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 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,118 --> 00:00:22,534 PROFESSOR: Our guest speaker today 9 00:00:22,534 --> 00:00:25,390 from Morgan Stanley, Ivan Masyukov. 10 00:00:25,390 --> 00:00:28,336 Dr. Ivan Masyukov. 11 00:00:28,336 --> 00:00:29,210 IVAN MASYUKOV: Hello. 12 00:00:29,210 --> 00:00:30,260 One, two, three. 13 00:00:30,260 --> 00:00:31,206 Can you hear me? 14 00:00:31,206 --> 00:00:34,280 PROFESSOR: And the microphone will just be recording you, 15 00:00:34,280 --> 00:00:35,940 but it doesn't broadcast you. 16 00:00:35,940 --> 00:00:36,970 IVAN MASYUKOV: Ah. 17 00:00:36,970 --> 00:00:37,470 Understood. 18 00:00:37,470 --> 00:00:37,970 All right. 19 00:00:37,970 --> 00:00:40,950 So I'm Ivan Masyukov. 20 00:00:40,950 --> 00:00:42,540 I work in Morgan Stanley. 21 00:00:42,540 --> 00:00:48,420 And my background is applied physics and mathematics 22 00:00:48,420 --> 00:00:52,970 from Moscow Institute of Physics and Technology. 23 00:00:52,970 --> 00:00:59,905 And today, the topic of the lecture 24 00:00:59,905 --> 00:01:03,860 is regularized pricing and risk models. 25 00:01:03,860 --> 00:01:07,600 So we will talk about typical pricing risk 26 00:01:07,600 --> 00:01:13,430 models for interest rate products, 27 00:01:13,430 --> 00:01:17,400 and the important aspect of adding 28 00:01:17,400 --> 00:01:20,340 some additional constraints, which means, 29 00:01:20,340 --> 00:01:22,280 like, adding some regularizers to the model. 30 00:01:24,900 --> 00:01:28,020 So we will start from bonds, which 31 00:01:28,020 --> 00:01:31,200 is probably the most simple interest rate 32 00:01:31,200 --> 00:01:33,110 product on the market. 33 00:01:33,110 --> 00:01:36,830 Then we will discuss swaps. 34 00:01:36,830 --> 00:01:39,980 We will build a yield curve. 35 00:01:39,980 --> 00:01:44,900 And we will see how yield curve models 36 00:01:44,900 --> 00:01:51,550 can be improved to satisfy needs of actual trader. 37 00:01:51,550 --> 00:01:55,220 And at the end, we'll look at the very nice example 38 00:01:55,220 --> 00:01:58,970 of ill-posed problem of calibrating 39 00:01:58,970 --> 00:02:01,050 the two-dimensional volatility surface 40 00:02:01,050 --> 00:02:05,690 necessary for volatility model-- Monte Carlo assimilation. 41 00:02:05,690 --> 00:02:11,074 And we will see how that problem can be solved. 42 00:02:11,074 --> 00:02:12,990 During the lecture, if you have any questions, 43 00:02:12,990 --> 00:02:14,800 please interrupt, OK? 44 00:02:18,490 --> 00:02:19,960 So what is bond? 45 00:02:19,960 --> 00:02:25,450 Bond is a security which is issued if someone 46 00:02:25,450 --> 00:02:27,200 like a borrower needs money. 47 00:02:27,200 --> 00:02:32,400 And it promises to pay some certain fixed amount 48 00:02:32,400 --> 00:02:35,200 of certain cash flows in the future, 49 00:02:35,200 --> 00:02:38,590 and request for some money up front for this. 50 00:02:38,590 --> 00:02:47,170 So typical bonds basically include same periodic payment-- 51 00:02:47,170 --> 00:02:50,720 let's say like every half year or every year 52 00:02:50,720 --> 00:02:55,190 until maturity, where at maturity the face value is 53 00:02:55,190 --> 00:02:58,540 paid, like the biggest sum of money. 54 00:02:58,540 --> 00:03:03,620 And again, during the beginning, the investor 55 00:03:03,620 --> 00:03:06,730 is asked to pay some up front. 56 00:03:06,730 --> 00:03:08,780 There are also zero-coupon bonds, 57 00:03:08,780 --> 00:03:12,070 which don't pay anything until the maturity. 58 00:03:14,840 --> 00:03:17,800 And there are very interesting perpetual bonds 59 00:03:17,800 --> 00:03:20,620 where basically you pay some money up front, 60 00:03:20,620 --> 00:03:24,560 and then you pay it back like infinitely-- which 61 00:03:24,560 --> 00:03:26,340 sounds like a good deal, but we will 62 00:03:26,340 --> 00:03:28,200 know how to price it right. 63 00:03:31,190 --> 00:03:34,730 So those are some diagrams. 64 00:03:34,730 --> 00:03:38,740 So the first one is the standard fixed-rate bond, 65 00:03:38,740 --> 00:03:47,900 where small green arrows represent a periodic payment. 66 00:03:47,900 --> 00:03:51,700 And there is a face value added on top of a periodic payment 67 00:03:51,700 --> 00:03:53,210 at the maturity of the bond. 68 00:03:56,680 --> 00:04:01,300 So this is a typical cash flow diagram used for analysis, OK? 69 00:04:01,300 --> 00:04:06,306 And so arrows up represent something that-- 70 00:04:06,306 --> 00:04:07,180 and it's green right? 71 00:04:07,180 --> 00:04:08,013 That is good for us. 72 00:04:08,013 --> 00:04:10,470 So it's something that we receive. 73 00:04:10,470 --> 00:04:14,680 And a red arrow facing down represents something 74 00:04:14,680 --> 00:04:16,095 that you have to pay. 75 00:04:16,095 --> 00:04:16,750 Right? 76 00:04:16,750 --> 00:04:25,110 So a zero coupon bond is something, as I said before, 77 00:04:25,110 --> 00:04:28,370 is something that you pay up front, 78 00:04:28,370 --> 00:04:34,250 and you get back a fixed amount of money in the future. 79 00:04:34,250 --> 00:04:36,500 What's interesting about this graph-- you 80 00:04:36,500 --> 00:04:41,630 can see that the green arrow has a bigger 81 00:04:41,630 --> 00:04:45,480 amplitude than the red one, which means that you kind 82 00:04:45,480 --> 00:04:50,620 of, every time you put like $100 now, right, 83 00:04:50,620 --> 00:04:52,482 you kind of expect that in return you 84 00:04:52,482 --> 00:04:53,440 get more in the future. 85 00:04:53,440 --> 00:04:55,340 Because if you don't get more in the future, 86 00:04:55,340 --> 00:04:57,509 you just don't get this money, don't put this money. 87 00:04:57,509 --> 00:04:58,800 You just keep it in the pocket. 88 00:05:01,330 --> 00:05:08,160 So as a result, you get the concept of time value of money. 89 00:05:08,160 --> 00:05:16,480 So tomorrow, $100 always will be more than just $100. 90 00:05:16,480 --> 00:05:21,980 And also, if you look at the graph of the fixed-rate coupon 91 00:05:21,980 --> 00:05:26,320 bond, and you sum all of the cash flows here, 92 00:05:26,320 --> 00:05:29,260 it looks like you get more than this red one. 93 00:05:29,260 --> 00:05:36,460 But again, there is, as further in the future 94 00:05:36,460 --> 00:05:41,170 the cash flow is, the more the kind of depreciation. 95 00:05:41,170 --> 00:05:44,930 And we call this depreciation a discount factor, OK? 96 00:05:44,930 --> 00:05:50,490 So basically the more in the future the cash 97 00:05:50,490 --> 00:05:53,980 flow is, the smaller the discount factor. 98 00:05:53,980 --> 00:05:55,550 And so for today the discount factor 99 00:05:55,550 --> 00:06:00,440 will be 1, for tomorrow it will be like 0.999, and so forth. 100 00:06:00,440 --> 00:06:04,760 And in 30 years, let's say, it will probably be like 0.1, 101 00:06:04,760 --> 00:06:07,755 depending on current rates in the market. 102 00:06:13,410 --> 00:06:20,370 So let's see how we can price the bond-- or not 103 00:06:20,370 --> 00:06:25,440 necessarily price, but compute a fair value 104 00:06:25,440 --> 00:06:28,840 of future cash flows. 105 00:06:28,840 --> 00:06:33,670 So our fair value of computed cash flows 106 00:06:33,670 --> 00:06:36,875 can be found if we have discount factors. 107 00:06:36,875 --> 00:06:39,690 So every discount factor at every cash flow 108 00:06:39,690 --> 00:06:43,130 in the future I-- which in this particular case 109 00:06:43,130 --> 00:06:47,540 will be a coupon times the face value-- 110 00:06:47,540 --> 00:06:49,930 should be multiplied by the discount factor. 111 00:06:49,930 --> 00:06:52,930 And then we also add a face value discount 112 00:06:52,930 --> 00:06:56,215 with a discount factor at the maturity of the instrument. 113 00:07:00,820 --> 00:07:06,085 So the way this product trades in the market 114 00:07:06,085 --> 00:07:13,530 is that people buy and sell bonds paying P, right? 115 00:07:13,530 --> 00:07:17,300 So it's very important to understand that for bonds, it's 116 00:07:17,300 --> 00:07:20,530 not something that we have cash flows which we kind of need 117 00:07:20,530 --> 00:07:21,660 to price. 118 00:07:21,660 --> 00:07:25,100 It's actually the price is already known. 119 00:07:25,100 --> 00:07:27,796 So it's very liquid. 120 00:07:27,796 --> 00:07:30,360 It's the result of activity in the market, 121 00:07:30,360 --> 00:07:32,860 meaning that there is very little uncertainty 122 00:07:32,860 --> 00:07:33,820 about the price. 123 00:07:33,820 --> 00:07:36,810 So this P is known. 124 00:07:36,810 --> 00:07:39,050 And as with all cash flows, it's something that's 125 00:07:39,050 --> 00:07:40,540 written in the contract, right? 126 00:07:40,540 --> 00:07:45,480 So it's something, we have fixed cash flows in the future. 127 00:07:45,480 --> 00:07:50,000 So it's always a question about what kind of model 128 00:07:50,000 --> 00:07:54,910 is useful for the discount factors. 129 00:07:54,910 --> 00:07:56,850 So we need a model for discounting. 130 00:08:00,090 --> 00:08:01,150 Any questions so far? 131 00:08:04,940 --> 00:08:11,230 So one of the simplest models is to use just one parameter 132 00:08:11,230 --> 00:08:16,470 to kind of cover all the discounting. 133 00:08:16,470 --> 00:08:20,300 And the discount factor can be represented 134 00:08:20,300 --> 00:08:27,600 as e minus y times t of i, t sub i, where 135 00:08:27,600 --> 00:08:33,226 y is some kind of-- it's called yield to maturity. 136 00:08:33,226 --> 00:08:34,809 Well, the reason why it's exponential, 137 00:08:34,809 --> 00:08:36,270 it's natural right? 138 00:08:36,270 --> 00:08:42,916 So if you have a 0.999 discount factor for today, 139 00:08:42,916 --> 00:08:44,290 and then we kind of say, OK, it's 140 00:08:44,290 --> 00:08:45,800 the same discount for tomorrow, we 141 00:08:45,800 --> 00:08:48,300 will have the same discounting for every other day. 142 00:08:48,300 --> 00:08:49,590 So we have to multiply them. 143 00:08:49,590 --> 00:08:52,240 As a result, the total discounting 144 00:08:52,240 --> 00:08:56,140 will be an exponential. 145 00:08:56,140 --> 00:09:00,630 So if our discount factors are like this, 146 00:09:00,630 --> 00:09:07,020 then our price basically can be represented 147 00:09:07,020 --> 00:09:09,770 as a linear combination of future cash flows, right? 148 00:09:09,770 --> 00:09:12,200 At this point, by the way, we kind of 149 00:09:12,200 --> 00:09:17,090 merge together the final coupon with the face value, 150 00:09:17,090 --> 00:09:20,470 and we'll just kind of be talking about the coupons 151 00:09:20,470 --> 00:09:24,190 only, about cash flows only. 152 00:09:24,190 --> 00:09:31,655 And so that's the formula for the bond price, is this. 153 00:09:31,655 --> 00:09:36,020 So basically, what's known on the market is P, right, 154 00:09:36,020 --> 00:09:40,420 which is a price that's-- that instrument is traded. 155 00:09:40,420 --> 00:09:43,910 We also have defined cash flows in the future. 156 00:09:43,910 --> 00:09:47,510 So we can solve for the yield. 157 00:09:47,510 --> 00:09:54,210 So essentially, if we know the bond price, 158 00:09:54,210 --> 00:09:57,710 we can find the bond yield, OK? 159 00:09:57,710 --> 00:10:02,170 And if we know the bond yield, we can find the bond price, OK? 160 00:10:02,170 --> 00:10:08,480 So typically, bonds are traded in terms of its price. 161 00:10:08,480 --> 00:10:11,630 But some bonds are traded in terms of yield. 162 00:10:11,630 --> 00:10:15,280 But again, this is like one-to-one. 163 00:10:15,280 --> 00:10:17,240 You can always go back and forth. 164 00:10:21,140 --> 00:10:27,490 What's important is-- what has economic value, 165 00:10:27,490 --> 00:10:31,680 right, is the bond price, OK, and the future cash 166 00:10:31,680 --> 00:10:34,200 amounts of cash flows. 167 00:10:34,200 --> 00:10:36,644 And when you talk about yield, it's 168 00:10:36,644 --> 00:10:37,810 not something that's traded. 169 00:10:37,810 --> 00:10:44,240 It's actually one of the ways to align future cash flows 170 00:10:44,240 --> 00:10:46,170 with the bond price. 171 00:10:46,170 --> 00:10:49,150 And that way assumes that we have, 172 00:10:49,150 --> 00:10:55,110 again, constant discounting for all time points in the future. 173 00:10:55,110 --> 00:10:58,700 And we will see that it may or may not be optimal case. 174 00:11:03,250 --> 00:11:08,180 So what's also important when we're talking about instrument 175 00:11:08,180 --> 00:11:11,380 price is to have the model of how that price changes 176 00:11:11,380 --> 00:11:13,610 if the market changes. 