1 00:00:00,070 --> 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 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:21,037 --> 00:00:22,870 DENIS GOROKHOV: So I work at Morgan Stanley. 9 00:00:22,870 --> 00:00:27,630 I run corporate treasury strategies at Morgan Stanley. 10 00:00:27,630 --> 00:00:29,580 So corporate treasury is the business unit 11 00:00:29,580 --> 00:00:32,195 that is responsible for issuing and risk management 12 00:00:32,195 --> 00:00:35,500 of Morgan Stanley debt. 13 00:00:35,500 --> 00:00:39,770 I also run desk strategies own the New York inflation desk. 14 00:00:39,770 --> 00:00:43,100 That's the business which is a part of the global interest 15 00:00:43,100 --> 00:00:47,970 rate business, which is responsible for trading 16 00:00:47,970 --> 00:00:49,870 derivatives linked to inflation. 17 00:00:49,870 --> 00:00:53,560 And today, I'm going to talk about the HJM model. 18 00:00:53,560 --> 00:00:59,600 So HJM model-- the abbreviation stands for Heath-Jarrow-Morton, 19 00:00:59,600 --> 00:01:03,730 these three individuals who discovered this framework 20 00:01:03,730 --> 00:01:05,620 in the beginning of 1990s. 21 00:01:05,620 --> 00:01:08,580 And this is a very general framework 22 00:01:08,580 --> 00:01:13,180 for pricing derivatives to interest rates and to credit. 23 00:01:13,180 --> 00:01:18,290 So on Wall Street, big banks make a substantial amount 24 00:01:18,290 --> 00:01:21,090 of money by trading all kinds of exotic products, 25 00:01:21,090 --> 00:01:22,190 exotic derivatives. 26 00:01:22,190 --> 00:01:26,130 And big banks like Morgan Stanley, 27 00:01:26,130 --> 00:01:31,130 like Goldman, JP Morgan-- trades thousands and thousands 28 00:01:31,130 --> 00:01:33,670 of different types of exotic derivatives. 29 00:01:33,670 --> 00:01:40,440 So a typical problem which the business faces 30 00:01:40,440 --> 00:01:45,570 is that new types of derivatives arrive all the time. 31 00:01:45,570 --> 00:01:47,470 So you need to be able to respond quickly 32 00:01:47,470 --> 00:01:50,030 to the demand from the clients. 33 00:01:50,030 --> 00:01:52,360 And you need to be able not just to tell 34 00:01:52,360 --> 00:01:53,700 the price of derivative. 35 00:01:53,700 --> 00:01:59,240 You need to be able also to risk manage this derivative. 36 00:01:59,240 --> 00:02:01,280 Because let's say if you sold an option, 37 00:02:01,280 --> 00:02:03,090 you've got some premium, if something 38 00:02:03,090 --> 00:02:05,990 goes not in your favor, you need to pay in the end. 39 00:02:05,990 --> 00:02:09,550 So you need to be able to hedge. 40 00:02:09,550 --> 00:02:13,590 And you can think about the HJM model, 41 00:02:13,590 --> 00:02:16,890 like this kind of framework, as something 42 00:02:16,890 --> 00:02:19,590 which is similar to theoretical physics in a way, right? 43 00:02:19,590 --> 00:02:26,000 So you get beautiful models-- it's like a solvable model. 44 00:02:26,000 --> 00:02:30,600 For example, let's say the hydrogen atom 45 00:02:30,600 --> 00:02:32,430 in quantum mechanics. 46 00:02:32,430 --> 00:02:34,890 So it's relatively straightforward to solve it, 47 00:02:34,890 --> 00:02:35,660 right? 48 00:02:35,660 --> 00:02:39,810 So we have an equation, which can be exactly solved. 49 00:02:39,810 --> 00:02:42,975 And we can find energy levels and understand this 50 00:02:42,975 --> 00:02:43,600 fairly quickly. 51 00:02:43,600 --> 00:02:48,330 But if you start going into more complex problems-- for example, 52 00:02:48,330 --> 00:02:49,970 you add one more electron and you 53 00:02:49,970 --> 00:02:53,090 have a helium atom-- it's already much more complicated. 54 00:02:53,090 --> 00:02:56,170 And then if you have complicated atoms or even molecules, 55 00:02:56,170 --> 00:02:57,620 it's unclear what to do. 56 00:02:57,620 --> 00:03:01,540 So people came up with approximate kind 57 00:03:01,540 --> 00:03:06,150 of methods, which allow nevertheless solve everything 58 00:03:06,150 --> 00:03:09,070 very accurately numerically. 59 00:03:09,070 --> 00:03:11,400 And HJM is a similar framework. 60 00:03:11,400 --> 00:03:16,050 So you can-- it allows to price all kinds 61 00:03:16,050 --> 00:03:17,920 of [INAUDIBLE] derivatives. 62 00:03:17,920 --> 00:03:26,150 And so it's very general. 63 00:03:26,150 --> 00:03:28,450 It's very flexible to incorporate new payoffs, 64 00:03:28,450 --> 00:03:32,010 all kinds of correlation between products and so on, so forth. 65 00:03:32,010 --> 00:03:38,130 And this HJM model-- [INAUDIBLE] natural [INAUDIBLE] 66 00:03:38,130 --> 00:03:41,830 more general framework like Monte Carlo simulation. 67 00:03:41,830 --> 00:03:46,135 And before actually going into details 68 00:03:46,135 --> 00:03:49,610 of pricing exotic interest rates and credit derivatives, 69 00:03:49,610 --> 00:03:52,190 let me just first explain how this framework appears 70 00:03:52,190 --> 00:03:55,420 in the most common type of derivatives, basically 71 00:03:55,420 --> 00:03:57,400 equity-linked product. 72 00:03:57,400 --> 00:03:59,610 So like a very, very simple example, right? 73 00:03:59,610 --> 00:04:04,580 So let's say if we have a derivatives desk at some firm, 74 00:04:04,580 --> 00:04:06,680 and they sell all kinds of products. 75 00:04:06,680 --> 00:04:08,720 Of course, ideally, let's say there's 76 00:04:08,720 --> 00:04:10,740 a client who wants to buy something from you. 77 00:04:10,740 --> 00:04:13,480 Of course, the easiest approach would be to find the client 78 00:04:13,480 --> 00:04:20,790 and do an opposite transaction with him, so that you're market 79 00:04:20,790 --> 00:04:22,695 neutral, at least in theory. 80 00:04:22,695 --> 00:04:24,070 So if you don't take into account 81 00:04:24,070 --> 00:04:25,290 counterparties and so on. 82 00:04:25,290 --> 00:04:27,740 However, it's rather difficult in general, 83 00:04:27,740 --> 00:04:30,590 so the portfolios are very complicated. 84 00:04:30,590 --> 00:04:33,170 And there's always some residual risk. 85 00:04:33,170 --> 00:04:35,700 So this is the cause of dynamic hedging. 86 00:04:35,700 --> 00:04:38,660 So for this example, very simple example, 87 00:04:38,660 --> 00:04:42,350 a dealer just sold a call option on a stock. 88 00:04:42,350 --> 00:04:44,800 And if you do this, then in principle, 89 00:04:44,800 --> 00:04:47,050 the amount of money which you can lose is unlimited. 90 00:04:47,050 --> 00:04:53,010 So you need to be able to hedge dynamically 91 00:04:53,010 --> 00:04:56,980 by trading underlying, for example, in this case. 92 00:04:56,980 --> 00:05:02,460 So just a brief illustration of the stock 93 00:05:02,460 --> 00:05:14,110 markets, you see how random it has been for the last 20 years 94 00:05:14,110 --> 00:05:14,800 or so. 95 00:05:14,800 --> 00:05:20,150 So first of all, this year, some kind of-- 96 00:05:20,150 --> 00:05:23,790 from beginnings of the 1990s to around 2000, 97 00:05:23,790 --> 00:05:25,500 we see really very sharp increase. 98 00:05:25,500 --> 00:05:28,950 And then we have dot-com bubble, and then we 99 00:05:28,950 --> 00:05:32,470 have the bank [INAUDIBLE] of 2008. 100 00:05:32,470 --> 00:05:35,820 And if you trade derivatives whose payoff depends, 101 00:05:35,820 --> 00:05:38,476 for example, on the FTSE 100 index, 102 00:05:38,476 --> 00:05:39,600 you should be very careful. 103 00:05:39,600 --> 00:05:40,100 All right? 104 00:05:40,100 --> 00:05:43,250 Because market can drop, and you need to be hedged. 105 00:05:43,250 --> 00:05:49,370 So you need to be able to come with some kind of good models 106 00:05:49,370 --> 00:05:51,380 which can recalibrate to the markets 107 00:05:51,380 --> 00:05:59,920 and which can truly risk manage your position. 108 00:05:59,920 --> 00:06:06,970 So the so the general idea of pricing derivatives 109 00:06:06,970 --> 00:06:14,460 is that one starts from some stochastic process. 110 00:06:14,460 --> 00:06:23,240 So in this example here, it's probably like the simplest 111 00:06:23,240 --> 00:06:26,140 possible-- nevertheless a very instructive-- model, 112 00:06:26,140 --> 00:06:30,050 which is essentially like these [INAUDIBLE] Black-Scholes 113 00:06:30,050 --> 00:06:39,710 formalism, which is where we have the stock, which follows 114 00:06:39,710 --> 00:06:41,390 the log-normal dynamics. 115 00:06:41,390 --> 00:06:42,370 I have a question. 116 00:06:42,370 --> 00:06:46,350 Do you have a pointer somewhere, or not? 117 00:06:46,350 --> 00:06:49,950 It's just easier-- OK, OK. 118 00:06:59,980 --> 00:07:00,900 PROFESSOR: Let's see. 119 00:07:00,900 --> 00:07:06,830 There's also a pen here, where you can use this. 120 00:07:06,830 --> 00:07:09,300 DENIS GOROKHOV: Oh, I see. 121 00:07:09,300 --> 00:07:11,020 PROFESSOR: Have you used this before? 122 00:07:11,020 --> 00:07:15,210 You press the color here that you want to use, say, 123 00:07:15,210 --> 00:07:17,138 and then you can draw. 124 00:07:17,138 --> 00:07:18,697 You press on the screen. 125 00:07:18,697 --> 00:07:19,780 DENIS GOROKHOV: Oh, I see. 126 00:07:19,780 --> 00:07:20,400 Excellent. 127 00:07:20,400 --> 00:07:22,320 That's even better. 128 00:07:22,320 --> 00:07:30,070 OK, so it seems like the market is very random. 129 00:07:30,070 --> 00:07:32,750 We need to be able to come up with some kind of dynamics. 130 00:07:32,750 --> 00:07:35,460 And it turns out that the log-normal dynamics 131 00:07:35,460 --> 00:07:38,630 is a very reasonable first approximation 132 00:07:38,630 --> 00:07:43,150 for the actual dynamics. 133 00:07:43,150 --> 00:07:46,180 So in this example, we have stochastic differential 134 00:07:46,180 --> 00:07:50,300 equation for the stock price. 135 00:07:50,300 --> 00:07:55,390 And it consists-- it's the sum of two terms. 136 00:07:55,390 --> 00:08:00,040 This is a drift, it's some kind of deterministic part 137 00:08:00,040 --> 00:08:02,380 of the stock price dynamics. 138 00:08:02,380 --> 00:08:04,750 And here, also, we have diffusion. 139 00:08:04,750 --> 00:08:10,940 So here, dB is the Brownian motion driving the stock, 140 00:08:10,940 --> 00:08:13,540 and S is the price of the stock here. 141 00:08:13,540 --> 00:08:15,780 Mu is the drift. 142 00:08:15,780 --> 00:08:19,710 And sigma is the volatility of the stock. 143 00:08:19,710 --> 00:08:23,660 Particularly, it shows the randomness. 144 00:08:23,660 --> 00:08:28,050 And it's the randomness impact on the stock price. 145 00:08:28,050 --> 00:08:37,539 And using this model, one can derive the Black-Scholes 146 00:08:37,539 --> 00:08:38,090 formula. 147 00:08:38,090 --> 00:08:40,510 And the Black-Scholes formula shows 148 00:08:40,510 --> 00:08:47,990 how to price derivatives whose payoff depends 149 00:08:47,990 --> 00:08:51,950 on the price of the stock. 150 00:08:51,950 --> 00:08:56,420 So here, if you look at this differential equation, 151 00:08:56,420 --> 00:08:57,840 then you can answer the question. 152 00:08:57,840 --> 00:09:02,260 Let's say if you started from some initial value 153 00:09:02,260 --> 00:09:04,290 for the stock at time t. 154 00:09:04,290 --> 00:09:11,440 And then we started the clock. 155 00:09:11,440 --> 00:09:15,730 Which are now to be at time capital T. 156 00:09:15,730 --> 00:09:21,167 And given that time T, then stock price is S_T. 157 00:09:21,167 --> 00:09:22,750 So what's the probability distribution 158 00:09:22,750 --> 00:09:26,200 for the stock at time T? 159 00:09:26,200 --> 00:09:33,820 So this kind of equation can be very easily solved. 160 00:09:33,820 --> 00:09:42,130 And one can obtain analytically the probability distribution 161 00:09:42,130 --> 00:09:44,530 function at any [INAUDIBLE] moment of time. 162 00:09:44,530 --> 00:09:48,010 So I mean, I just think I'll write a few equations, 163 00:09:48,010 --> 00:09:50,180 because it's very important to understand this. 164 00:09:50,180 --> 00:09:54,650 So I'm sure you probably have seen something like this 165 00:09:54,650 --> 00:09:59,630 already, but let me just show you the main ideas 166 00:09:59,630 --> 00:10:01,280 beyond this formula. 167 00:10:01,280 --> 00:10:06,060 So if you have a random process-- 168 00:10:06,060 --> 00:10:10,200 let's say A is some process, stochastic process, which 169 00:10:10,200 --> 00:10:10,890 is normal. 170 00:10:10,890 --> 00:10:13,290 So it follows some drift. 171 00:10:13,290 --> 00:10:18,831 Plus some volatility term. 172 00:10:18,831 --> 00:10:19,330 Right? 173 00:10:19,330 --> 00:10:21,610 So the difference between this equation 174 00:10:21,610 --> 00:10:24,370 is that I don't multiply by A here and A here. 175 00:10:24,370 --> 00:10:31,330 Especially, it's much simpler to solve. 176 00:10:31,330 --> 00:10:32,720 So the solution for this equation 177 00:10:32,720 --> 00:10:33,930 is very straightforward. 178 00:10:33,930 --> 00:10:39,604 So at any moment of time T, if you start at moment 0, 179 00:10:39,604 --> 00:10:42,020 the solution of the equation would be something like this. 180 00:10:42,020 --> 00:10:42,603 Drift-- right? 181 00:10:42,603 --> 00:10:44,150 I'm simply integrating. 182 00:10:44,150 --> 00:10:51,450 Plus-- and I assume that B of t is standard Brownian motion, 183 00:10:51,450 --> 00:10:54,950 so at time 0, it's 0. 184 00:10:54,950 --> 00:11:01,680 And then it's very easy to see now that... 185 00:11:08,550 --> 00:11:11,690 is equal to the Brownian motion. 186 00:11:11,690 --> 00:11:14,100 But this is nothing else. 187 00:11:14,100 --> 00:11:21,420 It's some random number, which is normally distributed, 188 00:11:21,420 --> 00:11:23,280 times square root of time. 189 00:11:23,280 --> 00:11:25,561 So epsilon is proportional to it. 190 00:11:28,650 --> 00:11:32,570 OK, so basically, this means that this is normally 191 00:11:32,570 --> 00:11:33,750 distributed. 192 00:11:33,750 --> 00:11:41,730 And its-- and probability distribution for this quantity 193 00:11:41,730 --> 00:11:44,240 is equal to-- we know it's exactly, right, 194 00:11:44,240 --> 00:11:46,975 because this is like a standard Gaussian distribution. 195 00:11:50,900 --> 00:11:58,410 And if you simply substitute A into here, 196 00:11:58,410 --> 00:12:02,520 then you will obtain the probability distribution 197 00:12:02,520 --> 00:12:05,260 for the actual quantity. 198 00:12:05,260 --> 00:12:08,420 And I'll just write it for the completeness. 199 00:12:11,230 --> 00:12:16,770 So basically, we obtain probability distribution 200 00:12:16,770 --> 00:12:25,020 for the standard variable. 201 00:12:25,020 --> 00:12:26,390 So this is straightforward. 202 00:12:26,390 --> 00:12:29,910 So the only difference between the case I'm doing here 203 00:12:29,910 --> 00:12:33,100 is that the dynamics is assumed to be log-normal. 204 00:12:33,100 --> 00:12:33,600 Right? 205 00:12:33,600 --> 00:12:35,660 And the interpretation is very simple. 206 00:12:35,660 --> 00:12:38,190 If it's normal, then the price of the stock 207 00:12:38,190 --> 00:12:39,912 can become negative. 208 00:12:39,912 --> 00:12:41,370 Which is just a financial nonsense. 209 00:12:41,370 --> 00:12:43,470 So the [INAUDIBLE] log-normal dynamics basically 210 00:12:43,470 --> 00:12:48,570 is a good first approximation. 211 00:12:48,570 --> 00:12:51,040 And in this case, what helps as a result is just 212 00:12:51,040 --> 00:12:52,740 known as Ito's lemma. 213 00:12:52,740 --> 00:12:54,595 So I just first of all write it, and then I 214 00:12:54,595 --> 00:12:57,590 will explain how you can obtain it. 215 00:12:57,590 --> 00:13:01,590 And if you look at this equation-- 216 00:13:01,590 --> 00:13:07,030 let me write it once again-- which is basically the drift 217 00:13:07,030 --> 00:13:17,780 plus-- then it turns out that, of course, since--- it-- 218 00:13:17,780 --> 00:13:26,080 intuitively it's clear that the dynamics of logarithm of S is-- 219 00:13:26,080 --> 00:13:32,010 dynamic of logarithm of S is normal. 220 00:13:32,010 --> 00:13:36,654 So essentially, we obtain something like this. 221 00:13:43,570 --> 00:13:51,360 So if you now substitute this into this, 222 00:13:51,360 --> 00:13:55,035 you locked in a very simple formula. 223 00:14:05,630 --> 00:14:08,540 OK, so here, I used the result, which 224 00:14:08,540 --> 00:14:12,230 is known as Ito's lemma, which I'm going to explain right now. 225 00:14:12,230 --> 00:14:17,290 Like how it was obtained-- basically, 226 00:14:17,290 --> 00:14:22,580 it tells us that when we differentiate the function 227 00:14:22,580 --> 00:14:24,240 of a stochastic variable. 228 00:14:24,240 --> 00:14:26,800 Then besides the trivial term, which is basically 229 00:14:26,800 --> 00:14:29,145 the first derivative times dS, there's 230 00:14:29,145 --> 00:14:31,200 an additional term, which is proportional 231 00:14:31,200 --> 00:14:32,740 to the second derivative. 232 00:14:32,740 --> 00:14:36,280 And it's non-stochastic, so I'll explain why it's this. 233 00:14:36,280 --> 00:14:38,300 But if you do it-- if you look at this equation, 234 00:14:38,300 --> 00:14:41,190 then you see essentially this formula. 235 00:14:41,190 --> 00:14:44,560 It's very, very similar to this formula. 236 00:14:44,560 --> 00:14:49,580 The only difference now is that alpha is just mu minus one half 237 00:14:49,580 --> 00:14:51,060 of sigma squared. 238 00:14:51,060 --> 00:14:55,520 So that's a how, if you iteratively use this solution, 239 00:14:55,520 --> 00:15:00,390 and simply substitute A by log S, 240 00:15:00,390 --> 00:15:02,150 you will come to this equation. 241 00:15:02,150 --> 00:15:04,860 So this is very important. 242 00:15:04,860 --> 00:15:15,447 So it's a very important effect, like-- yes? 243 00:15:15,447 --> 00:15:17,322 AUDIENCE: The fact that it can't be negative, 244 00:15:17,322 --> 00:15:21,120 does that exclude certain possibilities? 245 00:15:21,120 --> 00:15:24,810 When there's a normal Gaussian, can go negative or positive? 246 00:15:24,810 --> 00:15:30,094 DENIS GOROKHOV: Yes, but stock-- from a financial point of view, 247 00:15:30,094 --> 00:15:31,260 stock cannot be a liability. 