1 00:00:00,070 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:22,870 --> 00:00:24,270 PROFESSOR: Anyway, welcome today. 9 00:00:24,270 --> 00:00:27,720 Stefan Andreev is our guest speaker from Morgan Stanley. 10 00:00:27,720 --> 00:00:29,655 And as I understand you have a degree, 11 00:00:29,655 --> 00:00:31,952 a PhD Degree in chemical physics. 12 00:00:31,952 --> 00:00:33,660 STEFAN ANDREEV: In chemical physics, yes. 13 00:00:33,660 --> 00:00:35,697 And maybe I should go here. 14 00:00:35,697 --> 00:00:38,380 [LAUGHTER] 15 00:00:38,380 --> 00:00:41,063 PROFESSOR: And now he's in the world of finance. 16 00:00:41,063 --> 00:00:44,310 And we're here to benefit from your experience. 17 00:00:44,310 --> 00:00:46,880 STEFAN ANDREEV: Thank you very much for the introduction. 18 00:00:46,880 --> 00:00:47,400 Yeah. 19 00:00:47,400 --> 00:00:50,650 I went to school at Dartmouth College undergrad, 20 00:00:50,650 --> 00:00:53,970 and then up the street at Harvard for my PhD. 21 00:00:53,970 --> 00:00:58,260 And then I transitioned from science to finance. 22 00:00:58,260 --> 00:00:59,676 And for the last eight years, I've 23 00:00:59,676 --> 00:01:03,590 been working at Morgan Stanley, working with Vasily Strela, 24 00:01:03,590 --> 00:01:05,310 an instructor in the course. 25 00:01:08,740 --> 00:01:12,850 So today, what are we going to be talking about? 26 00:01:12,850 --> 00:01:20,370 Well, to give you a big-picture view of where our topic fits 27 00:01:20,370 --> 00:01:26,420 within the grand scheme of finance-- in general, 28 00:01:26,420 --> 00:01:29,860 there are really two big areas in my view. 29 00:01:29,860 --> 00:01:32,130 Well, there's probably more, but these 30 00:01:32,130 --> 00:01:34,690 are kind of most famous, I would say, areas 31 00:01:34,690 --> 00:01:38,390 where quantitative skills can be-- 32 00:01:38,390 --> 00:01:39,740 are very valuable in finance. 33 00:01:39,740 --> 00:01:42,820 And the two areas are-- one area is statistics, 34 00:01:42,820 --> 00:01:45,660 predictions, which is essentially, 35 00:01:45,660 --> 00:01:49,630 say given some historical behavior in the market, 36 00:01:49,630 --> 00:01:53,910 how do we predict what will happen in the future? 37 00:01:53,910 --> 00:01:56,620 And that's certainly a huge industry. 38 00:01:56,620 --> 00:02:00,640 People have made a ton of money applying quantitative concepts 39 00:02:00,640 --> 00:02:01,950 to that. 40 00:02:01,950 --> 00:02:04,760 But that's not what we're going to talk about today. 41 00:02:04,760 --> 00:02:07,540 What we're going to talk about is another very big area 42 00:02:07,540 --> 00:02:09,990 called pricing, which is pricing and hedging 43 00:02:09,990 --> 00:02:11,600 of complex instruments. 44 00:02:11,600 --> 00:02:18,350 And that area is really about essentially 45 00:02:18,350 --> 00:02:21,430 when you have a complex product that you don't really 46 00:02:21,430 --> 00:02:25,410 know the price of, but you know the prices of other products. 47 00:02:25,410 --> 00:02:27,710 And then you can use the other products 48 00:02:27,710 --> 00:02:32,290 to essentially replicate the payoff of your complex product. 49 00:02:32,290 --> 00:02:35,660 Then you can use mathematical techniques 50 00:02:35,660 --> 00:02:39,260 to essentially say, look, the main statement is hey, 51 00:02:39,260 --> 00:02:41,780 because I can replicate my payoff, 52 00:02:41,780 --> 00:02:45,850 and using products that I know the price of, 53 00:02:45,850 --> 00:02:49,640 then that means that I can say something 54 00:02:49,640 --> 00:02:51,890 about the price of my complex product. 55 00:02:51,890 --> 00:02:53,860 I can basically price it. 56 00:02:53,860 --> 00:02:55,830 And not only can I price it, but I 57 00:02:55,830 --> 00:02:58,710 can also-- when I give the price, I 58 00:02:58,710 --> 00:03:02,770 know that I can eliminate any uncertainty from owning 59 00:03:02,770 --> 00:03:05,510 the product by executing a hedging replication 60 00:03:05,510 --> 00:03:08,465 strategy-- at least theoretically speaking. 61 00:03:11,070 --> 00:03:14,200 So that's the area we're going to focus today. 62 00:03:14,200 --> 00:03:21,600 And our main focus is going to be on FX-- foreign exchange-- 63 00:03:21,600 --> 00:03:23,880 interest rates, and credit-- and in particular 64 00:03:23,880 --> 00:03:26,920 about credit-FX hybrid models. 65 00:03:26,920 --> 00:03:30,420 We're going to be talking about essentially what 66 00:03:30,420 --> 00:03:33,350 happens, why do we need credit-FX hybrid models, 67 00:03:33,350 --> 00:03:37,530 and going through an example of a simple one, 68 00:03:37,530 --> 00:03:39,629 and how to apply it. 69 00:03:39,629 --> 00:03:42,170 In particular-- and there's the mathematical techniques we're 70 00:03:42,170 --> 00:03:44,489 going to be using-- as I said, we 71 00:03:44,489 --> 00:03:47,030 are going to be talking about the risk-neutral pricing, which 72 00:03:47,030 --> 00:03:49,220 is essentially replication. 73 00:03:49,220 --> 00:03:54,350 And we're going to talk about how to use jump processes-- 74 00:03:54,350 --> 00:03:57,100 which you might have seen in other parts of your studies 75 00:03:57,100 --> 00:04:02,872 as Poisson processes-- to describe certain behaviors 76 00:04:02,872 --> 00:04:05,880 of price behavior that you cannot really describe very 77 00:04:05,880 --> 00:04:08,990 easily using pure diffusion Brownian motions that you 78 00:04:08,990 --> 00:04:13,250 probably have seen so far in the course. 79 00:04:13,250 --> 00:04:14,624 And why do we care about that? 80 00:04:14,624 --> 00:04:16,540 Well, there are certain financial applications 81 00:04:16,540 --> 00:04:17,540 where this is important. 82 00:04:17,540 --> 00:04:20,300 And in particular, something that 83 00:04:20,300 --> 00:04:24,150 happened in the last few years-- the sovereign crisis in Europe. 84 00:04:24,150 --> 00:04:26,259 And also, it has happen not just last year. 85 00:04:26,259 --> 00:04:28,550 This happened many times in other parts of the emerging 86 00:04:28,550 --> 00:04:29,170 markets. 87 00:04:29,170 --> 00:04:31,570 And given the emerging markets as my background, 88 00:04:31,570 --> 00:04:34,210 I've worked on these kind of models. 89 00:04:34,210 --> 00:04:38,550 And this is, when you have Greek bonds in Euros, 90 00:04:38,550 --> 00:04:42,940 and there's a potential for Greek default. And as we know, 91 00:04:42,940 --> 00:04:46,530 as you might have read in the news, 92 00:04:46,530 --> 00:04:49,380 there was really a big worry about what 93 00:04:49,380 --> 00:04:51,410 will happen to the Euro currency if there is 94 00:04:51,410 --> 00:04:53,300 a spate of sovereign defaults. 95 00:04:53,300 --> 00:04:56,514 And in fact, Euro currency did-- in anticipation 96 00:04:56,514 --> 00:04:58,305 of the possibility of default-- it actually 97 00:04:58,305 --> 00:05:05,170 did depreciate for while back in 2011 and 2012. 98 00:05:05,170 --> 00:05:07,710 Now it's pretty much back where it was before that, 99 00:05:07,710 --> 00:05:10,430 but it certainly-- there was a fear 100 00:05:10,430 --> 00:05:14,730 in the market-- which was also very, very obvious in terms 101 00:05:14,730 --> 00:05:19,370 of option prices-- that Euro currency could depreciate 102 00:05:19,370 --> 00:05:22,950 significantly if, in fact, a disorderly default did happen. 103 00:05:22,950 --> 00:05:26,790 Now it didn't happen, so that's good. 104 00:05:26,790 --> 00:05:28,510 But in other emerging markets in history, 105 00:05:28,510 --> 00:05:30,650 it has happened before. 106 00:05:30,650 --> 00:05:33,980 So it's not really an empty question. 107 00:05:33,980 --> 00:05:39,090 So foreign exchange-- how do we describe it in math finance? 108 00:05:39,090 --> 00:05:40,480 Well, we think of it as the price 109 00:05:40,480 --> 00:05:43,570 of a unit of foreign currency in dollars. 110 00:05:43,570 --> 00:05:46,870 In our presentation we're going to denote the spot FX 111 00:05:46,870 --> 00:05:49,830 rate, which is the current rate of exchange, by S. 112 00:05:49,830 --> 00:05:54,020 And here is a sample graph of euro-USD FX rates. 113 00:05:54,020 --> 00:05:56,740 You can see it looks like a random walk. 114 00:05:56,740 --> 00:05:59,300 It's very well described in normal circumstances 115 00:05:59,300 --> 00:06:00,000 as random walk. 116 00:06:03,850 --> 00:06:09,280 So one very fundamental property that connects FX and interest 117 00:06:09,280 --> 00:06:13,880 rates is the so-called FX forwards interest rate 118 00:06:13,880 --> 00:06:19,700 parity, which says if I have a certain amount of money-- 119 00:06:19,700 --> 00:06:27,710 in this example, $5 million, and I can invest it, 120 00:06:27,710 --> 00:06:30,200 there's two ways I can utilize this money. 121 00:06:30,200 --> 00:06:33,360 One way is to just invest it at a dollar 122 00:06:33,360 --> 00:06:34,620 kind of risk-free rate. 123 00:06:34,620 --> 00:06:37,700 And we're assuming here we have a risk-free rate-- 124 00:06:37,700 --> 00:06:39,150 the standard assumption. 125 00:06:39,150 --> 00:06:41,460 Or we can do something like we can take the money, 126 00:06:41,460 --> 00:06:44,112 exchange it into, say, euros, invest 127 00:06:44,112 --> 00:06:46,620 it using the euro risk-free rate, 128 00:06:46,620 --> 00:06:50,720 and then exchange it back into dollars. 129 00:06:50,720 --> 00:06:56,640 And this is essentially used to price FX forward contracts. 130 00:06:56,640 --> 00:06:59,890 So FX forward contracts are a contract 131 00:06:59,890 --> 00:07:02,410 that allow you to say, look, I'm going to agree with you, 132 00:07:02,410 --> 00:07:06,850 then in one month's time, I'm going to, say, give you 133 00:07:06,850 --> 00:07:11,110 4,108,405 euros, and you're going to give me back 134 00:07:11,110 --> 00:07:15,960 $5,170,000. 135 00:07:15,960 --> 00:07:18,839 It's essentially an agreement; it's a derivative contract. 136 00:07:18,839 --> 00:07:21,130 And if you see the-- if you have this forward contract, 137 00:07:21,130 --> 00:07:24,070 you can lock in, essentially, through conversion in euros. 138 00:07:24,070 --> 00:07:26,670 So you can lock in an effective dollar interest rate. 139 00:07:26,670 --> 00:07:32,060 So FX forwards can be essentially described fully 140 00:07:32,060 --> 00:07:34,700 by knowing the interest rates in each currency and the spot FX 141 00:07:34,700 --> 00:07:36,160 rate. 142 00:07:36,160 --> 00:07:39,220 Conversely, you can infer foreign interest rates 143 00:07:39,220 --> 00:07:42,790 knowing the FX forwards. 144 00:07:42,790 --> 00:07:43,780 They're very connected. 145 00:07:47,540 --> 00:07:48,620 Yes? 146 00:07:48,620 --> 00:07:50,070 AUDIENCE: In this example, there's no mispricing, 147 00:07:50,070 --> 00:07:51,236 so you get that same amount. 148 00:07:51,236 --> 00:07:52,257 Is that the idea? 149 00:07:52,257 --> 00:07:54,590 STEFAN ANDREEV: In this example, there is no mispricing. 150 00:07:54,590 --> 00:07:56,230 You get back the same amount. 151 00:07:56,230 --> 00:07:59,790 So we are assuming, essentially, there's no arbitrage. 152 00:07:59,790 --> 00:08:01,680 We're not assuming, but we're given-- 153 00:08:01,680 --> 00:08:04,880 if the prices were, indeed, if this interest rate-- 4.6% 154 00:08:04,880 --> 00:08:09,090 in Euros-- interest rate was 4.6, in dollars it's 3.4. 155 00:08:09,090 --> 00:08:12,200 And here are the current spots, which is 127 and the forward, 156 00:08:12,200 --> 00:08:16,790 125-- if these were, in fact, the observable mark 157 00:08:16,790 --> 00:08:20,460 quantities in the market, then there would be no arbitrage. 158 00:08:20,460 --> 00:08:23,640 And there is-- you're basically indifferent whether you invest 159 00:08:23,640 --> 00:08:26,390 the money in dollars, or you go the way 160 00:08:26,390 --> 00:08:28,980 of exchanging into Euros, and investing in Euros, and then 161 00:08:28,980 --> 00:08:31,190 back into dollars. 