1 00:00:00,060 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:17,835 at ocw.mit.edu. 8 00:00:25,950 --> 00:00:28,010 PETER CARR: So, I welcome comments or questions 9 00:00:28,010 --> 00:00:29,480 at any point during this talk. 10 00:00:29,480 --> 00:00:31,730 We have an hour and a half, and I have only 50 slides. 11 00:00:31,730 --> 00:00:33,630 So we should be OK. 12 00:00:33,630 --> 00:00:38,000 So, this is joint work with Jiming Yu, 13 00:00:38,000 --> 00:00:41,360 who's a colleague of mine in my group at Morgan Stanley. 14 00:00:41,360 --> 00:00:42,900 I head up the global market modeling 15 00:00:42,900 --> 00:00:43,930 team at Morgan Stanley. 16 00:00:43,930 --> 00:00:47,510 It's a group of about 70 PhDs, mostly, spread around 17 00:00:47,510 --> 00:00:49,140 the world. 18 00:00:49,140 --> 00:00:52,690 There's about 30 of us in New York, 19 00:00:52,690 --> 00:00:57,540 some in London, quite a few Budapest, and a few in Beijing. 20 00:00:57,540 --> 00:01:00,370 The title of this talk is Can We Recover? 21 00:01:00,370 --> 00:01:04,970 And it's meant as a triple entendre. 22 00:01:04,970 --> 00:01:09,020 So, it could refer to either the systemic risk arising 23 00:01:09,020 --> 00:01:14,740 from the credit crisis, or the main result in a recent paper 24 00:01:14,740 --> 00:01:18,880 by a professor here at MIT named Steve Ross in the Sloan School, 25 00:01:18,880 --> 00:01:23,210 or it could actually be the academic and practitioner 26 00:01:23,210 --> 00:01:27,910 reaction to this result. So, it's really 27 00:01:27,910 --> 00:01:30,520 about two and three. 28 00:01:30,520 --> 00:01:33,765 So, it's not about can we recover from the crisis. 29 00:01:38,750 --> 00:01:41,160 There's a professor at Sloan School named Stephen Ross. 30 00:01:41,160 --> 00:01:46,870 And he's very well-known in academic finance. 31 00:01:46,870 --> 00:01:49,060 Your professor was kind enough to mention that I won 32 00:01:49,060 --> 00:01:50,380 Financial Engineer of the Year. 33 00:01:50,380 --> 00:01:53,510 And that was two years ago. 34 00:01:53,510 --> 00:01:55,140 I was like the 20th winner. 35 00:01:55,140 --> 00:01:56,740 He was the second winner. 36 00:02:01,420 --> 00:02:05,550 The first winner was another MIT professor, Bob Merton. 37 00:02:05,550 --> 00:02:12,030 So anyway, he wrote a paper a couple years ago, 38 00:02:12,030 --> 00:02:15,300 and it's only now about to be published. 39 00:02:15,300 --> 00:02:17,360 So this is like typical in academic circles. 40 00:02:17,360 --> 00:02:22,290 It takes a long time for a paper to come out. 41 00:02:22,290 --> 00:02:25,440 And this paper is coming out in Journal of Finance. 42 00:02:25,440 --> 00:02:27,040 That's what JF stands for. 43 00:02:27,040 --> 00:02:30,530 And Journal of Finance is the main journal 44 00:02:30,530 --> 00:02:34,060 for the academic finance community. 45 00:02:34,060 --> 00:02:38,120 And the title of the paper is The Recovery Theorem. 46 00:02:38,120 --> 00:02:42,480 And that's also the title of the theorem one in his paper. 47 00:02:42,480 --> 00:02:44,810 And that theorem one we'll go over. 48 00:02:44,810 --> 00:02:49,360 And it gives a sufficient set of conditions under which, 49 00:02:49,360 --> 00:02:53,930 what Professor Ross calls "natural probabilities," 50 00:02:53,930 --> 00:02:56,780 at a point in time can be determined 51 00:02:56,780 --> 00:03:00,905 from-- OK mathematically, from exact knowledge 52 00:03:00,905 --> 00:03:03,030 of Arrow-Debreu security prices, which you probably 53 00:03:03,030 --> 00:03:04,071 don't know what they are. 54 00:03:04,071 --> 00:03:06,990 But less mathematically, we'll just 55 00:03:06,990 --> 00:03:10,580 say from market prices of derivatives. 56 00:03:10,580 --> 00:03:12,190 OK, so derivatives you've heard of, 57 00:03:12,190 --> 00:03:16,770 I'm sure-- things like options, for example, on stocks or stock 58 00:03:16,770 --> 00:03:20,340 indices, could be on currencies. 59 00:03:20,340 --> 00:03:22,760 So, imagine that you look at Bloomberg. 60 00:03:22,760 --> 00:03:25,600 Bloomberg publishes a whole bunch of prices. 61 00:03:25,600 --> 00:03:31,480 And the idea is that you take this information, 62 00:03:31,480 --> 00:03:35,290 and from it you're learning what the market believes 63 00:03:35,290 --> 00:03:38,610 are the probabilities concerning the future. 64 00:03:38,610 --> 00:03:43,960 And so, if the option is on S&P 500 stock index, 65 00:03:43,960 --> 00:03:47,350 then you're learning from options prices what 66 00:03:47,350 --> 00:03:49,950 the market believes are the likelihoods 67 00:03:49,950 --> 00:04:03,810 of various possible levels for the S&P 500. 68 00:04:03,810 --> 00:04:09,460 So, we take this information on Bloomberg and, truth be told, 69 00:04:09,460 --> 00:04:11,710 we use it along with some assumptions 70 00:04:11,710 --> 00:04:17,720 to extract these implied market probabilities. 71 00:04:17,720 --> 00:04:20,510 So, I want to tell you what those assumptions are. 72 00:04:20,510 --> 00:04:26,160 And so, the actual output of this analysis 73 00:04:26,160 --> 00:04:30,390 is a probability transition matrix. 74 00:04:30,390 --> 00:04:32,500 Or, if you do it in continuous time, 75 00:04:32,500 --> 00:04:35,550 you'd call it a-- in continuous state space, 76 00:04:35,550 --> 00:04:40,700 you'd call it a transition probability density function. 77 00:04:40,700 --> 00:04:45,370 So the key word there is "transition." 78 00:04:45,370 --> 00:04:46,940 And what transition means is you're 79 00:04:46,940 --> 00:04:49,840 getting not only the probabilities going 80 00:04:49,840 --> 00:04:55,830 from, say, the current S&P level to any one of several levels, 81 00:04:55,830 --> 00:04:58,330 but even the probabilities of going 82 00:04:58,330 --> 00:05:02,720 from some other level than we're presently at today 83 00:05:02,720 --> 00:05:05,287 to that range of levels. 84 00:05:05,287 --> 00:05:06,870 You could say, for example, the market 85 00:05:06,870 --> 00:05:09,320 believes that given that we're here now 86 00:05:09,320 --> 00:05:15,700 with S&P at, say, 1,500, that the probability of more 87 00:05:15,700 --> 00:05:22,165 than doubling is one half, for example-- 88 00:05:22,165 --> 00:05:23,290 which would be really high. 89 00:05:23,290 --> 00:05:28,070 But you know, I'm just picking numbers randomly here. 90 00:05:28,070 --> 00:05:33,530 And you can even say that if S&P were to drop instantaneously 91 00:05:33,530 --> 00:05:36,240 to half its level, that the probability of more 92 00:05:36,240 --> 00:05:40,930 than doubling from there is, say, one-third. 93 00:05:40,930 --> 00:05:44,125 So, you can answer questions like that. 94 00:05:44,125 --> 00:05:47,140 That's The output of this type of thinking. 95 00:05:51,160 --> 00:05:56,320 So there'll be three probability measures 96 00:05:56,320 --> 00:05:58,700 that we can be thinking about. 97 00:05:58,700 --> 00:06:01,160 And we'll call them P, Q, and R. And I'd 98 00:06:01,160 --> 00:06:03,510 like to tell you what each of them means. 99 00:06:03,510 --> 00:06:06,600 So P stands for physical probability measures. 100 00:06:06,600 --> 00:06:08,600 So the P is for physical. 101 00:06:08,600 --> 00:06:13,490 And think of that as the actual objective 102 00:06:13,490 --> 00:06:21,230 reality of future states for, say, S&P 500. 103 00:06:21,230 --> 00:06:28,160 So let's say God knows that, for example, the probability 104 00:06:28,160 --> 00:06:32,690 that S&P is up by the end of the year is one half. 105 00:06:32,690 --> 00:06:37,050 And we, unfortunately, not being God, don't know that. 106 00:06:37,050 --> 00:06:39,690 But let's say the philosophy is that. 107 00:06:39,690 --> 00:06:42,790 There is some sort of true probability of S&P 108 00:06:42,790 --> 00:06:45,350 being up at the end of the year. 109 00:06:45,350 --> 00:06:49,840 And let's say I used a half. 110 00:06:49,840 --> 00:06:51,960 Maybe it's 60%. 111 00:06:51,960 --> 00:06:53,690 If it is 60%, then the probability 112 00:06:53,690 --> 00:06:57,190 of S&P being down at the end of the year is 40%. 113 00:06:57,190 --> 00:07:07,430 And the point is P is meant to indicate 114 00:07:07,430 --> 00:07:11,200 the frequencies with which S&P 500 in my example 115 00:07:11,200 --> 00:07:14,230 takes on various values. 116 00:07:14,230 --> 00:07:16,730 Now, there's another probability measure 117 00:07:16,730 --> 00:07:19,770 that people in derivatives spend a lot of time working with. 118 00:07:19,770 --> 00:07:22,490 And that's called risk-neutral probability measure, 119 00:07:22,490 --> 00:07:25,020 and it's often denoted by a letter Q. 120 00:07:25,020 --> 00:07:29,149 So we'll denote it by Q. And the concept 121 00:07:29,149 --> 00:07:30,690 of a risk-neutral probability measure 122 00:07:30,690 --> 00:07:35,670 was also actually proposed by Steve Ross many years ago. 123 00:07:35,670 --> 00:07:47,540 And it's called risk neutral because when you're 124 00:07:47,540 --> 00:07:53,720 working with it, if you think about how fast prices 125 00:07:53,720 --> 00:07:58,480 appreciate over time, then they grow randomly. 126 00:07:58,480 --> 00:08:02,330 But on average, under this risk-neutral measure Q, 127 00:08:02,330 --> 00:08:05,250 the grow at the same rate as your bank balance would grow. 128 00:08:08,130 --> 00:08:11,150 So your bank balance, let's say, nowadays 129 00:08:11,150 --> 00:08:14,100 is growing at best at the rate of 1%. 130 00:08:14,100 --> 00:08:20,760 And when you look at how fast, historically, stocks 131 00:08:20,760 --> 00:08:26,060 have grown, it's actually much higher, on average, than 1%. 132 00:08:26,060 --> 00:08:28,020 It's more like about 9%. 133 00:08:28,020 --> 00:08:33,669 So we would call the difference between 9% and 1%-- 134 00:08:33,669 --> 00:08:37,559 we call that 8% differential risk premium. 135 00:08:37,559 --> 00:08:42,434 And let me just pretend there's no dividends 136 00:08:42,434 --> 00:08:46,080 to keep life simple when I say this. 137 00:08:46,080 --> 00:08:52,470 So now, this risk-neutral measure 138 00:08:52,470 --> 00:08:54,744 is kind of a fictitious probability 139 00:08:54,744 --> 00:08:56,160 measure in the sense that it's not 140 00:08:56,160 --> 00:08:59,890 describing the actual probabilities or frequencies 141 00:08:59,890 --> 00:09:04,140 of transitions, it's more a device, or a tool, 142 00:09:04,140 --> 00:09:07,930 or a trick that's handy. 143 00:09:07,930 --> 00:09:10,800 And one of its properties that causes it to earn the name 144 00:09:10,800 --> 00:09:15,770 risk-neutral probability measure is that when you look at how 145 00:09:15,770 --> 00:09:20,680 fast, say, S&P grows on average under this risk-neutral 146 00:09:20,680 --> 00:09:25,460 probability measure Q, it would be growing nowadays at 1%-- 147 00:09:25,460 --> 00:09:31,050 so the same as your bank balance is growing at. 148 00:09:31,050 --> 00:09:33,860 So the word risk-neutral is meant 149 00:09:33,860 --> 00:09:37,510 to indicate that the growth rate under this measure 150 00:09:37,510 --> 00:09:43,920 is consistent with investors in the economy being risk-neutral, 151 00:09:43,920 --> 00:09:46,500 meaning that they require no premium for bearing risk. 152 00:09:52,490 --> 00:09:54,910 Now there's a third probability measure 153 00:09:54,910 --> 00:09:58,360 that we're going to be talking about today that actually you 154 00:09:58,360 --> 00:10:00,860 won't find any literature on. 155 00:10:00,860 --> 00:10:05,150 And we're going to call it R. It seems like a natural letter 156 00:10:05,150 --> 00:10:08,280 to pick, having already gone through P and Q. 157 00:10:08,280 --> 00:10:14,550 And you can think of the R as standing 158 00:10:14,550 --> 00:10:17,417 for recovered probability measure. 159 00:10:17,417 --> 00:10:19,250 And it's going to be the probability measure 160 00:10:19,250 --> 00:10:23,710 that we get from market prices as I was talking about earlier. 161 00:10:23,710 --> 00:10:31,690 And the operational meaning of this R measure 162 00:10:31,690 --> 00:10:36,110 is it's capturing the market's beliefs regarding the future. 163 00:10:36,110 --> 00:10:40,530 But we allow for the possibility that the market could be wrong. 164 00:10:40,530 --> 00:10:43,980 So we're applying this to say houses and housing 165 00:10:43,980 --> 00:10:48,180 prices in, say, 2005-- it may well 166 00:10:48,180 --> 00:10:50,160 be that if we looked at Bloomberg 167 00:10:50,160 --> 00:10:54,280 and got prices of mortgage-backed securities, 168 00:10:54,280 --> 00:10:58,430 that we would extract an R probability measure that 169 00:10:58,430 --> 00:11:01,990 says housing prices are going to continue 170 00:11:01,990 --> 00:11:05,310 on their incessant upward trajectory. 171 00:11:05,310 --> 00:11:06,950 And, you know, we're going to keep 172 00:11:06,950 --> 00:11:10,520 growing at the rate of, say, 15% a year 173 00:11:10,520 --> 00:11:13,450 each year for the next 10 years, or something like that. 174 00:11:13,450 --> 00:11:17,650 So, that could be what the market's beliefs were back 175 00:11:17,650 --> 00:11:19,390 in 2005. 176 00:11:19,390 --> 00:11:21,520 And we know now that those beliefs were wrong, 177 00:11:21,520 --> 00:11:24,580 if that was what the market was inferring. 178 00:11:24,580 --> 00:11:28,150 So, I want to allow for at least the theoretical possibility 179 00:11:28,150 --> 00:11:30,290 that the market could be wrong. 180 00:11:30,290 --> 00:11:32,530 And so, that's why I'm drawing a distinction, 181 00:11:32,530 --> 00:11:35,140 let's say, between the R probability measure 182 00:11:35,140 --> 00:11:37,870 that captures the market's beliefs and the P probability 183 00:11:37,870 --> 00:11:41,250 measure that captures physical reality. 184 00:11:41,250 --> 00:11:45,040 So now, there's a lot of people in finance who simply cannot 185 00:11:45,040 --> 00:11:48,440 accept the possibility that the market could be wrong. 186 00:11:48,440 --> 00:11:51,680 And for those people-- the sort of true believers 187 00:11:51,680 --> 00:11:54,385 in market efficiency-- they are free to set R 188 00:11:54,385 --> 00:11:57,210 to P every time they see an R. But I want 189 00:11:57,210 --> 00:12:00,730 to allow for the possibility that what we recover 190 00:12:00,730 --> 00:12:03,600 is not physical probabilities, but simply the market beliefs. 