248 00:15:31,260 --> 00:15:31,990 Right? 249 00:15:31,990 --> 00:15:33,170 You buy a stock. 250 00:15:33,170 --> 00:15:35,110 This means basically, you pay some money. 251 00:15:35,110 --> 00:15:39,250 And you have basically some sort of, say, option 252 00:15:39,250 --> 00:15:41,750 on the profit of the company. 253 00:15:41,750 --> 00:15:43,950 So they can't charge you by default. 254 00:15:43,950 --> 00:15:47,034 So it can't go negative for the stock. 255 00:15:47,034 --> 00:15:48,450 Also, in principle, there might be 256 00:15:48,450 --> 00:15:53,070 derivatives, which can be both positive or negative payoff, 257 00:15:53,070 --> 00:15:54,375 but not the stock. 258 00:15:54,375 --> 00:15:58,260 So it's fundamental financial restriction. 259 00:15:58,260 --> 00:16:00,230 So very important thing. 260 00:16:00,230 --> 00:16:03,930 So if you talk about the stock dynamics and Black-Scholes 261 00:16:03,930 --> 00:16:07,455 formalism, it's very important that the probability 262 00:16:07,455 --> 00:16:10,520 distribution for the stock can be found exactly. 263 00:16:10,520 --> 00:16:12,632 And I'll just [INAUDIBLE] very briefly 264 00:16:12,632 --> 00:16:14,590 go, again, through the Black-Scholes formalism, 265 00:16:14,590 --> 00:16:17,140 it's very important just for understanding. 266 00:16:17,140 --> 00:16:21,640 And I believe there are a couple of things which, at least when 267 00:16:21,640 --> 00:16:24,040 I was studying this, it was not very clear to me, 268 00:16:24,040 --> 00:16:26,340 so I want to go to some more detail. 269 00:16:26,340 --> 00:16:32,840 So basically, here, this derivation 270 00:16:32,840 --> 00:16:35,160 is almost like every textbook. 271 00:16:35,160 --> 00:16:40,300 So the idea is that there is a very fundamental result 272 00:16:40,300 --> 00:16:42,490 in stochastic calculus. 273 00:16:42,490 --> 00:16:46,900 That if you have a stochastic function, function 274 00:16:46,900 --> 00:16:51,240 of stochastic variable S and time, then 275 00:16:51,240 --> 00:16:53,950 its differential can be written as the following form. 276 00:16:53,950 --> 00:16:56,660 So this is all very clear, right? 277 00:16:56,660 --> 00:16:59,250 This is standard calculus? 278 00:16:59,250 --> 00:17:00,960 It's straightforward. 279 00:17:00,960 --> 00:17:04,450 But there is an additional term that looks a bit suspicious. 280 00:17:04,450 --> 00:17:06,990 And I will explain what it actually 281 00:17:06,990 --> 00:17:08,550 means on the next slide. 282 00:17:08,550 --> 00:17:12,640 So a very important thing is that when you calculate dC, 283 00:17:12,640 --> 00:17:15,310 then you will obtain deterministic term which 284 00:17:15,310 --> 00:17:17,869 is proportional to the second derivative. 285 00:17:17,869 --> 00:17:21,416 And you see, there is no-- the fact is that you have here dt, 286 00:17:21,416 --> 00:17:23,790 basically this looks like it's an additional contribution 287 00:17:23,790 --> 00:17:25,869 to the drift. 288 00:17:25,869 --> 00:17:27,750 We view this as drift, and this is a drift. 289 00:17:27,750 --> 00:17:29,937 And there is no, any more, stochasticity. 290 00:17:29,937 --> 00:17:30,770 It's very important. 291 00:17:30,770 --> 00:17:32,980 This is like a crucial fact beyond the Black-Scholes 292 00:17:32,980 --> 00:17:38,030 formalism and the Monte Carlo method in finance. 293 00:17:38,030 --> 00:17:41,090 And then, the idea, you can read, for example, 294 00:17:41,090 --> 00:17:42,990 in Hull's book, its standard proof. 295 00:17:42,990 --> 00:17:53,050 So if we issue an option, then we hedge it-- 296 00:17:53,050 --> 00:17:57,090 by having a certain position in the underlying. 297 00:17:57,090 --> 00:18:00,370 So the idea is like this. 298 00:18:00,370 --> 00:18:02,860 Let's say I sold a call option of the stock. 299 00:18:02,860 --> 00:18:05,500 So when the stock market goes, up I make some money. 300 00:18:05,500 --> 00:18:07,820 And then, in the same time, I short the stock, 301 00:18:07,820 --> 00:18:09,520 so I lose money on my hedge. 302 00:18:09,520 --> 00:18:14,290 And wherever the market goes, I don't make or lose money. 303 00:18:14,290 --> 00:18:18,750 So that's the idea, basically, beyond hedging. 304 00:18:18,750 --> 00:18:23,830 And basically, what happens if I calculate 305 00:18:23,830 --> 00:18:27,990 the change in my portfolio, then since there 306 00:18:27,990 --> 00:18:31,380 is no risk involved, I assume I am perfectly hedged. 307 00:18:31,380 --> 00:18:36,420 Then I simply obtain the risk-free return. 308 00:18:36,420 --> 00:18:39,820 So r here is the risk-free interest rate. 309 00:18:39,820 --> 00:18:48,450 So if you simply look at this equation 310 00:18:48,450 --> 00:18:51,150 and substitute the Ito's lemma result here, 311 00:18:51,150 --> 00:18:53,830 then you obtain like a very simple equation, 312 00:18:53,830 --> 00:18:55,710 which is basically Black-Scholes differential 313 00:18:55,710 --> 00:19:00,490 equation for the stock-- for the price of the option. 314 00:19:00,490 --> 00:19:03,650 So this equation is very fundamental. 315 00:19:03,650 --> 00:19:07,160 And it's very elegant. 316 00:19:07,160 --> 00:19:12,790 So you can see although originally, right, 317 00:19:12,790 --> 00:19:17,986 if you started from something with some arbitrary risk, 318 00:19:17,986 --> 00:19:20,190 with some arbitrary drift mu. 319 00:19:20,190 --> 00:19:20,690 Right? 320 00:19:20,690 --> 00:19:22,440 Which is basically-- it could be anything. 321 00:19:22,440 --> 00:19:28,370 Which is that this drift mu drops out of the equation. 322 00:19:28,370 --> 00:19:33,120 And it depends on the interest rate. 323 00:19:33,120 --> 00:19:35,350 And this is a very interesting fact. 324 00:19:35,350 --> 00:19:49,040 So and this very interesting fact has to do with hedging. 325 00:19:49,040 --> 00:19:50,910 Again, you have position in an option, 326 00:19:50,910 --> 00:19:53,610 and you have an opposite position in underlying. 327 00:19:53,610 --> 00:19:57,350 And that's how the drift disappears. 328 00:19:57,350 --> 00:20:00,710 If you look at the movement of both the positions, 329 00:20:00,710 --> 00:20:04,670 then you see that there that the drift will disappear. 330 00:20:04,670 --> 00:20:08,500 So it's a very important and striking fact. 331 00:20:08,500 --> 00:20:14,700 And the second thing, which is truly a miracle, 332 00:20:14,700 --> 00:20:17,670 is that risk is eliminated completely. 333 00:20:17,670 --> 00:20:21,170 So this equation has absolutely no stochasticity. 334 00:20:21,170 --> 00:20:22,410 So you can just solve it. 335 00:20:22,410 --> 00:20:25,290 If you specify the option payoff, 336 00:20:25,290 --> 00:20:27,630 and if you know your volatility, which 337 00:20:27,630 --> 00:20:33,480 is a measure of your-- basically often how the stock fluctuates. 338 00:20:33,480 --> 00:20:35,700 And if you know the risk-free interest rate, 339 00:20:35,700 --> 00:20:38,000 you can just price the options. 340 00:20:38,000 --> 00:20:40,750 And this is a true miracle that occurs. 341 00:20:40,750 --> 00:20:43,180 And when I was studying this, I couldn't really 342 00:20:43,180 --> 00:20:44,890 understand this-- maybe because I 343 00:20:44,890 --> 00:20:46,600 was coming from theoretical physics, 344 00:20:46,600 --> 00:20:51,690 and all this result called Ito's lemma is buried somewhere 345 00:20:51,690 --> 00:20:55,605 in stochastic calculus. 346 00:20:55,605 --> 00:20:57,855 And I would just try to understand in [INAUDIBLE] what 347 00:20:57,855 --> 00:20:59,070 it all means. 348 00:20:59,070 --> 00:21:01,890 And let them just explain here, basically 349 00:21:01,890 --> 00:21:06,920 how one can understand this result, Ito's lemma, 350 00:21:06,920 --> 00:21:08,941 in a very simple term. 351 00:21:08,941 --> 00:21:09,690 [INAUDIBLE] terms. 352 00:21:09,690 --> 00:21:12,370 So let me just write-- let me remind you. 353 00:21:12,370 --> 00:21:15,220 So Ito's lemma basically tells the following-- once again, 354 00:21:15,220 --> 00:21:19,040 so if C is the function of stochastic variables, 355 00:21:19,040 --> 00:21:23,360 of stochastic variable S, then its differential 356 00:21:23,360 --> 00:21:27,060 is not just equal to some standard result from calculus. 357 00:21:27,060 --> 00:21:33,890 But we also get some kind of very exotic term, which 358 00:21:33,890 --> 00:21:39,770 is basically very nontrivial. 359 00:21:39,770 --> 00:21:43,050 And let me just try to explain to you how actually it appears. 360 00:21:43,050 --> 00:21:45,900 So just to understand this, I recommend everybody 361 00:21:45,900 --> 00:21:47,760 after this lecture, look at this derivation, 362 00:21:47,760 --> 00:21:51,510 because it really explains what this Ito's lemma means. 363 00:21:51,510 --> 00:21:53,710 So the idea is very simple. 364 00:21:53,710 --> 00:21:58,560 So let's start from electrons for the first principles. 365 00:21:58,560 --> 00:22:01,765 And let's say we have an interval of time, 366 00:22:01,765 --> 00:22:03,330 with length dt. 367 00:22:03,330 --> 00:22:06,240 And let's say we divide it into n intervals, 368 00:22:06,240 --> 00:22:10,522 and each interval length is dt prime. 369 00:22:10,522 --> 00:22:11,750 Right? 370 00:22:11,750 --> 00:22:17,410 And I assume that the ratio of dt over dt prime 371 00:22:17,410 --> 00:22:19,240 is sufficiently large. 372 00:22:21,870 --> 00:22:26,310 So first, we know that our stock, as we know, 373 00:22:26,310 --> 00:22:29,290 follows the log-normal dynamics. 374 00:22:29,290 --> 00:22:39,010 So this means that if I go from from time i to time i plus 1, 375 00:22:39,010 --> 00:22:43,010 here you need to exchange i and i minus 1. 376 00:22:43,010 --> 00:22:45,881 So then, you can always [INAUDIBLE] the following form, 377 00:22:45,881 --> 00:22:46,380 right? 378 00:22:46,380 --> 00:22:51,660 So S at time i plus 1 minus S at time i 379 00:22:51,660 --> 00:22:56,004 is equal to the drift term-- right? 380 00:22:56,004 --> 00:22:58,420 Which is a discrete version of the stochastic differential 381 00:22:58,420 --> 00:22:59,230 equation. 382 00:22:59,230 --> 00:23:00,840 Plus the randomness. 383 00:23:00,840 --> 00:23:03,630 So here again, sigma is volatility. 384 00:23:03,630 --> 00:23:05,880 It's the measure of how the stock fluctuates, 385 00:23:05,880 --> 00:23:09,070 which is the stock price, which is square root of dt, 386 00:23:09,070 --> 00:23:11,780 because Brownian motion fluctuation is 387 00:23:11,780 --> 00:23:13,530 proportional to the time. 388 00:23:13,530 --> 00:23:18,990 And also here, we have epsilon, and epsilon 389 00:23:18,990 --> 00:23:21,680 is a standard normal variable. 390 00:23:21,680 --> 00:23:23,630 Then-- OK, so we have this. 391 00:23:23,630 --> 00:23:25,440 This is pretty straightforward. 392 00:23:25,440 --> 00:23:27,730 Basically, I just throw stochastic differential 393 00:23:27,730 --> 00:23:29,010 equation on the latest. 394 00:23:29,010 --> 00:23:32,850 And I go from point i to point i plus 1. 395 00:23:32,850 --> 00:23:41,750 Now, let's see what it means for the price of the option. 396 00:23:41,750 --> 00:23:46,140 So again, so C is the price of the option. 397 00:23:46,140 --> 00:23:53,580 At time T, when the stock price is equal to S_(i+1). 398 00:23:53,580 --> 00:23:57,060 So the change in the option price 399 00:23:57,060 --> 00:23:59,110 is equal to-- like the first term, just 400 00:23:59,110 --> 00:24:01,450 something very standard, standard calculus. 401 00:24:01,450 --> 00:24:07,260 Plus the first derivatives and the difference in the stock 402 00:24:07,260 --> 00:24:10,260 price, plus I take the second-order term, 403 00:24:10,260 --> 00:24:12,890 this is the second derivative, and I 404 00:24:12,890 --> 00:24:18,840 have here S times i plus 1, minus S times i squared. 405 00:24:18,840 --> 00:24:21,950 So this is approximate, because I'm taking only the main terms. 406 00:24:21,950 --> 00:24:25,410 Or the other terms, given that both times dt and dt prime 407 00:24:25,410 --> 00:24:27,600 are very small, they can be neglected. 408 00:24:27,600 --> 00:24:30,640 So you can check it carefully at home if you want to. 409 00:24:30,640 --> 00:24:36,100 But I guarantee that there is no miracle here. 410 00:24:36,100 --> 00:24:37,720 Everything we need is here. 411 00:24:37,720 --> 00:24:42,150 Now let's do the following. 412 00:24:42,150 --> 00:24:43,274 So we have this equation. 413 00:24:43,274 --> 00:24:44,440 And let's look at this term. 414 00:24:44,440 --> 00:24:47,780 So this term, basically, is the cornerstone of the Ito's lemma. 415 00:24:47,780 --> 00:24:53,940 So let's take this equation for the difference 416 00:24:53,940 --> 00:24:56,270 and substitute into here. 417 00:24:56,270 --> 00:24:59,860 And you see here, again, you can look-- what is important, 418 00:24:59,860 --> 00:25:01,100 against the time scales. 419 00:25:01,100 --> 00:25:03,950 So dt prime is very small. 420 00:25:03,950 --> 00:25:09,841 Therefore, the term, which is random, dominates here. 421 00:25:09,841 --> 00:25:10,340 Right? 422 00:25:10,340 --> 00:25:11,923 Because [INAUDIBLE] square root of dt. 423 00:25:11,923 --> 00:25:15,220 And square root for small times is much bigger 424 00:25:15,220 --> 00:25:17,070 than the linear function. 425 00:25:17,070 --> 00:25:22,250 Therefore, we simply neglect this term compared this term. 426 00:25:22,250 --> 00:25:27,480 And with linear accuracy in dt prime, 427 00:25:27,480 --> 00:25:31,760 we can approximate this just by this term. 428 00:25:31,760 --> 00:25:36,800 Now, what to do-- so again, we wrote the same equation, 429 00:25:36,800 --> 00:25:40,100 that latest difference for the option price 430 00:25:40,100 --> 00:25:41,980 of two neighboring points. 431 00:25:41,980 --> 00:25:44,400 And what I'm doing right now, I have all this equation, 432 00:25:44,400 --> 00:25:48,560 and I will simply sum them-- basically from 0 to N. 433 00:25:48,560 --> 00:25:53,010 So let's say I have all these equations from 0 to N minus 1, 434 00:25:53,010 --> 00:25:54,630 and I sum them. 435 00:25:54,630 --> 00:25:56,340 And again, it's very straightforward 436 00:25:56,340 --> 00:25:58,610 and obtains the full equation. 437 00:25:58,610 --> 00:26:03,160 And again, what is very interesting is that we will 438 00:26:03,160 --> 00:26:05,600 obtain-- you look at this term. 439 00:26:05,600 --> 00:26:09,290 So this term is very complicated. 440 00:26:09,290 --> 00:26:10,780 It's essentially stochastic right? 441 00:26:10,780 --> 00:26:13,390 Because-- it looks like very stochastic. 442 00:26:13,390 --> 00:26:16,250 And because-- remember that this is 443 00:26:16,250 --> 00:26:18,000 the standard normal variable. 444 00:26:18,000 --> 00:26:19,890 And all of them are independent. 445 00:26:19,890 --> 00:26:26,030 So in principle, we have the sum of N 446 00:26:26,030 --> 00:26:28,630 independent normal variables squared. 447 00:26:28,630 --> 00:26:33,570 And it turns out-- it's really a very beautiful result, 448 00:26:33,570 --> 00:26:37,050 and I recommend everybody also do it at home, 449 00:26:37,050 --> 00:26:39,180 I try to show you right now on the blackboard-- 450 00:26:39,180 --> 00:26:43,130 that if you sum up all this epsilon squared, 451 00:26:43,130 --> 00:26:46,550 that in the limit when N goes to infinity, 452 00:26:46,550 --> 00:26:50,960 this term becomes deterministic. 453 00:26:50,960 --> 00:26:55,900 So let me just show you basically 454 00:26:55,900 --> 00:26:58,180 what exactly is meant. 455 00:26:58,180 --> 00:27:04,930 So what I mean by deterministic is that of course, 456 00:27:04,930 --> 00:27:14,890 if I have epsilon squared, then it's-- there is some 457 00:27:14,890 --> 00:27:16,440 probability distribution, right? 458 00:27:16,440 --> 00:27:21,250 It's distributed between 0 and infinity, right? 459 00:27:21,250 --> 00:27:23,270 So this is some kind of function. 460 00:27:23,270 --> 00:27:29,500 But my claim is that once I start 461 00:27:29,500 --> 00:27:37,220 adding more and more numbers-- and so on, and so on-- then 462 00:27:37,220 --> 00:27:42,170 this function will become more and more and more narrow. 463 00:27:42,170 --> 00:27:46,060 So it behaves like a deterministic random-- 464 00:27:46,060 --> 00:27:50,430 like a completely deterministic variable in the large N limit. 465 00:27:50,430 --> 00:27:55,850 And to do this, let me just write a very simple-- 466 00:27:55,850 --> 00:27:57,260 write explicitly of what I mean. 467 00:27:57,260 --> 00:28:04,521 So essentially, remember that we have the sum of variables. 468 00:28:04,521 --> 00:28:05,020 Right? 469 00:28:10,570 --> 00:28:16,010 And for us to show that it's become deterministic, 470 00:28:16,010 --> 00:28:19,830 we need to show that it squared-- 471 00:28:25,420 --> 00:28:33,790 The width of the distribution, which I defined as-- let's say 472 00:28:33,790 --> 00:28:35,170 you have a variable, right? 473 00:28:35,170 --> 00:28:40,340 And if I defined the dispersion in the following way, 474 00:28:40,340 --> 00:28:44,730 now I define here the dispersion for this random variable which 475 00:28:44,730 --> 00:28:47,695 is equal to the sum of epsilon squared. 476 00:28:47,695 --> 00:29:00,080 So if I write it here, then it turns out 477 00:29:00,080 --> 00:29:05,210 that each term in this equation is 478 00:29:05,210 --> 00:29:09,520 proportional to N squared, which is natural. 479 00:29:09,520 --> 00:29:12,870 But it turns out that the difference in the large N limit 480 00:29:12,870 --> 00:29:17,220 is proportional only to N. Therefore, 481 00:29:17,220 --> 00:29:23,640 if you have this variable, which-- if you sum up 482 00:29:23,640 --> 00:29:27,180 more and more terms, then we'll have a variable. 483 00:29:27,180 --> 00:29:30,470 We have a distribution for this variable, which 484 00:29:30,470 --> 00:29:32,080 is moving in this direction. 485 00:29:32,080 --> 00:29:35,280 And of course it moves this direction, 486 00:29:35,280 --> 00:29:39,960 but it becomes more and more and more narrow, basically. 