177 00:11:13,610 --> 00:11:20,530 So here, we're talking about sensitivity of the bond price 178 00:11:20,530 --> 00:11:21,130 to yield. 179 00:11:21,130 --> 00:11:25,630 And what is typically done is basically 180 00:11:25,630 --> 00:11:27,435 to normalize by the bond price itself. 181 00:11:29,950 --> 00:11:33,720 And then it's called bond duration. 182 00:11:33,720 --> 00:11:37,530 So the nice thing about normalizing 183 00:11:37,530 --> 00:11:40,980 is that the duration of that bond that you 184 00:11:40,980 --> 00:11:44,620 have in your portfolio doesn't really depend on how many bonds 185 00:11:44,620 --> 00:11:45,400 you have, right? 186 00:11:45,400 --> 00:11:50,330 So it's basically more like property of the bond itself, 187 00:11:50,330 --> 00:11:55,900 rather than how many bonds you have in your portfolio. 188 00:11:55,900 --> 00:11:58,640 So if you take the previous formula 189 00:11:58,640 --> 00:12:01,320 and take the derivative with respect to y, 190 00:12:01,320 --> 00:12:07,150 we get the following formula for duration. 191 00:12:07,150 --> 00:12:12,360 And we know what the price is, right? 192 00:12:12,360 --> 00:12:17,540 And we can rewrite this formula this way, 193 00:12:17,540 --> 00:12:27,680 which you see it's a sum of t_i's times 194 00:12:27,680 --> 00:12:31,370 some weights and divided by the sum of the weights. 195 00:12:31,370 --> 00:12:35,840 So it's essentially a weighted sum of time, OK? 196 00:12:35,840 --> 00:12:41,910 And those pieces of time, those moments of time 197 00:12:41,910 --> 00:12:46,580 is more important, as-- I mean the weights are 198 00:12:46,580 --> 00:12:51,870 proportional to present values of future cash flows. 199 00:12:51,870 --> 00:12:56,250 So that's why bond duration has a very nice kind 200 00:12:56,250 --> 00:12:57,265 of intuitive sense. 201 00:13:00,320 --> 00:13:04,280 As a result of that, and yeah, I forgot to mention one thing. 202 00:13:04,280 --> 00:13:07,540 So the duration is always negative, right? 203 00:13:07,540 --> 00:13:15,570 So we have a sign here, because if the bond price goes up, 204 00:13:15,570 --> 00:13:19,430 this means that the yields goes down, OK? 205 00:13:19,430 --> 00:13:24,610 And if the yield goes up, price goes down. 206 00:13:24,610 --> 00:13:26,450 And the explanation's very simple. 207 00:13:26,450 --> 00:13:29,120 So yield is kind of the same thing 208 00:13:29,120 --> 00:13:33,900 as interest rate on the market. 209 00:13:33,900 --> 00:13:37,270 So if those rates go up, this means 210 00:13:37,270 --> 00:13:40,540 that there will be more discounting in the future cash 211 00:13:40,540 --> 00:13:43,220 flows, they will be less valuable to me. 212 00:13:43,220 --> 00:13:47,770 So I'll be less willing to pay for those cash flows, OK? 213 00:13:47,770 --> 00:13:52,280 So it's kind of fundamental that relationship 214 00:13:52,280 --> 00:13:54,530 has a negative sign. 215 00:13:54,530 --> 00:13:59,790 So in case of a zero-coupon bond, 216 00:13:59,790 --> 00:14:02,900 we only have one cash flow in the future. 217 00:14:02,900 --> 00:14:07,720 So there is just one weight, and that weight 218 00:14:07,720 --> 00:14:10,220 is totally assigned to that last cash flow. 219 00:14:10,220 --> 00:14:18,780 So duration of zero-coupon bond equals to maturity. 220 00:14:18,780 --> 00:14:21,400 Duration of regular coupon bond depends, 221 00:14:21,400 --> 00:14:22,910 but it's always less than maturity, 222 00:14:22,910 --> 00:14:26,055 just because we'll have a weighted sum formula here. 223 00:14:32,090 --> 00:14:36,450 So essentially, that model for the bond duration 224 00:14:36,450 --> 00:14:44,370 kind of assumes that all rates-- so we have just one yield 225 00:14:44,370 --> 00:14:47,350 number for everything, so all rates go 226 00:14:47,350 --> 00:14:55,090 like in a parallel way, which was OK before the crisis, 227 00:14:55,090 --> 00:14:57,357 right, where kind of rates today are 228 00:14:57,357 --> 00:15:01,210 kind of similar to the rates expected in the future. 229 00:15:01,210 --> 00:15:02,600 But it's no longer the case. 230 00:15:02,600 --> 00:15:06,400 So the rates now, they're higher than like one year ago, 231 00:15:06,400 --> 00:15:09,300 but they're still much lower than expected in the future. 232 00:15:09,300 --> 00:15:12,720 So we expect that the rates will go very high. 233 00:15:12,720 --> 00:15:15,690 So the curve is very steep at the moment. 234 00:15:15,690 --> 00:15:21,254 So that model of just one number for everything 235 00:15:21,254 --> 00:15:22,170 might not be adequate. 236 00:15:22,170 --> 00:15:24,415 And we'll see how we can improve this situation. 237 00:15:29,250 --> 00:15:31,970 So it's worth mentioning the second derivative. 238 00:15:31,970 --> 00:15:34,230 We already spoke about the price, 239 00:15:34,230 --> 00:15:36,790 first derivative of the price with respect to yield, 240 00:15:36,790 --> 00:15:38,350 and a second. 241 00:15:38,350 --> 00:15:41,270 So for small changes in the yield, 242 00:15:41,270 --> 00:15:46,390 you can assume that it's linear, so it's 243 00:15:46,390 --> 00:15:48,120 OK to use just the first derivative. 244 00:15:48,120 --> 00:15:56,420 So second derivative will be necessary for larger movements 245 00:15:56,420 --> 00:15:59,270 of the market. 246 00:15:59,270 --> 00:16:07,557 Like as an example, if you're a trader, right, and the bond 247 00:16:07,557 --> 00:16:08,140 trades, right? 248 00:16:08,140 --> 00:16:10,390 So we call it a cash product. 249 00:16:10,390 --> 00:16:13,760 Means that you actually don't need any model to price it. 250 00:16:13,760 --> 00:16:17,060 You already have that price, OK? 251 00:16:17,060 --> 00:16:23,520 But if you try to explain like why you might have lost money 252 00:16:23,520 --> 00:16:26,470 today, right, and that always-- the trader 253 00:16:26,470 --> 00:16:29,910 always does that at the end of the day. 254 00:16:29,910 --> 00:16:32,090 And we always use first derivatives. 255 00:16:32,090 --> 00:16:36,380 And we try to explain it, but there is also unexplained, OK? 256 00:16:36,380 --> 00:16:39,870 And that unexplained can be quite high 257 00:16:39,870 --> 00:16:42,680 on this with large movements. 258 00:16:42,680 --> 00:16:45,720 So if you have like a term in your analytics 259 00:16:45,720 --> 00:16:48,010 for the bond convexity, that helps 260 00:16:48,010 --> 00:16:49,640 you to include the second derivative, 261 00:16:49,640 --> 00:16:55,045 and therefore make the second derivative smaller-- sorry-- 262 00:16:55,045 --> 00:16:56,610 the unexplained smaller. 263 00:17:02,170 --> 00:17:08,020 So let's now talk about interest rate swaps. 264 00:17:08,020 --> 00:17:12,410 So bond cash flow is basically a stream 265 00:17:12,410 --> 00:17:16,425 of fixed cash flow, which means that for certain dates 266 00:17:16,425 --> 00:17:18,300 it's just guaranteed that you will be getting 267 00:17:18,300 --> 00:17:22,800 $100 with certain periodicity. 268 00:17:22,800 --> 00:17:26,680 A swap means you exchange fixed payments 269 00:17:26,680 --> 00:17:28,430 with respect to some floating. 270 00:17:28,430 --> 00:17:33,060 And floating means that the amount of money that you'll 271 00:17:33,060 --> 00:17:36,870 be getting or paying, OK, receiving or paying, 272 00:17:36,870 --> 00:17:40,460 will depend on some market observable. 273 00:17:40,460 --> 00:17:43,920 So for interest rate swap, it will be typically-- and let's 274 00:17:43,920 --> 00:17:48,685 focus on the USD market, it will be a three month LIBOR rate. 275 00:17:48,685 --> 00:17:51,890 That rate is published daily, OK? 276 00:17:51,890 --> 00:17:56,300 And it's like if you need to go to the bank 277 00:17:56,300 --> 00:17:59,470 and get a three month CD with the money for the three months, 278 00:17:59,470 --> 00:18:01,130 that rate is already known. 279 00:18:01,130 --> 00:18:04,140 It's actually called LIBOR, because it's 280 00:18:04,140 --> 00:18:14,400 kind of between banks, and it's set at 11 AM London time. 281 00:18:14,400 --> 00:18:19,340 So as a result, we already know how 282 00:18:19,340 --> 00:18:23,800 to price cash flows in the future. 283 00:18:23,800 --> 00:18:32,460 So present value of the fixed stream of payments, as we know, 284 00:18:32,460 --> 00:18:34,480 will be like this. 285 00:18:34,480 --> 00:18:41,780 And there is a floating rate of cash flow as well. 286 00:18:41,780 --> 00:18:46,660 And the nice thing about the swap 287 00:18:46,660 --> 00:18:51,290 is that when you enter the swap, you don't pay any money, right? 288 00:18:51,290 --> 00:18:55,200 It's because you just kind of enter the agreement, 289 00:18:55,200 --> 00:18:59,710 rather than when you buy or sell a bond, 290 00:18:59,710 --> 00:19:02,010 there is some exchange of money. 291 00:19:02,010 --> 00:19:07,000 For swaps, swaps are designed the way such 292 00:19:07,000 --> 00:19:13,060 that-- so when you make this agreement, 293 00:19:13,060 --> 00:19:17,520 it's a certain moment of time, the fixed rate of the swap 294 00:19:17,520 --> 00:19:21,350 is picked in such a way that the present value 295 00:19:21,350 --> 00:19:28,260 of fixed minus the floating cash flows will be net to zero. 296 00:19:32,490 --> 00:19:36,730 So you can see, I mean, if we rewrite those equations, 297 00:19:36,730 --> 00:19:39,970 OK, we can see that the swap rate-- which 298 00:19:39,970 --> 00:19:43,890 is the most important quantity of the swap, 299 00:19:43,890 --> 00:19:46,080 and something that traders are basically 300 00:19:46,080 --> 00:19:47,350 are most concerned, right? 301 00:19:47,350 --> 00:19:50,830 So you first need to define what the swap is. 302 00:19:50,830 --> 00:19:54,325 And for USD, you are saying probably like 10 year swap, OK. 303 00:19:54,325 --> 00:19:55,200 And this is the rate. 304 00:19:55,200 --> 00:20:03,310 So the trader continuously kind of quotes bid and offer levels 305 00:20:03,310 --> 00:20:05,140 of the swap rate. 306 00:20:05,140 --> 00:20:07,930 So no one is talking about PVs and stuff like that. 307 00:20:07,930 --> 00:20:09,850 So it's always the swap rate. 308 00:20:09,850 --> 00:20:14,730 So that the swap rate is a weighted sum of forward rates. 309 00:20:14,730 --> 00:20:19,200 And it has a very nice intuitive explanation. 310 00:20:19,200 --> 00:20:27,970 So you have some stream of floating cash flows-- 311 00:20:27,970 --> 00:20:31,470 variable cash flows-- which at the moment, like, 312 00:20:31,470 --> 00:20:35,520 will probably be low now, but will be high in 10 years, 313 00:20:35,520 --> 00:20:38,940 will be much higher in 30 years. 314 00:20:38,940 --> 00:20:42,430 So the swap rate for this kind of environment 315 00:20:42,430 --> 00:20:44,320 will be kind of an average, right? 316 00:20:44,320 --> 00:20:49,815 And again, those weights depend on the discounting factors. 317 00:20:53,190 --> 00:20:59,190 So later, we will see that because we're 318 00:20:59,190 --> 00:21:02,925 talking about bond that having a fixed cash 319 00:21:02,925 --> 00:21:05,750 flows in the futures, and a swap that fixed in exchange 320 00:21:05,750 --> 00:21:13,300 of floating, swap can be hedged with bond, "hedged" meaning 321 00:21:13,300 --> 00:21:21,640 that-- you know what the term "hedged" means, no? 322 00:21:21,640 --> 00:21:25,990 Hedging means that if you have just, let's say, a swap, right? 323 00:21:25,990 --> 00:21:30,070 So if market changes, right, you can again lose money. 324 00:21:30,070 --> 00:21:38,250 So a typical task for the money-maker, 325 00:21:38,250 --> 00:21:42,790 trader, is to kind of offset that risk with something. 326 00:21:42,790 --> 00:21:46,430 Ideally, you sold one swap, you bought 327 00:21:46,430 --> 00:21:49,670 another swap the same way with a different rate. 328 00:21:49,670 --> 00:21:52,560 So you kind of locked in your profit. 329 00:21:52,560 --> 00:21:55,330 But you remain with a zero risk. 330 00:22:00,580 --> 00:22:02,960 So let's try to construct a yield curve. 331 00:22:02,960 --> 00:22:04,800 Why do we need the yield curve? 332 00:22:04,800 --> 00:22:11,360 So when we have, let's say, a series 333 00:22:11,360 --> 00:22:14,660 of swap with different maturities, right, 334 00:22:14,660 --> 00:22:18,450 all those swaps will start today, 335 00:22:18,450 --> 00:22:22,160 and usually, swap will have quarterly payment 336 00:22:22,160 --> 00:22:28,340 for the floating leg, and six month payments for the fixed 337 00:22:28,340 --> 00:22:31,160 leg, and you'll have different maturities. 