191 00:12:11,040 --> 00:12:12,800 And anyway, it's kind of semantics. 192 00:12:12,800 --> 00:12:16,250 It's good semantics if the probability measure we recover 193 00:12:16,250 --> 00:12:20,350 is the one Ross said we should get. 194 00:12:20,350 --> 00:12:21,200 R stands for Ross. 195 00:12:24,660 --> 00:12:26,400 So Ross calls the probability measure 196 00:12:26,400 --> 00:12:29,530 that we recover-- he calls them natural probability measures. 197 00:12:29,530 --> 00:12:32,686 And well, let's say, that suggests 198 00:12:32,686 --> 00:12:34,060 that the risk-neutral probability 199 00:12:34,060 --> 00:12:38,839 measures are unnatural, which I think is fair actually. 200 00:12:38,839 --> 00:12:40,630 Because when you hear the word probability, 201 00:12:40,630 --> 00:12:42,350 you tend to think about frequencies 202 00:12:42,350 --> 00:12:44,112 with which events occur. 203 00:12:44,112 --> 00:12:45,820 And the risk-neutral probability measures 204 00:12:45,820 --> 00:12:49,502 do not give you the frequencies with which events occur. 205 00:12:49,502 --> 00:12:51,460 What the risk-neutral probability measures give 206 00:12:51,460 --> 00:12:57,530 you is instead prices of so-called Arrow-Debreu 207 00:12:57,530 --> 00:12:58,810 securities. 208 00:12:58,810 --> 00:13:01,530 So, let me give you a sense of what that means. 209 00:13:01,530 --> 00:13:06,460 So say I tell you that the risk-neutral probability of S&P 210 00:13:06,460 --> 00:13:10,930 500 being up at the end of the year is 40%. 211 00:13:13,580 --> 00:13:16,280 Then how should you interpret that? 212 00:13:16,280 --> 00:13:18,510 Well, you should simply interpret it as this. 213 00:13:18,510 --> 00:13:21,830 Imagine that you can agree now to buy 214 00:13:21,830 --> 00:13:27,190 a security that pays $1 just if S&P 500 is up 215 00:13:27,190 --> 00:13:29,130 at the end of the year. 216 00:13:29,130 --> 00:13:33,150 And usually when you and I buy things, 217 00:13:33,150 --> 00:13:34,650 we buy them in a spot market. 218 00:13:34,650 --> 00:13:38,600 So we pay now for things. 219 00:13:38,600 --> 00:13:40,410 But sometimes your credit is good, 220 00:13:40,410 --> 00:13:45,000 and you can actually agree now to pay later. 221 00:13:45,000 --> 00:13:48,270 So, we're going to be thinking that you're agreeing now 222 00:13:48,270 --> 00:13:51,440 to pay later some fixed amount in return 223 00:13:51,440 --> 00:13:56,550 for the security that's going to pay $1 just if S&P 500 is up 224 00:13:56,550 --> 00:13:59,030 at the end of the year. 225 00:13:59,030 --> 00:14:02,940 And if I tell you that the risk-neutral probability of S&P 226 00:14:02,940 --> 00:14:08,190 500 being up by the end of the year is $0.40, 227 00:14:08,190 --> 00:14:12,980 what that means financially is that you agree now to pay $0.40 228 00:14:12,980 --> 00:14:15,180 at the end of the year for the security. 229 00:14:23,640 --> 00:14:26,660 So, you can imagine there'd be another security that 230 00:14:26,660 --> 00:14:33,140 pays $1 just if S&P 500 is down by the end of the year. 231 00:14:33,140 --> 00:14:41,455 And the only possible price that that security could have 232 00:14:41,455 --> 00:14:45,740 in an arbitrage-free world would be $0.60. 233 00:14:45,740 --> 00:14:49,520 Because if you were to buy both securities, 234 00:14:49,520 --> 00:14:52,330 then you get paid a total of $0.40 and $0.60. 235 00:14:52,330 --> 00:14:56,270 So you're agreeing now to pay $1 at the end of the year. 236 00:14:56,270 --> 00:14:59,460 And then having both securities, either S&P is up, 237 00:14:59,460 --> 00:15:00,930 or S&P is down. 238 00:15:00,930 --> 00:15:05,820 And so, you collect $1 from one of them and not the other. 239 00:15:05,820 --> 00:15:13,840 So if, for example, the one paying if S&P is up cost $0.40, 240 00:15:13,840 --> 00:15:17,585 while the one paying if S&P is down only cost $0.50, 241 00:15:17,585 --> 00:15:18,960 then there would be an arbitrage, 242 00:15:18,960 --> 00:15:21,350 which we would buy both securities, 243 00:15:21,350 --> 00:15:22,650 agree now to pay $0.90. 244 00:15:22,650 --> 00:15:25,140 And then get $1 for sure at the end of the period. 245 00:15:25,140 --> 00:15:27,230 So we'd be up $0.10 by the end of the year. 246 00:15:27,230 --> 00:15:27,730 Question-- 247 00:15:27,730 --> 00:15:29,688 AUDIENCE: These are similar to digital options? 248 00:15:29,688 --> 00:15:31,080 PETER CARR: Yes. 249 00:15:31,080 --> 00:15:32,439 It's more than similar. 250 00:15:32,439 --> 00:15:33,480 They are digital options. 251 00:15:33,480 --> 00:15:33,980 Yeah. 252 00:15:37,330 --> 00:15:38,477 So, that's right. 253 00:15:38,477 --> 00:15:40,310 So, that's another term, which I'll actually 254 00:15:40,310 --> 00:15:41,880 use on the next slide. 255 00:15:41,880 --> 00:15:43,550 So, that's exactly right. 256 00:15:43,550 --> 00:15:46,860 So, digital options is just too good a term. 257 00:15:46,860 --> 00:15:49,124 So economists, in order to obfuscate and look smart, 258 00:15:49,124 --> 00:15:50,540 call them Arrow-Debreu securities. 259 00:15:59,350 --> 00:16:02,170 So, continuing with the obfuscation, 260 00:16:02,170 --> 00:16:05,310 I want to tell you about a world with a representative agent. 261 00:16:09,480 --> 00:16:12,850 So, economists are fond of trying 262 00:16:12,850 --> 00:16:16,450 to formally model the market. 263 00:16:19,722 --> 00:16:21,450 You read the newspaper. 264 00:16:21,450 --> 00:16:23,360 Every day, you'll read something like market 265 00:16:23,360 --> 00:16:27,840 thought that stocks were no longer a good investment. 266 00:16:27,840 --> 00:16:28,840 So there was a sell-off. 267 00:16:32,720 --> 00:16:34,350 Market is a nice, short word to capture 268 00:16:34,350 --> 00:16:35,810 what people are thinking. 269 00:16:35,810 --> 00:16:37,760 And so economists, rather than say the market, 270 00:16:37,760 --> 00:16:40,400 will say there's a world where the representative agent-- 271 00:16:40,400 --> 00:16:44,930 So this representative agent is a fictitious investor 272 00:16:44,930 --> 00:16:51,570 who has all the mathematical properties that we 273 00:16:51,570 --> 00:16:54,055 give an investor, such as utility, function, 274 00:16:54,055 --> 00:16:57,190 and an endowment, and so on. 275 00:16:57,190 --> 00:17:00,930 And what makes this particular investor 276 00:17:00,930 --> 00:17:04,550 a representative agent is that this agent sort of 277 00:17:04,550 --> 00:17:08,680 finds that current prices are such that it's 278 00:17:08,680 --> 00:17:16,010 optimal to hold exactly what's available in the amount that 279 00:17:16,010 --> 00:17:18,569 is available. 280 00:17:18,569 --> 00:17:28,650 So if what's on offer is, let's say, some Google shares, 281 00:17:28,650 --> 00:17:31,410 and some Apple shares, and some IBM shares. 282 00:17:31,410 --> 00:17:36,070 And if we take the total market cap of Google, 283 00:17:36,070 --> 00:17:39,370 total market cap of Apple, total market cap of IBM, 284 00:17:39,370 --> 00:17:43,420 and, let's say, Apple's biggest. 285 00:17:43,420 --> 00:17:45,720 I don't actually know whether Google's bigger or IBM, 286 00:17:45,720 --> 00:17:47,650 but let's say it's Google, and then IBM. 287 00:17:50,920 --> 00:17:53,674 So let's just say Apple's biggest, then Google, then IBM. 288 00:17:53,674 --> 00:17:55,090 Well, this investor would actually 289 00:17:55,090 --> 00:17:57,100 find that it's optimal for him to have 290 00:17:57,100 --> 00:17:59,292 most of his money in Apple, second most 291 00:17:59,292 --> 00:18:01,000 of his money in Google, third most amount 292 00:18:01,000 --> 00:18:04,280 of his money in IBM, that's the representative agent. 293 00:18:04,280 --> 00:18:10,250 So, he's acting in the way the whole economy is acting. 294 00:18:13,610 --> 00:18:19,540 Well, I've been working in Wall Street now since 1996. 295 00:18:19,540 --> 00:18:21,022 I have yet to hear a trader tell me 296 00:18:21,022 --> 00:18:22,230 about a representative agent. 297 00:18:24,950 --> 00:18:28,780 Anyway, so although I understand what the words mean, 298 00:18:28,780 --> 00:18:32,840 and even the math, I wanted to present this material in a way 299 00:18:32,840 --> 00:18:35,770 that, let's say, at least quantitative traders 300 00:18:35,770 --> 00:18:38,450 could understand it. 301 00:18:38,450 --> 00:18:41,420 So I tried to get away from representative agents 302 00:18:41,420 --> 00:18:45,830 and present these ideas in the language 303 00:18:45,830 --> 00:18:50,670 that at least quants on Wall Street are familiar with. 304 00:18:56,390 --> 00:18:59,590 So, I won't be talking about a representative agent, 305 00:18:59,590 --> 00:19:04,320 and I will be talking instead about something that's probably 306 00:19:04,320 --> 00:19:08,010 not too familiar to you, but at least quants have heard of. 307 00:19:08,010 --> 00:19:11,770 And that would be something called numeraire portfolio. 308 00:19:11,770 --> 00:19:14,670 And it also goes by other names. 309 00:19:14,670 --> 00:19:17,180 Another name is growth optimal portfolio. 310 00:19:17,180 --> 00:19:18,940 And it even has a third name, which 311 00:19:18,940 --> 00:19:20,590 is called natural numeraire. 312 00:19:20,590 --> 00:19:22,530 And these are three different phrases 313 00:19:22,530 --> 00:19:25,290 that all describe the same mathematical object. 314 00:19:25,290 --> 00:19:32,050 And this mathematical object is a portfolio-- and more 315 00:19:32,050 --> 00:19:35,310 precisely, it's the value of a portfolio 316 00:19:35,310 --> 00:19:38,430 that has some nice properties. 317 00:19:38,430 --> 00:19:42,010 So the growth optimal portfolio indicates 318 00:19:42,010 --> 00:19:44,810 one of its properties. 319 00:19:44,810 --> 00:19:48,030 This portfolio has a very nice property, 320 00:19:48,030 --> 00:19:51,250 which is that in the long run-- meaning 321 00:19:51,250 --> 00:19:57,590 over an infinite horizon-- the growth rate of this portfolio 322 00:19:57,590 --> 00:19:59,430 is, first of all, random. 323 00:19:59,430 --> 00:20:04,020 But second, if you take the mean of that random growth rate, 324 00:20:04,020 --> 00:20:06,515 that mean is actually the largest possible 325 00:20:06,515 --> 00:20:07,390 among all portfolios. 326 00:20:11,130 --> 00:20:16,500 So, starting with Kelly in 1956, this particular portfolio 327 00:20:16,500 --> 00:20:21,830 with the largest mean growth rate over an infinite horizon 328 00:20:21,830 --> 00:20:24,760 receives a lot of attention. 329 00:20:24,760 --> 00:20:27,080 It's actually quite humorous, some of this attention 330 00:20:27,080 --> 00:20:28,650 that it's received. 331 00:20:28,650 --> 00:20:31,730 So, Kelly was a physicist who worked at Bell Labs. 332 00:20:31,730 --> 00:20:41,140 And he was actually a colleague of Shannon's at Bell Labs. 333 00:20:41,140 --> 00:20:44,110 So Shannon did his seminal work at Bell Labs, 334 00:20:44,110 --> 00:20:46,700 but actually came here after that. 335 00:20:46,700 --> 00:20:57,390 And his ideas really caught on-- and especially, I'd say, 336 00:20:57,390 --> 00:21:00,059 started the field of information science, 337 00:21:00,059 --> 00:21:01,100 we'll call it-- whatever. 338 00:21:03,850 --> 00:21:06,990 But Kelly was applying these ideas to finance. 339 00:21:06,990 --> 00:21:10,870 And certain financial economists were less than enthused 340 00:21:10,870 --> 00:21:14,350 about the application information of theory 341 00:21:14,350 --> 00:21:16,820 to finance. 342 00:21:16,820 --> 00:21:19,670 So, in particular, there was a financial economist here named 343 00:21:19,670 --> 00:21:23,590 Paul Samuelson who championed, I guess, 344 00:21:23,590 --> 00:21:30,000 the opposition to this Kelly criterion it's called. 345 00:21:30,000 --> 00:21:35,427 And so, I'll just tell you a short story. 346 00:21:35,427 --> 00:21:36,260 AUDIENCE: Excuse me. 347 00:21:36,260 --> 00:21:36,750 PETER CARR: Yeah. 348 00:21:36,750 --> 00:21:37,630 AUDIENCE: If I could just interject-- 349 00:21:37,630 --> 00:21:38,910 PETER CARR: Yeah, sure. 350 00:21:38,910 --> 00:21:41,105 AUDIENCE: We had mentioned in an earlier class 351 00:21:41,105 --> 00:21:42,920 the book Fortune's Formula. 352 00:21:42,920 --> 00:21:47,200 And this book goes into a lot of background and storytelling 353 00:21:47,200 --> 00:21:50,980 about this whole era and exchanges. 354 00:21:50,980 --> 00:21:51,980 PETER CARR: That's true. 355 00:21:51,980 --> 00:21:53,440 It's a fantastic book. 356 00:21:53,440 --> 00:21:54,380 I read it. 357 00:21:54,380 --> 00:21:55,827 I loved it. 358 00:21:55,827 --> 00:21:57,910 Especially if you're at MIT, you should definitely 359 00:21:57,910 --> 00:21:59,550 read this book. 360 00:21:59,550 --> 00:22:02,860 It talks about a lot of MIT professors, some of whom 361 00:22:02,860 --> 00:22:04,870 are still here, like Bob Merton. 362 00:22:08,440 --> 00:22:10,246 It's a quick, easy read. 363 00:22:10,246 --> 00:22:12,370 You don't even have to have a background in finance 364 00:22:12,370 --> 00:22:13,161 to really enjoy it. 365 00:22:16,440 --> 00:22:19,580 So you can read about the story I'm going 366 00:22:19,580 --> 00:22:21,850 to tell you now in that book. 367 00:22:21,850 --> 00:22:25,184 So the story is Samuelson grew a little tired, I guess, 368 00:22:25,184 --> 00:22:27,600 with trying to explain to these dumb information theorists 369 00:22:27,600 --> 00:22:30,490 that this Kelly criterion was not so great. 370 00:22:30,490 --> 00:22:33,116 So he published an article in a journal 371 00:22:33,116 --> 00:22:34,740 called Journal of Banking and Finance-- 372 00:22:34,740 --> 00:22:38,940 that's actually a finance journal-- where he explained 373 00:22:38,940 --> 00:22:41,930 why it wasn't necessarily such a good idea 374 00:22:41,930 --> 00:22:44,080 to hold this portfolio. 