487 00:29:39,960 --> 00:29:42,910 So as the limit of N tends to infinity, 488 00:29:42,910 --> 00:29:51,200 it becomes a deterministic. 489 00:29:51,200 --> 00:29:53,370 So I'd recommend everybody at home just do 490 00:29:53,370 --> 00:29:54,960 this very simple exercise. 491 00:29:54,960 --> 00:30:05,840 And you will see that essentially, the sum behaves 492 00:30:05,840 --> 00:30:09,000 as a deterministic quantity. 493 00:30:09,000 --> 00:30:20,740 So just to do this, you need to you 494 00:30:20,740 --> 00:30:25,330 need to know the very simple properties 495 00:30:25,330 --> 00:30:28,190 of the standard normal distribution. 496 00:30:28,190 --> 00:30:31,970 First of all, the average expected value of epsilon 497 00:30:31,970 --> 00:30:33,970 is equal to 1, right? 498 00:30:33,970 --> 00:30:35,380 For a standard normal variable. 499 00:30:35,380 --> 00:30:37,650 And also, you need to know that the fourth moment 500 00:30:37,650 --> 00:30:42,050 of the normal variable is equal to 3. 501 00:30:42,050 --> 00:30:45,710 So if you have this, then you can calculate this, 502 00:30:45,710 --> 00:30:47,590 which is trivial to calculate. 503 00:30:47,590 --> 00:30:55,217 And then you can come to this property that, once again, 504 00:30:55,217 --> 00:30:57,550 probability distribution function, in the large N limit, 505 00:30:57,550 --> 00:30:58,760 behaves deterministic. 506 00:30:58,760 --> 00:31:01,270 It essentially becomes like a delta function. 507 00:31:01,270 --> 00:31:04,025 So this is a very interesting result, 508 00:31:04,025 --> 00:31:09,460 because it basically explains why in the Black-Scholes 509 00:31:09,460 --> 00:31:15,950 equation, we have this very weird by deterministic term. 510 00:31:15,950 --> 00:31:21,170 And that's why the option pricing is possible. 511 00:31:21,170 --> 00:31:23,520 Because if you started pricing options-- 512 00:31:23,520 --> 00:31:25,870 like if you don't know anything about Black-Scholes, 513 00:31:25,870 --> 00:31:28,780 it might be that there's no price for the option, 514 00:31:28,780 --> 00:31:31,630 because it might be that although you do hedge, 515 00:31:31,630 --> 00:31:34,420 you still cannot eliminate your randomness completely. 516 00:31:34,420 --> 00:31:40,570 Maybe hedge helps you with just too narrow the distribution 517 00:31:40,570 --> 00:31:44,150 of your outcome, but we're just not guaranteed at all. 518 00:31:44,150 --> 00:31:51,167 So it's really very-- Ito's Lemma, which is usually 519 00:31:51,167 --> 00:31:52,750 in every book on derivatives, probably 520 00:31:52,750 --> 00:31:58,210 like the first equation ever written, 521 00:31:58,210 --> 00:31:59,890 basically is given without any proof. 522 00:31:59,890 --> 00:32:04,710 But this-- in reality, it's a very interesting limit. 523 00:32:04,710 --> 00:32:12,530 So it can be realized only if you have two different time 524 00:32:12,530 --> 00:32:13,300 scales. 525 00:32:13,300 --> 00:32:16,070 So the small time scale, which is dt prime 526 00:32:16,070 --> 00:32:19,410 is-- in the business sense, it corresponds 527 00:32:19,410 --> 00:32:21,180 to your hedging frequency. 528 00:32:21,180 --> 00:32:24,150 It's when you rebalance your hedging portfolio. 529 00:32:24,150 --> 00:32:28,110 And the time dt, there's a time scale dt, which 530 00:32:28,110 --> 00:32:30,710 is much bigger than dt prime. 531 00:32:30,710 --> 00:32:35,210 It's at the time at which you look at your portfolio. 532 00:32:35,210 --> 00:32:40,390 So only in this very weird limit, when dt over dt prime 533 00:32:40,390 --> 00:32:44,190 goes to infinity, you strictly have Ito's Lemma. 534 00:32:44,190 --> 00:32:46,805 So actually, if you look even like is most famous book 535 00:32:46,805 --> 00:32:48,050 on derivatives. 536 00:32:48,050 --> 00:32:51,510 If you look at this edition, you will see actually 537 00:32:51,510 --> 00:32:54,540 that the proof actually isn't correct. 538 00:32:54,540 --> 00:32:58,720 So just look at it and find what's wrong there. 539 00:32:58,720 --> 00:33:00,670 AUDIENCE: [INAUDIBLE] normal? 540 00:33:00,670 --> 00:33:03,265 DENIS GOROKHOV: Sorry? 541 00:33:03,265 --> 00:33:04,610 AUDIENCE: That's what it is? 542 00:33:04,610 --> 00:33:07,160 DENIS GOROKHOV: Yeah, this is what [INAUDIBLE] means. 543 00:33:07,160 --> 00:33:09,350 So if you use these two results here, 544 00:33:09,350 --> 00:33:17,630 you will see that your it's proportional only 545 00:33:17,630 --> 00:33:19,480 to N, not to N squared. 546 00:33:19,480 --> 00:33:27,280 So that's why your distribution becomes more and more narrow. 547 00:33:27,280 --> 00:33:31,950 Because when you sum up, what it means 548 00:33:31,950 --> 00:33:35,404 is you sum up more and more variables. 549 00:33:35,404 --> 00:33:37,320 Each of them was like random normal variables. 550 00:33:37,320 --> 00:33:40,650 So the average-- average goes like N. 551 00:33:40,650 --> 00:33:44,110 But the dispersion-- the dispersion, right? 552 00:33:44,110 --> 00:33:46,540 That's the standard deviation, right? 553 00:33:46,540 --> 00:33:49,640 You have a square root of N. That's 554 00:33:49,640 --> 00:33:53,130 why basically, square root of N over N is small. 555 00:33:53,130 --> 00:33:56,190 So by increasing N, basically you 556 00:33:56,190 --> 00:34:03,100 become more and more and more deterministic. 557 00:34:03,100 --> 00:34:06,541 So that's the main fact beyond Ito's lemma. 558 00:34:06,541 --> 00:34:07,540 So that's it's obtained. 559 00:34:07,540 --> 00:34:09,956 So I recommend everybody just look in detail, because this 560 00:34:09,956 --> 00:34:12,590 is the cornerstone of derivatives pricing theory, 561 00:34:12,590 --> 00:34:15,795 but at many books it's not really well-written. 562 00:34:18,520 --> 00:34:23,370 So when I was studying, it was like I couldn't 563 00:34:23,370 --> 00:34:24,510 understand for a while. 564 00:34:24,510 --> 00:34:26,989 So it took me a time just to understand. 565 00:34:26,989 --> 00:34:27,730 What else? 566 00:34:27,730 --> 00:34:30,219 And a very interesting thing now is 567 00:34:30,219 --> 00:34:32,650 that remember that we used Ito's lemma-- 568 00:34:32,650 --> 00:34:36,620 and basically, we are able to obtain this equation. 569 00:34:36,620 --> 00:34:46,120 And this equation is very well known in literature. 570 00:34:46,120 --> 00:34:47,860 It's very similar to the heat equation. 571 00:34:47,860 --> 00:34:51,929 And heat equation can be solved using standard methods. 572 00:34:51,929 --> 00:34:55,467 And I don't want to write any derivation here, 573 00:34:55,467 --> 00:34:56,800 it's relatively straightforward. 574 00:34:56,800 --> 00:34:59,190 Maybe a bit cumbersome but straightforward. 575 00:34:59,190 --> 00:35:04,987 And if payoff of your option at maturity 576 00:35:04,987 --> 00:35:07,570 is given by some function, which is not really important here. 577 00:35:07,570 --> 00:35:09,650 Because you can write a very general solution. 578 00:35:12,420 --> 00:35:17,270 So what is here is essentially Green's function 579 00:35:17,270 --> 00:35:19,960 of this equation. 580 00:35:19,960 --> 00:35:24,230 And this Green's function, if you look at this equation, 581 00:35:24,230 --> 00:35:29,200 is very similar to the probability distribution 582 00:35:29,200 --> 00:35:34,000 function, which we have on this slide in the very beginning. 583 00:35:34,000 --> 00:35:39,500 So this function is identical to this function, 584 00:35:39,500 --> 00:35:45,550 and the only difference is that the drift of stock 585 00:35:45,550 --> 00:35:47,000 in the real world disappears. 586 00:35:47,000 --> 00:35:52,960 And we are left only with the interest rate. 587 00:35:52,960 --> 00:36:01,240 And so this equation, which is again, also very important 588 00:36:01,240 --> 00:36:03,940 for the derivatives pricing, is how 589 00:36:03,940 --> 00:36:10,050 we come up with the whole idea of Monte Carlo simulation. 590 00:36:10,050 --> 00:36:13,490 So this is nothing else as a Green's function, which 591 00:36:13,490 --> 00:36:17,090 basically tells us how the stock evolves 592 00:36:17,090 --> 00:36:20,080 in the risk-neutral space. 593 00:36:20,080 --> 00:36:23,320 Risk-neutral space is essentially some kind 594 00:36:23,320 --> 00:36:26,610 of imaginary world, like [INAUDIBLE] world, 595 00:36:26,610 --> 00:36:39,102 where all the assets' drift is just the interest rate and not 596 00:36:39,102 --> 00:36:39,810 the actual drift. 597 00:36:39,810 --> 00:36:42,230 So it's very fundamental. 598 00:36:42,230 --> 00:36:45,180 So it's very important things that the drift in real world 599 00:36:45,180 --> 00:36:46,732 drops out of all the equations. 600 00:36:46,732 --> 00:36:49,190 So the only parameter which actually does matter for option 601 00:36:49,190 --> 00:36:50,992 pricing is volatility. 602 00:36:50,992 --> 00:36:53,450 So this parameter's relatively easier to understand, right? 603 00:36:53,450 --> 00:36:57,720 Because that's how much money your deterministic investment 604 00:36:57,720 --> 00:37:00,900 basically makes. 605 00:37:00,900 --> 00:37:03,190 So [INAUDIBLE] is the [INAUDIBLE] parameter. 606 00:37:03,190 --> 00:37:10,140 So naively, you could expect I need both mu-- let 607 00:37:10,140 --> 00:37:12,622 me just remind you what mu is. 608 00:37:12,622 --> 00:37:14,330 Mu and sigma, two independent parameters. 609 00:37:14,330 --> 00:37:16,663 But it turns out mu completely drops out of the picture. 610 00:37:16,663 --> 00:37:18,390 And this is because of dynamic hedging, 611 00:37:18,390 --> 00:37:20,360 because we hedged the position. 612 00:37:20,360 --> 00:37:26,556 And so now this equation-- since this 613 00:37:26,556 --> 00:37:28,680 is basically Green's function, and Green's function 614 00:37:28,680 --> 00:37:38,910 tells us what's the probability density 615 00:37:38,910 --> 00:37:41,380 of the stock at some time in the future, 616 00:37:41,380 --> 00:37:47,890 if the stock were at some point initially, 617 00:37:47,890 --> 00:37:56,720 then basically this means that we can simulate the stock 618 00:37:56,720 --> 00:37:57,570 dynamics. 619 00:37:57,570 --> 00:37:59,400 And we can price derivatives, like, 620 00:37:59,400 --> 00:38:01,020 using a very simple framework. 621 00:38:01,020 --> 00:38:03,210 So what do we do? 622 00:38:03,210 --> 00:38:06,540 We simply write the equation for the stock 623 00:38:06,540 --> 00:38:09,040 in the risk-neutral world. 624 00:38:09,040 --> 00:38:12,600 Remember, the difference is that instead of the actual drift 625 00:38:12,600 --> 00:38:17,369 of the stock, mu, we substitute here by the interest rate. 626 00:38:17,369 --> 00:38:19,660 And this is basically how much money, roughly speaking, 627 00:38:19,660 --> 00:38:21,280 the bank account makes. 628 00:38:21,280 --> 00:38:30,000 And what we do-- we start from some stock value at time 0, 629 00:38:30,000 --> 00:38:35,432 and then we simulate stock along different paths. 630 00:38:35,432 --> 00:38:36,890 So there are like three paths here. 631 00:38:36,890 --> 00:38:39,180 There could be like thousands. 632 00:38:39,180 --> 00:38:42,070 So now-- and you know, now, let's 633 00:38:42,070 --> 00:38:44,840 say we know the stock payoff at maturity. 634 00:38:44,840 --> 00:38:48,760 And what you do-- then the price of derivative is very simple. 635 00:38:48,760 --> 00:38:51,130 Essentially, you take the average of this payoff, 636 00:38:51,130 --> 00:38:52,047 over the distribution. 637 00:38:52,047 --> 00:38:53,838 And you know distribution, because you just 638 00:38:53,838 --> 00:38:55,060 simulated the stock price. 639 00:38:55,060 --> 00:39:00,249 And you just discount it with the interest rate. 640 00:39:00,249 --> 00:39:01,290 So it's extremely simple. 641 00:39:01,290 --> 00:39:04,720 So in principle, implementing this-- 642 00:39:04,720 --> 00:39:07,850 I'd say if you have package like MATLAB, 643 00:39:07,850 --> 00:39:11,290 it probably takes like maybe one hour at most, 644 00:39:11,290 --> 00:39:17,840 implement let's say pricing of Black-Scholes formula 645 00:39:17,840 --> 00:39:19,930 via a Monte Carlo simulation. 646 00:39:19,930 --> 00:39:22,930 So maybe if you have time, you can try this 647 00:39:22,930 --> 00:39:28,060 and see how your Monte Carlo solution converges 648 00:39:28,060 --> 00:39:32,650 to the exact result which was first obtained by Black-Scholes 649 00:39:32,650 --> 00:39:33,900 and Merton. 650 00:39:33,900 --> 00:39:37,420 So basically, this is a super powerful framework, 651 00:39:37,420 --> 00:39:39,560 which basically tells us something like this. 652 00:39:39,560 --> 00:39:43,190 So it's not applicable just to the stock prices, 653 00:39:43,190 --> 00:39:46,130 but it's also applicable to interest rate derivatives, 654 00:39:46,130 --> 00:39:49,460 credit derivatives, and foreign exchange derivatives, 655 00:39:49,460 --> 00:39:50,670 so on and so forth. 656 00:39:50,670 --> 00:39:52,190 Basically, the idea is like this. 657 00:39:52,190 --> 00:39:55,280 You have some-- the payoff of your derivative 658 00:39:55,280 --> 00:39:57,600 depends on various financial variables. 659 00:39:57,600 --> 00:40:01,670 And you simply simulate all of them in the risk-neutral world. 660 00:40:01,670 --> 00:40:02,170 Right? 661 00:40:02,170 --> 00:40:05,810 So you simulate all of them, and then you 662 00:40:05,810 --> 00:40:08,310 could calculate the average of the payoff. 663 00:40:08,310 --> 00:40:09,582 And you just discount it. 664 00:40:09,582 --> 00:40:11,290 And that's how you can price derivatives. 665 00:40:11,290 --> 00:40:16,340 So in principle, if you have a flexible IT infrastructure, 666 00:40:16,340 --> 00:40:20,090 like a financial institution, so you can implement it. 667 00:40:20,090 --> 00:40:23,910 And then you can price pretty much everything. 668 00:40:23,910 --> 00:40:27,790 That's basically how exotic derivatives 669 00:40:27,790 --> 00:40:33,310 are priced, whose prices are not easy to obtain 670 00:40:33,310 --> 00:40:35,450 using analytical methods. 671 00:40:35,450 --> 00:40:41,970 Which is the case for a large amount of derivatives. 672 00:40:41,970 --> 00:40:44,570 So this is the whole idea, right? 673 00:40:44,570 --> 00:40:50,270 So Monte Carlo simulation is a very fundamental concept. 674 00:40:50,270 --> 00:40:53,700 So we do the simulation in the risk-neutral world, 675 00:40:53,700 --> 00:40:56,315 and there are certain rules how to write these equations 676 00:40:56,315 --> 00:40:58,440 for different asset classes-- could be stock again, 677 00:40:58,440 --> 00:41:01,500 could be foreign exchange, could be credit, could be rates, 678 00:41:01,500 --> 00:41:02,250 whatever. 679 00:41:02,250 --> 00:41:07,530 And then you do some kind of sampling, you find average, 680 00:41:07,530 --> 00:41:10,270 and then basically you are done. 681 00:41:10,270 --> 00:41:11,970 So this is how it works with the stock, 682 00:41:11,970 --> 00:41:15,390 and let me just explain how to generalize 683 00:41:15,390 --> 00:41:18,030 all these ideas for the case of interest rates and credit 684 00:41:18,030 --> 00:41:18,900 derivatives. 685 00:41:18,900 --> 00:41:24,160 So and-- let me just start from the very basics of the interest 686 00:41:24,160 --> 00:41:25,750 rate derivatives. 687 00:41:25,750 --> 00:41:31,490 So of course the whole point of these derivatives 688 00:41:31,490 --> 00:41:39,700 is to allow financial institutions or individuals 689 00:41:39,700 --> 00:41:42,780 to manage their interest rate risk better. 690 00:41:42,780 --> 00:41:49,060 So businesses need money to run their business. 691 00:41:49,060 --> 00:41:52,590 So big institutions, big corporations, 692 00:41:52,590 --> 00:41:55,420 have billions, [INAUDIBLE] hundreds of billions of dollars 693 00:41:55,420 --> 00:41:57,645 of debt, and they know how to risk manage 694 00:41:57,645 --> 00:41:59,570 it, [INAUDIBLE] efficiently. 695 00:41:59,570 --> 00:42:04,474 And just to make money, and not even necessarily financial 696 00:42:04,474 --> 00:42:05,015 institutions. 697 00:42:13,356 --> 00:42:14,730 So of course if you borrow money, 698 00:42:14,730 --> 00:42:16,430 then you need to pay some interest. 699 00:42:16,430 --> 00:42:18,429 So you can think about interest rate derivatives 700 00:42:18,429 --> 00:42:21,620 as some kind of option on the stochastic interest, 701 00:42:21,620 --> 00:42:24,660 because let's say say today, you can borrow money at 5%. 702 00:42:24,660 --> 00:42:27,340 But tomorrow, this rate can change. 703 00:42:27,340 --> 00:42:33,800 So in order to control this uncertainty, 704 00:42:33,800 --> 00:42:38,470 you need to be able to buy some derivatives, 705 00:42:38,470 --> 00:42:41,305 just to hedge your exposure, for example. 706 00:42:41,305 --> 00:42:42,430 Or it might just speculate. 707 00:42:42,430 --> 00:42:45,040 Maybe you just have some view that rates will go up or down. 708 00:42:45,040 --> 00:42:47,740 So it depends on the type of investor or speculator, 709 00:42:47,740 --> 00:42:49,809 whatever. 710 00:42:49,809 --> 00:42:51,600 And so I mean this is a very simple concept 711 00:42:51,600 --> 00:42:52,880 of present value of money. 712 00:42:52,880 --> 00:42:55,290 If I have dollar today, it's definitely 713 00:42:55,290 --> 00:42:57,800 better than the dollar one year from now. 714 00:42:57,800 --> 00:43:00,130 Let's say I have a dollar, right? 715 00:43:08,860 --> 00:43:11,610 But I will get it only in one year from now. 716 00:43:11,610 --> 00:43:13,150 So how much does it cost? 717 00:43:13,150 --> 00:43:15,720 It's clear that if the interest rate is 5%, 718 00:43:15,720 --> 00:43:16,850 it roughly costs $0.95. 719 00:43:16,850 --> 00:43:17,700 Right? 720 00:43:17,700 --> 00:43:19,400 Because what do I do? 721 00:43:19,400 --> 00:43:25,430 If the interest rate is 5%, then I take $0.95, 722 00:43:25,430 --> 00:43:28,510 and I'd put it into bank account, and I'd make 5%. 723 00:43:28,510 --> 00:43:33,260 So I will get like $1 in one year from now. 724 00:43:33,260 --> 00:43:36,150 So there exists very important concept of the present value. 