162 00:08:31,190 --> 00:08:34,380 So in this example, the way I've presented, worked it out, 163 00:08:34,380 --> 00:08:35,309 there is no arbitrage. 164 00:08:35,309 --> 00:08:37,892 Now, if some of these numbers-- say the interest rate in Euros 165 00:08:37,892 --> 00:08:41,120 were 4% instead of 4.6, and all the other quantities were 166 00:08:41,120 --> 00:08:44,750 the same, then, in fact, there would be arbitrage. 167 00:08:44,750 --> 00:08:47,990 And you could make money by borrowing money 168 00:08:47,990 --> 00:08:50,915 in dollars and investing. 169 00:08:55,590 --> 00:08:57,040 I mean, the purpose of this slide 170 00:08:57,040 --> 00:08:59,160 is really to illustrate kind of hey, 171 00:08:59,160 --> 00:09:02,440 if there's no arbitrage, how one would actually 172 00:09:02,440 --> 00:09:04,280 compare-- how one would actually look 173 00:09:04,280 --> 00:09:07,140 for arbitrage in this example. 174 00:09:07,140 --> 00:09:09,670 This is-- again, this is a little bit of definition 175 00:09:09,670 --> 00:09:13,505 what are compound interest, interest rates. 176 00:09:16,260 --> 00:09:19,366 We're going to talk about instantaneous risk-free rates. 177 00:09:19,366 --> 00:09:21,240 We're going to, again, say they're risk-free. 178 00:09:21,240 --> 00:09:23,544 So basically, we know for sure we're 179 00:09:23,544 --> 00:09:24,710 going to get our money back. 180 00:09:24,710 --> 00:09:27,340 You can think of risk-free rates as the one 181 00:09:27,340 --> 00:09:31,170 that treasuries pay in real life, or the one 182 00:09:31,170 --> 00:09:33,490 that Federal Reserve guarantees on deposits. 183 00:09:33,490 --> 00:09:37,190 There's various examples of risk-free rates. 184 00:09:37,190 --> 00:09:39,522 And while, in practice, different risk-free rates 185 00:09:39,522 --> 00:09:41,980 can actually be different, so they're not really risk-free. 186 00:09:41,980 --> 00:09:45,482 But in our world right now, in our model, 187 00:09:45,482 --> 00:09:46,940 we're going to assume that there is 188 00:09:46,940 --> 00:09:50,050 such a thing as a risk-free rate for every currency, 189 00:09:50,050 --> 00:09:52,770 and it's unique. 190 00:09:52,770 --> 00:09:59,270 And now, as we talk about our dynamics of the FX process, 191 00:09:59,270 --> 00:10:01,310 what's really focused on here is an FX-- 192 00:10:01,310 --> 00:10:03,380 we're making an FX model. 193 00:10:03,380 --> 00:10:05,870 And we want to see-- in the previous example, 194 00:10:05,870 --> 00:10:09,525 we saw if you're given a FX rate and given some interest rates, 195 00:10:09,525 --> 00:10:10,900 here's what the FX forward really 196 00:10:10,900 --> 00:10:14,270 has to be in order to have no arbitrage. 197 00:10:14,270 --> 00:10:17,390 Well, now we are trying to describe. 198 00:10:17,390 --> 00:10:20,905 We tried to describe or define a process for the FX currency. 199 00:10:24,060 --> 00:10:27,290 Essentially, this kind of no-arbitrage condition 200 00:10:27,290 --> 00:10:33,060 leads to having certain constraints 201 00:10:33,060 --> 00:10:35,470 on what the stochastic differential 202 00:10:35,470 --> 00:10:36,870 equation has to be. 203 00:10:36,870 --> 00:10:39,600 So in this particular case, the constraint 204 00:10:39,600 --> 00:10:41,810 is that the drifts of the process 205 00:10:41,810 --> 00:10:45,320 has to be the differential in interest rates. 206 00:10:45,320 --> 00:10:48,917 So if one currency pays more than the other currency, 207 00:10:48,917 --> 00:10:51,250 obviously, people would want to invest in that currency. 208 00:10:51,250 --> 00:10:54,070 So that in order for no arbitrage to exist, 209 00:10:54,070 --> 00:10:56,930 there has to be an expectation that the currency that 210 00:10:56,930 --> 00:10:59,830 pays more would depreciate in the future, otherwise 211 00:10:59,830 --> 00:11:03,000 it would be an arbitrage. 212 00:11:03,000 --> 00:11:06,080 So if it doesn't depreciate, if you can kind of say, 213 00:11:06,080 --> 00:11:08,740 hey, this currency won't depreciate, 214 00:11:08,740 --> 00:11:11,080 then you can just always invest in that currency that 215 00:11:11,080 --> 00:11:15,330 pays higher interest rates and make money-- which, in fact, 216 00:11:15,330 --> 00:11:16,610 many people do. 217 00:11:16,610 --> 00:11:20,360 But again, that's a-- they're taking a certain risk. 218 00:11:20,360 --> 00:11:24,290 They're taking the risk that the currency will depreciate. 219 00:11:24,290 --> 00:11:26,700 So what do we actually want? 220 00:11:26,700 --> 00:11:32,460 What we want is to say-- we want to essentially-- 221 00:11:32,460 --> 00:11:35,500 and for the arbitrage conditions from before-- which 222 00:11:35,500 --> 00:11:38,870 is to say that my forward rate has to be essentially the spot 223 00:11:38,870 --> 00:11:46,660 rate-- well, this condition here has 224 00:11:46,660 --> 00:11:48,750 to be observed, essentially. 225 00:11:48,750 --> 00:11:50,780 And what does that mean? 226 00:11:50,780 --> 00:11:54,264 It means that my forward rate has to be my spot-- 227 00:11:54,264 --> 00:11:56,055 AUDIENCE: [INAUDIBLE] each other, you mean? 228 00:11:56,055 --> 00:11:56,530 STEFAN ANDREEV: What did you say? 229 00:11:56,530 --> 00:11:57,480 AUDIENCE: [INAUDIBLE] set those equal to each other? 230 00:11:57,480 --> 00:11:58,960 STEFAN ANDREEV: Yes. 231 00:11:58,960 --> 00:12:04,900 My forward rate has to equal to the spot rate times, 232 00:12:04,900 --> 00:12:08,251 essentially, the interest rate differential. 233 00:12:08,251 --> 00:12:16,460 If that is true, than in the previous set up, 234 00:12:16,460 --> 00:12:18,980 there will be no arbitrage. 235 00:12:18,980 --> 00:12:19,670 And why is that? 236 00:12:19,670 --> 00:12:23,000 Well, that's because the amount of money 237 00:12:23,000 --> 00:12:27,825 I earn on the domestic leg is e to the rd. 238 00:12:27,825 --> 00:12:31,730 The amount of money I earn on the foreign leg is e to the rf, 239 00:12:31,730 --> 00:12:35,030 but then I multiply by the forward, 240 00:12:35,030 --> 00:12:39,490 and that has to equal to the e to the rd. 241 00:12:39,490 --> 00:12:43,890 So this is a standard. 242 00:12:43,890 --> 00:12:49,040 This is a the most basic dynamic FX model 243 00:12:49,040 --> 00:12:51,180 that people use in industry. 244 00:12:51,180 --> 00:12:54,930 It's referred to as the Black-Scholes FX model. 245 00:12:54,930 --> 00:12:59,360 And the stock price-- you've seen stochastic models before. 246 00:12:59,360 --> 00:13:02,630 Usually, you see the stock price when people talk about options. 247 00:13:05,440 --> 00:13:10,470 In that case, this drift is just the risk-free interest rate. 248 00:13:10,470 --> 00:13:14,710 Well, here in FX, it's a differential of interest rates. 249 00:13:14,710 --> 00:13:18,010 Otherwise, it's very similar. 250 00:13:18,010 --> 00:13:23,070 So FX has some interesting properties. 251 00:13:23,070 --> 00:13:26,840 So we're gonna talk about the game. 252 00:13:26,840 --> 00:13:30,120 Before we go to the game, one question: 253 00:13:30,120 --> 00:13:34,360 can FX exchange rate ever be negative? 254 00:13:34,360 --> 00:13:36,369 What do you guys think? 255 00:13:36,369 --> 00:13:38,285 Can the dollar-euro exchange rate be negative? 256 00:13:42,600 --> 00:13:44,980 Any ideas? 257 00:13:44,980 --> 00:13:47,440 No, it's hard, because what does negative mean? 258 00:13:47,440 --> 00:13:52,970 It means I have to pay you money to give you euros. 259 00:13:52,970 --> 00:13:54,251 Why would you? 260 00:13:54,251 --> 00:13:56,000 You have to pay me money to give me euros. 261 00:13:56,000 --> 00:13:57,070 Nobody would do that. 262 00:13:57,070 --> 00:14:02,430 It can be 0, potentially, if dollars are worthless 263 00:14:02,430 --> 00:14:05,740 or something, but it cannot really be negative. 264 00:14:05,740 --> 00:14:13,150 So that's one reason why I wrote my SDE as a kind 265 00:14:13,150 --> 00:14:14,270 of a log-normal process. 266 00:14:14,270 --> 00:14:18,010 You recognize this by the form dS over S. 267 00:14:18,010 --> 00:14:22,480 So the changes in the FX are proportional to the value 268 00:14:22,480 --> 00:14:22,990 of the FX. 269 00:14:22,990 --> 00:14:27,350 So the process can never become negative. 270 00:14:27,350 --> 00:14:29,710 OK, so it can never become negative, 271 00:14:29,710 --> 00:14:31,010 but how big can it get? 272 00:14:33,880 --> 00:14:36,170 And the answer is, it can get very big. 273 00:14:36,170 --> 00:14:39,360 I mean, we have currencies, notably some currency 274 00:14:39,360 --> 00:14:41,962 like Zimbabwean dollars, that traded-- I don't know, 275 00:14:41,962 --> 00:14:44,420 I mean, I actually don't know what Zimbabwean dollars trade 276 00:14:44,420 --> 00:14:47,170 at, but I think it's somewhere in the billions 277 00:14:47,170 --> 00:14:50,421 of Zimbabwean dollars per dollar, so something. 278 00:14:50,421 --> 00:14:51,670 It's a really extreme example. 279 00:14:51,670 --> 00:14:54,400 It can really get extremely big. 280 00:14:54,400 --> 00:14:57,670 So there is now really upper bound, 281 00:14:57,670 --> 00:14:58,910 while there is a lower bound. 282 00:14:58,910 --> 00:15:01,610 So the distribution, as you can imagine has a skew. 283 00:15:01,610 --> 00:15:05,260 It's not symmetric around the average. 284 00:15:05,260 --> 00:15:07,650 It's limited on the lower side. 285 00:15:07,650 --> 00:15:09,490 It can go very high on the high side. 286 00:15:09,490 --> 00:15:12,405 And log-normal distribution has that property. 287 00:15:12,405 --> 00:15:14,280 Have you guys seen a log-normal distribution? 288 00:15:14,280 --> 00:15:18,490 You've talked about this stuff in the course before, right? 289 00:15:18,490 --> 00:15:21,020 So let's go back to our game. 290 00:15:23,620 --> 00:15:28,120 So our game is, we have assumptions. 291 00:15:28,120 --> 00:15:29,890 My assumptions are not to be realistic, 292 00:15:29,890 --> 00:15:31,280 but to make it simple. 293 00:15:31,280 --> 00:15:34,690 Let's assume that our dollars and euros exchange rate is one, 294 00:15:34,690 --> 00:15:37,050 so we can exchange one Euro for $1. 295 00:15:37,050 --> 00:15:41,840 Clearly not exactly the case, but let's make that assumption. 296 00:15:41,840 --> 00:15:44,120 And we also assume that the FX forwards 297 00:15:44,120 --> 00:15:46,100 is 1, which basically means that the interest 298 00:15:46,100 --> 00:15:50,400 rates in both currencies are the same. 299 00:15:50,400 --> 00:15:54,655 And now let's say I'm going to make you 300 00:15:54,655 --> 00:16:01,715 a bet that-- now dollars and euros is a volatile process. 301 00:16:01,715 --> 00:16:03,340 Right now, it's one, but in the future, 302 00:16:03,340 --> 00:16:05,890 it could be different from one-- could be higher or lower. 303 00:16:05,890 --> 00:16:10,950 So if dollar-euro FX process is more than one in one month, 304 00:16:10,950 --> 00:16:13,520 then you give me money. 305 00:16:13,520 --> 00:16:16,060 And then if it's less than one in one month, 306 00:16:16,060 --> 00:16:17,960 then I give you money. 307 00:16:17,960 --> 00:16:21,345 And we're going to have two payoffs, so two games. 308 00:16:23,855 --> 00:16:25,230 I don't know why it says "bet B." 309 00:16:25,230 --> 00:16:26,100 Should say just "bet." 310 00:16:26,100 --> 00:16:27,017 I'm sorry about that. 311 00:16:29,570 --> 00:16:33,230 That in the payoff A, basically you're 312 00:16:33,230 --> 00:16:36,890 going to give me $100 if I win, and I'm going 313 00:16:36,890 --> 00:16:38,990 to give you $100 if you win. 314 00:16:38,990 --> 00:16:43,040 And in payoff B, you're going to give me 100 euros if I win, 315 00:16:43,040 --> 00:16:48,420 and I'll give you 100 euros if you win. 316 00:16:48,420 --> 00:16:52,410 And the question is, which game would you prefer to play, 317 00:16:52,410 --> 00:16:55,420 or do you not care? 318 00:16:55,420 --> 00:17:01,140 So in each case, you kind of win, lose same number. 