375 00:22:44,080 --> 00:22:50,490 And in this article, every word he used was of one syllable, 376 00:22:50,490 --> 00:22:52,740 except the very last word of the article, 377 00:22:52,740 --> 00:22:59,550 where he managed to say that he has-- I can't even 378 00:22:59,550 --> 00:23:03,560 do it in one syllable-- OK, so just ignore 379 00:23:03,560 --> 00:23:04,790 my multi-syllabic words. 380 00:23:04,790 --> 00:23:07,090 But anyway, he says, I have managed 381 00:23:07,090 --> 00:23:15,100 to write an article with all words with just one syllable, 382 00:23:15,100 --> 00:23:20,847 except for this last syllable-- OK, I lost it-- sorry. 383 00:23:20,847 --> 00:23:22,430 But anyway, the last word in his thing 384 00:23:22,430 --> 00:23:27,340 was syllable itself, which is multi-syllabic-- or whatever. 385 00:23:27,340 --> 00:23:30,950 So anyway, it was kind of insane. 386 00:23:30,950 --> 00:23:32,300 So, let's move on. 387 00:23:32,300 --> 00:23:35,060 So this talk-- it has six parts. 388 00:23:35,060 --> 00:23:36,740 And we have an hour to go. 389 00:23:36,740 --> 00:23:39,010 So let's say we'll try to spend 10 minutes on each. 390 00:23:39,010 --> 00:23:40,006 AUDIENCE: [INAUDIBLE] 391 00:23:47,634 --> 00:23:48,300 PETER CARR: Yes. 392 00:23:51,080 --> 00:23:52,660 Well, that's a good question. 393 00:23:52,660 --> 00:23:56,080 So, it does have risk, first of all. 394 00:23:56,080 --> 00:23:57,540 It does have a lot of risk. 395 00:24:00,550 --> 00:24:03,480 It's not the riskiest, though. 396 00:24:03,480 --> 00:24:11,090 So some risk does not carry with it expected return. 397 00:24:11,090 --> 00:24:16,270 And so that's why it's not the riskiest-- but it's risky. 398 00:24:16,270 --> 00:24:19,310 So Samuelson's objections were precisely what 399 00:24:19,310 --> 00:24:23,490 you're getting at, that this is a fairly risky strategy. 400 00:24:23,490 --> 00:24:25,631 So, I'm glad you brought that up. 401 00:24:30,980 --> 00:24:31,480 OK. 402 00:24:31,480 --> 00:24:32,740 So there's six parts to the talk. 403 00:24:32,740 --> 00:24:34,698 I'm going to go over what Arrow-Debreu security 404 00:24:34,698 --> 00:24:38,100 prices are-- so again, they're digital options prices-- 405 00:24:38,100 --> 00:24:41,210 and their connection to market beliefs. 406 00:24:41,210 --> 00:24:43,360 I'll talk about this Ross recovery theorem. 407 00:24:43,360 --> 00:24:48,960 So in Ross's paper, which you can get on SSRN, 408 00:24:48,960 --> 00:24:54,510 he does everything in a setting that's called finite state 409 00:24:54,510 --> 00:24:55,540 Markov chains. 410 00:24:55,540 --> 00:24:58,640 And so that's mathematically simpler 411 00:24:58,640 --> 00:25:01,750 than what we use in practice. 412 00:25:01,750 --> 00:25:04,760 And I totally agree that when you try and introduce 413 00:25:04,760 --> 00:25:08,360 something, you do it in the simplest mathematical setting. 414 00:25:08,360 --> 00:25:10,070 So now that he's done that, I wanted 415 00:25:10,070 --> 00:25:14,040 to do it in a more familiar setting, which 416 00:25:14,040 --> 00:25:17,040 is a diffusion setting. 417 00:25:17,040 --> 00:25:20,104 A diffusion has an uncountably infinite number of states. 418 00:25:20,104 --> 00:25:21,520 And I still want to keep things as 419 00:25:21,520 --> 00:25:24,090 simple as possible while going beyond finite state 420 00:25:24,090 --> 00:25:24,760 Markov chains. 421 00:25:24,760 --> 00:25:27,020 So I work in a univariate diffusion setting. 422 00:25:27,020 --> 00:25:28,770 So there's only one source of uncertainty, 423 00:25:28,770 --> 00:25:30,240 which is the same as in Ross. 424 00:25:30,240 --> 00:25:37,250 And our technique is to get these results. 425 00:25:37,250 --> 00:25:39,440 It's based on something called change of numeraire. 426 00:25:39,440 --> 00:25:42,590 So numeraire is a technical term, actually, 427 00:25:42,590 --> 00:25:47,160 that describes an asset whose value is always positive. 428 00:25:47,160 --> 00:25:55,540 So there are securities whose values can have either sign. 429 00:25:55,540 --> 00:25:58,460 So, swaps are a classical example. 430 00:25:58,460 --> 00:26:03,070 So a swap is a security which at inception has zero value, 431 00:26:03,070 --> 00:26:03,990 actually. 432 00:26:03,990 --> 00:26:07,140 And then the moment after inception, the world changes, 433 00:26:07,140 --> 00:26:09,570 and the swap value either becomes positive 434 00:26:09,570 --> 00:26:10,550 or becomes negative. 435 00:26:13,520 --> 00:26:16,200 So a swap would not be eligible to be 436 00:26:16,200 --> 00:26:18,500 a numeraire because of that property 437 00:26:18,500 --> 00:26:21,020 that its value is real. 438 00:26:21,020 --> 00:26:24,030 On the other hand, if you take a stock, 439 00:26:24,030 --> 00:26:28,700 its price is always positive-- well, 440 00:26:28,700 --> 00:26:37,560 that's debatable actually-- so let's say let's not do stock. 441 00:26:37,560 --> 00:26:39,400 Let's do a treasury bond. 442 00:26:39,400 --> 00:26:43,230 A treasury bond-- US Treasury bond-- its price 443 00:26:43,230 --> 00:26:45,910 is always positive. 444 00:26:45,910 --> 00:26:48,100 The reason I want to shy away from stocks 445 00:26:48,100 --> 00:26:51,430 is because we take Lehman Brothers stock, for example. 446 00:26:51,430 --> 00:26:53,770 It's price was positive, then became zero. 447 00:26:53,770 --> 00:26:56,520 And actually, because Lehman's price 448 00:26:56,520 --> 00:27:01,654 became zero, Lehman's share you could not be a numeraire. 449 00:27:01,654 --> 00:27:04,070 So when I say that the numeraire value has to be positive, 450 00:27:04,070 --> 00:27:05,111 I mean strictly positive. 451 00:27:09,170 --> 00:27:11,164 And so anyway, there's this literature 452 00:27:11,164 --> 00:27:12,580 about how to change numeraire, how 453 00:27:12,580 --> 00:27:15,020 to go from one asset with positive value 454 00:27:15,020 --> 00:27:17,520 to another asset with positive value. 455 00:27:17,520 --> 00:27:20,780 And it's useful for understanding 456 00:27:20,780 --> 00:27:22,960 how this Ross recovery works. 457 00:27:22,960 --> 00:27:29,910 So, we apply it when we have a so-called time-homogeneous 458 00:27:29,910 --> 00:27:32,500 diffusion-- and I'll tell you what that means-- 459 00:27:32,500 --> 00:27:34,000 over a bounded state space. 460 00:27:34,000 --> 00:27:37,530 So bounded state space means that the set 461 00:27:37,530 --> 00:27:40,400 of values that the diffusion can take 462 00:27:40,400 --> 00:27:44,090 is in some finite interval. 463 00:27:44,090 --> 00:27:48,730 So if you're thinking about the uncertainty being, for example, 464 00:27:48,730 --> 00:27:53,170 S&P 500, then the natural lower bound for S&P 500 465 00:27:53,170 --> 00:27:54,950 would be zero. 466 00:27:54,950 --> 00:27:58,090 And you have to accept that there's a finite upper bound 467 00:27:58,090 --> 00:28:01,050 in order to apply our results. 468 00:28:01,050 --> 00:28:04,310 Now you know, personally, I have no problem 469 00:28:04,310 --> 00:28:08,120 saying the S&P 500 is bounded above by 20 trillion. 470 00:28:08,120 --> 00:28:12,190 OK, but some economists have actually said 471 00:28:12,190 --> 00:28:14,280 this is ridiculous. 472 00:28:14,280 --> 00:28:16,540 and challenged my work, and stuff 473 00:28:16,540 --> 00:28:20,540 like that for that assumption. 474 00:28:20,540 --> 00:28:22,300 So, because of those challenges. 475 00:28:22,300 --> 00:28:24,890 I have actually been trying to extend 476 00:28:24,890 --> 00:28:28,470 our work to an unbounded state space, where, let's say, 477 00:28:28,470 --> 00:28:32,210 the largest possible value for S&P 500 would be infinity. 478 00:28:32,210 --> 00:28:37,280 And I've found, actually, that it's not that easy. 479 00:28:37,280 --> 00:28:42,600 And so sometimes, I can make it work, and sometimes I cannot. 480 00:28:42,600 --> 00:28:45,740 So, when we get there, I'll explain some examples that work 481 00:28:45,740 --> 00:28:47,310 and some examples that don't. 482 00:28:47,310 --> 00:28:50,950 So this last section is kind of incomplete, this sixth section. 483 00:28:50,950 --> 00:28:53,080 And so, basically, I've got examples that fail, 484 00:28:53,080 --> 00:28:54,770 examples that succeed. 485 00:28:54,770 --> 00:28:57,640 But I don't have a general theory. 486 00:28:57,640 --> 00:29:00,100 So there'll be different assumptions 487 00:29:00,100 --> 00:29:01,410 in different parts of the talk. 488 00:29:01,410 --> 00:29:03,529 But within a section, there's only one set 489 00:29:03,529 --> 00:29:04,570 of assumptions operating, 490 00:29:04,570 --> 00:29:05,510 AUDIENCE: Excuse me. 491 00:29:05,510 --> 00:29:06,260 PETER CARR: Yeah. 492 00:29:06,260 --> 00:29:11,417 AUDIENCE: [INAUDIBLE] the value of anything is [INAUDIBLE]. 493 00:29:11,417 --> 00:29:12,250 [INTERPOSING VOICES] 494 00:29:15,004 --> 00:29:16,670 PETER CARR: That's been my response too. 495 00:29:16,670 --> 00:29:18,462 So the universe is bounded. 496 00:29:18,462 --> 00:29:19,920 And it's growing, but it's bounded. 497 00:29:23,150 --> 00:29:24,410 So, I agree. 498 00:29:24,410 --> 00:29:27,100 You know, I'm on your side on this. 499 00:29:27,100 --> 00:29:29,270 I'm just telling you what I've been told. 500 00:29:29,270 --> 00:29:30,140 Yeah. 501 00:29:30,140 --> 00:29:33,650 So, I'm working on it anyway, just so they can shut up. 502 00:29:33,650 --> 00:29:34,770 But, anyway-- 503 00:29:34,770 --> 00:29:36,602 AUDIENCE: Actually, I have some comments 504 00:29:36,602 --> 00:29:38,900 on the issue of the numeraire. 505 00:29:38,900 --> 00:29:40,620 You'll tell me how connected this 506 00:29:40,620 --> 00:29:44,930 is-- but with the Kelly criterion, one 507 00:29:44,930 --> 00:29:48,880 of the origins of that is if you have a gambling 508 00:29:48,880 --> 00:29:51,720 opportunity where it's favorable, 509 00:29:51,720 --> 00:29:55,650 how much of your bankroll should you bet on that gamble? 510 00:29:55,650 --> 00:29:59,860 And basically, the Kelly criterion 511 00:29:59,860 --> 00:30:01,610 tells you what proportion of your bankroll 512 00:30:01,610 --> 00:30:02,990 you should invest at all times. 513 00:30:02,990 --> 00:30:04,570 You should never bet everything. 514 00:30:04,570 --> 00:30:08,080 And if you do bet everything, you lose everything, 515 00:30:08,080 --> 00:30:09,670 and you're done. 516 00:30:09,670 --> 00:30:14,630 So, the issue with the numeraire portfolio and never 517 00:30:14,630 --> 00:30:16,690 being able to go down to zero, in the sense 518 00:30:16,690 --> 00:30:19,680 that you can never go bankrupt. 519 00:30:19,680 --> 00:30:24,530 And so, assumptions of being able to always 520 00:30:24,530 --> 00:30:27,850 rebalance your portfolio-- 521 00:30:27,850 --> 00:30:29,820 PETER CARR: So, just give you a flavor 522 00:30:29,820 --> 00:30:34,600 of what this numeraire portfolio is-- 523 00:30:34,600 --> 00:30:37,160 you're betting a constant fraction of your wealth 524 00:30:37,160 --> 00:30:37,995 in every security. 525 00:30:40,970 --> 00:30:42,220 So let's just keep it simple. 526 00:30:42,220 --> 00:30:44,170 There's only two securities. 527 00:30:44,170 --> 00:30:46,480 One is risky, and the other's riskless. 528 00:30:46,480 --> 00:30:50,700 And so you might be betting putting 40% of your wealth 529 00:30:50,700 --> 00:30:55,410 in the risky one, and 60% then in the riskless one. 530 00:30:55,410 --> 00:30:57,100 And that's when you start. 531 00:30:57,100 --> 00:31:00,180 So you have $100, and you put $40 in the risky one, 532 00:31:00,180 --> 00:31:03,040 and $60 in the riskless one. 533 00:31:03,040 --> 00:31:07,440 And then, time moves forward. 534 00:31:07,440 --> 00:31:12,500 And let's say the price of the risky one changes. 535 00:31:12,500 --> 00:31:16,780 Then when you revalue using the new price, 536 00:31:16,780 --> 00:31:23,930 it's unlikely that 40% of your wealth is in the risky one. 537 00:31:23,930 --> 00:31:27,000 So in fact, if the price went up of the risky one, 538 00:31:27,000 --> 00:31:30,520 you'll have more than 40% of your wealth in that risky one. 539 00:31:30,520 --> 00:31:33,660 So you need to sell some of that risky one. 540 00:31:33,660 --> 00:31:36,180 And then the money you get, you put into the riskless one. 541 00:31:39,720 --> 00:31:41,650 And so, every time the price changes, 542 00:31:41,650 --> 00:31:43,940 you need to trade, theoretically, in order 543 00:31:43,940 --> 00:31:49,940 to maintain a constant fraction of 40% of your wealth invested 544 00:31:49,940 --> 00:31:51,010 in this risky asset. 545 00:31:55,410 --> 00:32:00,430 So we assume zero transactions cost when we do this analysis. 546 00:32:00,430 --> 00:32:02,360 Because there are positive transactions cost. 547 00:32:05,030 --> 00:32:06,710 One should take that into account. 548 00:32:06,710 --> 00:32:08,460 And there is literature on how to do that. 549 00:32:12,749 --> 00:32:14,790 So, I won't be formally entertaining transactions 550 00:32:14,790 --> 00:32:17,610 cost in this talk. 551 00:32:17,610 --> 00:32:21,620 There's work here at MIT, actually, on doing that. 552 00:32:27,880 --> 00:32:30,810 For the question of how should you invest, 553 00:32:30,810 --> 00:32:34,330 it feels like it's a complication that won't change 554 00:32:34,330 --> 00:32:38,350 anything qualitative about-- it'd definitely change 555 00:32:38,350 --> 00:32:43,060 how frequently you trade, but it wouldn't, let's say, 556 00:32:43,060 --> 00:32:45,900 it's unclear how it would change your initial investment 557 00:32:45,900 --> 00:32:47,930 across bets. 558 00:32:52,440 --> 00:32:54,860 So, let's begin with part one. 559 00:32:54,860 --> 00:32:58,350 So we have the digital options, or also called binary options. 560 00:32:58,350 --> 00:32:59,610 That's another term. 561 00:32:59,610 --> 00:33:06,610 And they trade, actually, in FX markets-- so foreign exchange. 562 00:33:06,610 --> 00:33:10,540 And they pay one unit of some currencies, 563 00:33:10,540 --> 00:33:15,587 so say dollar-- If an event comes true. 564 00:33:15,587 --> 00:33:17,670 So it might be that you're looking at dollar/euro. 565 00:33:17,670 --> 00:33:22,260 And if by the end of the year, dollar/euro exceeds 2, 566 00:33:22,260 --> 00:33:23,394 then you get $1. 567 00:33:23,394 --> 00:33:24,310 Otherwise, you get $0. 