725 00:43:36,150 --> 00:43:39,740 Or like time value of money. 726 00:43:39,740 --> 00:43:43,080 Depending on where in the future you 727 00:43:43,080 --> 00:43:47,480 are, how much money it costs today. 728 00:43:47,480 --> 00:43:51,305 And people talk about-- it's very often a fundamental notion 729 00:43:51,305 --> 00:43:55,100 of the fixed income derivative, is the discount factor. 730 00:43:55,100 --> 00:43:57,290 So it essentially tell you that OK, 731 00:43:57,290 --> 00:44:00,720 if you have one dollar today, it costs one dollar. 732 00:44:00,720 --> 00:44:04,820 But if you have one dollar in the future, 733 00:44:04,820 --> 00:44:06,900 basically it costs something else. 734 00:44:06,900 --> 00:44:14,640 So this is a very important notion in finance. 735 00:44:14,640 --> 00:44:17,750 So I'll tell a little bit more how [INAUDIBLE] 736 00:44:17,750 --> 00:44:20,270 them together, this functional [INAUDIBLE]. 737 00:44:20,270 --> 00:44:23,740 So another very important thing in the interest rate 738 00:44:23,740 --> 00:44:25,660 derivatives is the forward rate. 739 00:44:25,660 --> 00:44:27,917 So remember, okay, so we have discount factor. 740 00:44:27,917 --> 00:44:30,000 And the very important thing about discount factor 741 00:44:30,000 --> 00:44:31,270 is it should start at 1. 742 00:44:31,270 --> 00:44:33,440 Because a dollar today is a dollar. 743 00:44:33,440 --> 00:44:35,600 There is no uncertainty, right? 744 00:44:35,600 --> 00:44:38,250 Thus it's clear that this function 745 00:44:38,250 --> 00:44:42,870 should be decaying, or at least non-increasing, with time. 746 00:44:42,870 --> 00:44:44,750 So that's why it's very convenient 747 00:44:44,750 --> 00:44:51,600 to parametrize this kind of function with forward rates. 748 00:44:51,600 --> 00:44:56,270 So this is some positive forward rates. 749 00:44:56,270 --> 00:44:58,430 And [INAUDIBLE] very convenient. 750 00:44:58,430 --> 00:45:00,170 And remember, let's say, in the example 751 00:45:00,170 --> 00:45:09,500 below, like on this page, if all maturities earn 5%, 752 00:45:09,500 --> 00:45:14,320 then this is simply 5% a year. 753 00:45:14,320 --> 00:45:22,510 So for this example, basically your forward rate is just flat. 754 00:45:22,510 --> 00:45:24,330 OK, so if this is an example-- and when 755 00:45:24,330 --> 00:45:26,038 you talk about interest rate derivatives, 756 00:45:26,038 --> 00:45:31,030 it's very convenient to model the dynamics 757 00:45:31,030 --> 00:45:31,990 of the forward rates. 758 00:45:31,990 --> 00:45:38,690 So again, it's very different from the stock, 759 00:45:38,690 --> 00:45:41,150 because it's got an additional dimension. 760 00:45:41,150 --> 00:45:44,060 So if you model the stock dynamics, 761 00:45:44,060 --> 00:45:45,170 it's just a point process. 762 00:45:45,170 --> 00:45:45,670 Right? 763 00:45:45,670 --> 00:45:49,710 Let's say it's $100 today, and then you start modeling. 764 00:45:49,710 --> 00:45:53,620 Next, they'll go to $95, could go to $105, so on and so forth. 765 00:45:53,620 --> 00:45:59,240 But interest rates, it's more about curve. 766 00:45:59,240 --> 00:46:01,990 So it has an extra dimension-- it's a one-dimensional object. 767 00:46:01,990 --> 00:46:03,239 And the reason is very simple. 768 00:46:03,239 --> 00:46:06,340 In general, let's say if you borrow money for one year, 769 00:46:06,340 --> 00:46:09,840 then let's say you pay one percent. 770 00:46:09,840 --> 00:46:12,550 But if you borrow for two years, it 771 00:46:12,550 --> 00:46:16,010 might be that you borrow it for 2%, and so on. 772 00:46:16,010 --> 00:46:18,120 So there's a concept of the yield curve. 773 00:46:18,120 --> 00:46:22,650 And here basically tells us how much different maturities make. 774 00:46:22,650 --> 00:46:24,754 So in a typical situation, with your curve, 775 00:46:24,754 --> 00:46:26,170 if you don't have some [INAUDIBLE] 776 00:46:26,170 --> 00:46:28,524 of recession, which sometimes happens, 777 00:46:28,524 --> 00:46:29,690 it's usually upward sloping. 778 00:46:29,690 --> 00:46:33,170 This basically means if you borrow money for longer term, 779 00:46:33,170 --> 00:46:34,730 you pay higher interest. 780 00:46:34,730 --> 00:46:36,430 You can see it very easily. 781 00:46:36,430 --> 00:46:38,750 Like for those who have mortgages right there, 782 00:46:38,750 --> 00:46:45,460 it's always like 15-year mortgage rate is lower 783 00:46:45,460 --> 00:46:48,660 than 30-year mortgage rate. 784 00:46:48,660 --> 00:46:51,369 And just here I just show-- to give you 785 00:46:51,369 --> 00:46:53,910 a [INAUDIBLE], of where we are right now in terms of interest 786 00:46:53,910 --> 00:47:01,127 rates, basically I just show you the yield of a 10-year US 787 00:47:01,127 --> 00:47:01,710 Treasury note. 788 00:47:01,710 --> 00:47:03,360 So what is 10-year Treasury note? 789 00:47:03,360 --> 00:47:05,660 Basically, the US government borrows money 790 00:47:05,660 --> 00:47:07,320 to finance its activities. 791 00:47:07,320 --> 00:47:09,570 And then it works like this. 792 00:47:09,570 --> 00:47:11,190 Let's say I'm an investor. 793 00:47:11,190 --> 00:47:13,860 I'm giving the US government $100. 794 00:47:13,860 --> 00:47:17,440 And then every year, like for the next 10 years-- 795 00:47:17,440 --> 00:47:19,320 more exactly, like twice a year-- 796 00:47:19,320 --> 00:47:22,460 let's say they are paying me some coupon. 797 00:47:22,460 --> 00:47:26,360 Let's say if the interest rate per year is 5%, 798 00:47:26,360 --> 00:47:29,490 this means that if I give the US government $100, 799 00:47:29,490 --> 00:47:34,550 then the government pays me $2.50 every half a year. 800 00:47:34,550 --> 00:47:38,550 And at the very end, in 10 years from now, 801 00:47:38,550 --> 00:47:43,330 they must return $100, the notional. 802 00:47:43,330 --> 00:47:46,500 And basically, if you look again how stochastic the rates are 803 00:47:46,500 --> 00:47:48,120 right and what kind of environment 804 00:47:48,120 --> 00:47:53,710 we are in right now, you can see that over the last about 50 805 00:47:53,710 --> 00:47:58,010 years, we see very interesting picture. 806 00:47:58,010 --> 00:48:02,070 From about '60s to about '80, '82, 807 00:48:02,070 --> 00:48:05,320 we can see a tremendous increase in interest rates. 808 00:48:05,320 --> 00:48:09,290 And this is something which looks very unbelievable right 809 00:48:09,290 --> 00:48:10,140 now. 810 00:48:10,140 --> 00:48:12,030 So this problem nowadays. 811 00:48:12,030 --> 00:48:21,330 If one takes, let's say a mortgage, 812 00:48:21,330 --> 00:48:25,790 now a 30-year mortgage is maybe 4%, 4.5% nowadays. 813 00:48:25,790 --> 00:48:30,670 But let's say here, about 30 years ago, it 814 00:48:30,670 --> 00:48:33,140 was like a [INAUDIBLE] interest rate-- very high inflation. 815 00:48:33,140 --> 00:48:35,780 And mortgage rates were in double digits. 816 00:48:35,780 --> 00:48:37,940 It was not uncommon to pay like 15% 817 00:48:37,940 --> 00:48:39,790 if you would take mortgage somewhere here. 818 00:48:39,790 --> 00:48:41,490 So the rates were increasing. 819 00:48:41,490 --> 00:48:44,520 But since then, we live in a very different environment, 820 00:48:44,520 --> 00:48:47,110 when interest rates gradually go and go down. 821 00:48:47,110 --> 00:48:51,180 So essentially, here, basically it shows in 1980, 822 00:48:51,180 --> 00:48:54,540 the US Government would pay 12% a year 823 00:48:54,540 --> 00:48:58,020 each year to borrow money for 10 years. 824 00:48:58,020 --> 00:49:02,990 So at the end of 2012, it paid less than 2%-- just 1.7%. 825 00:49:02,990 --> 00:49:06,300 So there like a very clear trend, you know? 826 00:49:06,300 --> 00:49:07,870 Something's going down. 827 00:49:07,870 --> 00:49:12,100 So in recent years, there is some kind of uptick here. 828 00:49:12,100 --> 00:49:16,840 But you know, we always get some kind of situation here. 829 00:49:16,840 --> 00:49:17,820 So where are we going? 830 00:49:17,820 --> 00:49:20,440 Nobody knows. 831 00:49:20,440 --> 00:49:22,810 But really, we're in this situation where interest rates 832 00:49:22,810 --> 00:49:24,272 are extremely low. 833 00:49:24,272 --> 00:49:25,730 It was nothing like this, basically 834 00:49:25,730 --> 00:49:27,520 for the last 50 years. 835 00:49:27,520 --> 00:49:33,801 So it's very unusual, and you have these very low interest 836 00:49:33,801 --> 00:49:34,300 rates. 837 00:49:34,300 --> 00:49:36,180 This means that the economy is very weak, 838 00:49:36,180 --> 00:49:39,790 because this means there's not much demand on borrowing, 839 00:49:39,790 --> 00:49:40,340 right? 840 00:49:40,340 --> 00:49:42,840 Because corporations, like individuals, 841 00:49:42,840 --> 00:49:48,350 they don't want to borrow a lot, because once [INAUDIBLE] again, 842 00:49:48,350 --> 00:49:49,940 like supply-demand, right? 843 00:49:49,940 --> 00:49:53,210 Because if you want to borrow, basically you're 844 00:49:53,210 --> 00:49:55,580 willing to pay higher rate. 845 00:49:55,580 --> 00:49:59,280 So also, of course another reason for this 846 00:49:59,280 --> 00:50:03,060 is because-- we live in a very unusual environment, 847 00:50:03,060 --> 00:50:07,430 because the government interferes a lot on the market. 848 00:50:07,430 --> 00:50:11,820 So they're trying to make the rates as low as possible, 849 00:50:11,820 --> 00:50:14,760 just to make the interest rates burden for corporations, 850 00:50:14,760 --> 00:50:18,620 for private individuals as small as possible. 851 00:50:18,620 --> 00:50:20,600 And hopefully, we'll go out of this recession. 852 00:50:20,600 --> 00:50:23,470 But as I said, this is very singular, very 853 00:50:23,470 --> 00:50:26,900 unusual environment-- just to understand what's going on. 854 00:50:26,900 --> 00:50:30,935 And there a whole world of interest rate-- yes? 855 00:50:30,935 --> 00:50:33,410 AUDIENCE: But it pays to invest in a non-productive access, 856 00:50:33,410 --> 00:50:36,300 like real estate, which are expected 857 00:50:36,300 --> 00:50:39,874 to rise with time, without, for example, [INAUDIBLE]. 858 00:50:43,250 --> 00:50:45,825 Doesn't it skew whatever investment 859 00:50:45,825 --> 00:50:50,686 is made toward assets which are expected to rise with time? 860 00:50:50,686 --> 00:50:52,390 It may not be productive access-- 861 00:50:52,390 --> 00:50:54,910 DENIS GOROKHOV: Yes, yes, but right now, I 862 00:50:54,910 --> 00:50:59,290 mean I think even right now, lots of people 863 00:50:59,290 --> 00:51:00,900 are just scared to buy real estate. 864 00:51:00,900 --> 00:51:04,360 You never know what's going on, right? 865 00:51:04,360 --> 00:51:07,960 Because prices are still pretty high, so 866 00:51:07,960 --> 00:51:10,254 who knows what will happen? 867 00:51:10,254 --> 00:51:10,920 So you're right. 868 00:51:10,920 --> 00:51:14,240 There is some kind of psychology [INAUDIBLE]. 869 00:51:14,240 --> 00:51:18,040 But many people who bought like 2006, whatever-- like before, 870 00:51:18,040 --> 00:51:19,890 they basically lost tons of money. 871 00:51:19,890 --> 00:51:21,910 You never know. 872 00:51:21,910 --> 00:51:24,860 So it's like when you buy some assets, 873 00:51:24,860 --> 00:51:26,140 you've got some finance. 874 00:51:26,140 --> 00:51:27,560 Let's say fixed rate finance. 875 00:51:27,560 --> 00:51:29,268 So you know how much you're going to pay, 876 00:51:29,268 --> 00:51:32,630 but where is the guarantee that, you know-- 877 00:51:32,630 --> 00:51:37,120 I mean, long term, it goes up, of course, 878 00:51:37,120 --> 00:51:40,940 but long-term basically means tens of years. 879 00:51:40,940 --> 00:51:43,884 But if you look at the real estate prices, for the last, 880 00:51:43,884 --> 00:51:44,800 whatever, seven years. 881 00:51:47,690 --> 00:51:50,880 We are going up right now, but still, we didn't go through 882 00:51:50,880 --> 00:51:51,840 for the minimum. 883 00:51:51,840 --> 00:51:54,750 Like the [INAUDIBLE] maximum, which you had before, 884 00:51:54,750 --> 00:51:55,360 basically. 885 00:51:55,360 --> 00:52:00,190 So you never know. 886 00:52:00,190 --> 00:52:05,041 Yes, and so there's a whole world of interest rate 887 00:52:05,041 --> 00:52:05,540 derivatives. 888 00:52:05,540 --> 00:52:08,530 So I'm just very briefly explaining what it all means. 889 00:52:08,530 --> 00:52:12,230 So usually-- here I mentioned it's all about Treasury. 890 00:52:12,230 --> 00:52:14,380 So it's all like government-- it's 891 00:52:14,380 --> 00:52:17,440 kind of yield implied from the government bonds. 892 00:52:17,440 --> 00:52:20,510 But usually, all the derivatives are linked to another 893 00:52:20,510 --> 00:52:22,750 very famous rate, which is called LIBOR. 894 00:52:22,750 --> 00:52:24,310 And LIBOR-- roughly speaking, it's 895 00:52:24,310 --> 00:52:29,190 a short-term rate at which financial institutions 896 00:52:29,190 --> 00:52:34,160 in London borrow money from each other on an unsecured basis. 897 00:52:34,160 --> 00:52:36,540 So there's a lot of caveats here on this definition, 898 00:52:36,540 --> 00:52:39,590 but that's roughly what that means. 899 00:52:39,590 --> 00:52:43,170 And there is like a fundamental derivative in the interest rate 900 00:52:43,170 --> 00:52:45,000 world is a LIBOR swap. 901 00:52:45,000 --> 00:52:47,250 So the standard USD LIBOR swap is something like this, 902 00:52:47,250 --> 00:52:47,750 basically. 903 00:52:47,750 --> 00:52:51,520 It's paying-- once a three months, 904 00:52:51,520 --> 00:52:55,980 it's paying three months LIBOR rate. 905 00:52:55,980 --> 00:52:57,500 And so this is stochastic, right? 906 00:52:57,500 --> 00:53:02,550 So basically, every day, there is this certain procedure, 907 00:53:02,550 --> 00:53:06,390 which tells us what this LIBOR, this short-term borrowing rate 908 00:53:06,390 --> 00:53:08,580 is. 909 00:53:08,580 --> 00:53:11,030 And in exchange for this, if you're 910 00:53:11,030 --> 00:53:14,000 paying out this LIBOR swap, this LIBOR rate, 911 00:53:14,000 --> 00:53:16,634 you are receiving the fixed rate, which is diminished. 912 00:53:16,634 --> 00:53:18,800 So this is like fundamental interest rate basically. 913 00:53:18,800 --> 00:53:21,302 It's like, essentially, if you believe that rates will go up 914 00:53:21,302 --> 00:53:23,010 and you just want to speculate, basically 915 00:53:23,010 --> 00:53:25,780 you're trying to be long LIBOR and short fixed rate, 916 00:53:25,780 --> 00:53:27,330 and vice versa. 917 00:53:27,330 --> 00:53:31,290 So this is a very important instrument for pricing. 918 00:53:31,290 --> 00:53:33,420 And it's all kinds of derivatives 919 00:53:33,420 --> 00:53:34,980 linked to this LIBOR rate. 920 00:53:34,980 --> 00:53:40,360 For example, you can talk about a swaption. 921 00:53:40,360 --> 00:53:41,160 What is a swaption? 922 00:53:41,160 --> 00:53:44,450 Swaption is a derivative to enter an interest rate 923 00:53:44,450 --> 00:53:46,192 swap in the future. 924 00:53:46,192 --> 00:53:47,900 Remember like in the equity option world, 925 00:53:47,900 --> 00:53:51,110 let's say if I have a call option on a stock, that's 926 00:53:51,110 --> 00:53:54,070 the right to buy a stock at a fixed 927 00:53:54,070 --> 00:53:57,580 price-- it's fixed today-- like at some time in the future. 928 00:53:57,580 --> 00:53:59,310 Here, this is basically the same idea. 929 00:53:59,310 --> 00:54:02,140 If you're here today, at sometime in the future 930 00:54:02,140 --> 00:54:03,980 you can enter a swap, a kind of contract, 931 00:54:03,980 --> 00:54:07,450 which pays various legs and there 932 00:54:07,450 --> 00:54:10,150 is some price given for today. 933 00:54:10,150 --> 00:54:13,600 And there are also all kinds of false derivatives. 934 00:54:13,600 --> 00:54:15,490 You can talk about rates. 935 00:54:15,490 --> 00:54:20,080 Basically you can buy or sell options on a particular LIBOR 936 00:54:20,080 --> 00:54:21,370 rate. 937 00:54:21,370 --> 00:54:23,730 Or there's also cancel-able swaps, 938 00:54:23,730 --> 00:54:26,460 which basically are you can enter a swap, 939 00:54:26,460 --> 00:54:29,490 but if you don't want to pay, like, let's say, high rate 940 00:54:29,490 --> 00:54:30,690 anymore, you can cancel it. 941 00:54:30,690 --> 00:54:33,350 Of course, it's affecting the price so on and so forth. 942 00:54:33,350 --> 00:54:36,700 So, very important idea if you think about all these 943 00:54:36,700 --> 00:54:39,500 that it turns out that when you price all these derivatives, 944 00:54:39,500 --> 00:54:45,280 they all depend-- Their price depends on these discount 945 00:54:45,280 --> 00:54:46,550 factors. 946 00:54:46,550 --> 00:54:49,299 And the discount factors depend on these forward rates, 947 00:54:49,299 --> 00:54:51,090 which is basically trivial parametrization. 948 00:54:51,090 --> 00:54:53,915 But it's very important, very convenient, to work 949 00:54:53,915 --> 00:54:55,380 with these forward rates. 950 00:54:55,380 --> 00:54:59,170 And when we model interest rate derivatives, using Monte Carlo 951 00:54:59,170 --> 00:55:02,140 simulations, and there are no analytical models available, 952 00:55:02,140 --> 00:55:05,950 then [INAUDIBLE] model of dynamics of forward rates. 953 00:55:05,950 --> 00:55:09,700 And you can ask a question. 954 00:55:09,700 --> 00:55:16,060 So how can we get, basically, this curve in practice, 955 00:55:16,060 --> 00:55:17,190 or this curve? 956 00:55:17,190 --> 00:55:26,350 And it turns out that the swap market tells us 957 00:55:26,350 --> 00:55:27,860 how to obtain this curve. 958 00:55:27,860 --> 00:55:33,290 So here I show some quotes, real market quotes, 959 00:55:33,290 --> 00:55:35,610 for interest rate swap of different maturities. 960 00:55:35,610 --> 00:55:37,600 Let's say two years, three years, four years, 961 00:55:37,600 --> 00:55:39,010 and so on and so forth. 