00:07:51,720 a stock that's declining in value, 155 00:07:51,720 --> 00:07:58,490 then it may end up that d much smaller than u, and 1 minus p 156 00:07:58,490 --> 00:08:00,680 is bigger than p, which means that you're 157 00:08:00,680 --> 00:08:03,770 more likely to be going down than you are going up. 158 00:08:03,770 --> 00:08:05,207 So it's pretty general. 159 00:08:05,207 --> 00:08:06,640 Yeah? 160 00:08:06,640 --> 00:08:10,245 AUDIENCE: u is bigger than 1 and d is lower than 1, or you could 161 00:08:10,245 --> 00:08:10,856 [INAUDIBLE]? 162 00:08:10,856 --> 00:08:12,230 PROFESSOR: It doesn't have to be. 163 00:08:12,230 --> 00:08:15,560 But typically you would think that in the up state, 164 00:08:15,560 --> 00:08:17,762 it's going to be bigger than 1, and the down state, 165 00:08:17,762 --> 00:08:18,720 it will be less than 1. 166 00:08:18,720 --> 00:08:21,110 But it doesn't have to be. 167 00:08:21,110 --> 00:08:25,610 What I'm going to normalize it to be is u is greater than d. 168 00:08:25,610 --> 00:08:28,281 And later on, we may make some other economic assumptions 169 00:08:28,281 --> 00:08:30,530 that I'll come to that will tell you a little bit more 170 00:08:30,530 --> 00:08:32,000 about what u and d are. 171 00:08:36,059 --> 00:08:40,770 Now, if it's true that the stock price can only 172 00:08:40,770 --> 00:08:45,050 take on two values tomorrow, then it 173 00:08:45,050 --> 00:08:49,700 stands to reason that the option can only take on two values 174 00:08:49,700 --> 00:08:51,470 tomorrow. 175 00:08:51,470 --> 00:08:52,700 And those are the two values. 176 00:08:52,700 --> 00:08:56,060 It's going to be Cu and Cd. 177 00:08:56,060 --> 00:09:00,320 Cu is where the stock price goes up to u times S0. 178 00:09:00,320 --> 00:09:04,040 Therefore, the option's going to be worth u S0 minus K, 0, 179 00:09:04,040 --> 00:09:06,200 maximum of those two. 180 00:09:06,200 --> 00:09:10,340 And similarly, if it turns out that the stock price goes down 181 00:09:10,340 --> 00:09:13,510 tomorrow, then the option is worth this tomorrow. 182 00:09:16,700 --> 00:09:18,530 Two values for the stock tomorrow 183 00:09:18,530 --> 00:09:23,030 implies two values for the option tomorrow. 184 00:09:23,030 --> 00:09:26,430 Any questions about that? 185 00:09:26,430 --> 00:09:31,680 OK, so having said that, we can now 186 00:09:31,680 --> 00:09:38,100 proceed to ask the question, given this simple framework, 187 00:09:38,100 --> 00:09:43,270 what should the option price today depend on? 188 00:09:43,270 --> 00:09:46,720 It's going to be a function of a bunch of parameters. 189 00:09:46,720 --> 00:09:50,040 So what should it depend upon? 190 00:09:50,040 --> 00:09:55,210 Well, the parameters that are given are these-- 191 00:09:55,210 --> 00:10:00,630 the stock price today, the strike price, u and d, 192 00:10:00,630 --> 00:10:05,610 p, and the interest rate between today and tomorrow. 193 00:10:05,610 --> 00:10:08,099 Those are the only parameters that we have. 194 00:10:08,099 --> 00:10:08,640 These are it. 195 00:10:08,640 --> 00:10:11,310 This is everything. 196 00:10:11,310 --> 00:10:15,650 It's going to turn out that with the simple framework that 197 00:10:15,650 --> 00:10:21,500 I've put down, we will be able to derive a closed-form 198 00:10:21,500 --> 00:10:25,970 analytical expression for what the option price has to be 199 00:10:25,970 --> 00:10:28,010 today-- 200 00:10:28,010 --> 00:10:28,910 C0. 201 00:10:28,910 --> 00:10:31,845 I'm going to do that for you in just a minute. 202 00:10:31,845 --> 00:10:33,970 But it's going to turn out that that option pricing 203 00:10:33,970 --> 00:10:37,960 formula, that f of stuff, is going 204 00:10:37,960 --> 00:10:42,800 to depend on all of these parameters except for one. 205 00:10:42,800 --> 00:10:46,190 One of these parameters is going to drop out. 206 00:10:46,190 --> 00:10:50,760 In other words, one of these parameters is redundant. 207 00:10:50,760 --> 00:10:53,070 And anybody want to take a guess as to what 208 00:10:53,070 --> 00:10:54,450 that parameter might be? 209 00:10:54,450 --> 00:10:57,570 What parameter do you think might not 210 00:10:57,570 --> 00:11:00,260 matter for pricing an option? 211 00:11:00,260 --> 00:11:01,035 Yeah, Terry. 212 00:11:01,035 --> 00:11:02,762 AUDIENCE: The interest rate, the r? 213 00:11:02,762 --> 00:11:03,970 PROFESSOR: The interest rate. 214 00:11:03,970 --> 00:11:10,060 Well, that's a good guess, but that's not the case. 215 00:11:10,060 --> 00:11:11,880 That's what I would have guessed, 216 00:11:11,880 --> 00:11:14,280 because that seems to be the thing that 217 00:11:14,280 --> 00:11:17,430 should matter the least, given how important all 218 00:11:17,430 --> 00:11:19,122 of these other parameters are. 219 00:11:19,122 --> 00:11:20,580 Anybody want to take another guess? 220 00:11:20,580 --> 00:11:21,200 Yeah, Ken. 221 00:11:21,200 --> 00:11:22,449 AUDIENCE: Today's stock price. 222 00:11:22,449 --> 00:11:23,850 PROFESSOR: Today's stock price. 223 00:11:23,850 --> 00:11:24,940 That's another good guess. 224 00:11:24,940 --> 00:11:26,520 [LAUGHTER] 225 00:11:26,520 --> 00:11:30,660 Although that's not correct, because in both the case 226 00:11:30,660 --> 00:11:33,060 of the interest rate and today's stock price, 227 00:11:33,060 --> 00:11:35,400 you could ask the question, suppose the stock price 228 00:11:35,400 --> 00:11:38,370 were at $1,000 versus $10. 229 00:11:38,370 --> 00:11:40,400 That would matter, wouldn't it? 230 00:11:40,400 --> 00:11:43,800 Or if the interest rate were at 20% versus 1%, 231 00:11:43,800 --> 00:11:46,200 that should matter, shouldn't it? 232 00:11:46,200 --> 00:11:47,850 And it does. 233 00:11:47,850 --> 00:11:51,690 In fact, if you look at every single one of these parameters, 234 00:11:51,690 --> 00:11:57,030 none of them looks like they're unnecessary. 235 00:11:57,030 --> 00:11:59,080 It looks like all of them are required. 236 00:11:59,080 --> 00:12:01,468 Yeah, John. 237 00:12:01,468 --> 00:12:03,910 AUDIENCE: [INAUDIBLE] 238 00:12:03,910 --> 00:12:06,330 PROFESSOR: Well, the strike price remains the same, 239 00:12:06,330 --> 00:12:08,790 but the thing is that the question is whether or not 240 00:12:08,790 --> 00:12:11,680 the value of the option depends on the strike price. 241 00:12:11,680 --> 00:12:13,950 And if the option is, for example, in the money 242 00:12:13,950 --> 00:12:15,908 or out of the money, you would expect that that 243 00:12:15,908 --> 00:12:17,346 would make a big difference. 244 00:12:17,346 --> 00:12:19,130 AUDIENCE: [INAUDIBLE] 245 00:12:19,130 --> 00:12:20,420 PROFESSOR: Right. 246 00:12:20,420 --> 00:12:22,910 In fact, let me tell you that there 247 00:12:22,910 --> 00:12:26,390 is no good answer to this, because all of these parameters 248 00:12:26,390 --> 00:12:28,130 look like they belong. 249 00:12:28,130 --> 00:12:31,250 But I want to tell you that one of them will not. 250 00:12:31,250 --> 00:12:32,960 One of them will not be in here. 251 00:12:32,960 --> 00:12:38,300 And this is going to be a major source of both confusion 252 00:12:38,300 --> 00:12:42,770 and illumination for what really depends on-- 253 00:12:42,770 --> 00:12:44,627 what option pricing really depends on. 254 00:12:44,627 --> 00:12:47,210 All right, so let me just show you how we're going to do this. 255 00:12:47,210 --> 00:12:50,030 Let me illustrate to you the method. 256 00:12:50,030 --> 00:12:53,720 And we're going to do this in the exact same way 257 00:12:53,720 --> 00:12:57,830 that we've priced virtually everything under the sun. 258 00:12:57,830 --> 00:13:01,310 We're going to use an arbitrage argument. 259 00:13:01,310 --> 00:13:03,650 I'm going to construct a portfolio 260 00:13:03,650 --> 00:13:08,940 that will have the identical payoff to the option, 261 00:13:08,940 --> 00:13:12,090 and therefore if the portfolio has the exact same cash 262 00:13:12,090 --> 00:13:15,180 flows as the option, then the cost 263 00:13:15,180 --> 00:13:17,070 of constructing that portfolio has 264 00:13:17,070 --> 00:13:20,860 to be the price of the option. 265 00:13:20,860 --> 00:13:23,850 Moreover, if it's not, you're going 266 00:13:23,850 --> 00:13:27,250 to be very happy, because that will mean that there 267 00:13:27,250 --> 00:13:28,810 is an arbitrage opportunity. 268 00:13:28,810 --> 00:13:30,820 That is, there's money to be made. 269 00:13:30,820 --> 00:13:33,970 If this theory fails, then you're 270 00:13:33,970 --> 00:13:37,780 going to be able to get rich beyond your wildest dreams. 271 00:13:37,780 --> 00:13:42,254 So we're hoping for a violation of this. 272 00:13:42,254 --> 00:13:43,420 So let's see how we do that. 273 00:13:45,990 --> 00:13:49,250 I want you to now forget about the option for a moment, 274 00:13:49,250 --> 00:13:51,980 and I want you to imagine that at time 0, 275 00:13:51,980 --> 00:13:56,890 we construct a portfolio consisting of stocks 276 00:13:56,890 --> 00:14:00,740 and riskless bonds, in particular delta 277 00:14:00,740 --> 00:14:06,610 shares of stocks and B dollars of riskless bonds-- 278 00:14:06,610 --> 00:14:10,920 riskless in terms of default. 279 00:14:10,920 --> 00:14:20,530 And the total cost of this portfolio today, time 0, 280 00:14:20,530 --> 00:14:23,140 is simply equal to the price per share times 281 00:14:23,140 --> 00:14:27,400 the number of shares of stock, so that's S0 times delta, 282 00:14:27,400 --> 00:14:30,910 plus the value of the bonds that I'm 283 00:14:30,910 --> 00:14:33,190 buying-- the market value of the bonds 284 00:14:33,190 --> 00:14:35,650 that I'm buying today, or selling. 285 00:14:35,650 --> 00:14:38,050 So B could be a positive or negative number. 286 00:14:38,050 --> 00:14:41,860 Delta could be a positive or a negative number. 287 00:14:41,860 --> 00:14:48,110 And that's my cost today, time 0. 288 00:14:48,110 --> 00:14:50,070 Now, I want to look at the payoff 289 00:14:50,070 --> 00:14:52,050 tomorrow for this portfolio. 290 00:14:52,050 --> 00:14:54,040 So V1 is the payoff for the portfolio. 291 00:14:54,040 --> 00:14:56,010 That's what it's worth tomorrow. 292 00:14:56,010 --> 00:15:02,450 And V1 is going to be given by the value of the stocks 293 00:15:02,450 --> 00:15:04,440 and the value of the bonds. 294 00:15:04,440 --> 00:15:06,410 Now, the stocks are going to be-- 295 00:15:06,410 --> 00:15:07,590 there are two possibilities. 296 00:15:07,590 --> 00:15:10,820 Either the stock goes up or the stock goes down. 297 00:15:10,820 --> 00:15:14,340 And if it goes up, it'll be worth u S0 times delta, 298 00:15:14,340 --> 00:15:17,510 and if it goes down it'll be worth d S0 times delta. 299 00:15:17,510 --> 00:15:21,410 I don't know whether it'll go up or down, but whatever it does, 300 00:15:21,410 --> 00:15:24,170 this is the value tomorrow. 301 00:15:24,170 --> 00:15:26,090 Now, what about my bond portfolio? 302 00:15:26,090 --> 00:15:29,750 Well, I bought B bonds, and r now 303 00:15:29,750 --> 00:15:32,830 is the gross rate of return, the gross interest rate, 304 00:15:32,830 --> 00:15:37,040 so it's a number like 1.03 or 1.05. 