568 00:33:28,740 --> 00:33:31,780 So there would be a price in the FX markets. 569 00:33:31,780 --> 00:33:34,740 And it would be a spot price typically-- so meaning 570 00:33:34,740 --> 00:33:36,012 you have to pay now for it. 571 00:33:38,990 --> 00:33:46,620 Let's let A, for arrow, be the price today of such a security. 572 00:33:46,620 --> 00:33:50,970 And the subscripts on A are j given i. 573 00:33:50,970 --> 00:33:54,450 So, the idea is that you can think of yourself 574 00:33:54,450 --> 00:33:57,180 as in a finite-state setting. 575 00:33:57,180 --> 00:33:59,150 There's various discrete levels of say, 576 00:33:59,150 --> 00:34:03,310 dollar/euro that we have that can be possible today. 577 00:34:03,310 --> 00:34:05,760 And there's also various discrete levels for dollar/euro 578 00:34:05,760 --> 00:34:07,200 by the end of the year. 579 00:34:07,200 --> 00:34:10,989 And i indicates the state we're in. 580 00:34:10,989 --> 00:34:14,100 So maybe dollar euro is $2 per euro right now. 581 00:34:16,719 --> 00:34:21,030 And j indicates the state we can go to. 582 00:34:21,030 --> 00:34:22,989 Maybe we can go to $3 per euro. 583 00:34:29,315 --> 00:34:34,250 So in my example, A_(3|2) would be the price of an Arrow-Debreu 584 00:34:34,250 --> 00:34:39,323 security, given that the current dollar euro exchange rate is $2 585 00:34:39,323 --> 00:34:45,300 per euro, and it pays $1 just if dollar/euro transitions from $2 586 00:34:45,300 --> 00:34:47,130 per euro to $3 per euro. 587 00:34:58,920 --> 00:35:04,420 So the idea is we have discrete states. 588 00:35:04,420 --> 00:35:07,300 And let's say these are values that 589 00:35:07,300 --> 00:35:11,390 are possible at the end of the year. 590 00:35:11,390 --> 00:35:13,710 And the example I just went through-- you're 591 00:35:13,710 --> 00:35:20,290 getting $1 just if it's $3 per euro at the end of the year. 592 00:35:20,290 --> 00:35:22,440 So the height of that vertical line is one. 593 00:35:29,420 --> 00:35:34,360 Now, I'll just comment that this is 594 00:35:34,360 --> 00:35:39,510 a slightly exotic option, in the sense that-- 595 00:35:39,510 --> 00:35:40,670 let's call it exotic. 596 00:35:40,670 --> 00:35:42,960 It's slightly exotic. 597 00:35:42,960 --> 00:35:46,570 So in contrast with exotics, there's this term "vanilla." 598 00:35:46,570 --> 00:35:49,480 OK, and it actually indicates a flavor of ice cream. 599 00:35:52,010 --> 00:35:53,610 So, we have this terminology which 600 00:35:53,610 --> 00:35:54,500 you get used to after awhile. 601 00:35:54,500 --> 00:35:57,041 And you can't understand when you talk to a man on the street 602 00:35:57,041 --> 00:36:00,100 why they don't understand what a vanilla option is. 603 00:36:00,100 --> 00:36:09,960 So a vanilla option is a payoff that looks like this-- 604 00:36:09,960 --> 00:36:12,250 so it's a hockey stick payoff. 605 00:36:12,250 --> 00:36:15,580 And that's the payoff from a call option. 606 00:36:15,580 --> 00:36:21,930 And it turns that there is a portfolio involving options 607 00:36:21,930 --> 00:36:24,830 at three different strikes that can perfectly 608 00:36:24,830 --> 00:36:28,830 replicate the payoff to this Arrow-Debreu security. 609 00:36:28,830 --> 00:36:33,900 And so, here is a payoff from a single option struck at two. 610 00:36:33,900 --> 00:36:40,480 And I'll just say that if I had changed the strike 611 00:36:40,480 --> 00:36:42,533 to, say, be three, then it would look like that. 612 00:36:48,690 --> 00:36:51,460 Now, you can combine options in your portfolio. 613 00:36:51,460 --> 00:36:55,410 So you could, for example, buy a call struck at two. 614 00:36:55,410 --> 00:37:00,570 And then you can furthermore sell two calls struck at three. 615 00:37:00,570 --> 00:37:03,350 So if you sell, on top of that, two calls struck at three, 616 00:37:03,350 --> 00:37:05,710 you end up creating a portfolio that goes like this. 617 00:37:10,380 --> 00:37:13,960 And so, they can go negative in value. 618 00:37:13,960 --> 00:37:19,240 So if you not only buy one call struck at two, 619 00:37:19,240 --> 00:37:22,340 sell two calls struck at three, but furthermore, 620 00:37:22,340 --> 00:37:26,040 buy one call struck at four, then you 621 00:37:26,040 --> 00:37:29,390 end up with this payoff, which the payoff is called 622 00:37:29,390 --> 00:37:30,640 a butterfly spread payoff. 623 00:37:30,640 --> 00:37:33,405 Because the picture is meant to remind you of a butterfly. 624 00:37:36,430 --> 00:37:42,970 And notice that if the only possible values for the FX rate 625 00:37:42,970 --> 00:37:46,600 were $1 per euro or $2 per euro, or $3 or $4, or $5, 626 00:37:46,600 --> 00:37:48,110 if that were the world. 627 00:37:48,110 --> 00:37:53,650 Then notice that when you formed that portfolio, 628 00:37:53,650 --> 00:37:56,150 the only positive payoff you can get from it 629 00:37:56,150 --> 00:37:58,970 is $1 just if dollar euros at 3. 630 00:38:03,800 --> 00:38:05,840 You can synthesize a Arrow-Debreu security 631 00:38:05,840 --> 00:38:08,880 using a butterfly spread. 632 00:38:08,880 --> 00:38:11,260 So, this was pointed out many years ago. 633 00:38:15,200 --> 00:38:17,040 So even if the FX market were, let's say, 634 00:38:17,040 --> 00:38:19,920 not directly giving us the prices of digital options, 635 00:38:19,920 --> 00:38:24,630 we could from vanilla options extract the implicit price 636 00:38:24,630 --> 00:38:25,370 of a digital. 637 00:38:32,560 --> 00:38:36,380 And what you would learn from vanilla options 638 00:38:36,380 --> 00:38:42,040 is what the market is charging for the digital, given 639 00:38:42,040 --> 00:38:46,920 that, let's say, we're presently at $2 per euro. 640 00:38:46,920 --> 00:38:51,360 And what you would not learn from these options 641 00:38:51,360 --> 00:38:55,660 prices is what the price of the security 642 00:38:55,660 --> 00:39:00,430 will be should we today have the exchange rate 643 00:39:00,430 --> 00:39:01,625 change to some other value. 644 00:39:05,550 --> 00:39:07,890 However, you can make assumptions 645 00:39:07,890 --> 00:39:10,970 as have what the options prices will be 646 00:39:10,970 --> 00:39:13,145 were today's exchange rate different. 647 00:39:16,840 --> 00:39:18,745 So, that's commonly done in practice. 648 00:39:21,520 --> 00:39:23,860 So a common assumption, for example, 649 00:39:23,860 --> 00:39:27,980 is that the probability of transitioning 650 00:39:27,980 --> 00:39:35,090 from two to three-- so moving up by half-- so you're moving up 651 00:39:35,090 --> 00:39:39,010 by half of two to three-- is the same 652 00:39:39,010 --> 00:39:41,160 if you were at any other level. 653 00:39:41,160 --> 00:39:44,000 So for example, if you were at four, 654 00:39:44,000 --> 00:39:46,636 then the probability of going to six 655 00:39:46,636 --> 00:39:48,010 would be whatever the probability 656 00:39:48,010 --> 00:39:50,080 is of going from two to three. 657 00:39:50,080 --> 00:39:52,510 Because if you're at four, the probability of going up 658 00:39:52,510 --> 00:39:56,990 by half of four to six-- that's the assumption. 659 00:39:56,990 --> 00:39:59,400 OK, so that's called sticky delta, 660 00:39:59,400 --> 00:40:01,720 and it's a common assumption. 661 00:40:01,720 --> 00:40:03,450 So if you make that assumption, then you 662 00:40:03,450 --> 00:40:08,400 can take the information at just today's level. 663 00:40:08,400 --> 00:40:13,950 And like, let's say, you know all the digitals from two, 664 00:40:13,950 --> 00:40:16,480 and you can make that assumption. 665 00:40:16,480 --> 00:40:22,090 Let's say the probability of a given percentage change 666 00:40:22,090 --> 00:40:24,430 is invariant to the starting level. 667 00:40:24,430 --> 00:40:28,452 And then you can, from that, figure out 668 00:40:28,452 --> 00:40:30,910 what the probability of going from four-- a different level 669 00:40:30,910 --> 00:40:33,980 than we're at today-- is to all these different levels. 670 00:40:33,980 --> 00:40:37,640 So you can go from a vector bit of information 671 00:40:37,640 --> 00:40:40,310 that the market is giving you to a matrix. 672 00:40:40,310 --> 00:40:43,420 And that matrix is called transition matrix. 673 00:40:43,420 --> 00:40:49,020 And so, we're going to, in this talk, 674 00:40:49,020 --> 00:40:52,500 assume that somebody's made such an assumption. 675 00:40:52,500 --> 00:40:55,120 And so, you actually know this matrix. 676 00:40:55,120 --> 00:40:58,510 So you actually know, as a starting point, 677 00:40:58,510 --> 00:41:02,440 what the prices are of these Arrow-Debreu securities 678 00:41:02,440 --> 00:41:05,900 or binary options starting from any level 679 00:41:05,900 --> 00:41:06,960 and going to any level. 680 00:41:18,650 --> 00:41:24,800 I think in order to get through my whole talk, 681 00:41:24,800 --> 00:41:26,915 I'm going to skip these slides. 682 00:41:26,915 --> 00:41:28,290 Because they're kind of like just 683 00:41:28,290 --> 00:41:32,270 being very precise about what some terms mean 684 00:41:32,270 --> 00:41:34,540 that aren't going to be that important 685 00:41:34,540 --> 00:41:36,520 for the overall story. 686 00:41:36,520 --> 00:41:41,065 So OK, let's go to this slide. 687 00:41:44,390 --> 00:41:47,570 So we think of there being just a single source of uncertainty 688 00:41:47,570 --> 00:41:49,680 X, which could be dollar/euro. 689 00:41:49,680 --> 00:41:54,550 And we imagine that we have this matrix of Arrow-Debreu security 690 00:41:54,550 --> 00:41:56,060 prices. 691 00:41:56,060 --> 00:41:58,580 We know every number in this matrix. 692 00:41:58,580 --> 00:42:04,750 And we ask what does the market believe about transitions 693 00:42:04,750 --> 00:42:08,010 from any place to any place? 694 00:42:08,010 --> 00:42:10,020 What does the market believe is the frequency 695 00:42:10,020 --> 00:42:11,110 of these transitions? 696 00:42:11,110 --> 00:42:14,860 Now, suppose that the number that's 697 00:42:14,860 --> 00:42:18,260 indicating the price of the Arrow-Debreu security going 698 00:42:18,260 --> 00:42:24,130 from two to three-- suppose that number is, say, 0.1. 699 00:42:24,130 --> 00:42:25,830 Now what does it mean? 700 00:42:25,830 --> 00:42:30,920 It just means that you pay $0.10 today for security paying $1 701 00:42:30,920 --> 00:42:33,480 just if you go from two to three. 702 00:42:33,480 --> 00:42:35,280 That's all it means. 703 00:42:35,280 --> 00:42:38,890 Now you can ask what is the frequency with which you 704 00:42:38,890 --> 00:42:41,190 go from two to three? 705 00:42:41,190 --> 00:42:44,700 It need not be 10%. 706 00:42:44,700 --> 00:42:50,450 There's at least two reasons why the $0.10 price could differ 707 00:42:50,450 --> 00:42:54,400 from the probability of going from where you are to where you 708 00:42:54,400 --> 00:42:56,680 get paid. 709 00:42:56,680 --> 00:43:00,030 One such reason is simply time value of money. 710 00:43:00,030 --> 00:43:04,650 So if you were to buy all these Arrow-Debreu securities, 711 00:43:04,650 --> 00:43:08,420 the one paying off-- one for every state, 712 00:43:08,420 --> 00:43:13,500 you'll find that the total cost is less than 1, 713 00:43:13,500 --> 00:43:16,000 even though the payoff, for sure, 714 00:43:16,000 --> 00:43:18,652 is one from the portfolio. 715 00:43:18,652 --> 00:43:20,860 And that's simply because of the time value of money. 716 00:43:20,860 --> 00:43:24,640 So when you put $1 in the bank today, 717 00:43:24,640 --> 00:43:28,130 you actually get more than $1 back 718 00:43:28,130 --> 00:43:29,960 when you pull out at the end of the year. 719 00:43:33,140 --> 00:43:34,680 And if you do the inverse problem-- 720 00:43:34,680 --> 00:43:36,305 how much do you have to put in the bank 721 00:43:36,305 --> 00:43:38,880 today in order to have $1 at the end of the year? 722 00:43:38,880 --> 00:43:44,030 It might be $0.95. 723 00:43:44,030 --> 00:43:45,780 So that's called time value of money. 724 00:43:45,780 --> 00:43:48,740 And so, just the fact that you have to pay now 725 00:43:48,740 --> 00:43:49,990 for the Arrow-Debreu security. 726 00:43:49,990 --> 00:43:54,160 And you only get paid off at the end of the year. 727 00:43:54,160 --> 00:44:03,090 That causes this price of $0.10 to be lower. 728 00:44:03,090 --> 00:44:06,200 So that's just discounting for time. 729 00:44:06,200 --> 00:44:07,660 The interest rates are positive. 730 00:44:07,660 --> 00:44:09,070 So that's one effect. 731 00:44:09,070 --> 00:44:15,130 Now there's another effect, which is called risk aversion. 732 00:44:15,130 --> 00:44:25,940 So risk aversion is the thought that even if the interest rate 733 00:44:25,940 --> 00:44:31,392 was 0, to abstract away from the effect I just described, 734 00:44:31,392 --> 00:44:37,780 that it still may be the case that a $0.10 price paid 735 00:44:37,780 --> 00:44:41,600 for an Arrow-Debreu security transitioning from two to three 736 00:44:41,600 --> 00:44:43,321 is different from the probability of such 737 00:44:43,321 --> 00:44:45,320 a transition, the real-world probability of such 738 00:44:45,320 --> 00:44:47,260 a transition. 739 00:44:47,260 --> 00:44:55,350 Because, for example, it may be quite desirable to get money 740 00:44:55,350 --> 00:45:01,420 in that state, in which case $0.10 is over the real-world 741 00:45:01,420 --> 00:45:03,500 probability. 742 00:45:03,500 --> 00:45:06,980 Or it could be the opposite that maybe it's not desirable to get 743 00:45:06,980 --> 00:45:09,930 money in that state, in which case $0.10 is under 744 00:45:09,930 --> 00:45:11,055 the real-world probability. 745 00:45:14,080 --> 00:45:23,760 So give you a more concrete example-- 746 00:45:23,760 --> 00:45:26,860 let's say something that is maybe a little closer to home 747 00:45:26,860 --> 00:45:31,920 is-- let's say this is S&P 500, and I 748 00:45:31,920 --> 00:45:34,540 know the values are very different than the numbers I've 749 00:45:34,540 --> 00:45:36,140 indicated here-- but let's just forget 750 00:45:36,140 --> 00:45:37,181 about the actual numbers. 