319 00:17:03,660 --> 00:17:06,130 So I want to see hands. 320 00:17:06,130 --> 00:17:08,609 Who wants to play game A? 321 00:17:11,450 --> 00:17:14,329 Come on guys, wake up. 322 00:17:14,329 --> 00:17:16,300 Who wants to play game A? 323 00:17:16,300 --> 00:17:18,159 I mean if you don't know, you just-- 324 00:17:18,159 --> 00:17:19,450 lets say you like Euros better. 325 00:17:19,450 --> 00:17:21,740 You can say this is not really graded, so it's OK. 326 00:17:25,790 --> 00:17:27,740 OK, nobody really knows what to play? 327 00:17:27,740 --> 00:17:29,002 Like how about game B? 328 00:17:29,002 --> 00:17:30,210 Anybody wants to play game B? 329 00:17:30,210 --> 00:17:31,790 OK, you guys want to play? 330 00:17:31,790 --> 00:17:37,930 Three people for game-- four-- game A nobody still? 331 00:17:37,930 --> 00:17:39,500 Same person for game A and B? 332 00:17:39,500 --> 00:17:40,910 All right. 333 00:17:40,910 --> 00:17:42,730 OK, two people-- so now-- 334 00:17:42,730 --> 00:17:44,170 AUDIENCE: Behavioral science says 335 00:17:44,170 --> 00:17:48,360 people are reluctant to lose-- more reluctant to lose. 336 00:17:48,360 --> 00:17:50,620 STEFAN ANDREEV: That's true, that is true. 337 00:17:50,620 --> 00:17:53,580 However, so people are reluctant to lose. 338 00:17:53,580 --> 00:17:57,000 And I said, look, the FX forward in one month is 1. 339 00:17:57,000 --> 00:18:00,890 So you can actually-- that's the market price that in one month, 340 00:18:00,890 --> 00:18:01,760 FX forward is 1. 341 00:18:01,760 --> 00:18:05,470 And this-- our bet-- kind of the strike is 1. 342 00:18:05,470 --> 00:18:06,884 So our bet level is 1. 343 00:18:06,884 --> 00:18:08,300 So you can kind of say, well, this 344 00:18:08,300 --> 00:18:11,460 looks like kind of fair game. 345 00:18:11,460 --> 00:18:13,090 So I don't expect to win or lose much, 346 00:18:13,090 --> 00:18:15,406 but I'm just reluctant to do it. 347 00:18:15,406 --> 00:18:16,530 And I can get that feeling. 348 00:18:16,530 --> 00:18:18,840 That's the risk aversion aspect of it. 349 00:18:18,840 --> 00:18:22,730 But if you were forced to make a bet, 350 00:18:22,730 --> 00:18:25,950 question is, which one would you prefer? 351 00:18:25,950 --> 00:18:30,180 So I understand you might not want to play. 352 00:18:30,180 --> 00:18:33,119 But I'll say, OK, so you guys don't 353 00:18:33,119 --> 00:18:34,410 seem to be in the mood to play. 354 00:18:34,410 --> 00:18:35,033 That's fine. 355 00:18:37,880 --> 00:18:42,080 Let's look at some scenarios. 356 00:18:42,080 --> 00:18:47,370 So let's say in one month, dollar-euro goes to 1.25. 357 00:18:47,370 --> 00:18:49,810 In bet A, I lose $100. 358 00:18:49,810 --> 00:18:52,200 In bet B, I lose 100 euros. 359 00:18:52,200 --> 00:18:57,700 So bet A-- actually, you lose 100 euros, 360 00:18:57,700 --> 00:19:04,260 not-- so bet A for you, you are $25 better than bet B. 361 00:19:04,260 --> 00:19:08,210 And in the second case, if dollar-euro is 0.75, 362 00:19:08,210 --> 00:19:14,830 you make $100, or you make 100 euros in bet B. In that case, 363 00:19:14,830 --> 00:19:19,610 you also-- bet A is $25 better. 364 00:19:19,610 --> 00:19:23,180 So it doesn't matter what happens. 365 00:19:23,180 --> 00:19:25,600 Bet A seems to be the better case. 366 00:19:25,600 --> 00:19:30,595 So if you're, like, our dear professor here, then you 367 00:19:30,595 --> 00:19:32,220 don't like to lose, then you probably 368 00:19:32,220 --> 00:19:34,455 are going to choose bet A, I assume, right? 369 00:19:34,455 --> 00:19:35,330 That's a better deal. 370 00:19:37,742 --> 00:19:39,200 And that's kind of strange, though. 371 00:19:39,200 --> 00:19:42,370 I mean, like both payouts were symmetric-- 372 00:19:42,370 --> 00:19:46,250 so it's 100 euros, 100 euros, $100, $100. 373 00:19:46,250 --> 00:19:49,100 Why is it one is better than the other? 374 00:19:49,100 --> 00:19:55,640 Well it's like what really happens 375 00:19:55,640 --> 00:19:59,880 is the units of the bet-- the value of those units 376 00:19:59,880 --> 00:20:02,020 depend on whether you win or lose. 377 00:20:02,020 --> 00:20:06,050 So it's not like if I was betting using acorns, 378 00:20:06,050 --> 00:20:09,760 then you get two acorns or I get two acorns, 379 00:20:09,760 --> 00:20:12,430 then actually, it might be a fair bet. 380 00:20:12,430 --> 00:20:15,060 But because I'm betting in euros and dollars, and the value 381 00:20:15,060 --> 00:20:19,010 of these things-- the relative value changes based 382 00:20:19,010 --> 00:20:21,920 on the actual whether you win or lose, 383 00:20:21,920 --> 00:20:26,960 then the game is not symmetric any more. 384 00:20:26,960 --> 00:20:31,280 So the reason I wanted to take you through this game 385 00:20:31,280 --> 00:20:33,980 is because there are a lot of cases in finance 386 00:20:33,980 --> 00:20:36,610 where people make bets. 387 00:20:36,610 --> 00:20:39,270 But then the value of what you get 388 00:20:39,270 --> 00:20:41,830 depends on whether you win or you lose. 389 00:20:41,830 --> 00:20:47,720 And that has an effect on the value of the bet. 390 00:20:47,720 --> 00:20:50,430 And in particular, the case we're going to talk about today 391 00:20:50,430 --> 00:20:54,770 is one of these cases, which is the credit FX. 392 00:20:54,770 --> 00:20:58,045 That's why we need to credit-FX quanto models. 393 00:20:58,045 --> 00:21:01,250 To give you an illustration from finance, 394 00:21:01,250 --> 00:21:02,510 let's take Italy bonds. 395 00:21:02,510 --> 00:21:06,639 So Italy issues bonds both in dollars and in euros. 396 00:21:06,639 --> 00:21:08,180 Why does it issue in both currencies? 397 00:21:08,180 --> 00:21:10,460 Because Italy has to issue a lot of bonds. 398 00:21:10,460 --> 00:21:15,270 And they need to find as many investors as they can. 399 00:21:15,270 --> 00:21:17,220 And some investors want to buy euro bonds, 400 00:21:17,220 --> 00:21:19,053 and some investors want to buy dollar bonds. 401 00:21:19,053 --> 00:21:22,330 And they want to access both bases of investors. 402 00:21:22,330 --> 00:21:25,810 Now, these bonds they cross-default, 403 00:21:25,810 --> 00:21:29,900 meaning if Italy defaults on one bond, all of its bonds 404 00:21:29,900 --> 00:21:34,880 default together, including the euros and the dollar bonds. 405 00:21:34,880 --> 00:21:42,050 So then there is a notion of credit spread, 406 00:21:42,050 --> 00:21:44,520 which is the measure of how risky Italy is. 407 00:21:44,520 --> 00:21:46,260 So you can take euro bonds, and you 408 00:21:46,260 --> 00:21:47,760 can say, well, how much premium does 409 00:21:47,760 --> 00:21:50,145 Italy pay over German bonds? 410 00:21:50,145 --> 00:21:52,020 Let's assume that German bonds are risk-free, 411 00:21:52,020 --> 00:21:54,370 which is the standard assumption for euros, 412 00:21:54,370 --> 00:22:00,270 that the German bonds-- Germany is the main underlying 413 00:22:00,270 --> 00:22:02,070 economic force for the Euro. 414 00:22:02,070 --> 00:22:04,770 They're kind of risk-free bonds. 415 00:22:04,770 --> 00:22:07,600 And Italy pays a certain spread over euros. 416 00:22:07,600 --> 00:22:12,160 Same thing for dollars-- it pays a certain spread over USA. 417 00:22:12,160 --> 00:22:13,650 So if Italy wants to borrow money, 418 00:22:13,650 --> 00:22:15,220 they have to pay a higher interest rate, 419 00:22:15,220 --> 00:22:17,553 just like if you want to borrow money for student loans, 420 00:22:17,553 --> 00:22:21,350 you have to pay higher interest rate than the Fed. 421 00:22:21,350 --> 00:22:24,730 And the size of that spread is in the market. 422 00:22:24,730 --> 00:22:29,240 It determines how risky of a borrower you are. 423 00:22:29,240 --> 00:22:32,450 Well, it turns out that these spreads are not 424 00:22:32,450 --> 00:22:34,130 the same in both currencies. 425 00:22:34,130 --> 00:22:37,981 One currency has a higher spread than other currencies. 426 00:22:37,981 --> 00:22:39,480 That's kind of an interesting thing. 427 00:22:39,480 --> 00:22:42,510 So there is two questions really: 428 00:22:42,510 --> 00:22:44,310 when the spreads are not the same, 429 00:22:44,310 --> 00:22:47,880 which currency would Italy prefer to issue bonds in? 430 00:22:47,880 --> 00:22:51,047 And which currency do investors prefer to buy bonds in? 431 00:22:54,530 --> 00:22:59,170 So this is kind of similar to the previous game we played, 432 00:22:59,170 --> 00:23:03,330 because if you're an investor trying to buy bonds, well, 433 00:23:03,330 --> 00:23:06,950 if Italy defaults, then chances are 434 00:23:06,950 --> 00:23:09,830 the euro is not doing so well. 435 00:23:09,830 --> 00:23:12,030 So you would lose money. 436 00:23:12,030 --> 00:23:14,100 And if you have euro bonds, you would lose euros. 437 00:23:14,100 --> 00:23:16,940 If you have dollar bonds, you lose dollars. 438 00:23:16,940 --> 00:23:20,430 On the other hand, if Italy does not default and pays you back, 439 00:23:20,430 --> 00:23:23,890 then chances are, euros are not doing that bad. 440 00:23:23,890 --> 00:23:26,910 So you would actually be making euros and dollars. 441 00:23:26,910 --> 00:23:29,810 So it's an interesting-- it's kind 442 00:23:29,810 --> 00:23:32,090 of a similar dynamic going on. 443 00:23:35,410 --> 00:23:36,910 So there's the same kind of question 444 00:23:36,910 --> 00:23:40,450 that I asked before-- USD, euros, are equal in both. 445 00:23:40,450 --> 00:23:41,880 So what do you think now? 446 00:23:41,880 --> 00:23:44,120 Now that we've gone through an example, maybe 447 00:23:44,120 --> 00:23:47,890 we'll have a higher participation in my pop quiz. 448 00:23:47,890 --> 00:23:54,290 Who thinks that USD bonds have a higher credit spread, 449 00:23:54,290 --> 00:24:02,830 and who thinks-- so A. Vote for A. One, two. 450 00:24:02,830 --> 00:24:05,180 So who thinks that euro bonds will have a higher credit 451 00:24:05,180 --> 00:24:07,880 spread? 452 00:24:07,880 --> 00:24:14,850 OK, one, all right-- so two to one. 453 00:24:14,850 --> 00:24:17,240 I think the two to one wins. 454 00:24:17,240 --> 00:24:23,237 All right, I must say, you guys seem that-- maybe it's 455 00:24:23,237 --> 00:24:24,445 the format of the auditorium. 456 00:24:24,445 --> 00:24:26,460 People don't like to raise their hands too much, 457 00:24:26,460 --> 00:24:28,744 or maybe they're afraid that they're being filmed. 458 00:24:28,744 --> 00:24:30,106 [LAUGHS] 459 00:24:31,470 --> 00:24:33,710 OK. 460 00:24:33,710 --> 00:24:40,760 Well, how are we going to do this? 461 00:24:40,760 --> 00:24:43,332 How are we going to answer this question? 462 00:24:43,332 --> 00:24:44,790 Before I give you the answer, we're 463 00:24:44,790 --> 00:24:45,956 going to go through a slide. 464 00:24:45,956 --> 00:24:49,430 Well, first we're going to say, well, FX rates are volatile. 465 00:24:49,430 --> 00:24:51,600 There is volatility, as we said before. 466 00:24:51,600 --> 00:24:56,260 So now we're going to-- in order to compare euro bonds to dollar 467 00:24:56,260 --> 00:24:59,440 bonds, we need to really come up with a strategy 468 00:24:59,440 --> 00:25:03,080 to replicate one with the other, and then look at the price-- 469 00:25:03,080 --> 00:25:07,700 look at how much do we need to buy one to replicate the other. 470 00:25:07,700 --> 00:25:11,020 If we're able to come up with such a replication strategy, 471 00:25:11,020 --> 00:25:12,640 then we can immediately say, hey, 472 00:25:12,640 --> 00:25:20,160 if you need 150 euro bonds to replicate 100 dollar bonds, 473 00:25:20,160 --> 00:25:23,330 then that means that the euro bonds have to be cheaper. 