305 00:15:37,040 --> 00:15:39,170 And the reason that I'm switching notation 306 00:15:39,170 --> 00:15:42,610 is I'm following the notation used originally by Cox, Ross, 307 00:15:42,610 --> 00:15:46,370 and Rubenstein, so I apologize for the kind 308 00:15:46,370 --> 00:15:49,760 of cognitive dissonance that this may generate. 309 00:15:49,760 --> 00:15:53,720 But this r, the way that Cox, Ross, and Rubenstein wrote it, 310 00:15:53,720 --> 00:15:56,160 was meant to be a gross rate of return. 311 00:15:56,160 --> 00:15:58,910 So you'll never see a 1 plus r, because this-- 312 00:15:58,910 --> 00:16:02,859 in their framework, because this already contains the 1. 313 00:16:02,859 --> 00:16:04,650 So just keep that in the back of your mind, 314 00:16:04,650 --> 00:16:07,730 and make a note of that. r is the gross interest rate. 315 00:16:07,730 --> 00:16:10,680 It's 1 plus the net interest rate, 316 00:16:10,680 --> 00:16:16,491 so it's a number like 1.03 for a 3% rate of return. 317 00:16:16,491 --> 00:16:17,990 Actually nowadays, it should be more 318 00:16:17,990 --> 00:16:23,667 like 1.01 for short-term interest rates, or less. 319 00:16:23,667 --> 00:16:25,250 Now, you'll notice that whether or not 320 00:16:25,250 --> 00:16:27,230 stocks go up or down has no impact 321 00:16:27,230 --> 00:16:28,640 on your riskless borrowing. 322 00:16:28,640 --> 00:16:31,070 You're going to get r times B no matter what, 323 00:16:31,070 --> 00:16:34,320 or you're going to owe r times B if B 324 00:16:34,320 --> 00:16:38,360 was a negative number, in both cases, because it's riskless. 325 00:16:38,360 --> 00:16:40,340 It has nothing to do with whether or not 326 00:16:40,340 --> 00:16:43,700 stocks go up or down. 327 00:16:43,700 --> 00:16:45,860 OK, now here's what I want you to do. 328 00:16:45,860 --> 00:16:51,530 I want you to select a specific amount of stocks and bonds 329 00:16:51,530 --> 00:16:58,350 at date zero in order to make two things true. 330 00:16:58,350 --> 00:17:05,369 I want you to select delta and B so as to satisfy these two 331 00:17:05,369 --> 00:17:07,700 equations. 332 00:17:07,700 --> 00:17:10,990 I want you to pick delta and B so that in the up state, 333 00:17:10,990 --> 00:17:16,089 you get Cu and in the down state you get Cd. 334 00:17:16,089 --> 00:17:17,261 Now, what are Cu and Cd? 335 00:17:17,261 --> 00:17:18,219 Remember what they are? 336 00:17:18,219 --> 00:17:20,950 They're the value of the call option in the up 337 00:17:20,950 --> 00:17:22,990 state and the down state. 338 00:17:22,990 --> 00:17:24,329 And you know that in advance. 339 00:17:24,329 --> 00:17:26,079 You know what those two possibilities are. 340 00:17:26,079 --> 00:17:27,940 You don't know which one's going to occur, 341 00:17:27,940 --> 00:17:31,570 but you know that if the up state occurs, it'll be Cu, 342 00:17:31,570 --> 00:17:36,300 and if the down state occurs, it'll be Cd. 343 00:17:36,300 --> 00:17:42,540 So I want you to find two numbers, delta and B, 344 00:17:42,540 --> 00:17:45,370 that make those two equations true. 345 00:17:45,370 --> 00:17:47,935 Can you always do that? 346 00:17:47,935 --> 00:17:49,560 How do you know you can always do that? 347 00:17:53,420 --> 00:17:58,180 OK, can you always find a delta and a B 348 00:17:58,180 --> 00:18:03,980 to make those two relationships true? 349 00:18:03,980 --> 00:18:04,741 Yeah. 350 00:18:04,741 --> 00:18:07,200 AUDIENCE: You have two equations and two variables. 351 00:18:07,200 --> 00:18:11,060 PROFESSOR: Ah, you have two linear equations 352 00:18:11,060 --> 00:18:13,310 in two unknowns. 353 00:18:13,310 --> 00:18:15,960 And from basic high school algebra, 354 00:18:15,960 --> 00:18:19,550 you know that unless those two linear equations are 355 00:18:19,550 --> 00:18:24,820 multiples of each other, you can always find one-- 356 00:18:24,820 --> 00:18:29,430 exactly one solution that satisfies those two 357 00:18:29,430 --> 00:18:32,200 equations in two unknowns. 358 00:18:32,200 --> 00:18:37,910 Kind of a handy feature about linear equations. 359 00:18:37,910 --> 00:18:41,120 So as long as these two equations are said to be 360 00:18:41,120 --> 00:18:43,912 linearly independent-- that's a fancy way of saying that 361 00:18:43,912 --> 00:18:45,620 they're actually two different equations, 362 00:18:45,620 --> 00:18:48,690 they're not multiples of each other-- 363 00:18:48,690 --> 00:18:50,790 as long as these two equations are not 364 00:18:50,790 --> 00:18:53,070 multiples of each other, you can always 365 00:18:53,070 --> 00:18:57,900 find two numbers, delta and B, to make that true. 366 00:18:57,900 --> 00:18:58,890 And here they are. 367 00:18:58,890 --> 00:19:01,000 Those are the two numbers, delta star and B star. 368 00:19:01,000 --> 00:19:02,801 I solved the equation for you, not 369 00:19:02,801 --> 00:19:04,300 that you couldn't do it on your own, 370 00:19:04,300 --> 00:19:09,560 but for convenience, there it is. 371 00:19:09,560 --> 00:19:11,900 So let me tell you what we've done. 372 00:19:11,900 --> 00:19:16,990 We've put together a portfolio of stocks and bonds at date 0 373 00:19:16,990 --> 00:19:22,900 such that at time 1, the value of this portfolio 374 00:19:22,900 --> 00:19:26,950 is always equal to the value of the call 375 00:19:26,950 --> 00:19:32,960 option in no matter what state of the world actually occurs. 376 00:19:32,960 --> 00:19:36,870 Well, by the principle of arbitrage, 377 00:19:36,870 --> 00:19:41,610 what this tells us is that the cost of putting together 378 00:19:41,610 --> 00:19:47,490 this portfolio that replicates the call option's cash flows, 379 00:19:47,490 --> 00:19:49,860 the value of that portfolio at date 0 380 00:19:49,860 --> 00:19:54,680 must equal the price of a call option. 381 00:19:54,680 --> 00:19:58,350 So we're done. 382 00:19:58,350 --> 00:20:04,860 The solution of what is the call option price at date 0, 383 00:20:04,860 --> 00:20:06,900 it's given by this formula right here. 384 00:20:06,900 --> 00:20:11,530 There it is-- a closed-form solution. 385 00:20:11,530 --> 00:20:13,860 Now, before we beat up on it and say, 386 00:20:13,860 --> 00:20:16,590 gee, there's only two possibilities, 387 00:20:16,590 --> 00:20:18,210 life is more complicated than that, 388 00:20:18,210 --> 00:20:21,180 and also there's only one period, let's not-- 389 00:20:21,180 --> 00:20:23,050 let's not beat up on it just yet. 390 00:20:23,050 --> 00:20:26,340 Let's take a look to see whether or not this makes sense 391 00:20:26,340 --> 00:20:29,760 and whether we agree and understand 392 00:20:29,760 --> 00:20:32,070 that if, in fact, the assumptions are true, 393 00:20:32,070 --> 00:20:35,190 that this is indeed the price of an option. 394 00:20:35,190 --> 00:20:38,010 Because this is a pretty remarkable formula. 395 00:20:38,010 --> 00:20:41,190 It's a remarkable formula for its simplicity, 396 00:20:41,190 --> 00:20:45,060 and for the fact that we actually 397 00:20:45,060 --> 00:20:49,280 have been able to derive it explicitly. 398 00:20:49,280 --> 00:20:51,030 Now, the other amazing thing is that there 399 00:20:51,030 --> 00:20:53,340 is a missing parameter here. 400 00:20:53,340 --> 00:20:56,040 Now you see what the missing parameter is. 401 00:20:56,040 --> 00:21:01,410 What this formula doesn't depend on 402 00:21:01,410 --> 00:21:07,000 is the probability of the thing going up or down. 403 00:21:07,000 --> 00:21:08,820 Now, that's astonishing. 404 00:21:08,820 --> 00:21:13,320 It's astonishing because what it says is that you and I, 405 00:21:13,320 --> 00:21:17,370 we can disagree on whether General Electric is going 406 00:21:17,370 --> 00:21:21,250 to go up tomorrow or down tomorrow, 407 00:21:21,250 --> 00:21:24,460 and yet we still are going to agree 408 00:21:24,460 --> 00:21:27,580 on what the value of a General Electric call option 409 00:21:27,580 --> 00:21:29,710 is tomorrow. 410 00:21:29,710 --> 00:21:32,580 That's a remarkable fact, and it has 411 00:21:32,580 --> 00:21:36,810 to do with a very deep, deep phenomenon 412 00:21:36,810 --> 00:21:41,310 going on in option pricing, which is that option pricing is 413 00:21:41,310 --> 00:21:47,340 all about pricing the relative magnitude of the security 414 00:21:47,340 --> 00:21:50,190 relative to the stock price. 415 00:21:50,190 --> 00:21:55,020 And once we understand the basic features of the stock price, 416 00:21:55,020 --> 00:21:57,990 like whether or not it can go up or down 417 00:21:57,990 --> 00:22:04,800 by u or d, that's more important than the actual probabilities 418 00:22:04,800 --> 00:22:06,810 of u and d. 419 00:22:06,810 --> 00:22:11,670 So this expression-- and when you fill in for Cu and Cd, 420 00:22:11,670 --> 00:22:17,760 you can plug in for that maximum of u S0 minus k, 0, 421 00:22:17,760 --> 00:22:20,130 you'll see that there's no p in there as well. 422 00:22:23,260 --> 00:22:26,606 So any questions about this? 423 00:22:26,606 --> 00:22:27,105 Yeah. 424 00:22:27,800 --> 00:22:30,100 AUDIENCE: You just said that we could 425 00:22:30,100 --> 00:22:34,780 disagree on what we-- if the stock will go up or down, 426 00:22:34,780 --> 00:22:35,436 [INAUDIBLE]. 427 00:22:35,436 --> 00:22:38,060 PROFESSOR: Yeah, the probability of it going up or down, right. 428 00:22:38,060 --> 00:22:42,290 AUDIENCE: And what if we disagree on the actual number? 429 00:22:42,290 --> 00:22:44,520 PROFESSOR: Then we will disagree on the option price. 430 00:22:44,520 --> 00:22:47,494 So we have to agree on the u and the d. 431 00:22:47,494 --> 00:22:48,470 AUDIENCE: But if we-- 432 00:22:48,470 --> 00:22:50,236 PROFESSOR: Sorry-- yeah, the u and the d. 433 00:22:50,236 --> 00:22:52,569 AUDIENCE: [INAUDIBLE] we will disagree the option price, 434 00:22:52,569 --> 00:22:55,649 or that's why this market is possible, 435 00:22:55,649 --> 00:22:57,581 because someone will think it will go up-- 436 00:22:57,581 --> 00:22:59,490 PROFESSOR: No, no, it's not the reason 437 00:22:59,490 --> 00:23:01,220 that the market will be possible. 438 00:23:01,220 --> 00:23:03,260 The possibility of the market actually 439 00:23:03,260 --> 00:23:05,450 does depend on whether or not there's 440 00:23:05,450 --> 00:23:07,730 a demand for this particular kind of payoff. 441 00:23:07,730 --> 00:23:10,505 But that doesn't necessarily hinge on the u or the d. 442 00:23:10,505 --> 00:23:12,760 In other words, we can agree on the u and the d, 443 00:23:12,760 --> 00:23:15,440 but it turns out that you think that the price is going 444 00:23:15,440 --> 00:23:18,270 to go up, therefore, you want to have that kind of a call option 445 00:23:18,270 --> 00:23:18,770 bet. 446 00:23:18,770 --> 00:23:20,460 I think the price is going to go down, 447 00:23:20,460 --> 00:23:21,830 so I'm happy to sell it to you, because I think I'm 448 00:23:21,830 --> 00:23:23,450 going to get a good deal on it. 449 00:23:23,450 --> 00:23:25,760 So we disagree on the p. 450 00:23:25,760 --> 00:23:27,546 You think that there's a high p. 451 00:23:27,546 --> 00:23:29,600 I think it's a low p. 452 00:23:29,600 --> 00:23:31,160 That's what drives the market. 453 00:23:31,160 --> 00:23:34,880 And the beauty of this particular setup 454 00:23:34,880 --> 00:23:40,020 is that it tells you that you can actually agree on a price, 455 00:23:40,020 --> 00:23:41,790 but you have very different reasons 456 00:23:41,790 --> 00:23:43,920 for engaging in the transaction. 