WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:21.118 --> 00:00:22.534 PROFESSOR: Our guest speaker today 00:00:22.534 --> 00:00:25.390 from Morgan Stanley, Ivan Masyukov. 00:00:25.390 --> 00:00:28.336 Dr. Ivan Masyukov. 00:00:28.336 --> 00:00:29.210 IVAN MASYUKOV: Hello. 00:00:29.210 --> 00:00:30.260 One, two, three. 00:00:30.260 --> 00:00:31.206 Can you hear me? 00:00:31.206 --> 00:00:34.280 PROFESSOR: And the microphone will just be recording you, 00:00:34.280 --> 00:00:35.940 but it doesn't broadcast you. 00:00:35.940 --> 00:00:36.970 IVAN MASYUKOV: Ah. 00:00:36.970 --> 00:00:37.470 Understood. 00:00:37.470 --> 00:00:37.970 All right. 00:00:37.970 --> 00:00:40.950 So I'm Ivan Masyukov. 00:00:40.950 --> 00:00:42.540 I work in Morgan Stanley. 00:00:42.540 --> 00:00:48.420 And my background is applied physics and mathematics 00:00:48.420 --> 00:00:52.970 from Moscow Institute of Physics and Technology. 00:00:52.970 --> 00:00:59.905 And today, the topic of the lecture 00:00:59.905 --> 00:01:03.860 is regularized pricing and risk models. 00:01:03.860 --> 00:01:07.600 So we will talk about typical pricing risk 00:01:07.600 --> 00:01:13.430 models for interest rate products, 00:01:13.430 --> 00:01:17.400 and the important aspect of adding 00:01:17.400 --> 00:01:20.340 some additional constraints, which means, 00:01:20.340 --> 00:01:22.280 like, adding some regularizers to the model. 00:01:24.900 --> 00:01:28.020 So we will start from bonds, which 00:01:28.020 --> 00:01:31.200 is probably the most simple interest rate 00:01:31.200 --> 00:01:33.110 product on the market. 00:01:33.110 --> 00:01:36.830 Then we will discuss swaps. 00:01:36.830 --> 00:01:39.980 We will build a yield curve. 00:01:39.980 --> 00:01:44.900 And we will see how yield curve models 00:01:44.900 --> 00:01:51.550 can be improved to satisfy needs of actual trader. 00:01:51.550 --> 00:01:55.220 And at the end, we'll look at the very nice example 00:01:55.220 --> 00:01:58.970 of ill-posed problem of calibrating 00:01:58.970 --> 00:02:01.050 the two-dimensional volatility surface 00:02:01.050 --> 00:02:05.690 necessary for volatility model-- Monte Carlo assimilation. 00:02:05.690 --> 00:02:11.074 And we will see how that problem can be solved. 00:02:11.074 --> 00:02:12.990 During the lecture, if you have any questions, 00:02:12.990 --> 00:02:14.800 please interrupt, OK? 00:02:18.490 --> 00:02:19.960 So what is bond? 00:02:19.960 --> 00:02:25.450 Bond is a security which is issued if someone 00:02:25.450 --> 00:02:27.200 like a borrower needs money. 00:02:27.200 --> 00:02:32.400 And it promises to pay some certain fixed amount 00:02:32.400 --> 00:02:35.200 of certain cash flows in the future, 00:02:35.200 --> 00:02:38.590 and request for some money up front for this. 00:02:38.590 --> 00:02:47.170 So typical bonds basically include same periodic payment-- 00:02:47.170 --> 00:02:50.720 let's say like every half year or every year 00:02:50.720 --> 00:02:55.190 until maturity, where at maturity the face value is 00:02:55.190 --> 00:02:58.540 paid, like the biggest sum of money. 00:02:58.540 --> 00:03:03.620 And again, during the beginning, the investor 00:03:03.620 --> 00:03:06.730 is asked to pay some up front. 00:03:06.730 --> 00:03:08.780 There are also zero-coupon bonds, 00:03:08.780 --> 00:03:12.070 which don't pay anything until the maturity. 00:03:14.840 --> 00:03:17.800 And there are very interesting perpetual bonds 00:03:17.800 --> 00:03:20.620 where basically you pay some money up front, 00:03:20.620 --> 00:03:24.560 and then you pay it back like infinitely-- which 00:03:24.560 --> 00:03:26.340 sounds like a good deal, but we will 00:03:26.340 --> 00:03:28.200 know how to price it right. 00:03:31.190 --> 00:03:34.730 So those are some diagrams. 00:03:34.730 --> 00:03:38.740 So the first one is the standard fixed-rate bond, 00:03:38.740 --> 00:03:47.900 where small green arrows represent a periodic payment. 00:03:47.900 --> 00:03:51.700 And there is a face value added on top of a periodic payment 00:03:51.700 --> 00:03:53.210 at the maturity of the bond. 00:03:56.680 --> 00:04:01.300 So this is a typical cash flow diagram used for analysis, OK? 00:04:01.300 --> 00:04:06.306 And so arrows up represent something that-- 00:04:06.306 --> 00:04:07.180 and it's green right? 00:04:07.180 --> 00:04:08.013 That is good for us. 00:04:08.013 --> 00:04:10.470 So it's something that we receive. 00:04:10.470 --> 00:04:14.680 And a red arrow facing down represents something 00:04:14.680 --> 00:04:16.095 that you have to pay. 00:04:16.095 --> 00:04:16.750 Right? 00:04:16.750 --> 00:04:25.110 So a zero coupon bond is something, as I said before, 00:04:25.110 --> 00:04:28.370 is something that you pay up front, 00:04:28.370 --> 00:04:34.250 and you get back a fixed amount of money in the future. 00:04:34.250 --> 00:04:36.500 What's interesting about this graph-- you 00:04:36.500 --> 00:04:41.630 can see that the green arrow has a bigger 00:04:41.630 --> 00:04:45.480 amplitude than the red one, which means that you kind 00:04:45.480 --> 00:04:50.620 of, every time you put like $100 now, right, 00:04:50.620 --> 00:04:52.482 you kind of expect that in return you 00:04:52.482 --> 00:04:53.440 get more in the future. 00:04:53.440 --> 00:04:55.340 Because if you don't get more in the future, 00:04:55.340 --> 00:04:57.509 you just don't get this money, don't put this money. 00:04:57.509 --> 00:04:58.800 You just keep it in the pocket. 00:05:01.330 --> 00:05:08.160 So as a result, you get the concept of time value of money. 00:05:08.160 --> 00:05:16.480 So tomorrow, $100 always will be more than just $100. 00:05:16.480 --> 00:05:21.980 And also, if you look at the graph of the fixed-rate coupon 00:05:21.980 --> 00:05:26.320 bond, and you sum all of the cash flows here, 00:05:26.320 --> 00:05:29.260 it looks like you get more than this red one. 00:05:29.260 --> 00:05:36.460 But again, there is, as further in the future 00:05:36.460 --> 00:05:41.170 the cash flow is, the more the kind of depreciation. 00:05:41.170 --> 00:05:44.930 And we call this depreciation a discount factor, OK? 00:05:44.930 --> 00:05:50.490 So basically the more in the future the cash 00:05:50.490 --> 00:05:53.980 flow is, the smaller the discount factor. 00:05:53.980 --> 00:05:55.550 And so for today the discount factor 00:05:55.550 --> 00:06:00.440 will be 1, for tomorrow it will be like 0.999, and so forth. 00:06:00.440 --> 00:06:04.760 And in 30 years, let's say, it will probably be like 0.1, 00:06:04.760 --> 00:06:07.755 depending on current rates in the market. 00:06:13.410 --> 00:06:20.370 So let's see how we can price the bond-- or not 00:06:20.370 --> 00:06:25.440 necessarily price, but compute a fair value 00:06:25.440 --> 00:06:28.840 of future cash flows. 00:06:28.840 --> 00:06:33.670 So our fair value of computed cash flows 00:06:33.670 --> 00:06:36.875 can be found if we have discount factors. 00:06:36.875 --> 00:06:39.690 So every discount factor at every cash flow 00:06:39.690 --> 00:06:43.130 in the future I-- which in this particular case 00:06:43.130 --> 00:06:47.540 will be a coupon times the face value-- 00:06:47.540 --> 00:06:49.930 should be multiplied by the discount factor. 00:06:49.930 --> 00:06:52.930 And then we also add a face value discount 00:06:52.930 --> 00:06:56.215 with a discount factor at the maturity of the instrument. 00:07:00.820 --> 00:07:06.085 So the way this product trades in the market 00:07:06.085 --> 00:07:13.530 is that people buy and sell bonds paying P, right? 00:07:13.530 --> 00:07:17.300 So it's very important to understand that for bonds, it's 00:07:17.300 --> 00:07:20.530 not something that we have cash flows which we kind of need 00:07:20.530 --> 00:07:21.660 to price. 00:07:21.660 --> 00:07:25.100 It's actually the price is already known. 00:07:25.100 --> 00:07:27.796 So it's very liquid. 00:07:27.796 --> 00:07:30.360 It's the result of activity in the market, 00:07:30.360 --> 00:07:32.860 meaning that there is very little uncertainty 00:07:32.860 --> 00:07:33.820 about the price. 00:07:33.820 --> 00:07:36.810 So this P is known. 00:07:36.810 --> 00:07:39.050 And as with all cash flows, it's something that's 00:07:39.050 --> 00:07:40.540 written in the contract, right? 00:07:40.540 --> 00:07:45.480 So it's something, we have fixed cash flows in the future. 00:07:45.480 --> 00:07:50.000 So it's always a question about what kind of model 00:07:50.000 --> 00:07:54.910 is useful for the discount factors. 00:07:54.910 --> 00:07:56.850 So we need a model for discounting. 00:08:00.090 --> 00:08:01.150 Any questions so far? 00:08:04.940 --> 00:08:11.230 So one of the simplest models is to use just one parameter 00:08:11.230 --> 00:08:16.470 to kind of cover all the discounting. 00:08:16.470 --> 00:08:20.300 And the discount factor can be represented 00:08:20.300 --> 00:08:27.600 as e minus y times t of i, t sub i, where 00:08:27.600 --> 00:08:33.226 y is some kind of-- it's called yield to maturity. 00:08:33.226 --> 00:08:34.809 Well, the reason why it's exponential, 00:08:34.809 --> 00:08:36.270 it's natural right? 00:08:36.270 --> 00:08:42.916 So if you have a 0.999 discount factor for today, 00:08:42.916 --> 00:08:44.290 and then we kind of say, OK, it's 00:08:44.290 --> 00:08:45.800 the same discount for tomorrow, we 00:08:45.800 --> 00:08:48.300 will have the same discounting for every other day. 00:08:48.300 --> 00:08:49.590 So we have to multiply them. 00:08:49.590 --> 00:08:52.240 As a result, the total discounting 00:08:52.240 --> 00:08:56.140 will be an exponential. 00:08:56.140 --> 00:09:00.630 So if our discount factors are like this, 00:09:00.630 --> 00:09:07.020 then our price basically can be represented 00:09:07.020 --> 00:09:09.770 as a linear combination of future cash flows, right? 00:09:09.770 --> 00:09:12.200 At this point, by the way, we kind of 00:09:12.200 --> 00:09:17.090 merge together the final coupon with the face value, 00:09:17.090 --> 00:09:20.470 and we'll just kind of be talking about the coupons 00:09:20.470 --> 00:09:24.190 only, about cash flows only. 00:09:24.190 --> 00:09:31.655 And so that's the formula for the bond price, is this. 00:09:31.655 --> 00:09:36.020 So basically, what's known on the market is P, right, 00:09:36.020 --> 00:09:40.420 which is a price that's-- that instrument is traded. 00:09:40.420 --> 00:09:43.910 We also have defined cash flows in the future. 00:09:43.910 --> 00:09:47.510 So we can solve for the yield. 00:09:47.510 --> 00:09:54.210 So essentially, if we know the bond price, 00:09:54.210 --> 00:09:57.710 we can find the bond yield, OK? 00:09:57.710 --> 00:10:02.170 And if we know the bond yield, we can find the bond price, OK? 00:10:02.170 --> 00:10:08.480 So typically, bonds are traded in terms of its price. 00:10:08.480 --> 00:10:11.630 But some bonds are traded in terms of yield. 00:10:11.630 --> 00:10:15.280 But again, this is like one-to-one. 00:10:15.280 --> 00:10:17.240 You can always go back and forth. 00:10:21.140 --> 00:10:27.490 What's important is-- what has economic value, 00:10:27.490 --> 00:10:31.680 right, is the bond price, OK, and the future cash 00:10:31.680 --> 00:10:34.200 amounts of cash flows. 00:10:34.200 --> 00:10:36.644 And when you talk about yield, it's 00:10:36.644 --> 00:10:37.810 not something that's traded. 00:10:37.810 --> 00:10:44.240 It's actually one of the ways to align future cash flows 00:10:44.240 --> 00:10:46.170 with the bond price. 00:10:46.170 --> 00:10:49.150 And that way assumes that we have, 00:10:49.150 --> 00:10:55.110 again, constant discounting for all time points in the future. 00:10:55.110 --> 00:10:58.700 And we will see that it may or may not be optimal case. 00:11:03.250 --> 00:11:08.180 So what's also important when we're talking about instrument 00:11:08.180 --> 00:11:11.380 price is to have the model of how that price changes 00:11:11.380 --> 00:11:13.610 if the market changes. 