474 00:25:23,330 --> 00:25:26,740 That's basically the replication argument. 475 00:25:26,740 --> 00:25:31,320 So you can try to do that by piecing together bonds, 476 00:25:31,320 --> 00:25:33,847 or we can use the powerful tools of mathematical finance 477 00:25:33,847 --> 00:25:35,430 that you've been learning about, which 478 00:25:35,430 --> 00:25:38,010 is all about replication and pricing. 479 00:25:38,010 --> 00:25:39,910 And the three steps are: we're going 480 00:25:39,910 --> 00:25:42,210 to analyze the payoffs of the instruments, 481 00:25:42,210 --> 00:25:45,000 and we're going to write some model, 482 00:25:45,000 --> 00:25:47,440 a model for FX and for credit, and we're 483 00:25:47,440 --> 00:25:49,047 going to price those bonds. 484 00:25:49,047 --> 00:25:50,880 And then we're going to look at the results, 485 00:25:50,880 --> 00:25:54,220 and try to understand the problem intuitively. 486 00:25:54,220 --> 00:25:58,040 And that's basically what we do, pretty much. 487 00:25:58,040 --> 00:26:03,260 That's what option quants do on Wall Street all the time. 488 00:26:03,260 --> 00:26:07,510 So here's the answer: dollar versus euro 489 00:26:07,510 --> 00:26:09,410 spreads from the marketplace. 490 00:26:09,410 --> 00:26:12,730 So usually what happens in these kind of questions in finances, 491 00:26:12,730 --> 00:26:14,920 you kind of have an answer, and then you 492 00:26:14,920 --> 00:26:17,330 try to compute a model that explains the difference. 493 00:26:17,330 --> 00:26:21,190 So that's what we're going to do now. 494 00:26:21,190 --> 00:26:27,720 Well, the USD spreads are actually lower-- USD bond 495 00:26:27,720 --> 00:26:28,980 spreads are actually lower. 496 00:26:39,650 --> 00:26:47,170 Now, so there is really-- when we're 497 00:26:47,170 --> 00:26:51,790 talking about bonds, risky bonds, there's two states. 498 00:26:51,790 --> 00:26:57,130 They are either performing, or they are non-performing 499 00:26:57,130 --> 00:27:01,300 and in default. And we're going to go here 500 00:27:01,300 --> 00:27:05,440 through an example of two bonds. 501 00:27:05,440 --> 00:27:09,514 And we're going to use two zero-coupon bonds, which 502 00:27:09,514 --> 00:27:10,805 essentially have zero recovery. 503 00:27:13,490 --> 00:27:19,150 And the idea there is really to make the question simple 504 00:27:19,150 --> 00:27:21,490 so we can analyze it better. 505 00:27:21,490 --> 00:27:23,330 But you don't lose a lot of generality 506 00:27:23,330 --> 00:27:26,800 by saying zero-coupon versus coupon. 507 00:27:26,800 --> 00:27:28,880 It's not-- the answer, the intuition 508 00:27:28,880 --> 00:27:30,770 would be exactly the same. 509 00:27:30,770 --> 00:27:32,980 So let's say we have two zero-coupon bonds, 510 00:27:32,980 --> 00:27:35,640 same maturity, they pay 100 on maturity. 511 00:27:35,640 --> 00:27:39,420 And by the way, bonds-- I don't know how much you guys have-- I 512 00:27:39,420 --> 00:27:40,240 say these things. 513 00:27:40,240 --> 00:27:41,540 I'm very familiar with them. 514 00:27:41,540 --> 00:27:42,956 Bonds are nothing more than loans. 515 00:27:42,956 --> 00:27:47,460 So zero-coupon bond means I give you some amount of money, 516 00:27:47,460 --> 00:27:54,220 and at some pre-agreed maturity, you're going to pay me 100. 517 00:27:54,220 --> 00:27:56,880 So let's say I give you 80 cents, one year from now, 518 00:27:56,880 --> 00:27:58,880 you pay me 100. 519 00:27:58,880 --> 00:28:00,900 And I call this a zero-coupon bond, 520 00:28:00,900 --> 00:28:03,090 because you don't pay me any intervening coupons. 521 00:28:03,090 --> 00:28:05,260 There's no interest payments, but just I 522 00:28:05,260 --> 00:28:08,080 pay you less money now, and you pay me more at maturity. 523 00:28:10,820 --> 00:28:15,380 OK, so we know that bond U pays $100. 524 00:28:15,380 --> 00:28:17,420 Bond E pays 100 euros. 525 00:28:17,420 --> 00:28:21,380 And let's say we denote the prices-- price of U is Pu, 526 00:28:21,380 --> 00:28:23,490 price of E is Pu. 527 00:28:23,490 --> 00:28:27,370 Our spot FX rate, we're are going to call it St; 528 00:28:27,370 --> 00:28:30,960 our FX forward, Ft. 529 00:28:30,960 --> 00:28:36,070 Now we're going to have kind of a simple arbitrage strategy. 530 00:28:36,070 --> 00:28:42,620 Well, let's say if we can sell 100 times Ft dollar bonds 531 00:28:42,620 --> 00:28:48,630 and with the proceeds, buy 100-- we're 532 00:28:48,630 --> 00:28:52,160 going to get-- if we sell 1,000 dollar bonds, 533 00:28:52,160 --> 00:28:54,210 we're going to get this much proceeds. 534 00:28:54,210 --> 00:28:56,610 There's the price. 535 00:28:56,610 --> 00:29:04,860 And if we buy 1,000 euro bonds-- so we 536 00:29:04,860 --> 00:29:07,780 can enter into an FX forward contract 537 00:29:07,780 --> 00:29:14,060 for 100,000 euros for maturity T at zero cost. 538 00:29:14,060 --> 00:29:19,380 All right, so let's see how this strategy actually pays out. 539 00:29:19,380 --> 00:29:32,230 Well, what happens is you-- there's 100,000 euros. 540 00:29:32,230 --> 00:29:35,480 You get 100,000 euros for selling the euro bonds. 541 00:29:35,480 --> 00:29:39,300 You pay 100,000 times Ft dollars, say dollar bonds. 542 00:29:39,300 --> 00:29:44,950 There is an FX forward contract, and at maturity, you 543 00:29:44,950 --> 00:29:50,222 can exchange this $100,000 for 100,000 euros using this FX 544 00:29:50,222 --> 00:29:50,930 forward contract. 545 00:29:50,930 --> 00:29:52,900 You already agreed to do that. 546 00:29:52,900 --> 00:29:55,530 So your FX forward actually exactly 547 00:29:55,530 --> 00:29:59,600 hedges-- you can basically use the proceeds of these bonds 548 00:29:59,600 --> 00:30:01,850 to-- you can exchange the proceeds 549 00:30:01,850 --> 00:30:04,450 at zero cost at maturity, because you have entered 550 00:30:04,450 --> 00:30:06,470 into the FX forward contract. 551 00:30:06,470 --> 00:30:09,660 So your net payoff is 0. 552 00:30:09,660 --> 00:30:11,500 So that means that the prices of these bonds 553 00:30:11,500 --> 00:30:14,070 have to be the same. 554 00:30:14,070 --> 00:30:15,840 But what if they're not? 555 00:30:15,840 --> 00:30:19,110 What if Ft, which is in this case 556 00:30:19,110 --> 00:30:22,120 is 1, forward contract-- what if the price in dollars 557 00:30:22,120 --> 00:30:27,020 is different from exchange rate times the price in euros? 558 00:30:27,020 --> 00:30:31,800 Well, in that case, you can say, well, there is an arbitrage. 559 00:30:31,800 --> 00:30:36,170 And you'll be right, if you would 560 00:30:36,170 --> 00:30:41,030 be able to make money if, in fact, the bonds performed. 561 00:30:41,030 --> 00:30:44,780 But what happens if there is a default? 562 00:30:44,780 --> 00:30:51,139 If there's default, that wouldn't really necessarily be 563 00:30:51,139 --> 00:30:52,680 the case, because if there's default, 564 00:30:52,680 --> 00:30:54,513 these bonds don't pay anything, and you just 565 00:30:54,513 --> 00:30:56,559 have an FX forward contract. 566 00:30:56,559 --> 00:30:58,100 And this FX forward contract is going 567 00:30:58,100 --> 00:31:00,330 to be worth something after default, 568 00:31:00,330 --> 00:31:04,440 especially if the FX rate depends, 569 00:31:04,440 --> 00:31:10,099 like jumps, upon default. So arbitrage, again, 570 00:31:10,099 --> 00:31:11,390 is-- so you start with 0 money. 571 00:31:11,390 --> 00:31:14,620 You make money if there's nonzero probability. 572 00:31:14,620 --> 00:31:20,490 And let's say in this particular case, 573 00:31:20,490 --> 00:31:24,030 the strategy-- payoff in case of default with 25% recovery rate 574 00:31:24,030 --> 00:31:26,760 is-- you actually have only-- 25% 575 00:31:26,760 --> 00:31:29,440 means you only have a quarter of the payoff now at maturity, 576 00:31:29,440 --> 00:31:30,580 if default occurs. 577 00:31:30,580 --> 00:31:33,960 But you have a hedge for the full 100,000. 578 00:31:33,960 --> 00:31:36,430 Your FX forward is for full 100,000. 579 00:31:36,430 --> 00:31:39,560 So for 25,000 of it, you can use the FX forward 580 00:31:39,560 --> 00:31:40,760 to exchange money. 581 00:31:40,760 --> 00:31:43,390 For the remaining 75,000 you just 582 00:31:43,390 --> 00:31:45,210 have an FX forward outright. 583 00:31:45,210 --> 00:31:49,270 So if FX moved against you, you would lose money. 584 00:31:49,270 --> 00:31:53,495 So that's why the strategy is not necessarily an arbitrage. 585 00:31:53,495 --> 00:31:55,870 And that's why the two prices of the dollar and euro bond 586 00:31:55,870 --> 00:31:59,480 are not necessarily related to each other. 587 00:31:59,480 --> 00:32:04,240 They don't have to be equal, because, in fact, there 588 00:32:04,240 --> 00:32:05,520 is a possibility of default. 589 00:32:05,520 --> 00:32:08,070 And you cannot really directly hedge. 590 00:32:08,070 --> 00:32:11,290 You cannot really construct an arbitrage strategy by using FX 591 00:32:11,290 --> 00:32:15,190 forwards and the bonds together so easily. 592 00:32:15,190 --> 00:32:18,690 You have to take into account what happens if default occurs. 593 00:32:23,280 --> 00:32:25,000 OK, so give an example. 594 00:32:25,000 --> 00:32:27,850 What happens upon FX when default occurs? 595 00:32:27,850 --> 00:32:30,970 Well, one of the most recent defaults 596 00:32:30,970 --> 00:32:33,840 of a country-- of a big country that has its own currency-- 597 00:32:33,840 --> 00:32:37,800 is Argentina, 2001. 598 00:32:37,800 --> 00:32:44,400 And when it defaulted, the Argentinian peso skyrocketed. 599 00:32:44,400 --> 00:32:47,180 Here is the graph of the price series. 600 00:32:47,180 --> 00:32:54,790 So if you had an FX forward contract that essentially-- 601 00:32:54,790 --> 00:32:57,170 if you had a position where you were left with a naked FX 602 00:32:57,170 --> 00:32:59,860 forward contract, where you were receiving pesos and paying 603 00:32:59,860 --> 00:33:02,640 dollars in the event of default, you 604 00:33:02,640 --> 00:33:05,350 would have lost a lot of money when the default happened. 605 00:33:05,350 --> 00:33:07,200 It would have really gone against you. 606 00:33:07,200 --> 00:33:10,460 And this is, by the way, this is a massive move. 607 00:33:10,460 --> 00:33:12,080 And the Argentinian peso still is not 608 00:33:12,080 --> 00:33:18,480 recovered from that default. So can we do better? 609 00:33:18,480 --> 00:33:20,720 What do we actually-- what should we be 610 00:33:20,720 --> 00:33:22,040 doing when we're hedging this? 611 00:33:22,040 --> 00:33:24,210 And the answer is, again, we have 612 00:33:24,210 --> 00:33:30,120 to apply mathematical models to really try to come up 613 00:33:30,120 --> 00:33:32,500 with a replication strategy. 614 00:33:32,500 --> 00:33:34,237 So what is the main features of a model 615 00:33:34,237 --> 00:33:35,320 that will help me do this? 616 00:33:35,320 --> 00:33:38,650 Well, first I need to model a credit default, the credit 617 00:33:38,650 --> 00:33:39,970 default event. 618 00:33:39,970 --> 00:33:41,980 I need to have this in my model. 619 00:33:41,980 --> 00:33:43,940 And I need to have something which 620 00:33:43,940 --> 00:33:48,840 says FX has to move upon default. 621 00:33:48,840 --> 00:33:51,990 And then we're going to construct a complete market. 622 00:33:51,990 --> 00:33:55,140 Then we're going to define some simple dynamics on our exchange 623 00:33:55,140 --> 00:33:58,240 rate and on our defaults, and we're 624 00:33:58,240 --> 00:34:01,980 going to try to price for bonds. 