338 00:22:31,160 --> 00:22:35,900 But if you try to kind of get discount factors 339 00:22:35,900 --> 00:22:38,530 from that information, you will see 340 00:22:38,530 --> 00:22:43,180 that you can get those discount factors only for certain dates, 341 00:22:43,180 --> 00:22:44,000 OK? 342 00:22:44,000 --> 00:22:48,300 But the typical situation is that given 343 00:22:48,300 --> 00:22:51,580 on some liquid market instruments, 344 00:22:51,580 --> 00:22:55,760 you want to price your entire portfolio, which 345 00:22:55,760 --> 00:22:57,890 has continuous spectrum of cash flows 346 00:22:57,890 --> 00:23:01,870 from now to 30 years, 40 years. 347 00:23:01,870 --> 00:23:07,860 And also, for typical swap portfolio 348 00:23:07,860 --> 00:23:12,360 that I personally deal with on a daily basis 349 00:23:12,360 --> 00:23:15,230 contains hundreds of thousands of swaps. 350 00:23:15,230 --> 00:23:17,630 Every swap has many cash flows. 351 00:23:17,630 --> 00:23:21,710 So you need something that can, based 352 00:23:21,710 --> 00:23:26,900 on discrete information of reliable liquid instruments 353 00:23:26,900 --> 00:23:30,290 on the market, draw the line. 354 00:23:30,290 --> 00:23:33,210 Can basically construct the curve. 355 00:23:33,210 --> 00:23:36,600 Which means that you can, so that you 356 00:23:36,600 --> 00:23:39,530 are able to get discount factors for any potential day 357 00:23:39,530 --> 00:23:41,990 in the future, or you can compute forward rate 358 00:23:41,990 --> 00:23:45,380 for every date in the future. 359 00:23:45,380 --> 00:23:48,570 So the first step to construct a yield curve 360 00:23:48,570 --> 00:23:53,590 is to select input instruments for calibration. 361 00:23:53,590 --> 00:23:55,050 So you have a set of instruments, 362 00:23:55,050 --> 00:23:56,780 and a new set of input quotes. 363 00:23:56,780 --> 00:23:59,840 Then you also need to kind of decide 364 00:23:59,840 --> 00:24:02,000 what kind of properties of that line will be. 365 00:24:02,000 --> 00:24:09,970 So you can say, OK, first of all, 366 00:24:09,970 --> 00:24:13,010 you need to decide what quantity will be interpolated. 367 00:24:13,010 --> 00:24:15,770 It could be daily discount factors, 368 00:24:15,770 --> 00:24:19,820 or daily forward rates, or maybe three-month forward rates. 369 00:24:19,820 --> 00:24:22,220 Then you select the spline type. 370 00:24:22,220 --> 00:24:24,480 So I'm not sure if you're familiar with the splines. 371 00:24:24,480 --> 00:24:29,860 Probably you heard about cubic spline, right? 372 00:24:29,860 --> 00:24:32,860 So there are different types of splines, and some of them 373 00:24:32,860 --> 00:24:37,190 are better and some of them are worse for different situation. 374 00:24:37,190 --> 00:24:38,940 And you also need to decide like what 375 00:24:38,940 --> 00:24:41,750 that will be the node points for the spline itself. 376 00:24:41,750 --> 00:24:42,920 OK. 377 00:24:42,920 --> 00:24:45,050 And then, as a final step, so you 378 00:24:45,050 --> 00:24:47,420 have some mathematical quantity, which 379 00:24:47,420 --> 00:24:50,310 is mathematical object where you know what the line is, 380 00:24:50,310 --> 00:24:51,980 and you have control points. 381 00:24:51,980 --> 00:24:54,100 And you need to adjust your control points such 382 00:24:54,100 --> 00:25:00,220 that when you reprice your instruments, 383 00:25:00,220 --> 00:25:02,800 those instruments are repriced exactly to the same quotes 384 00:25:02,800 --> 00:25:06,216 that you find on the market. 385 00:25:06,216 --> 00:25:07,445 You have a question. 386 00:25:07,445 --> 00:25:11,791 AUDIENCE: Is that spline, again, is it just like a-- 387 00:25:11,791 --> 00:25:12,832 IVAN MASYUKOV: All right. 388 00:25:12,832 --> 00:25:13,332 So let me. 389 00:25:22,890 --> 00:25:23,390 All right. 390 00:25:23,390 --> 00:25:26,645 So this is a picture of the cubic spline. 391 00:25:31,070 --> 00:25:41,940 So spline is a way to draw a smooth curve. 392 00:25:41,940 --> 00:25:44,050 This is an example of the cubic spline. 393 00:25:44,050 --> 00:25:48,080 So you start to define your node points. 394 00:25:48,080 --> 00:25:53,150 Your node points in this case are 1, 10, 20, 40, 395 00:25:53,150 --> 00:25:56,320 80, 160, and 240, right? 396 00:25:56,320 --> 00:26:00,000 And then for every one of those intervals, 397 00:26:00,000 --> 00:26:04,600 OK, the functional form of the shape of this curve 398 00:26:04,600 --> 00:26:09,840 is a cubic polynomial, OK? 399 00:26:09,840 --> 00:26:14,690 Well, if you just do cubic polynomial for every interval 400 00:26:14,690 --> 00:26:17,680 without kind of putting additional constraints, 401 00:26:17,680 --> 00:26:21,580 you can have all kinds of boundary effects, like jumps, 402 00:26:21,580 --> 00:26:27,480 kinks, and other things, because we want our cubic curve-- 403 00:26:27,480 --> 00:26:33,820 cubic spline-- to be meaningful, right? 404 00:26:33,820 --> 00:26:36,130 So we want to maintain, to preserve 405 00:26:36,130 --> 00:26:42,250 maximum number of derivatives for every node point. 406 00:26:42,250 --> 00:26:44,050 So we're not going to check. 407 00:26:44,050 --> 00:26:49,400 But believe me, this curve, it is 408 00:26:49,400 --> 00:26:53,900 a cubic polynomial for every one of those interval. 409 00:26:53,900 --> 00:26:58,960 And also, we have two continuous derivative at every node point, 410 00:26:58,960 --> 00:27:04,100 because for the n degree of the spline, 411 00:27:04,100 --> 00:27:05,720 you always have like n minus 1. 412 00:27:05,720 --> 00:27:08,860 You can have n minus 1. 413 00:27:08,860 --> 00:27:12,640 So the same thing, a spline can be represented 414 00:27:12,640 --> 00:27:15,980 in terms of B-splines. 415 00:27:15,980 --> 00:27:18,720 B-spline is a new type of spline. 416 00:27:18,720 --> 00:27:24,750 It's just as a representation which is more intuitive, 417 00:27:24,750 --> 00:27:27,020 I should say. 418 00:27:27,020 --> 00:27:32,520 So all universe of the curves with those node points, 419 00:27:32,520 --> 00:27:37,950 with two continuous derivatives, can 420 00:27:37,950 --> 00:27:40,910 be represented as a linear combination of those basis 421 00:27:40,910 --> 00:27:41,670 functions. 422 00:27:41,670 --> 00:27:43,660 So B-spline, I mean, if you're interested, 423 00:27:43,660 --> 00:27:48,650 you should probably, we're not going to discuss it in details, 424 00:27:48,650 --> 00:27:51,579 but it's nice separate kind of topic about how 425 00:27:51,579 --> 00:27:52,620 to build those B-splines. 426 00:27:55,670 --> 00:28:00,530 But essentially, what's nice about those B-splines-- 427 00:28:00,530 --> 00:28:04,790 and "B," as you probably already understood, B is basis, right? 428 00:28:04,790 --> 00:28:06,200 So you have like basis functions. 429 00:28:06,200 --> 00:28:09,610 So those functions look like bell shapes. 430 00:28:09,610 --> 00:28:13,600 They are non-zero on some sub-interval. 431 00:28:13,600 --> 00:28:19,310 On every interval, it will be a cubic polynomial. 432 00:28:19,310 --> 00:28:29,860 Everyone will have always two continuous derivatives. 433 00:28:29,860 --> 00:28:32,070 As a result, in any linear combination 434 00:28:32,070 --> 00:28:35,400 of those-- which the first curve is-- 435 00:28:35,400 --> 00:28:37,430 will also have that property. 436 00:28:50,530 --> 00:28:51,030 OK. 437 00:28:51,030 --> 00:28:54,900 So now we, yeah, so the calibrate 438 00:28:54,900 --> 00:28:58,550 means that we basically have some solver 439 00:28:58,550 --> 00:29:03,610 to make sure that our swaps with the rates for those maturities 440 00:29:03,610 --> 00:29:06,420 actually repriced at par. 441 00:29:06,420 --> 00:29:10,510 At par means that the PV is zero. 442 00:29:10,510 --> 00:29:15,120 This is a typical example of the yield curve instruments. 443 00:29:15,120 --> 00:29:21,320 And IRS stands for "interest rate swap," 444 00:29:21,320 --> 00:29:25,250 and we have maturities from one year to 30 years. 445 00:29:25,250 --> 00:29:36,540 And the quotes are of 0.33% up to 2.67%. 446 00:29:36,540 --> 00:29:40,210 So you can see that that's actually, 447 00:29:40,210 --> 00:29:44,190 that's from my one-year-old presentation. 448 00:29:44,190 --> 00:29:47,030 Rates are quite high these days. 449 00:29:47,030 --> 00:29:52,060 So this is an example of the yield curve graph. 450 00:29:52,060 --> 00:30:01,290 So again, so those are the rates from 0.3 to 3.5. 451 00:30:01,290 --> 00:30:06,120 And the shape of the curve is not flat at all, right? 452 00:30:06,120 --> 00:30:08,130 So it's actually pretty steep. 453 00:30:08,130 --> 00:30:13,000 So for the first five years, it's very steep. 454 00:30:13,000 --> 00:30:15,760 Then it reaches the plateau. 455 00:30:15,760 --> 00:30:20,720 And then there is some feature there, 456 00:30:20,720 --> 00:30:27,360 probably because of some behavior in the 20-year region. 457 00:30:27,360 --> 00:30:30,330 So three-month forward rate is the LIBOR rate. 458 00:30:30,330 --> 00:30:36,300 LIBOR is the rate for the three month. 459 00:30:36,300 --> 00:30:38,880 It's mostly kind of common. 460 00:30:38,880 --> 00:30:43,840 And the reason why is because the standard interest rate, 461 00:30:43,840 --> 00:30:48,100 USD swap has a three-month frequency 462 00:30:48,100 --> 00:30:51,530 for payment on the floating leg. 463 00:30:51,530 --> 00:30:54,600 So if you're talking about floating rates, 464 00:30:54,600 --> 00:30:55,860 is always three months. 465 00:30:55,860 --> 00:30:56,850 And it's always LIBOR. 466 00:31:00,080 --> 00:31:01,970 So because we've already built the curve, 467 00:31:01,970 --> 00:31:07,660 now let's see how we can improve the situation with a bond. 468 00:31:07,660 --> 00:31:11,430 So we have the curve, so we have the discount factors, right? 469 00:31:11,430 --> 00:31:15,620 And we see that those discount factors cannot be obtained 470 00:31:15,620 --> 00:31:18,620 on the assumption that you have just one parameter yield 471 00:31:18,620 --> 00:31:22,450 for everything, because the curve we know is not flat. 472 00:31:22,450 --> 00:31:31,450 So if we just try to price it using those discount factors, 473 00:31:31,450 --> 00:31:34,040 try to get a fair price, we probably 474 00:31:34,040 --> 00:31:35,930 won't match the market observables. 475 00:31:35,930 --> 00:31:38,140 So we need some extra term. 476 00:31:38,140 --> 00:31:41,700 And again, here we can use it in a similar form 477 00:31:41,700 --> 00:31:44,330 as we did it for the yield. 478 00:31:44,330 --> 00:31:48,910 But right now it's going to be a small correction to the yield 479 00:31:48,910 --> 00:31:54,140 curve, rather than kind of really rough assumption 480 00:31:54,140 --> 00:31:58,140 about that the curve is flat, OK? 481 00:31:58,140 --> 00:32:10,010 So typically, if the curve magnitude is, let's say, 3%. 482 00:32:10,010 --> 00:32:10,510 OK? 483 00:32:10,510 --> 00:32:14,490 So the spread is probably 100 times lower. 484 00:32:14,490 --> 00:32:20,245 So having a nice correction is always better, right? 485 00:32:23,850 --> 00:32:31,090 And another nice feature is that of this approach for the bond, 486 00:32:31,090 --> 00:32:34,350 like if we already build our yield curve model, 487 00:32:34,350 --> 00:32:37,935 and we know sensitivities of our portfolio 488 00:32:37,935 --> 00:32:41,290 to inputs of the curves, which then transition 489 00:32:41,290 --> 00:32:46,410 into like differences in discount factors, 490 00:32:46,410 --> 00:32:48,690 we can easily apply that to the bond. 491 00:32:48,690 --> 00:32:53,050 We can first find what this spread parameter 492 00:32:53,050 --> 00:32:59,380 is, to solve for s knowing P, which is very liquid market 493 00:32:59,380 --> 00:33:01,070 tradable. 494 00:33:01,070 --> 00:33:04,919 And then we can kind of use consistent model for the bonds 495 00:33:04,919 --> 00:33:06,210 and the swaps in our portfolio. 496 00:33:08,780 --> 00:33:10,637 Any questions? 497 00:33:10,637 --> 00:33:11,220 AUDIENCE: Yes. 498 00:33:11,220 --> 00:33:15,392 So what does the bond rate tell us about the bond? 499 00:33:19,740 --> 00:33:22,880 IVAN MASYUKOV: That's a very good question. 500 00:33:22,880 --> 00:33:31,770 So it might tell us something like bond liquidity, 501 00:33:31,770 --> 00:33:32,980 for example. 502 00:33:32,980 --> 00:33:34,575 Like if it's not liquid, or there 503 00:33:34,575 --> 00:33:42,280 is some-- so it may be related to the bond itself. 