457 00:23:43,920 --> 00:23:49,330 And then you will have markets for this particular security. 458 00:23:49,330 --> 00:23:51,520 Now, I want to go through and look at this formula 459 00:23:51,520 --> 00:23:54,130 and try to understand it. 460 00:23:54,130 --> 00:23:58,300 First of all, we see that this formula is a weighted average 461 00:23:58,300 --> 00:24:00,730 of the value of the call option in the up 462 00:24:00,730 --> 00:24:02,980 state and the down state. 463 00:24:02,980 --> 00:24:05,270 It's a weighted average. 464 00:24:05,270 --> 00:24:08,300 And this part inside the bracket you 465 00:24:08,300 --> 00:24:11,870 can think of as a weighted average of the outcome. 466 00:24:11,870 --> 00:24:18,045 But then you discount it back to the 0th period using the one 467 00:24:18,045 --> 00:24:18,920 period interest rate. 468 00:24:18,920 --> 00:24:21,590 And again, remember this is not meant 469 00:24:21,590 --> 00:24:24,670 to be a perpetuity kind of expression. 470 00:24:24,670 --> 00:24:27,170 This r is a gross interest rate, so it 471 00:24:27,170 --> 00:24:30,440 is equivalent to our old 1 over 1 plus 472 00:24:30,440 --> 00:24:36,560 r, where the r that we used is the net interest rate. 473 00:24:36,560 --> 00:24:40,280 Here, because of the Cox, Ross, and Rubenstein notation, 474 00:24:40,280 --> 00:24:44,080 this is meant to be the gross interest rate. 475 00:24:44,080 --> 00:24:48,510 So this looks like a present value, 476 00:24:48,510 --> 00:24:51,120 because whatever is inside the bracket, 477 00:24:51,120 --> 00:24:54,090 you can think of as some weighted average 478 00:24:54,090 --> 00:25:02,280 of the value at date 1, and then this brings it back to date 0. 479 00:25:02,280 --> 00:25:04,200 But now let's look at the weighted average. 480 00:25:04,200 --> 00:25:09,720 The weights r minus d and u minus d, those-- 481 00:25:09,720 --> 00:25:15,550 it turns out that this plus this adds up to 1, 482 00:25:15,550 --> 00:25:18,520 so indeed, it is a weighted average. 483 00:25:18,520 --> 00:25:23,560 When you multiply by theta and 1 minus theta, 484 00:25:23,560 --> 00:25:25,030 the weights add up to 1. 485 00:25:25,030 --> 00:25:28,180 You're basically taking a weighted average. 486 00:25:28,180 --> 00:25:31,970 But I want to argue that it's more than just 487 00:25:31,970 --> 00:25:34,060 a simple weighted average. 488 00:25:34,060 --> 00:25:38,600 I'm gonna argue that these weights are always 489 00:25:38,600 --> 00:25:41,050 non-negative. 490 00:25:41,050 --> 00:25:44,890 So in fact, this looks like not just a weighted average, 491 00:25:44,890 --> 00:25:50,020 this looks an awful lot like a kind of an expected value, 492 00:25:50,020 --> 00:25:52,210 like a probability weighted average. 493 00:25:52,210 --> 00:25:54,170 This looks like a probability. 494 00:25:54,170 --> 00:25:56,980 It's not a probability, but I want 495 00:25:56,980 --> 00:25:59,740 to argue that this number is always non-negative, 496 00:25:59,740 --> 00:26:00,799 and they add up to 1. 497 00:26:00,799 --> 00:26:02,965 So when you've got two numbers that are not negative 498 00:26:02,965 --> 00:26:04,798 and they add up to 1, you can interpret them 499 00:26:04,798 --> 00:26:07,960 as a probability. 500 00:26:07,960 --> 00:26:11,440 Now, what's the argument for why this number is always 501 00:26:11,440 --> 00:26:13,480 going to be non-negative? 502 00:26:13,480 --> 00:26:17,380 The condition that's required for these numbers 503 00:26:17,380 --> 00:26:21,100 to be non-negative is that the interest rate 504 00:26:21,100 --> 00:26:24,910 r, the risk-free rate, is strictly 505 00:26:24,910 --> 00:26:28,420 contained in between u and d. 506 00:26:28,420 --> 00:26:32,340 So you've got u here, d here. 507 00:26:32,340 --> 00:26:34,070 r has to be in the middle. 508 00:26:34,070 --> 00:26:37,600 And when that's the case, then you've 509 00:26:37,600 --> 00:26:41,030 got these things looking like probabilities. 510 00:26:41,030 --> 00:26:43,480 Now, the question is, is that a reasonable assumption? 511 00:26:43,480 --> 00:26:49,150 Is it reasonable to assume that d is less 512 00:26:49,150 --> 00:26:52,270 than r is less than u? 513 00:26:52,270 --> 00:26:54,880 Can anybody give me some intuition 514 00:26:54,880 --> 00:26:58,400 for why that makes economic sense? 515 00:26:58,400 --> 00:27:00,199 It has nothing to do with mathematics. 516 00:27:00,199 --> 00:27:02,490 The mathematics couldn't care less as to whether or not 517 00:27:02,490 --> 00:27:03,600 that inequality held. 518 00:27:03,600 --> 00:27:04,227 Brian? 519 00:27:04,227 --> 00:27:08,560 AUDIENCE: If the downside was less than the rate, 520 00:27:08,560 --> 00:27:11,452 then you'd just automatically buy the security. 521 00:27:11,452 --> 00:27:12,160 PROFESSOR: Right. 522 00:27:12,160 --> 00:27:15,940 AUDIENCE: And if the upside was less than the risk-free rate, 523 00:27:15,940 --> 00:27:18,060 they'd you'd just go into the risk-free bills. 524 00:27:18,060 --> 00:27:18,830 PROFESSOR: Right. 525 00:27:18,830 --> 00:27:20,010 That's exactly right. 526 00:27:20,010 --> 00:27:22,290 That's a very important economic insight. 527 00:27:22,290 --> 00:27:23,740 Let me go through that slowly. 528 00:27:23,740 --> 00:27:28,830 So Brian, you said if r is less than the downside, then 529 00:27:28,830 --> 00:27:31,070 what happens in that case? 530 00:27:31,070 --> 00:27:34,020 AUDIENCE: Then you'd want to buy the stock, the security. 531 00:27:34,020 --> 00:27:38,190 PROFESSOR: If the stock in its worst possible state 532 00:27:38,190 --> 00:27:41,820 offers more than T-bills, why would 533 00:27:41,820 --> 00:27:43,290 you ever want to buy T-bills? 534 00:27:43,290 --> 00:27:44,730 In fact, you wouldn't. 535 00:27:44,730 --> 00:27:46,559 And if that were true, then what would 536 00:27:46,559 --> 00:27:47,850 happen to the price of T-bills? 537 00:27:51,670 --> 00:27:53,525 The price of T-bills-- 538 00:27:53,525 --> 00:27:54,650 AUDIENCE: It would go down. 539 00:27:54,650 --> 00:27:56,250 PROFESSOR: It would go to 0. 540 00:27:56,250 --> 00:27:59,240 Nobody would hold it, and therefore the value of it 541 00:27:59,240 --> 00:28:00,020 would go to zero. 542 00:28:00,020 --> 00:28:02,370 It would not exist any longer. 543 00:28:02,370 --> 00:28:04,010 So if we're going to assume that there 544 00:28:04,010 --> 00:28:06,560 exists riskless borrowing, that can't be true. 545 00:28:06,560 --> 00:28:08,847 We can't have r over here. 546 00:28:08,847 --> 00:28:10,430 Now, what about the other side, Brian? 547 00:28:10,430 --> 00:28:12,560 What happens if r is over here? 548 00:28:12,560 --> 00:28:13,895 What did you say? 549 00:28:13,895 --> 00:28:16,020 AUDIENCE: Then you'd want to go into the risk-free. 550 00:28:16,020 --> 00:28:16,250 PROFESSOR: Right. 551 00:28:16,250 --> 00:28:18,083 You would never hold the stock, because even 552 00:28:18,083 --> 00:28:21,020 in the best possible world for the stock, 553 00:28:21,020 --> 00:28:24,650 you would not be able to get as good a return as T-bills, 554 00:28:24,650 --> 00:28:27,650 in which case the value of the stock would go to zero, 555 00:28:27,650 --> 00:28:30,020 and therefore there'd be no more stocks in the economy. 556 00:28:30,020 --> 00:28:33,830 The only situation where you can have stocks and T-bills 557 00:28:33,830 --> 00:28:36,650 coexisting in this simple world-- 558 00:28:36,650 --> 00:28:42,210 the only case where that's true is if this inequality held. 559 00:28:42,210 --> 00:28:45,500 That's the economics of this pricing formula. 560 00:28:45,500 --> 00:28:47,830 It has nothing to do with math. 561 00:28:47,830 --> 00:28:49,080 It's the economics. 562 00:28:49,080 --> 00:28:51,780 And the economics tells you that these things 563 00:28:51,780 --> 00:28:52,980 have to be non-negative. 564 00:28:52,980 --> 00:28:54,730 That's good, because that suggests 565 00:28:54,730 --> 00:28:57,420 that the price of the call option at date 0 566 00:28:57,420 --> 00:29:01,680 can never be negative, because these guys, Cu and Cd, 567 00:29:01,680 --> 00:29:04,870 are non-negative, and 1 over r is non-negative. 568 00:29:04,870 --> 00:29:07,977 So if it turns out that the weights can never be negative, 569 00:29:07,977 --> 00:29:09,810 then you know that you've got something that 570 00:29:09,810 --> 00:29:11,199 really is a pricing formula. 571 00:29:11,199 --> 00:29:12,990 You're never going to punch in some numbers 572 00:29:12,990 --> 00:29:18,140 and get out a formula that says this thing is worth minus 2. 573 00:29:18,140 --> 00:29:20,510 But more importantly, it suggests 574 00:29:20,510 --> 00:29:23,480 that there is a probability interpretation. 575 00:29:23,480 --> 00:29:26,930 But the probability is not the mathematical probability 576 00:29:26,930 --> 00:29:27,780 that matters. 577 00:29:27,780 --> 00:29:31,160 It is the economic probability. 578 00:29:31,160 --> 00:29:34,970 And there is a term for this particular probability. 579 00:29:34,970 --> 00:29:39,950 This is known as the risk-neutral probabilities 580 00:29:39,950 --> 00:29:43,910 of the particular economy that we've created. 581 00:29:43,910 --> 00:29:46,400 And it turns out that these probabilities 582 00:29:46,400 --> 00:29:49,010 can be used to price not just options, 583 00:29:49,010 --> 00:29:50,430 but anything under the sun. 584 00:29:50,430 --> 00:29:52,880 So there's a very, very important 585 00:29:52,880 --> 00:29:56,030 property and very deep property that we can't go into here, 586 00:29:56,030 --> 00:29:59,930 but you'll cover in 15 437, about the so-called risk 587 00:29:59,930 --> 00:30:03,200 neutral probabilities. 588 00:30:03,200 --> 00:30:04,980 But now we've got a formula here. 589 00:30:04,980 --> 00:30:07,340 This is a bona fide pricing formula. 590 00:30:07,340 --> 00:30:09,680 And the beauty of it is that if it 591 00:30:09,680 --> 00:30:12,530 is violated-- if it is violated but the assumptions are 592 00:30:12,530 --> 00:30:15,860 correct, then there is a way to create a money 593 00:30:15,860 --> 00:30:20,180 machine, an arbitrage, either by buying the cheap stuff 594 00:30:20,180 --> 00:30:23,840 and shorting the expensive, or vice versa, 595 00:30:23,840 --> 00:30:27,890 in the case where the signs are flipped. 596 00:30:27,890 --> 00:30:29,480 So here's the argument. 597 00:30:29,480 --> 00:30:33,770 Suppose that C is greater than V. Then here's the arbitrage. 598 00:30:33,770 --> 00:30:36,650 Suppose it's less, and then you basically 599 00:30:36,650 --> 00:30:38,540 construct the opposite arbitrage. 600 00:30:38,540 --> 00:30:42,170 Therefore the cost of the option has to be equal to the value 601 00:30:42,170 --> 00:30:42,920 that we computed. 602 00:30:42,920 --> 00:30:45,135 Yeah, [INAUDIBLE]. 603 00:30:45,135 --> 00:30:51,100 AUDIENCE: Is this function sensitive [INAUDIBLE] 604 00:30:51,100 --> 00:30:53,150 PROFESSOR: Well, I mean, you tell me. 605 00:30:53,150 --> 00:30:56,250 It's a convex combination of these two things. 606 00:30:56,250 --> 00:30:57,410 So in that sense-- 607 00:30:57,410 --> 00:30:59,850 AUDIENCE: [INAUDIBLE] 608 00:30:59,850 --> 00:31:01,340 PROFESSOR: That's right, exactly. 