751 00:45:39,860 --> 00:45:43,880 So the point is let's suppose that it's equally likely, 752 00:45:43,880 --> 00:45:47,890 in terms of true probabilities, to go from two to three 753 00:45:47,890 --> 00:45:49,550 as it is to go from two to one. 754 00:45:49,550 --> 00:45:52,320 So we have two Arrow-Debreu securities. 755 00:45:52,320 --> 00:45:53,240 One struck at one. 756 00:45:53,240 --> 00:45:55,150 The other struck at three. 757 00:45:55,150 --> 00:45:57,840 And I'm telling you that it's equally likely 758 00:45:57,840 --> 00:46:02,080 that you go up by one as it is to go down by one. 759 00:46:02,080 --> 00:46:04,080 Now you can ask the question does it necessarily 760 00:46:04,080 --> 00:46:09,140 mean that the prices of these securities that pay $1 761 00:46:09,140 --> 00:46:10,260 are the same? 762 00:46:10,260 --> 00:46:13,020 And the answer is no, not necessarily. 763 00:46:13,020 --> 00:46:15,760 And actually, the sort of standard thinking 764 00:46:15,760 --> 00:46:20,880 in financial academic circles is that for S&P 500, 765 00:46:20,880 --> 00:46:24,680 it would cost more to buy this Arrow-Debreu security than it 766 00:46:24,680 --> 00:46:28,360 would cost to buy that one, even though everyone 767 00:46:28,360 --> 00:46:32,470 agrees that it's equally likely to get paid from each of them. 768 00:46:32,470 --> 00:46:36,850 And the reason that it's thought to cost more 769 00:46:36,850 --> 00:46:38,310 to buy this one than it is to buy 770 00:46:38,310 --> 00:46:42,110 that one is because this one has an insurance value. 771 00:46:42,110 --> 00:46:45,520 So the thinking is that on average, people are long 772 00:46:45,520 --> 00:46:49,160 the stocks in the stock market, and that that 773 00:46:49,160 --> 00:46:52,340 means that they're really upset when the stocks fall. 774 00:46:52,340 --> 00:46:54,660 And so they really like this one that 775 00:46:54,660 --> 00:46:59,390 ends up paying should the stock market fall from two to one, 776 00:46:59,390 --> 00:47:03,320 whereas this one, while it's nice to get money, 777 00:47:03,320 --> 00:47:05,900 let's say you're already fairly wealthy from the fact 778 00:47:05,900 --> 00:47:09,590 that you're owning stocks and the stock market went up. 779 00:47:09,590 --> 00:47:13,110 So you'll pay a positive amount for this security, 780 00:47:13,110 --> 00:47:15,240 but not as much as you pay for this one. 781 00:47:19,199 --> 00:47:20,490 So that's called risk aversion. 782 00:47:24,200 --> 00:47:27,270 So what we want to do is go from the prices that 783 00:47:27,270 --> 00:47:31,660 are contaminated, let's say, by time value of money effects 784 00:47:31,660 --> 00:47:32,970 and by risk aversion effects. 785 00:47:32,970 --> 00:47:36,420 And we want to cleanse them of that contamination 786 00:47:36,420 --> 00:47:39,130 and try to extract what the market believes 787 00:47:39,130 --> 00:47:41,680 are the frequencies of the future states. 788 00:47:45,079 --> 00:47:47,370 So I'll tell you that this was thought to be impossible 789 00:47:47,370 --> 00:47:50,710 before the Ross paper, and in fact, 790 00:47:50,710 --> 00:47:53,440 without making assumptions, it is impossible. 791 00:47:53,440 --> 00:47:56,640 So all Ross did is make some assumptions that 792 00:47:56,640 --> 00:48:01,880 are thought to be fairly mild by some, including me. 793 00:48:01,880 --> 00:48:05,600 And so he essentially, in essence, 794 00:48:05,600 --> 00:48:07,330 showed the power of some assumptions. 795 00:48:07,330 --> 00:48:08,920 That's one way of thinking about it. 796 00:48:15,670 --> 00:48:19,440 So again, let's denote by R the recovered probability 797 00:48:19,440 --> 00:48:21,030 measure which will tell us the market 798 00:48:21,030 --> 00:48:24,910 beliefs about the frequencies future state. 799 00:48:24,910 --> 00:48:28,060 And we don't know R when we start. 800 00:48:28,060 --> 00:48:31,050 What we do know is these Arrow-Debreu security prices, 801 00:48:31,050 --> 00:48:32,280 I'm assuming. 802 00:48:32,280 --> 00:48:34,840 And we'll denote those by A for Arrow. 803 00:48:38,930 --> 00:48:42,717 So what Ross's paper does is it says, you know A. 804 00:48:42,717 --> 00:48:45,050 And if you're willing to make the following assumptions, 805 00:48:45,050 --> 00:48:47,280 then you'll know R. So what are the assumptions? 806 00:48:47,280 --> 00:48:48,780 Well, before I tell you assumptions, 807 00:48:48,780 --> 00:48:51,240 I have to tell you some terminology so that you 808 00:48:51,240 --> 00:48:53,670 understand the assumptions. 809 00:48:53,670 --> 00:48:58,660 So he'll work with a pricing matrix A, which we've actually 810 00:48:58,660 --> 00:48:59,800 been going through. 811 00:48:59,800 --> 00:49:02,530 So that's the Arrow-Debreu security prices 812 00:49:02,530 --> 00:49:05,315 index by starting state and final state, which 813 00:49:05,315 --> 00:49:08,500 we'll call x is starting state, y is final state. 814 00:49:08,500 --> 00:49:10,970 Then there'll be the desired output 815 00:49:10,970 --> 00:49:14,100 from this analysis, which he calls natural probability 816 00:49:14,100 --> 00:49:15,320 transition matrix. 817 00:49:15,320 --> 00:49:19,310 So these are the markets beliefs for every starting value 818 00:49:19,310 --> 00:49:22,180 x and for every final value y. 819 00:49:22,180 --> 00:49:25,180 And then there'll be something called pricing kernel, 820 00:49:25,180 --> 00:49:28,610 which is literally the ratio of these Arrow-Debreu security 821 00:49:28,610 --> 00:49:35,140 prices to these output natural probabilities. 822 00:49:35,140 --> 00:49:37,790 So, if you want to get an understanding of what 823 00:49:37,790 --> 00:49:40,290 this pricing kernel is, you can think of it 824 00:49:40,290 --> 00:49:46,220 as an attempt to capture the effects 825 00:49:46,220 --> 00:49:50,100 from time value of money and from risk aversion. 826 00:49:50,100 --> 00:49:53,050 So think of it as a normalization. 827 00:49:53,050 --> 00:49:55,970 You start with A, and A is actually 828 00:49:55,970 --> 00:49:57,170 affected by three things. 829 00:49:57,170 --> 00:50:00,940 It's affected by the unknown real world 830 00:50:00,940 --> 00:50:04,466 probabilities-- or at least markets beliefs of them. 831 00:50:04,466 --> 00:50:06,530 A is also affected by a second thing, 832 00:50:06,530 --> 00:50:08,200 which is time value of money. 833 00:50:08,200 --> 00:50:11,300 And A is affected by a third thing, which is risk aversion. 834 00:50:11,300 --> 00:50:15,760 So if we take A and divide by P, then we're 835 00:50:15,760 --> 00:50:19,320 normalizing for the first effect, the frequencies. 836 00:50:19,320 --> 00:50:23,200 And so we're left with just the combined effect 837 00:50:23,200 --> 00:50:26,110 from time value of money and from risk aversion. 838 00:50:28,890 --> 00:50:34,870 And so, let's say, if interest rate were zero and people 839 00:50:34,870 --> 00:50:40,990 were risk neutral, then we would actually expect A to equal P. 840 00:50:40,990 --> 00:50:42,775 And so this ratio would be just constant. 841 00:50:48,800 --> 00:50:53,630 So Ross talks about a world with a representative investor. 842 00:50:53,630 --> 00:51:01,650 And essentially, this is an assumption-- this equation 843 00:51:01,650 --> 00:51:02,590 you're seeing here. 844 00:51:06,950 --> 00:51:10,070 It's an assumption on the form that a function 845 00:51:10,070 --> 00:51:11,170 of two variables takes. 846 00:51:11,170 --> 00:51:14,600 So phi, first of all, is a positive function. 847 00:51:14,600 --> 00:51:18,760 So phi is positive, as opposed to-- so phi cannot take 848 00:51:18,760 --> 00:51:21,680 negative values because both A and P are positive. 849 00:51:21,680 --> 00:51:27,130 And phi is a function of two variables-- x and y. 850 00:51:27,130 --> 00:51:29,850 And what this assumption is doing is it's saying, 851 00:51:29,850 --> 00:51:33,520 well, let's put structure on this function phi 852 00:51:33,520 --> 00:51:36,450 because it'll help us to find it if we put the structure. 853 00:51:36,450 --> 00:51:39,380 So this is the first key assumption actually-- 854 00:51:39,380 --> 00:51:44,210 that the function of two variables x and y actually 855 00:51:44,210 --> 00:51:47,680 has the form on the right, which, for a moment, 856 00:51:47,680 --> 00:51:49,764 just ignore the delta for a moment. 857 00:51:49,764 --> 00:51:51,930 And then you can see that what you have on the right 858 00:51:51,930 --> 00:51:54,570 if you ignore delta, if you think of delta as one, 859 00:51:54,570 --> 00:51:56,840 is you have a function of y. 860 00:51:56,840 --> 00:51:59,000 And then you have the same function of x. 861 00:51:59,000 --> 00:52:02,489 So it's written in a convoluted way with this U prime, and c, 862 00:52:02,489 --> 00:52:03,280 and all that stuff. 863 00:52:03,280 --> 00:52:07,340 But if delta's one, then you have a fraction whose numerator 864 00:52:07,340 --> 00:52:09,190 is a function of y and whose denominator 865 00:52:09,190 --> 00:52:10,710 is the same function, but of x. 866 00:52:17,550 --> 00:52:20,590 In essence, what that does is it reduces 867 00:52:20,590 --> 00:52:23,950 the dimensionality of the thing we're searching for by a lot. 868 00:52:23,950 --> 00:52:27,170 So we started by searching for a function phi of two variables. 869 00:52:27,170 --> 00:52:29,970 And we, by this assumption, reduced the search 870 00:52:29,970 --> 00:52:31,840 to a function of one variable, which 871 00:52:31,840 --> 00:52:34,030 is, say, the function in the numerator, which 872 00:52:34,030 --> 00:52:35,988 is the same as the function in the denominator. 873 00:52:38,640 --> 00:52:41,170 So, now let's bring back delta. 874 00:52:41,170 --> 00:52:45,460 And delta's a scalar here, and it's a positive scalar. 875 00:52:45,460 --> 00:52:47,970 And so we need to search for that as well. 876 00:52:47,970 --> 00:52:50,080 So in the end, we reduce the search 877 00:52:50,080 --> 00:52:54,850 to a function of one variable and a scalar delta. 878 00:52:54,850 --> 00:52:59,700 So the economic meaning of, first of all, 879 00:52:59,700 --> 00:53:01,550 the function of one variable is-- 880 00:53:01,550 --> 00:53:03,930 it's called marginal utility. 881 00:53:03,930 --> 00:53:07,410 And it's meant to indicate how much happiness you 882 00:53:07,410 --> 00:53:11,210 get from each additional unit of consumption. 883 00:53:11,210 --> 00:53:14,250 So it's the typical-- what we think 884 00:53:14,250 --> 00:53:22,420 it looks like as a function of c-- U prime as a function of c 885 00:53:22,420 --> 00:53:24,180 is thought to typically look like that. 886 00:53:24,180 --> 00:53:26,760 So it's positive, meaning every unit of consumption 887 00:53:26,760 --> 00:53:27,840 makes you happy. 888 00:53:27,840 --> 00:53:31,750 And it's actually declining, meaning 889 00:53:31,750 --> 00:53:34,366 the first unit of consumption makes you real happy. 890 00:53:34,366 --> 00:53:35,740 Then the next unit of consumption 891 00:53:35,740 --> 00:53:38,310 still brings some happiness, but not as much, and so on. 892 00:53:41,750 --> 00:53:43,870 So that's the kind of function we're looking for. 893 00:53:43,870 --> 00:53:46,990 U prime as a function of c. 894 00:53:46,990 --> 00:53:50,520 He won't actually find U prime as a function of c. 895 00:53:50,520 --> 00:53:56,840 He'll find the composition of U prime with a function c of y. 896 00:54:01,920 --> 00:54:05,030 Keep that in mind. 897 00:54:05,030 --> 00:54:06,710 Then, there's that delta. 898 00:54:06,710 --> 00:54:10,260 And that's, again, a positive scalar. 899 00:54:10,260 --> 00:54:14,540 And it's meant to capture time value of money. 900 00:54:14,540 --> 00:54:21,497 And so, that's like the y is the state at the end of the period. 901 00:54:21,497 --> 00:54:23,580 And x is the state at the beginning of the period. 902 00:54:23,580 --> 00:54:26,230 And so, that's why delta's associated with the numerator, 903 00:54:26,230 --> 00:54:27,790 not the denominator. 904 00:54:27,790 --> 00:54:30,020 So delta would be a number like 0.9. 905 00:54:30,020 --> 00:54:35,340 And that indicates how much discount you give to, 906 00:54:35,340 --> 00:54:38,890 let's say happiness received in the future, rather than now. 907 00:54:42,760 --> 00:54:50,380 Now, here's a quote from Ross's paper that is his Theorem 1. 908 00:54:50,380 --> 00:54:52,150 That's called the recovery theorem. 909 00:54:52,150 --> 00:54:56,480 And the only thing is I changed the letters 910 00:54:56,480 --> 00:54:58,840 to conform with the letters I'm using, 911 00:54:58,840 --> 00:55:00,224 rather than the ones he used. 912 00:55:00,224 --> 00:55:01,890 And that's because his choice of letters 913 00:55:01,890 --> 00:55:06,520 is completely unnatural to me and most people. 914 00:55:06,520 --> 00:55:10,440 So I don't even want to tell you what he used. 915 00:55:10,440 --> 00:55:13,980 So anyway, whereas I tried to choose letters that make sense. 916 00:55:13,980 --> 00:55:15,160 So I used A for Debreu-- 917 00:55:20,381 --> 00:55:21,880 So anyway, he says, you have a world 918 00:55:21,880 --> 00:55:23,046 with a representative agent. 919 00:55:23,046 --> 00:55:28,630 So that's actually this restriction 920 00:55:28,630 --> 00:55:31,130 that we talked about on the last slide. 921 00:55:31,130 --> 00:55:34,750 And then he says, if the pricing matrix-- which 922 00:55:34,750 --> 00:55:37,980 is the Arrow-Debreu security prices-- is positive-- 923 00:55:37,980 --> 00:55:39,640 which means that all entries in it 924 00:55:39,640 --> 00:55:43,020 are strictly above zero-- or irreducible-- 925 00:55:43,020 --> 00:55:46,060 which means that some entries have zeros, 926 00:55:46,060 --> 00:55:48,700 with the rest being positive, and there's 927 00:55:48,700 --> 00:55:50,990 some structure, which we need not get into where 928 00:55:50,990 --> 00:55:56,840 the zeros are-- then there exists a unique solution 929 00:55:56,840 --> 00:56:01,880 of the problem of finding P, which P is actually 930 00:56:01,880 --> 00:56:02,750 market beliefs. 931 00:56:02,750 --> 00:56:05,630 And I've been calling that R often. 932 00:56:05,630 --> 00:56:08,160 So anyway, I slipped a bit there and called it P. 933 00:56:08,160 --> 00:56:11,950 So anyway, that's market beliefs about the frequencies 934 00:56:11,950 --> 00:56:13,170 of future states. 