00:11:13.610 --> 00:11:20.530 So here, we're talking about sensitivity of the bond price 00:11:20.530 --> 00:11:21.130 to yield. 00:11:21.130 --> 00:11:25.630 And what is typically done is basically 00:11:25.630 --> 00:11:27.435 to normalize by the bond price itself. 00:11:29.950 --> 00:11:33.720 And then it's called bond duration. 00:11:33.720 --> 00:11:37.530 So the nice thing about normalizing 00:11:37.530 --> 00:11:40.980 is that the duration of that bond that you 00:11:40.980 --> 00:11:44.620 have in your portfolio doesn't really depend on how many bonds 00:11:44.620 --> 00:11:45.400 you have, right? 00:11:45.400 --> 00:11:50.330 So it's basically more like property of the bond itself, 00:11:50.330 --> 00:11:55.900 rather than how many bonds you have in your portfolio. 00:11:55.900 --> 00:11:58.640 So if you take the previous formula 00:11:58.640 --> 00:12:01.320 and take the derivative with respect to y, 00:12:01.320 --> 00:12:07.150 we get the following formula for duration. 00:12:07.150 --> 00:12:12.360 And we know what the price is, right? 00:12:12.360 --> 00:12:17.540 And we can rewrite this formula this way, 00:12:17.540 --> 00:12:27.680 which you see it's a sum of t_i's times 00:12:27.680 --> 00:12:31.370 some weights and divided by the sum of the weights. 00:12:31.370 --> 00:12:35.840 So it's essentially a weighted sum of time, OK? 00:12:35.840 --> 00:12:41.910 And those pieces of time, those moments of time 00:12:41.910 --> 00:12:46.580 is more important, as-- I mean the weights are 00:12:46.580 --> 00:12:51.870 proportional to present values of future cash flows. 00:12:51.870 --> 00:12:56.250 So that's why bond duration has a very nice kind 00:12:56.250 --> 00:12:57.265 of intuitive sense. 00:13:00.320 --> 00:13:04.280 As a result of that, and yeah, I forgot to mention one thing. 00:13:04.280 --> 00:13:07.540 So the duration is always negative, right? 00:13:07.540 --> 00:13:15.570 So we have a sign here, because if the bond price goes up, 00:13:15.570 --> 00:13:19.430 this means that the yields goes down, OK? 00:13:19.430 --> 00:13:24.610 And if the yield goes up, price goes down. 00:13:24.610 --> 00:13:26.450 And the explanation's very simple. 00:13:26.450 --> 00:13:29.120 So yield is kind of the same thing 00:13:29.120 --> 00:13:33.900 as interest rate on the market. 00:13:33.900 --> 00:13:37.270 So if those rates go up, this means 00:13:37.270 --> 00:13:40.540 that there will be more discounting in the future cash 00:13:40.540 --> 00:13:43.220 flows, they will be less valuable to me. 00:13:43.220 --> 00:13:47.770 So I'll be less willing to pay for those cash flows, OK? 00:13:47.770 --> 00:13:52.280 So it's kind of fundamental that relationship 00:13:52.280 --> 00:13:54.530 has a negative sign. 00:13:54.530 --> 00:13:59.790 So in case of a zero-coupon bond, 00:13:59.790 --> 00:14:02.900 we only have one cash flow in the future. 00:14:02.900 --> 00:14:07.720 So there is just one weight, and that weight 00:14:07.720 --> 00:14:10.220 is totally assigned to that last cash flow. 00:14:10.220 --> 00:14:18.780 So duration of zero-coupon bond equals to maturity. 00:14:18.780 --> 00:14:21.400 Duration of regular coupon bond depends, 00:14:21.400 --> 00:14:22.910 but it's always less than maturity, 00:14:22.910 --> 00:14:26.055 just because we'll have a weighted sum formula here. 00:14:32.090 --> 00:14:36.450 So essentially, that model for the bond duration 00:14:36.450 --> 00:14:44.370 kind of assumes that all rates-- so we have just one yield 00:14:44.370 --> 00:14:47.350 number for everything, so all rates go 00:14:47.350 --> 00:14:55.090 like in a parallel way, which was OK before the crisis, 00:14:55.090 --> 00:14:57.357 right, where kind of rates today are 00:14:57.357 --> 00:15:01.210 kind of similar to the rates expected in the future. 00:15:01.210 --> 00:15:02.600 But it's no longer the case. 00:15:02.600 --> 00:15:06.400 So the rates now, they're higher than like one year ago, 00:15:06.400 --> 00:15:09.300 but they're still much lower than expected in the future. 00:15:09.300 --> 00:15:12.720 So we expect that the rates will go very high. 00:15:12.720 --> 00:15:15.690 So the curve is very steep at the moment. 00:15:15.690 --> 00:15:21.254 So that model of just one number for everything 00:15:21.254 --> 00:15:22.170 might not be adequate. 00:15:22.170 --> 00:15:24.415 And we'll see how we can improve this situation. 00:15:29.250 --> 00:15:31.970 So it's worth mentioning the second derivative. 00:15:31.970 --> 00:15:34.230 We already spoke about the price, 00:15:34.230 --> 00:15:36.790 first derivative of the price with respect to yield, 00:15:36.790 --> 00:15:38.350 and a second. 00:15:38.350 --> 00:15:41.270 So for small changes in the yield, 00:15:41.270 --> 00:15:46.390 you can assume that it's linear, so it's 00:15:46.390 --> 00:15:48.120 OK to use just the first derivative. 00:15:48.120 --> 00:15:56.420 So second derivative will be necessary for larger movements 00:15:56.420 --> 00:15:59.270 of the market. 00:15:59.270 --> 00:16:07.557 Like as an example, if you're a trader, right, and the bond 00:16:07.557 --> 00:16:08.140 trades, right? 00:16:08.140 --> 00:16:10.390 So we call it a cash product. 00:16:10.390 --> 00:16:13.760 Means that you actually don't need any model to price it. 00:16:13.760 --> 00:16:17.060 You already have that price, OK? 00:16:17.060 --> 00:16:23.520 But if you try to explain like why you might have lost money 00:16:23.520 --> 00:16:26.470 today, right, and that always-- the trader 00:16:26.470 --> 00:16:29.910 always does that at the end of the day. 00:16:29.910 --> 00:16:32.090 And we always use first derivatives. 00:16:32.090 --> 00:16:36.380 And we try to explain it, but there is also unexplained, OK? 00:16:36.380 --> 00:16:39.870 And that unexplained can be quite high 00:16:39.870 --> 00:16:42.680 on this with large movements. 00:16:42.680 --> 00:16:45.720 So if you have like a term in your analytics 00:16:45.720 --> 00:16:48.010 for the bond convexity, that helps 00:16:48.010 --> 00:16:49.640 you to include the second derivative, 00:16:49.640 --> 00:16:55.045 and therefore make the second derivative smaller-- sorry-- 00:16:55.045 --> 00:16:56.610 the unexplained smaller. 00:17:02.170 --> 00:17:08.020 So let's now talk about interest rate swaps. 00:17:08.020 --> 00:17:12.410 So bond cash flow is basically a stream 00:17:12.410 --> 00:17:16.425 of fixed cash flow, which means that for certain dates 00:17:16.425 --> 00:17:18.300 it's just guaranteed that you will be getting 00:17:18.300 --> 00:17:22.800 $100 with certain periodicity. 00:17:22.800 --> 00:17:26.680 A swap means you exchange fixed payments 00:17:26.680 --> 00:17:28.430 with respect to some floating. 00:17:28.430 --> 00:17:33.060 And floating means that the amount of money that you'll 00:17:33.060 --> 00:17:36.870 be getting or paying, OK, receiving or paying, 00:17:36.870 --> 00:17:40.460 will depend on some market observable. 00:17:40.460 --> 00:17:43.920 So for interest rate swap, it will be typically-- and let's 00:17:43.920 --> 00:17:48.685 focus on the USD market, it will be a three month LIBOR rate. 00:17:48.685 --> 00:17:51.890 That rate is published daily, OK? 00:17:51.890 --> 00:17:56.300 And it's like if you need to go to the bank 00:17:56.300 --> 00:17:59.470 and get a three month CD with the money for the three months, 00:17:59.470 --> 00:18:01.130 that rate is already known. 00:18:01.130 --> 00:18:04.140 It's actually called LIBOR, because it's 00:18:04.140 --> 00:18:14.400 kind of between banks, and it's set at 11 AM London time. 00:18:14.400 --> 00:18:19.340 So as a result, we already know how 00:18:19.340 --> 00:18:23.800 to price cash flows in the future. 00:18:23.800 --> 00:18:32.460 So present value of the fixed stream of payments, as we know, 00:18:32.460 --> 00:18:34.480 will be like this. 00:18:34.480 --> 00:18:41.780 And there is a floating rate of cash flow as well. 00:18:41.780 --> 00:18:46.660 And the nice thing about the swap 00:18:46.660 --> 00:18:51.290 is that when you enter the swap, you don't pay any money, right? 00:18:51.290 --> 00:18:55.200 It's because you just kind of enter the agreement, 00:18:55.200 --> 00:18:59.710 rather than when you buy or sell a bond, 00:18:59.710 --> 00:19:02.010 there is some exchange of money. 00:19:02.010 --> 00:19:07.000 For swaps, swaps are designed the way such 00:19:07.000 --> 00:19:13.060 that-- so when you make this agreement, 00:19:13.060 --> 00:19:17.520 it's a certain moment of time, the fixed rate of the swap 00:19:17.520 --> 00:19:21.350 is picked in such a way that the present value 00:19:21.350 --> 00:19:28.260 of fixed minus the floating cash flows will be net to zero. 00:19:32.490 --> 00:19:36.730 So you can see, I mean, if we rewrite those equations, 00:19:36.730 --> 00:19:39.970 OK, we can see that the swap rate-- which 00:19:39.970 --> 00:19:43.890 is the most important quantity of the swap, 00:19:43.890 --> 00:19:46.080 and something that traders are basically 00:19:46.080 --> 00:19:47.350 are most concerned, right? 00:19:47.350 --> 00:19:50.830 So you first need to define what the swap is. 00:19:50.830 --> 00:19:54.325 And for USD, you are saying probably like 10 year swap, OK. 00:19:54.325 --> 00:19:55.200 And this is the rate. 00:19:55.200 --> 00:20:03.310 So the trader continuously kind of quotes bid and offer levels 00:20:03.310 --> 00:20:05.140 of the swap rate. 00:20:05.140 --> 00:20:07.930 So no one is talking about PVs and stuff like that. 00:20:07.930 --> 00:20:09.850 So it's always the swap rate. 00:20:09.850 --> 00:20:14.730 So that the swap rate is a weighted sum of forward rates. 00:20:14.730 --> 00:20:19.200 And it has a very nice intuitive explanation. 00:20:19.200 --> 00:20:27.970 So you have some stream of floating cash flows-- 00:20:27.970 --> 00:20:31.470 variable cash flows-- which at the moment, like, 00:20:31.470 --> 00:20:35.520 will probably be low now, but will be high in 10 years, 00:20:35.520 --> 00:20:38.940 will be much higher in 30 years. 00:20:38.940 --> 00:20:42.430 So the swap rate for this kind of environment 00:20:42.430 --> 00:20:44.320 will be kind of an average, right? 00:20:44.320 --> 00:20:49.815 And again, those weights depend on the discounting factors. 00:20:53.190 --> 00:20:59.190 So later, we will see that because we're 00:20:59.190 --> 00:21:02.925 talking about bond that having a fixed cash 00:21:02.925 --> 00:21:05.750 flows in the futures, and a swap that fixed in exchange 00:21:05.750 --> 00:21:13.300 of floating, swap can be hedged with bond, "hedged" meaning 00:21:13.300 --> 00:21:21.640 that-- you know what the term "hedged" means, no? 00:21:21.640 --> 00:21:25.990 Hedging means that if you have just, let's say, a swap, right? 00:21:25.990 --> 00:21:30.070 So if market changes, right, you can again lose money. 00:21:30.070 --> 00:21:38.250 So a typical task for the money-maker, 00:21:38.250 --> 00:21:42.790 trader, is to kind of offset that risk with something. 00:21:42.790 --> 00:21:46.430 Ideally, you sold one swap, you bought 00:21:46.430 --> 00:21:49.670 another swap the same way with a different rate. 00:21:49.670 --> 00:21:52.560 So you kind of locked in your profit. 00:21:52.560 --> 00:21:55.330 But you remain with a zero risk. 00:22:00.580 --> 00:22:02.960 So let's try to construct a yield curve. 00:22:02.960 --> 00:22:04.800 Why do we need the yield curve? 00:22:04.800 --> 00:22:11.360 So when we have, let's say, a series 00:22:11.360 --> 00:22:14.660 of swap with different maturities, right, 00:22:14.660 --> 00:22:18.450 all those swaps will start today, 00:22:18.450 --> 00:22:22.160 and usually, swap will have quarterly payment 00:22:22.160 --> 00:22:28.340 for the floating leg, and six month payments for the fixed 00:22:28.340 --> 00:22:31.160 leg, and you'll have different maturities. 