504 00:33:42,280 --> 00:33:45,590 And sometimes we kind of think that the bond 505 00:33:45,590 --> 00:33:48,720 is riskless, which means that-- especially if it's 506 00:33:48,720 --> 00:33:53,710 issued by US government, which if we can assume 507 00:33:53,710 --> 00:33:58,600 that those cash flows in the futures are guaranteed, right, 508 00:33:58,600 --> 00:34:01,961 then I basically will be willing to bring 509 00:34:01,961 --> 00:34:04,210 a certain amount of money and discount factors, right? 510 00:34:04,210 --> 00:34:07,610 But if you tell me that you will pay me that in the future, 511 00:34:07,610 --> 00:34:11,020 I won't be so certain, right? 512 00:34:11,020 --> 00:34:16,199 So I'll need to add some kind of credit spread 513 00:34:16,199 --> 00:34:18,980 to that-- we call it credit spreads-- as a result. 514 00:34:18,980 --> 00:34:23,590 It's the credit spread will kind of propagate to the spread 515 00:34:23,590 --> 00:34:24,469 number. 516 00:34:24,469 --> 00:34:28,090 On the other hand, if the bond is really US government-issued, 517 00:34:28,090 --> 00:34:32,909 and is considered to be guaranteed, 518 00:34:32,909 --> 00:34:36,100 then it may be a feature of the swap, OK? 519 00:34:36,100 --> 00:34:41,469 Where just because of some liquidity situations 520 00:34:41,469 --> 00:34:43,770 in swap market-- like all of a sudden, 521 00:34:43,770 --> 00:34:47,670 let's say, all option traders on the street 522 00:34:47,670 --> 00:34:49,909 needed this 10-year swap, OK? 523 00:34:49,909 --> 00:34:54,429 Because they kind of need to hedge certain very popular 524 00:34:54,429 --> 00:34:58,380 products-- volatility products-- they'll start to buy it, 525 00:34:58,380 --> 00:35:03,200 that spread will change. 526 00:35:03,200 --> 00:35:04,780 But what's even more interesting, 527 00:35:04,780 --> 00:35:08,000 that spread is tradable by itself, OK? 528 00:35:08,000 --> 00:35:12,910 So you can go to the market and you trade the spread. 529 00:35:12,910 --> 00:35:18,290 Moreover, let's look, like, 10-year situation. 530 00:35:18,290 --> 00:35:24,690 So you have 10-year bond on the market. 531 00:35:24,690 --> 00:35:29,340 You have tradable swap, and you have tradable spread. 532 00:35:29,340 --> 00:35:32,021 So the question is which one is the most liquid? 533 00:35:32,021 --> 00:35:32,770 What do you think? 534 00:35:38,300 --> 00:35:42,520 The most liquid is the bond, of course. 535 00:35:42,520 --> 00:35:45,890 It has much more liquidity. 536 00:35:45,890 --> 00:35:47,730 Surprisingly, the second one. 537 00:35:47,730 --> 00:35:52,430 It's the spread between the 10-year swap 538 00:35:52,430 --> 00:35:54,690 and the bond is traded in the market. 539 00:35:54,690 --> 00:35:58,470 So there's more transaction on the spread 540 00:35:58,470 --> 00:36:00,910 rather compared to the swap. 541 00:36:00,910 --> 00:36:03,970 As a result, when we built our curves, 542 00:36:03,970 --> 00:36:08,540 we're not taking like 10-year swap from the market. 543 00:36:08,540 --> 00:36:09,040 OK. 544 00:36:09,040 --> 00:36:13,360 We actually take the yield and the spread. 545 00:36:13,360 --> 00:36:16,696 And that's how we define the most kind 546 00:36:16,696 --> 00:36:18,380 of reliable level of the swap. 547 00:36:18,380 --> 00:36:20,450 Of course, we could have just take 548 00:36:20,450 --> 00:36:23,340 whatever we observe for the 10-year swap, 549 00:36:23,340 --> 00:36:25,790 but it could be off. 550 00:36:25,790 --> 00:36:29,160 And also, if you observe, there will be more bid-offer spread 551 00:36:29,160 --> 00:36:29,660 as well. 552 00:36:35,650 --> 00:36:40,700 So as an example, let's try to shift 553 00:36:40,700 --> 00:36:45,780 one of the inputs of the curve by one basis point. 554 00:36:45,780 --> 00:36:50,260 And that will result in this kind of deviation 555 00:36:50,260 --> 00:36:57,640 of forward trades, which will be combination of basis splines. 556 00:36:57,640 --> 00:36:59,680 But what's interesting first of all, 557 00:36:59,680 --> 00:37:01,925 it's kind of complicated [INAUDIBLE] behavior. 558 00:37:01,925 --> 00:37:07,120 The reason why is because you are saying that nothing changed 559 00:37:07,120 --> 00:37:09,330 before the nine year, like nothing 560 00:37:09,330 --> 00:37:13,640 changed after ninth year, but just point in between. 561 00:37:13,640 --> 00:37:17,030 So in order to calibrate to that kind of weird condition, 562 00:37:17,030 --> 00:37:22,310 right, you need to have a ripple here. 563 00:37:22,310 --> 00:37:27,590 But what also is more important is that by shifting one year 564 00:37:27,590 --> 00:37:32,460 basis point by one basis point, that the amplitude 565 00:37:32,460 --> 00:37:37,370 of shifts in the curve reaches 14 basis points. 566 00:37:37,370 --> 00:37:41,550 So not sure if you're familiar, but it's an ill-posed problem 567 00:37:41,550 --> 00:37:42,050 right? 568 00:37:42,050 --> 00:37:45,800 So small changes in your inputs can cause large variations 569 00:37:45,800 --> 00:37:46,590 in your outputs. 570 00:37:49,450 --> 00:37:51,230 This is a very important slide. 571 00:37:51,230 --> 00:37:56,100 So the first column is, again, we 572 00:37:56,100 --> 00:37:59,230 saw those are our instruments, quotes, 573 00:37:59,230 --> 00:38:01,420 and this is the risk of the portfolio. 574 00:38:01,420 --> 00:38:05,100 That's something that a trader needs no matter what. 575 00:38:05,100 --> 00:38:07,930 It basically shows you what happens 576 00:38:07,930 --> 00:38:11,570 on the market if different-- what 577 00:38:11,570 --> 00:38:13,250 will be the change in your portfolio 578 00:38:13,250 --> 00:38:14,740 if the market changes. 579 00:38:14,740 --> 00:38:18,220 So the meaning of the number-- for example, for the five 580 00:38:18,220 --> 00:38:25,700 year-- is that if five year rate moves up by one basis point, 581 00:38:25,700 --> 00:38:28,845 we'll lose minus 700K. 582 00:38:36,390 --> 00:38:41,250 We also marked here yellow points 583 00:38:41,250 --> 00:38:44,530 that are more liquid than the others. 584 00:38:44,530 --> 00:38:47,940 So now a typical situation is that you need 585 00:38:47,940 --> 00:38:49,880 to hedge your portfolio, right? 586 00:38:49,880 --> 00:38:57,800 So you need to liquidate your risk, I'm basically saying, 587 00:38:57,800 --> 00:39:00,720 OK, given the model that we have, 588 00:39:00,720 --> 00:39:07,030 I want its value to be insensitive to any movements 589 00:39:07,030 --> 00:39:08,600 on the market. 590 00:39:08,600 --> 00:39:12,170 So for that purpose, what you can do, you can go 591 00:39:12,170 --> 00:39:20,290 and you can buy as many one-year swaps as plus 200, 592 00:39:20,290 --> 00:39:27,490 as many two-year swaps which would be the risk of minus 1.3, 593 00:39:27,490 --> 00:39:30,640 and so forth, right? 594 00:39:30,640 --> 00:39:35,550 Then that always cost you money right? 595 00:39:35,550 --> 00:39:41,180 And that money is kind of proportional to bid-offer 596 00:39:41,180 --> 00:39:42,770 of the particular instruments. 597 00:39:42,770 --> 00:39:45,880 And that bid-offer is smaller for liquid instruments 598 00:39:45,880 --> 00:39:51,450 and larger for less liquid instruments. 599 00:39:51,450 --> 00:39:53,520 So if you multiply by the diff-- we 600 00:39:53,520 --> 00:39:55,270 can see that if we want to hedge our risk, 601 00:39:55,270 --> 00:39:58,300 it's going to be quite expensive. 602 00:39:58,300 --> 00:40:00,610 It will cost us 3.6 million dollars. 603 00:40:04,037 --> 00:40:05,480 Any questions so far? 604 00:40:11,820 --> 00:40:23,440 So traders never hedge every bucket in the risk. 605 00:40:23,440 --> 00:40:25,890 Bucket means every line here. 606 00:40:25,890 --> 00:40:30,480 So you always see some numbers, but if you 607 00:40:30,480 --> 00:40:33,210 try to make every number here zero, 608 00:40:33,210 --> 00:40:36,530 which means that if you trade seven here, 609 00:40:36,530 --> 00:40:38,460 you also could try to go to the market, 610 00:40:38,460 --> 00:40:43,890 find the offsetting seven here, you'll have to pay too much 611 00:40:43,890 --> 00:40:50,030 and you won't be profitable. 612 00:40:50,030 --> 00:40:54,210 So what the traders do if someone ask for the seven year, 613 00:40:54,210 --> 00:40:57,070 they make this transaction, but they go, then 614 00:40:57,070 --> 00:40:59,680 hedge it from the more liquid points, which 615 00:40:59,680 --> 00:41:01,520 is less expensive to buy. 616 00:41:05,030 --> 00:41:07,740 So we need a better model for hedging. 617 00:41:07,740 --> 00:41:12,820 And a general formulation of the model is presented here. 618 00:41:12,820 --> 00:41:22,380 So we have portfolio risk which is a just the vector here, 619 00:41:22,380 --> 00:41:24,030 right? 620 00:41:24,030 --> 00:41:29,950 And we have hedging. 621 00:41:29,950 --> 00:41:33,610 Portfolio risk is basically if you have candidates 622 00:41:33,610 --> 00:41:37,320 of instruments that you can use for portfolio hedging, again, 623 00:41:37,320 --> 00:41:40,820 the risk will be represented in this format in terms 624 00:41:40,820 --> 00:41:44,505 of sensitivities to swap rates. 625 00:41:47,050 --> 00:41:54,160 And we have weights of that hedging portfolio 626 00:41:54,160 --> 00:41:55,970 that we need to find, obviously. 627 00:41:55,970 --> 00:41:57,440 So you have this hedging portfolio. 628 00:41:57,440 --> 00:42:02,190 You multiply H by x, you get risk of this hedging portfolio. 629 00:42:02,190 --> 00:42:05,050 You add it to the risk of your portfolio. 630 00:42:05,050 --> 00:42:07,177 And then, what we need to minimize, 631 00:42:07,177 --> 00:42:08,760 you don't need to minimize everything. 632 00:42:08,760 --> 00:42:10,301 But you need to if they give you, OK? 633 00:42:10,301 --> 00:42:11,630 What can happen on the market? 634 00:42:11,630 --> 00:42:14,300 What are the typical modes of the market? 635 00:42:14,300 --> 00:42:19,530 And so essentially you kind of define your market 636 00:42:19,530 --> 00:42:24,810 scenarios, which can be found in a different way. 637 00:42:24,810 --> 00:42:27,520 So one of the ways to approach that problem 638 00:42:27,520 --> 00:42:33,260 is to use principal component analysis. 639 00:42:33,260 --> 00:42:37,670 I know you already are familiar with SVD. 640 00:42:37,670 --> 00:42:50,370 So if D is data of market movements in matrix-- 641 00:42:50,370 --> 00:42:57,240 then any matrix can be decomposed using SVD. 642 00:42:57,240 --> 00:43:04,350 And we can then look at this spectrum of this decomposition, 643 00:43:04,350 --> 00:43:09,940 looking at those eigenvalues, and just 644 00:43:09,940 --> 00:43:13,160 pick the ones that look high enough for us, 645 00:43:13,160 --> 00:43:14,760 and just keep that number. 646 00:43:14,760 --> 00:43:20,200 And let's, for example, we find that we really investigated 647 00:43:20,200 --> 00:43:29,180 this market, and we found that there are just 648 00:43:29,180 --> 00:43:32,130 five components that drive the market, 649 00:43:32,130 --> 00:43:36,130 and the rest is just so little that it's meaningless, right? 650 00:43:36,130 --> 00:43:38,200 On every day, and we are certain that it's just 651 00:43:38,200 --> 00:43:42,820 five components, five modes of market moments. 652 00:43:42,820 --> 00:43:48,060 Then, if we have a curve that consists of 20 points, 653 00:43:48,060 --> 00:43:50,600 we don't need to hedge every swap 654 00:43:50,600 --> 00:43:52,550 with its corresponding maturity. 655 00:43:52,550 --> 00:43:57,100 We can just pick five swaps that are 656 00:43:57,100 --> 00:43:59,030 liquid enough and cheap enough for us 657 00:43:59,030 --> 00:44:00,720 to hedge, and just use them. 658 00:44:08,990 --> 00:44:11,720 So let's look now at those typical graphs 659 00:44:11,720 --> 00:44:16,190 of those principal components. 660 00:44:16,190 --> 00:44:19,210 X-axis is the swap maturity in years, 661 00:44:19,210 --> 00:44:22,380 and y is some kind of relative, let's think 662 00:44:22,380 --> 00:44:23,920 of that as basis points. 663 00:44:23,920 --> 00:44:26,760 So blue line is the first component 664 00:44:26,760 --> 00:44:30,610 which is the prevalent. 665 00:44:30,610 --> 00:44:34,600 And it kind of, you can see that swap rates, 666 00:44:34,600 --> 00:44:37,090 they're basically flattish after 10 years, 667 00:44:37,090 --> 00:44:40,500 but the first component is pretty steep. 668 00:44:40,500 --> 00:44:47,760 And what it says as well is that the main behavior of the market 669 00:44:47,760 --> 00:44:50,010 is that rates now do not move, but they 670 00:44:50,010 --> 00:44:51,340 will move in the future. 671 00:44:51,340 --> 00:44:53,750 And that's basically because Fed is in a hold, right? 