609 00:31:01,340 --> 00:31:02,960 Yeah. 610 00:31:02,960 --> 00:31:07,410 AUDIENCE: And this is not going to [INAUDIBLE] 611 00:31:07,410 --> 00:31:08,900 PROFESSOR: Yeah. 612 00:31:08,900 --> 00:31:12,472 AUDIENCE: The u and d are determined by the market-- 613 00:31:12,472 --> 00:31:13,055 PROFESSOR: No. 614 00:31:13,055 --> 00:31:14,270 AUDIENCE: [INAUDIBLE] 615 00:31:14,270 --> 00:31:14,870 PROFESSOR: No. 616 00:31:14,870 --> 00:31:16,560 That's a modeling assumption. 617 00:31:16,560 --> 00:31:20,630 So in advance, we agree what u and d are. 618 00:31:20,630 --> 00:31:24,710 Now in a minute, I'm going to start relaxing 619 00:31:24,710 --> 00:31:26,865 all of these assumptions. 620 00:31:26,865 --> 00:31:28,490 But before we do that, I want make sure 621 00:31:28,490 --> 00:31:30,110 we all agree on what this says. 622 00:31:30,110 --> 00:31:30,900 Yeah. 623 00:31:30,900 --> 00:31:33,000 AUDIENCE: So the p is missing, as you said. 624 00:31:33,000 --> 00:31:33,625 PROFESSOR: Yes. 625 00:31:33,625 --> 00:31:35,460 AUDIENCE: But isn't that-- 626 00:31:35,460 --> 00:31:38,070 isn't that embedded in Cu and Cd, because you 627 00:31:38,070 --> 00:31:40,350 rely on a market price for Cu and Cd? 628 00:31:40,350 --> 00:31:44,640 PROFESSOR: No, there is no market price for Cu and Cd. 629 00:31:44,640 --> 00:31:48,307 Let's go back and take a look at what Cu and the Cd are. 630 00:31:51,290 --> 00:31:53,220 That's not a market price. 631 00:31:53,220 --> 00:31:55,310 This is not a market price. 632 00:31:55,310 --> 00:32:00,560 Cu is basically the outcome of u times S0 minus K, 633 00:32:00,560 --> 00:32:05,080 and Cd is the outcome of d S0 minus K, 0. 634 00:32:05,080 --> 00:32:07,650 That's not a market price. 635 00:32:07,650 --> 00:32:12,390 We have to agree in advance on what the possible outcomes are. 636 00:32:12,390 --> 00:32:14,640 But once we agree on those outcomes, 637 00:32:14,640 --> 00:32:17,280 everything follows from that. 638 00:32:17,280 --> 00:32:18,820 There's no market price here. 639 00:32:18,820 --> 00:32:20,880 The only market price is S0. 640 00:32:20,880 --> 00:32:22,890 That is the market price. 641 00:32:22,890 --> 00:32:24,000 That is determined today. 642 00:32:24,000 --> 00:32:28,290 But fortunately, that market price we observe. 643 00:32:28,290 --> 00:32:29,900 We can see it. 644 00:32:29,900 --> 00:32:37,150 Now, there is a link between the market price, u 645 00:32:37,150 --> 00:32:49,750 and d, p, because if the stock price today is worth $20, 646 00:32:49,750 --> 00:32:53,570 and tomorrow we say that there are two possibilities, 647 00:32:53,570 --> 00:33:00,530 either it's $30 or $10, then that tells you 648 00:33:00,530 --> 00:33:06,680 that p sort of has to be somewhere around 0.5. 649 00:33:06,680 --> 00:33:07,660 We may disagree. 650 00:33:07,660 --> 00:33:09,260 I may think it's 0.55. 651 00:33:09,260 --> 00:33:12,230 You may think it's 0.45, whatever. 652 00:33:12,230 --> 00:33:17,100 But when we aggregate all of our expectations, 653 00:33:17,100 --> 00:33:22,400 we come up with $20 for the stock today. 654 00:33:22,400 --> 00:33:24,770 So it's all related. 655 00:33:24,770 --> 00:33:26,670 It's in there. 656 00:33:26,670 --> 00:33:29,250 But we don't need to make an assumption explicitly 657 00:33:29,250 --> 00:33:30,360 for what p is. 658 00:33:30,360 --> 00:33:34,050 That is the power of this kind of option pricing approach. 659 00:33:34,050 --> 00:33:34,690 [INAUDIBLE] 660 00:33:34,690 --> 00:33:41,183 AUDIENCE: [INAUDIBLE] the reason you get the market here is we 661 00:33:41,183 --> 00:33:43,348 agree on everything except that I think that 662 00:33:43,348 --> 00:33:44,310 the higher [INAUDIBLE]. 663 00:33:44,310 --> 00:33:44,550 PROFESSOR: Yeah. 664 00:33:44,550 --> 00:33:46,032 AUDIENCE: --it will go up, and you think 665 00:33:46,032 --> 00:33:47,140 it will go down [INAUDIBLE]. 666 00:33:47,140 --> 00:33:47,540 PROFESSOR: Right. 667 00:33:47,540 --> 00:33:48,956 AUDIENCE: What if we fundamentally 668 00:33:48,956 --> 00:33:50,820 don't agree on u and d? 669 00:33:50,820 --> 00:33:52,530 PROFESSOR: Oh, then we have a problem. 670 00:33:52,530 --> 00:33:57,230 We need to assume a particular u and d that we can agree on. 671 00:33:57,230 --> 00:33:58,560 So let me turn to that now. 672 00:33:58,560 --> 00:34:00,570 Let me turn to the extension of this. 673 00:34:00,570 --> 00:34:04,320 So what I've derived is a one-period pricing model-- 674 00:34:04,320 --> 00:34:05,880 very, very simple. 675 00:34:05,880 --> 00:34:10,650 It turns out that you can do a multiperiod pricing model. 676 00:34:10,650 --> 00:34:13,409 And this multiperiod generalization 677 00:34:13,409 --> 00:34:15,480 is given by this. 678 00:34:15,480 --> 00:34:17,489 What is that multiperiod generalization? 679 00:34:17,489 --> 00:34:21,150 Basically you have-- let me see if I have the diagram here-- 680 00:34:24,040 --> 00:34:26,320 the multiperiod generalization is simply 681 00:34:26,320 --> 00:34:33,650 that you now have a bunch of possibilities, 682 00:34:33,650 --> 00:34:36,350 and you are figuring out what the price of the option 683 00:34:36,350 --> 00:34:42,159 is at date 0 when it pays off at date capital N, 684 00:34:42,159 --> 00:34:44,230 or lowercase n in this case-- 685 00:34:44,230 --> 00:34:46,310 n periods. 686 00:34:46,310 --> 00:34:50,000 And you can use exactly the same arbitrage argument that I just 687 00:34:50,000 --> 00:34:51,650 showed you, but it's a little bit more 688 00:34:51,650 --> 00:34:55,070 complicated now because you've got multiple branches. 689 00:34:55,070 --> 00:34:57,920 But it's still, at every step of the way, a binomial 690 00:34:57,920 --> 00:34:59,930 or a Bernoulli trial. 691 00:34:59,930 --> 00:35:04,950 And so in a multiperiod setting, you get a binomial tree. 692 00:35:04,950 --> 00:35:08,700 Now, the reason that this is such a powerful extension 693 00:35:08,700 --> 00:35:13,500 is that nowhere have I specified what a period is. 694 00:35:13,500 --> 00:35:15,990 I just said it's a period, today versus tomorrow. 695 00:35:15,990 --> 00:35:19,680 But it could be today versus three minutes from now, 696 00:35:19,680 --> 00:35:24,720 or three femtoseconds from now, or three years from now. 697 00:35:24,720 --> 00:35:26,630 I haven't specified. 698 00:35:26,630 --> 00:35:29,280 So if you say we can't agree on a u 699 00:35:29,280 --> 00:35:32,820 and a d, fine, let's not agree on a u and a d. 700 00:35:32,820 --> 00:35:37,410 Let's agree that between now and five minutes from now, 701 00:35:37,410 --> 00:35:41,320 there are 256 possible outcomes for the stock price. 702 00:35:41,320 --> 00:35:42,819 Do you agree on that? 703 00:35:42,819 --> 00:35:44,110 You think we can agree on that? 704 00:35:44,110 --> 00:35:45,860 Is that something that's easy to agree on? 705 00:35:45,860 --> 00:35:47,940 Well, if that's the case, then all 706 00:35:47,940 --> 00:35:51,450 I need to do is to have enough steps between now 707 00:35:51,450 --> 00:35:55,470 and five minutes from now to have 256 possibilities. 708 00:35:55,470 --> 00:35:57,810 And by the way, I chose that number specifically 709 00:35:57,810 --> 00:36:01,470 as a power of 2 because with these kinds of branches, 710 00:36:01,470 --> 00:36:04,320 it's actually very easy to be able to get 711 00:36:04,320 --> 00:36:07,500 that kind of a tree, with that many branches. 712 00:36:07,500 --> 00:36:10,860 So now you see that the u and the d, that's 713 00:36:10,860 --> 00:36:12,840 not relevant, because we can make 714 00:36:12,840 --> 00:36:14,340 it as small as you would like. 715 00:36:14,340 --> 00:36:16,860 If you would like to have it really, really fine, 716 00:36:16,860 --> 00:36:21,180 I can get it down to double precision, 32 decimal places, 717 00:36:21,180 --> 00:36:25,460 by basically taking one period to be a millisecond. 718 00:36:25,460 --> 00:36:28,610 And this binomial option pricing formula 719 00:36:28,610 --> 00:36:32,850 will apply exactly in the same way. 720 00:36:32,850 --> 00:36:35,780 It turns out that when you let the number of periods 721 00:36:35,780 --> 00:36:40,170 go to infinity, and at the same time, 722 00:36:40,170 --> 00:36:41,820 you control the u and the d and make 723 00:36:41,820 --> 00:36:44,140 them smaller and smaller and smaller and smaller 724 00:36:44,140 --> 00:36:49,350 so as to be able to get a tree that is reasonably realistic, 725 00:36:49,350 --> 00:36:51,452 you know what you get? 726 00:36:51,452 --> 00:36:54,370 You get the Black-Scholes formula. 727 00:36:54,370 --> 00:36:58,090 The pricing formula that you get is a solution 728 00:36:58,090 --> 00:36:59,980 to this parabolic partial differential 729 00:36:59,980 --> 00:37:03,010 equation with the following boundary conditions. 730 00:37:03,010 --> 00:37:09,260 And so using the simple binomial two-step kind of process, when 731 00:37:09,260 --> 00:37:12,020 you let it go to infinity and you 732 00:37:12,020 --> 00:37:13,970 shrink the probabilities and the u 733 00:37:13,970 --> 00:37:18,390 and the d to make it more and more refined, 734 00:37:18,390 --> 00:37:21,400 you get the Black-Scholes formula. 735 00:37:21,400 --> 00:37:23,440 This is something that Black and Scholes 736 00:37:23,440 --> 00:37:25,870 never, never contemplated. 737 00:37:25,870 --> 00:37:28,980 So this is a completely different approach 738 00:37:28,980 --> 00:37:33,600 that allows you to reach the exact same conclusion, which 739 00:37:33,600 --> 00:37:35,910 is a startling one. 740 00:37:35,910 --> 00:37:38,730 Now, as I told you at the beginning, 741 00:37:38,730 --> 00:37:42,960 when people apply option pricing formulas, most of the time 742 00:37:42,960 --> 00:37:44,400 they do not do this. 743 00:37:44,400 --> 00:37:47,520 They do not solve the heat equation. 744 00:37:47,520 --> 00:37:50,170 What they do is that. 745 00:37:50,170 --> 00:37:51,910 They do a binomial tree. 746 00:37:51,910 --> 00:37:55,030 The reason is because in order to solve these PDEs, 747 00:37:55,030 --> 00:37:59,650 except in a very, very small number of textbook example 748 00:37:59,650 --> 00:38:03,190 cases, you can't solve this analytically anyway. 749 00:38:03,190 --> 00:38:04,840 You can't get a formula. 750 00:38:04,840 --> 00:38:06,581 You have to solve it numerically. 751 00:38:06,581 --> 00:38:08,830 And so if you're going to go to the trouble of solving 752 00:38:08,830 --> 00:38:11,746 these differential equations numerically, 753 00:38:11,746 --> 00:38:13,870 you may as well just do the binomial option pricing 754 00:38:13,870 --> 00:38:17,390 formula, because that's numerical as well. 755 00:38:17,390 --> 00:38:20,800 And it's a lot simpler computationally to be 756 00:38:20,800 --> 00:38:22,420 able to do that binomial tree. 757 00:38:22,420 --> 00:38:25,030 By the way, for those of you computing 758 00:38:25,030 --> 00:38:29,680 fans who like to think about parallel processing, 759 00:38:29,680 --> 00:38:32,950 these kinds of binomials trees are extraordinarily 760 00:38:32,950 --> 00:38:34,810 easy to parallelize. 