962 00:55:39,010 --> 00:55:42,420 And then if you add this number and this number, 963 00:55:42,420 --> 00:55:46,970 then you obtain the swap rate. 964 00:55:49,930 --> 00:55:53,470 So if you take these swap rates, then it 965 00:55:53,470 --> 00:55:58,290 turns out that you can show very easily that if you 966 00:55:58,290 --> 00:56:00,220 know all these numbers, then you will 967 00:56:00,220 --> 00:56:07,200 be able to obtain this curve in a pretty unique way. 968 00:56:07,200 --> 00:56:12,940 So because of this market of swaps-- 969 00:56:12,940 --> 00:56:15,560 so once again, if you add these two numbers here, 970 00:56:15,560 --> 00:56:21,420 then basically it tells you that, for example, 971 00:56:21,420 --> 00:56:23,580 for this instrument, let's say, five years. 972 00:56:23,580 --> 00:56:26,180 For the next five years, I'm going to pay roughly 973 00:56:26,180 --> 00:56:31,760 like 0.75% a year. 974 00:56:31,760 --> 00:56:32,260 Right. 975 00:56:32,260 --> 00:56:34,110 So these two payments, basically, 976 00:56:34,110 --> 00:56:38,720 correspond to like 0.75% in exchange for the LIBOR payment, 977 00:56:38,720 --> 00:56:39,220 right? 978 00:56:39,220 --> 00:56:41,770 So if I enter a swap-- so I know that the I will 979 00:56:41,770 --> 00:56:44,380 be paying fixed-- but I'll receive 980 00:56:44,380 --> 00:56:48,970 floating, which is random, because we 981 00:56:48,970 --> 00:56:50,380 don't know what it is. 982 00:56:50,380 --> 00:56:55,340 And [INAUDIBLE] is a pretty complicated concept. 983 00:57:00,030 --> 00:57:03,360 The idea is very simple. 984 00:57:03,360 --> 00:57:05,460 So basically the swap market allows 985 00:57:05,460 --> 00:57:10,970 you to obtain this discount factor-- basically 986 00:57:10,970 --> 00:57:15,900 this function-- which tells you how much your dollar 987 00:57:15,900 --> 00:57:19,530 in the future is today. 988 00:57:19,530 --> 00:57:21,400 So if you know how much a dollar is, 989 00:57:21,400 --> 00:57:24,600 then you know how much C dollars, basically, cost. 990 00:57:28,510 --> 00:57:30,490 Then basically, let's say you have C dollars. 991 00:57:30,490 --> 00:57:33,380 Then you simply multiply them by the discount factor, 992 00:57:33,380 --> 00:57:37,220 and that's what the present value of your fixed rate 993 00:57:37,220 --> 00:57:37,900 payment is. 994 00:57:37,900 --> 00:57:39,400 So remember that finance [INAUDIBLE] 995 00:57:39,400 --> 00:57:40,430 very important things. 996 00:57:40,430 --> 00:57:43,885 In finance, at least in the derivative world, 997 00:57:43,885 --> 00:57:49,285 we typically-- what is called PV or present value of all 998 00:57:49,285 --> 00:57:50,910 our future payments, right? 999 00:57:50,910 --> 00:57:53,840 So we have some future liability, which 1000 00:57:53,840 --> 00:57:55,252 is something very complicated. 1001 00:57:55,252 --> 00:57:57,210 I say, I'll pay you something very complicated, 1002 00:57:57,210 --> 00:57:59,260 pay off in 10 years from now. 1003 00:57:59,260 --> 00:58:02,120 But we are trying to understand how much it's worth today. 1004 00:58:02,120 --> 00:58:05,620 Because idea for this business is clients come to the bank. 1005 00:58:05,620 --> 00:58:07,530 And they say, I want this derivative. 1006 00:58:07,530 --> 00:58:08,680 You sell this derivative. 1007 00:58:08,680 --> 00:58:10,360 You charge the money right now, and you 1008 00:58:10,360 --> 00:58:11,825 spend this money on hedging. 1009 00:58:11,825 --> 00:58:13,950 Of course, you try to charge them a little bit more 1010 00:58:13,950 --> 00:58:15,650 because you need to still make living. 1011 00:58:15,650 --> 00:58:18,780 But in [INAUDIBLE] basically is like you've spent 1012 00:58:18,780 --> 00:58:20,530 most of your money on hedging. 1013 00:58:23,840 --> 00:58:25,975 But you to try to come up with a number today. 1014 00:58:28,710 --> 00:58:31,730 Here's, again, a very simple example. 1015 00:58:31,730 --> 00:58:38,150 So if you know, once again, how much your dollar is 1016 00:58:38,150 --> 00:58:41,640 in the future, then you can present 1017 00:58:41,640 --> 00:58:43,720 value, PV, every payment. 1018 00:58:43,720 --> 00:58:47,900 So let's say in 10 years from now, d is equal to 0.5, 1019 00:58:47,900 --> 00:58:52,450 then if you payoff's $1,000, the present value is equal to $500. 1020 00:58:52,450 --> 00:58:55,090 Because, again, the argument is very simple, right? 1021 00:58:55,090 --> 00:58:59,200 You take $500 today and invest for 10 years, 1022 00:58:59,200 --> 00:59:01,060 and you get $1,000 in the future. 1023 00:59:01,060 --> 00:59:03,180 This is the replication argument. 1024 00:59:03,180 --> 00:59:06,378 Another very important thing here, 1025 00:59:06,378 --> 00:59:07,877 is that if you have an interest rate 1026 00:59:07,877 --> 00:59:14,910 swap, which is paying LIBOR. 1027 00:59:14,910 --> 00:59:17,060 And let's say on a notional. 1028 00:59:17,060 --> 00:59:21,060 Let's say I pay you LIBOR, which is some rate which 1029 00:59:21,060 --> 00:59:22,910 is measured in percent. 1030 00:59:22,910 --> 00:59:25,630 LIBOR is like a 1% a year, for example. 1031 00:59:25,630 --> 00:59:27,920 Then notional of the swap is $1 million 1032 00:59:27,920 --> 00:59:33,640 which means that the floating rate payment is based on $1 1033 00:59:33,640 --> 00:59:37,630 million times 1% is $10,000. 1034 00:59:37,630 --> 00:59:40,880 So it turns out that very interesting thing 1035 00:59:40,880 --> 00:59:53,020 is that if you pay LIBOR rate and if you 1036 00:59:53,020 --> 00:59:55,010 pay the notional at the very end, 1037 00:59:55,010 --> 00:59:59,880 then the present value of this is equal to the notional. 1038 00:59:59,880 --> 01:00:03,120 So it's the beauty of floating rate is security. 1039 01:00:06,090 --> 01:00:12,380 [INAUDIBLE] is basically that if you pay the current market 1040 01:00:12,380 --> 01:00:16,030 rate all the time, then the price of your security 1041 01:00:16,030 --> 01:00:17,535 is always equal to the notional. 1042 01:00:19,020 --> 01:00:22,670 It's very nice fact which is also fundamental here. 1043 01:00:22,670 --> 01:00:25,524 And very interesting thing would happen 1044 01:00:25,524 --> 01:00:27,690 after crisis is that all the derivatives have become 1045 01:00:27,690 --> 01:00:29,217 what's called collateralized. 1046 01:00:29,217 --> 01:00:31,050 So you need to post some money all the time. 1047 01:00:31,050 --> 01:00:32,966 So there's another concept of OIS discounting, 1048 01:00:32,966 --> 01:00:35,292 which I don't talk about here. 1049 01:00:35,292 --> 01:00:37,250 The main idea which you need to understand here 1050 01:00:37,250 --> 01:00:41,090 is that we have this function, like discount function, which 1051 01:00:41,090 --> 01:00:44,690 shows us again how much the dollar is worth in the future. 1052 01:00:44,690 --> 01:00:48,630 And using this function, we can price all kinds of swaps. 1053 01:00:48,630 --> 01:00:55,490 So we can PV the value of the swap today using this. 1054 01:00:55,490 --> 01:00:59,120 So the idea of interest rate derivatives 1055 01:00:59,120 --> 01:01:01,510 it's all about dynamics of the yield curve. 1056 01:01:01,510 --> 01:01:04,510 It's basically how your discount function 1057 01:01:04,510 --> 01:01:10,480 or how your yields, future yields, evolve. 1058 01:01:10,480 --> 01:01:14,100 The whole idea is similar to the stock. 1059 01:01:14,100 --> 01:01:20,640 So again, at time 0 you start from some curve. 1060 01:01:20,640 --> 01:01:23,220 For example, something like this, right? 1061 01:01:23,220 --> 01:01:26,740 From some curve which is shown here. 1062 01:01:26,740 --> 01:01:29,130 And then it stopped evolving and you 1063 01:01:29,130 --> 01:01:32,340 want to be able to model it mathematically and price 1064 01:01:32,340 --> 01:01:33,820 all kinds of derivatives. 1065 01:01:33,820 --> 01:01:37,800 So there is like a very interesting difference 1066 01:01:37,800 --> 01:01:45,720 between stock options and interest rate options 1067 01:01:45,720 --> 01:01:49,000 because for the stock options, we know the price today. 1068 01:01:49,000 --> 01:01:50,760 If it's a liquid stock, it's just known. 1069 01:01:50,760 --> 01:01:52,440 We know what it's trading right now. 1070 01:01:52,440 --> 01:01:55,110 But for the yield curve, it's different. 1071 01:01:55,110 --> 01:01:58,550 We first need to take the swap markets quotes 1072 01:01:58,550 --> 01:02:00,690 and do what is called bootstrapping 1073 01:02:00,690 --> 01:02:02,330 to get the function d of t. 1074 01:02:02,330 --> 01:02:03,940 The next step, we need to specify 1075 01:02:03,940 --> 01:02:06,720 the volatility of different forward rates in the future 1076 01:02:06,720 --> 01:02:12,570 and we need to come up with some kind of dynamics 1077 01:02:12,570 --> 01:02:15,790 which describes the future dynamics of forward rates. 1078 01:02:15,790 --> 01:02:17,590 And then once we have this, we can 1079 01:02:17,590 --> 01:02:21,500 use the Monte Carlo framework to price all kinds of derivatives. 1080 01:02:21,500 --> 01:02:25,435 So before I start talking about the HJM framework here, 1081 01:02:25,435 --> 01:02:26,810 I just want to mention that there 1082 01:02:26,810 --> 01:02:33,100 are some other more simple models which are historically 1083 01:02:33,100 --> 01:02:36,175 appear before the HJM model which basically describe 1084 01:02:36,175 --> 01:02:39,285 the dynamics of the short rate. 1085 01:02:39,285 --> 01:02:42,750 And so the most famous ones are the Ho-Lee model, 1086 01:02:42,750 --> 01:02:46,250 Hull-White model, and so-called CIR model. 1087 01:02:46,250 --> 01:02:48,890 And basically, the idea is that if you 1088 01:02:48,890 --> 01:02:54,070 have this function for forward rates-- which I wrote here. 1089 01:02:54,070 --> 01:02:57,480 So they describe dynamics, instantaneous dynamics, 1090 01:02:57,480 --> 01:02:58,950 of this rate. 1091 01:02:58,950 --> 01:03:02,270 So instead of modeling the whole curve, 1092 01:03:02,270 --> 01:03:06,570 you model only just this short rate and so on. 1093 01:03:06,570 --> 01:03:09,620 So some of these models are particular case 1094 01:03:09,620 --> 01:03:10,960 of the HJM model. 1095 01:03:10,960 --> 01:03:12,350 Some of them are not. 1096 01:03:12,350 --> 01:03:17,410 But just to mention. 1097 01:03:17,410 --> 01:03:27,130 And basically the idea, then, of the interest rate derivatives, 1098 01:03:27,130 --> 01:03:30,870 for example, let's say I want to price an option that in five 1099 01:03:30,870 --> 01:03:33,080 years from now, I enter a particular interest rate 1100 01:03:33,080 --> 01:03:38,740 swap which pays 5% on the fixed leg and receives LIBOR. 1101 01:03:38,740 --> 01:03:44,180 So I need to model the dynamic of future yields. 1102 01:03:44,180 --> 01:03:49,050 And remember, it's a very important thing that, again, 1103 01:03:49,050 --> 01:03:57,710 because we have the curve, now we 1104 01:03:57,710 --> 01:04:00,090 have two different times here. 1105 01:04:00,090 --> 01:04:02,620 For the stock derivatives, we just basically write dynamics, 1106 01:04:02,620 --> 01:04:04,320 d of S_t is equal to something. 1107 01:04:04,320 --> 01:04:06,380 And t is just basically instantaneous time. 1108 01:04:06,380 --> 01:04:12,470 Here t stands for instantaneous time. 1109 01:04:12,470 --> 01:04:20,940 And T, capital T, stands for the future time. 1110 01:04:20,940 --> 01:04:21,560 Here. 1111 01:04:21,560 --> 01:04:23,450 So essentially if you're here, you're 1112 01:04:23,450 --> 01:04:26,459 looking at the forward rate somewhere here. 1113 01:04:26,459 --> 01:04:28,375 And then you basically describe with dynamics. 1114 01:04:31,840 --> 01:04:34,030 I don't want to go into details, but again, 1115 01:04:34,030 --> 01:04:38,520 using this very fundamental result in pricing theory 1116 01:04:38,520 --> 01:04:45,510 like Ito's Lemma, you can derive the equation for this drift. 1117 01:04:45,510 --> 01:04:48,910 So the problem is it turns out it's always 1118 01:04:48,910 --> 01:04:51,090 the case in the Monte Carlo simulation. 1119 01:04:51,090 --> 01:04:55,290 So you [INAUDIBLE] some time equation and you have drift 1120 01:04:55,290 --> 01:04:56,980 and you have volatility. 1121 01:04:56,980 --> 01:05:00,390 So it turns out that this drift, the real time 1122 01:05:00,390 --> 01:05:03,500 drift, because you hedge, drops out of your equation. 1123 01:05:03,500 --> 01:05:05,520 And it turns out that for the interest rate, 1124 01:05:05,520 --> 01:05:07,035 there is some complication. 1125 01:05:07,035 --> 01:05:11,980 In the risk-neutral world, this real-world drift [INAUDIBLE] 1126 01:05:11,980 --> 01:05:16,160 by some equation which depends on sigma. 1127 01:05:16,160 --> 01:05:18,140 So if you do the calculation, then you 1128 01:05:18,140 --> 01:05:24,140 will see that in the risk-neutral world, 1129 01:05:24,140 --> 01:05:25,910 if you [INAUDIBLE] of following form, 1130 01:05:25,910 --> 01:05:28,150 which is some non-local equation. 1131 01:05:28,150 --> 01:05:29,500 But it is what it is. 1132 01:05:29,500 --> 01:05:30,830 So it's very straightforward. 1133 01:05:30,830 --> 01:05:34,750 I encourage you just to, if you have time, to go through this 1134 01:05:34,750 --> 01:05:38,740 and really understand how it works. 1135 01:05:38,740 --> 01:05:44,130 But now once we have this, the model for interest rate 1136 01:05:44,130 --> 01:05:46,450 derivatives is very simple. 1137 01:05:46,450 --> 01:05:50,176 And remember that in the stock world-- 1138 01:05:50,176 --> 01:05:54,872 let me go back just to this equation. 1139 01:05:54,872 --> 01:05:56,830 So we started from some stochastic differential 1140 01:05:56,830 --> 01:05:58,040 equation. 1141 01:05:58,040 --> 01:06:00,890 And then we simulate different paths. 1142 01:06:00,890 --> 01:06:05,260 And then basically we average over the pay-off 1143 01:06:05,260 --> 01:06:08,170 here at maturity of the derivative, 1144 01:06:08,170 --> 01:06:10,990 when actually we do the payment. 1145 01:06:10,990 --> 01:06:14,300 And here the situation is very similar. 1146 01:06:14,300 --> 01:06:19,090 So we have some initial curve which we 1147 01:06:19,090 --> 01:06:21,310 obtain from the market today. 1148 01:06:21,310 --> 01:06:27,580 And this curve dynamics is described by this equation. 1149 01:06:27,580 --> 01:06:30,290 Then we have distribution of this curve in the future, 1150 01:06:30,290 --> 01:06:32,600 and then you can price all kinds of derivatives. 1151 01:06:32,600 --> 01:06:35,270 So again, it's a very fundamental framework. 1152 01:06:35,270 --> 01:06:37,060 So very general. 1153 01:06:37,060 --> 01:06:42,160 So once the curve and the volatility are known, 1154 01:06:42,160 --> 01:06:46,000 you simply run this simulation and you get your pay-off. 1155 01:06:46,000 --> 01:06:49,200 So basically that's how it works. 1156 01:06:53,860 --> 01:06:58,280 And now another example, which is basically-- 1157 01:06:58,280 --> 01:07:04,190 of this HJM model, is basically credit derivatives. 1158 01:07:04,190 --> 01:07:07,220 So I don't have much time, but just 1159 01:07:07,220 --> 01:07:13,210 mention-- I'll go very briefly what's going on. 1160 01:07:13,210 --> 01:07:17,090 So if you give money just to someone, 1161 01:07:17,090 --> 01:07:19,630 like to the corporations, then there 1162 01:07:19,630 --> 01:07:24,510 is a probability that you won't get your money back. 1163 01:07:24,510 --> 01:07:27,255 So corporations issue bonds, financial instruments 1164 01:07:27,255 --> 01:07:28,460 to raise capital. 1165 01:07:28,460 --> 01:07:31,110 It's, again, very similar to the US treasuries. 1166 01:07:31,110 --> 01:07:34,390 And so you give them $100 and they pay you 5% < every year. 1167 01:07:34,390 --> 01:07:37,430 And then let's say in 10 years, if it's a 10 year bond, 1168 01:07:37,430 --> 01:07:39,377 they are supposed to give your money back. 1169 01:07:39,377 --> 01:07:40,460 But this might not happen. 1170 01:07:40,460 --> 01:07:43,340 Corporations default because they make their own decisions. 1171 01:07:43,340 --> 01:07:45,400 Like something went wrong with economy, 1172 01:07:45,400 --> 01:07:46,820 and so on and so forth. 1173 01:07:46,820 --> 01:07:47,940 It happens. 1174 01:07:47,940 --> 01:07:51,360 So there is some risk which is indicated here. 1175 01:07:51,360 --> 01:07:53,020 We just call it default risk. 1176 01:07:53,020 --> 01:07:55,870 So corporations or private individuals, 1177 01:07:55,870 --> 01:08:00,710 they have a right to default. So they can default. 1178 01:08:00,710 --> 01:08:06,650 And this is reflected in the coupons which they pay. 1179 01:08:06,650 --> 01:08:09,910 So for the US government at the end of 2012. 1180 01:08:09,910 --> 01:08:15,210 A 10 year bond would pay just 1.7% a year. 1181 01:08:15,210 --> 01:08:18,000 Again, we are in extremely low environment which 1182 01:08:18,000 --> 01:08:19,630 looks like almost nothing. 1183 01:08:19,630 --> 01:08:26,069 And remember that even if you're an investor and if you 1184 01:08:26,069 --> 01:08:29,895 buy this bond, then you get your 1% interest 1185 01:08:29,895 --> 01:08:31,770 but then you need to pay taxes on the profit. 1186 01:08:31,770 --> 01:08:34,890 So the return is really very small. 1187 01:08:34,890 --> 01:08:39,060 So then, of course, if you're an investor, then OK. 1188 01:08:39,060 --> 01:08:43,020 The US government securities are assumed to be risk-free, 1189 01:08:43,020 --> 01:08:44,819 so you won't be able to lose money. 1190 01:08:44,819 --> 01:08:47,859 So this is a very important benchmark. 1191 01:08:47,859 --> 01:08:50,365 But then you can buy bonds of corporations. 1192 01:08:50,365 --> 01:08:53,840 But, of course, to compensate for possible default, 1193 01:08:53,840 --> 01:08:56,460 they pay higher coupon. 1194 01:08:56,460 --> 01:08:59,160 For example, at the end of 2012, Morgan Stanley bonds 1195 01:08:59,160 --> 01:09:02,189 would pay around, let's say 5% a year. 1196 01:09:02,189 --> 01:09:03,140 Significantly higher. 