00:22:31.160 --> 00:22:35.900 But if you try to kind of get discount factors 00:22:35.900 --> 00:22:38.530 from that information, you will see 00:22:38.530 --> 00:22:43.180 that you can get those discount factors only for certain dates, 00:22:43.180 --> 00:22:44.000 OK? 00:22:44.000 --> 00:22:48.300 But the typical situation is that given 00:22:48.300 --> 00:22:51.580 on some liquid market instruments, 00:22:51.580 --> 00:22:55.760 you want to price your entire portfolio, which 00:22:55.760 --> 00:22:57.890 has continuous spectrum of cash flows 00:22:57.890 --> 00:23:01.870 from now to 30 years, 40 years. 00:23:01.870 --> 00:23:07.860 And also, for typical swap portfolio 00:23:07.860 --> 00:23:12.360 that I personally deal with on a daily basis 00:23:12.360 --> 00:23:15.230 contains hundreds of thousands of swaps. 00:23:15.230 --> 00:23:17.630 Every swap has many cash flows. 00:23:17.630 --> 00:23:21.710 So you need something that can, based 00:23:21.710 --> 00:23:26.900 on discrete information of reliable liquid instruments 00:23:26.900 --> 00:23:30.290 on the market, draw the line. 00:23:30.290 --> 00:23:33.210 Can basically construct the curve. 00:23:33.210 --> 00:23:36.600 Which means that you can, so that you 00:23:36.600 --> 00:23:39.530 are able to get discount factors for any potential day 00:23:39.530 --> 00:23:41.990 in the future, or you can compute forward rate 00:23:41.990 --> 00:23:45.380 for every date in the future. 00:23:45.380 --> 00:23:48.570 So the first step to construct a yield curve 00:23:48.570 --> 00:23:53.590 is to select input instruments for calibration. 00:23:53.590 --> 00:23:55.050 So you have a set of instruments, 00:23:55.050 --> 00:23:56.780 and a new set of input quotes. 00:23:56.780 --> 00:23:59.840 Then you also need to kind of decide 00:23:59.840 --> 00:24:02.000 what kind of properties of that line will be. 00:24:02.000 --> 00:24:09.970 So you can say, OK, first of all, 00:24:09.970 --> 00:24:13.010 you need to decide what quantity will be interpolated. 00:24:13.010 --> 00:24:15.770 It could be daily discount factors, 00:24:15.770 --> 00:24:19.820 or daily forward rates, or maybe three-month forward rates. 00:24:19.820 --> 00:24:22.220 Then you select the spline type. 00:24:22.220 --> 00:24:24.480 So I'm not sure if you're familiar with the splines. 00:24:24.480 --> 00:24:29.860 Probably you heard about cubic spline, right? 00:24:29.860 --> 00:24:32.860 So there are different types of splines, and some of them 00:24:32.860 --> 00:24:37.190 are better and some of them are worse for different situation. 00:24:37.190 --> 00:24:38.940 And you also need to decide like what 00:24:38.940 --> 00:24:41.750 that will be the node points for the spline itself. 00:24:41.750 --> 00:24:42.920 OK. 00:24:42.920 --> 00:24:45.050 And then, as a final step, so you 00:24:45.050 --> 00:24:47.420 have some mathematical quantity, which 00:24:47.420 --> 00:24:50.310 is mathematical object where you know what the line is, 00:24:50.310 --> 00:24:51.980 and you have control points. 00:24:51.980 --> 00:24:54.100 And you need to adjust your control points such 00:24:54.100 --> 00:25:00.220 that when you reprice your instruments, 00:25:00.220 --> 00:25:02.800 those instruments are repriced exactly to the same quotes 00:25:02.800 --> 00:25:06.216 that you find on the market. 00:25:06.216 --> 00:25:07.445 You have a question. 00:25:07.445 --> 00:25:11.791 AUDIENCE: Is that spline, again, is it just like a-- 00:25:11.791 --> 00:25:12.832 IVAN MASYUKOV: All right. 00:25:12.832 --> 00:25:13.332 So let me. 00:25:22.890 --> 00:25:23.390 All right. 00:25:23.390 --> 00:25:26.645 So this is a picture of the cubic spline. 00:25:31.070 --> 00:25:41.940 So spline is a way to draw a smooth curve. 00:25:41.940 --> 00:25:44.050 This is an example of the cubic spline. 00:25:44.050 --> 00:25:48.080 So you start to define your node points. 00:25:48.080 --> 00:25:53.150 Your node points in this case are 1, 10, 20, 40, 00:25:53.150 --> 00:25:56.320 80, 160, and 240, right? 00:25:56.320 --> 00:26:00.000 And then for every one of those intervals, 00:26:00.000 --> 00:26:04.600 OK, the functional form of the shape of this curve 00:26:04.600 --> 00:26:09.840 is a cubic polynomial, OK? 00:26:09.840 --> 00:26:14.690 Well, if you just do cubic polynomial for every interval 00:26:14.690 --> 00:26:17.680 without kind of putting additional constraints, 00:26:17.680 --> 00:26:21.580 you can have all kinds of boundary effects, like jumps, 00:26:21.580 --> 00:26:27.480 kinks, and other things, because we want our cubic curve-- 00:26:27.480 --> 00:26:33.820 cubic spline-- to be meaningful, right? 00:26:33.820 --> 00:26:36.130 So we want to maintain, to preserve 00:26:36.130 --> 00:26:42.250 maximum number of derivatives for every node point. 00:26:42.250 --> 00:26:44.050 So we're not going to check. 00:26:44.050 --> 00:26:49.400 But believe me, this curve, it is 00:26:49.400 --> 00:26:53.900 a cubic polynomial for every one of those interval. 00:26:53.900 --> 00:26:58.960 And also, we have two continuous derivative at every node point, 00:26:58.960 --> 00:27:04.100 because for the n degree of the spline, 00:27:04.100 --> 00:27:05.720 you always have like n minus 1. 00:27:05.720 --> 00:27:08.860 You can have n minus 1. 00:27:08.860 --> 00:27:12.640 So the same thing, a spline can be represented 00:27:12.640 --> 00:27:15.980 in terms of B-splines. 00:27:15.980 --> 00:27:18.720 B-spline is a new type of spline. 00:27:18.720 --> 00:27:24.750 It's just as a representation which is more intuitive, 00:27:24.750 --> 00:27:27.020 I should say. 00:27:27.020 --> 00:27:32.520 So all universe of the curves with those node points, 00:27:32.520 --> 00:27:37.950 with two continuous derivatives, can 00:27:37.950 --> 00:27:40.910 be represented as a linear combination of those basis 00:27:40.910 --> 00:27:41.670 functions. 00:27:41.670 --> 00:27:43.660 So B-spline, I mean, if you're interested, 00:27:43.660 --> 00:27:48.650 you should probably, we're not going to discuss it in details, 00:27:48.650 --> 00:27:51.579 but it's nice separate kind of topic about how 00:27:51.579 --> 00:27:52.620 to build those B-splines. 00:27:55.670 --> 00:28:00.530 But essentially, what's nice about those B-splines-- 00:28:00.530 --> 00:28:04.790 and "B," as you probably already understood, B is basis, right? 00:28:04.790 --> 00:28:06.200 So you have like basis functions. 00:28:06.200 --> 00:28:09.610 So those functions look like bell shapes. 00:28:09.610 --> 00:28:13.600 They are non-zero on some sub-interval. 00:28:13.600 --> 00:28:19.310 On every interval, it will be a cubic polynomial. 00:28:19.310 --> 00:28:29.860 Everyone will have always two continuous derivatives. 00:28:29.860 --> 00:28:32.070 As a result, in any linear combination 00:28:32.070 --> 00:28:35.400 of those-- which the first curve is-- 00:28:35.400 --> 00:28:37.430 will also have that property. 00:28:50.530 --> 00:28:51.030 OK. 00:28:51.030 --> 00:28:54.900 So now we, yeah, so the calibrate 00:28:54.900 --> 00:28:58.550 means that we basically have some solver 00:28:58.550 --> 00:29:03.610 to make sure that our swaps with the rates for those maturities 00:29:03.610 --> 00:29:06.420 actually repriced at par. 00:29:06.420 --> 00:29:10.510 At par means that the PV is zero. 00:29:10.510 --> 00:29:15.120 This is a typical example of the yield curve instruments. 00:29:15.120 --> 00:29:21.320 And IRS stands for "interest rate swap," 00:29:21.320 --> 00:29:25.250 and we have maturities from one year to 30 years. 00:29:25.250 --> 00:29:36.540 And the quotes are of 0.33% up to 2.67%. 00:29:36.540 --> 00:29:40.210 So you can see that that's actually, 00:29:40.210 --> 00:29:44.190 that's from my one-year-old presentation. 00:29:44.190 --> 00:29:47.030 Rates are quite high these days. 00:29:47.030 --> 00:29:52.060 So this is an example of the yield curve graph. 00:29:52.060 --> 00:30:01.290 So again, so those are the rates from 0.3 to 3.5. 00:30:01.290 --> 00:30:06.120 And the shape of the curve is not flat at all, right? 00:30:06.120 --> 00:30:08.130 So it's actually pretty steep. 00:30:08.130 --> 00:30:13.000 So for the first five years, it's very steep. 00:30:13.000 --> 00:30:15.760 Then it reaches the plateau. 00:30:15.760 --> 00:30:20.720 And then there is some feature there, 00:30:20.720 --> 00:30:27.360 probably because of some behavior in the 20-year region. 00:30:27.360 --> 00:30:30.330 So three-month forward rate is the LIBOR rate. 00:30:30.330 --> 00:30:36.300 LIBOR is the rate for the three month. 00:30:36.300 --> 00:30:38.880 It's mostly kind of common. 00:30:38.880 --> 00:30:43.840 And the reason why is because the standard interest rate, 00:30:43.840 --> 00:30:48.100 USD swap has a three-month frequency 00:30:48.100 --> 00:30:51.530 for payment on the floating leg. 00:30:51.530 --> 00:30:54.600 So if you're talking about floating rates, 00:30:54.600 --> 00:30:55.860 is always three months. 00:30:55.860 --> 00:30:56.850 And it's always LIBOR. 00:31:00.080 --> 00:31:01.970 So because we've already built the curve, 00:31:01.970 --> 00:31:07.660 now let's see how we can improve the situation with a bond. 00:31:07.660 --> 00:31:11.430 So we have the curve, so we have the discount factors, right? 00:31:11.430 --> 00:31:15.620 And we see that those discount factors cannot be obtained 00:31:15.620 --> 00:31:18.620 on the assumption that you have just one parameter yield 00:31:18.620 --> 00:31:22.450 for everything, because the curve we know is not flat. 00:31:22.450 --> 00:31:31.450 So if we just try to price it using those discount factors, 00:31:31.450 --> 00:31:34.040 try to get a fair price, we probably 00:31:34.040 --> 00:31:35.930 won't match the market observables. 00:31:35.930 --> 00:31:38.140 So we need some extra term. 00:31:38.140 --> 00:31:41.700 And again, here we can use it in a similar form 00:31:41.700 --> 00:31:44.330 as we did it for the yield. 00:31:44.330 --> 00:31:48.910 But right now it's going to be a small correction to the yield 00:31:48.910 --> 00:31:54.140 curve, rather than kind of really rough assumption 00:31:54.140 --> 00:31:58.140 about that the curve is flat, OK? 00:31:58.140 --> 00:32:10.010 So typically, if the curve magnitude is, let's say, 3%. 00:32:10.010 --> 00:32:10.510 OK? 00:32:10.510 --> 00:32:14.490 So the spread is probably 100 times lower. 00:32:14.490 --> 00:32:20.245 So having a nice correction is always better, right? 00:32:23.850 --> 00:32:31.090 And another nice feature is that of this approach for the bond, 00:32:31.090 --> 00:32:34.350 like if we already build our yield curve model, 00:32:34.350 --> 00:32:37.935 and we know sensitivities of our portfolio 00:32:37.935 --> 00:32:41.290 to inputs of the curves, which then transition 00:32:41.290 --> 00:32:46.410 into like differences in discount factors, 00:32:46.410 --> 00:32:48.690 we can easily apply that to the bond. 00:32:48.690 --> 00:32:53.050 We can first find what this spread parameter 00:32:53.050 --> 00:32:59.380 is, to solve for s knowing P, which is very liquid market 00:32:59.380 --> 00:33:01.070 tradable. 00:33:01.070 --> 00:33:04.919 And then we can kind of use consistent model for the bonds 00:33:04.919 --> 00:33:06.210 and the swaps in our portfolio. 00:33:08.780 --> 00:33:10.637 Any questions? 00:33:10.637 --> 00:33:11.220 AUDIENCE: Yes. 00:33:11.220 --> 00:33:15.392 So what does the bond rate tell us about the bond? 00:33:19.740 --> 00:33:22.880 IVAN MASYUKOV: That's a very good question. 00:33:22.880 --> 00:33:31.770 So it might tell us something like bond liquidity, 00:33:31.770 --> 00:33:32.980 for example. 00:33:32.980 --> 00:33:34.575 Like if it's not liquid, or there 00:33:34.575 --> 00:33:42.280 is some-- so it may be related to the bond itself. 