672 00:44:53,750 --> 00:44:57,030 So they kind of stimulate the market in a way such 673 00:44:57,030 --> 00:45:02,030 that the rate remains the same until sometime in the future. 674 00:45:02,030 --> 00:45:10,440 Mode number two is a kind of like tilting situation. 675 00:45:10,440 --> 00:45:13,050 Mode number three is more complex. 676 00:45:13,050 --> 00:45:15,770 And we'll have several other modes here as well. 677 00:45:19,910 --> 00:45:25,950 So now, following our previous general approach 678 00:45:25,950 --> 00:45:27,920 to the problem, we formulated here 679 00:45:27,920 --> 00:45:35,030 as-- so we have PCA factors here in P. 680 00:45:35,030 --> 00:45:38,390 And now, because the number of factors that we selected 681 00:45:38,390 --> 00:45:40,140 is the number of hedging instruments-- 682 00:45:40,140 --> 00:45:42,200 we no longer need to minimize. 683 00:45:42,200 --> 00:45:44,890 We can always feed, which you can always achieve 684 00:45:44,890 --> 00:45:46,527 like perfect minimization. 685 00:45:46,527 --> 00:45:47,860 We can always achieve that zero. 686 00:45:47,860 --> 00:45:52,610 So that's why we formulate it as zero. 687 00:45:52,610 --> 00:45:54,765 So solving that problem here. 688 00:45:58,230 --> 00:45:58,730 Yeah. 689 00:45:58,730 --> 00:46:01,930 And the hedging matrix, this is an example of hedging matrix. 690 00:46:01,930 --> 00:46:09,260 So what that matrix says is that if I take one year swap 691 00:46:09,260 --> 00:46:11,450 and put in my portfolio, empty portfolio, 692 00:46:11,450 --> 00:46:17,500 and then they apply my model, I'll have just sensitivity 693 00:46:17,500 --> 00:46:19,310 to that particular swap. 694 00:46:19,310 --> 00:46:24,580 Which kind of makes sense, because since you 695 00:46:24,580 --> 00:46:29,020 use the same instruments to calibrate your yield curve, 696 00:46:29,020 --> 00:46:33,770 right, then it should be sensitive to itself only. 697 00:46:33,770 --> 00:46:37,690 That's why that matrix is just ones for itself, 698 00:46:37,690 --> 00:46:40,880 and zeroes otherwise. 699 00:46:40,880 --> 00:46:47,860 So then, as a result, we get this matrix. 700 00:46:47,860 --> 00:46:50,690 So same portfolio that we had before. 701 00:46:50,690 --> 00:46:58,250 This is our PCA matrix that translates our risk 702 00:46:58,250 --> 00:47:07,286 into those few numbers, right? 703 00:47:07,286 --> 00:47:12,730 And because we know it translates our risk-- low risk, 704 00:47:12,730 --> 00:47:17,350 in terms of many curve inputs-- into just five most 705 00:47:17,350 --> 00:47:21,230 liquid ones, which is 1, 2, 5, 10, and 30. 706 00:47:21,230 --> 00:47:24,300 As a result, our translated risk, 707 00:47:24,300 --> 00:47:29,770 which tells us what we need to do to hedge our portfolio, 708 00:47:29,770 --> 00:47:31,095 is just those numbers. 709 00:47:31,095 --> 00:47:35,990 And now, if we take a bid-offer charge, 0.1 basis 710 00:47:35,990 --> 00:47:38,420 points for those, and multiply, we 711 00:47:38,420 --> 00:47:40,160 get numbers which are orders of magnitude 712 00:47:40,160 --> 00:47:42,060 smaller than we got before, right? 713 00:47:42,060 --> 00:47:46,760 So we probably get something like there were 400. 714 00:47:46,760 --> 00:47:51,220 It's not 4 million-- 3.6 million anymore. 715 00:47:51,220 --> 00:47:54,650 That's exactly what traders do. 716 00:47:54,650 --> 00:47:58,190 And different traders have different opinions 717 00:47:58,190 --> 00:47:59,745 of what dynamics of the market is. 718 00:48:04,770 --> 00:48:06,390 But they always have some model. 719 00:48:14,060 --> 00:48:15,360 So disadvantages. 720 00:48:15,360 --> 00:48:20,120 So PCA model is something that just formally attuned 721 00:48:20,120 --> 00:48:22,690 to historical data. 722 00:48:22,690 --> 00:48:30,960 I always say that if you take kind of scramble your swap 723 00:48:30,960 --> 00:48:34,140 maturities in your model, and you do your computations, 724 00:48:34,140 --> 00:48:36,450 and you kind of unscramble them, you 725 00:48:36,450 --> 00:48:42,820 get exactly the same result. Which means that in PCA model, 726 00:48:42,820 --> 00:48:44,940 you don't put any constraints on that. 727 00:48:47,510 --> 00:48:49,400 Two year is very close to one year, 728 00:48:49,400 --> 00:48:51,560 and two year is between one year and five year. 729 00:48:55,250 --> 00:48:58,680 So PCA model or hedging coefficients of that matrix 730 00:48:58,680 --> 00:49:06,730 is not very stable-- especially for recent modes in the market. 731 00:49:06,730 --> 00:49:11,780 Also, because SVDs kind of is the least squares 732 00:49:11,780 --> 00:49:14,530 approximation, it's very sensitive to outliers. 733 00:49:14,530 --> 00:49:19,150 So there is just one event on the market that kind of one day 734 00:49:19,150 --> 00:49:21,510 happens, something like rates went up, 735 00:49:21,510 --> 00:49:24,160 and then it went down significantly, 736 00:49:24,160 --> 00:49:30,260 it may have unnecessarily high influence on the outputs. 737 00:49:30,260 --> 00:49:32,720 And if those coefficients change daily, 738 00:49:32,720 --> 00:49:36,410 right, then again, it may be too costly. 739 00:49:36,410 --> 00:49:39,810 And quite often we just overfitting to historical data. 740 00:49:39,810 --> 00:49:42,700 So we're saying, OK, what can I do. 741 00:49:42,700 --> 00:49:44,600 I just take historical data, and I 742 00:49:44,600 --> 00:49:50,390 prove that my model works, would have worked for the last three 743 00:49:50,390 --> 00:49:54,180 years, or the last three months, but that 744 00:49:54,180 --> 00:49:57,298 doesn't mean that it will work for the next three months. 745 00:50:04,830 --> 00:50:09,290 If we kind of try to put some additional constraints 746 00:50:09,290 --> 00:50:12,456 or additional thoughts about what this behavior should be, 747 00:50:12,456 --> 00:50:13,580 this may improve situation. 748 00:50:16,290 --> 00:50:21,420 So PCA interpretation is that risk matrix 749 00:50:21,420 --> 00:50:28,050 is a linear combination of principal components producing 750 00:50:28,050 --> 00:50:30,520 a shift on one hedging instrument at a time. 751 00:50:35,780 --> 00:50:41,640 Now the question is, let's forget about historical, OK? 752 00:50:41,640 --> 00:50:43,300 Is there any other approach? 753 00:50:43,300 --> 00:50:46,619 We know historical is noisy, and it's kind of first step 754 00:50:46,619 --> 00:50:47,910 if you want to build the model. 755 00:50:47,910 --> 00:50:51,250 But can we do something better? 756 00:50:51,250 --> 00:50:54,422 And the answer is yes. 757 00:50:54,422 --> 00:51:01,570 So we can say that we have our yield curve 758 00:51:01,570 --> 00:51:04,880 in terms of forward rates. 759 00:51:04,880 --> 00:51:07,487 And typically, when we build this curve, 760 00:51:07,487 --> 00:51:08,695 we observe that it is smooth. 761 00:51:11,650 --> 00:51:15,130 It's smooth not only because we use smooth splines, 762 00:51:15,130 --> 00:51:25,520 but also because if there is no certainty about some event 763 00:51:25,520 --> 00:51:32,420 in 10 years from now, there is no reason to kind of expect 764 00:51:32,420 --> 00:51:36,280 there will be spike or some non-smooth feature 765 00:51:36,280 --> 00:51:39,940 in the forward rate space. 766 00:51:39,940 --> 00:51:44,880 So what we can do is that we can try 767 00:51:44,880 --> 00:51:50,010 to minimize those equations where Jacobian 768 00:51:50,010 --> 00:51:56,010 is a matrix translating shifts of yield curve inputs 769 00:51:56,010 --> 00:51:58,580 into movements of forward trades. 770 00:51:58,580 --> 00:52:04,000 So essentially, we will try to penalize non-smoothness. 771 00:52:09,160 --> 00:52:11,760 And the solution will be like this, 772 00:52:11,760 --> 00:52:17,210 with some kind of-- so we'll be adding a penalty, OK. 773 00:52:17,210 --> 00:52:20,470 And penalty will be a small regularization parameter. 774 00:52:25,930 --> 00:52:31,070 So this is, as an example, that's what we get. 775 00:52:35,500 --> 00:52:37,530 Again, here in that model. 776 00:52:40,640 --> 00:52:47,200 You can view this matrix as if one year rate moves, 777 00:52:47,200 --> 00:52:51,360 what it basically-- so your drivers are 1, 2, 5, 10s, 778 00:52:51,360 --> 00:52:51,860 and 30s. 779 00:52:51,860 --> 00:52:56,100 So that's your drivers. 780 00:52:56,100 --> 00:52:59,230 Knowing the movements of your drivers, 781 00:52:59,230 --> 00:53:02,090 what would be the response to your swap rates? 782 00:53:02,090 --> 00:53:06,150 And you know that it will always be one to itself, 783 00:53:06,150 --> 00:53:08,490 right, as you see here. 784 00:53:08,490 --> 00:53:11,390 And in between, it will be kind of a smooth functions. 785 00:53:21,900 --> 00:53:26,960 So let's take, this moment, a broader view 786 00:53:26,960 --> 00:53:31,290 at what the pricing model does. 787 00:53:31,290 --> 00:53:36,180 And we have a pricing engine, essentially. 788 00:53:36,180 --> 00:53:38,860 It's a way, if you have all model parameters-- including 789 00:53:38,860 --> 00:53:42,490 curves, volatility, the surface, everything, right? 790 00:53:42,490 --> 00:53:50,050 And in order for those parameters 791 00:53:50,050 --> 00:53:52,950 to be consistent with the benchmark prices, 792 00:53:52,950 --> 00:53:58,310 you need some calibration engine which 793 00:53:58,310 --> 00:54:02,190 matches market observables to the ones that's 794 00:54:02,190 --> 00:54:08,970 been repriced by the model output of the pricing engine. 795 00:54:08,970 --> 00:54:19,300 And once you make sure that benchmark prices of your model 796 00:54:19,300 --> 00:54:25,520 equal or are kind of close enough to benchmark prices 797 00:54:25,520 --> 00:54:28,360 observed in the market, you calibrated the model, 798 00:54:28,360 --> 00:54:35,160 then you can essentially price your portfolio, 799 00:54:35,160 --> 00:54:36,610 and get values and risk. 800 00:54:43,780 --> 00:54:50,610 So let's look at one of the nice examples of how 801 00:54:50,610 --> 00:54:57,060 that pricing engine and pricing and calibration process works. 802 00:54:57,060 --> 00:55:07,070 We'll look at HJM model, which is used to price volatility 803 00:55:07,070 --> 00:55:08,820 products. 804 00:55:08,820 --> 00:55:13,470 So we're not going to go into too many details about this, 805 00:55:13,470 --> 00:55:19,920 but this is equations of evolution of forward rates 806 00:55:19,920 --> 00:55:24,510 that we need for simulation, for Monte Carlo simulation. 807 00:55:24,510 --> 00:55:27,851 What we're saying here is this change of the forward rates-- 808 00:55:27,851 --> 00:55:29,600 because forward rates is the quantity that 809 00:55:29,600 --> 00:55:33,420 is being assimilated-- has some drift, OK? 810 00:55:33,420 --> 00:55:35,460 Because dt is time. 811 00:55:35,460 --> 00:55:41,720 And also, it has some dependence on the forward rates 812 00:55:41,720 --> 00:55:43,560 to the power of beta, right? 813 00:55:43,560 --> 00:55:50,860 So if it's log-normal model, beta will be one. 814 00:55:50,860 --> 00:55:54,270 If it's normal model, beta will be zero. 815 00:55:54,270 --> 00:55:57,080 But in general, it's different. 816 00:55:57,080 --> 00:55:59,580 Then we have volatility surface, right, 817 00:55:59,580 --> 00:56:04,500 which kind of gives you what the number of volatility 818 00:56:04,500 --> 00:56:09,880 to use for this calendar and forward time. 819 00:56:09,880 --> 00:56:14,812 And we have correlation and factor structure which we're 820 00:56:14,812 --> 00:56:16,020 not going to talk about here. 821 00:56:16,020 --> 00:56:19,990 And this is Brownian motions. 822 00:56:19,990 --> 00:56:24,820 So we're not going to go any more complex like this. 823 00:56:24,820 --> 00:56:28,580 We'll just start looking at nice, two-dimensional surfaces 824 00:56:28,580 --> 00:56:35,570 here, and see what are the problems of calibrating 825 00:56:35,570 --> 00:56:36,770 the volatility surface. 826 00:56:40,800 --> 00:56:45,062 Just to give you a diagram of when 827 00:56:45,062 --> 00:56:47,020 we look at the surface, what different elements 828 00:56:47,020 --> 00:56:48,670 of that surface mean. 829 00:56:54,060 --> 00:56:55,650 It's a triangular surface. 830 00:56:55,650 --> 00:57:00,250 You have a calendar time, right, and you have a forward time. 831 00:57:00,250 --> 00:57:05,660 So your simulation starts at the first vertical line. 832 00:57:05,660 --> 00:57:10,500 So you have forward rates here as calibrated 833 00:57:10,500 --> 00:57:12,360 from the curve as of today. 834 00:57:12,360 --> 00:57:17,740 So those are square boxes here, square elements. 