625 00:34:01,980 --> 00:34:07,055 So how do we do that? 626 00:34:10,710 --> 00:34:15,830 Well, what we-- generally, again, 627 00:34:15,830 --> 00:34:19,409 how we're going to use the models, 628 00:34:19,409 --> 00:34:23,250 we're going to define an SDE, like I just defined initially 629 00:34:23,250 --> 00:34:25,210 a dS over S of something. 630 00:34:25,210 --> 00:34:28,139 And I'm going to solve this SDE either analytically 631 00:34:28,139 --> 00:34:29,480 or numerically. 632 00:34:29,480 --> 00:34:32,730 And what's important, the way we actually 633 00:34:32,730 --> 00:34:34,290 use these models in trading, we're 634 00:34:34,290 --> 00:34:37,110 going to look at how the price of each instrument 635 00:34:37,110 --> 00:34:39,280 depends on the hedging instrument. 636 00:34:39,280 --> 00:34:41,090 And that is going to define my hedge ratio 637 00:34:41,090 --> 00:34:43,239 or my replicating strategy. 638 00:34:43,239 --> 00:34:45,719 That's really kind of the main part. 639 00:34:45,719 --> 00:34:48,130 It's really hedging and evaluation and pricing 640 00:34:48,130 --> 00:34:50,510 are the same-- right and left hands. 641 00:34:50,510 --> 00:34:52,260 We're really talking about the same thing. 642 00:34:52,260 --> 00:34:54,344 You cannot really price without hedging. 643 00:34:54,344 --> 00:34:56,469 And pricing without hedging is kind of meaningless, 644 00:34:56,469 --> 00:34:57,750 in some sense. 645 00:34:57,750 --> 00:35:00,280 Pricing represents the price of a hedging strategy. 646 00:35:03,160 --> 00:35:08,040 OK, how do we-- basic credit model, how do we model default? 647 00:35:08,040 --> 00:35:11,690 Well the standard model in finance for default 648 00:35:11,690 --> 00:35:14,520 is to define the default events. 649 00:35:14,520 --> 00:35:16,150 And we say well, this default event 650 00:35:16,150 --> 00:35:19,140 arrives as a discrete event. 651 00:35:19,140 --> 00:35:23,860 And it arrives at the time tau, which is a random time. 652 00:35:23,860 --> 00:35:27,230 And we're going to model the tau, the time, 653 00:35:27,230 --> 00:35:31,310 as a Poisson process, which means that we don't know when 654 00:35:31,310 --> 00:35:33,300 it's going to come, but we know something 655 00:35:33,300 --> 00:35:36,930 about the probabilities of when it's coming. 656 00:35:36,930 --> 00:35:39,900 And the Poisson process has an intensity. 657 00:35:39,900 --> 00:35:42,450 The intensity in this case is h. 658 00:35:42,450 --> 00:35:45,310 And basically, the meaning of intensity 659 00:35:45,310 --> 00:35:49,980 means the probability of the default time not the arriving 660 00:35:49,980 --> 00:35:56,480 by time capital T is e to the minus h times capital 661 00:35:56,480 --> 00:35:57,950 T minus little t. 662 00:35:57,950 --> 00:35:59,860 Little t means now. 663 00:35:59,860 --> 00:36:01,550 Let's say we're saying at time t, 664 00:36:01,550 --> 00:36:03,477 we know the default has not arrived. 665 00:36:03,477 --> 00:36:05,310 Here is the probability the default will not 666 00:36:05,310 --> 00:36:08,426 arrive by some time t later. 667 00:36:08,426 --> 00:36:10,800 So in our model, we're going to make a simple assumption. 668 00:36:10,800 --> 00:36:15,320 Let's say constant hazard rate, and we can, 669 00:36:15,320 --> 00:36:18,420 since we know the probability of the default time not arriving 670 00:36:18,420 --> 00:36:21,020 after a certain time capital T-- that's like a cumulative 671 00:36:21,020 --> 00:36:21,990 distribution-- 672 00:36:21,990 --> 00:36:25,350 we can also find the probability density the default 673 00:36:25,350 --> 00:36:30,060 time happens at some time capital T, 674 00:36:30,060 --> 00:36:35,940 or around some time epsilon around capital T. 675 00:36:35,940 --> 00:36:38,950 It's just the derivative of the cumulative distribution. 676 00:36:41,660 --> 00:36:45,320 And corollary is that the probability density 677 00:36:45,320 --> 00:36:47,340 of the default at any given time is 678 00:36:47,340 --> 00:36:53,190 h, which is essentially the limit of capital T going 679 00:36:53,190 --> 00:36:53,910 to little t. 680 00:36:57,130 --> 00:37:01,670 So now in our model, what happens to FX rate? 681 00:37:01,670 --> 00:37:06,010 Well FX rate is going to be denoted by S. And FX rate right 682 00:37:06,010 --> 00:37:11,070 after default would be equal to FX rate before default times 683 00:37:11,070 --> 00:37:15,600 e to the power J. And J essentially 684 00:37:15,600 --> 00:37:17,650 is our kind of percent devaluation, 685 00:37:17,650 --> 00:37:19,070 you can think of it. 686 00:37:19,070 --> 00:37:23,430 So it's kind of like a percent devaluation. 687 00:37:23,430 --> 00:37:27,830 So J can go from minus infinity to infinity. 688 00:37:27,830 --> 00:37:32,300 If J is 0, then that means is there's no devaluation. 689 00:37:32,300 --> 00:37:38,980 So you can see the log of St basically jumps by J. 690 00:37:38,980 --> 00:37:43,660 So in a log-normal process, the log of St is normal, 691 00:37:43,660 --> 00:37:46,660 and essentially, it's just a shift 692 00:37:46,660 --> 00:37:49,080 of the normal distribution. 693 00:37:49,080 --> 00:37:52,550 OK, so how do we describe this? 694 00:37:52,550 --> 00:37:55,680 We define a jump from default Poisson process with intensity 695 00:37:55,680 --> 00:37:59,590 h, as on the board. 696 00:37:59,590 --> 00:38:04,370 And our FX dynamics-- and I apologize for the small 697 00:38:04,370 --> 00:38:13,880 script-- is that our d log of S will have some drift, mu_t dt, 698 00:38:13,880 --> 00:38:17,712 and then a jump process J*dN. 699 00:38:17,712 --> 00:38:20,170 So this is slightly different from what you've seen so far. 700 00:38:20,170 --> 00:38:23,440 So far you've seen Brownian motions. 701 00:38:23,440 --> 00:38:25,530 This is J*dN. 702 00:38:25,530 --> 00:38:27,730 This is now a jump process. 703 00:38:27,730 --> 00:38:35,060 Now what we want, again, we want still 704 00:38:35,060 --> 00:38:37,850 our standard no-arbitrage condition to remain constant. 705 00:38:37,850 --> 00:38:40,260 And from before, we had a condition 706 00:38:40,260 --> 00:38:49,380 that expected value of S of T has to be S of 0 times 707 00:38:49,380 --> 00:39:04,880 e to the rf minus rd times T. So that still has to be the case. 708 00:39:04,880 --> 00:39:07,220 And in our case, we're going to assume 709 00:39:07,220 --> 00:39:15,700 that rf and rd are both 0. 710 00:39:15,700 --> 00:39:19,954 So in our case, we're going to ask-- basically 711 00:39:19,954 --> 00:39:21,870 zero interest rate environment to make, again, 712 00:39:21,870 --> 00:39:23,330 the model simple. 713 00:39:23,330 --> 00:39:27,590 Then we just want the expected value of S_T to be S_0. 714 00:39:27,590 --> 00:39:29,302 So how do we achieve that? 715 00:39:29,302 --> 00:39:30,760 Well, we need to show, essentially, 716 00:39:30,760 --> 00:39:36,850 that this mu, the drift, has to equal to this expression 717 00:39:36,850 --> 00:39:41,060 here, h times 1 e to the J. That's 718 00:39:41,060 --> 00:39:42,380 known as the compensator term. 719 00:39:42,380 --> 00:39:44,630 And you can think-- you can imagine this as a formula. 720 00:39:44,630 --> 00:39:46,610 Like, if I have a Poisson process that 721 00:39:46,610 --> 00:39:49,390 has a possibility of jumping up, then 722 00:39:49,390 --> 00:39:50,800 in order for that Poisson process 723 00:39:50,800 --> 00:39:54,600 to be on average to be equal to the initial value, 724 00:39:54,600 --> 00:39:57,720 it has to be kind of trending down most of the time. 725 00:39:57,720 --> 00:40:01,830 And then, so that when the possibility of jump is there, 726 00:40:01,830 --> 00:40:04,710 the average of the two can be 0. 727 00:40:04,710 --> 00:40:10,260 So that's known as a compensator term of the Poisson process. 728 00:40:10,260 --> 00:40:14,450 OK, so we can go through and derive 729 00:40:14,450 --> 00:40:18,830 how do we get-- what we want to do 730 00:40:18,830 --> 00:40:24,260 is, we're going to check that this form actually does indeed 731 00:40:24,260 --> 00:40:27,980 give you that expectation, does satisfy the condition 732 00:40:27,980 --> 00:40:29,970 for the expectation. 733 00:40:29,970 --> 00:40:39,030 OK, so again, we start with dS_t is mu dt. 734 00:40:39,030 --> 00:40:51,180 So in our case, it's going to be h times 1 minus e^j times 735 00:40:51,180 --> 00:40:58,465 the indicator function of tau bigger than T, plus J dN_t. 736 00:41:02,210 --> 00:41:14,860 OK, so we're-- not dS_t, sorry. 737 00:41:14,860 --> 00:41:18,030 This is d log of S_t. 738 00:41:18,030 --> 00:41:21,790 OK, so now what do we want to do? 739 00:41:21,790 --> 00:41:24,860 We want to integrate this equation. 740 00:41:24,860 --> 00:41:29,040 So essentially, what we're going to do is write integral from 0 741 00:41:29,040 --> 00:41:32,295 to capital T of d of log S_t. 742 00:41:32,295 --> 00:41:35,120 We integrate both sides-- integral 743 00:41:35,120 --> 00:41:48,340 from 0 to capital T, h 1 minus e to the J, tau is bigger than t, 744 00:41:48,340 --> 00:41:55,770 times-- here has a dt in here-- times dt plus integral from 0 745 00:41:55,770 --> 00:41:57,950 to t of J dN_t. 746 00:42:00,920 --> 00:42:09,080 OK, so then this I just gives me essentially the log 747 00:42:09,080 --> 00:42:13,470 of S_T over S_0. 748 00:42:13,470 --> 00:42:17,270 This is just basic calculus. 749 00:42:17,270 --> 00:42:26,510 And then here we have-- we can-- this indicator function 750 00:42:26,510 --> 00:42:30,880 just says if tau is bigger than t, it's 1. 751 00:42:30,880 --> 00:42:33,820 If tau is less than t, then it's 0. 752 00:42:33,820 --> 00:42:36,940 That's basically what it is. 753 00:42:36,940 --> 00:42:41,370 So I know that essentially, this is only 1 754 00:42:41,370 --> 00:42:42,500 when t is less than tau. 755 00:42:42,500 --> 00:42:45,660 So my integral goes from 0 to tau now. 756 00:42:45,660 --> 00:42:49,500 I can replace this from an integral from 0 to tau. 757 00:42:49,500 --> 00:42:51,580 And I can take out the indicator function now. 758 00:42:55,590 --> 00:43:00,670 Of h, 1 minus e to the J, dt. 759 00:43:00,670 --> 00:43:13,300 And then I can say, well, what if tau is bigger 760 00:43:13,300 --> 00:43:27,660 than-- there is also a possibility here that tau is-- 761 00:43:27,660 --> 00:43:31,660 this is tau is less than capital T. 762 00:43:31,660 --> 00:43:33,400 And there's also a possibility that tau 763 00:43:33,400 --> 00:43:36,685 is greater than capital T. In which case, 764 00:43:36,685 --> 00:43:38,060 if tau is greater than capital T, 765 00:43:38,060 --> 00:43:42,270 this integral is there without any indicator functions. 766 00:43:42,270 --> 00:43:47,710 So again, integral from 0 to capital T of h 1 minus 767 00:43:47,710 --> 00:43:53,410 e to the J dt times indicator function, 768 00:43:53,410 --> 00:43:55,650 tau being greater than capital T. 769 00:43:55,650 --> 00:44:02,850 So I kind of divided this, counting both possibilities 770 00:44:02,850 --> 00:44:04,920 separately, essentially. 771 00:44:04,920 --> 00:44:08,680 And now the second part, integral from 0 772 00:44:08,680 --> 00:44:11,050 to capital T of J dN_t. 773 00:44:11,050 --> 00:44:15,690 Now, N is-- what was N? 774 00:44:15,690 --> 00:44:24,520 Well, N of t is essentially-- it starts out 775 00:44:24,520 --> 00:44:32,870 as 0 for t less than tau, and then becomes 1 for t 776 00:44:32,870 --> 00:44:33,630 bigger than tau. 777 00:44:36,640 --> 00:44:43,570 So this integral is just-- J is a constant, 778 00:44:43,570 --> 00:44:45,730 so it's just J times N of t. 779 00:44:48,710 --> 00:44:50,220 And this is capital T here. 780 00:44:50,220 --> 00:44:54,340 And by the way, all these derivations are posted 781 00:44:54,340 --> 00:44:57,970 on the notes, so you don't necessarily have to worry 782 00:44:57,970 --> 00:45:05,000 if you can't-- can I can move this board up? 783 00:45:05,000 --> 00:45:05,634 Not really. 