00:33:42.280 --> 00:33:45.590 And sometimes we kind of think that the bond 00:33:45.590 --> 00:33:48.720 is riskless, which means that-- especially if it's 00:33:48.720 --> 00:33:53.710 issued by US government, which if we can assume 00:33:53.710 --> 00:33:58.600 that those cash flows in the futures are guaranteed, right, 00:33:58.600 --> 00:34:01.961 then I basically will be willing to bring 00:34:01.961 --> 00:34:04.210 a certain amount of money and discount factors, right? 00:34:04.210 --> 00:34:07.610 But if you tell me that you will pay me that in the future, 00:34:07.610 --> 00:34:11.020 I won't be so certain, right? 00:34:11.020 --> 00:34:16.199 So I'll need to add some kind of credit spread 00:34:16.199 --> 00:34:18.980 to that-- we call it credit spreads-- as a result. 00:34:18.980 --> 00:34:23.590 It's the credit spread will kind of propagate to the spread 00:34:23.590 --> 00:34:24.469 number. 00:34:24.469 --> 00:34:28.090 On the other hand, if the bond is really US government-issued, 00:34:28.090 --> 00:34:32.909 and is considered to be guaranteed, 00:34:32.909 --> 00:34:36.100 then it may be a feature of the swap, OK? 00:34:36.100 --> 00:34:41.469 Where just because of some liquidity situations 00:34:41.469 --> 00:34:43.770 in swap market-- like all of a sudden, 00:34:43.770 --> 00:34:47.670 let's say, all option traders on the street 00:34:47.670 --> 00:34:49.909 needed this 10-year swap, OK? 00:34:49.909 --> 00:34:54.429 Because they kind of need to hedge certain very popular 00:34:54.429 --> 00:34:58.380 products-- volatility products-- they'll start to buy it, 00:34:58.380 --> 00:35:03.200 that spread will change. 00:35:03.200 --> 00:35:04.780 But what's even more interesting, 00:35:04.780 --> 00:35:08.000 that spread is tradable by itself, OK? 00:35:08.000 --> 00:35:12.910 So you can go to the market and you trade the spread. 00:35:12.910 --> 00:35:18.290 Moreover, let's look, like, 10-year situation. 00:35:18.290 --> 00:35:24.690 So you have 10-year bond on the market. 00:35:24.690 --> 00:35:29.340 You have tradable swap, and you have tradable spread. 00:35:29.340 --> 00:35:32.021 So the question is which one is the most liquid? 00:35:32.021 --> 00:35:32.770 What do you think? 00:35:38.300 --> 00:35:42.520 The most liquid is the bond, of course. 00:35:42.520 --> 00:35:45.890 It has much more liquidity. 00:35:45.890 --> 00:35:47.730 Surprisingly, the second one. 00:35:47.730 --> 00:35:52.430 It's the spread between the 10-year swap 00:35:52.430 --> 00:35:54.690 and the bond is traded in the market. 00:35:54.690 --> 00:35:58.470 So there's more transaction on the spread 00:35:58.470 --> 00:36:00.910 rather compared to the swap. 00:36:00.910 --> 00:36:03.970 As a result, when we built our curves, 00:36:03.970 --> 00:36:08.540 we're not taking like 10-year swap from the market. 00:36:08.540 --> 00:36:09.040 OK. 00:36:09.040 --> 00:36:13.360 We actually take the yield and the spread. 00:36:13.360 --> 00:36:16.696 And that's how we define the most kind 00:36:16.696 --> 00:36:18.380 of reliable level of the swap. 00:36:18.380 --> 00:36:20.450 Of course, we could have just take 00:36:20.450 --> 00:36:23.340 whatever we observe for the 10-year swap, 00:36:23.340 --> 00:36:25.790 but it could be off. 00:36:25.790 --> 00:36:29.160 And also, if you observe, there will be more bid-offer spread 00:36:29.160 --> 00:36:29.660 as well. 00:36:35.650 --> 00:36:40.700 So as an example, let's try to shift 00:36:40.700 --> 00:36:45.780 one of the inputs of the curve by one basis point. 00:36:45.780 --> 00:36:50.260 And that will result in this kind of deviation 00:36:50.260 --> 00:36:57.640 of forward trades, which will be combination of basis splines. 00:36:57.640 --> 00:36:59.680 But what's interesting first of all, 00:36:59.680 --> 00:37:01.925 it's kind of complicated [INAUDIBLE] behavior. 00:37:01.925 --> 00:37:07.120 The reason why is because you are saying that nothing changed 00:37:07.120 --> 00:37:09.330 before the nine year, like nothing 00:37:09.330 --> 00:37:13.640 changed after ninth year, but just point in between. 00:37:13.640 --> 00:37:17.030 So in order to calibrate to that kind of weird condition, 00:37:17.030 --> 00:37:22.310 right, you need to have a ripple here. 00:37:22.310 --> 00:37:27.590 But what also is more important is that by shifting one year 00:37:27.590 --> 00:37:32.460 basis point by one basis point, that the amplitude 00:37:32.460 --> 00:37:37.370 of shifts in the curve reaches 14 basis points. 00:37:37.370 --> 00:37:41.550 So not sure if you're familiar, but it's an ill-posed problem 00:37:41.550 --> 00:37:42.050 right? 00:37:42.050 --> 00:37:45.800 So small changes in your inputs can cause large variations 00:37:45.800 --> 00:37:46.590 in your outputs. 00:37:49.450 --> 00:37:51.230 This is a very important slide. 00:37:51.230 --> 00:37:56.100 So the first column is, again, we 00:37:56.100 --> 00:37:59.230 saw those are our instruments, quotes, 00:37:59.230 --> 00:38:01.420 and this is the risk of the portfolio. 00:38:01.420 --> 00:38:05.100 That's something that a trader needs no matter what. 00:38:05.100 --> 00:38:07.930 It basically shows you what happens 00:38:07.930 --> 00:38:11.570 on the market if different-- what 00:38:11.570 --> 00:38:13.250 will be the change in your portfolio 00:38:13.250 --> 00:38:14.740 if the market changes. 00:38:14.740 --> 00:38:18.220 So the meaning of the number-- for example, for the five 00:38:18.220 --> 00:38:25.700 year-- is that if five year rate moves up by one basis point, 00:38:25.700 --> 00:38:28.845 we'll lose minus 700K. 00:38:36.390 --> 00:38:41.250 We also marked here yellow points 00:38:41.250 --> 00:38:44.530 that are more liquid than the others. 00:38:44.530 --> 00:38:47.940 So now a typical situation is that you need 00:38:47.940 --> 00:38:49.880 to hedge your portfolio, right? 00:38:49.880 --> 00:38:57.800 So you need to liquidate your risk, I'm basically saying, 00:38:57.800 --> 00:39:00.720 OK, given the model that we have, 00:39:00.720 --> 00:39:07.030 I want its value to be insensitive to any movements 00:39:07.030 --> 00:39:08.600 on the market. 00:39:08.600 --> 00:39:12.170 So for that purpose, what you can do, you can go 00:39:12.170 --> 00:39:20.290 and you can buy as many one-year swaps as plus 200, 00:39:20.290 --> 00:39:27.490 as many two-year swaps which would be the risk of minus 1.3, 00:39:27.490 --> 00:39:30.640 and so forth, right? 00:39:30.640 --> 00:39:35.550 Then that always cost you money right? 00:39:35.550 --> 00:39:41.180 And that money is kind of proportional to bid-offer 00:39:41.180 --> 00:39:42.770 of the particular instruments. 00:39:42.770 --> 00:39:45.880 And that bid-offer is smaller for liquid instruments 00:39:45.880 --> 00:39:51.450 and larger for less liquid instruments. 00:39:51.450 --> 00:39:53.520 So if you multiply by the diff-- we 00:39:53.520 --> 00:39:55.270 can see that if we want to hedge our risk, 00:39:55.270 --> 00:39:58.300 it's going to be quite expensive. 00:39:58.300 --> 00:40:00.610 It will cost us 3.6 million dollars. 00:40:04.037 --> 00:40:05.480 Any questions so far? 00:40:11.820 --> 00:40:23.440 So traders never hedge every bucket in the risk. 00:40:23.440 --> 00:40:25.890 Bucket means every line here. 00:40:25.890 --> 00:40:30.480 So you always see some numbers, but if you 00:40:30.480 --> 00:40:33.210 try to make every number here zero, 00:40:33.210 --> 00:40:36.530 which means that if you trade seven here, 00:40:36.530 --> 00:40:38.460 you also could try to go to the market, 00:40:38.460 --> 00:40:43.890 find the offsetting seven here, you'll have to pay too much 00:40:43.890 --> 00:40:50.030 and you won't be profitable. 00:40:50.030 --> 00:40:54.210 So what the traders do if someone ask for the seven year, 00:40:54.210 --> 00:40:57.070 they make this transaction, but they go, then 00:40:57.070 --> 00:40:59.680 hedge it from the more liquid points, which 00:40:59.680 --> 00:41:01.520 is less expensive to buy. 00:41:05.030 --> 00:41:07.740 So we need a better model for hedging. 00:41:07.740 --> 00:41:12.820 And a general formulation of the model is presented here. 00:41:12.820 --> 00:41:22.380 So we have portfolio risk which is a just the vector here, 00:41:22.380 --> 00:41:24.030 right? 00:41:24.030 --> 00:41:29.950 And we have hedging. 00:41:29.950 --> 00:41:33.610 Portfolio risk is basically if you have candidates 00:41:33.610 --> 00:41:37.320 of instruments that you can use for portfolio hedging, again, 00:41:37.320 --> 00:41:40.820 the risk will be represented in this format in terms 00:41:40.820 --> 00:41:44.505 of sensitivities to swap rates. 00:41:47.050 --> 00:41:54.160 And we have weights of that hedging portfolio 00:41:54.160 --> 00:41:55.970 that we need to find, obviously. 00:41:55.970 --> 00:41:57.440 So you have this hedging portfolio. 00:41:57.440 --> 00:42:02.190 You multiply H by x, you get risk of this hedging portfolio. 00:42:02.190 --> 00:42:05.050 You add it to the risk of your portfolio. 00:42:05.050 --> 00:42:07.177 And then, what we need to minimize, 00:42:07.177 --> 00:42:08.760 you don't need to minimize everything. 00:42:08.760 --> 00:42:10.301 But you need to if they give you, OK? 00:42:10.301 --> 00:42:11.630 What can happen on the market? 00:42:11.630 --> 00:42:14.300 What are the typical modes of the market? 00:42:14.300 --> 00:42:19.530 And so essentially you kind of define your market 00:42:19.530 --> 00:42:24.810 scenarios, which can be found in a different way. 00:42:24.810 --> 00:42:27.520 So one of the ways to approach that problem 00:42:27.520 --> 00:42:33.260 is to use principal component analysis. 00:42:33.260 --> 00:42:37.670 I know you already are familiar with SVD. 00:42:37.670 --> 00:42:50.370 So if D is data of market movements in matrix-- 00:42:50.370 --> 00:42:57.240 then any matrix can be decomposed using SVD. 00:42:57.240 --> 00:43:04.350 And we can then look at this spectrum of this decomposition, 00:43:04.350 --> 00:43:09.940 looking at those eigenvalues, and just 00:43:09.940 --> 00:43:13.160 pick the ones that look high enough for us, 00:43:13.160 --> 00:43:14.760 and just keep that number. 00:43:14.760 --> 00:43:20.200 And let's, for example, we find that we really investigated 00:43:20.200 --> 00:43:29.180 this market, and we found that there are just 00:43:29.180 --> 00:43:32.130 five components that drive the market, 00:43:32.130 --> 00:43:36.130 and the rest is just so little that it's meaningless, right? 00:43:36.130 --> 00:43:38.200 On every day, and we are certain that it's just 00:43:38.200 --> 00:43:42.820 five components, five modes of market moments. 00:43:42.820 --> 00:43:48.060 Then, if we have a curve that consists of 20 points, 00:43:48.060 --> 00:43:50.600 we don't need to hedge every swap 00:43:50.600 --> 00:43:52.550 with its corresponding maturity. 00:43:52.550 --> 00:43:57.100 We can just pick five swaps that are 00:43:57.100 --> 00:43:59.030 liquid enough and cheap enough for us 00:43:59.030 --> 00:44:00.720 to hedge, and just use them. 00:44:08.990 --> 00:44:11.720 So let's look now at those typical graphs 00:44:11.720 --> 00:44:16.190 of those principal components. 00:44:16.190 --> 00:44:19.210 X-axis is the swap maturity in years, 00:44:19.210 --> 00:44:22.380 and y is some kind of relative, let's think 00:44:22.380 --> 00:44:23.920 of that as basis points. 00:44:23.920 --> 00:44:26.760 So blue line is the first component 00:44:26.760 --> 00:44:30.610 which is the prevalent. 00:44:30.610 --> 00:44:34.600 And it kind of, you can see that swap rates, 00:44:34.600 --> 00:44:37.090 they're basically flattish after 10 years, 00:44:37.090 --> 00:44:40.500 but the first component is pretty steep. 00:44:40.500 --> 00:44:47.760 And what it says as well is that the main behavior of the market 00:44:47.760 --> 00:44:50.010 is that rates now do not move, but they 00:44:50.010 --> 00:44:51.340 will move in the future. 