835 00:57:17,740 --> 00:57:21,240 So you need to kind of transition 836 00:57:21,240 --> 00:57:23,940 from the first line to the second step 837 00:57:23,940 --> 00:57:27,370 using Monte Carlo simulation. 838 00:57:27,370 --> 00:57:33,750 And that's when, for every arrow here, 839 00:57:33,750 --> 00:57:36,890 you need the volatility number. 840 00:57:36,890 --> 00:57:40,994 Then, once you did your Monte Carlo 841 00:57:40,994 --> 00:57:42,410 simulation for the second one, you 842 00:57:42,410 --> 00:57:44,230 need ones for the third one. 843 00:57:44,230 --> 00:57:48,700 And again you need data-- which volatility to use, OK? 844 00:57:48,700 --> 00:57:51,280 So essentially, the surface that we'll 845 00:57:51,280 --> 00:57:54,160 be looking at on next slides is essentially 846 00:57:54,160 --> 00:57:56,840 representing the numbers necessary for this transition's 847 00:57:56,840 --> 00:58:00,250 volatility for every arrow. 848 00:58:00,250 --> 00:58:08,580 So to explain, there are different areas here. 849 00:58:08,580 --> 00:58:18,170 Like, for example, those will be-- if one step is one here, 850 00:58:18,170 --> 00:58:25,760 OK, we're talking about this, it will be the forward rates that 851 00:58:25,760 --> 00:58:28,800 will be observed in two years. 852 00:58:28,800 --> 00:58:34,045 That one will be observed as of now for like one year from now. 853 00:58:34,045 --> 00:58:36,630 But again, in two years. 854 00:58:36,630 --> 00:58:44,240 And so those rates are essential to compute the swap, 855 00:58:44,240 --> 00:58:45,930 forward swap rate. 856 00:58:45,930 --> 00:58:50,730 And if we do our Monte Carlo simulation, 857 00:58:50,730 --> 00:58:53,370 that's essential information that we 858 00:58:53,370 --> 00:58:57,130 need to compute the price for the option 859 00:58:57,130 --> 00:59:01,180 on the swap, which we're not going to discuss here. 860 00:59:01,180 --> 00:59:05,320 But just an example, it shows that for different instruments 861 00:59:05,320 --> 00:59:09,885 observed in the market you have quite overlapping 862 00:59:09,885 --> 00:59:10,760 areas of sensitivity. 863 00:59:16,940 --> 00:59:23,440 So this is a typical example of the volatility surface 864 00:59:23,440 --> 00:59:26,860 where this is calendar time, this is forward time. 865 00:59:26,860 --> 00:59:33,100 And it has spikes for certain regions. 866 00:59:33,100 --> 00:59:34,460 But in general, it's smooth. 867 00:59:38,880 --> 00:59:42,090 So why this problem is challenging. 868 00:59:45,900 --> 00:59:48,740 If we try to compute the triangle matrix, which 869 00:59:48,740 --> 00:59:53,190 has dimension of 240 by 240-- the reason why is 240 870 00:59:53,190 --> 00:59:58,510 is because every element is for three months, OK? 871 00:59:58,510 --> 01:00:06,080 But we need up to 60 years of data, 872 01:00:06,080 --> 01:00:12,110 which means that it's 60 by 4, which 873 01:00:12,110 --> 01:00:14,630 is the number of quarters, is 240 by 240. 874 01:00:14,630 --> 01:00:18,310 If you just need triangle elements, it's 200K elements. 875 01:00:18,310 --> 01:00:23,360 So if you try to calibrate everything at the same time, 876 01:00:23,360 --> 01:00:26,710 and you formally try to solve your problem, 877 01:00:26,710 --> 01:00:33,130 you kind of needed to store, at least to build a matrix of 28K 878 01:00:33,130 --> 01:00:35,230 by 28K. 879 01:00:35,230 --> 01:00:40,060 And we just don't have memory for this. 880 01:00:40,060 --> 01:00:41,840 And we also have very small number 881 01:00:41,840 --> 01:00:47,660 of calibration instruments only in terms of swaptions or caps, 882 01:00:47,660 --> 01:00:49,360 which are typical volatility products. 883 01:00:49,360 --> 01:00:51,890 We just have a relatively small number. 884 01:00:51,890 --> 01:00:55,330 So it's an underdetermined problem. 885 01:00:55,330 --> 01:00:58,840 We also, as we saw on the previous example, 886 01:00:58,840 --> 01:01:05,030 areas of sensitivity of different instruments overlap. 887 01:01:05,030 --> 01:01:08,320 And it's an ill-posed inverse problems which 888 01:01:08,320 --> 01:01:13,750 produces unstable solutions. 889 01:01:13,750 --> 01:01:18,760 And no matter what we do, right, the resulting surface 890 01:01:18,760 --> 01:01:19,940 should be nice, right? 891 01:01:19,940 --> 01:01:20,940 Should look nice, right? 892 01:01:20,940 --> 01:01:27,320 Because if it has spikes in some points in the future, then 893 01:01:27,320 --> 01:01:29,690 we either have an economic reason for this, 894 01:01:29,690 --> 01:01:32,640 or we claim that this is something that's not realistic. 895 01:01:37,640 --> 01:01:41,650 So this is how we approach the problem. 896 01:01:41,650 --> 01:01:44,540 So the first step, we represent our volatility surface. 897 01:01:44,540 --> 01:01:49,520 And here, even though volatility surface is two-dimensional 898 01:01:49,520 --> 01:01:56,710 we just kind of assign a number for every of those elements, 899 01:01:56,710 --> 01:02:00,750 OK, and then represent the surface as a vector, OK? 900 01:02:00,750 --> 01:02:03,160 Whereas saying that the new surface v 901 01:02:03,160 --> 01:02:10,255 will be some initial state plus a linear combination 902 01:02:10,255 --> 01:02:12,230 of basis functions. 903 01:02:12,230 --> 01:02:15,730 And basis functions should correspond 904 01:02:15,730 --> 01:02:22,240 to some reasonable functions, OK? 905 01:02:22,240 --> 01:02:25,180 But the nice feature is that number of basis functions 906 01:02:25,180 --> 01:02:30,467 will be much smaller than the number of elements 907 01:02:30,467 --> 01:02:32,380 that we need to calibrate. 908 01:02:32,380 --> 01:02:35,970 But we will be very formal here. 909 01:02:35,970 --> 01:02:38,850 And we'll try to use same number of basis functions 910 01:02:38,850 --> 01:02:41,930 as we have our input instruments. 911 01:02:41,930 --> 01:02:45,540 So in case we had 50 input instruments, 912 01:02:45,540 --> 01:02:50,210 we select basis functions also the number 50. 913 01:02:50,210 --> 01:02:59,390 So we will use typical Newton-Raphson approach here. 914 01:02:59,390 --> 01:03:09,160 We will compute sensitivities of input of all instruments 915 01:03:09,160 --> 01:03:14,350 to perturbations of a volatility surface, OK? 916 01:03:14,350 --> 01:03:16,310 We'll build this Jacobian matrix. 917 01:03:16,310 --> 01:03:28,780 And then, if we made the reasonable assumptions 918 01:03:28,780 --> 01:03:35,730 about what those basis functions are, 919 01:03:35,730 --> 01:03:39,790 then we can invert our square Jacobian. 920 01:03:39,790 --> 01:03:41,680 And again, the reason why it's square 921 01:03:41,680 --> 01:03:46,010 is because we selected same number of basis functions 922 01:03:46,010 --> 01:03:48,420 as the number of input instruments. 923 01:03:48,420 --> 01:03:50,830 It's actually quite common approach, 924 01:03:50,830 --> 01:03:55,450 but it's very often is wrong approach. 925 01:03:55,450 --> 01:03:56,980 It produces unstable results. 926 01:03:56,980 --> 01:04:01,110 And we will see why. 927 01:04:01,110 --> 01:04:03,510 So we converge to exact solution, 928 01:04:03,510 --> 01:04:06,850 but now the volatility surface looks like this. 929 01:04:06,850 --> 01:04:09,870 It looks less like a volatility surface, 930 01:04:09,870 --> 01:04:14,820 but more like Manhattan skyline. 931 01:04:14,820 --> 01:04:16,530 So you have a Hudson River here, and you 932 01:04:16,530 --> 01:04:20,550 have some buildings right? 933 01:04:20,550 --> 01:04:25,910 Obviously, even though it calibrates exactly, right? 934 01:04:25,910 --> 01:04:27,660 And you could go and price your portfolio, 935 01:04:27,660 --> 01:04:32,090 but probably prices for instruments 936 01:04:32,090 --> 01:04:34,780 in the portfolio that are not input 937 01:04:34,780 --> 01:04:38,380 instruments for calibration would be meaningless. 938 01:04:38,380 --> 01:04:44,280 Because the reason why we need this surface to be smooth 939 01:04:44,280 --> 01:04:47,560 is because for similar instruments 940 01:04:47,560 --> 01:04:49,770 for similar products in your portfolio 941 01:04:49,770 --> 01:04:51,830 you kind of expect similar prices, right? 942 01:04:51,830 --> 01:04:53,770 So if your volatility jumps, that's 943 01:04:53,770 --> 01:04:56,490 something that just contradicts with this assumption. 944 01:05:00,570 --> 01:05:04,480 So now how can we improve the situation? 945 01:05:04,480 --> 01:05:09,420 So we can try to use our basis functions which 946 01:05:09,420 --> 01:05:11,590 were selected in terms of piecewise constant shift 947 01:05:11,590 --> 01:05:13,130 of different areas. 948 01:05:13,130 --> 01:05:18,180 We can use a smooth version of those plans. 949 01:05:18,180 --> 01:05:22,930 But again, the result looks better, but still 950 01:05:22,930 --> 01:05:23,810 is not good enough. 951 01:05:26,720 --> 01:05:31,380 And just to demonstrate that this is an ill-posed problem-- 952 01:05:31,380 --> 01:05:33,390 an ill-posed problem is something 953 01:05:33,390 --> 01:05:37,370 that small changes of your inputs 954 01:05:37,370 --> 01:05:41,000 results in insane changes in your output. 955 01:05:41,000 --> 01:05:43,640 And this is a typical example. 956 01:05:43,640 --> 01:05:46,860 So keep all the instruments the same, OK? 957 01:05:46,860 --> 01:05:50,350 We just change by 1%-- which is not a big number-- 958 01:05:50,350 --> 01:05:57,850 of the five year by 10 year swaption, 959 01:05:57,850 --> 01:06:01,360 results in a quite large change of the volatility surface. 960 01:06:01,360 --> 01:06:02,910 But look also at the shape right? 961 01:06:02,910 --> 01:06:05,650 So it's really kind of you look at one building 962 01:06:05,650 --> 01:06:08,370 with an antenna, and another building, right? 963 01:06:08,370 --> 01:06:12,660 So it's very unreasonable change of the volatility surface. 964 01:06:17,530 --> 01:06:22,070 So we can use ill-posedness to our advantage. 965 01:06:22,070 --> 01:06:27,430 So basically, at this point we say, 966 01:06:27,430 --> 01:06:30,300 well, it's not a requirement to calibrate exactly, 967 01:06:30,300 --> 01:06:32,800 just because every instruments that 968 01:06:32,800 --> 01:06:35,750 is an input of collaboration actually has some tolerance. 969 01:06:35,750 --> 01:06:38,990 So even there is no point to calibrate it exactly. 970 01:06:41,980 --> 01:06:48,830 So because we know that small variations in inputs 971 01:06:48,830 --> 01:06:52,010 can be large variations of outputs, 972 01:06:52,010 --> 01:06:55,590 we can put some constraints on the outputs. 973 01:06:55,590 --> 01:06:57,140 And actually, that may not cost us 974 01:06:57,140 --> 01:07:01,130 much in terms of not being able to calibrate exactly, 975 01:07:01,130 --> 01:07:09,420 but produce much more meaningful result. 976 01:07:09,420 --> 01:07:15,880 And just to be absolutely sure that our output 977 01:07:15,880 --> 01:07:20,046 result, our surface is smooth, we 978 01:07:20,046 --> 01:07:27,250 can use basis functions that are smooth to begin with. 979 01:07:27,250 --> 01:07:30,840 So we'll use B-splines, but those will be two-dimensional. 980 01:07:30,840 --> 01:07:34,950 And we'll talk a little bit more about this. 981 01:07:34,950 --> 01:07:41,790 And it's not a requirement for us to have as many basis 982 01:07:41,790 --> 01:07:46,540 functions as we have instruments, because we 983 01:07:46,540 --> 01:07:47,860 can put some other constraints. 984 01:07:47,860 --> 01:07:50,770 Like, for example, we can put smoothness or gradient 985 01:07:50,770 --> 01:07:52,610 smoothness to the surface. 986 01:07:52,610 --> 01:07:56,970 So let's pick some relatively high but reasonable number 987 01:07:56,970 --> 01:07:59,400 of functions-- could be more than the number of input 988 01:07:59,400 --> 01:08:01,590 instruments-- and see what we can do. 989 01:08:01,590 --> 01:08:03,997 So first of all, let's build our basis 990 01:08:03,997 --> 01:08:05,080 functions for the surface. 991 01:08:05,080 --> 01:08:07,300 So this slide we already just saw, 992 01:08:07,300 --> 01:08:13,520 that we selected to use B-splines, which is 993 01:08:13,520 --> 01:08:15,630 very convenient to work with. 994 01:08:15,630 --> 01:08:18,630 This is a one-dimensional. 995 01:08:18,630 --> 01:08:21,229 This is the way we build them, typically. 996 01:08:21,229 --> 01:08:24,830 So we use the Cox-de Boor recursion formula. 997 01:08:24,830 --> 01:08:27,430 You start from linear, right? 998 01:08:27,430 --> 01:08:29,590 Then you apply that formula. 999 01:08:29,590 --> 01:08:34,370 You transition to the basis set of the second order. 