784 00:45:14,670 --> 00:45:17,580 So I'm going to do one more line. 785 00:45:17,580 --> 00:45:19,020 I'm going to erase this top line. 786 00:45:25,960 --> 00:45:29,590 So we get to here, and there's one more step, which is now 787 00:45:29,590 --> 00:45:32,750 to actually do the integration. 788 00:45:32,750 --> 00:45:38,290 We're going to have log of S_T over S_0. 789 00:45:38,290 --> 00:45:44,940 Well, two things-- now, if tau is less than T-- 790 00:45:44,940 --> 00:45:49,950 so default happened before capital T-- then what is N_t? 791 00:45:49,950 --> 00:45:52,910 N_t is going to be 1. 792 00:45:52,910 --> 00:46:03,520 So I can say this equals to h tau times 1 minus e to the J. 793 00:46:03,520 --> 00:46:11,460 This is the first-- this integral now-- plus J. 794 00:46:11,460 --> 00:46:16,410 So this is if tau is less than T. 795 00:46:16,410 --> 00:46:21,670 And then if tau is bigger than T, then this term is 0. 796 00:46:21,670 --> 00:46:24,890 This is a term that's for tau bigger than T. 797 00:46:24,890 --> 00:46:29,250 This is just a constant, so it just becomes h times capital T, 798 00:46:29,250 --> 00:46:35,090 1 minus e to the J times indicator function of tau 799 00:46:35,090 --> 00:46:39,430 bigger than or equal to T. 800 00:46:39,430 --> 00:46:43,220 And we can then exponentiate both sides. 801 00:46:43,220 --> 00:46:48,830 And it becomes-- use the magic of the blackboard. 802 00:46:48,830 --> 00:46:49,980 You can erase. 803 00:46:49,980 --> 00:46:54,480 S_T equals S_0 times the exponential of this. 804 00:46:58,050 --> 00:47:06,160 So I have-- this is what my exchange rate is going to be, 805 00:47:06,160 --> 00:47:11,140 essentially, at time capital T. Now, what was I trying to do? 806 00:47:11,140 --> 00:47:15,860 I was trying to do this-- to compute this expectation. 807 00:47:15,860 --> 00:47:17,360 With the computed expectation, now I 808 00:47:17,360 --> 00:47:19,650 have to integrate over the probability 809 00:47:19,650 --> 00:47:21,880 distribution of tau. 810 00:47:21,880 --> 00:47:25,200 Now remember, probability distribution of tau 811 00:47:25,200 --> 00:47:27,930 is a Poisson process. 812 00:47:27,930 --> 00:47:38,940 So we have essentially-- I'll write it here-- phi of 0, 813 00:47:38,940 --> 00:47:45,520 t is just h times e to the minus ht. 814 00:47:45,520 --> 00:47:49,290 That's kind of the probability density of tau. 815 00:47:49,290 --> 00:47:52,780 So now what I need to do is essentially, 816 00:47:52,780 --> 00:48:00,480 the expectation of S_T is just the integral from 0 to infinity 817 00:48:00,480 --> 00:48:06,990 of S of tau, times phi(0, tau) d tau. 818 00:48:06,990 --> 00:48:11,085 So here is my S of T. You can think of this S of T 819 00:48:11,085 --> 00:48:17,370 of tau for time tau. 820 00:48:17,370 --> 00:48:21,280 So this is for a given time tau, I know what my value of S of T 821 00:48:21,280 --> 00:48:22,110 is. 822 00:48:22,110 --> 00:48:24,080 So I can do this integral. 823 00:48:24,080 --> 00:48:27,270 And now we're going to do it. 824 00:48:27,270 --> 00:48:29,680 So what is going to be the first term? 825 00:48:29,680 --> 00:48:38,230 So exponential of S_T-- not exponential, expectation of S_T 826 00:48:38,230 --> 00:48:45,600 is going to be-- it's going to be 827 00:48:45,600 --> 00:48:51,010 integral from 0 to capital T. It's going to have two terms. 828 00:48:51,010 --> 00:48:54,740 First, I'm going to integrate from 0 to capital T. 829 00:48:54,740 --> 00:48:58,880 And then I'm going to integrate from capital T to infinity. 830 00:48:58,880 --> 00:49:01,670 I'll split this integral into two parts. 831 00:49:01,670 --> 00:49:06,990 And from 0 to capital T, I have essentially-- 832 00:49:06,990 --> 00:49:12,870 h times e to the minus h*tau. 833 00:49:12,870 --> 00:49:15,910 And this is my density function. 834 00:49:15,910 --> 00:49:18,250 And then I'm going to plug that in here. 835 00:49:18,250 --> 00:49:20,760 So this is for tau being less than T, 836 00:49:20,760 --> 00:49:25,430 so it's basically this first term. 837 00:49:25,430 --> 00:49:29,630 I'm going to divide it by 0 here, to make it easy. 838 00:49:29,630 --> 00:49:35,580 So first term is going to be e to the h*tau times 1 minus e 839 00:49:35,580 --> 00:49:50,250 to the J plus J. OK, and so this is d tau. 840 00:49:50,250 --> 00:49:52,760 So this is the first part from 0 to T. 841 00:49:52,760 --> 00:49:56,190 And the second part is essentially the integral 842 00:49:56,190 --> 00:50:02,270 from capital T to infinity for tau 843 00:50:02,270 --> 00:50:06,140 being bigger than capital T. Now that's actually-- this part 844 00:50:06,140 --> 00:50:08,270 here does not depend on tau. 845 00:50:08,270 --> 00:50:09,100 It's a constant. 846 00:50:09,100 --> 00:50:13,110 So it would be just h, capital T, 1 minus e 847 00:50:13,110 --> 00:50:19,000 to the J times-- what's the probability of tau being bigger 848 00:50:19,000 --> 00:50:20,610 than capital T? 849 00:50:20,610 --> 00:50:23,380 That's just the cumulative probability distribution 850 00:50:23,380 --> 00:50:28,430 we saw before, just e to the minus hT. 851 00:50:28,430 --> 00:50:32,890 That's the probability that tau is bigger than T. 852 00:50:32,890 --> 00:50:42,820 OK so it's e to the hT, 1 minus e to the J times 853 00:50:42,820 --> 00:50:44,820 e to the minus hT. 854 00:50:44,820 --> 00:50:50,620 So I can now simplify this expression somewhat. 855 00:50:50,620 --> 00:50:55,460 You can see that, say, this term and this term, this term 856 00:50:55,460 --> 00:50:57,880 and this term go away. 857 00:50:57,880 --> 00:51:03,110 And also this term and this go away. 858 00:51:03,110 --> 00:51:16,180 So I'm left with the integral from 0 to T of, essentially, 859 00:51:16,180 --> 00:51:31,410 h times e to minus h*tau e to the J times e to the J. 860 00:51:31,410 --> 00:51:36,720 So you can think of this as h times e to the J times e 861 00:51:36,720 --> 00:51:40,430 to the minus h*tau e to the J d tau, 862 00:51:40,430 --> 00:51:51,440 plus e to the minus h capital T times e to the J. So this is-- 863 00:51:51,440 --> 00:51:54,710 if I think if h e to the J as this is the constant in front 864 00:51:54,710 --> 00:52:01,450 of tau, this is just a standard integral of exponential, 865 00:52:01,450 --> 00:52:05,440 so this just becomes, essentially, 866 00:52:05,440 --> 00:52:17,210 e to the minus hT e to the J, minus 1 plus e to the minus hT 867 00:52:17,210 --> 00:52:20,010 e to the J. 868 00:52:20,010 --> 00:52:24,310 And these two terms are going to cancel out. 869 00:52:24,310 --> 00:52:26,860 And I'm going to have 1. 870 00:52:26,860 --> 00:52:31,650 So again, the ratio of e to the S_T over S_0 just gives you 1. 871 00:52:31,650 --> 00:52:34,110 So all this is just to kind of show you 872 00:52:34,110 --> 00:52:36,820 a little bit how you work with jump processes, 873 00:52:36,820 --> 00:52:39,280 and take expectations. 874 00:52:39,280 --> 00:52:42,140 It's not-- nothing you haven't seen in terms of math. 875 00:52:42,140 --> 00:52:45,320 It's just slightly different from Brownian motions. 876 00:52:45,320 --> 00:52:48,110 But still the same idea-- you have dN 877 00:52:48,110 --> 00:52:51,000 and you have a compensator term. 878 00:52:51,000 --> 00:52:53,340 So this here proves that, essentially, my drift 879 00:52:53,340 --> 00:52:55,970 guess that I started with, in fact 880 00:52:55,970 --> 00:52:58,410 does make my expectation 0. 881 00:53:01,070 --> 00:53:03,300 OK, so what have we done so far? 882 00:53:03,300 --> 00:53:05,620 We've defined dynamics for log of S 883 00:53:05,620 --> 00:53:08,610 with jump on default, defined probability density. 884 00:53:08,610 --> 00:53:14,180 And now we have to derive the dynamics of S, price euro 885 00:53:14,180 --> 00:53:17,970 bonds, hedge ratios, and so on. 886 00:53:17,970 --> 00:53:27,620 OK so log of S dynamics, we-- I apologize again 887 00:53:27,620 --> 00:53:34,272 for the small font-- here we have the log S dynamics. 888 00:53:34,272 --> 00:53:43,405 Applying Ito's lemma, there is an equivalent-- Ito's lemma 889 00:53:43,405 --> 00:53:46,870 you know from Brownian motion, but there is another one 890 00:53:46,870 --> 00:53:50,900 for Poisson processes, as well. 891 00:53:50,900 --> 00:53:54,340 And that is-- Ito's lemma is like the chain rule. 892 00:53:54,340 --> 00:54:00,560 So if you know the process for some log of S, 893 00:54:00,560 --> 00:54:03,610 how do you find the process for S itself? 894 00:54:03,610 --> 00:54:12,130 Well, in this case, what's going to happen is our dS over S 895 00:54:12,130 --> 00:54:20,130 is going to be the same drift-- h times 1 minus e to the J, 896 00:54:20,130 --> 00:54:30,440 tau is less than T, dT-- sorry, T less than tau-- plus e 897 00:54:30,440 --> 00:54:40,360 to the J minus 1, so J minus 1, dN, dN_t. 898 00:54:40,360 --> 00:54:51,548 So that's really the derivation of the-- 899 00:54:51,548 --> 00:54:57,680 that's the final result for S. Now, how do we get to this? 900 00:54:57,680 --> 00:55:03,962 Well maybe I should-- I can write Ito's lemma. 901 00:55:03,962 --> 00:55:04,670 What does it say? 902 00:55:04,670 --> 00:55:08,750 Ito's lemma basically says that if we have dX_t 903 00:55:08,750 --> 00:55:15,870 is equal mu dt plus J dN, then-- and you have a function 904 00:55:15,870 --> 00:55:23,580 Y of t, which is f of X_t, then dY 905 00:55:23,580 --> 00:55:48,880 is df/dx mu dT plus f of X_t plus J minus f of X_t dN_t. 906 00:55:51,580 --> 00:56:00,890 So this is the kind of the term that 907 00:56:00,890 --> 00:56:04,860 is kind of an analog of the convexity term in your Brownian 908 00:56:04,860 --> 00:56:08,890 motion Ito's lemma, but it's now for jump processes. 909 00:56:08,890 --> 00:56:11,900 So this f of X_t plus J and f of X_t-- so what 910 00:56:11,900 --> 00:56:17,210 happens, essentially, so you have some function f, and X_t 911 00:56:17,210 --> 00:56:19,700 plus J is what happens if a jump happens. 912 00:56:19,700 --> 00:56:22,350 And X_t is before the jump, so the effect 913 00:56:22,350 --> 00:56:24,760 of the jump on the function. 914 00:56:24,760 --> 00:56:27,730 That's what this term is. 915 00:56:27,730 --> 00:56:30,720 That's like the convexity term. 916 00:56:30,720 --> 00:56:33,310 I think of as a convexity term. 917 00:56:33,310 --> 00:56:34,530 I don't know how it's called. 918 00:56:34,530 --> 00:56:38,290 Maybe more mathematical minds here might. 919 00:56:38,290 --> 00:56:42,510 So in our case, if you look at the top equation, 920 00:56:42,510 --> 00:56:47,250 our function is just essentially the exponent. 921 00:56:47,250 --> 00:56:51,390 And what happens is when the function goes up by J 922 00:56:51,390 --> 00:56:55,850 is that the exponent goes e to the J minus e to 0. 923 00:56:55,850 --> 00:56:58,470 That's what this term is. 924 00:56:58,470 --> 00:57:04,470 OK, so that's how you write the equation. 925 00:57:04,470 --> 00:57:12,560 And now the SDE, solving the SDE generally 926 00:57:12,560 --> 00:57:15,050 means write down what S is. 927 00:57:15,050 --> 00:57:18,740 So we have S of t. 928 00:57:18,740 --> 00:57:25,207 In our case, it's going to be S of little t. 929 00:57:25,207 --> 00:57:26,290 I'm not going to write it. 930 00:57:26,290 --> 00:57:27,600 You have it on the board. 931 00:57:27,600 --> 00:57:34,700 I think we're going to get late, so hurry up a little bit. 932 00:57:34,700 --> 00:57:40,280 We're going to the next part, which is the pricing exercise. 933 00:57:40,280 --> 00:57:43,140 So we have two bonds-- zero-coupon, zero-recovery 934 00:57:43,140 --> 00:57:43,970 bonds. 935 00:57:43,970 --> 00:57:45,335 One pays $1. 936 00:57:45,335 --> 00:57:47,870 The other pays one euro. 937 00:57:47,870 --> 00:57:51,400 So how are we going to price this? 938 00:57:51,400 --> 00:57:52,710 We have to use our model. 