00:44:51.340 --> 00:44:53.750 And that's basically because Fed is in a hold, right? 00:44:53.750 --> 00:44:57.030 So they kind of stimulate the market in a way such 00:44:57.030 --> 00:45:02.030 that the rate remains the same until sometime in the future. 00:45:02.030 --> 00:45:10.440 Mode number two is a kind of like tilting situation. 00:45:10.440 --> 00:45:13.050 Mode number three is more complex. 00:45:13.050 --> 00:45:15.770 And we'll have several other modes here as well. 00:45:19.910 --> 00:45:25.950 So now, following our previous general approach 00:45:25.950 --> 00:45:27.920 to the problem, we formulated here 00:45:27.920 --> 00:45:35.030 as-- so we have PCA factors here in P. 00:45:35.030 --> 00:45:38.390 And now, because the number of factors that we selected 00:45:38.390 --> 00:45:40.140 is the number of hedging instruments-- 00:45:40.140 --> 00:45:42.200 we no longer need to minimize. 00:45:42.200 --> 00:45:44.890 We can always feed, which you can always achieve 00:45:44.890 --> 00:45:46.527 like perfect minimization. 00:45:46.527 --> 00:45:47.860 We can always achieve that zero. 00:45:47.860 --> 00:45:52.610 So that's why we formulate it as zero. 00:45:52.610 --> 00:45:54.765 So solving that problem here. 00:45:58.230 --> 00:45:58.730 Yeah. 00:45:58.730 --> 00:46:01.930 And the hedging matrix, this is an example of hedging matrix. 00:46:01.930 --> 00:46:09.260 So what that matrix says is that if I take one year swap 00:46:09.260 --> 00:46:11.450 and put in my portfolio, empty portfolio, 00:46:11.450 --> 00:46:17.500 and then they apply my model, I'll have just sensitivity 00:46:17.500 --> 00:46:19.310 to that particular swap. 00:46:19.310 --> 00:46:24.580 Which kind of makes sense, because since you 00:46:24.580 --> 00:46:29.020 use the same instruments to calibrate your yield curve, 00:46:29.020 --> 00:46:33.770 right, then it should be sensitive to itself only. 00:46:33.770 --> 00:46:37.690 That's why that matrix is just ones for itself, 00:46:37.690 --> 00:46:40.880 and zeroes otherwise. 00:46:40.880 --> 00:46:47.860 So then, as a result, we get this matrix. 00:46:47.860 --> 00:46:50.690 So same portfolio that we had before. 00:46:50.690 --> 00:46:58.250 This is our PCA matrix that translates our risk 00:46:58.250 --> 00:47:07.286 into those few numbers, right? 00:47:07.286 --> 00:47:12.730 And because we know it translates our risk-- low risk, 00:47:12.730 --> 00:47:17.350 in terms of many curve inputs-- into just five most 00:47:17.350 --> 00:47:21.230 liquid ones, which is 1, 2, 5, 10, and 30. 00:47:21.230 --> 00:47:24.300 As a result, our translated risk, 00:47:24.300 --> 00:47:29.770 which tells us what we need to do to hedge our portfolio, 00:47:29.770 --> 00:47:31.095 is just those numbers. 00:47:31.095 --> 00:47:35.990 And now, if we take a bid-offer charge, 0.1 basis 00:47:35.990 --> 00:47:38.420 points for those, and multiply, we 00:47:38.420 --> 00:47:40.160 get numbers which are orders of magnitude 00:47:40.160 --> 00:47:42.060 smaller than we got before, right? 00:47:42.060 --> 00:47:46.760 So we probably get something like there were 400. 00:47:46.760 --> 00:47:51.220 It's not 4 million-- 3.6 million anymore. 00:47:51.220 --> 00:47:54.650 That's exactly what traders do. 00:47:54.650 --> 00:47:58.190 And different traders have different opinions 00:47:58.190 --> 00:47:59.745 of what dynamics of the market is. 00:48:04.770 --> 00:48:06.390 But they always have some model. 00:48:14.060 --> 00:48:15.360 So disadvantages. 00:48:15.360 --> 00:48:20.120 So PCA model is something that just formally attuned 00:48:20.120 --> 00:48:22.690 to historical data. 00:48:22.690 --> 00:48:30.960 I always say that if you take kind of scramble your swap 00:48:30.960 --> 00:48:34.140 maturities in your model, and you do your computations, 00:48:34.140 --> 00:48:36.450 and you kind of unscramble them, you 00:48:36.450 --> 00:48:42.820 get exactly the same result. Which means that in PCA model, 00:48:42.820 --> 00:48:44.940 you don't put any constraints on that. 00:48:47.510 --> 00:48:49.400 Two year is very close to one year, 00:48:49.400 --> 00:48:51.560 and two year is between one year and five year. 00:48:55.250 --> 00:48:58.680 So PCA model or hedging coefficients of that matrix 00:48:58.680 --> 00:49:06.730 is not very stable-- especially for recent modes in the market. 00:49:06.730 --> 00:49:11.780 Also, because SVDs kind of is the least squares 00:49:11.780 --> 00:49:14.530 approximation, it's very sensitive to outliers. 00:49:14.530 --> 00:49:19.150 So there is just one event on the market that kind of one day 00:49:19.150 --> 00:49:21.510 happens, something like rates went up, 00:49:21.510 --> 00:49:24.160 and then it went down significantly, 00:49:24.160 --> 00:49:30.260 it may have unnecessarily high influence on the outputs. 00:49:30.260 --> 00:49:32.720 And if those coefficients change daily, 00:49:32.720 --> 00:49:36.410 right, then again, it may be too costly. 00:49:36.410 --> 00:49:39.810 And quite often we just overfitting to historical data. 00:49:39.810 --> 00:49:42.700 So we're saying, OK, what can I do. 00:49:42.700 --> 00:49:44.600 I just take historical data, and I 00:49:44.600 --> 00:49:50.390 prove that my model works, would have worked for the last three 00:49:50.390 --> 00:49:54.180 years, or the last three months, but that 00:49:54.180 --> 00:49:57.298 doesn't mean that it will work for the next three months. 00:50:04.830 --> 00:50:09.290 If we kind of try to put some additional constraints 00:50:09.290 --> 00:50:12.456 or additional thoughts about what this behavior should be, 00:50:12.456 --> 00:50:13.580 this may improve situation. 00:50:16.290 --> 00:50:21.420 So PCA interpretation is that risk matrix 00:50:21.420 --> 00:50:28.050 is a linear combination of principal components producing 00:50:28.050 --> 00:50:30.520 a shift on one hedging instrument at a time. 00:50:35.780 --> 00:50:41.640 Now the question is, let's forget about historical, OK? 00:50:41.640 --> 00:50:43.300 Is there any other approach? 00:50:43.300 --> 00:50:46.619 We know historical is noisy, and it's kind of first step 00:50:46.619 --> 00:50:47.910 if you want to build the model. 00:50:47.910 --> 00:50:51.250 But can we do something better? 00:50:51.250 --> 00:50:54.422 And the answer is yes. 00:50:54.422 --> 00:51:01.570 So we can say that we have our yield curve 00:51:01.570 --> 00:51:04.880 in terms of forward rates. 00:51:04.880 --> 00:51:07.487 And typically, when we build this curve, 00:51:07.487 --> 00:51:08.695 we observe that it is smooth. 00:51:11.650 --> 00:51:15.130 It's smooth not only because we use smooth splines, 00:51:15.130 --> 00:51:25.520 but also because if there is no certainty about some event 00:51:25.520 --> 00:51:32.420 in 10 years from now, there is no reason to kind of expect 00:51:32.420 --> 00:51:36.280 there will be spike or some non-smooth feature 00:51:36.280 --> 00:51:39.940 in the forward rate space. 00:51:39.940 --> 00:51:44.880 So what we can do is that we can try 00:51:44.880 --> 00:51:50.010 to minimize those equations where Jacobian 00:51:50.010 --> 00:51:56.010 is a matrix translating shifts of yield curve inputs 00:51:56.010 --> 00:51:58.580 into movements of forward trades. 00:51:58.580 --> 00:52:04.000 So essentially, we will try to penalize non-smoothness. 00:52:09.160 --> 00:52:11.760 And the solution will be like this, 00:52:11.760 --> 00:52:17.210 with some kind of-- so we'll be adding a penalty, OK. 00:52:17.210 --> 00:52:20.470 And penalty will be a small regularization parameter. 00:52:25.930 --> 00:52:31.070 So this is, as an example, that's what we get. 00:52:35.500 --> 00:52:37.530 Again, here in that model. 00:52:40.640 --> 00:52:47.200 You can view this matrix as if one year rate moves, 00:52:47.200 --> 00:52:51.360 what it basically-- so your drivers are 1, 2, 5, 10s, 00:52:51.360 --> 00:52:51.860 and 30s. 00:52:51.860 --> 00:52:56.100 So that's your drivers. 00:52:56.100 --> 00:52:59.230 Knowing the movements of your drivers, 00:52:59.230 --> 00:53:02.090 what would be the response to your swap rates? 00:53:02.090 --> 00:53:06.150 And you know that it will always be one to itself, 00:53:06.150 --> 00:53:08.490 right, as you see here. 00:53:08.490 --> 00:53:11.390 And in between, it will be kind of a smooth functions. 00:53:21.900 --> 00:53:26.960 So let's take, this moment, a broader view 00:53:26.960 --> 00:53:31.290 at what the pricing model does. 00:53:31.290 --> 00:53:36.180 And we have a pricing engine, essentially. 00:53:36.180 --> 00:53:38.860 It's a way, if you have all model parameters-- including 00:53:38.860 --> 00:53:42.490 curves, volatility, the surface, everything, right? 00:53:42.490 --> 00:53:50.050 And in order for those parameters 00:53:50.050 --> 00:53:52.950 to be consistent with the benchmark prices, 00:53:52.950 --> 00:53:58.310 you need some calibration engine which 00:53:58.310 --> 00:54:02.190 matches market observables to the ones that's 00:54:02.190 --> 00:54:08.970 been repriced by the model output of the pricing engine. 00:54:08.970 --> 00:54:19.300 And once you make sure that benchmark prices of your model 00:54:19.300 --> 00:54:25.520 equal or are kind of close enough to benchmark prices 00:54:25.520 --> 00:54:28.360 observed in the market, you calibrated the model, 00:54:28.360 --> 00:54:35.160 then you can essentially price your portfolio, 00:54:35.160 --> 00:54:36.610 and get values and risk. 00:54:43.780 --> 00:54:50.610 So let's look at one of the nice examples of how 00:54:50.610 --> 00:54:57.060 that pricing engine and pricing and calibration process works. 00:54:57.060 --> 00:55:07.070 We'll look at HJM model, which is used to price volatility 00:55:07.070 --> 00:55:08.820 products. 00:55:08.820 --> 00:55:13.470 So we're not going to go into too many details about this, 00:55:13.470 --> 00:55:19.920 but this is equations of evolution of forward rates 00:55:19.920 --> 00:55:24.510 that we need for simulation, for Monte Carlo simulation. 00:55:24.510 --> 00:55:27.851 What we're saying here is this change of the forward rates-- 00:55:27.851 --> 00:55:29.600 because forward rates is the quantity that 00:55:29.600 --> 00:55:33.420 is being assimilated-- has some drift, OK? 00:55:33.420 --> 00:55:35.460 Because dt is time. 00:55:35.460 --> 00:55:41.720 And also, it has some dependence on the forward rates 00:55:41.720 --> 00:55:43.560 to the power of beta, right? 00:55:43.560 --> 00:55:50.860 So if it's log-normal model, beta will be one. 00:55:50.860 --> 00:55:54.270 If it's normal model, beta will be zero. 00:55:54.270 --> 00:55:57.080 But in general, it's different. 00:55:57.080 --> 00:55:59.580 Then we have volatility surface, right, 00:55:59.580 --> 00:56:04.500 which kind of gives you what the number of volatility 00:56:04.500 --> 00:56:09.880 to use for this calendar and forward time. 00:56:09.880 --> 00:56:14.812 And we have correlation and factor structure which we're 00:56:14.812 --> 00:56:16.020 not going to talk about here. 00:56:16.020 --> 00:56:19.990 And this is Brownian motions. 00:56:19.990 --> 00:56:24.820 So we're not going to go any more complex like this. 00:56:24.820 --> 00:56:28.580 We'll just start looking at nice, two-dimensional surfaces 00:56:28.580 --> 00:56:35.570 here, and see what are the problems of calibrating 00:56:35.570 --> 00:56:36.770 the volatility surface. 00:56:40.800 --> 00:56:45.062 Just to give you a diagram of when 00:56:45.062 --> 00:56:47.020 we look at the surface, what different elements 00:56:47.020 --> 00:56:48.670 of that surface mean. 00:56:54.060 --> 00:56:55.650 It's a triangular surface. 00:56:55.650 --> 00:57:00.250 You have a calendar time, right, and you have a forward time. 00:57:00.250 --> 00:57:05.660 So your simulation starts at the first vertical line. 00:57:05.660 --> 00:57:10.500 So you have forward rates here as calibrated 00:57:10.500 --> 00:57:12.360 from the curve as of today. 