935 00:56:13,170 --> 00:56:15,600 He'll also get as an output the delta, 936 00:56:15,600 --> 00:56:19,590 which is the positive scalar telling you the market's time 937 00:56:19,590 --> 00:56:20,380 value of money. 938 00:56:20,380 --> 00:56:24,010 And finally, this pricing kernel phi, 939 00:56:24,010 --> 00:56:35,397 which is the ratio of A to P. So, 940 00:56:35,397 --> 00:56:37,980 what you're supposed to realize, even though he didn't say it, 941 00:56:37,980 --> 00:56:42,107 is that as a result-- well, OK. 942 00:56:42,107 --> 00:56:43,190 So he did say it actually. 943 00:56:43,190 --> 00:56:45,230 You're finding P. I think that's the main thing. 944 00:56:49,630 --> 00:56:54,440 He's actually saying, if you make these assumptions, 945 00:56:54,440 --> 00:57:03,370 surprisingly, there's only one possible real world or market 946 00:57:03,370 --> 00:57:06,909 beliefs that are consistent with the data 947 00:57:06,909 --> 00:57:07,950 and the assumptions made. 948 00:57:12,240 --> 00:57:15,240 To give you a sense of what the importance of this result 949 00:57:15,240 --> 00:57:19,260 is-- so prior to his paper-- I mean, 950 00:57:19,260 --> 00:57:21,300 people have been interested in trying 951 00:57:21,300 --> 00:57:25,560 to infer from market prices what the market believes. 952 00:57:25,560 --> 00:57:27,860 But they always thought that you had 953 00:57:27,860 --> 00:57:34,300 to supply some parameters that capture market risk aversion. 954 00:57:34,300 --> 00:57:36,100 So for example, common approach is 955 00:57:36,100 --> 00:57:40,520 to assume that you have a representative investor. 956 00:57:40,520 --> 00:57:42,210 And that they have a particular type 957 00:57:42,210 --> 00:57:45,420 of utility function called constant relative risk 958 00:57:45,420 --> 00:57:46,280 aversion. 959 00:57:46,280 --> 00:57:48,375 And there's a parameter in that utility function. 960 00:57:48,375 --> 00:57:50,750 And you had to specify the numerical value that parameter 961 00:57:50,750 --> 00:57:57,800 takes before you could learn the market's beliefs from prices. 962 00:57:57,800 --> 00:58:00,130 And no one ever felt very comfortable 963 00:58:00,130 --> 00:58:02,270 specifying that parameter. 964 00:58:02,270 --> 00:58:10,300 So what Ross essentially did is he managed to essentially do 965 00:58:10,300 --> 00:58:15,080 the identification non-parametrically, where 966 00:58:15,080 --> 00:58:17,330 you don't have to supply any parameters. 967 00:58:17,330 --> 00:58:25,040 And so you essentially just have to buy his assumptions. 968 00:58:25,040 --> 00:58:31,480 You don't have to do any work to actually go from market prices 969 00:58:31,480 --> 00:58:33,070 to market's beliefs. 970 00:58:40,490 --> 00:58:42,160 OK, so let's skip these remarks. 971 00:58:44,790 --> 00:58:45,290 Yeah. 972 00:58:45,290 --> 00:58:49,170 AUDIENCE: Can you elaborate on the fact 973 00:58:49,170 --> 00:58:52,460 that risk aversion does enter in? 974 00:58:55,032 --> 00:58:55,740 PETER CARR: Yeah. 975 00:58:55,740 --> 00:58:57,350 So the exact statement is you don't 976 00:58:57,350 --> 00:59:02,440 have to supply a parameter that describes 977 00:59:02,440 --> 00:59:05,910 the amount of the market's risk aversion. 978 00:59:05,910 --> 00:59:11,060 Rather you have to accept this assumption-- 979 00:59:11,060 --> 00:59:14,030 and I'll show you-- this assumption 980 00:59:14,030 --> 00:59:16,230 about the structure of phi. 981 00:59:16,230 --> 00:59:21,200 OK, so if you just accept that this function of two variables 982 00:59:21,200 --> 00:59:26,220 doesn't have the full amount of degrees of freedom 983 00:59:26,220 --> 00:59:28,670 that an arbitrary function two variables has, 984 00:59:28,670 --> 00:59:30,140 it has a reduced number of degrees 985 00:59:30,140 --> 00:59:33,880 of freedom implicit on the right-hand side. 986 00:59:33,880 --> 00:59:38,140 So remembering that x is actually 987 00:59:38,140 --> 00:59:42,730 just a vector of finite length and so is y, 988 00:59:42,730 --> 00:59:45,460 then think of the left-hand side as having degrees of freedom n 989 00:59:45,460 --> 00:59:47,400 squared. 990 00:59:47,400 --> 00:59:49,310 And on the right-hand side, you're 991 00:59:49,310 --> 00:59:54,070 looking for the numerator function is just 992 00:59:54,070 --> 00:59:56,060 a vector of length n. 993 00:59:56,060 --> 01:00:00,600 And the denominator function is the same function, so 994 01:00:00,600 --> 01:00:01,570 the same vector. 995 01:00:01,570 --> 01:00:04,140 And then there's also this delta. 996 01:00:04,140 --> 01:00:06,330 So let's say on the left-hand side, 997 01:00:06,330 --> 01:00:09,450 you're describing something that without restriction is 998 01:00:09,450 --> 01:00:10,880 of order n squared. 999 01:00:10,880 --> 01:00:14,410 So let's say n is 10, so it has 100 degrees of freedom. 1000 01:00:14,410 --> 01:00:15,890 And on the right-hand side, you're 1001 01:00:15,890 --> 01:00:20,140 describing a vector of length 10 along with a scalar-- 1002 01:00:20,140 --> 01:00:21,800 so 11 degrees of freedom. 1003 01:00:21,800 --> 01:00:23,370 So you have to accept that you're 1004 01:00:23,370 --> 01:00:26,890 willing to before you place any restriction, 1005 01:00:26,890 --> 01:00:28,290 it's 100 degrees of freedom. 1006 01:00:28,290 --> 01:00:29,540 Now you make your restriction. 1007 01:00:29,540 --> 01:00:30,040 It's 11. 1008 01:00:30,040 --> 01:00:31,540 You have to accept that. 1009 01:00:31,540 --> 01:00:39,255 And if you do, then he'll tell you the 11 entries. 1010 01:00:39,255 --> 01:00:39,780 That's it. 1011 01:00:39,780 --> 01:00:41,321 So you don't have to supply anything. 1012 01:00:43,560 --> 01:00:45,310 So I haven't told you how we'll find them. 1013 01:00:45,310 --> 01:00:46,672 That's probably what you're asking-- how the hell 1014 01:00:46,672 --> 01:00:47,590 will you get to 11? 1015 01:00:47,590 --> 01:00:48,960 OK, so I haven't shown you that. 1016 01:00:48,960 --> 01:00:49,460 Yes? 1017 01:00:49,460 --> 01:00:52,750 AUDIENCE: Just really quickly, the c change 1018 01:00:52,750 --> 01:00:55,200 as a function of time and spot price? 1019 01:00:55,200 --> 01:00:58,880 PETER CARR: c is not a function of time, 1020 01:00:58,880 --> 01:01:01,710 to answer your question. 1021 01:01:01,710 --> 01:01:08,910 And then, the argument of c could be a price. 1022 01:01:08,910 --> 01:01:11,150 It's allowed to be a price. 1023 01:01:11,150 --> 01:01:12,710 OK? 1024 01:01:12,710 --> 01:01:16,620 So, that's how you should think of it. 1025 01:01:16,620 --> 01:01:19,670 So there's a lot of time homogeneity in everything 1026 01:01:19,670 --> 01:01:20,540 he does here. 1027 01:01:20,540 --> 01:01:24,000 So he'll never let anything depend on time, actually, 1028 01:01:24,000 --> 01:01:25,030 to answer your question. 1029 01:01:31,820 --> 01:01:36,930 So, I still haven't shown you how he did it. 1030 01:01:36,930 --> 01:01:38,320 He uses Perron-Frobenius theorem. 1031 01:01:41,620 --> 01:01:43,710 I don't actually have slides on how you actually 1032 01:01:43,710 --> 01:01:46,990 calculate the 11 entries. 1033 01:01:46,990 --> 01:01:49,960 So I think I just have to refer you to the paper. 1034 01:01:49,960 --> 01:01:52,860 But he relies on something called 1035 01:01:52,860 --> 01:01:54,250 Perron-Frobenius theorem. 1036 01:01:54,250 --> 01:02:00,010 And I'm going to show you how we-- my co-author 1037 01:02:00,010 --> 01:02:08,040 and I-- actually calculate the analog 1038 01:02:08,040 --> 01:02:09,750 of that 11-dimensional unknown. 1039 01:02:12,840 --> 01:02:16,670 So we're going to work in a continuous setting, where 1040 01:02:16,670 --> 01:02:19,520 instead of looking for a vector and a scalar, 1041 01:02:19,520 --> 01:02:22,080 we're going to look for a function, and a scalar, 1042 01:02:22,080 --> 01:02:23,970 and a function of one variable. 1043 01:02:28,530 --> 01:02:31,120 So you'll get a sense of how to do it from ours. 1044 01:02:31,120 --> 01:02:33,130 And essentially, if you discretize what we do, 1045 01:02:33,130 --> 01:02:34,088 you'll get what he did. 1046 01:02:37,670 --> 01:02:41,220 Let's forget these remarks, and let's forget these. 1047 01:02:41,220 --> 01:02:45,410 And so now, we'll get into some theory 1048 01:02:45,410 --> 01:02:46,910 about changing numeraire. 1049 01:02:46,910 --> 01:02:53,095 So this is a backdrop to how my co-author and I proceed. 1050 01:02:59,280 --> 01:03:02,720 So again, a numeraire is a portfolio whose value 1051 01:03:02,720 --> 01:03:04,790 is always strictly positive. 1052 01:03:04,790 --> 01:03:08,986 And there is a well-developed theory 1053 01:03:08,986 --> 01:03:11,360 in derivatives pricing about how to change the numeraire. 1054 01:03:16,080 --> 01:03:20,500 We're going to use that theory to understand what Ross did. 1055 01:03:20,500 --> 01:03:24,590 So we start with an economy with a so-called money market 1056 01:03:24,590 --> 01:03:25,620 account. 1057 01:03:25,620 --> 01:03:28,650 And so that's a theoretical construct 1058 01:03:28,650 --> 01:03:34,900 that's pretty familiar to most of us, and it's a bank account. 1059 01:03:34,900 --> 01:03:38,160 So we're going to be working now in continuous time. 1060 01:03:38,160 --> 01:03:41,830 So imagine that time, which is continuous, is on this axis. 1061 01:03:41,830 --> 01:03:43,710 And then we're sitting here today, 1062 01:03:43,710 --> 01:03:45,580 and we put some money into the bank. 1063 01:03:45,580 --> 01:03:49,160 And being poor, we only put $1 in. 1064 01:03:49,160 --> 01:03:55,400 So then we ask, looking forward, how will this money in our bank 1065 01:03:55,400 --> 01:03:56,660 change? 1066 01:03:56,660 --> 01:04:01,680 Well, they do still pay a positive interest rate, 1067 01:04:01,680 --> 01:04:04,030 and it's awfully small, but it's positive. 1068 01:04:04,030 --> 01:04:05,920 And so it'll go up. 1069 01:04:05,920 --> 01:04:08,680 And they change the rate actually. 1070 01:04:08,680 --> 01:04:12,290 So now, maybe it's 0.5%, but next week, Chase 1071 01:04:12,290 --> 01:04:15,980 might decide to give you 1%, in which case it goes up faster. 1072 01:04:15,980 --> 01:04:17,940 And then they might the week after give you 2%, 1073 01:04:17,940 --> 01:04:18,705 it goes up faster. 1074 01:04:18,705 --> 01:04:20,530 Then they might go back to 0.5%. 1075 01:04:20,530 --> 01:04:24,150 So that's one possible path for your money market account 1076 01:04:24,150 --> 01:04:25,060 balance. 1077 01:04:25,060 --> 01:04:27,950 And we don't know the future. 1078 01:04:27,950 --> 01:04:30,811 We know how much we're getting over this first little bit 1079 01:04:30,811 --> 01:04:31,310 of time. 1080 01:04:31,310 --> 01:04:33,510 But they could actually decide to pay 1081 01:04:33,510 --> 01:04:37,051 less over the second period, and then the third, or something 1082 01:04:37,051 --> 01:04:37,550 like that. 1083 01:04:37,550 --> 01:04:40,899 OK, so it's increasing and it's random. 1084 01:04:40,899 --> 01:04:42,690 So that's the money market account balance. 1085 01:04:42,690 --> 01:04:48,660 It's considered as an increasing, random process. 1086 01:04:48,660 --> 01:04:50,835 And actually, there's nothing in the math that 1087 01:04:50,835 --> 01:04:55,260 requires it to be increasing if some really cheap bank-- 1088 01:04:55,260 --> 01:04:57,480 like Bank of America tried this actually-- 1089 01:04:57,480 --> 01:05:00,100 charge a negative rate. 1090 01:05:00,100 --> 01:05:03,164 Then it would actually go down with a negative rate. 1091 01:05:03,164 --> 01:05:04,330 But it wouldn't go negative. 1092 01:05:04,330 --> 01:05:06,280 So it's still counts as a numeraire. 1093 01:05:09,240 --> 01:05:11,760 So anyway, that's allowed, as an aside. 1094 01:05:14,439 --> 01:05:16,230 OK, so we've got this money market account. 1095 01:05:19,410 --> 01:05:23,610 So the growth rate is called r, and that's just real-valued. 1096 01:05:23,610 --> 01:05:25,480 And then we also have risky asset. 1097 01:05:25,480 --> 01:05:29,060 So we'll have a total of n risky assets. 1098 01:05:29,060 --> 01:05:32,690 And then we're going to say there's 1099 01:05:32,690 --> 01:05:35,985 no arbitrage between the n risky assets and the one money market 1100 01:05:35,985 --> 01:05:36,485 account. 1101 01:05:43,290 --> 01:05:46,030 The idea is that we look at Bloomberg's prices 1102 01:05:46,030 --> 01:05:49,570 for these n plus 1 assets, we're able to extract 1103 01:05:49,570 --> 01:05:51,670 the Arrow-Debreu security prices. 1104 01:05:51,670 --> 01:05:52,390 That's the idea. 1105 01:05:55,610 --> 01:05:59,960 What I'm assuming is that what we're extracting 1106 01:05:59,960 --> 01:06:04,330 is consistent with the idea that the uncertainty that's 1107 01:06:04,330 --> 01:06:07,070 driving everything here is a diffusion, meaning 1108 01:06:07,070 --> 01:06:15,620 that the uncertainty has sample paths that are continuous, 1109 01:06:15,620 --> 01:06:18,300 but they're allowed to be fairly jagged. 1110 01:06:18,300 --> 01:06:20,520 So diffusions actually have continuous but 1111 01:06:20,520 --> 01:06:22,820 non-differentiable sample paths. 1112 01:06:22,820 --> 01:06:24,600 And we're going to assume that. 1113 01:06:24,600 --> 01:06:26,930 So this is a common assumption. 1114 01:06:26,930 --> 01:06:32,130 This basically got its start here at MIT. 1115 01:06:32,130 --> 01:06:37,750 And diffusions were first used in a finance context 1116 01:06:37,750 --> 01:06:42,090 back in 1965 when both Samuelson and McKean were here. 1117 01:06:42,090 --> 01:06:43,640 So McKean is a probabilist. 1118 01:06:43,640 --> 01:06:47,940 He's now at NYU where I teach, and he's still active. 1119 01:06:47,940 --> 01:06:52,680 And diffusions are widely used. 1120 01:06:52,680 --> 01:06:55,380 So they really got a big boost in 1973 1121 01:06:55,380 --> 01:06:58,897 when Black-Scholes and Merton, who were all here, 1122 01:06:58,897 --> 01:07:01,480 used the diffusion to describe the price of a stock underlying 1123 01:07:01,480 --> 01:07:02,570 an option. 