1197 01:09:07,394 --> 01:09:08,852 Some governments are right now very 1198 01:09:08,852 --> 01:09:11,210 close to default. So some time ago, 1199 01:09:11,210 --> 01:09:14,380 for example, when Morgan Stanley bonds would pay 5% a year. 1200 01:09:14,380 --> 01:09:17,560 But say, Greece bonds would pay 25%, 30% a year. 1201 01:09:17,560 --> 01:09:23,359 Because nobody knows what's going to happen there. 1202 01:09:23,359 --> 01:09:26,880 It's clear that the economy is not in good shape 1203 01:09:26,880 --> 01:09:30,610 and it all depends on the bailouts. 1204 01:09:30,610 --> 01:09:32,939 Or these bailouts are conditioned, for example, 1205 01:09:32,939 --> 01:09:36,080 that the right government-- if you'll 1206 01:09:36,080 --> 01:09:38,429 be in power and the [INAUDIBLE] is unclear. 1207 01:09:38,429 --> 01:09:39,720 So there's lots of uncertainty. 1208 01:09:44,670 --> 01:09:47,300 Such uncertainties, that's why, essentially, the yield-- 1209 01:09:47,300 --> 01:09:50,689 investors tell you would require very high yield. 1210 01:09:50,689 --> 01:09:56,210 And in the credit derivatives, the fundamental instrument, 1211 01:09:56,210 --> 01:09:58,490 is credit default swap. 1212 01:09:58,490 --> 01:10:03,340 So if you have a risky bond, then in order 1213 01:10:03,340 --> 01:10:07,500 to protect from default you can go, let's say to a bank, 1214 01:10:07,500 --> 01:10:09,160 and buy a credit default swap. 1215 01:10:09,160 --> 01:10:11,000 It basically means that if you hold 1216 01:10:11,000 --> 01:10:16,010 a bond and default happens, then the seller of this protection 1217 01:10:16,010 --> 01:10:17,780 will compensate you for the loss. 1218 01:10:17,780 --> 01:10:23,270 For example, let's say you bought a bond at $100. 1219 01:10:23,270 --> 01:10:26,330 And then, let's say, in one year the corporation defaults. 1220 01:10:26,330 --> 01:10:29,200 And then what happens in this event? 1221 01:10:29,200 --> 01:10:31,342 Then court. 1222 01:10:31,342 --> 01:10:32,450 Court happens. 1223 01:10:32,450 --> 01:10:35,710 And the judge decides how much money is recovered. 1224 01:10:35,710 --> 01:10:38,550 And this money is distributed to the bond investors. 1225 01:10:38,550 --> 01:10:41,530 They're first in the queue. 1226 01:10:41,530 --> 01:10:46,930 And then if, let's say, $0.70 on the dollar were recovered, 1227 01:10:46,930 --> 01:10:54,600 then the default swap will pay you $32 which you lost. 1228 01:10:54,600 --> 01:11:03,100 And very fundamental concept in the world of credit derivatives 1229 01:11:03,100 --> 01:11:07,580 is market implied survival probability. 1230 01:11:07,580 --> 01:11:12,990 So in principle, credit default swaps 1231 01:11:12,990 --> 01:11:15,570 are available for different entities. 1232 01:11:15,570 --> 01:11:16,830 Let's say like Morgan Stanley. 1233 01:11:16,830 --> 01:11:19,360 It could be Verizon. 1234 01:11:19,360 --> 01:11:21,580 Could be AT&T and so on and so forth. 1235 01:11:23,860 --> 01:11:29,244 And [INAUDIBLE] require different payments. 1236 01:11:29,244 --> 01:11:30,910 For example, let's say if credit default 1237 01:11:30,910 --> 01:11:32,930 swap for Morgan Stanley, probably 1238 01:11:32,930 --> 01:11:36,020 is like 5 year maturity, you pay around 100 basis points. 1239 01:11:36,020 --> 01:11:39,140 And if there is some-- like Greece, 1240 01:11:39,140 --> 01:11:43,162 probably, you pay like 500, maybe 1,000 basis points 1241 01:11:43,162 --> 01:11:44,120 or something like this. 1242 01:11:44,120 --> 01:11:46,000 So market differentiates. 1243 01:11:46,000 --> 01:11:52,360 And based on this, you can then do a very simple calculation. 1244 01:11:52,360 --> 01:12:00,060 And you consider, it's very easy to come with a concept 1245 01:12:00,060 --> 01:12:01,450 of the survival probability. 1246 01:12:01,450 --> 01:12:07,580 Roughly speaking if, let's say, default protection 1247 01:12:07,580 --> 01:12:11,840 on some reference entity is worth 1% a year. 1248 01:12:11,840 --> 01:12:15,640 And then what do we see? 1249 01:12:15,640 --> 01:12:18,940 Then with probability 99% a year, you will get your money. 1250 01:12:18,940 --> 01:12:23,350 If probability 1% per year, you will get nothing. 1251 01:12:23,350 --> 01:12:25,190 So you can think about it like this. 1252 01:12:25,190 --> 01:12:29,010 This means you can say the probability to default 1253 01:12:29,010 --> 01:12:33,200 is roughly 1% a year, in this case. 1254 01:12:33,200 --> 01:12:39,390 And then we could talk about survival probabilities, 1255 01:12:39,390 --> 01:12:40,960 which is basically one [INAUDIBLE] 1256 01:12:40,960 --> 01:12:43,240 default probability. 1257 01:12:43,240 --> 01:12:45,994 And you can then come up with the concept 1258 01:12:45,994 --> 01:12:47,910 of survival probabilities, which you can again 1259 01:12:47,910 --> 01:12:53,530 parametrize with forward rates which are called hazard rates. 1260 01:12:53,530 --> 01:12:56,026 So credit derivatives, in a sense, 1261 01:12:56,026 --> 01:12:57,900 they're similar to interest rate derivatives. 1262 01:12:57,900 --> 01:13:00,260 Remember, in the case of interest rate derivatives, 1263 01:13:00,260 --> 01:13:04,250 we were talking about discount factors. 1264 01:13:04,250 --> 01:13:06,540 So this is like the present value. 1265 01:13:06,540 --> 01:13:08,419 Present value of money. 1266 01:13:08,419 --> 01:13:10,710 In terms of world of credit derivatives-- besides this, 1267 01:13:10,710 --> 01:13:13,536 because of course interest rates are also 1268 01:13:13,536 --> 01:13:15,160 very important for credit derivatives-- 1269 01:13:15,160 --> 01:13:17,110 we talk about survival probability. 1270 01:13:17,110 --> 01:13:21,030 Today it's equal to 1, but then it decays. 1271 01:13:21,030 --> 01:13:23,530 And let's say if you have a US government, 1272 01:13:23,530 --> 01:13:25,670 basically it always stay at one. 1273 01:13:25,670 --> 01:13:27,900 And let's say if it's like Morgan Stanley, 1274 01:13:27,900 --> 01:13:29,520 it goes like this. 1275 01:13:29,520 --> 01:13:32,305 If it's some distressed European sovereign, 1276 01:13:32,305 --> 01:13:33,857 it will go like this. 1277 01:13:33,857 --> 01:13:35,690 So basically it's market-implied probability 1278 01:13:35,690 --> 01:13:40,470 of default based on the credit default swap market. 1279 01:13:40,470 --> 01:13:44,060 And the idea of the HJM model for the credit derivatives 1280 01:13:44,060 --> 01:13:49,795 is that-- similar to the dynamic of forward rates in interest 1281 01:13:49,795 --> 01:13:54,320 rate case-- you simply describe the dynamics of hazard rates 1282 01:13:54,320 --> 01:13:58,300 which parametrize your survival probabilities. 1283 01:13:58,300 --> 01:14:05,040 And now let me see. 1284 01:14:05,040 --> 01:14:09,390 Let me show an example of very important type of derivatives, 1285 01:14:09,390 --> 01:14:13,250 which are priced using credit models. 1286 01:14:13,250 --> 01:14:17,980 Let's talk about the corporate callable bonds. 1287 01:14:17,980 --> 01:14:25,880 So it's a very simple instrument. 1288 01:14:25,880 --> 01:14:27,240 Again, I'm a corporation. 1289 01:14:27,240 --> 01:14:28,820 I borrow $100 from you. 1290 01:14:28,820 --> 01:14:33,240 And let's say I'm paying you 5% every year. 1291 01:14:33,240 --> 01:14:39,040 But I have the right at any time-- 1292 01:14:39,040 --> 01:14:45,460 or, let's say, once in three months-- return you this $100, 1293 01:14:45,460 --> 01:14:46,940 and basically close the deal. 1294 01:14:46,940 --> 01:14:49,200 So why is that so valuable for the corporations? 1295 01:14:49,200 --> 01:14:51,470 Because today's environment is such 1296 01:14:51,470 --> 01:14:54,030 that I borrow at a very high rate. 1297 01:14:54,030 --> 01:14:58,320 In this example, let's say I am paying 5% a year. 1298 01:14:58,320 --> 01:15:03,770 And I issued a 10 year bond and there's $100 million notional. 1299 01:15:03,770 --> 01:15:06,400 So basically this means that every year, I 1300 01:15:06,400 --> 01:15:09,180 am paying to the investor 5%. 1301 01:15:09,180 --> 01:15:10,640 $5 million. 1302 01:15:10,640 --> 01:15:12,780 But let's say I'm paying 5%. 1303 01:15:12,780 --> 01:15:14,800 I need this money to run my business and so on. 1304 01:15:14,800 --> 01:15:19,270 So it's some burden, but usually all the corporations 1305 01:15:19,270 --> 01:15:21,580 have significant amount of debt. 1306 01:15:21,580 --> 01:15:24,620 So it's good to have debt if you know how to manage it. 1307 01:15:24,620 --> 01:15:27,530 Now let's say in three years from now, situation changed. 1308 01:15:27,530 --> 01:15:32,100 So now I can borrow money for seven years, 1309 01:15:32,100 --> 01:15:35,800 because initially I issued the bond for 10 years. 1310 01:15:35,800 --> 01:15:37,920 And now I have seven years remaining, 1311 01:15:37,920 --> 01:15:45,450 but it turns out I can issue just a 3%. 1312 01:15:45,450 --> 01:15:46,870 Basically this means if I do this, 1313 01:15:46,870 --> 01:15:51,980 if I exercise my call option, then I will save 5 minus 3-- 1314 01:15:51,980 --> 01:15:55,690 2%-- times $100 million times seven years. 1315 01:15:55,690 --> 01:15:57,660 So it's $14 million. 1316 01:15:57,660 --> 01:16:02,350 So that's kind of why callable debt, it's good to issue it, 1317 01:16:02,350 --> 01:16:04,414 because you can save money. 1318 01:16:04,414 --> 01:16:06,580 It's very similar to what's happening right now also 1319 01:16:06,580 --> 01:16:08,670 for private individuals. 1320 01:16:08,670 --> 01:16:13,760 Because in recent year or couple of years, 1321 01:16:13,760 --> 01:16:16,240 there was lot of refinancing activity in the US. 1322 01:16:16,240 --> 01:16:20,640 Remember rates are at historical low right now. 1323 01:16:20,640 --> 01:16:22,450 So rates are going down, down, down. 1324 01:16:22,450 --> 01:16:26,630 So let's say if you took out a mortgage here at 6%, 1325 01:16:26,630 --> 01:16:30,030 it was like you could refinance at here, for example, 1326 01:16:30,030 --> 01:16:31,170 the same mortgage. 1327 01:16:31,170 --> 01:16:33,570 You could [INAUDIBLE] like at 3.5%. 1328 01:16:33,570 --> 01:16:36,630 So the same [INAUDIBLE] has happened to corporations. 1329 01:16:36,630 --> 01:16:44,330 So in the US, by default, all mortgages are callable. 1330 01:16:44,330 --> 01:16:46,330 And basically by default, everybody 1331 01:16:46,330 --> 01:16:48,010 has a right to refinance. 1332 01:16:48,010 --> 01:16:50,330 So it's not like you issue a 30 year bond 1333 01:16:50,330 --> 01:16:53,010 and then even you're paying a huge coupon, 1334 01:16:53,010 --> 01:16:54,950 even you can refinance lower percent-- which 1335 01:16:54,950 --> 01:16:56,866 might be the case for corporation, by the way. 1336 01:16:56,866 --> 01:17:05,880 But by law in the US, all the mortgages can be refinanced. 1337 01:17:05,880 --> 01:17:07,130 So basically, that's the idea. 1338 01:17:10,730 --> 01:17:16,882 So if you price this kind of instrument as callable bond 1339 01:17:16,882 --> 01:17:18,340 then you need to take into account, 1340 01:17:18,340 --> 01:17:23,380 of course, the interest rate risk because you need 1341 01:17:23,380 --> 01:17:26,770 to understand what is the current level of interest rate 1342 01:17:26,770 --> 01:17:27,840 you can charge. 1343 01:17:27,840 --> 01:17:31,860 And also you need to take into account 1344 01:17:31,860 --> 01:17:35,060 the quality of the issuer. 1345 01:17:35,060 --> 01:17:37,130 So if, let's say again, Greece. 1346 01:17:37,130 --> 01:17:40,960 Or, let's say, Morgan Stanley issue debt right now, 1347 01:17:40,960 --> 01:17:47,060 then Morgan Stanley would pay significantly less. 1348 01:17:47,060 --> 01:17:50,030 It's all [INAUDIBLE] on the fair market. 1349 01:17:50,030 --> 01:17:51,950 [INAUDIBLE] result and subsidies. 1350 01:17:51,950 --> 01:17:54,870 And, of course, Morgan Stanley would pay significantly less 1351 01:17:54,870 --> 01:17:58,090 in the interest because for the case of Greece, 1352 01:17:58,090 --> 01:18:01,570 it is a much higher default risk. 1353 01:18:01,570 --> 01:18:06,750 And as I mentioned, the idea is that you, 1354 01:18:06,750 --> 01:18:11,500 in the world of credit derivatives, 1355 01:18:11,500 --> 01:18:14,400 there is the concept of hazard rates 1356 01:18:14,400 --> 01:18:19,500 which, again, some curve which shows how risky the issuer is 1357 01:18:19,500 --> 01:18:21,590 at some point in the future. 1358 01:18:21,590 --> 01:18:27,110 And here I show the dynamics for the forward rates, 1359 01:18:27,110 --> 01:18:30,270 and here is the dynamics of hazard rates. 1360 01:18:30,270 --> 01:18:33,270 It shows you, basically, how risky the issuer is. 1361 01:18:33,270 --> 01:18:45,160 And then using similar approach-- I show, give you 1362 01:18:45,160 --> 01:18:47,610 as an exercise-- you can prove again-- 1363 01:18:47,610 --> 01:18:53,030 it turns out if you know the volatility of hazard rates, 1364 01:18:53,030 --> 01:19:01,270 then you know how to simulate the dynamics of hazard rates. 1365 01:19:01,270 --> 01:19:06,300 So essentially, it's the dynamics of all this. 1366 01:19:06,300 --> 01:19:11,150 So again, it's the idea-- let me go back just to the stock 1367 01:19:11,150 --> 01:19:13,750 case-- again, it's the idea, it's very simple. 1368 01:19:13,750 --> 01:19:18,550 So you have all the dynamic variables 1369 01:19:18,550 --> 01:19:20,900 like rates and [INAUDIBLE], in this case. 1370 01:19:20,900 --> 01:19:24,100 Then what you do, you simulate that in risk-neutral world. 1371 01:19:24,100 --> 01:19:25,690 You have different path. 1372 01:19:25,690 --> 01:19:27,760 And then you simply average over the pay-off. 1373 01:19:27,760 --> 01:19:32,449 So this is the beauty of the risk-neutral pricing. 1374 01:19:32,449 --> 01:19:34,865 There is a visual framework which is basically implemented 1375 01:19:34,865 --> 01:19:36,500 at all the major banks. 1376 01:19:36,500 --> 01:19:39,890 Which is really like the right approach 1377 01:19:39,890 --> 01:19:42,010 to price very exotic derivatives for which 1378 01:19:42,010 --> 01:19:46,830 it's very hard to find the exact analytical formulas. 1379 01:19:46,830 --> 01:19:56,800 And let me show you one example of securities 1380 01:19:56,800 --> 01:20:00,400 which are issued by big banks. 1381 01:20:00,400 --> 01:20:05,840 And that's where this HJM model and Monte Carlo simulation 1382 01:20:05,840 --> 01:20:08,612 are used all the time because the pay-offs are 1383 01:20:08,612 --> 01:20:09,320 very complicated. 1384 01:20:12,550 --> 01:20:17,744 And example of such a product is called structured note. 1385 01:20:17,744 --> 01:20:18,910 So what's a structured note? 1386 01:20:18,910 --> 01:20:27,480 It's-- again, corporations need to raise money just to run this 1387 01:20:27,480 --> 01:20:28,380 business. 1388 01:20:28,380 --> 01:20:32,459 But, of course, I cannot just get this money for free. 1389 01:20:32,459 --> 01:20:33,625 I need to pay some interest. 1390 01:20:36,460 --> 01:20:42,620 And again, if you look at what happened last year. 1391 01:20:42,620 --> 01:20:47,285 Again, at the end of last year, for example a US 20 year bond 1392 01:20:47,285 --> 01:20:49,260 would pay 1.7%. 1393 01:20:49,260 --> 01:20:51,840 And if you also pay all the taxes, 1394 01:20:51,840 --> 01:20:56,050 then you probably get something like 1.1%. 1395 01:20:56,050 --> 01:20:58,010 And this might be even lower than inflation. 1396 01:20:58,010 --> 01:21:00,790 So investors, especially long-term investors, 1397 01:21:00,790 --> 01:21:04,730 they are not interested in investing in the US treasuries 1398 01:21:04,730 --> 01:21:10,240 because although it's risk free, but there's no return. 1399 01:21:10,240 --> 01:21:12,900 So you want to generate some money. 1400 01:21:12,900 --> 01:21:14,570 So what can you do, then? 1401 01:21:14,570 --> 01:21:17,350 OK, so you don't want to invest into treasuries. 1402 01:21:17,350 --> 01:21:22,420 So then you can try to find some corporate bonds. 1403 01:21:22,420 --> 01:21:29,670 Again, corporates are risky compared to the United States 1404 01:21:29,670 --> 01:21:30,170 government. 1405 01:21:34,720 --> 01:21:41,390 So typical coupon paid by the corporate bonds 1406 01:21:41,390 --> 01:21:42,920 would be higher. 1407 01:21:42,920 --> 01:21:48,551 So let's say 5% for a non-distressed typical US 1408 01:21:48,551 --> 01:21:49,050 corporation. 1409 01:21:52,170 --> 01:21:53,670 But again, 5%. 1410 01:21:53,670 --> 01:22:00,380 Then you need to pay, let's say, 30% tax top of this. 1411 01:22:00,380 --> 01:22:02,290 So you're left 3.3%. 1412 01:22:02,290 --> 01:22:05,660 There's inflation and so on and so forth. 1413 01:22:05,660 --> 01:22:09,380 So it still looks like a low return. 1414 01:22:09,380 --> 01:22:11,250 Of course, [INAUDIBLE] below, you 1415 01:22:11,250 --> 01:22:18,850 can buy some distressed bonds, say from Greece or maybe 1416 01:22:18,850 --> 01:22:21,690 from some distressed corporations, which 1417 01:22:21,690 --> 01:22:23,070 is a much higher. 1418 01:22:23,070 --> 01:22:25,820 But it becomes more like gambling. 1419 01:22:28,720 --> 01:22:30,250 There's so much uncertainties there, 1420 01:22:30,250 --> 01:22:32,440 so it's more like you can get very high return, 1421 01:22:32,440 --> 01:22:39,040 but you can lose everything because basically you're 1422 01:22:39,040 --> 01:22:41,730 bearing very high credit risk. 1423 01:22:41,730 --> 01:22:43,160 So what to do in this situation? 1424 01:22:43,160 --> 01:22:48,110 Turns out that banks issue very special securities called 1425 01:22:48,110 --> 01:22:55,850 structured notes which are very attractive to some investors. 1426 01:22:55,850 --> 01:22:59,170 So let's say Morgan Stanley-- but instead 1427 01:22:59,170 --> 01:23:02,950 of issuing vanilla bond, I am issuing-- and at 5%, 1428 01:23:02,950 --> 01:23:07,720 let's say for 10 years-- I issue a bond which pays 10% a year. 1429 01:23:07,720 --> 01:23:08,860 So much higher coupon. 1430 01:23:08,860 --> 01:23:13,800 But I pay you 10% only if certain market conditions 1431 01:23:13,800 --> 01:23:15,290 are satisfied. 