939 00:57:52,710 --> 00:57:54,140 We have a model for the FX rate. 940 00:57:54,140 --> 00:57:55,690 We have a model for credit. 941 00:57:55,690 --> 00:58:00,569 So we price both bonds in dollars. 942 00:58:00,569 --> 00:58:02,360 What is the price in dollars for each bond? 943 00:58:02,360 --> 00:58:04,690 And the ratio of prices kind of gives you 944 00:58:04,690 --> 00:58:07,070 the ratio of the notionals in your hedge portfolio, 945 00:58:07,070 --> 00:58:10,140 if you want to hedge one against the other. 946 00:58:10,140 --> 00:58:14,555 So it's a zero-coupon bond. 947 00:58:17,630 --> 00:58:20,070 So I wrote here the dollar bond price is this. 948 00:58:20,070 --> 00:58:22,580 So why do I write it like that? 949 00:58:22,580 --> 00:58:27,230 Well, it's a zero-coupon bond. 950 00:58:27,230 --> 00:58:35,220 So what a zero-coupon bond says is at maturity, it pays 1. 951 00:58:35,220 --> 00:58:39,190 So we have something where the payoff at time T 952 00:58:39,190 --> 00:58:46,370 is either 1 if tau is bigger than T, 953 00:58:46,370 --> 00:58:52,760 or 0 if tau is less than T. 954 00:58:52,760 --> 00:58:56,770 OK, so now what is my price? 955 00:58:56,770 --> 00:59:01,120 Well, I know that standard pricing theory tells me 956 00:59:01,120 --> 00:59:13,130 that the price of time little t is equal to expectation 957 00:59:13,130 --> 00:59:18,800 of a price at time big T. And you can kind of 958 00:59:18,800 --> 00:59:20,940 say there is a money market account. 959 00:59:20,940 --> 00:59:22,940 But money market accounts, in our case, 960 00:59:22,940 --> 00:59:26,000 is just 1, because interest rates are 0. 961 00:59:26,000 --> 00:59:28,630 So that's really just the case. 962 00:59:28,630 --> 00:59:29,860 That's just true. 963 00:59:29,860 --> 00:59:31,680 So now the expectation of this-- well, 964 00:59:31,680 --> 00:59:34,780 that's just equal to the expectation of an indicator 965 00:59:34,780 --> 00:59:39,250 function of tau bigger than T, which just equals 966 00:59:39,250 --> 00:59:50,080 to the probability of tau bigger than T. So if that's true, 967 00:59:50,080 --> 00:59:51,140 we know what that is. 968 00:59:51,140 --> 00:59:52,900 That's just the probability-- that's 969 00:59:52,900 --> 00:59:58,180 the cumulative probability function e to the minus hT. 970 00:59:58,180 --> 01:00:00,720 That's why the price of the bond in dollars 971 01:00:00,720 --> 01:00:02,530 has to be e to the minus hT. 972 01:00:06,420 --> 01:00:15,800 Euro bond price-- same idea, except euro bond 973 01:00:15,800 --> 01:00:18,390 price in dollars is that. 974 01:00:18,390 --> 01:00:23,260 So why is the euro bond price in dollars like that? 975 01:00:23,260 --> 01:00:32,070 Well, the euro bond price in dollars, again, 976 01:00:32,070 --> 01:00:33,840 what is the payoff? 977 01:00:33,840 --> 01:00:36,880 Same payoff, except the payoff is in euros, right? 978 01:00:36,880 --> 01:00:40,660 So if I want to do the payoff of my bond in dollars-- so this, 979 01:00:40,660 --> 01:00:43,590 I'm going to call this the euro bond. 980 01:00:43,590 --> 01:00:46,480 But the payoff now, if I want to do it in dollars, 981 01:00:46,480 --> 01:00:48,100 is not really 1. 982 01:00:48,100 --> 01:00:56,610 It's 1 times S of T, and 0 times S of T. 983 01:00:56,610 --> 01:00:58,400 That's really my payoff. 984 01:00:58,400 --> 01:01:03,370 So then the expectation here is not just 1, but actually 985 01:01:03,370 --> 01:01:06,660 S of T. So now I have something where 986 01:01:06,660 --> 01:01:12,170 I have to take the expectation of S of T, 987 01:01:12,170 --> 01:01:16,060 essentially, at maturity. 988 01:01:16,060 --> 01:01:24,080 My bond price in euros is equal to the expectation of S of T. 989 01:01:24,080 --> 01:01:27,130 And what is my expectation of S of T? 990 01:01:27,130 --> 01:01:34,010 Well, it's e to the minus hT times e to the J. 991 01:01:34,010 --> 01:01:36,930 And that's the expectation of S of T 992 01:01:36,930 --> 01:01:40,990 the indicator function of tau bigger than T, right? 993 01:01:40,990 --> 01:01:43,920 So not just-- the expectation of S of T is S of 0, 994 01:01:43,920 --> 01:01:47,150 but the expectation of S of T times indicator 995 01:01:47,150 --> 01:01:50,470 function only in the cases of tau bigger than capital T. Now, 996 01:01:50,470 --> 01:01:52,170 that's not 0. 997 01:01:52,170 --> 01:01:59,010 That's basically this-- e to the minus hT times e to the J. 998 01:01:59,010 --> 01:02:07,650 OK, so what can we do? 999 01:02:07,650 --> 01:02:10,490 Well, we construct a-- what we should do 1000 01:02:10,490 --> 01:02:12,470 is we construct a portfolio at time 1001 01:02:12,470 --> 01:02:15,410 equals 0, which is we sell one dollar bond, 1002 01:02:15,410 --> 01:02:19,220 and we buy this much amount here of euro bonds. 1003 01:02:19,220 --> 01:02:23,420 And the portfolio value at time equals t equals 0 is 0. 1004 01:02:28,530 --> 01:02:32,460 Basically, you can take-- so e to the hT, the first bond, 1005 01:02:32,460 --> 01:02:34,340 you would get e to the minus hT. 1006 01:02:34,340 --> 01:02:37,085 And from the second amount would cost you e to the minus hT 1007 01:02:37,085 --> 01:02:38,410 to buy. 1008 01:02:38,410 --> 01:02:42,950 That's how I've chosen these scaling factors. 1009 01:02:42,950 --> 01:02:46,420 We start a portfolio which costs 0. 1010 01:02:46,420 --> 01:02:50,210 And I should probably-- I'm going to go back here, 1011 01:02:50,210 --> 01:02:55,826 and going to write down the notionals, 1012 01:02:55,826 --> 01:02:56,700 because we lost them. 1013 01:03:01,120 --> 01:03:04,114 So how many dollar bonds do we have? 1014 01:03:04,114 --> 01:03:06,610 We have minus 1. 1015 01:03:06,610 --> 01:03:10,560 And how many euro bonds do we have? 1016 01:03:10,560 --> 01:03:20,610 We have e to the minus hT times 1 1017 01:03:20,610 --> 01:03:27,050 minus e to the J. This is how many bonds we have. 1018 01:03:27,050 --> 01:03:31,440 OK, so some time delta T later, what 1019 01:03:31,440 --> 01:03:32,810 happens to our bond prices? 1020 01:03:32,810 --> 01:03:35,050 Well, we know what the bond prices are. 1021 01:03:35,050 --> 01:03:40,190 The only thing that changed was that some time expired. 1022 01:03:40,190 --> 01:03:43,550 So now instead of capital T, we have T minus delta T 1023 01:03:43,550 --> 01:03:45,537 to expiration. 1024 01:03:45,537 --> 01:03:47,620 So these are the bond prices if we didn't default. 1025 01:03:47,620 --> 01:03:51,390 Of course, if we defaulted, then the bond prices are 0. 1026 01:03:51,390 --> 01:03:56,540 So obviously, if we defaulted, since both bond prices are 0, 1027 01:03:56,540 --> 01:03:58,530 we started with the a portfolio that's worth 0. 1028 01:03:58,530 --> 01:04:01,029 If default happened, now we have a portfolio that's worth 0. 1029 01:04:01,029 --> 01:04:03,660 So nothing changed, right? 1030 01:04:03,660 --> 01:04:08,740 So the key part is, OK, now what if default didn't happen? 1031 01:04:08,740 --> 01:04:10,402 Would we have the same price as well? 1032 01:04:10,402 --> 01:04:11,610 That's what we want to check. 1033 01:04:11,610 --> 01:04:14,250 And if we have the same price, both in the case of default 1034 01:04:14,250 --> 01:04:19,120 and in the case of no default, then that means we have, 1035 01:04:19,120 --> 01:04:23,064 essentially, a replicated portfolio-- a hedged portfolio. 1036 01:04:26,700 --> 01:04:29,710 OK, so what is the value of the bonds 1037 01:04:29,710 --> 01:04:31,010 if default did not happen? 1038 01:04:31,010 --> 01:04:35,970 Again, we have these is a dollar bonds here, 1039 01:04:35,970 --> 01:04:40,800 and these are the euro bonds, and this is my FX rate. 1040 01:04:40,800 --> 01:04:42,030 Why did my FX rate move? 1041 01:04:42,030 --> 01:04:44,279 Well, because default did not happen, 1042 01:04:44,279 --> 01:04:45,320 so a jump did not happen. 1043 01:04:45,320 --> 01:04:47,278 But still I had my drift, my compensator drift, 1044 01:04:47,278 --> 01:04:51,290 so FX drifts in the opposite direction. 1045 01:04:51,290 --> 01:05:00,050 OK, so the dollar bonds-- dollar bond 1046 01:05:00,050 --> 01:05:04,160 that was one bond, minus 1 bond, and the price. 1047 01:05:04,160 --> 01:05:06,440 So the value of the dollar bond is just minus 1048 01:05:06,440 --> 01:05:11,710 e to the minus h T minus delta T. What about the Euro bonds? 1049 01:05:11,710 --> 01:05:16,410 Well, the Euro bonds-- here is the number of bonds we have. 1050 01:05:16,410 --> 01:05:20,590 This is divided by S_0, by the way. 1051 01:05:20,590 --> 01:05:24,700 In our case, S_0 is 1, so it doesn't matter. 1052 01:05:24,700 --> 01:05:27,730 Price of each bond, again, we take 1053 01:05:27,730 --> 01:05:33,880 that from-- the price of each bond comes from this formula. 1054 01:05:33,880 --> 01:05:38,400 And then the FX rate-- multiply by the FX rate. 1055 01:05:38,400 --> 01:05:41,340 And then when you actually multiply all these guys out, 1056 01:05:41,340 --> 01:05:46,300 you end up with, essentially, the value 1057 01:05:46,300 --> 01:05:48,680 in dollars of you euro bond equals, again, 1058 01:05:48,680 --> 01:05:50,610 the value of your dollar bonds. 1059 01:05:50,610 --> 01:05:53,390 So we started out with a portfolio that was worth 0, 1060 01:05:53,390 --> 01:05:55,910 and then some time delta T later, 1061 01:05:55,910 --> 01:06:02,550 it's worth 0 again, both in the case of default 1062 01:06:02,550 --> 01:06:04,470 and in the case of no default. 1063 01:06:04,470 --> 01:06:06,340 So there's no arbitrage. 1064 01:06:06,340 --> 01:06:08,360 In some sense, not terribly surprising, 1065 01:06:08,360 --> 01:06:10,110 because we actually derived these prices 1066 01:06:10,110 --> 01:06:11,776 based on the assumption of no arbitrage. 1067 01:06:11,776 --> 01:06:13,240 But it's a good check. 1068 01:06:13,240 --> 01:06:16,410 It kind of tells you, hey, if I actually 1069 01:06:16,410 --> 01:06:19,930 follow this model to hedge, I'm really going to be hedged. 1070 01:06:19,930 --> 01:06:23,450 And I'm going to be hedged not just when default occurs, 1071 01:06:23,450 --> 01:06:25,070 or only if default does not occur, 1072 01:06:25,070 --> 01:06:27,260 but I'm hedged in both situations-- 1073 01:06:27,260 --> 01:06:30,610 if default occurs and default does not occur. 1074 01:06:30,610 --> 01:06:32,280 And you can't really do that unless you 1075 01:06:32,280 --> 01:06:39,070 have models that actually are hybrid models-- that 1076 01:06:39,070 --> 01:06:42,090 allow you to mix and match-- to basically describe 1077 01:06:42,090 --> 01:06:44,335 both the current event and the FX process. 1078 01:06:47,660 --> 01:06:50,550 So that's kind of the usefulness. 1079 01:06:50,550 --> 01:06:56,090 And the hedging strategy you can see-- it's 1080 01:06:56,090 --> 01:07:02,590 interesting that the hedging strategy-- the hedge ratio 1081 01:07:02,590 --> 01:07:04,695 depends on the credit riskiness. 1082 01:07:04,695 --> 01:07:08,200 So how much bonds we bought depends on J. First 1083 01:07:08,200 --> 01:07:10,110 it depends on h, the credit riskiness. 1084 01:07:10,110 --> 01:07:12,955 And it also depends on J, the jump size. 1085 01:07:12,955 --> 01:07:14,340 So it really depends. 1086 01:07:14,340 --> 01:07:16,840 How many bonds you use-- how many Euro bonds you 1087 01:07:16,840 --> 01:07:19,140 buy to hedge your dollar bonds, it 1088 01:07:19,140 --> 01:07:20,930 depends on both the probability of default 1089 01:07:20,930 --> 01:07:24,530 and on the jump size. 1090 01:07:24,530 --> 01:07:30,570 So that's what I mean by it depends on credit riskiness. 1091 01:07:30,570 --> 01:07:35,180 It's also dynamic, in the sense that for a given amount 1092 01:07:35,180 --> 01:07:38,010 of dollar bonds, the amount of euro bonds you need to sell 1093 01:07:38,010 --> 01:07:46,430 is going to vary as FX and time goes forward. 