00:57:12.360 --> 00:57:17.740 So those are square boxes here, square elements. 00:57:17.740 --> 00:57:21.240 So you need to kind of transition 00:57:21.240 --> 00:57:23.940 from the first line to the second step 00:57:23.940 --> 00:57:27.370 using Monte Carlo simulation. 00:57:27.370 --> 00:57:33.750 And that's when, for every arrow here, 00:57:33.750 --> 00:57:36.890 you need the volatility number. 00:57:36.890 --> 00:57:40.994 Then, once you did your Monte Carlo 00:57:40.994 --> 00:57:42.410 simulation for the second one, you 00:57:42.410 --> 00:57:44.230 need ones for the third one. 00:57:44.230 --> 00:57:48.700 And again you need data-- which volatility to use, OK? 00:57:48.700 --> 00:57:51.280 So essentially, the surface that we'll 00:57:51.280 --> 00:57:54.160 be looking at on next slides is essentially 00:57:54.160 --> 00:57:56.840 representing the numbers necessary for this transition's 00:57:56.840 --> 00:58:00.250 volatility for every arrow. 00:58:00.250 --> 00:58:08.580 So to explain, there are different areas here. 00:58:08.580 --> 00:58:18.170 Like, for example, those will be-- if one step is one here, 00:58:18.170 --> 00:58:25.760 OK, we're talking about this, it will be the forward rates that 00:58:25.760 --> 00:58:28.800 will be observed in two years. 00:58:28.800 --> 00:58:34.045 That one will be observed as of now for like one year from now. 00:58:34.045 --> 00:58:36.630 But again, in two years. 00:58:36.630 --> 00:58:44.240 And so those rates are essential to compute the swap, 00:58:44.240 --> 00:58:45.930 forward swap rate. 00:58:45.930 --> 00:58:50.730 And if we do our Monte Carlo simulation, 00:58:50.730 --> 00:58:53.370 that's essential information that we 00:58:53.370 --> 00:58:57.130 need to compute the price for the option 00:58:57.130 --> 00:59:01.180 on the swap, which we're not going to discuss here. 00:59:01.180 --> 00:59:05.320 But just an example, it shows that for different instruments 00:59:05.320 --> 00:59:09.885 observed in the market you have quite overlapping 00:59:09.885 --> 00:59:10.760 areas of sensitivity. 00:59:16.940 --> 00:59:23.440 So this is a typical example of the volatility surface 00:59:23.440 --> 00:59:26.860 where this is calendar time, this is forward time. 00:59:26.860 --> 00:59:33.100 And it has spikes for certain regions. 00:59:33.100 --> 00:59:34.460 But in general, it's smooth. 00:59:38.880 --> 00:59:42.090 So why this problem is challenging. 00:59:45.900 --> 00:59:48.740 If we try to compute the triangle matrix, which 00:59:48.740 --> 00:59:53.190 has dimension of 240 by 240-- the reason why is 240 00:59:53.190 --> 00:59:58.510 is because every element is for three months, OK? 00:59:58.510 --> 01:00:06.080 But we need up to 60 years of data, 01:00:06.080 --> 01:00:12.110 which means that it's 60 by 4, which 01:00:12.110 --> 01:00:14.630 is the number of quarters, is 240 by 240. 01:00:14.630 --> 01:00:18.310 If you just need triangle elements, it's 200K elements. 01:00:18.310 --> 01:00:23.360 So if you try to calibrate everything at the same time, 01:00:23.360 --> 01:00:26.710 and you formally try to solve your problem, 01:00:26.710 --> 01:00:33.130 you kind of needed to store, at least to build a matrix of 28K 01:00:33.130 --> 01:00:35.230 by 28K. 01:00:35.230 --> 01:00:40.060 And we just don't have memory for this. 01:00:40.060 --> 01:00:41.840 And we also have very small number 01:00:41.840 --> 01:00:47.660 of calibration instruments only in terms of swaptions or caps, 01:00:47.660 --> 01:00:49.360 which are typical volatility products. 01:00:49.360 --> 01:00:51.890 We just have a relatively small number. 01:00:51.890 --> 01:00:55.330 So it's an underdetermined problem. 01:00:55.330 --> 01:00:58.840 We also, as we saw on the previous example, 01:00:58.840 --> 01:01:05.030 areas of sensitivity of different instruments overlap. 01:01:05.030 --> 01:01:08.320 And it's an ill-posed inverse problems which 01:01:08.320 --> 01:01:13.750 produces unstable solutions. 01:01:13.750 --> 01:01:18.760 And no matter what we do, right, the resulting surface 01:01:18.760 --> 01:01:19.940 should be nice, right? 01:01:19.940 --> 01:01:20.940 Should look nice, right? 01:01:20.940 --> 01:01:27.320 Because if it has spikes in some points in the future, then 01:01:27.320 --> 01:01:29.690 we either have an economic reason for this, 01:01:29.690 --> 01:01:32.640 or we claim that this is something that's not realistic. 01:01:37.640 --> 01:01:41.650 So this is how we approach the problem. 01:01:41.650 --> 01:01:44.540 So the first step, we represent our volatility surface. 01:01:44.540 --> 01:01:49.520 And here, even though volatility surface is two-dimensional 01:01:49.520 --> 01:01:56.710 we just kind of assign a number for every of those elements, 01:01:56.710 --> 01:02:00.750 OK, and then represent the surface as a vector, OK? 01:02:00.750 --> 01:02:03.160 Whereas saying that the new surface v 01:02:03.160 --> 01:02:10.255 will be some initial state plus a linear combination 01:02:10.255 --> 01:02:12.230 of basis functions. 01:02:12.230 --> 01:02:15.730 And basis functions should correspond 01:02:15.730 --> 01:02:22.240 to some reasonable functions, OK? 01:02:22.240 --> 01:02:25.180 But the nice feature is that number of basis functions 01:02:25.180 --> 01:02:30.467 will be much smaller than the number of elements 01:02:30.467 --> 01:02:32.380 that we need to calibrate. 01:02:32.380 --> 01:02:35.970 But we will be very formal here. 01:02:35.970 --> 01:02:38.850 And we'll try to use same number of basis functions 01:02:38.850 --> 01:02:41.930 as we have our input instruments. 01:02:41.930 --> 01:02:45.540 So in case we had 50 input instruments, 01:02:45.540 --> 01:02:50.210 we select basis functions also the number 50. 01:02:50.210 --> 01:02:59.390 So we will use typical Newton-Raphson approach here. 01:02:59.390 --> 01:03:09.160 We will compute sensitivities of input of all instruments 01:03:09.160 --> 01:03:14.350 to perturbations of a volatility surface, OK? 01:03:14.350 --> 01:03:16.310 We'll build this Jacobian matrix. 01:03:16.310 --> 01:03:28.780 And then, if we made the reasonable assumptions 01:03:28.780 --> 01:03:35.730 about what those basis functions are, 01:03:35.730 --> 01:03:39.790 then we can invert our square Jacobian. 01:03:39.790 --> 01:03:41.680 And again, the reason why it's square 01:03:41.680 --> 01:03:46.010 is because we selected same number of basis functions 01:03:46.010 --> 01:03:48.420 as the number of input instruments. 01:03:48.420 --> 01:03:50.830 It's actually quite common approach, 01:03:50.830 --> 01:03:55.450 but it's very often is wrong approach. 01:03:55.450 --> 01:03:56.980 It produces unstable results. 01:03:56.980 --> 01:04:01.110 And we will see why. 01:04:01.110 --> 01:04:03.510 So we converge to exact solution, 01:04:03.510 --> 01:04:06.850 but now the volatility surface looks like this. 01:04:06.850 --> 01:04:09.870 It looks less like a volatility surface, 01:04:09.870 --> 01:04:14.820 but more like Manhattan skyline. 01:04:14.820 --> 01:04:16.530 So you have a Hudson River here, and you 01:04:16.530 --> 01:04:20.550 have some buildings right? 01:04:20.550 --> 01:04:25.910 Obviously, even though it calibrates exactly, right? 01:04:25.910 --> 01:04:27.660 And you could go and price your portfolio, 01:04:27.660 --> 01:04:32.090 but probably prices for instruments 01:04:32.090 --> 01:04:34.780 in the portfolio that are not input 01:04:34.780 --> 01:04:38.380 instruments for calibration would be meaningless. 01:04:38.380 --> 01:04:44.280 Because the reason why we need this surface to be smooth 01:04:44.280 --> 01:04:47.560 is because for similar instruments 01:04:47.560 --> 01:04:49.770 for similar products in your portfolio 01:04:49.770 --> 01:04:51.830 you kind of expect similar prices, right? 01:04:51.830 --> 01:04:53.770 So if your volatility jumps, that's 01:04:53.770 --> 01:04:56.490 something that just contradicts with this assumption. 01:05:00.570 --> 01:05:04.480 So now how can we improve the situation? 01:05:04.480 --> 01:05:09.420 So we can try to use our basis functions which 01:05:09.420 --> 01:05:11.590 were selected in terms of piecewise constant shift 01:05:11.590 --> 01:05:13.130 of different areas. 01:05:13.130 --> 01:05:18.180 We can use a smooth version of those plans. 01:05:18.180 --> 01:05:22.930 But again, the result looks better, but still 01:05:22.930 --> 01:05:23.810 is not good enough. 01:05:26.720 --> 01:05:31.380 And just to demonstrate that this is an ill-posed problem-- 01:05:31.380 --> 01:05:33.390 an ill-posed problem is something 01:05:33.390 --> 01:05:37.370 that small changes of your inputs 01:05:37.370 --> 01:05:41.000 results in insane changes in your output. 01:05:41.000 --> 01:05:43.640 And this is a typical example. 01:05:43.640 --> 01:05:46.860 So keep all the instruments the same, OK? 01:05:46.860 --> 01:05:50.350 We just change by 1%-- which is not a big number-- 01:05:50.350 --> 01:05:57.850 of the five year by 10 year swaption, 01:05:57.850 --> 01:06:01.360 results in a quite large change of the volatility surface. 01:06:01.360 --> 01:06:02.910 But look also at the shape right? 01:06:02.910 --> 01:06:05.650 So it's really kind of you look at one building 01:06:05.650 --> 01:06:08.370 with an antenna, and another building, right? 01:06:08.370 --> 01:06:12.660 So it's very unreasonable change of the volatility surface. 01:06:17.530 --> 01:06:22.070 So we can use ill-posedness to our advantage. 01:06:22.070 --> 01:06:27.430 So basically, at this point we say, 01:06:27.430 --> 01:06:30.300 well, it's not a requirement to calibrate exactly, 01:06:30.300 --> 01:06:32.800 just because every instruments that 01:06:32.800 --> 01:06:35.750 is an input of collaboration actually has some tolerance. 01:06:35.750 --> 01:06:38.990 So even there is no point to calibrate it exactly. 01:06:41.980 --> 01:06:48.830 So because we know that small variations in inputs 01:06:48.830 --> 01:06:52.010 can be large variations of outputs, 01:06:52.010 --> 01:06:55.590 we can put some constraints on the outputs. 01:06:55.590 --> 01:06:57.140 And actually, that may not cost us 01:06:57.140 --> 01:07:01.130 much in terms of not being able to calibrate exactly, 01:07:01.130 --> 01:07:09.420 but produce much more meaningful result. 01:07:09.420 --> 01:07:15.880 And just to be absolutely sure that our output 01:07:15.880 --> 01:07:20.046 result, our surface is smooth, we 01:07:20.046 --> 01:07:27.250 can use basis functions that are smooth to begin with. 01:07:27.250 --> 01:07:30.840 So we'll use B-splines, but those will be two-dimensional. 01:07:30.840 --> 01:07:34.950 And we'll talk a little bit more about this. 01:07:34.950 --> 01:07:41.790 And it's not a requirement for us to have as many basis 01:07:41.790 --> 01:07:46.540 functions as we have instruments, because we 01:07:46.540 --> 01:07:47.860 can put some other constraints. 01:07:47.860 --> 01:07:50.770 Like, for example, we can put smoothness or gradient 01:07:50.770 --> 01:07:52.610 smoothness to the surface. 01:07:52.610 --> 01:07:56.970 So let's pick some relatively high but reasonable number 01:07:56.970 --> 01:07:59.400 of functions-- could be more than the number of input 01:07:59.400 --> 01:08:01.590 instruments-- and see what we can do. 01:08:01.590 --> 01:08:03.997 So first of all, let's build our basis 01:08:03.997 --> 01:08:05.080 functions for the surface. 01:08:05.080 --> 01:08:07.300 So this slide we already just saw, 01:08:07.300 --> 01:08:13.520 that we selected to use B-splines, which is 01:08:13.520 --> 01:08:15.630 very convenient to work with. 01:08:15.630 --> 01:08:18.630 This is a one-dimensional. 01:08:18.630 --> 01:08:21.229 This is the way we build them, typically. 