1432 01:23:15,290 --> 01:23:18,770 So let's say market condition like this. 1433 01:23:18,770 --> 01:23:21,740 30 year swap rate is higher than two year swap rate. 1434 01:23:24,670 --> 01:23:29,230 Let's go back to the picture which I drew. 1435 01:23:29,230 --> 01:23:35,650 So essentially this means that if you borrow money, then 1436 01:23:35,650 --> 01:23:38,240 the short term borrowing rate is smaller 1437 01:23:38,240 --> 01:23:40,480 than the long term borrowing rate, which 1438 01:23:40,480 --> 01:23:42,420 usually is the case. 1439 01:23:42,420 --> 01:23:45,820 So basically, let's assume I pay you 10% percent 1440 01:23:45,820 --> 01:23:47,580 if two conditions are satisfied. 1441 01:23:47,580 --> 01:23:51,810 1% is the 30 year borrowing rate in the economy right 1442 01:23:51,810 --> 01:23:54,650 now is higher than two year borrowing 1443 01:23:54,650 --> 01:23:56,630 rate, which is this condition. 1444 01:23:56,630 --> 01:23:58,520 Plus this second condition. 1445 01:23:58,520 --> 01:24:01,210 S&P 500 index is higher than 880. 1446 01:24:03,380 --> 01:24:08,430 So now if these conditions are satisfied, 1447 01:24:08,430 --> 01:24:11,320 then the investor will get 10%. 1448 01:24:11,320 --> 01:24:13,190 If one of these conditions breaks down, 1449 01:24:13,190 --> 01:24:15,950 the investor would get nothing. 1450 01:24:15,950 --> 01:24:17,570 So there are many investors who would 1451 01:24:17,570 --> 01:24:21,270 like to bear this kind of risk because they 1452 01:24:21,270 --> 01:24:26,870 have certain view on how the economy would develop. 1453 01:24:26,870 --> 01:24:29,600 Because right now, for example, S&P 500 index 1454 01:24:29,600 --> 01:24:36,780 is pretty close to 2,000. 1455 01:24:36,780 --> 01:24:39,990 So it's very unlikely that it'll go down 1456 01:24:39,990 --> 01:24:45,250 by the factor of two, which is 880. 1457 01:24:45,250 --> 01:24:46,920 So it's very low probability. 1458 01:24:50,200 --> 01:24:53,570 And then also investor believes that this will never happen. 1459 01:24:53,570 --> 01:24:55,170 So we always will be in the economy 1460 01:24:55,170 --> 01:25:01,770 where it's still more expensive to borrow long-term 1461 01:25:01,770 --> 01:25:03,420 than short term. 1462 01:25:03,420 --> 01:25:06,030 So in this case, it turns out that the coupon 1463 01:25:06,030 --> 01:25:07,290 can be enhanced. 1464 01:25:07,290 --> 01:25:09,950 This is a whole idea of the structured note. 1465 01:25:09,950 --> 01:25:14,390 So instead of setting like a plain coupon, 5%, 1466 01:25:14,390 --> 01:25:17,190 I am selling [INAUDIBLE] the derivative. 1467 01:25:17,190 --> 01:25:19,810 And if investors like it, it's kind of gambling 1468 01:25:19,810 --> 01:25:23,870 but in educated way because there's 1469 01:25:23,870 --> 01:25:27,370 certain economic meaning of these conditions. 1470 01:25:27,370 --> 01:25:29,300 But this can get high return. 1471 01:25:29,300 --> 01:25:35,319 And this is a very popular way of financing because it turns 1472 01:25:35,319 --> 01:25:37,610 out that investors are buying this kind of instruments, 1473 01:25:37,610 --> 01:25:38,651 but they are very unique. 1474 01:25:41,927 --> 01:25:43,010 There's a lot very liquid. 1475 01:25:43,010 --> 01:25:45,230 Therefore when issue this kind of instrument, 1476 01:25:45,230 --> 01:25:48,920 even if you price it correctly using all the models, 1477 01:25:48,920 --> 01:25:51,140 the bank or financial institution 1478 01:25:51,140 --> 01:25:55,570 which issues these instruments can make some extra money. 1479 01:25:55,570 --> 01:26:00,140 So effectively it's cheaper to issue these instruments 1480 01:26:00,140 --> 01:26:02,330 than to issue vanilla bonds. 1481 01:26:02,330 --> 01:26:05,300 And all of these big banks, they have all the machinery 1482 01:26:05,300 --> 01:26:08,110 to risk manage this kind of [INAUDIBLE] derivatives. 1483 01:26:08,110 --> 01:26:11,070 So they know what they are doing. 1484 01:26:11,070 --> 01:26:13,160 So they sell this kind of product, 1485 01:26:13,160 --> 01:26:14,910 and they're hedging their exposure. 1486 01:26:14,910 --> 01:26:18,540 And they realize some profit because you 1487 01:26:18,540 --> 01:26:21,480 can't identify how much [INAUDIBLE] instrument is. 1488 01:26:21,480 --> 01:26:22,439 So it's good for banks. 1489 01:26:22,439 --> 01:26:24,313 And it's also good for investors because they 1490 01:26:24,313 --> 01:26:26,360 are looking for this kind of yield enhancement. 1491 01:26:26,360 --> 01:26:29,400 They want to have a higher yield. 1492 01:26:29,400 --> 01:26:31,600 And they are taking-- and they're 1493 01:26:31,600 --> 01:26:34,730 willing to take this risk. 1494 01:26:34,730 --> 01:26:39,300 But again, it's an educated risk because like, 1495 01:26:39,300 --> 01:26:41,310 this condition, for example, here, they 1496 01:26:41,310 --> 01:26:43,170 have a very clear economical meaning. 1497 01:26:43,170 --> 01:26:45,710 So if an investor understands what's going on, 1498 01:26:45,710 --> 01:26:48,889 then it's a reasonable risk. 1499 01:26:48,889 --> 01:26:50,680 And, of course, what do you do in this case 1500 01:26:50,680 --> 01:26:52,450 if you want to model something like this? 1501 01:26:52,450 --> 01:26:55,510 Then it's very complicated to find 1502 01:26:55,510 --> 01:26:58,100 any kind of analytic approximations 1503 01:26:58,100 --> 01:27:00,360 here in the real world. 1504 01:27:00,360 --> 01:27:02,120 So what do we do? 1505 01:27:02,120 --> 01:27:04,560 We simulate the stock market price. 1506 01:27:04,560 --> 01:27:11,170 We simulate the 30 year yield and 10 year yield. 1507 01:27:11,170 --> 01:27:14,570 And we simulate Morgan Stanley's credit spread. 1508 01:27:14,570 --> 01:27:17,020 And we do it all simultaneously, at the same time. 1509 01:27:17,020 --> 01:27:20,620 And then we see in the Monte Carlo simulation 1510 01:27:20,620 --> 01:27:24,420 if this condition is satisfied for every coupon date, 1511 01:27:24,420 --> 01:27:27,140 then we're paying 10%. 1512 01:27:27,140 --> 01:27:29,630 If something is broken, then we are paying 0. 1513 01:27:29,630 --> 01:27:31,660 So if we simulate many, many paths like this 1514 01:27:31,660 --> 01:27:34,920 and then we calculate the average value of it. 1515 01:27:34,920 --> 01:27:36,630 And then we come up with the price 1516 01:27:36,630 --> 01:27:39,520 and then we quote this price to the investor. 1517 01:27:39,520 --> 01:27:41,961 And again, I say, these products are very nonstandard. 1518 01:27:41,961 --> 01:27:42,460 That's fine. 1519 01:27:42,460 --> 01:27:46,070 You can make some extra money. 1520 01:27:46,070 --> 01:27:50,240 And as a firm, you save money because it's 1521 01:27:50,240 --> 01:27:53,420 cheaper than to issue plain vanilla bonds. 1522 01:27:53,420 --> 01:27:57,020 And just to give you the idea where 1523 01:27:57,020 --> 01:27:58,550 we are in terms of numbers. 1524 01:27:58,550 --> 01:28:05,430 So here there is a graph of difference 1525 01:28:05,430 --> 01:28:08,600 between 30 year borrowing rate and two year borrowing 1526 01:28:08,600 --> 01:28:14,060 rate for the last decade. 1527 01:28:14,060 --> 01:28:17,070 So you see, this difference always positive. 1528 01:28:17,070 --> 01:28:22,260 It was negative only very shortly for some time 1529 01:28:22,260 --> 01:28:25,640 around 2005, 2006. 1530 01:28:25,640 --> 01:28:28,380 So it's very interesting thing. 1531 01:28:28,380 --> 01:28:34,975 So when you price derivative, then there's 1532 01:28:34,975 --> 01:28:38,100 a notion of market-implied numbers. 1533 01:28:38,100 --> 01:28:42,235 It turns out if you look at how different instruments are 1534 01:28:42,235 --> 01:28:44,210 priced on the market, then the probability-- 1535 01:28:47,320 --> 01:28:49,600 Then you can ask a question: What 1536 01:28:49,600 --> 01:28:56,200 is the probability that this-- Let's 1537 01:28:56,200 --> 01:29:00,530 say if I run, for example, this stochastically for the last 10 1538 01:29:00,530 --> 01:29:05,720 years, then how-- what the probability 1539 01:29:05,720 --> 01:29:10,929 that this difference is positive. 1540 01:29:10,929 --> 01:29:12,637 And then it turns out probability is only 1541 01:29:12,637 --> 01:29:13,720 80% percent. 1542 01:29:13,720 --> 01:29:18,010 Whereas in reality, it was realized only for a few days. 1543 01:29:18,010 --> 01:29:21,280 So it's significantly lower. 1544 01:29:21,280 --> 01:29:24,420 So basically, then, the investor says like this. 1545 01:29:24,420 --> 01:29:27,810 So market give me the discount, like 80%. 1546 01:29:27,810 --> 01:29:30,950 But I know that this almost never happen in the past. 1547 01:29:30,950 --> 01:29:34,860 Therefore I believe that it will not happen in the future. 1548 01:29:34,860 --> 01:29:37,680 Maybe it will happen, but I will still 1549 01:29:37,680 --> 01:29:39,530 make some extra money because of this. 1550 01:29:39,530 --> 01:29:43,420 So basically we have [INAUDIBLE] enhancement by a factor 1 1551 01:29:43,420 --> 01:29:44,540 divided by 0.8. 1552 01:29:44,540 --> 01:29:47,690 1.25%. 1553 01:29:47,690 --> 01:29:49,830 Second thing is about S&P 500. 1554 01:29:49,830 --> 01:29:53,720 If you look at the history of this index, which is basically 1555 01:29:53,720 --> 01:29:56,960 the main US market index, then you 1556 01:29:56,960 --> 01:30:07,300 see that it was historically above 880 level for 94 days out 1557 01:30:07,300 --> 01:30:08,930 of 100 days. 1558 01:30:08,930 --> 01:30:10,750 So very, very high probability. 1559 01:30:10,750 --> 01:30:16,194 But the market implies this will be the case only in 75% case. 1560 01:30:16,194 --> 01:30:17,860 The credit investor would say like this. 1561 01:30:17,860 --> 01:30:22,890 OK, now S&P 500 is around 1,800. 1562 01:30:22,890 --> 01:30:26,340 So what the probability it's going to drop below 880? 1563 01:30:26,340 --> 01:30:27,890 Of course there is some probability, 1564 01:30:27,890 --> 01:30:30,360 but if it's going to happen because it will mean 1565 01:30:30,360 --> 01:30:32,570 a very serious recession, and it looks 1566 01:30:32,570 --> 01:30:34,640 like the economy is improving. 1567 01:30:34,640 --> 01:30:36,880 The market might drop down, but maybe 1568 01:30:36,880 --> 01:30:40,890 to the level of 1,500, 1,400. 1569 01:30:40,890 --> 01:30:42,130 But not that low. 1570 01:30:42,130 --> 01:30:48,570 Therefore the investor believes that he, by taking this risk, 1571 01:30:48,570 --> 01:30:52,640 he will again get a higher coupon. 1572 01:30:52,640 --> 01:30:54,960 So [INAUDIBLE] very popular instruments which are 1573 01:30:54,960 --> 01:30:58,754 solely price by Monte Carlo simulation, which-- we have 1574 01:30:58,754 --> 01:30:59,920 big businesses, for example, 1575 01:30:59,920 --> 01:31:04,200 like Morgan Stanley, whose goal is to raise capital 1576 01:31:04,200 --> 01:31:08,680 by selling these exotic products and hedging them using 1577 01:31:08,680 --> 01:31:11,170 the Monte Carlo framework. 1578 01:31:11,170 --> 01:31:13,900 And if the interest rates are crucial for dynamics, 1579 01:31:13,900 --> 01:31:18,260 then we use the HJM model for simulating interest rates. 1580 01:31:18,260 --> 01:31:21,210 So that's everything I wanted to tell you 1581 01:31:21,210 --> 01:31:23,790 about today, so thank you very much. 1582 01:31:23,790 --> 01:31:25,770 [APPLAUSE] 1583 01:31:28,245 --> 01:31:29,730 DENIS GOROKHOV: Yeah? 1584 01:31:29,730 --> 01:31:31,710 AUDIENCE: [INAUDIBLE] simulation. 1585 01:31:31,710 --> 01:31:34,927 Is there some choice-- you might make certain choices 1586 01:31:34,927 --> 01:31:36,560 based on historical precedence? 1587 01:31:36,560 --> 01:31:38,310 DENIS GOROKHOV: It's a very good question. 1588 01:31:38,310 --> 01:31:40,200 So, in reality. 1589 01:31:40,200 --> 01:31:41,930 So here's what happens. 1590 01:31:41,930 --> 01:31:48,140 So let's go to a very simple case of stock prices. 1591 01:31:48,140 --> 01:31:53,710 So again, r here basically is just the borrowing rate. 1592 01:31:53,710 --> 01:31:55,960 It's like, let's say, whatever the bank account gives. 1593 01:31:55,960 --> 01:31:56,960 Which is known. 1594 01:31:56,960 --> 01:31:59,210 So the only parameter which isn't known is volatility. 1595 01:31:59,210 --> 01:32:04,170 So usually, you have liquid stocks, for example. 1596 01:32:04,170 --> 01:32:06,020 Like IBM, Apple. 1597 01:32:06,020 --> 01:32:10,650 Then there are a lot of derivatives traded, 1598 01:32:10,650 --> 01:32:12,180 which are very liquid. 1599 01:32:12,180 --> 01:32:18,902 This means that you can imply this sigma from the price 1600 01:32:18,902 --> 01:32:21,030 of liquid derivatives. 1601 01:32:21,030 --> 01:32:24,630 Because you know, for example, that this particular option-- 1602 01:32:24,630 --> 01:32:27,580 let's say today Apple traded at 600-- 1603 01:32:27,580 --> 01:32:31,750 and you know that at the money option, so option with a strike 1604 01:32:31,750 --> 01:32:35,320 600, in one year now, for example, it's worth whatever. 1605 01:32:35,320 --> 01:32:37,250 Like $50, for example. 1606 01:32:37,250 --> 01:32:40,420 By knowing this, you can imply this sigma. 1607 01:32:40,420 --> 01:32:42,170 So the whole idea is like this. 1608 01:32:42,170 --> 01:32:45,540 So you take very liquid derivatives, like standard call 1609 01:32:45,540 --> 01:32:49,340 options, and you imply this sigma. 1610 01:32:49,340 --> 01:32:51,240 And then you use this model to price 1611 01:32:51,240 --> 01:32:53,561 really truly exotic derivatives, which are not vitally 1612 01:32:53,561 --> 01:32:54,060 available. 1613 01:32:54,060 --> 01:32:55,393 That's how big banks make money. 1614 01:32:55,393 --> 01:32:56,910 Because we know how to price them. 1615 01:32:56,910 --> 01:32:58,830 We have clients come in. 1616 01:32:58,830 --> 01:33:03,060 And we see the prices of very liquid instruments 1617 01:33:03,060 --> 01:33:04,550 and we buy them to hedge. 1618 01:33:04,550 --> 01:33:07,770 So very often what we do is that we do some very complicated 1619 01:33:07,770 --> 01:33:11,190 deal, but we have an ability to off-load it 1620 01:33:11,190 --> 01:33:14,514 into simpler contracts, which we know how to price. 1621 01:33:14,514 --> 01:33:15,180 That's the idea. 1622 01:33:15,780 --> 01:33:18,470 And the same is true for all the other derivatives, 1623 01:33:18,470 --> 01:33:20,670 from credit derivatives or [INAUDIBLE] derivatives. 1624 01:33:20,670 --> 01:33:23,510 So you try to imply the sigma from the market. 1625 01:33:23,510 --> 01:33:26,280 If there is no way to do this-- which 1626 01:33:26,280 --> 01:33:29,424 is very often the case for credit derivatives because 1627 01:33:29,424 --> 01:33:31,090 for the credit derivatives, credit vol-- 1628 01:33:31,090 --> 01:33:36,780 is not very liquid, not liquidly traded. 1629 01:33:36,780 --> 01:33:39,350 Then the best thing that you can do 1630 01:33:39,350 --> 01:33:42,450 is to take historical estimates. 1631 01:33:42,450 --> 01:33:46,750 So we also do this. 1632 01:33:46,750 --> 01:33:49,134 There is nothing else. 1633 01:33:49,134 --> 01:33:49,990 Yeah. 1634 01:33:49,990 --> 01:33:50,506 Yeah? 1635 01:33:50,506 --> 01:33:51,922 AUDIENCE: On your last slide where 1636 01:33:51,922 --> 01:33:56,028 you talked about the implied frequency of the S&P 500 1637 01:33:56,028 --> 01:33:57,966 being lower than 880? 1638 01:33:57,966 --> 01:33:58,840 DENIS GOROKHOV: Yeah. 1639 01:33:58,840 --> 01:34:00,548 AUDIENCE: Was that from historical quotes 1640 01:34:00,548 --> 01:34:02,380 or current quotes? 1641 01:34:02,380 --> 01:34:06,640 DENIS GOROKHOV: OK This number, I 1642 01:34:06,640 --> 01:34:17,180 think, if you go to the end of 2012 and go back to 2002. 1643 01:34:17,180 --> 01:34:18,580 10 years into the past. 1644 01:34:18,580 --> 01:34:28,250 Then I think it was above 880 in 94% of case. 1645 01:34:28,250 --> 01:34:29,630 We can go back. 1646 01:34:29,630 --> 01:34:33,240 So remember, just to the slide I showed in the very beginning. 1647 01:34:36,459 --> 01:34:37,250 Here it was, right. 1648 01:34:37,250 --> 01:34:40,290 So 880 is somewhere here. 1649 01:34:40,290 --> 01:34:42,290 2012 is here. 1650 01:34:42,290 --> 01:34:45,630 You go back 2002. 1651 01:34:45,630 --> 01:34:52,550 It was below 880 around 2000, internet bubble. 1652 01:34:52,550 --> 01:34:59,380 And around, say, 2008, 2009 when we had major banking crisis. 1653 01:34:59,380 --> 01:35:00,490 [INAUDIBLE] just now. 1654 01:35:00,490 --> 01:35:04,340 So you can see probability is not 1655 01:35:04,340 --> 01:35:05,820 very high based on historicals. 1656 01:35:05,820 --> 01:35:07,778 These kind of people believe that in the future 1657 01:35:07,778 --> 01:35:09,780 it might happen, but then the stock 1658 01:35:09,780 --> 01:35:12,760 will go back again because the government will intervene 1659 01:35:12,760 --> 01:35:13,720 and so on and so forth. 1660 01:35:13,720 --> 01:35:15,770 That's the way of thinking of these investors 1661 01:35:15,770 --> 01:35:18,140 who invest into structured notes like this. 1662 01:35:18,140 --> 01:35:21,980 AUDIENCE: So for the implied frequency, 1663 01:35:21,980 --> 01:35:23,180 that's from the current-- 1664 01:35:23,180 --> 01:35:23,540 DENIS GOROKHOV: Exactly. 1665 01:35:23,540 --> 01:35:24,300 AUDIENCE: --option prices-- 1666 01:35:24,300 --> 01:35:25,280 DENIS GOROKHOV: Exactly. 1667 01:35:25,280 --> 01:35:25,750 Exactly. 1668 01:35:25,750 --> 01:35:26,250 Exactly. 1669 01:35:26,250 --> 01:35:28,580 Exactly. 1670 01:35:28,580 --> 01:35:31,950 So now that's how historical was obtained. 1671 01:35:31,950 --> 01:35:33,672 Ah, let me see. 1672 01:35:40,462 --> 01:35:40,962 [INAUDIBLE] 1673 01:35:46,690 --> 01:35:47,190 Yeah. 1674 01:35:47,190 --> 01:35:48,654 Well, let me see. 1675 01:35:55,000 --> 01:35:55,590 So, yes. 1676 01:35:55,590 --> 01:35:56,470 So it's like this. 1677 01:35:56,470 --> 01:36:02,840 So you're today and you have your Monte Carlo model. 