761 00:38:34,810 --> 00:38:36,700 So if you thought about the old days, where 762 00:38:36,700 --> 00:38:38,290 you had a connection machine that 763 00:38:38,290 --> 00:38:40,810 was developed by Danny Hillis, you 764 00:38:40,810 --> 00:38:44,020 had 64,000 processors in parallel. 765 00:38:44,020 --> 00:38:45,670 You can actually make use of that 766 00:38:45,670 --> 00:38:47,500 by implementing a binomial tree. 767 00:38:47,500 --> 00:38:49,060 Nowadays we've got grid computing. 768 00:38:49,060 --> 00:38:52,450 The most recent advance is to be able to use both hardware 769 00:38:52,450 --> 00:38:55,600 and software to do distributed computing. 770 00:38:55,600 --> 00:38:58,510 The binomial tree is ideally suited 771 00:38:58,510 --> 00:38:59,810 for being able to do that. 772 00:38:59,810 --> 00:39:02,230 So you can evaluate extraordinarily complex 773 00:39:02,230 --> 00:39:08,030 derivatives very, very quickly using this kind of a framework. 774 00:39:08,030 --> 00:39:11,165 So you're not giving up a lot by the u and the d, 775 00:39:11,165 --> 00:39:14,170 because we can make the u and the d as fine as possible 776 00:39:14,170 --> 00:39:16,420 so that ultimately we would all say, yeah, enough. 777 00:39:16,420 --> 00:39:18,070 I agree, all right, leave me alone. 778 00:39:18,070 --> 00:39:20,110 I don't want any more binomial trees. 779 00:39:20,110 --> 00:39:22,350 This is complicated enough. 780 00:39:22,350 --> 00:39:24,910 256 of them over a five-minute interval 781 00:39:24,910 --> 00:39:28,780 is enough for all practical purposes. 782 00:39:28,780 --> 00:39:30,740 Yeah. 783 00:39:30,740 --> 00:39:36,124 AUDIENCE: [INAUDIBLE] if that cannot be solved, 784 00:39:36,124 --> 00:39:38,245 why was it so important? 785 00:39:38,245 --> 00:39:40,310 PROFESSOR: Oh, no, this can be solved. 786 00:39:40,310 --> 00:39:43,834 The solution of this equation is the Black-Scholes formula. 787 00:39:43,834 --> 00:39:46,000 What I said cannot be solved is when you have a more 788 00:39:46,000 --> 00:39:47,530 complicated security. 789 00:39:47,530 --> 00:39:49,630 So for example, the option pricing 790 00:39:49,630 --> 00:39:53,050 formula that we looked at with the simple plain vanilla call 791 00:39:53,050 --> 00:39:56,420 and put option, that's relatively straightforward. 792 00:39:56,420 --> 00:39:59,920 But think about something like a mortgage-backed security 793 00:39:59,920 --> 00:40:04,000 that has all sorts of conversion features and knockout features, 794 00:40:04,000 --> 00:40:07,450 and other types of legal restrictions, 795 00:40:07,450 --> 00:40:10,600 as well as certain rights and requirements. 796 00:40:10,600 --> 00:40:13,120 Then it's not so easy. 797 00:40:13,120 --> 00:40:14,930 It looks much more complicated. 798 00:40:14,930 --> 00:40:17,980 For example, this particular coefficient 799 00:40:17,980 --> 00:40:20,020 that multiplies this second derivative 800 00:40:20,020 --> 00:40:22,510 ends up being a highly non-linear function, not just 801 00:40:22,510 --> 00:40:23,740 a quadratic. 802 00:40:23,740 --> 00:40:26,380 Or this piece here becomes a nonlinear function, 803 00:40:26,380 --> 00:40:29,890 or the boundary conditions are kind of weird. 804 00:40:29,890 --> 00:40:32,560 In that case, you can't solve it analytically. 805 00:40:32,560 --> 00:40:35,068 You have to use numerical methods to solve it. 806 00:40:35,068 --> 00:40:37,060 AUDIENCE: [INAUDIBLE] 807 00:40:37,060 --> 00:40:39,520 PROFESSOR: This is just the arbitrage condition 808 00:40:39,520 --> 00:40:44,750 that says that the solution C will give you a null arbitrage 809 00:40:44,750 --> 00:40:47,680 price for the call option. 810 00:40:47,680 --> 00:40:53,271 So the equivalent of this PDE, partial differential equation, 811 00:40:53,271 --> 00:40:53,770 is-- 812 00:40:56,280 --> 00:41:02,530 go back-- is this, the simultaneous equation 813 00:41:02,530 --> 00:41:07,520 up there and down here, and then this expression 814 00:41:07,520 --> 00:41:10,300 that says that the price of the option 815 00:41:10,300 --> 00:41:13,600 has to be given by this particular portfolio. 816 00:41:13,600 --> 00:41:18,550 That's what the PDE looks like in continuous time, 817 00:41:18,550 --> 00:41:20,635 or when you have an infinite number of time steps. 818 00:41:23,980 --> 00:41:25,970 So it is not-- 819 00:41:25,970 --> 00:41:27,790 that's absolutely a good question, 820 00:41:27,790 --> 00:41:30,400 because this is solvable. 821 00:41:30,400 --> 00:41:34,570 But very quickly, when you change the terms of a contract, 822 00:41:34,570 --> 00:41:36,920 it turns out that it's very hard to model. 823 00:41:36,920 --> 00:41:37,420 Yes. 824 00:41:39,780 --> 00:41:41,670 AUDIENCE: Question about the random walk. 825 00:41:41,670 --> 00:41:42,400 PROFESSOR: Yes. 826 00:41:42,400 --> 00:41:46,050 AUDIENCE: Can you just briefly mention how that feeds 827 00:41:46,050 --> 00:41:49,920 into the final answer, and how it will change things if it's-- 828 00:41:49,920 --> 00:41:53,200 PROFESSOR: Well, the random walk hypothesis is implicit in here, 829 00:41:53,200 --> 00:41:56,160 because I've got a coin toss. 830 00:41:56,160 --> 00:42:00,670 And the coin toss is independent period by period. 831 00:42:00,670 --> 00:42:03,620 If the coin toss is not independent, 832 00:42:03,620 --> 00:42:06,020 then that's the wrong formula. 833 00:42:06,020 --> 00:42:09,230 In other words, you don't have a simple binomial distribution 834 00:42:09,230 --> 00:42:11,540 if you don't have IID coin tosses. 835 00:42:11,540 --> 00:42:14,180 The random walk is basically the assumption 836 00:42:14,180 --> 00:42:17,510 of IID coin tosses-- independently and identically 837 00:42:17,510 --> 00:42:18,120 distributed. 838 00:42:18,120 --> 00:42:19,730 That's what IID stands for-- 839 00:42:19,730 --> 00:42:21,480 IID coin tosses. 840 00:42:21,480 --> 00:42:23,885 So that's where the Bachelier assumption came in. 841 00:42:23,885 --> 00:42:27,470 In order for Bachelier to derive the heat equation, 842 00:42:27,470 --> 00:42:30,410 or some variant of the heat equation, 843 00:42:30,410 --> 00:42:34,340 he was implicitly assuming that what happens in one period 844 00:42:34,340 --> 00:42:37,730 for the stock has no bearing on what happens in next period. 845 00:42:37,730 --> 00:42:42,080 If stock prices are correlated over time, 846 00:42:42,080 --> 00:42:44,240 then these formulas do not work. 847 00:42:44,240 --> 00:42:46,460 You need a different kind of formula. 848 00:42:46,460 --> 00:42:47,900 It's actually not that far off. 849 00:42:47,900 --> 00:42:52,190 You can derive an expression for an option pricing formula 850 00:42:52,190 --> 00:42:53,600 with correlated returns. 851 00:42:53,600 --> 00:42:56,720 In fact, professor Wang and I published a paper, 852 00:42:56,720 --> 00:42:58,820 I think it's maybe close to 10 years ago, 853 00:42:58,820 --> 00:43:01,160 where we worked out that case. 854 00:43:01,160 --> 00:43:03,220 But up until then, most people assumed 855 00:43:03,220 --> 00:43:05,760 that stock prices are not correlated, 856 00:43:05,760 --> 00:43:09,890 so the Brownian motion or random walk idea fit in very nicely 857 00:43:09,890 --> 00:43:12,080 with this binomial. 858 00:43:12,080 --> 00:43:15,410 If they're correlated, then you no longer have IID Bernoulli 859 00:43:15,410 --> 00:43:17,690 trials, you have a Markov chain, and you 860 00:43:17,690 --> 00:43:19,190 have to use Markov pricing in order 861 00:43:19,190 --> 00:43:22,820 to be able to get this formula. 862 00:43:22,820 --> 00:43:25,730 If you're interested in this, I urge you to take 15 437, 863 00:43:25,730 --> 00:43:28,000 because that's where we go into it in much more depth. 864 00:43:28,000 --> 00:43:28,872 Yeah. 865 00:43:28,872 --> 00:43:34,420 AUDIENCE: [INAUDIBLE] use the binomial coin to value options, 866 00:43:34,420 --> 00:43:37,540 and we see a range of prices. 867 00:43:39,050 --> 00:43:39,830 PROFESSOR: Yeah. 868 00:43:39,830 --> 00:43:41,881 AUDIENCE: So how do we approach that? 869 00:43:41,881 --> 00:43:44,470 Do we take some kind of average? 870 00:43:44,470 --> 00:43:47,832 Is this common, or do we receive a specific u and a d each time? 871 00:43:47,832 --> 00:43:49,956 I mean, I imagine it could be a range [INAUDIBLE].. 872 00:43:49,956 --> 00:43:51,060 PROFESSOR: No, no. 873 00:43:51,060 --> 00:43:52,760 So the way that you would apply this 874 00:43:52,760 --> 00:43:55,100 is that you would, first of all, pick 875 00:43:55,100 --> 00:43:58,460 the number of periods that are appropriate to the problem 876 00:43:58,460 --> 00:43:59,240 at hand. 877 00:43:59,240 --> 00:44:01,070 So if you have got an option that's 878 00:44:01,070 --> 00:44:03,920 expiring in three months, then typically, 879 00:44:03,920 --> 00:44:07,130 if you did it on a daily basis or an hourly basis, 880 00:44:07,130 --> 00:44:08,840 that would be more than enough. 881 00:44:08,840 --> 00:44:11,780 And then you would assume that there would be a u and a d 882 00:44:11,780 --> 00:44:16,040 in order to match the approximate outcomes that you 883 00:44:16,040 --> 00:44:17,060 would expect. 884 00:44:17,060 --> 00:44:20,010 And then out of that, you would actually get a number. 885 00:44:20,010 --> 00:44:24,750 So this, this C0, when you plug in all of these parameters, 886 00:44:24,750 --> 00:44:30,810 you actually get a number, like $30.25. 887 00:44:30,810 --> 00:44:33,160 That's the price of the option. 888 00:44:33,160 --> 00:44:35,050 And of course if you change the parameters, 889 00:44:35,050 --> 00:44:38,020 you change the strike price, the interest rate changes, the u 890 00:44:38,020 --> 00:44:40,540 and the d changes, that will change the value of the option 891 00:44:40,540 --> 00:44:41,622 price as well. 892 00:44:41,622 --> 00:44:44,520 AUDIENCE: [INAUDIBLE] every now and then, 893 00:44:44,520 --> 00:44:46,900 [INAUDIBLE] to receive a range from you, and a range-- 894 00:44:46,900 --> 00:44:48,280 PROFESSOR: No, no, no. 895 00:44:48,280 --> 00:44:50,440 What you do is you start off with an assumption 896 00:44:50,440 --> 00:44:53,200 for what u and d exactly are. 897 00:44:53,200 --> 00:44:57,190 Not a range, but actually if it goes up, it goes up by 1.05. 898 00:44:57,190 --> 00:45:01,120 If it goes down it goes down by 0.92. 899 00:45:01,120 --> 00:45:02,673 Yeah. 900 00:45:02,673 --> 00:45:04,374 AUDIENCE: [INAUDIBLE] 901 00:45:04,374 --> 00:45:06,040 PROFESSOR: Oh, well, it varies depending 902 00:45:06,040 --> 00:45:09,620 on the particular instrument that you're trying to price. 903 00:45:09,620 --> 00:45:13,030 So-- well, no, what I mean is options on what stock? 904 00:45:13,030 --> 00:45:16,250 So in other words, with any kind of option pricing formula, 905 00:45:16,250 --> 00:45:18,250 you actually have to calibrate these parameters. 906 00:45:18,250 --> 00:45:20,416 So you have to figure out what the interest rate is, 907 00:45:20,416 --> 00:45:22,540 and then typically what is done is you assume 908 00:45:22,540 --> 00:45:26,440 a particular grid, and then use a u and a d that will capture 909 00:45:26,440 --> 00:45:27,830 all the elements of that grid. 910 00:45:27,830 --> 00:45:30,010 So for example, let's assume that u 911 00:45:30,010 --> 00:45:37,130 is 25 basis points plus 1, and d is 1 minus 25 basis points. 