1000 01:08:34,370 --> 01:08:40,550 And then the next iteration, you have the third order. 1001 01:08:40,550 --> 01:08:43,370 And those ones will be built. 1002 01:08:43,370 --> 01:08:51,660 So now, if you take those basis functions in one dimension, 1003 01:08:51,660 --> 01:08:54,890 and same basis functions in the other dimension, 1004 01:08:54,890 --> 01:08:58,700 and then you compute the kind of 10s of products, 1005 01:08:58,700 --> 01:09:01,880 like you multiply them one by one, 1006 01:09:01,880 --> 01:09:05,000 then you get basis functions with shapes like this. 1007 01:09:11,830 --> 01:09:14,920 Which means that no matter what we do, right, 1008 01:09:14,920 --> 01:09:19,859 like, because every basis function makes sense, 1009 01:09:19,859 --> 01:09:23,859 then any linear combinations will also be good enough. 1010 01:09:23,859 --> 01:09:26,850 So to formulate the problem is very simple. 1011 01:09:26,850 --> 01:09:31,606 So we're saying, OK, the quotes produced by our model 1012 01:09:31,606 --> 01:09:33,939 should be close enough to what's observed in the market, 1013 01:09:33,939 --> 01:09:35,090 with some weights again. 1014 01:09:35,090 --> 01:09:37,060 But we don't require any more that those 1015 01:09:37,060 --> 01:09:39,779 are calibrated exactly. 1016 01:09:39,779 --> 01:09:44,647 We are going to put some penalty function to the change 1017 01:09:44,647 --> 01:09:45,730 of the volatility surface. 1018 01:09:49,260 --> 01:09:52,910 And we're going to put some penalty function 1019 01:09:52,910 --> 01:09:58,620 to the volatility surface itself. 1020 01:09:58,620 --> 01:10:03,980 So those are vectors, right? 1021 01:10:03,980 --> 01:10:06,890 And L_1 and L_2 are matrices. 1022 01:10:06,890 --> 01:10:11,490 So just to give you an example what those matrices should be, 1023 01:10:11,490 --> 01:10:16,440 like if you are talking about smoothness, 1024 01:10:16,440 --> 01:10:20,410 if you want to penalize the gradient of the vector, right, 1025 01:10:20,410 --> 01:10:25,560 then the matrix will consist of rows of one 1026 01:10:25,560 --> 01:10:26,840 and following minus 1. 1027 01:10:26,840 --> 01:10:29,490 So what you're saying, OK, if I want 1028 01:10:29,490 --> 01:10:32,490 to penalize the difference between this and the next. 1029 01:10:32,490 --> 01:10:35,220 And you do that for every element, OK? 1030 01:10:35,220 --> 01:10:41,890 And the penalty kind of consists of all penalties that you have. 1031 01:10:41,890 --> 01:10:51,490 So here we just formulate our problem right? 1032 01:10:51,490 --> 01:10:57,680 So we want once-- because we've had the Jacobian-- we want 1033 01:10:57,680 --> 01:11:03,300 to price things close enough. 1034 01:11:03,300 --> 01:11:05,740 And there are two penalty terms here 1035 01:11:05,740 --> 01:11:11,440 with the different regularization parameters. 1036 01:11:11,440 --> 01:11:17,020 So once we have this, we can just, using linear algebra, 1037 01:11:17,020 --> 01:11:20,190 the solution is defined here. 1038 01:11:20,190 --> 01:11:29,130 And this is resulting calibration, which 1039 01:11:29,130 --> 01:11:31,990 we see is nice and smooth. 1040 01:11:41,440 --> 01:11:46,530 So if we take the analysis of the calibration inverse 1041 01:11:46,530 --> 01:11:51,060 problem, let's do that using our linear algebra 1042 01:11:51,060 --> 01:11:54,080 tools to understand where the problem is coming from. 1043 01:11:54,080 --> 01:12:00,020 OK, so A is a matrix, translates our model parameters 1044 01:12:00,020 --> 01:12:01,330 to market observables. 1045 01:12:01,330 --> 01:12:05,380 And there is some error there-- epsilon, OK? 1046 01:12:05,380 --> 01:12:12,620 So you can see that your solution 1047 01:12:12,620 --> 01:12:19,950 is a linear combination of singular values divided 1048 01:12:19,950 --> 01:12:21,627 by the singular values, OK? 1049 01:12:21,627 --> 01:12:32,820 So if your values are high, OK, then that's not a problem. 1050 01:12:32,820 --> 01:12:38,280 The problem is that once you get very small singular values, OK, 1051 01:12:38,280 --> 01:12:42,770 the deviation of v_i's can result 1052 01:12:42,770 --> 01:12:45,682 in the large deviation of your reconstructed result. 1053 01:12:45,682 --> 01:12:47,140 And that's when you have a problem. 1054 01:12:52,860 --> 01:12:56,020 So this is described on this slide. 1055 01:12:56,020 --> 01:13:04,840 So "ill-posed" is that small noise may be significantly 1056 01:13:04,840 --> 01:13:08,360 amplified by small singular values. 1057 01:13:08,360 --> 01:13:13,700 And if you have a problem when you don't know how good it is, 1058 01:13:13,700 --> 01:13:15,820 and whether you can trust it or not, 1059 01:13:15,820 --> 01:13:20,560 so it's a very standard approach, 1060 01:13:20,560 --> 01:13:23,010 you compute the condition number, 1061 01:13:23,010 --> 01:13:27,490 which is the ratio of the maximum to minimum singular 1062 01:13:27,490 --> 01:13:28,550 values. 1063 01:13:28,550 --> 01:13:32,040 And if the number is high, which means 1064 01:13:32,040 --> 01:13:37,120 that there are some very insignificant modes 1065 01:13:37,120 --> 01:13:42,000 in your input data that can cause substantial changes 1066 01:13:42,000 --> 01:13:42,680 in your output. 1067 01:13:42,680 --> 01:13:44,730 And if you know that, it's not comforting right? 1068 01:13:44,730 --> 01:13:48,720 So if that mode actually doesn't present in reality, 1069 01:13:48,720 --> 01:13:50,216 then that's fine. 1070 01:13:50,216 --> 01:13:51,590 But there is no guarantee, right? 1071 01:13:51,590 --> 01:13:55,038 See, if that happens, then your model basically blows up. 1072 01:14:00,150 --> 01:14:06,770 And that slide displays exactly that noiseless situation, 1073 01:14:06,770 --> 01:14:11,230 where it looks like if you don't have any noise 1074 01:14:11,230 --> 01:14:14,960 and if your model is perfect, then 1075 01:14:14,960 --> 01:14:17,000 you're always able to calibrate exactly 1076 01:14:17,000 --> 01:14:19,350 to the market observables. 1077 01:14:19,350 --> 01:14:21,800 But it's never the case right? 1078 01:14:21,800 --> 01:14:24,760 So there's always uncertainty to the numbers 1079 01:14:24,760 --> 01:14:27,650 that you're calibrating to. 1080 01:14:27,650 --> 01:14:33,200 And your model is not always perfect. 1081 01:14:37,700 --> 01:14:44,530 So very standard technique for that particular problem 1082 01:14:44,530 --> 01:14:48,680 is the Tikhonov regularization, which, 1083 01:14:48,680 --> 01:14:54,680 when you solve your ill-posed problem, as trying 1084 01:14:54,680 --> 01:15:01,374 to minimize x minus y, you add some penalty to the amplitude 1085 01:15:01,374 --> 01:15:03,120 to your solution. 1086 01:15:03,120 --> 01:15:05,620 Which essentially saying, OK, give me 1087 01:15:05,620 --> 01:15:08,090 something reasonable, but something that's not blown-up. 1088 01:15:12,040 --> 01:15:19,430 If you go through this linear algebra, 1089 01:15:19,430 --> 01:15:24,350 to see how that lambda parameter in the Tikhonov regularization 1090 01:15:24,350 --> 01:15:30,760 affects the weights of the SVD kind of representation 1091 01:15:30,760 --> 01:15:36,580 of your solution, we now see that small singular values 1092 01:15:36,580 --> 01:15:39,600 is no longer a problem, just because we're not 1093 01:15:39,600 --> 01:15:42,840 dividing by the small number, but actually we 1094 01:15:42,840 --> 01:15:46,041 are kind of limited by some regularization parameter. 1095 01:15:49,410 --> 01:15:53,470 And typically, when you apply that regularization, 1096 01:15:53,470 --> 01:15:58,470 your model no longer gives you a perfect match, right? 1097 01:15:58,470 --> 01:16:02,083 But the result is much more meaningful, and more stable. 1098 01:16:09,780 --> 01:16:19,060 Another approach to the problem is-- 1099 01:16:19,060 --> 01:16:20,960 and before we go to that slide. 1100 01:16:20,960 --> 01:16:23,790 So kind of Tikhonov regularization we 1101 01:16:23,790 --> 01:16:29,120 used for surface calibration. 1102 01:16:29,120 --> 01:16:31,060 Here, a standard Tikhonov regularization 1103 01:16:31,060 --> 01:16:33,840 is something that you just penalize 1104 01:16:33,840 --> 01:16:35,640 the amplitude of the solution itself. 1105 01:16:35,640 --> 01:16:37,973 But it doesn't have to be the amplitude of the solution. 1106 01:16:37,973 --> 01:16:42,290 It can be some linear combination of your solution. 1107 01:16:42,290 --> 01:16:45,990 And in terms of calibrating of volatility surface, 1108 01:16:45,990 --> 01:16:49,090 we didn't apply penalty to the reconstructed volatility, 1109 01:16:49,090 --> 01:16:54,200 but we say it's not that the amplitude of the solution 1110 01:16:54,200 --> 01:16:55,630 that we don't like. 1111 01:16:55,630 --> 01:16:57,260 We don't like non-smoothness. 1112 01:16:57,260 --> 01:17:00,870 So let's penalize the derivatives of the surface 1113 01:17:00,870 --> 01:17:02,088 in the different angles. 1114 01:17:07,070 --> 01:17:12,290 Another approach would be to use a truncated SVD, where 1115 01:17:12,290 --> 01:17:14,810 we say, OK, so we did our singular value decomposition. 1116 01:17:14,810 --> 01:17:18,610 We're looking at the spectrum of singular values, 1117 01:17:18,610 --> 01:17:23,790 and we find that some numbers look nice and large, some very 1118 01:17:23,790 --> 01:17:25,000 small. 1119 01:17:25,000 --> 01:17:28,210 And then we just skip the small ones. 1120 01:17:28,210 --> 01:17:35,220 It's very similar to the PCA approach for the risk 1121 01:17:35,220 --> 01:17:38,160 management that we saw before, where 1122 01:17:38,160 --> 01:17:40,900 we just selected five principal components 1123 01:17:40,900 --> 01:17:42,720 and we ignored the rest. 1124 01:17:42,720 --> 01:17:47,030 As a result, the model is much more robust. 1125 01:17:47,030 --> 01:17:49,300 And by doing this, we essentially 1126 01:17:49,300 --> 01:17:51,414 truncate the null space of the model. 1127 01:17:51,414 --> 01:17:52,830 If you're familiar with this, it's 1128 01:17:52,830 --> 01:18:05,894 basically the space that has very small singular values. 1129 01:18:12,670 --> 01:18:21,190 So what regularized models gives you is that improved stability. 1130 01:18:21,190 --> 01:18:24,125 It's absolutely essential for ill-conditioned problems. 1131 01:18:28,480 --> 01:18:34,950 And it's a more realistic and meaningful result 1132 01:18:34,950 --> 01:18:39,860 at the expense of some beauty to fit exactly the data, 1133 01:18:39,860 --> 01:18:44,940 but that's something that is quite often acceptable. 1134 01:18:44,940 --> 01:18:47,550 It might cause a biased solution, 1135 01:18:47,550 --> 01:18:51,510 meaning that your solution again may not be exact. 1136 01:18:51,510 --> 01:18:55,320 It might be biased towards some better result. 1137 01:18:55,320 --> 01:19:00,480 For example, if you apply smoothness constraint, 1138 01:19:00,480 --> 01:19:03,810 the solution would kind of assume a little bit 1139 01:19:03,810 --> 01:19:06,960 more smooth result than it actually is. 1140 01:19:06,960 --> 01:19:10,590 But that's acceptable. 1141 01:19:10,590 --> 01:19:15,840 And the bias, again, can be minimized 1142 01:19:15,840 --> 01:19:20,210 by reasonable selection of what quantity you actually 1143 01:19:20,210 --> 01:19:21,200 don't like. 1144 01:19:21,200 --> 01:19:23,820 Again, you can say, oh, during calibration 1145 01:19:23,820 --> 01:19:27,067 of our volatilities of our surface we could have said, 1146 01:19:27,067 --> 01:19:27,650 you know what? 1147 01:19:27,650 --> 01:19:32,470 Let's just open the textbook and see what's the regularization. 1148 01:19:32,470 --> 01:19:33,850 OK, we find Tikhonov. 1149 01:19:33,850 --> 01:19:36,040 We start to penalize the amplitude. 1150 01:19:36,040 --> 01:19:37,890 Then the result won't be good. 1151 01:19:37,890 --> 01:19:40,570 So we need to think about and say, OK, 1152 01:19:40,570 --> 01:19:42,080 what exactly we don't like. 1153 01:19:42,080 --> 01:19:45,480 Like, for example, is like absolutely flat volatility 1154 01:19:45,480 --> 01:19:46,760 surface good for us? 1155 01:19:46,760 --> 01:19:49,240 And we'll find, yeah, that that's actually fine. 1156 01:19:49,240 --> 01:19:52,690 Then, if we said that then penalizing the amplitude 1157 01:19:52,690 --> 01:19:54,440 doesn't make sense, so we need to penalize 1158 01:19:54,440 --> 01:19:57,790 something that is a deviation from that perfect flat 1159 01:19:57,790 --> 01:19:59,030 solution. 1160 01:19:59,030 --> 01:20:02,560 Or to be more kind of precise, we 1161 01:20:02,560 --> 01:20:07,990 penalize like the derivative in different directions. 