01:08:21.229 --> 01:08:24.830 So we use the Cox-de Boor recursion formula. 01:08:24.830 --> 01:08:27.430 You start from linear, right? 01:08:27.430 --> 01:08:29.590 Then you apply that formula. 01:08:29.590 --> 01:08:34.370 You transition to the basis set of the second order. 01:08:34.370 --> 01:08:40.550 And then the next iteration, you have the third order. 01:08:40.550 --> 01:08:43.370 And those ones will be built. 01:08:43.370 --> 01:08:51.660 So now, if you take those basis functions in one dimension, 01:08:51.660 --> 01:08:54.890 and same basis functions in the other dimension, 01:08:54.890 --> 01:08:58.700 and then you compute the kind of 10s of products, 01:08:58.700 --> 01:09:01.880 like you multiply them one by one, 01:09:01.880 --> 01:09:05.000 then you get basis functions with shapes like this. 01:09:11.830 --> 01:09:14.920 Which means that no matter what we do, right, 01:09:14.920 --> 01:09:19.859 like, because every basis function makes sense, 01:09:19.859 --> 01:09:23.859 then any linear combinations will also be good enough. 01:09:23.859 --> 01:09:26.850 So to formulate the problem is very simple. 01:09:26.850 --> 01:09:31.606 So we're saying, OK, the quotes produced by our model 01:09:31.606 --> 01:09:33.939 should be close enough to what's observed in the market, 01:09:33.939 --> 01:09:35.090 with some weights again. 01:09:35.090 --> 01:09:37.060 But we don't require any more that those 01:09:37.060 --> 01:09:39.779 are calibrated exactly. 01:09:39.779 --> 01:09:44.647 We are going to put some penalty function to the change 01:09:44.647 --> 01:09:45.730 of the volatility surface. 01:09:49.260 --> 01:09:52.910 And we're going to put some penalty function 01:09:52.910 --> 01:09:58.620 to the volatility surface itself. 01:09:58.620 --> 01:10:03.980 So those are vectors, right? 01:10:03.980 --> 01:10:06.890 And L_1 and L_2 are matrices. 01:10:06.890 --> 01:10:11.490 So just to give you an example what those matrices should be, 01:10:11.490 --> 01:10:16.440 like if you are talking about smoothness, 01:10:16.440 --> 01:10:20.410 if you want to penalize the gradient of the vector, right, 01:10:20.410 --> 01:10:25.560 then the matrix will consist of rows of one 01:10:25.560 --> 01:10:26.840 and following minus 1. 01:10:26.840 --> 01:10:29.490 So what you're saying, OK, if I want 01:10:29.490 --> 01:10:32.490 to penalize the difference between this and the next. 01:10:32.490 --> 01:10:35.220 And you do that for every element, OK? 01:10:35.220 --> 01:10:41.890 And the penalty kind of consists of all penalties that you have. 01:10:41.890 --> 01:10:51.490 So here we just formulate our problem right? 01:10:51.490 --> 01:10:57.680 So we want once-- because we've had the Jacobian-- we want 01:10:57.680 --> 01:11:03.300 to price things close enough. 01:11:03.300 --> 01:11:05.740 And there are two penalty terms here 01:11:05.740 --> 01:11:11.440 with the different regularization parameters. 01:11:11.440 --> 01:11:17.020 So once we have this, we can just, using linear algebra, 01:11:17.020 --> 01:11:20.190 the solution is defined here. 01:11:20.190 --> 01:11:29.130 And this is resulting calibration, which 01:11:29.130 --> 01:11:31.990 we see is nice and smooth. 01:11:41.440 --> 01:11:46.530 So if we take the analysis of the calibration inverse 01:11:46.530 --> 01:11:51.060 problem, let's do that using our linear algebra 01:11:51.060 --> 01:11:54.080 tools to understand where the problem is coming from. 01:11:54.080 --> 01:12:00.020 OK, so A is a matrix, translates our model parameters 01:12:00.020 --> 01:12:01.330 to market observables. 01:12:01.330 --> 01:12:05.380 And there is some error there-- epsilon, OK? 01:12:05.380 --> 01:12:12.620 So you can see that your solution 01:12:12.620 --> 01:12:19.950 is a linear combination of singular values divided 01:12:19.950 --> 01:12:21.627 by the singular values, OK? 01:12:21.627 --> 01:12:32.820 So if your values are high, OK, then that's not a problem. 01:12:32.820 --> 01:12:38.280 The problem is that once you get very small singular values, OK, 01:12:38.280 --> 01:12:42.770 the deviation of v_i's can result 01:12:42.770 --> 01:12:45.682 in the large deviation of your reconstructed result. 01:12:45.682 --> 01:12:47.140 And that's when you have a problem. 01:12:52.860 --> 01:12:56.020 So this is described on this slide. 01:12:56.020 --> 01:13:04.840 So "ill-posed" is that small noise may be significantly 01:13:04.840 --> 01:13:08.360 amplified by small singular values. 01:13:08.360 --> 01:13:13.700 And if you have a problem when you don't know how good it is, 01:13:13.700 --> 01:13:15.820 and whether you can trust it or not, 01:13:15.820 --> 01:13:20.560 so it's a very standard approach, 01:13:20.560 --> 01:13:23.010 you compute the condition number, 01:13:23.010 --> 01:13:27.490 which is the ratio of the maximum to minimum singular 01:13:27.490 --> 01:13:28.550 values. 01:13:28.550 --> 01:13:32.040 And if the number is high, which means 01:13:32.040 --> 01:13:37.120 that there are some very insignificant modes 01:13:37.120 --> 01:13:42.000 in your input data that can cause substantial changes 01:13:42.000 --> 01:13:42.680 in your output. 01:13:42.680 --> 01:13:44.730 And if you know that, it's not comforting right? 01:13:44.730 --> 01:13:48.720 So if that mode actually doesn't present in reality, 01:13:48.720 --> 01:13:50.216 then that's fine. 01:13:50.216 --> 01:13:51.590 But there is no guarantee, right? 01:13:51.590 --> 01:13:55.038 See, if that happens, then your model basically blows up. 01:14:00.150 --> 01:14:06.770 And that slide displays exactly that noiseless situation, 01:14:06.770 --> 01:14:11.230 where it looks like if you don't have any noise 01:14:11.230 --> 01:14:14.960 and if your model is perfect, then 01:14:14.960 --> 01:14:17.000 you're always able to calibrate exactly 01:14:17.000 --> 01:14:19.350 to the market observables. 01:14:19.350 --> 01:14:21.800 But it's never the case right? 01:14:21.800 --> 01:14:24.760 So there's always uncertainty to the numbers 01:14:24.760 --> 01:14:27.650 that you're calibrating to. 01:14:27.650 --> 01:14:33.200 And your model is not always perfect. 01:14:37.700 --> 01:14:44.530 So very standard technique for that particular problem 01:14:44.530 --> 01:14:48.680 is the Tikhonov regularization, which, 01:14:48.680 --> 01:14:54.680 when you solve your ill-posed problem, as trying 01:14:54.680 --> 01:15:01.374 to minimize x minus y, you add some penalty to the amplitude 01:15:01.374 --> 01:15:03.120 to your solution. 01:15:03.120 --> 01:15:05.620 Which essentially saying, OK, give me 01:15:05.620 --> 01:15:08.090 something reasonable, but something that's not blown-up. 01:15:12.040 --> 01:15:19.430 If you go through this linear algebra, 01:15:19.430 --> 01:15:24.350 to see how that lambda parameter in the Tikhonov regularization 01:15:24.350 --> 01:15:30.760 affects the weights of the SVD kind of representation 01:15:30.760 --> 01:15:36.580 of your solution, we now see that small singular values 01:15:36.580 --> 01:15:39.600 is no longer a problem, just because we're not 01:15:39.600 --> 01:15:42.840 dividing by the small number, but actually we 01:15:42.840 --> 01:15:46.041 are kind of limited by some regularization parameter. 01:15:49.410 --> 01:15:53.470 And typically, when you apply that regularization, 01:15:53.470 --> 01:15:58.470 your model no longer gives you a perfect match, right? 01:15:58.470 --> 01:16:02.083 But the result is much more meaningful, and more stable. 01:16:09.780 --> 01:16:19.060 Another approach to the problem is-- 01:16:19.060 --> 01:16:20.960 and before we go to that slide. 01:16:20.960 --> 01:16:23.790 So kind of Tikhonov regularization we 01:16:23.790 --> 01:16:29.120 used for surface calibration. 01:16:29.120 --> 01:16:31.060 Here, a standard Tikhonov regularization 01:16:31.060 --> 01:16:33.840 is something that you just penalize 01:16:33.840 --> 01:16:35.640 the amplitude of the solution itself. 01:16:35.640 --> 01:16:37.973 But it doesn't have to be the amplitude of the solution. 01:16:37.973 --> 01:16:42.290 It can be some linear combination of your solution. 01:16:42.290 --> 01:16:45.990 And in terms of calibrating of volatility surface, 01:16:45.990 --> 01:16:49.090 we didn't apply penalty to the reconstructed volatility, 01:16:49.090 --> 01:16:54.200 but we say it's not that the amplitude of the solution 01:16:54.200 --> 01:16:55.630 that we don't like. 01:16:55.630 --> 01:16:57.260 We don't like non-smoothness. 01:16:57.260 --> 01:17:00.870 So let's penalize the derivatives of the surface 01:17:00.870 --> 01:17:02.088 in the different angles. 01:17:07.070 --> 01:17:12.290 Another approach would be to use a truncated SVD, where 01:17:12.290 --> 01:17:14.810 we say, OK, so we did our singular value decomposition. 01:17:14.810 --> 01:17:18.610 We're looking at the spectrum of singular values, 01:17:18.610 --> 01:17:23.790 and we find that some numbers look nice and large, some very 01:17:23.790 --> 01:17:25.000 small. 01:17:25.000 --> 01:17:28.210 And then we just skip the small ones. 01:17:28.210 --> 01:17:35.220 It's very similar to the PCA approach for the risk 01:17:35.220 --> 01:17:38.160 management that we saw before, where 01:17:38.160 --> 01:17:40.900 we just selected five principal components 01:17:40.900 --> 01:17:42.720 and we ignored the rest. 01:17:42.720 --> 01:17:47.030 As a result, the model is much more robust. 01:17:47.030 --> 01:17:49.300 And by doing this, we essentially 01:17:49.300 --> 01:17:51.414 truncate the null space of the model. 01:17:51.414 --> 01:17:52.830 If you're familiar with this, it's 01:17:52.830 --> 01:18:05.894 basically the space that has very small singular values. 01:18:12.670 --> 01:18:21.190 So what regularized models gives you is that improved stability. 01:18:21.190 --> 01:18:24.125 It's absolutely essential for ill-conditioned problems. 01:18:28.480 --> 01:18:34.950 And it's a more realistic and meaningful result 01:18:34.950 --> 01:18:39.860 at the expense of some beauty to fit exactly the data, 01:18:39.860 --> 01:18:44.940 but that's something that is quite often acceptable. 01:18:44.940 --> 01:18:47.550 It might cause a biased solution, 01:18:47.550 --> 01:18:51.510 meaning that your solution again may not be exact. 01:18:51.510 --> 01:18:55.320 It might be biased towards some better result. 01:18:55.320 --> 01:19:00.480 For example, if you apply smoothness constraint, 01:19:00.480 --> 01:19:03.810 the solution would kind of assume a little bit 01:19:03.810 --> 01:19:06.960 more smooth result than it actually is. 01:19:06.960 --> 01:19:10.590 But that's acceptable. 01:19:10.590 --> 01:19:15.840 And the bias, again, can be minimized 01:19:15.840 --> 01:19:20.210 by reasonable selection of what quantity you actually 01:19:20.210 --> 01:19:21.200 don't like. 01:19:21.200 --> 01:19:23.820 Again, you can say, oh, during calibration 01:19:23.820 --> 01:19:27.067 of our volatilities of our surface we could have said, 01:19:27.067 --> 01:19:27.650 you know what? 01:19:27.650 --> 01:19:32.470 Let's just open the textbook and see what's the regularization. 01:19:32.470 --> 01:19:33.850 OK, we find Tikhonov. 01:19:33.850 --> 01:19:36.040 We start to penalize the amplitude. 01:19:36.040 --> 01:19:37.890 Then the result won't be good. 01:19:37.890 --> 01:19:40.570 So we need to think about and say, OK, 01:19:40.570 --> 01:19:42.080 what exactly we don't like. 01:19:42.080 --> 01:19:45.480 Like, for example, is like absolutely flat volatility 01:19:45.480 --> 01:19:46.760 surface good for us? 01:19:46.760 --> 01:19:49.240 And we'll find, yeah, that that's actually fine. 01:19:49.240 --> 01:19:52.690 Then, if we said that then penalizing the amplitude 01:19:52.690 --> 01:19:54.440 doesn't make sense, so we need to penalize 01:19:54.440 --> 01:19:57.790 something that is a deviation from that perfect flat 01:19:57.790 --> 01:19:59.030 solution. 