912 00:45:37,130 --> 00:45:40,150 So that means you can capture stock price movements that 913 00:45:40,150 --> 00:45:42,760 go up by 25 basis points or down, 914 00:45:42,760 --> 00:45:46,240 and you assume a number of n in order 915 00:45:46,240 --> 00:45:48,700 to get that tree to be as fine as you 916 00:45:48,700 --> 00:45:53,110 would like for the particular time that you're pricing it at. 917 00:45:53,110 --> 00:45:58,000 So in other words, if I use 25 basis points and n equal to 1, 918 00:45:58,000 --> 00:46:01,990 that means that I can capture a situation where, 919 00:46:01,990 --> 00:46:06,370 at maturity, the stock price goes up or down by 25 basis 920 00:46:06,370 --> 00:46:08,230 points. 921 00:46:08,230 --> 00:46:10,690 If I now go four periods, then I can 922 00:46:10,690 --> 00:46:14,200 capture a situation where the stock price goes up by 1% 923 00:46:14,200 --> 00:46:18,100 or down by 1% in 25-basis-point increments. 924 00:46:18,100 --> 00:46:21,725 And if I want more refinements, then I keep going, 925 00:46:21,725 --> 00:46:24,160 let n get bigger and bigger and bigger. 926 00:46:24,160 --> 00:46:28,120 And then whatever that is, that final number of nodes 927 00:46:28,120 --> 00:46:33,646 will be the possible stock price values. 928 00:46:33,646 --> 00:46:38,012 AUDIENCE: [INAUDIBLE] historical data on the specific stock to-- 929 00:46:38,012 --> 00:46:39,720 PROFESSOR: You would use historical data. 930 00:46:39,720 --> 00:46:42,053 You would use historical-- because the way you calibrate 931 00:46:42,053 --> 00:46:45,460 this is you can show that the expected value-- 932 00:46:45,460 --> 00:46:49,260 so the expected value of S1 is just 933 00:46:49,260 --> 00:46:54,450 equal to the probability of u S0 plus 1 minus 934 00:46:54,450 --> 00:46:57,150 probability of d S0. 935 00:46:57,150 --> 00:46:58,740 So you've got the expected value. 936 00:46:58,740 --> 00:47:02,550 Calculate the variance of S1, and you'll 937 00:47:02,550 --> 00:47:05,610 get another expression with u and d and p, 938 00:47:05,610 --> 00:47:07,320 and then you simply use historical data 939 00:47:07,320 --> 00:47:10,200 to match the parameters and pick them 940 00:47:10,200 --> 00:47:13,260 so that they give you a reasonable approximation 941 00:47:13,260 --> 00:47:14,830 to reality. 942 00:47:14,830 --> 00:47:16,830 AUDIENCE: [INAUDIBLE] doesn't continue to behave 943 00:47:16,830 --> 00:47:18,830 as the history-- 944 00:47:18,830 --> 00:47:19,484 PROFESSOR: Yes. 945 00:47:19,484 --> 00:47:20,650 AUDIENCE: --so the options-- 946 00:47:20,650 --> 00:47:21,190 PROFESSOR: Yeah. 947 00:47:21,190 --> 00:47:21,890 AUDIENCE: --don't match. 948 00:47:21,890 --> 00:47:22,500 PROFESSOR: Absolutely. 949 00:47:22,500 --> 00:47:23,950 That's always the case, isn't it? 950 00:47:23,950 --> 00:47:27,270 In other words, if you don't have IID, 951 00:47:27,270 --> 00:47:28,860 you're going to get a problem. 952 00:47:28,860 --> 00:47:31,830 But remember, it doesn't depend upon the p. 953 00:47:31,830 --> 00:47:35,250 And so in that sense, if there's a change in p, 954 00:47:35,250 --> 00:47:38,670 as long as the u and the d are appropriate, 955 00:47:38,670 --> 00:47:41,970 you'll still be able to capture the value of the option. 956 00:47:41,970 --> 00:47:42,749 Question. 957 00:47:42,749 --> 00:47:44,186 AUDIENCE: I'm trying to figure out 958 00:47:44,186 --> 00:47:47,060 the analogy with the [INAUDIBLE].. 959 00:47:47,060 --> 00:47:47,802 PROFESSOR: Yeah. 960 00:47:47,802 --> 00:47:52,540 AUDIENCE: So I understand how it works in temperature. 961 00:47:52,540 --> 00:47:56,260 What would be here that [INAUDIBLE].. 962 00:47:56,260 --> 00:47:58,590 PROFESSOR: Let me-- let me not talk about that now, 963 00:47:58,590 --> 00:48:00,690 because I suspect that while you may be interested 964 00:48:00,690 --> 00:48:02,670 and a couple of other people, we probably 965 00:48:02,670 --> 00:48:04,870 don't have everybody being physicists here. 966 00:48:04,870 --> 00:48:05,900 So we'll talk about that afterwards. 967 00:48:05,900 --> 00:48:07,525 And also, that's something that, again, 968 00:48:07,525 --> 00:48:09,270 in 437, they may touch upon. 969 00:48:09,270 --> 00:48:12,870 But I want to keep moving along, because this is already 970 00:48:12,870 --> 00:48:16,170 more complicated than the nature of what 971 00:48:16,170 --> 00:48:17,994 I want to cover in this course. 972 00:48:17,994 --> 00:48:19,410 So let me get back to you on that, 973 00:48:19,410 --> 00:48:21,159 but we can talk about it afterwards. 974 00:48:21,159 --> 00:48:22,450 Any other questions about this? 975 00:48:22,450 --> 00:48:23,250 Yeah. 976 00:48:23,250 --> 00:48:25,250 AUDIENCE: I have a question about volatility, 977 00:48:25,250 --> 00:48:28,250 and how it is going to play in the equation. 978 00:48:28,250 --> 00:48:30,070 Like for example, I have two scenarios. 979 00:48:30,070 --> 00:48:35,170 They all, in three months, could go up or down by u or d. 980 00:48:35,170 --> 00:48:38,922 But the volatility of those to scenarios vary dramatically. 981 00:48:38,922 --> 00:48:39,630 PROFESSOR: Right. 982 00:48:39,630 --> 00:48:40,912 AUDIENCE: So how does-- 983 00:48:40,912 --> 00:48:42,870 PROFESSOR: How does volatility enter into this. 984 00:48:42,870 --> 00:48:43,828 That's a good question. 985 00:48:43,828 --> 00:48:46,320 Well, what do you think volatility 986 00:48:46,320 --> 00:48:48,740 is captured by in this simple Bernoulli trial? 987 00:48:49,332 --> 00:48:51,040 AUDIENCE: The difference between u and d. 988 00:48:51,040 --> 00:48:52,290 PROFESSOR: Exactly, exactly. 989 00:48:52,290 --> 00:48:58,660 Volatility is a measure of the spread between u and d. 990 00:48:58,660 --> 00:49:01,030 Holding other things equal-- 991 00:49:01,030 --> 00:49:03,730 by that, I mean holding the current stock price equal, 992 00:49:03,730 --> 00:49:06,580 holding the probability p equal-- 993 00:49:06,580 --> 00:49:12,700 so fixing that, as I increase the spread between u and d, 994 00:49:12,700 --> 00:49:14,270 I'm increasing the volatility. 995 00:49:16,890 --> 00:49:21,840 And if there's one thing that we see that matters 996 00:49:21,840 --> 00:49:24,750 is the spread between u and d. 997 00:49:30,730 --> 00:49:35,170 So if the spread between you and d increases, 998 00:49:35,170 --> 00:49:39,400 that actually will have an impact on this formula, 999 00:49:39,400 --> 00:49:42,190 and you have to work out the effects, which 1000 00:49:42,190 --> 00:49:43,790 is a very easy thing to do. 1001 00:49:43,790 --> 00:49:45,580 You can even do this in a spreadsheet. 1002 00:49:45,580 --> 00:49:49,330 But you can show that as the volatility increases, 1003 00:49:49,330 --> 00:49:54,070 the value of the call option is actually increasing. 1004 00:49:54,070 --> 00:49:56,770 So take a look at that, and you'll 1005 00:49:56,770 --> 00:50:01,610 see that it behaves the way that we think it should. 1006 00:50:01,610 --> 00:50:06,760 OK, other questions? 1007 00:50:06,760 --> 00:50:11,740 OK, well, so I'll leave I'll leave it 1008 00:50:11,740 --> 00:50:14,230 at this point, which is to say that the derivatives 1009 00:50:14,230 --> 00:50:17,000 literature is huge. 1010 00:50:17,000 --> 00:50:20,890 And it has really spawned a number of different not only 1011 00:50:20,890 --> 00:50:24,610 securities, but also different methods 1012 00:50:24,610 --> 00:50:28,780 for hedging and managing your portfolios, 1013 00:50:28,780 --> 00:50:33,250 to the point where really, derivatives are everywhere. 1014 00:50:33,250 --> 00:50:36,880 And there are some examples that I've given you here, 1015 00:50:36,880 --> 00:50:39,910 but this is an area which is considered 1016 00:50:39,910 --> 00:50:42,400 rocket science because of the analytics 1017 00:50:42,400 --> 00:50:43,810 that are so demanding. 1018 00:50:43,810 --> 00:50:46,720 So this is a natural area for students here at MIT 1019 00:50:46,720 --> 00:50:49,510 to be involved in, but it's certainly not the only area. 1020 00:50:49,510 --> 00:50:53,380 And ultimately, what's important about derivatives is not just 1021 00:50:53,380 --> 00:50:56,660 the pricing and the hedging, but rather the application. 1022 00:50:56,660 --> 00:50:59,440 So the fact that we spend a fair bit of time 1023 00:50:59,440 --> 00:51:01,750 at the beginning of this lecture talking 1024 00:51:01,750 --> 00:51:07,480 about payoff diagrams, that wasn't just for completeness. 1025 00:51:07,480 --> 00:51:09,730 That really is one of the most important aspects, 1026 00:51:09,730 --> 00:51:13,360 is how you use options in order to tailor 1027 00:51:13,360 --> 00:51:16,900 the kinds of risks and return profiles 1028 00:51:16,900 --> 00:51:18,190 that you'd like to have. 1029 00:51:18,190 --> 00:51:19,930 And now that you know how to price them, 1030 00:51:19,930 --> 00:51:22,000 you can have a very clear sense of 1031 00:51:22,000 --> 00:51:24,130 whether or not they are appropriate 1032 00:51:24,130 --> 00:51:25,732 from a risk-return tradeoff. 1033 00:51:25,732 --> 00:51:27,190 But they are very different, as you 1034 00:51:27,190 --> 00:51:30,070 can see, from the securities that we've done. 1035 00:51:30,070 --> 00:51:32,260 However, having done it, having now 1036 00:51:32,260 --> 00:51:35,750 priced options and other derivatives, 1037 00:51:35,750 --> 00:51:39,780 which are really relatively straightforward extensions, 1038 00:51:39,780 --> 00:51:42,360 we've now been able to price virtually 1039 00:51:42,360 --> 00:51:46,470 99% of all the securities that you would ever run into. 1040 00:51:46,470 --> 00:51:47,790 We've done stocks. 1041 00:51:47,790 --> 00:51:48,720 We've done bonds. 1042 00:51:48,720 --> 00:51:52,260 We've done futures, forwards, and now options, 1043 00:51:52,260 --> 00:51:54,810 so there really isn't any other kind of financial security 1044 00:51:54,810 --> 00:51:58,470 out there that you could possibly come across that you 1045 00:51:58,470 --> 00:51:59,820 don't know how to price. 1046 00:51:59,820 --> 00:52:03,120 You may not realize it yet, and the purpose 1047 00:52:03,120 --> 00:52:05,250 of the second half of the course is 1048 00:52:05,250 --> 00:52:08,730 to introduce risk and show you how to use all of these methods 1049 00:52:08,730 --> 00:52:10,890 to price all of the other securities 1050 00:52:10,890 --> 00:52:12,440 that you will come into contact with. 1051 00:52:12,440 --> 00:52:15,390 And then, of course, in 402 and other finance courses, 1052 00:52:15,390 --> 00:52:17,480 you'll see that much more closely. 1053 00:52:17,480 --> 00:52:21,530 So for example, a revolving credit agreement, 1054 00:52:21,530 --> 00:52:26,720 a sinking fund debt issue, a credit default swap, 1055 00:52:26,720 --> 00:52:29,070 an interest rate swap-- 1056 00:52:29,070 --> 00:52:33,840 all of these securities are mixtures of the securities 1057 00:52:33,840 --> 00:52:35,730 that we've seen till now. 1058 00:52:35,730 --> 00:52:38,490 And the pricing method in all of these cases 1059 00:52:38,490 --> 00:52:43,080 is exactly the same, which is identify the cash flows, 1060 00:52:43,080 --> 00:52:47,070 come up with another portfolio that has the same cash flows, 1061 00:52:47,070 --> 00:52:49,860 but where you know how to construct it, 1062 00:52:49,860 --> 00:52:51,780 therefore the price of that security 1063 00:52:51,780 --> 00:52:53,760 has to be equal to the price of the thing 1064 00:52:53,760 --> 00:52:56,160 that you're trying to value. 