1162 01:20:07,990 --> 01:20:13,540 So this kind of concludes my presentation today. 1163 01:20:13,540 --> 01:20:18,580 And there are some useful links if you 1164 01:20:18,580 --> 01:20:22,546 want to get more information. 1165 01:20:25,997 --> 01:20:28,462 Thank you. 1166 01:20:28,462 --> 01:20:29,448 Any question? 1167 01:20:34,880 --> 01:20:35,838 AUDIENCE: Yes. 1168 01:20:35,838 --> 01:20:41,814 So regarding the techniques that you use for fitting 1169 01:20:41,814 --> 01:20:46,794 function that you are using spline techniques. 1170 01:20:46,794 --> 01:20:50,480 What other techniques-- is the spline 1171 01:20:50,480 --> 01:20:52,370 the best technique you use? 1172 01:20:56,220 --> 01:20:57,970 IVAN MASYUKOV: Well, spline is, yes. 1173 01:20:57,970 --> 01:21:01,450 So a spline or interpolation is the same. 1174 01:21:01,450 --> 01:21:03,350 So we're always talking about interpolation. 1175 01:21:03,350 --> 01:21:07,080 So you have some limited number of inputs, 1176 01:21:07,080 --> 01:21:08,560 and you want to draw in between. 1177 01:21:08,560 --> 01:21:11,420 So there are just two words for this, 1178 01:21:11,420 --> 01:21:15,120 which is a interpolation-- or spline-- which I consider to be 1179 01:21:15,120 --> 01:21:16,600 the same thing in general. 1180 01:21:26,310 --> 01:21:30,050 AUDIENCE: I have a question about the interpolation graph 1181 01:21:30,050 --> 01:21:32,525 that you had where the following was very smooth. 1182 01:21:35,495 --> 01:21:39,046 When you, as an expert in this, look at that graph 1183 01:21:39,046 --> 01:21:46,090 and see, I guess, just some odd shapes 1184 01:21:46,090 --> 01:21:53,222 at certain parts of the curve, how do you interpret that? 1185 01:21:53,222 --> 01:21:56,980 And do you assess that that's a feature of the current term 1186 01:21:56,980 --> 01:22:00,764 market liquidity conditions, or possibly just a mathematical-- 1187 01:22:00,764 --> 01:22:02,180 IVAN MASYUKOV: Well, first of all, 1188 01:22:02,180 --> 01:22:09,680 I mean, that grid is done like for-- every element is 1189 01:22:09,680 --> 01:22:10,685 a three-month. 1190 01:22:10,685 --> 01:22:12,700 But what's traded on the market? 1191 01:22:12,700 --> 01:22:15,620 Like, typical maturities are three-month, 1192 01:22:15,620 --> 01:22:19,710 maybe half-year, one year, two year, five year, 10 year. 1193 01:22:19,710 --> 01:22:25,590 So we should kind of rescale it in a logarithmic scale, 1194 01:22:25,590 --> 01:22:27,920 or-- you know what I'm talking about? 1195 01:22:27,920 --> 01:22:31,010 And then, if you do that, then this peak 1196 01:22:31,010 --> 01:22:32,460 doesn't look a peak anymore. 1197 01:22:32,460 --> 01:22:35,450 So the reason why it looks like a feature to you 1198 01:22:35,450 --> 01:22:39,630 is because it's quite sharper than this guy, right? 1199 01:22:39,630 --> 01:22:44,990 But that's because you have many more detailed instruments there 1200 01:22:44,990 --> 01:22:46,060 compared to here. 1201 01:22:49,570 --> 01:22:52,191 And that's the reason why we selected our basis functions 1202 01:22:52,191 --> 01:22:52,690 like this. 1203 01:22:52,690 --> 01:22:56,890 So we selected more node density in the front, 1204 01:22:56,890 --> 01:23:01,270 just because there are more instruments in the front 1205 01:23:01,270 --> 01:23:03,210 than at the end. 1206 01:23:03,210 --> 01:23:08,620 So we want our spline to be more detailed at the beginning, 1207 01:23:08,620 --> 01:23:12,200 and kind of just nice and smooth at the end. 1208 01:23:12,200 --> 01:23:16,990 So that's why those basis functions-- which correspond 1209 01:23:16,990 --> 01:23:20,900 very well to actual instruments that we have in the portfolio-- 1210 01:23:20,900 --> 01:23:25,820 can produce, first of all, you see this spike here too, right? 1211 01:23:25,820 --> 01:23:30,250 So is it like you can compare this guy to this guy. 1212 01:23:30,250 --> 01:23:36,010 But essentially, it's just they have similar magnitude, 1213 01:23:36,010 --> 01:23:39,000 but we don't have enough instruments in this area 1214 01:23:39,000 --> 01:23:41,720 to support any sharper features. 1215 01:23:41,720 --> 01:23:47,540 So I don't see any problems with this graph. 1216 01:23:47,540 --> 01:23:52,150 But traders, because they look at this every day, right? 1217 01:23:52,150 --> 01:23:54,150 And then they calibrate, and they see a feature, 1218 01:23:54,150 --> 01:23:57,190 and then they immediately kind of trying to think. 1219 01:23:57,190 --> 01:24:00,920 OK, if you see something that you know is not typical-- 1220 01:24:00,920 --> 01:24:04,110 and that sense of typical/not typical 1221 01:24:04,110 --> 01:24:07,490 comes with years of experience-- then they 1222 01:24:07,490 --> 01:24:09,870 try to arbitrage this. 1223 01:24:09,870 --> 01:24:13,840 Because if there is a spike in the surface, 1224 01:24:13,840 --> 01:24:15,980 it's very likely that it will disappear soon. 1225 01:24:19,820 --> 01:24:21,260 AUDIENCE: So if this is the model 1226 01:24:21,260 --> 01:24:26,317 for volatility, that's used in modeling for swaptions, right? 1227 01:24:26,317 --> 01:24:27,784 IVAN MASYUKOV: Yeah. 1228 01:24:27,784 --> 01:24:29,495 AUDIENCE: So if you were to actually try 1229 01:24:29,495 --> 01:24:33,163 to go about making a trade out of some discrepancy 1230 01:24:33,163 --> 01:24:35,363 that you see, can you kind of describe how 1231 01:24:35,363 --> 01:24:36,815 you'd do that with swaption? 1232 01:24:36,815 --> 01:24:38,564 Do you use basically like regular options? 1233 01:24:42,547 --> 01:24:44,380 IVAN MASYUKOV: I don't have the screen here. 1234 01:24:44,380 --> 01:24:50,010 But essentially what traders do, right? 1235 01:24:50,010 --> 01:24:56,130 So this is, I mean that calibrator we actually use. 1236 01:24:56,130 --> 01:24:58,400 And it's a real-time calibrator. 1237 01:24:58,400 --> 01:25:00,640 The reason why it can be real time 1238 01:25:00,640 --> 01:25:04,470 is because there is just simple linear algebra there. 1239 01:25:04,470 --> 01:25:08,000 Most of stuff like A transpose A can be pre-calculated. 1240 01:25:08,000 --> 01:25:11,500 So it can be real time. 1241 01:25:11,500 --> 01:25:13,920 So they can see this volatility surface 1242 01:25:13,920 --> 01:25:20,290 moving while-- we connect to actual market data. 1243 01:25:20,290 --> 01:25:25,230 And once they see there is anomaly on the market, 1244 01:25:25,230 --> 01:25:27,970 there is something traded which they believe is wrong, 1245 01:25:27,970 --> 01:25:30,300 OK, they're just make an advantage of that. 1246 01:25:30,300 --> 01:25:35,000 They just make a trade, that would be swaption, for example, 1247 01:25:35,000 --> 01:25:39,240 or maybe some other trade more exotic than swaptions. 1248 01:25:39,240 --> 01:25:41,320 But again, having like the dependence 1249 01:25:41,320 --> 01:25:46,320 on the particular instruments, which kind of would 1250 01:25:46,320 --> 01:25:48,840 express your position that this will change 1251 01:25:48,840 --> 01:25:50,290 like in a day or so. 1252 01:25:50,290 --> 01:25:55,660 So that's exactly how the desk makes money. 1253 01:25:55,660 --> 01:25:58,220 So we call it relative value analysis. 1254 01:25:58,220 --> 01:26:01,030 So if we have a tool like that, you have a model. 1255 01:26:01,030 --> 01:26:04,500 You have your input instruments, and you have some regularizing 1256 01:26:04,500 --> 01:26:06,150 terms of it could be smoothness, it 1257 01:26:06,150 --> 01:26:09,910 could be a PCA, it could be combination of PCA, right? 1258 01:26:09,910 --> 01:26:13,200 But then that additional information 1259 01:26:13,200 --> 01:26:17,410 allows you to find anomalies in the market. 1260 01:26:17,410 --> 01:26:19,250 Once you find those anomalies, you 1261 01:26:19,250 --> 01:26:22,670 can take advantage of them-- provided that your model is 1262 01:26:22,670 --> 01:26:24,480 robust enough. 1263 01:26:24,480 --> 01:26:27,970 And if you are saying, well, I am kind of doing well, 1264 01:26:27,970 --> 01:26:31,350 and I'm calibrate well with just some smoothness assumption 1265 01:26:31,350 --> 01:26:33,410 about smoothness of the forward rates, 1266 01:26:33,410 --> 01:26:35,850 there is nothing more fundamental than this. 1267 01:26:35,850 --> 01:26:41,980 So if your model is based on fundamental principle, 1268 01:26:41,980 --> 01:26:44,990 you can expect that it will be more stable in the future, 1269 01:26:44,990 --> 01:26:45,820 rather than PCA. 1270 01:26:45,820 --> 01:26:47,795 Because for PCA, you just kind of say, 1271 01:26:47,795 --> 01:26:49,590 OK, I took the time interval. 1272 01:26:49,590 --> 01:26:55,090 I kind of did my regression analysis, whatever. 1273 01:26:55,090 --> 01:26:57,800 But that doesn't mean that the market will continue 1274 01:26:57,800 --> 01:26:59,400 to do the same in the future. 1275 01:27:02,865 --> 01:27:04,845 AUDIENCE: I have a question. 1276 01:27:04,845 --> 01:27:07,815 [INAUDIBLE] marketer. 1277 01:27:07,815 --> 01:27:10,785 And then we would try to price the bond at that premium. 1278 01:27:10,785 --> 01:27:14,313 You mentioned that actually bond is the most liquid instrument 1279 01:27:14,313 --> 01:27:15,260 in the market. 1280 01:27:15,260 --> 01:27:18,383 So why not you do the area around by inverse bond 1281 01:27:18,383 --> 01:27:22,889 to derive the discounting factor from a bond [INAUDIBLE]. 1282 01:27:22,889 --> 01:27:24,930 IVAN MASYUKOV: Well, that's a very good question. 1283 01:27:24,930 --> 01:27:30,150 So basically we could have done that, OK. 1284 01:27:30,150 --> 01:27:33,730 And some firms do that. 1285 01:27:33,730 --> 01:27:36,190 The problem is that with swaps, we 1286 01:27:36,190 --> 01:27:42,010 kind of have those swaps today, they kind of 1287 01:27:42,010 --> 01:27:44,560 roll every day, OK? 1288 01:27:44,560 --> 01:27:48,164 They like, the swap today starts today, swap tomorrow 1289 01:27:48,164 --> 01:27:49,580 starts tomorrow, things like that. 1290 01:27:49,580 --> 01:27:51,550 But bonds do not. 1291 01:27:51,550 --> 01:27:56,750 OK so they basically, there is like on-the-run bond, which 1292 01:27:56,750 --> 01:28:00,510 is the most liquid one, which once the new ten-year bond is 1293 01:28:00,510 --> 01:28:04,060 issued, then that bond becomes off-the-run. 1294 01:28:04,060 --> 01:28:05,790 It's still traded, but then everyone 1295 01:28:05,790 --> 01:28:07,800 switches to on-the-run. 1296 01:28:07,800 --> 01:28:13,300 So you don't have a nice continuous spectrum of bonds. 1297 01:28:13,300 --> 01:28:15,670 You have kind of concentrations between on-the-runs 1298 01:28:15,670 --> 01:28:17,400 and off-the-runs. 1299 01:28:17,400 --> 01:28:21,960 And if you want to draw the curve for all of them, 1300 01:28:21,960 --> 01:28:24,990 you typically cannot do the perfect fit. 1301 01:28:24,990 --> 01:28:29,160 You kind of need to do the least squares. 1302 01:28:29,160 --> 01:28:33,310 So it's just more convenient to do it in the swap. 1303 01:28:33,310 --> 01:28:38,730 But once we build the swap curve, swap trader typically-- 1304 01:28:38,730 --> 01:28:41,170 I should say always use bonds for hedging, 1305 01:28:41,170 --> 01:28:44,080 just because bonds are much more liquid. 1306 01:28:44,080 --> 01:28:48,220 Then we project bonds to the swap curve 1307 01:28:48,220 --> 01:28:51,240 rather than swaps to the bond curve, which is hard to build. 1308 01:28:52,206 --> 01:28:53,977 AUDIENCE: So in this case, when they 1309 01:28:53,977 --> 01:28:57,036 switch from on-the-run, off-the-run, [INAUDIBLE]? 1310 01:29:01,069 --> 01:29:01,860 IVAN MASYUKOV: Yes. 1311 01:29:01,860 --> 01:29:05,700 So your curve won't be stable, just because those roll 1312 01:29:05,700 --> 01:29:09,690 effects-- we call those "roll effects"-- 1313 01:29:09,690 --> 01:29:12,580 which means that there is something substantial changes 1314 01:29:12,580 --> 01:29:13,170 on the market. 1315 01:29:13,170 --> 01:29:14,900 And there may be such a big demand 1316 01:29:14,900 --> 01:29:21,650 for this new bond on the market, that will make your curve, that 1317 01:29:21,650 --> 01:29:23,700 won't look nice. 1318 01:29:23,700 --> 01:29:27,410 So there are also traders that just trade bonds. 1319 01:29:27,410 --> 01:29:32,470 And those typically don't have curves. 1320 01:29:32,470 --> 01:29:36,638 They rely on some PCA models, or some other things. 1321 01:29:39,851 --> 01:29:40,850 PROFESSOR: Thanks again. 1322 01:29:40,850 --> 01:29:42,610 IVAN MASYUKOV: Thank you.