1124 01:07:02,570 --> 01:07:06,440 And since then, they've just been used extensively 1125 01:07:06,440 --> 01:07:07,580 in finance. 1126 01:07:07,580 --> 01:07:12,440 So Merton, who's here, really, I'd say, 1127 01:07:12,440 --> 01:07:14,730 pioneered the use of them in finance. 1128 01:07:29,840 --> 01:07:33,210 So there's this uncertainty X is probably 1129 01:07:33,210 --> 01:07:37,300 mysterious to you, hence the name X. 1130 01:07:37,300 --> 01:07:39,180 So it's like, you get to choose what 1131 01:07:39,180 --> 01:07:40,380 it is, is kind of the idea. 1132 01:07:40,380 --> 01:07:42,400 So this is theory. 1133 01:07:42,400 --> 01:07:46,990 And it's not trying to be overly specific so that you can 1134 01:07:46,990 --> 01:07:50,310 apply it in different contexts. 1135 01:07:50,310 --> 01:07:52,920 But you'd like to know at least some examples, I'm sure. 1136 01:07:52,920 --> 01:07:56,940 So one example would be X is the level of S&P 500. 1137 01:07:56,940 --> 01:08:02,470 A different example would be X is actually an interest rate. 1138 01:08:02,470 --> 01:08:05,650 So let's say the benchmark 30 year yield. 1139 01:08:08,280 --> 01:08:11,710 X could instead be a shorter-term interest rate, 1140 01:08:11,710 --> 01:08:18,200 something called OIS-- overnight index swap-- is 1141 01:08:18,200 --> 01:08:20,540 a possible choice for x. 1142 01:08:23,439 --> 01:08:25,120 When I apply Ross's stuff, that's 1143 01:08:25,120 --> 01:08:28,340 how I choose X, as a short-term interest rate. 1144 01:08:32,850 --> 01:08:36,350 In general, let's say I developed a theory that 1145 01:08:36,350 --> 01:08:38,410 says the short-rate of some function of X. 1146 01:08:38,410 --> 01:08:40,300 And when I actually apply it, the function 1147 01:08:40,300 --> 01:08:41,200 is the identity map. 1148 01:08:51,160 --> 01:08:56,520 The mathematics says that if there's no arbitrage, 1149 01:08:56,520 --> 01:09:00,063 then there exists-- as we're assuming-- then 1150 01:09:00,063 --> 01:09:02,229 there exists this so-called risk-neutral probability 1151 01:09:02,229 --> 01:09:06,890 measure that I talked about earlier and denoted by Q. 1152 01:09:06,890 --> 01:09:09,600 It's related but not equal to the Arrow-Debreu security 1153 01:09:09,600 --> 01:09:10,439 prices. 1154 01:09:10,439 --> 01:09:14,700 So if you were to just imagine that instead 1155 01:09:14,700 --> 01:09:17,620 of buying these Arrow-Debreu security prices in a spot 1156 01:09:17,620 --> 01:09:20,350 market, if you instead bought them in a forward market 1157 01:09:20,350 --> 01:09:23,050 where you actually pay when they mature, 1158 01:09:23,050 --> 01:09:27,310 then those Arrow-Debreu security prices in the forward market 1159 01:09:27,310 --> 01:09:31,609 would be Q. So Q and A are really close. 1160 01:09:31,609 --> 01:09:36,392 So the measure A need not integrate to one, 1161 01:09:36,392 --> 01:09:38,350 and that's just due to the time value of money, 1162 01:09:38,350 --> 01:09:39,935 and that's because you're paying in the spot market. 1163 01:09:39,935 --> 01:09:41,590 If you're actually paying in the forward market, 1164 01:09:41,590 --> 01:09:43,881 then you don't have to worry about time value of money. 1165 01:09:43,881 --> 01:09:46,422 And so then, the measure Q does integrate to one. 1166 01:09:46,422 --> 01:09:48,380 So that's why we call it a probability measure. 1167 01:09:53,300 --> 01:09:55,570 Under this probability measure Q, 1168 01:09:55,570 --> 01:09:59,290 the expected return on all assets is the risk-free rate. 1169 01:09:59,290 --> 01:10:02,910 So that's what that actually says, although you're probably 1170 01:10:02,910 --> 01:10:05,350 not seeing that this is literally the expected 1171 01:10:05,350 --> 01:10:08,340 return-- well more precisely, it's expected price change. 1172 01:10:08,340 --> 01:10:09,810 So the expected price change is-- 1173 01:10:09,810 --> 01:10:12,462 what that means is expected price change is 1174 01:10:12,462 --> 01:10:13,920 the risk-free rate times the price. 1175 01:10:13,920 --> 01:10:16,510 That's what that says. 1176 01:10:16,510 --> 01:10:18,550 So if you divide both sides by the spot price 1177 01:10:18,550 --> 01:10:21,290 when it's positive, then you'll get the expected return 1178 01:10:21,290 --> 01:10:23,170 is equal to risk-free rate. 1179 01:10:23,170 --> 01:10:25,280 And we're doing things in continuous time here. 1180 01:10:25,280 --> 01:10:27,960 So we're working with diffusions. 1181 01:10:27,960 --> 01:10:30,420 And you may or may not have been introduced 1182 01:10:30,420 --> 01:10:33,260 to diffusions at this stage in your mathematical career. 1183 01:10:33,260 --> 01:10:36,590 But mathematically, one way to describe diffusion 1184 01:10:36,590 --> 01:10:39,500 is via the infinitesimal generator. 1185 01:10:39,500 --> 01:10:42,750 So this is a differential operator 1186 01:10:42,750 --> 01:10:46,620 that's first order in time, second order in space. 1187 01:10:46,620 --> 01:10:50,090 And let's just say this is formally how mathematicians 1188 01:10:50,090 --> 01:10:53,250 think about this type of thing. 1189 01:10:53,250 --> 01:10:56,370 What I've drawn here is a single sample path of diffusion. 1190 01:10:56,370 --> 01:10:58,830 There's definitely possibility of other sample paths. 1191 01:10:58,830 --> 01:11:00,788 These actually are an infinite number of paths. 1192 01:11:00,788 --> 01:11:04,760 But they're all continuous and nowhere differentiable. 1193 01:11:22,454 --> 01:11:24,120 I want to just kind of give you a flavor 1194 01:11:24,120 --> 01:11:27,350 of how you change numeraires. 1195 01:11:27,350 --> 01:11:29,850 So we started with the numeraire being the money 1196 01:11:29,850 --> 01:11:31,530 market account-- this guy. 1197 01:11:31,530 --> 01:11:33,289 And the idea is we're going to switch 1198 01:11:33,289 --> 01:11:34,330 to a different numeraire. 1199 01:11:39,640 --> 01:11:42,790 What we're mainly interested in figuring out 1200 01:11:42,790 --> 01:11:58,960 is what are the drifts of assets when we measure their values 1201 01:11:58,960 --> 01:12:00,530 in a different numeraire. 1202 01:12:00,530 --> 01:12:04,480 So I've kind of given you a sense of what this is about. 1203 01:12:04,480 --> 01:12:11,250 So you could hold IBM, and every time you get a gain, 1204 01:12:11,250 --> 01:12:14,790 you could put that gain in your local bank-- Chase-- 1205 01:12:14,790 --> 01:12:18,660 and see how fast your bank balance grows 1206 01:12:18,660 --> 01:12:23,130 as you're putting all your gains in IBM in the bank. 1207 01:12:23,130 --> 01:12:28,630 And you'll get a certain growth rate from that strategy. 1208 01:12:28,630 --> 01:12:30,980 Now, you could try a different strategy where 1209 01:12:30,980 --> 01:12:32,670 you take your gains from IBM. 1210 01:12:32,670 --> 01:12:36,600 And you actually ship them off over to a British bank, which 1211 01:12:36,600 --> 01:12:40,000 is denominated in pounds, and see how fast 1212 01:12:40,000 --> 01:12:41,960 that bank balance grows. 1213 01:12:41,960 --> 01:12:45,160 And there's no reason that the two bank balances-- 1214 01:12:45,160 --> 01:12:46,800 the American one and the British one-- 1215 01:12:46,800 --> 01:12:48,580 need to grow at the same rate. 1216 01:12:48,580 --> 01:12:51,940 Because they're denominated in different currencies. 1217 01:12:51,940 --> 01:12:55,410 So we're basically interested to know, 1218 01:12:55,410 --> 01:12:57,540 given that we know how fast, let's say, 1219 01:12:57,540 --> 01:12:59,830 the American bank balance would grow, 1220 01:12:59,830 --> 01:13:01,940 we want to know how fast the British bank 1221 01:13:01,940 --> 01:13:03,380 balance would grow. 1222 01:13:03,380 --> 01:13:07,130 And what affects the growth rate of the British bank balance 1223 01:13:07,130 --> 01:13:09,770 is the covariance, actually, between 1224 01:13:09,770 --> 01:13:15,450 the dollar/pound exchange rate and IBM. 1225 01:13:15,450 --> 01:13:17,510 So remember, we're investing in IBM 1226 01:13:17,510 --> 01:13:20,030 and we're putting gains in either an American bank 1227 01:13:20,030 --> 01:13:22,340 or a British bank. 1228 01:13:22,340 --> 01:13:24,550 So IBM stock prices in dollars. 1229 01:13:24,550 --> 01:13:28,110 And so there's no issues with putting IBM's gains 1230 01:13:28,110 --> 01:13:30,400 in an American bank. 1231 01:13:30,400 --> 01:13:31,960 But there's actually a subtle effect 1232 01:13:31,960 --> 01:13:35,490 that happens when you put IBM's gains in a British bank, which 1233 01:13:35,490 --> 01:13:37,830 the subtle effect is there's this random exchange rate 1234 01:13:37,830 --> 01:13:39,240 dollars per pound. 1235 01:13:39,240 --> 01:13:43,730 And suppose that there's some correlation, for whatever 1236 01:13:43,730 --> 01:13:49,440 reason, between dollars per pound and IBM. 1237 01:13:49,440 --> 01:13:52,450 So suppose the correlation's the following form-- 1238 01:13:52,450 --> 01:13:54,780 every time IBM goes up, the dollar 1239 01:13:54,780 --> 01:13:57,540 gets weaker against the pound. 1240 01:13:57,540 --> 01:14:01,800 So in other words, what happens is IBM goes up, you go hooray, 1241 01:14:01,800 --> 01:14:02,910 I'm rich. 1242 01:14:02,910 --> 01:14:04,370 I got all these dollars. 1243 01:14:04,370 --> 01:14:07,660 I'm going to go put them in a British bank account. 1244 01:14:11,390 --> 01:14:16,280 But suppose, unluckily for you, every time IBM goes up, 1245 01:14:16,280 --> 01:14:18,180 the dollar weakens against the pound. 1246 01:14:18,180 --> 01:14:25,500 And so, you cannot buy so many pounds as a result. 1247 01:14:25,500 --> 01:14:28,780 So contrast that with the opposite situation where 1248 01:14:28,780 --> 01:14:31,170 when IBM goes up, the dollar strengthens as opposed 1249 01:14:31,170 --> 01:14:32,310 to weakens. 1250 01:14:32,310 --> 01:14:35,220 Then you can buy lots and lots of pounds with your IBM gains. 1251 01:14:37,820 --> 01:14:40,600 So the correlation between the dollar pound exchange 1252 01:14:40,600 --> 01:14:44,510 rate and IBM affects how fast your British bank 1253 01:14:44,510 --> 01:14:45,980 balance would grow. 1254 01:14:45,980 --> 01:14:48,950 And that's actually like the key point. 1255 01:14:57,440 --> 01:14:59,840 So this would be well-known to anybody-- 1256 01:14:59,840 --> 01:15:01,090 especially an FX client. 1257 01:15:05,210 --> 01:15:12,000 So what we're actually going to do 1258 01:15:12,000 --> 01:15:16,550 is find a numeraire such that the growth 1259 01:15:16,550 --> 01:15:21,080 rate of the balance in that numeraire 1260 01:15:21,080 --> 01:15:26,030 is actually the real-world drift of the underlying. 1261 01:15:26,030 --> 01:15:30,200 So the idea is let's say that I told you 1262 01:15:30,200 --> 01:15:32,660 at the beginning of this talk that historically stocks 1263 01:15:32,660 --> 01:15:34,340 grow at 9% on average. 1264 01:15:38,560 --> 01:15:40,900 Our starting point here in this part of the talk 1265 01:15:40,900 --> 01:15:45,680 is that we're starting from this risk-neutral measure Q, which, 1266 01:15:45,680 --> 01:15:49,220 by definition, is the property that stocks would grow only 1267 01:15:49,220 --> 01:15:49,720 at 1%. 1268 01:15:52,680 --> 01:15:54,390 So what we're actually going to do 1269 01:15:54,390 --> 01:16:00,980 is go find some numeraire which will be correlated 1270 01:16:00,980 --> 01:16:08,890 with the stocks, such that when we put our stock 1271 01:16:08,890 --> 01:16:13,590 gains in that numeraire, we end up growing at 9%, 1272 01:16:13,590 --> 01:16:14,215 rather than 1%. 1273 01:16:20,080 --> 01:16:25,060 That's the way we think about things. 1274 01:16:25,060 --> 01:16:28,136 And the key is to find that numeraire that 1275 01:16:28,136 --> 01:16:28,885 has that property. 1276 01:16:31,940 --> 01:16:33,450 I'm going to go fast now-- there's 1277 01:16:33,450 --> 01:16:40,130 a paper by John Long where he shows that that numeraire that 1278 01:16:40,130 --> 01:16:45,130 converts a risk-free growth rate into the real-world growth rate 1279 01:16:45,130 --> 01:16:46,390 always exists. 1280 01:16:46,390 --> 01:16:48,890 And he gave it a name, and he called it numeraire portfolio. 1281 01:16:48,890 --> 01:16:51,250 It has another name-- growth optimal portfolio-- 1282 01:16:51,250 --> 01:16:52,670 that Kelly was talking about. 1283 01:17:03,670 --> 01:17:06,670 So there's a reference if you're interested in following up 1284 01:17:06,670 --> 01:17:09,280 on this material. 1285 01:17:12,320 --> 01:17:17,434 So the theory says that there always 1286 01:17:17,434 --> 01:17:19,100 exists this numeraire called John Long's 1287 01:17:19,100 --> 01:17:21,170 numeraire portfolio, such that if you park 1288 01:17:21,170 --> 01:17:25,530 your gains in this numeraire, you end up growing 1289 01:17:25,530 --> 01:17:27,210 at the real-world drift. 1290 01:17:27,210 --> 01:17:31,180 And so, let's say all we got to do 1291 01:17:31,180 --> 01:17:35,460 to find that real-world drift is go find this special numeraire. 1292 01:17:39,310 --> 01:17:43,140 So this part of the talk is about making some assumptions 1293 01:17:43,140 --> 01:17:48,340 that lead to an identification of that particular numeraire-- 1294 01:17:48,340 --> 01:17:49,260 John Long's numeraire. 1295 01:17:54,680 --> 01:17:57,180 We're going to continue to work with diffusions. 1296 01:17:57,180 --> 01:18:03,090 And now we're going to also impose time homogeneity 1297 01:18:03,090 --> 01:18:04,610 like Ross was doing. 1298 01:18:04,610 --> 01:18:07,500 So let's say when I was just talking about numeraire, 1299 01:18:07,500 --> 01:18:09,400 I was allowing time inhomogeneity. 1300 01:18:09,400 --> 01:18:12,160 But now we're going to go time homogeneous. 1301 01:18:14,472 --> 01:18:16,430 I haven't really been introducing the notation, 1302 01:18:16,430 --> 01:18:19,760 but a(x, t) is the diffusion coefficient of the state 1303 01:18:19,760 --> 01:18:20,940 variable x. 1304 01:18:20,940 --> 01:18:24,330 And now it's just being assumed to be a function of x only. 1305 01:18:24,330 --> 01:18:27,180 So b^Q(x, t) was the drift coefficient of x. 1306 01:18:27,180 --> 01:18:29,570 And now it's a function of x only. 1307 01:18:29,570 --> 01:18:32,820 r(x, t) was the function linking the short interest rate 1308 01:18:32,820 --> 01:18:34,030 to the state variable x. 1309 01:18:34,030 --> 01:18:36,310 And now, it's a function of x only. 1310 01:18:36,310 --> 01:18:40,025 And finally, sigma_L(x, t) was the volatility of John 1311 01:18:40,025 --> 01:18:41,195 Long's numeraire portfolio. 