1678 01:36:02,840 --> 01:36:06,800 And you simulated going forward for 10 years 1679 01:36:06,800 --> 01:36:13,050 and you see what the probability to be below 880. 1680 01:36:13,050 --> 01:36:15,190 And actually, much higher because usually 1681 01:36:15,190 --> 01:36:21,770 the market is extremely risk-averse. 1682 01:36:21,770 --> 01:36:24,980 So if you're buying a deep out of the money option 1683 01:36:24,980 --> 01:36:28,690 you usually-- there is-- everybody requires premium. 1684 01:36:28,690 --> 01:36:32,220 Because if this happens, if you don't really like 1685 01:36:32,220 --> 01:36:36,020 charge enough money, basically that you're out of business. 1686 01:36:36,020 --> 01:36:38,200 That is how I obtain this number, what, 75%. 1687 01:36:40,840 --> 01:36:41,340 Whatever. 1688 01:36:41,340 --> 01:36:42,753 OK. 1689 01:36:42,753 --> 01:36:44,637 Yeah? 1690 01:36:44,637 --> 01:36:49,710 AUDIENCE: So is the pricing of these more exotic products 1691 01:36:49,710 --> 01:36:52,030 totally reliant upon Monte Carlo, 1692 01:36:52,030 --> 01:36:54,240 or are there other techniques? 1693 01:36:54,240 --> 01:36:57,090 DENIS GOROKHOV: I mean, usually it's Monte Carlo. 1694 01:36:57,090 --> 01:37:02,010 So there are some derivatives where analytical approximations 1695 01:37:02,010 --> 01:37:02,940 are available. 1696 01:37:02,940 --> 01:37:06,880 For example, for interest rate derivative. 1697 01:37:06,880 --> 01:37:09,910 Swaps are like a very simple linear product. 1698 01:37:09,910 --> 01:37:12,170 To price them, you need discount function. 1699 01:37:12,170 --> 01:37:13,560 So it's just arithmetic. 1700 01:37:13,560 --> 01:37:16,040 Of course, it's all done, just simple arithmetic. 1701 01:37:16,040 --> 01:37:18,630 For swaptions, standard swaptions, 1702 01:37:18,630 --> 01:37:20,690 there is a model called SABR model which 1703 01:37:20,690 --> 01:37:22,670 allows some kind of semi-analytical solutions, 1704 01:37:22,670 --> 01:37:25,900 which are approximate but of high quality. 1705 01:37:25,900 --> 01:37:27,140 Then you can do it. 1706 01:37:27,140 --> 01:37:29,170 But there are different schools of thought. 1707 01:37:29,170 --> 01:37:32,380 Because with some approximations, 1708 01:37:32,380 --> 01:37:35,070 which might fail for some if maturity is very long, 1709 01:37:35,070 --> 01:37:38,430 or it's very-- very deep out of the money option. 1710 01:37:38,430 --> 01:37:43,120 So very often what traders do, even if their official numbers 1711 01:37:43,120 --> 01:37:47,520 are only-- more simplified models which kind of has 1712 01:37:47,520 --> 01:37:50,150 some formula, they still round the Monte Carlo simulation 1713 01:37:50,150 --> 01:37:52,260 for the whole portfolio to understand 1714 01:37:52,260 --> 01:37:56,994 what the most complicated model, like in terms 1715 01:37:56,994 --> 01:37:58,660 of your present value of your portfolio, 1716 01:37:58,660 --> 01:38:00,182 in terms of the risk. 1717 01:38:00,182 --> 01:38:01,640 But, of course, this kind of double 1718 01:38:01,640 --> 01:38:04,730 range accruals, which are just [INAUDIBLE]. 1719 01:38:04,730 --> 01:38:07,240 It's impossible to build any meaningful analytical model. 1720 01:38:07,240 --> 01:38:11,230 You can do something, but you won't 1721 01:38:11,230 --> 01:38:12,540 be able to be competitive. 1722 01:38:12,540 --> 01:38:16,392 It's just all Monte Carlo simulation. 1723 01:38:16,392 --> 01:38:19,380 AUDIENCE: So you said usually this whole simulation 1724 01:38:19,380 --> 01:38:22,190 process takes an hour on a MATLAB program? 1725 01:38:24,389 --> 01:38:25,180 DENIS GOROKHOV: No. 1726 01:38:25,180 --> 01:38:26,040 No. 1727 01:38:26,040 --> 01:38:27,770 It takes probably like one hour just 1728 01:38:27,770 --> 01:38:30,580 to write the whole program, because it's very simple. 1729 01:38:30,580 --> 01:38:33,450 So what you do, you have Brownian motion. 1730 01:38:33,450 --> 01:38:34,260 But what you do. 1731 01:38:34,260 --> 01:38:36,170 MATLAB generates Brownian motion. 1732 01:38:36,170 --> 01:38:37,660 So you just do it. 1733 01:38:37,660 --> 01:38:41,060 And then you write the change in your price 1734 01:38:41,060 --> 01:38:43,470 is equal to your drift, which you know, 1735 01:38:43,470 --> 01:38:44,720 plus some random number. 1736 01:38:44,720 --> 01:38:47,510 And you basically just simulate different path. 1737 01:38:47,510 --> 01:38:49,600 And then if you price a call option, 1738 01:38:49,600 --> 01:38:54,920 you know the distribution of your stock prices, 1739 01:38:54,920 --> 01:38:56,920 let's say, in one year from now with maturities. 1740 01:38:56,920 --> 01:38:57,919 And you just do average. 1741 01:38:57,919 --> 01:39:00,270 So it might take someone 15 minutes 1742 01:39:00,270 --> 01:39:01,910 to write this kind of program. 1743 01:39:01,910 --> 01:39:05,350 This is so you can verify numerically the accuracy 1744 01:39:05,350 --> 01:39:08,810 of the-- verify numerically Black-Scholes formula, 1745 01:39:08,810 --> 01:39:09,750 for example. 1746 01:39:09,750 --> 01:39:11,745 But the idea fits very simple here. 1747 01:39:11,745 --> 01:39:13,620 But, of course, for these complicated models, 1748 01:39:13,620 --> 01:39:16,920 which you-- for term structure process like HJM, 1749 01:39:16,920 --> 01:39:19,890 because it's already one-dimensional object. 1750 01:39:19,890 --> 01:39:23,070 And, of course, it's much more complicated 1751 01:39:23,070 --> 01:39:26,200 because besides pricing, you need 1752 01:39:26,200 --> 01:39:28,400 to have this idea of calibration as you mentioned, 1753 01:39:28,400 --> 01:39:32,180 because these volatilities are not just usually historical. 1754 01:39:32,180 --> 01:39:34,467 They're implied from other instruments. 1755 01:39:34,467 --> 01:39:36,050 So what you do in practice, like this. 1756 01:39:36,050 --> 01:39:38,510 So if you have liquid instruments, liquid options, 1757 01:39:38,510 --> 01:39:40,370 you have the model. 1758 01:39:40,370 --> 01:39:42,210 But the model has unknown parameters. 1759 01:39:42,210 --> 01:39:43,890 First, we do the calibration. 1760 01:39:43,890 --> 01:39:45,870 So we make sure that our model prices all 1761 01:39:45,870 --> 01:39:47,260 the simple instruments. 1762 01:39:47,260 --> 01:39:49,950 And then we take the derivative whose price 1763 01:39:49,950 --> 01:39:55,660 is unknown because it's just something very complicated. 1764 01:39:55,660 --> 01:39:57,580 And then we just price it, but our model's 1765 01:39:57,580 --> 01:39:59,180 calibrated to simple derivatives. 1766 01:39:59,180 --> 01:40:01,510 And this tells us-- then this model, 1767 01:40:01,510 --> 01:40:04,290 after pricing it and running sensitivity 1768 01:40:04,290 --> 01:40:05,730 with respect to market parameters, 1769 01:40:05,730 --> 01:40:08,580 tells us how to hedge it. 1770 01:40:08,580 --> 01:40:11,149 That's the idea. 1771 01:40:11,149 --> 01:40:12,940 AUDIENCE: You ought to do post-hoc analyses 1772 01:40:12,940 --> 01:40:17,615 to see how the models did in the past so you can adjust them. 1773 01:40:17,615 --> 01:40:18,490 DENIS GOROKHOV: Yeah. 1774 01:40:18,490 --> 01:40:19,556 Yeah. 1775 01:40:19,556 --> 01:40:21,722 AUDIENCE: Is that a big part of what you have to do? 1776 01:40:25,874 --> 01:40:27,540 DENIS GOROKHOV: I will say in general we 1777 01:40:27,540 --> 01:40:29,110 are moving to this direction. 1778 01:40:29,110 --> 01:40:35,020 In general, of course, for Monte Carlo-- from the [INAUDIBLE] 1779 01:40:35,020 --> 01:40:37,630 point of view for complicated Monte Carlo model, 1780 01:40:37,630 --> 01:40:40,070 it's very difficult to do technically. 1781 01:40:40,070 --> 01:40:42,500 It's very difficult. But if you do it, 1782 01:40:42,500 --> 01:40:46,240 you cannot afford simple models like for swaps and so on. 1783 01:40:46,240 --> 01:40:47,674 AUDIENCE: --historical experience 1784 01:40:47,674 --> 01:40:50,439 with the projection that you made. 1785 01:40:50,439 --> 01:40:51,230 DENIS GOROKHOV: No. 1786 01:40:51,230 --> 01:40:53,160 But the situation, it's very different. 1787 01:40:53,160 --> 01:40:55,730 So we don't make any predictions here, remember. 1788 01:40:55,730 --> 01:40:57,090 It's risk-neutral pricing. 1789 01:40:57,090 --> 01:40:58,980 Just no prediction here. 1790 01:40:58,980 --> 01:41:01,460 What we do here is like this. 1791 01:41:04,300 --> 01:41:07,110 If we are a bank and we want to trade 1792 01:41:07,110 --> 01:41:09,110 all kinds of very exotic derivatives 1793 01:41:09,110 --> 01:41:13,440 which nobody knows how to price but 1794 01:41:13,440 --> 01:41:16,910 we have clients who want to buy them with different reasons. 1795 01:41:16,910 --> 01:41:19,170 Might want to speculate or they want 1796 01:41:19,170 --> 01:41:23,650 to manage their risk exposure and so on and so forth. 1797 01:41:23,650 --> 01:41:26,990 So nobody knows except for like 10, 20 banks, 1798 01:41:26,990 --> 01:41:27,740 how to price them. 1799 01:41:27,740 --> 01:41:29,345 Because this is like, you need to have infrastructure. 1800 01:41:29,345 --> 01:41:31,160 You need to know how to do this. 1801 01:41:31,160 --> 01:41:34,070 Then you need to have some business channels, 1802 01:41:34,070 --> 01:41:35,500 how to off-load this risk. 1803 01:41:35,500 --> 01:41:38,110 So this is some very exotic products. 1804 01:41:38,110 --> 01:41:41,330 So now the idea of dynamic hedging is like this. 1805 01:41:41,330 --> 01:41:43,520 Remember, in the case of Black-Scholes. 1806 01:41:47,440 --> 01:41:51,480 You buy an option and then you hedge it 1807 01:41:51,480 --> 01:41:54,240 by holding a certain amount of the underlying. 1808 01:41:54,240 --> 01:41:56,120 So you don't make any money. 1809 01:41:56,120 --> 01:42:04,170 But you want to make sure that whatever happens to the market, 1810 01:42:04,170 --> 01:42:05,110 you're fully hedged. 1811 01:42:05,110 --> 01:42:07,680 So the market moves here, you don't make any money. 1812 01:42:07,680 --> 01:42:13,070 The market moves down, you don't make any money. 1813 01:42:13,070 --> 01:42:16,210 So the way how you make money in this situation, basically 1814 01:42:16,210 --> 01:42:20,930 this Black-Scholes formula, in this case, the price which 1815 01:42:20,930 --> 01:42:24,740 you charge for the option is the price 1816 01:42:24,740 --> 01:42:26,560 of executing the hedging strategy. 1817 01:42:26,560 --> 01:42:28,500 So if you charge a little bit more, 1818 01:42:28,500 --> 01:42:31,850 this is extra money which you can make. 1819 01:42:31,850 --> 01:42:32,980 So it's very different. 1820 01:42:32,980 --> 01:42:36,500 So what you just mentioned is like proprietary business. 1821 01:42:36,500 --> 01:42:39,710 Big banks, they are not supposed to do this. 1822 01:42:39,710 --> 01:42:41,944 It's more like a hedge fund world. 1823 01:42:41,944 --> 01:42:42,860 Very different models. 1824 01:42:42,860 --> 01:42:48,220 What we do, we try to manage big portfolios of derivatives, 1825 01:42:48,220 --> 01:42:50,010 all kinds of derivatives. 1826 01:42:50,010 --> 01:42:56,010 And we try to price them and charge a little bit extra so 1827 01:42:56,010 --> 01:42:57,510 that we can make our living. 1828 01:42:57,510 --> 01:43:00,420 But on the other hand, we don't take any risk. 1829 01:43:00,420 --> 01:43:02,300 That's the idea. 1830 01:43:02,300 --> 01:43:04,170 [INAUDIBLE] just models. 1831 01:43:04,170 --> 01:43:07,090 So from point of view in terms of testing historically, 1832 01:43:07,090 --> 01:43:08,470 you can still ask a question. 1833 01:43:08,470 --> 01:43:13,930 Let's say if I go back 10 years. 1834 01:43:13,930 --> 01:43:16,080 And let's say 10 years ago, I would sell, 1835 01:43:16,080 --> 01:43:18,840 for example, this stock option. 1836 01:43:18,840 --> 01:43:22,320 And for the next 10 years, using historical data, 1837 01:43:22,320 --> 01:43:26,170 I see basically how-- my model then 1838 01:43:26,170 --> 01:43:29,790 tells me what my Greeks or like what my sensitivity 1839 01:43:29,790 --> 01:43:32,716 with respect to the underlying-- what 1840 01:43:32,716 --> 01:43:34,590 my sensitivity with respect to underlying is. 1841 01:43:34,590 --> 01:43:36,860 And then you can ask a question. 1842 01:43:36,860 --> 01:43:40,950 How was this delta H performing historically? 1843 01:43:40,950 --> 01:43:44,730 Which is a reasonable question because maybe you 1844 01:43:44,730 --> 01:43:47,730 assume that the model pretty much continuous. 1845 01:43:47,730 --> 01:43:50,110 But maybe if your dynamics is very jerky, 1846 01:43:50,110 --> 01:43:53,120 then you can just lose money because you just don't take 1847 01:43:53,120 --> 01:43:55,030 into account these effects. 1848 01:43:55,030 --> 01:43:57,494 This is an example of historical analysis which we may run, 1849 01:43:57,494 --> 01:43:59,410 but it has nothing to do with prediction here. 1850 01:43:59,410 --> 01:44:00,743 So it's a whole different world. 1851 01:44:00,743 --> 01:44:02,480 So it's risk-neutral pricing. 1852 01:44:02,480 --> 01:44:03,680 So we don't take any risk. 1853 01:44:03,680 --> 01:44:04,751 That's the whole idea. 1854 01:44:04,751 --> 01:44:07,000 But due to the fact that derivatives are very complex, 1855 01:44:07,000 --> 01:44:12,690 even in this case, still banks bear some residual risk, 1856 01:44:12,690 --> 01:44:16,430 because remember we cannot exactly off-load it, the risk. 1857 01:44:16,430 --> 01:44:19,730 So we still have some assumptions 1858 01:44:19,730 --> 01:44:22,520 that we can re-balance our position dynamically and move 1859 01:44:22,520 --> 01:44:25,820 forward, basically, and not lose money. 1860 01:44:25,820 --> 01:44:28,206 That's the idea of it. 1861 01:44:28,206 --> 01:44:33,970 AUDIENCE: I have a question about the Monte Carlo pricing. 1862 01:44:33,970 --> 01:44:37,460 You can set up the Monte Carlo using implied parameters 1863 01:44:37,460 --> 01:44:39,685 from current prices of various derivatives 1864 01:44:39,685 --> 01:44:43,672 in the market, which gives you a good baseline price. 1865 01:44:43,672 --> 01:44:46,395 I'm wondering what other Monte Carlos 1866 01:44:46,395 --> 01:44:53,200 do you do to have a robust estimate of pricing, hedging 1867 01:44:53,200 --> 01:44:55,690 cost. 1868 01:44:55,690 --> 01:44:58,310 I would think that there would be, I don't know, 1869 01:44:58,310 --> 01:45:04,846 maybe some stress scenarios in the market or alternatives. 1870 01:45:04,846 --> 01:45:08,480 You probably don't just do one Monte Carlo 1871 01:45:08,480 --> 01:45:10,850 study with current parameters. 1872 01:45:10,850 --> 01:45:13,040 You probably have different sets. 1873 01:45:13,040 --> 01:45:15,950 And I'm wondering how extensive is that? 1874 01:45:17,255 --> 01:45:18,380 DENIS GOROKHOV: Absolutely. 1875 01:45:18,380 --> 01:45:18,963 You are right. 1876 01:45:18,963 --> 01:45:21,020 So if you just do the Monte Carlo, 1877 01:45:21,020 --> 01:45:23,140 then you just know the price. 1878 01:45:23,140 --> 01:45:27,115 But price is nothing, because dynamic hedging, 1879 01:45:27,115 --> 01:45:28,490 all this business of derivatives, 1880 01:45:28,490 --> 01:45:30,550 it's not just about how much it's right now, 1881 01:45:30,550 --> 01:45:33,380 but what to do if the market behaves this way. 1882 01:45:33,380 --> 01:45:36,120 So of course you could collate all your Greeks. 1883 01:45:36,120 --> 01:45:37,140 That's very important. 1884 01:45:37,140 --> 01:45:41,160 But Greeks is like, say, your delta. 1885 01:45:41,160 --> 01:45:42,340 It's all about linear terms. 1886 01:45:42,340 --> 01:45:44,460 So of course it's a very important thing. 1887 01:45:44,460 --> 01:45:46,160 What happen to the portfolio, let's 1888 01:45:46,160 --> 01:45:48,960 say, if there is a very sharp, for example, 1889 01:45:48,960 --> 01:45:50,180 jump in interest rate. 1890 01:45:50,180 --> 01:45:55,950 So let's say, what happens if rates jump forward by 1%. 1891 01:45:55,950 --> 01:45:57,590 Or if they jump down. 1892 01:45:57,590 --> 01:46:02,390 What happens if volatility in a particular time, 1893 01:46:02,390 --> 01:46:04,320 region, for example, blows up. 1894 01:46:04,320 --> 01:46:06,040 You run all these kind of analyses. 1895 01:46:06,040 --> 01:46:09,380 So it's big departments at the banks who 1896 01:46:09,380 --> 01:46:10,685 look at all this kind of risks. 1897 01:46:10,685 --> 01:46:15,919 So it all comes to one business unit 1898 01:46:15,919 --> 01:46:17,710 which looks all kinds of risks of the firm. 1899 01:46:20,490 --> 01:46:27,150 It's a very big thing for the bank. 1900 01:46:27,150 --> 01:46:29,190 This notion of stress test. 1901 01:46:29,190 --> 01:46:31,780 Basically right now, all of the banks 1902 01:46:31,780 --> 01:46:33,980 are very heavily regulated by the government. 1903 01:46:33,980 --> 01:46:36,210 So the government can tell us what happens. 1904 01:46:36,210 --> 01:46:38,060 For example, for the whole bank-- 1905 01:46:38,060 --> 01:46:40,477 not just for a particular desk which trades, 1906 01:46:40,477 --> 01:46:41,310 whatever, swaptions. 1907 01:46:41,310 --> 01:46:44,800 What happens to all your bank, to all kinds of cash flows 1908 01:46:44,800 --> 01:46:46,430 which you can have if, let's say, 1909 01:46:46,430 --> 01:46:49,110 interest rates jump by 100% percent. 1910 01:46:49,110 --> 01:46:54,870 We have a huge group of people, quants, IT, risk managers, 1911 01:46:54,870 --> 01:46:56,870 who are looking at all these numbers trying 1912 01:46:56,870 --> 01:46:57,900 to understand it. 1913 01:46:57,900 --> 01:47:02,340 And for a big bank, very non-trivial problem, actually. 1914 01:47:02,340 --> 01:47:04,195 So it's very good point. 1915 01:47:04,195 --> 01:47:08,190 But, of course, we do as good as we can. 1916 01:47:08,190 --> 01:47:10,022 Yeah. 1917 01:47:10,022 --> 01:47:11,347 AUDIENCE: Well, thanks again. 1918 01:47:11,347 --> 01:47:12,930 And for a little time afterwards for-- 1919 01:47:12,930 --> 01:47:13,530 [APPLAUSE] 1920 01:47:13,530 --> 01:47:14,723 DENIS GOROKHOV: Thank you.