1065 00:52:56,160 --> 00:53:00,030 That's the basic principle in virtually all financial pricing 1066 00:53:00,030 --> 00:53:01,360 applications. 1067 00:53:01,360 --> 00:53:04,020 So once you understand these concepts, 1068 00:53:04,020 --> 00:53:06,810 you can literally price anything under the sun, 1069 00:53:06,810 --> 00:53:10,420 and all you need between now and then is practice, practice, 1070 00:53:10,420 --> 00:53:13,990 practice in doing that. 1071 00:53:13,990 --> 00:53:17,560 All right, so that wraps up the lecture on derivatives. 1072 00:53:17,560 --> 00:53:22,030 And now I want to turn to risk and reward, 1073 00:53:22,030 --> 00:53:25,330 because up until now, we've really 1074 00:53:25,330 --> 00:53:28,810 talked about risk in an indirect way, 1075 00:53:28,810 --> 00:53:33,040 and I want to talk about it in a much more direct fashion 1076 00:53:33,040 --> 00:53:36,640 by looking at measures of risk. 1077 00:53:36,640 --> 00:53:38,530 So what I want to do now is to turn 1078 00:53:38,530 --> 00:53:42,700 to a little bit of statistical background 1079 00:53:42,700 --> 00:53:44,350 to talk about risk and return. 1080 00:53:44,350 --> 00:53:46,270 I want to motivate it first, and then 1081 00:53:46,270 --> 00:53:47,853 give you the measures that we're going 1082 00:53:47,853 --> 00:53:49,990 to use for capturing risk and return, 1083 00:53:49,990 --> 00:53:53,290 and then apply it to stocks, and get a sense of what 1084 00:53:53,290 --> 00:53:56,730 kinds of anomalies are out there that we should be aware of. 1085 00:53:56,730 --> 00:53:58,480 And then I'm going to take these measures, 1086 00:53:58,480 --> 00:54:02,980 and then tell you how to come up with the one number 1087 00:54:02,980 --> 00:54:08,240 that I've had to put off for the first half of the semester, 1088 00:54:08,240 --> 00:54:10,900 which is the cost of capital-- 1089 00:54:10,900 --> 00:54:14,140 the required rate of return, the risk-adjusted rate of return. 1090 00:54:14,140 --> 00:54:16,120 We are now going to get to a point 1091 00:54:16,120 --> 00:54:19,030 where we can actually identify what that number is, 1092 00:54:19,030 --> 00:54:21,340 and how to make that risk adjustment. 1093 00:54:21,340 --> 00:54:23,870 So that's where we're going. 1094 00:54:23,870 --> 00:54:29,400 Now, to give you a quick summary of where we are, as I told you, 1095 00:54:29,400 --> 00:54:32,010 we've priced all of these different securities. 1096 00:54:32,010 --> 00:54:34,230 But underlying all of these prices 1097 00:54:34,230 --> 00:54:37,290 is a kind of a net present value calculation where we're 1098 00:54:37,290 --> 00:54:39,690 taking some kind of a payoff or expected payoff 1099 00:54:39,690 --> 00:54:41,997 and discounting it at a particular rate, 1100 00:54:41,997 --> 00:54:44,580 and we need to figure out what that appropriate rate of return 1101 00:54:44,580 --> 00:54:45,360 is. 1102 00:54:45,360 --> 00:54:47,220 I've said before that that rate of return 1103 00:54:47,220 --> 00:54:49,380 is determined by the marketplace. 1104 00:54:49,380 --> 00:54:52,900 But what we want to know is how. 1105 00:54:52,900 --> 00:54:55,290 How does the market do that? 1106 00:54:55,290 --> 00:54:58,440 Because unless we understand a little bit better what 1107 00:54:58,440 --> 00:55:01,080 that mechanism is, we won't be in a position 1108 00:55:01,080 --> 00:55:04,200 to be able to say that the particular market that we're 1109 00:55:04,200 --> 00:55:08,250 using is either working very well or completely out 1110 00:55:08,250 --> 00:55:11,130 to lunch and crazy. 1111 00:55:11,130 --> 00:55:14,490 So we need to deconstruct the process by which 1112 00:55:14,490 --> 00:55:17,280 the market gets to that. 1113 00:55:17,280 --> 00:55:20,280 In order to do that, we have to go back even farther 1114 00:55:20,280 --> 00:55:22,920 and peel back the onion and ask the question, 1115 00:55:22,920 --> 00:55:29,010 how do people measure risk, and how do they engage in risk 1116 00:55:29,010 --> 00:55:30,540 taking behavior? 1117 00:55:30,540 --> 00:55:34,164 So we have to do a little bit more work in figuring out 1118 00:55:34,164 --> 00:55:35,580 these different kinds of measures, 1119 00:55:35,580 --> 00:55:39,810 and then talking explicitly about how individuals actually 1120 00:55:39,810 --> 00:55:42,840 incorporate that into their world view. 1121 00:55:42,840 --> 00:55:45,480 Along the way, we're going to ask questions like, 1122 00:55:45,480 --> 00:55:47,850 is the market efficient, and how do 1123 00:55:47,850 --> 00:55:52,470 we measure the performance of portfolio managers? 1124 00:55:52,470 --> 00:55:56,790 This past year, the typical portfolio manager 1125 00:55:56,790 --> 00:56:01,530 has lost about 30% to 40%. 1126 00:56:01,530 --> 00:56:05,130 That's a pretty devastating kind of return. 1127 00:56:05,130 --> 00:56:09,960 And in that environment, if you found a portfolio manager that 1128 00:56:09,960 --> 00:56:14,160 ended up losing you 10%, you might think, gee, 1129 00:56:14,160 --> 00:56:16,770 that's pretty good. 1130 00:56:16,770 --> 00:56:18,030 Does that really make sense? 1131 00:56:18,030 --> 00:56:21,180 Is it ever the case that we want to congratulate a portfolio 1132 00:56:21,180 --> 00:56:23,490 manager for losing money for us? 1133 00:56:23,490 --> 00:56:26,010 We have to answer that question in the context of how 1134 00:56:26,010 --> 00:56:29,220 you figure out what an appropriate or fair rate 1135 00:56:29,220 --> 00:56:30,150 of return is. 1136 00:56:30,150 --> 00:56:32,660 So that's what we're going to be doing. 1137 00:56:32,660 --> 00:56:35,390 Now, to do that, I need to develop 1138 00:56:35,390 --> 00:56:36,680 a little bit of new notation. 1139 00:56:36,680 --> 00:56:38,690 And so the notation that I'm going to develop 1140 00:56:38,690 --> 00:56:42,200 is to talk about returns that are 1141 00:56:42,200 --> 00:56:45,440 inclusive of any kind of distributions, like dividends. 1142 00:56:45,440 --> 00:56:48,740 So when I talk about the returns of equities, 1143 00:56:48,740 --> 00:56:53,060 I'm going to be talking explicitly about a return that 1144 00:56:53,060 --> 00:56:55,940 includes the dividend. 1145 00:56:55,940 --> 00:56:57,500 And so the concept that we're going 1146 00:56:57,500 --> 00:56:59,690 to be working on, for the most part, 1147 00:56:59,690 --> 00:57:05,130 for the next half of this course is the expected rate of return. 1148 00:57:05,130 --> 00:57:08,420 We obviously will be talking about realized returns, 1149 00:57:08,420 --> 00:57:10,700 but from a portfolio management perspective, 1150 00:57:10,700 --> 00:57:13,660 we're going to be focusing not just on what happened this year 1151 00:57:13,660 --> 00:57:15,530 or what happened last year, but we're 1152 00:57:15,530 --> 00:57:18,710 going to be focusing on the average rate of return 1153 00:57:18,710 --> 00:57:22,950 that we would expect over the course of the next five years. 1154 00:57:22,950 --> 00:57:25,100 We're going to be looking at excess returns, which 1155 00:57:25,100 --> 00:57:31,100 is in excess of the net risk-free rate, little rf. 1156 00:57:31,100 --> 00:57:35,390 And what we refer to as a risk premium is simply 1157 00:57:35,390 --> 00:57:38,810 the average rate of return of a risky security 1158 00:57:38,810 --> 00:57:40,290 minus the risk-free rate. 1159 00:57:40,290 --> 00:57:44,920 So the excess return you can think of as a realization 1160 00:57:44,920 --> 00:57:46,480 of that risk premium. 1161 00:57:46,480 --> 00:57:49,402 But on average over a long period of time, 1162 00:57:49,402 --> 00:57:51,610 the number that we're going to be concerned with most 1163 00:57:51,610 --> 00:57:55,240 is this risk premium number, the average rate of return 1164 00:57:55,240 --> 00:57:57,100 minus the risk-free rate. 1165 00:57:57,100 --> 00:57:59,860 Over the course of the last 100 years or so, 1166 00:57:59,860 --> 00:58:04,000 US equity markets have provided an average rate 1167 00:58:04,000 --> 00:58:10,120 of return minus the risk-free rate on the order of 7%. 1168 00:58:10,120 --> 00:58:13,630 That's pretty good, but that's a long-run average. 1169 00:58:13,630 --> 00:58:17,740 The realized excess rate of return this year 1170 00:58:17,740 --> 00:58:19,890 is horrible, so I'm not even going 1171 00:58:19,890 --> 00:58:23,050 to talk about what that number is, but it's bad. 1172 00:58:23,050 --> 00:58:25,390 But do you see the difference between this year's rate 1173 00:58:25,390 --> 00:58:28,420 of return versus the long-run average? 1174 00:58:28,420 --> 00:58:30,910 And we can talk about both of them, 1175 00:58:30,910 --> 00:58:34,240 but we're going to use different techniques for each. 1176 00:58:34,240 --> 00:58:38,320 So the technique for talking about the statistical aspects 1177 00:58:38,320 --> 00:58:41,560 of returns will be from the language of statistics. 1178 00:58:41,560 --> 00:58:44,290 We're going to talk about the expected rate of return. 1179 00:58:44,290 --> 00:58:47,890 I'm going to use the Greek letter mu to denote that. 1180 00:58:47,890 --> 00:58:51,310 We're gonna also talk about the riskiness of returns, which 1181 00:58:51,310 --> 00:58:56,080 I'm going to use the variance and the standard deviation 1182 00:58:56,080 --> 00:58:57,880 to proxy for. 1183 00:58:57,880 --> 00:59:01,870 So the variance is simply the expected value 1184 00:59:01,870 --> 00:59:05,830 of the squared excess return. 1185 00:59:05,830 --> 00:59:09,910 That gives you a sense of the fluctuations around the mean. 1186 00:59:09,910 --> 00:59:12,220 And the standard deviation is the square root 1187 00:59:12,220 --> 00:59:13,030 of the variance. 1188 00:59:13,030 --> 00:59:14,980 And we use the standard deviation simply 1189 00:59:14,980 --> 00:59:16,960 because that's in the same units. 1190 00:59:16,960 --> 00:59:20,380 It's in units of percent per year, 1191 00:59:20,380 --> 00:59:24,496 whereas the variance is in units of percentage points squared 1192 00:59:24,496 --> 00:59:25,870 per year, so it's a little easier 1193 00:59:25,870 --> 00:59:29,380 to deal with the standard deviation. 1194 00:59:29,380 --> 00:59:35,890 And those concepts are the theoretical or population 1195 00:59:35,890 --> 00:59:39,310 values of the underlying securities 1196 00:59:39,310 --> 00:59:40,670 that we're going to look at. 1197 00:59:40,670 --> 00:59:43,510 We also want to look at the historical estimates, 1198 00:59:43,510 --> 00:59:45,550 and the historical estimates are given 1199 00:59:45,550 --> 00:59:47,900 by the sample counterparts. 1200 00:59:47,900 --> 00:59:52,319 So this is the sample mean, the sample variance, and the sample 1201 00:59:52,319 --> 00:59:53,110 standard deviation. 1202 00:59:53,110 --> 00:59:56,740 You should all remember this from your DMD class. 1203 00:59:56,740 --> 01:00:00,290 But if not, we'll have the TAs go over it during recitation. 