01:19:59.030 --> 01:20:02.560 Or to be more kind of precise, we 01:20:02.560 --> 01:20:07.990 penalize like the derivative in different directions. 01:20:07.990 --> 01:20:13.540 So this kind of concludes my presentation today. 01:20:13.540 --> 01:20:18.580 And there are some useful links if you 01:20:18.580 --> 01:20:22.546 want to get more information. 01:20:25.997 --> 01:20:28.462 Thank you. 01:20:28.462 --> 01:20:29.448 Any question? 01:20:34.880 --> 01:20:35.838 AUDIENCE: Yes. 01:20:35.838 --> 01:20:41.814 So regarding the techniques that you use for fitting 01:20:41.814 --> 01:20:46.794 function that you are using spline techniques. 01:20:46.794 --> 01:20:50.480 What other techniques-- is the spline 01:20:50.480 --> 01:20:52.370 the best technique you use? 01:20:56.220 --> 01:20:57.970 IVAN MASYUKOV: Well, spline is, yes. 01:20:57.970 --> 01:21:01.450 So a spline or interpolation is the same. 01:21:01.450 --> 01:21:03.350 So we're always talking about interpolation. 01:21:03.350 --> 01:21:07.080 So you have some limited number of inputs, 01:21:07.080 --> 01:21:08.560 and you want to draw in between. 01:21:08.560 --> 01:21:11.420 So there are just two words for this, 01:21:11.420 --> 01:21:15.120 which is a interpolation-- or spline-- which I consider to be 01:21:15.120 --> 01:21:16.600 the same thing in general. 01:21:26.310 --> 01:21:30.050 AUDIENCE: I have a question about the interpolation graph 01:21:30.050 --> 01:21:32.525 that you had where the following was very smooth. 01:21:35.495 --> 01:21:39.046 When you, as an expert in this, look at that graph 01:21:39.046 --> 01:21:46.090 and see, I guess, just some odd shapes 01:21:46.090 --> 01:21:53.222 at certain parts of the curve, how do you interpret that? 01:21:53.222 --> 01:21:56.980 And do you assess that that's a feature of the current term 01:21:56.980 --> 01:22:00.764 market liquidity conditions, or possibly just a mathematical-- 01:22:00.764 --> 01:22:02.180 IVAN MASYUKOV: Well, first of all, 01:22:02.180 --> 01:22:09.680 I mean, that grid is done like for-- every element is 01:22:09.680 --> 01:22:10.685 a three-month. 01:22:10.685 --> 01:22:12.700 But what's traded on the market? 01:22:12.700 --> 01:22:15.620 Like, typical maturities are three-month, 01:22:15.620 --> 01:22:19.710 maybe half-year, one year, two year, five year, 10 year. 01:22:19.710 --> 01:22:25.590 So we should kind of rescale it in a logarithmic scale, 01:22:25.590 --> 01:22:27.920 or-- you know what I'm talking about? 01:22:27.920 --> 01:22:31.010 And then, if you do that, then this peak 01:22:31.010 --> 01:22:32.460 doesn't look a peak anymore. 01:22:32.460 --> 01:22:35.450 So the reason why it looks like a feature to you 01:22:35.450 --> 01:22:39.630 is because it's quite sharper than this guy, right? 01:22:39.630 --> 01:22:44.990 But that's because you have many more detailed instruments there 01:22:44.990 --> 01:22:46.060 compared to here. 01:22:49.570 --> 01:22:52.191 And that's the reason why we selected our basis functions 01:22:52.191 --> 01:22:52.690 like this. 01:22:52.690 --> 01:22:56.890 So we selected more node density in the front, 01:22:56.890 --> 01:23:01.270 just because there are more instruments in the front 01:23:01.270 --> 01:23:03.210 than at the end. 01:23:03.210 --> 01:23:08.620 So we want our spline to be more detailed at the beginning, 01:23:08.620 --> 01:23:12.200 and kind of just nice and smooth at the end. 01:23:12.200 --> 01:23:16.990 So that's why those basis functions-- which correspond 01:23:16.990 --> 01:23:20.900 very well to actual instruments that we have in the portfolio-- 01:23:20.900 --> 01:23:25.820 can produce, first of all, you see this spike here too, right? 01:23:25.820 --> 01:23:30.250 So is it like you can compare this guy to this guy. 01:23:30.250 --> 01:23:36.010 But essentially, it's just they have similar magnitude, 01:23:36.010 --> 01:23:39.000 but we don't have enough instruments in this area 01:23:39.000 --> 01:23:41.720 to support any sharper features. 01:23:41.720 --> 01:23:47.540 So I don't see any problems with this graph. 01:23:47.540 --> 01:23:52.150 But traders, because they look at this every day, right? 01:23:52.150 --> 01:23:54.150 And then they calibrate, and they see a feature, 01:23:54.150 --> 01:23:57.190 and then they immediately kind of trying to think. 01:23:57.190 --> 01:24:00.920 OK, if you see something that you know is not typical-- 01:24:00.920 --> 01:24:04.110 and that sense of typical/not typical 01:24:04.110 --> 01:24:07.490 comes with years of experience-- then they 01:24:07.490 --> 01:24:09.870 try to arbitrage this. 01:24:09.870 --> 01:24:13.840 Because if there is a spike in the surface, 01:24:13.840 --> 01:24:15.980 it's very likely that it will disappear soon. 01:24:19.820 --> 01:24:21.260 AUDIENCE: So if this is the model 01:24:21.260 --> 01:24:26.317 for volatility, that's used in modeling for swaptions, right? 01:24:26.317 --> 01:24:27.784 IVAN MASYUKOV: Yeah. 01:24:27.784 --> 01:24:29.495 AUDIENCE: So if you were to actually try 01:24:29.495 --> 01:24:33.163 to go about making a trade out of some discrepancy 01:24:33.163 --> 01:24:35.363 that you see, can you kind of describe how 01:24:35.363 --> 01:24:36.815 you'd do that with swaption? 01:24:36.815 --> 01:24:38.564 Do you use basically like regular options? 01:24:42.547 --> 01:24:44.380 IVAN MASYUKOV: I don't have the screen here. 01:24:44.380 --> 01:24:50.010 But essentially what traders do, right? 01:24:50.010 --> 01:24:56.130 So this is, I mean that calibrator we actually use. 01:24:56.130 --> 01:24:58.400 And it's a real-time calibrator. 01:24:58.400 --> 01:25:00.640 The reason why it can be real time 01:25:00.640 --> 01:25:04.470 is because there is just simple linear algebra there. 01:25:04.470 --> 01:25:08.000 Most of stuff like A transpose A can be pre-calculated. 01:25:08.000 --> 01:25:11.500 So it can be real time. 01:25:11.500 --> 01:25:13.920 So they can see this volatility surface 01:25:13.920 --> 01:25:20.290 moving while-- we connect to actual market data. 01:25:20.290 --> 01:25:25.230 And once they see there is anomaly on the market, 01:25:25.230 --> 01:25:27.970 there is something traded which they believe is wrong, 01:25:27.970 --> 01:25:30.300 OK, they're just make an advantage of that. 01:25:30.300 --> 01:25:35.000 They just make a trade, that would be swaption, for example, 01:25:35.000 --> 01:25:39.240 or maybe some other trade more exotic than swaptions. 01:25:39.240 --> 01:25:41.320 But again, having like the dependence 01:25:41.320 --> 01:25:46.320 on the particular instruments, which kind of would 01:25:46.320 --> 01:25:48.840 express your position that this will change 01:25:48.840 --> 01:25:50.290 like in a day or so. 01:25:50.290 --> 01:25:55.660 So that's exactly how the desk makes money. 01:25:55.660 --> 01:25:58.220 So we call it relative value analysis. 01:25:58.220 --> 01:26:01.030 So if we have a tool like that, you have a model. 01:26:01.030 --> 01:26:04.500 You have your input instruments, and you have some regularizing 01:26:04.500 --> 01:26:06.150 terms of it could be smoothness, it 01:26:06.150 --> 01:26:09.910 could be a PCA, it could be combination of PCA, right? 01:26:09.910 --> 01:26:13.200 But then that additional information 01:26:13.200 --> 01:26:17.410 allows you to find anomalies in the market. 01:26:17.410 --> 01:26:19.250 Once you find those anomalies, you 01:26:19.250 --> 01:26:22.670 can take advantage of them-- provided that your model is 01:26:22.670 --> 01:26:24.480 robust enough. 01:26:24.480 --> 01:26:27.970 And if you are saying, well, I am kind of doing well, 01:26:27.970 --> 01:26:31.350 and I'm calibrate well with just some smoothness assumption 01:26:31.350 --> 01:26:33.410 about smoothness of the forward rates, 01:26:33.410 --> 01:26:35.850 there is nothing more fundamental than this. 01:26:35.850 --> 01:26:41.980 So if your model is based on fundamental principle, 01:26:41.980 --> 01:26:44.990 you can expect that it will be more stable in the future, 01:26:44.990 --> 01:26:45.820 rather than PCA. 01:26:45.820 --> 01:26:47.795 Because for PCA, you just kind of say, 01:26:47.795 --> 01:26:49.590 OK, I took the time interval. 01:26:49.590 --> 01:26:55.090 I kind of did my regression analysis, whatever. 01:26:55.090 --> 01:26:57.800 But that doesn't mean that the market will continue 01:26:57.800 --> 01:26:59.400 to do the same in the future. 01:27:02.865 --> 01:27:04.845 AUDIENCE: I have a question. 01:27:04.845 --> 01:27:07.815 [INAUDIBLE] marketer. 01:27:07.815 --> 01:27:10.785 And then we would try to price the bond at that premium. 01:27:10.785 --> 01:27:14.313 You mentioned that actually bond is the most liquid instrument 01:27:14.313 --> 01:27:15.260 in the market. 01:27:15.260 --> 01:27:18.383 So why not you do the area around by inverse bond 01:27:18.383 --> 01:27:22.889 to derive the discounting factor from a bond [INAUDIBLE]. 01:27:22.889 --> 01:27:24.930 IVAN MASYUKOV: Well, that's a very good question. 01:27:24.930 --> 01:27:30.150 So basically we could have done that, OK. 01:27:30.150 --> 01:27:33.730 And some firms do that. 01:27:33.730 --> 01:27:36.190 The problem is that with swaps, we 01:27:36.190 --> 01:27:42.010 kind of have those swaps today, they kind of 01:27:42.010 --> 01:27:44.560 roll every day, OK? 01:27:44.560 --> 01:27:48.164 They like, the swap today starts today, swap tomorrow 01:27:48.164 --> 01:27:49.580 starts tomorrow, things like that. 01:27:49.580 --> 01:27:51.550 But bonds do not. 01:27:51.550 --> 01:27:56.750 OK so they basically, there is like on-the-run bond, which 01:27:56.750 --> 01:28:00.510 is the most liquid one, which once the new ten-year bond is 01:28:00.510 --> 01:28:04.060 issued, then that bond becomes off-the-run. 01:28:04.060 --> 01:28:05.790 It's still traded, but then everyone 01:28:05.790 --> 01:28:07.800 switches to on-the-run. 01:28:07.800 --> 01:28:13.300 So you don't have a nice continuous spectrum of bonds. 01:28:13.300 --> 01:28:15.670 You have kind of concentrations between on-the-runs 01:28:15.670 --> 01:28:17.400 and off-the-runs. 01:28:17.400 --> 01:28:21.960 And if you want to draw the curve for all of them, 01:28:21.960 --> 01:28:24.990 you typically cannot do the perfect fit. 01:28:24.990 --> 01:28:29.160 You kind of need to do the least squares. 01:28:29.160 --> 01:28:33.310 So it's just more convenient to do it in the swap. 01:28:33.310 --> 01:28:38.730 But once we build the swap curve, swap trader typically-- 01:28:38.730 --> 01:28:41.170 I should say always use bonds for hedging, 01:28:41.170 --> 01:28:44.080 just because bonds are much more liquid. 01:28:44.080 --> 01:28:48.220 Then we project bonds to the swap curve 01:28:48.220 --> 01:28:51.240 rather than swaps to the bond curve, which is hard to build. 01:28:52.206 --> 01:28:53.977 AUDIENCE: So in this case, when they 01:28:53.977 --> 01:28:57.036 switch from on-the-run, off-the-run, [INAUDIBLE]? 01:29:01.069 --> 01:29:01.860 IVAN MASYUKOV: Yes. 01:29:01.860 --> 01:29:05.700 So your curve won't be stable, just because those roll 01:29:05.700 --> 01:29:09.690 effects-- we call those "roll effects"-- 01:29:09.690 --> 01:29:12.580 which means that there is something substantial changes 01:29:12.580 --> 01:29:13.170 on the market. 01:29:13.170 --> 01:29:14.900 And there may be such a big demand 01:29:14.900 --> 01:29:21.650 for this new bond on the market, that will make your curve, that 01:29:21.650 --> 01:29:23.700 won't look nice. 01:29:23.700 --> 01:29:27.410 So there are also traders that just trade bonds. 01:29:27.410 --> 01:29:32.470 And those typically don't have curves. 01:29:32.470 --> 01:29:36.638 They rely on some PCA models, or some other things. 01:29:39.851 --> 01:29:40.850 PROFESSOR: Thanks again. 01:29:40.850 --> 01:29:42.610 IVAN MASYUKOV: Thank you.