1312 01:18:41,195 --> 01:18:42,820 And again, that's a function of x only. 1313 01:18:45,640 --> 01:18:48,330 So anyway, another assumption that we're 1314 01:18:48,330 --> 01:18:51,110 going to impose now in order to determine uniquely what 1315 01:18:51,110 --> 01:18:53,460 this numeraire portfolio value is 1316 01:18:53,460 --> 01:18:57,850 is to require that the diffusion that's driving everything 1317 01:18:57,850 --> 01:18:59,510 live in a bounded interval. 1318 01:18:59,510 --> 01:19:04,140 So essentially, the sample paths all 1319 01:19:04,140 --> 01:19:07,600 have to be bounded below by some constant, which 1320 01:19:07,600 --> 01:19:09,240 could be negative, and have to be 1321 01:19:09,240 --> 01:19:12,170 bounded above by some constant, which again could be negative. 1322 01:19:17,800 --> 01:19:20,010 We make all those assumptions, and we move on. 1323 01:19:20,010 --> 01:19:23,140 And so in the end, what have we been assuming? 1324 01:19:23,140 --> 01:19:27,029 So we're assuming that there's a single source of uncertainty X. 1325 01:19:27,029 --> 01:19:28,820 And that it's a time-homogeneous diffusion. 1326 01:19:28,820 --> 01:19:30,950 So that's this middle equation here. 1327 01:19:30,950 --> 01:19:34,790 And so that says changes in X have a predictable part, 1328 01:19:34,790 --> 01:19:36,680 which is b^Q(X) dt. 1329 01:19:36,680 --> 01:19:40,300 And they have an unpredictable part, which is a of X dW. 1330 01:19:40,300 --> 01:19:43,530 So W there is standard Brownian motion. 1331 01:19:43,530 --> 01:19:46,950 And since I'm big on mnemonics, you 1332 01:19:46,950 --> 01:19:49,950 might ask why does W stand for standard Brownian motion? 1333 01:19:49,950 --> 01:19:52,080 And that's because W actually stands 1334 01:19:52,080 --> 01:19:54,572 for Wiener process-- Norbert Wiener being an MIT 1335 01:19:54,572 --> 01:19:55,155 mathematician. 1336 01:20:00,430 --> 01:20:03,920 And the W is a standard notation for this kind of thing. 1337 01:20:07,920 --> 01:20:11,360 As an aside, when Bob Merton was here working out all this stuff 1338 01:20:11,360 --> 01:20:14,190 for the first time in the late '60s, 1339 01:20:14,190 --> 01:20:16,650 he knew the standard notation for standard Brownian motion 1340 01:20:16,650 --> 01:20:18,920 was W. But it turns out in finance, 1341 01:20:18,920 --> 01:20:22,380 the standard notation for wealth is also W. 1342 01:20:22,380 --> 01:20:25,460 And he wanted to work on stochastic wealth dynamics. 1343 01:20:25,460 --> 01:20:28,200 And so he had to choose should I use the letter W for wealth, 1344 01:20:28,200 --> 01:20:30,790 or should I use the letter W for Wiener process? 1345 01:20:30,790 --> 01:20:33,090 And he chose W for wealth, which meant 1346 01:20:33,090 --> 01:20:35,730 he had to pick a different letter for Wiener process. 1347 01:20:35,730 --> 01:20:40,000 And so he actually chose the letter Z. 1348 01:20:40,000 --> 01:20:42,017 And you'll have to ask him why he chose 1349 01:20:42,017 --> 01:20:44,600 that letter, because it doesn't stand for anything as far as I 1350 01:20:44,600 --> 01:20:48,290 know, except that actually the sample paths of a Wiener 1351 01:20:48,290 --> 01:20:51,770 process look very jagged, so if you turn your head, 1352 01:20:51,770 --> 01:20:59,570 you might be able to see a Z. 1353 01:20:59,570 --> 01:21:01,860 So another assumption is that we're 1354 01:21:01,860 --> 01:21:04,590 going to restrict the possible dynamics of the numeraire 1355 01:21:04,590 --> 01:21:06,140 portfolio's value. 1356 01:21:06,140 --> 01:21:08,910 So we're going to let L denote the value 1357 01:21:08,910 --> 01:21:10,140 of this numeraire portfolio. 1358 01:21:10,140 --> 01:21:13,910 And the mnemonic here is that John Long 1359 01:21:13,910 --> 01:21:17,030 invented this concept, so we're calling it L for Long. 1360 01:21:17,030 --> 01:21:19,380 Now it's unfortunate that the inventor of this concept 1361 01:21:19,380 --> 01:21:20,540 was named Long, actually. 1362 01:21:20,540 --> 01:21:24,310 Because in finance, the word "long" indicates 1363 01:21:24,310 --> 01:21:30,935 that for a security with a non-negative payoff, 1364 01:21:30,935 --> 01:21:32,310 if you're long, then you're going 1365 01:21:32,310 --> 01:21:33,604 to be receiving that payoff. 1366 01:21:33,604 --> 01:21:36,020 As you pay money now, you're going to receive that payoff. 1367 01:21:36,020 --> 01:21:37,650 It's the opposite of short, where 1368 01:21:37,650 --> 01:21:40,750 if you're short a security with a non-negative payoff, 1369 01:21:40,750 --> 01:21:43,060 then actually you get money now and you have 1370 01:21:43,060 --> 01:21:47,000 to deliver that payoff later. 1371 01:21:47,000 --> 01:21:51,620 So as it happens, this numeraire portfolio 1372 01:21:51,620 --> 01:21:54,010 has multiple positions in it. 1373 01:21:54,010 --> 01:21:55,510 And the signs of the positions are 1374 01:21:55,510 --> 01:21:59,190 allowed to be real-- so positives and negatives. 1375 01:21:59,190 --> 01:22:02,140 So it's kind of a misnomer. 1376 01:22:02,140 --> 01:22:03,592 I say Long's numeraire portfolio, 1377 01:22:03,592 --> 01:22:05,300 and everyone thinks the positions in them 1378 01:22:05,300 --> 01:22:06,136 are all positive. 1379 01:22:06,136 --> 01:22:07,760 It's not true-- so they're real-valued. 1380 01:22:12,160 --> 01:22:15,810 The kind of problem here is that we've 1381 01:22:15,810 --> 01:22:18,460 put the structure on the value L, John Long's 1382 01:22:18,460 --> 01:22:32,690 numeraire portfolio, namely that L is a continuous process, 1383 01:22:32,690 --> 01:22:37,452 but it's not quite a diffusion in itself. 1384 01:22:37,452 --> 01:22:41,550 The only thing you can say is that the pair X and L 1385 01:22:41,550 --> 01:22:44,179 are a bivariate diffusion. 1386 01:22:44,179 --> 01:22:45,720 If you bring this L over to the side, 1387 01:22:45,720 --> 01:22:50,160 you can see the coefficients for dL depend on L and X-- 1388 01:22:50,160 --> 01:22:53,230 and same thing with the volatility part. 1389 01:22:53,230 --> 01:22:56,770 So anyway, we place the structure. 1390 01:22:56,770 --> 01:23:02,290 And the idea is that we know, from looking at Bloomberg, 1391 01:23:02,290 --> 01:23:06,950 what the risk-neutral drift of X is-- that's b^Q(X). 1392 01:23:06,950 --> 01:23:08,160 We know that function. 1393 01:23:08,160 --> 01:23:10,150 We know what the diffusion coefficient of X is. 1394 01:23:10,150 --> 01:23:11,830 That's the function A of X. 1395 01:23:11,830 --> 01:23:14,250 We know what the risk-neutral drift of L 1396 01:23:14,250 --> 01:23:16,820 is-- that's that function r of X. 1397 01:23:16,820 --> 01:23:20,387 But we don't know the volatility of John Long's 1398 01:23:20,387 --> 01:23:21,220 numeraire portfolio. 1399 01:23:21,220 --> 01:23:23,540 That's the function sigma_L of X. 1400 01:23:23,540 --> 01:23:27,740 And if only we could find it, we would actually 1401 01:23:27,740 --> 01:23:32,520 know how to determine the real-world drift. 1402 01:23:32,520 --> 01:23:38,810 And remember I was saying if IBM and you could 1403 01:23:38,810 --> 01:23:43,320 put an American bank account, and let's say there 1404 01:23:43,320 --> 01:23:44,660 was certain growth rate there. 1405 01:23:44,660 --> 01:23:46,660 And then if instead you were putting those gains 1406 01:23:46,660 --> 01:23:49,320 in a British bank account, you'd achieve a different growth 1407 01:23:49,320 --> 01:23:50,220 rate. 1408 01:23:50,220 --> 01:23:58,070 And I was stressing that the correlation of dollar/pound 1409 01:23:58,070 --> 01:24:01,310 with IBM was important for determining that growth rate. 1410 01:24:01,310 --> 01:24:03,370 And I stand by that. 1411 01:24:03,370 --> 01:24:05,070 When you're in a one-factor world, 1412 01:24:05,070 --> 01:24:07,280 that correlation can only be one. 1413 01:24:07,280 --> 01:24:09,290 And so that's what's happening here. 1414 01:24:09,290 --> 01:24:12,500 We're in a one-factor world, and that correlation is one. 1415 01:24:12,500 --> 01:24:18,550 And the other thing that affects the growth rate, though, 1416 01:24:18,550 --> 01:24:21,340 of your British bank account balance 1417 01:24:21,340 --> 01:24:23,710 is actually the volatility of the exchange rate. 1418 01:24:23,710 --> 01:24:27,060 So what actually matters is the covariance 1419 01:24:27,060 --> 01:24:30,940 between the British exchange rate and IBM. 1420 01:24:30,940 --> 01:24:33,260 That covariance depends on both the correlation 1421 01:24:33,260 --> 01:24:39,415 and the volatility of the FX rate. 1422 01:24:39,415 --> 01:24:41,540 So you can think of the FX rate as here John Long's 1423 01:24:41,540 --> 01:24:42,820 numeraire portfolio. 1424 01:24:42,820 --> 01:24:46,546 And so that sigma_L is sort of the key. 1425 01:24:46,546 --> 01:24:48,920 It's like we've set things up so we know the correlation, 1426 01:24:48,920 --> 01:24:51,840 but we still don't know the covariance. 1427 01:24:51,840 --> 01:24:53,340 And that's what's actually relevant. 1428 01:24:53,340 --> 01:24:54,940 So as soon as we get the sigma_L, 1429 01:24:54,940 --> 01:24:56,023 we'll know the covariance. 1430 01:24:58,100 --> 01:24:59,000 So we'll be in shape. 1431 01:25:01,550 --> 01:25:05,279 So we got to find that volatility function sigma_L. 1432 01:25:05,279 --> 01:25:06,945 And now I know many of you have classes, 1433 01:25:06,945 --> 01:25:08,790 so I'm going to have to start moving. 1434 01:25:08,790 --> 01:25:11,840 AUDIENCE: Now, Peter, people will have access 1435 01:25:11,840 --> 01:25:13,563 to these slides afterwards. 1436 01:25:13,563 --> 01:25:16,477 And so, I'm just seeing you've got another 15 slides left. 1437 01:25:16,477 --> 01:25:18,060 PETER CARR: Yes, well actually, you'll 1438 01:25:18,060 --> 01:25:20,185 be glad to know that five of those are disclaimers. 1439 01:25:23,800 --> 01:25:24,970 If I could move along-- 1440 01:25:24,970 --> 01:25:27,020 AUDIENCE: But the point is to what-- 1441 01:25:27,020 --> 01:25:28,870 PETER CARR: The key is towards the end. 1442 01:25:28,870 --> 01:25:30,100 Yes, absolutely. 1443 01:25:30,100 --> 01:25:30,810 We're very close. 1444 01:25:30,810 --> 01:25:31,310 OK. 1445 01:25:31,310 --> 01:25:34,160 So I'll be done in two minutes. 1446 01:25:34,160 --> 01:25:36,155 So basically, where we are now is 1447 01:25:36,155 --> 01:25:37,780 we're going to make one more assumption 1448 01:25:37,780 --> 01:25:39,821 that the value of John Long's numeraire portfolio 1449 01:25:39,821 --> 01:25:41,929 is a function of X and D. OK then, let's say 1450 01:25:41,929 --> 01:25:43,220 we've made all our assumptions. 1451 01:25:43,220 --> 01:25:46,310 And where it goes is that the assumptions 1452 01:25:46,310 --> 01:25:49,310 imply that this value function splits 1453 01:25:49,310 --> 01:25:51,300 into an unknown positive function of x, 1454 01:25:51,300 --> 01:25:53,500 and an unknown positive function of time. 1455 01:25:53,500 --> 01:25:56,990 And when you kind of further analyze, 1456 01:25:56,990 --> 01:26:00,150 you find that the unknown function of time 1457 01:26:00,150 --> 01:26:02,000 is an exponential function of time. 1458 01:26:02,000 --> 01:26:06,050 And the unknown function of x solves an ordinary differential 1459 01:26:06,050 --> 01:26:11,000 equation of this kind. 1460 01:26:11,000 --> 01:26:13,510 So this is called a Sturm-Liouville problem. 1461 01:26:13,510 --> 01:26:16,415 And it turns out that Sturm and Liouville 1462 01:26:16,415 --> 01:26:18,915 were the only mathematicians I've mentioned in this talk who 1463 01:26:18,915 --> 01:26:20,150 were not at MIT. 1464 01:26:20,150 --> 01:26:24,300 And they actually solved this problem. 1465 01:26:24,300 --> 01:26:27,400 And one of the things they show is 1466 01:26:27,400 --> 01:26:31,170 that when you're searching for functions pi and scalars 1467 01:26:31,170 --> 01:26:34,320 lambda that solve this problem, there's 1468 01:26:34,320 --> 01:26:40,580 only one solution that delivers you a positive function pi. 1469 01:26:40,580 --> 01:26:42,240 And so this is how you get uniqueness. 1470 01:26:45,410 --> 01:26:47,690 Remember I was saying back with 11-- 1471 01:26:47,690 --> 01:26:50,680 so we're searching for like a 10-vector and a scalar. 1472 01:26:50,680 --> 01:26:52,340 Now the 10 vector is a function. 1473 01:26:52,340 --> 01:26:56,320 And that function is pi, and the scalar's lambda. 1474 01:26:56,320 --> 01:26:58,072 So the point is is that the math implies 1475 01:26:58,072 --> 01:26:59,780 there's a unique solution to the problem. 1476 01:26:59,780 --> 01:27:02,640 So we learn the volatility of the numeraire portfolio 1477 01:27:02,640 --> 01:27:03,240 in the end. 1478 01:27:03,240 --> 01:27:05,690 And then we learn the drifts of everything 1479 01:27:05,690 --> 01:27:08,280 you want to know under the market's beliefs. 1480 01:27:08,280 --> 01:27:11,230 So that's the gist of it. 1481 01:27:11,230 --> 01:27:13,720 So then there's been work on trying to extend 1482 01:27:13,720 --> 01:27:16,210 to unbounded intervals. 1483 01:27:16,210 --> 01:27:19,750 And basically, in the famous Black-Scholes model, 1484 01:27:19,750 --> 01:27:24,010 this effort fails, whereas in the less famous but still 1485 01:27:24,010 --> 01:27:28,000 important Cox-Ingersoll-Ross model, this effort succeeds. 1486 01:27:28,000 --> 01:27:31,270 So the sort of punchline is that when 1487 01:27:31,270 --> 01:27:36,040 it comes to unbounded state space, the theory's open. 1488 01:27:36,040 --> 01:27:38,090 So if there's a grad student in the room who 1489 01:27:38,090 --> 01:27:40,660 wants a good dissertation problem, this is it. 1490 01:27:40,660 --> 01:27:41,160 OK. 1491 01:27:41,160 --> 01:27:43,410 So that's all I wanted to say today. 1492 01:27:43,410 --> 01:27:44,960 Thanks.