1204 01:00:00,290 --> 01:00:02,680 You can also look in the appendix of Brealey, Myers, 1205 01:00:02,680 --> 01:00:07,330 and Allen, and they'll provide a little review about this. 1206 01:00:07,330 --> 01:00:09,250 Now, there are lots of other statistics, 1207 01:00:09,250 --> 01:00:12,280 and the only one that I'm gonna spend time on 1208 01:00:12,280 --> 01:00:13,870 is the correlation. 1209 01:00:13,870 --> 01:00:15,550 There's the median instead of the mean. 1210 01:00:15,550 --> 01:00:18,790 You can look at skewness, which way the distribution leans. 1211 01:00:18,790 --> 01:00:22,060 But what we're going to look at in just a little 1212 01:00:22,060 --> 01:00:26,950 while is correlation, which is how closely do 1213 01:00:26,950 --> 01:00:31,600 the returns of two investments move together. 1214 01:00:31,600 --> 01:00:33,760 If they move together a lot, then we 1215 01:00:33,760 --> 01:00:37,750 say that they're highly correlated, or co-related. 1216 01:00:37,750 --> 01:00:40,540 And if they don't move together a lot, 1217 01:00:40,540 --> 01:00:42,300 they're not very highly correlated. 1218 01:00:42,300 --> 01:00:45,680 And in some cases, if they move in opposite directions, 1219 01:00:45,680 --> 01:00:48,120 we say that they're negatively correlated. 1220 01:00:48,120 --> 01:00:50,830 So correlation, as most of you already know, 1221 01:00:50,830 --> 01:00:53,950 is a statistic that's a number between minus 1 and 1, 1222 01:00:53,950 --> 01:00:57,040 or minus 100% and 100%, that measures 1223 01:00:57,040 --> 01:01:02,860 the degree of association between these two securities. 1224 01:01:02,860 --> 01:01:05,680 We're going to be making use of correlations a lot 1225 01:01:05,680 --> 01:01:08,050 in the coming couple of lectures to try 1226 01:01:08,050 --> 01:01:11,590 to get a sense of whether or not an investment is going to help 1227 01:01:11,590 --> 01:01:14,920 you diversify your overall portfolio, 1228 01:01:14,920 --> 01:01:17,140 or if an investment is only going to add 1229 01:01:17,140 --> 01:01:19,660 to the risks of your portfolio. 1230 01:01:19,660 --> 01:01:22,370 And you can guess as to how we're going to measure that. 1231 01:01:22,370 --> 01:01:26,950 If the new investment is either zero correlated 1232 01:01:26,950 --> 01:01:29,740 or negatively correlated with your current portfolio, 1233 01:01:29,740 --> 01:01:33,460 that's going to help in terms of dampening your fluctuations. 1234 01:01:33,460 --> 01:01:37,030 But if the two investments move at the same time, that's 1235 01:01:37,030 --> 01:01:38,650 not only going to not help, that's 1236 01:01:38,650 --> 01:01:41,080 going to actually add to your risks. 1237 01:01:41,080 --> 01:01:42,520 And you don't want that, at least 1238 01:01:42,520 --> 01:01:46,157 not without the proper reward. 1239 01:01:46,157 --> 01:01:47,740 So that's a brief preview of how we're 1240 01:01:47,740 --> 01:01:50,020 going to use these statistics. 1241 01:01:50,020 --> 01:01:52,930 And you get some examples here about what 1242 01:01:52,930 --> 01:01:54,570 correlation looks like. 1243 01:01:54,570 --> 01:01:58,300 Here I've plotted four different scatter graphs 1244 01:01:58,300 --> 01:02:01,960 of the return of one asset on the x-axis 1245 01:02:01,960 --> 01:02:05,020 and the return of another asset on the y-axis. 1246 01:02:05,020 --> 01:02:07,540 And the dots represent those pairs 1247 01:02:07,540 --> 01:02:11,780 of returns for different assumptions about correlation. 1248 01:02:11,780 --> 01:02:14,710 So the upper left-hand scatter graph 1249 01:02:14,710 --> 01:02:17,770 is a graph where there's no correlation. 1250 01:02:17,770 --> 01:02:19,540 The correlation is zero. 1251 01:02:19,540 --> 01:02:21,880 The scatter graph on the lower left 1252 01:02:21,880 --> 01:02:24,210 is where there's very high positive correlation-- 1253 01:02:24,210 --> 01:02:27,490 80% correlation between the two. 1254 01:02:27,490 --> 01:02:30,820 And the scatter graph on the lower right 1255 01:02:30,820 --> 01:02:35,770 is where there's a negative 50% correlation. 1256 01:02:35,770 --> 01:02:38,530 So we're going to use correlation, 1257 01:02:38,530 --> 01:02:43,030 along with mean and variance, to try to put together 1258 01:02:43,030 --> 01:02:47,460 good collections of securities-- in other words, good portfolios 1259 01:02:47,460 --> 01:02:48,690 of securities. 1260 01:02:48,690 --> 01:02:52,710 And by doing that, we're going to show that we can actually 1261 01:02:52,710 --> 01:02:58,050 construct some very attractive kinds of investments using 1262 01:02:58,050 --> 01:02:59,976 relatively simple information. 1263 01:02:59,976 --> 01:03:01,350 But at the same time, we're going 1264 01:03:01,350 --> 01:03:04,980 to use that insight to then deconstruct 1265 01:03:04,980 --> 01:03:06,960 how to come up with the appropriate risk 1266 01:03:06,960 --> 01:03:10,750 adjustment for cost of capital calculations. 1267 01:03:10,750 --> 01:03:13,230 Now, there's a review here about normal distributions 1268 01:03:13,230 --> 01:03:15,000 and confidence intervals, and I'd 1269 01:03:15,000 --> 01:03:17,790 like you to go over that, either on your own or with the TAs 1270 01:03:17,790 --> 01:03:19,390 during recitations. 1271 01:03:19,390 --> 01:03:21,840 We're going to be using these kinds of concepts 1272 01:03:21,840 --> 01:03:23,700 to try to measure the risk and return 1273 01:03:23,700 --> 01:03:25,690 of various different investments. 1274 01:03:25,690 --> 01:03:28,950 Here's an example of General Motors' monthly returns. 1275 01:03:28,950 --> 01:03:34,170 That's a histogram in blue, and the line, the dark line, 1276 01:03:34,170 --> 01:03:38,160 is the assumed normal distribution that 1277 01:03:38,160 --> 01:03:40,830 has the same mean and variance. 1278 01:03:40,830 --> 01:03:43,380 And you could see that it looks like it's 1279 01:03:43,380 --> 01:03:47,580 sort of a good approximation, but there are actually 1280 01:03:47,580 --> 01:03:52,650 little bits of extra probability stuck out here and stuck out 1281 01:03:52,650 --> 01:03:56,140 here that don't exactly correspond to normal. 1282 01:03:56,140 --> 01:03:58,740 In other words, the assumption of normality 1283 01:03:58,740 --> 01:04:00,900 would say that the probability of getting 1284 01:04:00,900 --> 01:04:05,280 a return of minus 15% is relatively low, 1285 01:04:05,280 --> 01:04:10,230 then getting a return less than minus 20% is exceedingly low. 1286 01:04:10,230 --> 01:04:12,630 But the reality is different. 1287 01:04:12,630 --> 01:04:18,870 There are risks of having much lower returns in the data. 1288 01:04:18,870 --> 01:04:20,855 And after this year, I can tell you 1289 01:04:20,855 --> 01:04:23,760 that these tails are going to be fatter. 1290 01:04:23,760 --> 01:04:28,939 So this is meant to be an approximation, not reality. 1291 01:04:28,939 --> 01:04:30,480 The approximation is what we're going 1292 01:04:30,480 --> 01:04:34,590 to go over in this course, and in the very last lecture, 1293 01:04:34,590 --> 01:04:37,687 I want to tell you how good that approximation is. 1294 01:04:37,687 --> 01:04:40,020 And then I'm going to tell you about a number of courses 1295 01:04:40,020 --> 01:04:42,660 you might want to take that focus on getting 1296 01:04:42,660 --> 01:04:45,240 that last 5% right. 1297 01:04:45,240 --> 01:04:49,740 So 95% of the distribution is captured by what I'm going 1298 01:04:49,740 --> 01:04:52,717 to teach you in this course, but if you want to get the other 5% 1299 01:04:52,717 --> 01:04:55,050 right-- and by the way, if you're going into investments 1300 01:04:55,050 --> 01:04:58,330 as a profession, it's all about that 5%-- 1301 01:04:58,330 --> 01:05:02,850 then you'll want to take 15 433, investments. 1302 01:05:02,850 --> 01:05:06,029 So with that as the basic preamble, 1303 01:05:06,029 --> 01:05:08,320 let me tell you what I'm going to talk about next time, 1304 01:05:08,320 --> 01:05:09,870 since we're almost out of time. 1305 01:05:09,870 --> 01:05:11,820 What we're gonna do next time is I'm 1306 01:05:11,820 --> 01:05:13,960 going to talk about the US stock market. 1307 01:05:13,960 --> 01:05:17,160 I'm gonna talk about volatility, about predictability, 1308 01:05:17,160 --> 01:05:19,290 and then I'm going to talk a bit about the notion 1309 01:05:19,290 --> 01:05:21,360 of efficient markets, and try to describe 1310 01:05:21,360 --> 01:05:24,720 to you what kinds of properties we expect 1311 01:05:24,720 --> 01:05:26,334 from typical investments. 1312 01:05:26,334 --> 01:05:28,500 And we're actually going to go through some numbers. 1313 01:05:28,500 --> 01:05:32,670 I'm gonna show you some examples of basic statistics 1314 01:05:32,670 --> 01:05:34,650 for the stock market that will give you 1315 01:05:34,650 --> 01:05:38,880 a sense of how things have behaved over the last 50 years. 1316 01:05:38,880 --> 01:05:41,400 And what you'll get a sense of is that in some cases, 1317 01:05:41,400 --> 01:05:43,050 there is a lot of predictability. 1318 01:05:43,050 --> 01:05:45,000 There are certain things that we can count on. 1319 01:05:45,000 --> 01:05:48,360 For example, these are stock market returns 1320 01:05:48,360 --> 01:05:50,640 from 1946 to 2001. 1321 01:05:50,640 --> 01:05:54,180 This is monthly data, monthly returns 1322 01:05:54,180 --> 01:05:58,990 of the S&P 500 over a fairly long period of time. 1323 01:05:58,990 --> 01:06:02,760 And this might sort of look like a typical person's EKG 1324 01:06:02,760 --> 01:06:04,260 over the last few weeks. 1325 01:06:04,260 --> 01:06:07,020 Not surprisingly, there were periods where 1326 01:06:07,020 --> 01:06:09,240 we had some pretty bad returns. 1327 01:06:09,240 --> 01:06:11,340 We're going to see another one of these things 1328 01:06:11,340 --> 01:06:14,610 as well over the more recent period. 1329 01:06:14,610 --> 01:06:17,160 But when you look at this thing, you then 1330 01:06:17,160 --> 01:06:19,800 begin to appreciate that what we're living through now, 1331 01:06:19,800 --> 01:06:22,680 while it's bad and it's scary, it's 1332 01:06:22,680 --> 01:06:26,640 not at all unusual or completely unheard of. 1333 01:06:26,640 --> 01:06:28,620 There are periods in the stock market 1334 01:06:28,620 --> 01:06:30,720 where we've seen really big swings. 1335 01:06:30,720 --> 01:06:32,550 And by the way, this is just the US. 1336 01:06:32,550 --> 01:06:36,270 If I had shown you some emerging market returns, 1337 01:06:36,270 --> 01:06:38,020 it would go off the screen. 1338 01:06:38,020 --> 01:06:40,120 So we're going to talk about that next time. 1339 01:06:40,120 --> 01:06:42,270 And out of all of this chaos, we're 1340 01:06:42,270 --> 01:06:45,396 going to distill a very important relationship. 1341 01:06:45,396 --> 01:06:46,770 We're going to ultimately come up 1342 01:06:46,770 --> 01:06:50,460 with a simple linear equation that shows you 1343 01:06:50,460 --> 01:06:52,890 how to make that risk adjustment between the expected 1344 01:06:52,890 --> 01:06:56,140 return and the underlying risk of a portfolio. 1345 01:06:56,140 --> 01:06:58,890 So that's coming up, and we'll do that on Wednesday. 1346 01:06:58,890 --> 01:07:01,180 All right, see you then.