1094 01:07:46,430 --> 01:07:51,490 As you can see, if you have one day before expiration, 1095 01:07:51,490 --> 01:07:52,990 the hedge ratio of the two are going 1096 01:07:52,990 --> 01:07:55,970 to be different than one year before expiration. 1097 01:07:55,970 --> 01:07:59,180 So you have to be rebalancing your portfolio continuously. 1098 01:07:59,180 --> 01:08:01,610 Which is not-- again, not unusual. 1099 01:08:01,610 --> 01:08:05,680 If you're hedging an option, they also have to rebalance. 1100 01:08:05,680 --> 01:08:09,760 But it's different from, say, a static replication strategy, 1101 01:08:09,760 --> 01:08:12,260 where you say, I'm going to buy x amount of euro bonds, 1102 01:08:12,260 --> 01:08:13,801 x amount of dollar bonds, and I won't 1103 01:08:13,801 --> 01:08:15,519 have to ever worry about it. 1104 01:08:15,519 --> 01:08:16,560 It's not really the case. 1105 01:08:16,560 --> 01:08:20,160 Here you're saying, well, I buy this ratio of bonds, 1106 01:08:20,160 --> 01:08:21,689 and if default does not happen, I'm 1107 01:08:21,689 --> 01:08:23,450 going to have to readjust my ratio. 1108 01:08:23,450 --> 01:08:25,859 Because the original ratio took into account 1109 01:08:25,859 --> 01:08:27,601 the probability of default happening. 1110 01:08:27,601 --> 01:08:29,100 And if default did not happen, now I 1111 01:08:29,100 --> 01:08:31,689 have some information-- extra information. 1112 01:08:31,689 --> 01:08:34,350 And now I have to readjust my ratio to reflect that. 1113 01:08:36,899 --> 01:08:42,950 So what happens if recovery is bigger than 0? 1114 01:08:42,950 --> 01:08:46,729 And by the way, how much time do we have-- a quick check? 1115 01:08:46,729 --> 01:08:48,479 PROFESSOR: We have till 4 o'clock. 1116 01:08:48,479 --> 01:08:49,270 STEFAN ANDREEV: OK. 1117 01:08:49,270 --> 01:08:51,850 So we have about 12 minutes, 10 minutes. 1118 01:08:51,850 --> 01:08:54,310 OK, Good. 1119 01:08:54,310 --> 01:08:57,689 So what happens in case the recovery is bigger than 0? 1120 01:09:01,300 --> 01:09:05,550 Well, if recovery is bigger than 0, 1121 01:09:05,550 --> 01:09:08,979 we can go through this exercise that we did, again, 1122 01:09:08,979 --> 01:09:10,680 the pricing exercise, and see what 1123 01:09:10,680 --> 01:09:14,520 happens to our bond prices. 1124 01:09:14,520 --> 01:09:21,060 So let's do this for dollars and euro bonds, 1125 01:09:21,060 --> 01:09:23,939 just to give an example of some of the complexity that 1126 01:09:23,939 --> 01:09:28,770 can arise when you start making the model more realistic. 1127 01:09:28,770 --> 01:09:31,899 Because usually bonds do not have zero recovery. 1128 01:09:31,899 --> 01:09:35,430 So then we assume that our payoff of the zero coupon, zero 1129 01:09:35,430 --> 01:09:40,100 recovery bonds was 1 if default doesn't happen, 0 1130 01:09:40,100 --> 01:09:41,010 if default happens. 1131 01:09:41,010 --> 01:09:46,229 Now, it's going to be the payoff of dollar bond at time 1132 01:09:46,229 --> 01:09:55,530 T is going to be 1 if default did not happen, 1133 01:09:55,530 --> 01:10:05,840 so if tau is bigger than T, and R if default was less than T. 1134 01:10:05,840 --> 01:10:10,910 OK, so now when we price our expectation, 1135 01:10:10,910 --> 01:10:14,180 it's going to be like this. 1136 01:10:17,480 --> 01:10:21,190 P of little t would be just expectation 1137 01:10:21,190 --> 01:10:26,680 at time little t of-- or let's say in this case, 1138 01:10:26,680 --> 01:10:29,660 I'll call expectation the initial price of 0-- 1139 01:10:29,660 --> 01:10:36,760 the expectation of P of capital T, which 1140 01:10:36,760 --> 01:10:43,090 is equal to expectation of essentially 1141 01:10:43,090 --> 01:10:53,930 1 of tau bigger than T plus R 1 of tau less than T. 1142 01:10:53,930 --> 01:11:00,550 Well, what we have here is essentially-- 1143 01:11:00,550 --> 01:11:07,600 so you can think we have this first guy is going 1144 01:11:07,600 --> 01:11:18,570 to be e to the-- if tau bigger than T, it's e to the minus hT. 1145 01:11:18,570 --> 01:11:22,940 And the second guy plus R times the probability 1146 01:11:22,940 --> 01:11:25,940 of tau being less than T, is 1 minus the probability 1147 01:11:25,940 --> 01:11:31,966 of tau being bigger than T, so 1 minus e to the minus hT. 1148 01:11:31,966 --> 01:11:48,260 Which essentially gives you R plus e to the minus hT times 1 1149 01:11:48,260 --> 01:11:55,810 minus R. So that's how you derive the dollar bond price. 1150 01:11:55,810 --> 01:12:00,510 And for the euro bond price, you would do the same thing, 1151 01:12:00,510 --> 01:12:04,900 except now these will be multiplied by the FX rate. 1152 01:12:04,900 --> 01:12:12,720 And now the FX rate-- the tricky thing about the FX rate 1153 01:12:12,720 --> 01:12:15,560 is that the FX rate jumps on default. 1154 01:12:15,560 --> 01:12:19,450 So it's not going to be the same number. 1155 01:12:19,450 --> 01:12:26,110 So in this case, P_T-- this is for one kind of dollar unit-- 1156 01:12:26,110 --> 01:12:30,100 it's 1 times S of T and R times S of T. 1157 01:12:30,100 --> 01:12:35,150 So now we have P little T-- this is 1158 01:12:35,150 --> 01:12:44,910 for euros-- the price at time 0 of the euro bond divided 1159 01:12:44,910 --> 01:12:56,600 by S_0, that equals to expected value time 0 of S of T 1160 01:12:56,600 --> 01:13:09,481 of tau bigger than T plus R S of T times tau less than T. Well, 1161 01:13:09,481 --> 01:13:09,980 OK. 1162 01:13:13,200 --> 01:13:15,290 The first part, S of T, tau bigger 1163 01:13:15,290 --> 01:13:18,780 than T, that was like the zero-coupon bond price. 1164 01:13:18,780 --> 01:13:27,600 So that's just essentially, the-- in order to really, 1165 01:13:27,600 --> 01:13:31,370 I would say, guess this well, we have 1166 01:13:31,370 --> 01:13:35,290 to go back to what was S of T. So if we go back 1167 01:13:35,290 --> 01:13:45,540 to the equation for S of T, let me write that. 1168 01:13:45,540 --> 01:14:02,100 So S of T is S of little t times e to the hT, 1 minus e 1169 01:14:02,100 --> 01:14:37,790 to the J plus J times 1 tau bigger than T, 1170 01:14:37,790 --> 01:14:44,980 this is h tau, tau less than T, and this 1171 01:14:44,980 --> 01:14:49,670 is-- if tau is less than T, and then 1172 01:14:49,670 --> 01:15:03,630 times e to the hT, 1 minus e to the J, tau bigger than T. 1173 01:15:03,630 --> 01:15:07,590 So if default has not occurred, S of T 1174 01:15:07,590 --> 01:15:19,215 is S of 0-- in this case, S of T is S of 0 times this term. 1175 01:15:19,215 --> 01:15:21,490 And if default has occurred, then it's 1176 01:15:21,490 --> 01:15:23,040 S of 0 times this term. 1177 01:15:23,040 --> 01:15:29,680 So the two terms are the same, except for the J part. 1178 01:15:29,680 --> 01:15:33,165 OK, so now when we try to do this expectation, 1179 01:15:33,165 --> 01:15:34,540 here we're in the situation where 1180 01:15:34,540 --> 01:15:36,730 tau-- where default has not occurred, 1181 01:15:36,730 --> 01:15:42,910 so our FX rate is essentially S_0 times the second term. 1182 01:15:42,910 --> 01:15:51,990 So we have expectation of S_0 times-- well, 1183 01:15:51,990 --> 01:15:57,390 and we're kind of dividing by S_0, so S_0 drops out. 1184 01:15:57,390 --> 01:16:11,380 e to the hT times 1 minus e to the J. OK, 1185 01:16:11,380 --> 01:16:22,260 and when tau bigger than T. That's the first expectation. 1186 01:16:22,260 --> 01:16:26,360 And the second one, the expectation 1187 01:16:26,360 --> 01:16:38,080 of-- so we put this R times the expectation of-- now here we 1188 01:16:38,080 --> 01:16:43,190 have tau is less than T. So we're going to have our S of T 1189 01:16:43,190 --> 01:16:46,590 is the first part only would be true. 1190 01:16:46,590 --> 01:16:49,840 Second part would be 1, so that would 1191 01:16:49,840 --> 01:17:03,896 be the formula-- e to the h tau, 1 minus e to the J plus J times 1192 01:17:03,896 --> 01:17:08,510 1, tau less than T. 1193 01:17:08,510 --> 01:17:21,440 So this e to the J term that you see here 1194 01:17:21,440 --> 01:17:25,330 in the euro price, that comes from this term here. 1195 01:17:28,030 --> 01:17:31,450 So how do I do this expectation? 1196 01:17:31,450 --> 01:17:35,950 Well to do this expectation, again, you 1197 01:17:35,950 --> 01:17:40,000 have to do an integral, essentially, 1198 01:17:40,000 --> 01:17:43,530 over the interval from 0 to infinity of the probability 1199 01:17:43,530 --> 01:17:45,260 density. 1200 01:17:45,260 --> 01:17:47,450 Since tau here is bigger than T, I'm really 1201 01:17:47,450 --> 01:17:50,900 integrating from T to infinity. 1202 01:17:50,900 --> 01:17:58,080 So this here is just a constant. 1203 01:17:58,080 --> 01:18:05,490 So this first term-- I'll write it here. 1204 01:18:05,490 --> 01:18:16,180 So you have P_0 over S_0, the first term would 1205 01:18:16,180 --> 01:18:23,620 be e to the hT, 1 minus e to the J. 1206 01:18:23,620 --> 01:18:26,750 And it's going to be integral from big T 1207 01:18:26,750 --> 01:18:29,250 to infinity of the partial differential function, 1208 01:18:29,250 --> 01:18:32,520 so that is just e to the minus hT. 1209 01:18:32,520 --> 01:18:34,020 So this looks like something we've 1210 01:18:34,020 --> 01:18:38,290 already done before in the previous calculation. 1211 01:18:38,290 --> 01:18:46,850 And then the second term is R times-- now 1212 01:18:46,850 --> 01:18:51,840 we're integrating from 0 to tau. 1213 01:18:51,840 --> 01:19:00,990 So this would be integrating from 0 to T, e to the h*tau, 1214 01:19:00,990 --> 01:19:08,660 1 minus e to the J plus J-- I can do like this, e to the J. 1215 01:19:08,660 --> 01:19:11,100 Let's put it like that. 1216 01:19:11,100 --> 01:19:19,730 And times h times e to the minus h*tau d tau-- 1217 01:19:19,730 --> 01:19:26,090 this part being the distribution function, 1218 01:19:26,090 --> 01:19:27,880 probability distribution function. 1219 01:19:27,880 --> 01:19:32,290 So again, we have this guy cancels this. 1220 01:19:32,290 --> 01:19:36,450 And what we're left with-- first term 1221 01:19:36,450 --> 01:19:49,750 gives us e to the minus hT e to the J plus R times h times 1222 01:19:49,750 --> 01:19:52,230 e to the J times tau. 1223 01:19:52,230 --> 01:19:54,610 This is, again, an exponent function. 1224 01:19:54,610 --> 01:20:03,296 So we have e to the hT, e to the J minus 1. 1225 01:20:07,480 --> 01:20:08,842 That's true. 1226 01:20:15,920 --> 01:20:23,530 Oh, sorry, there's a minus sign here in front of this. 1227 01:20:23,530 --> 01:20:25,960 The reason there's a minus sign is 1228 01:20:25,960 --> 01:20:29,670 we have minus h e to the J times tau, 1229 01:20:29,670 --> 01:20:31,791 and so we have to put a minus here in front 1230 01:20:31,791 --> 01:20:32,790 when we do the integral. 1231 01:20:32,790 --> 01:20:37,730 So there is a minus here in front. 1232 01:20:37,730 --> 01:20:43,439 So this thing just basically reduces to that expression 1233 01:20:43,439 --> 01:20:43,980 on the board. 1234 01:20:46,670 --> 01:20:52,910 So that's basically-- so this is how 1235 01:20:52,910 --> 01:20:57,320 we expand the problem to having non-zero recoveries. 1236 01:20:57,320 --> 01:21:03,670 What you could do for your final paper, if you decide 1237 01:21:03,670 --> 01:21:07,750 to do a final paper on this topic, 1238 01:21:07,750 --> 01:21:14,900 is to extend the model one step further, and say, in our model, 1239 01:21:14,900 --> 01:21:20,510 our FX rates jumped, but did not have any diffusive elements. 1240 01:21:20,510 --> 01:21:36,600 It was just-- our equation was d log of S was mu dT plus J dN_t. 1241 01:21:36,600 --> 01:21:39,460 That was our SDE for log of S. 1242 01:21:39,460 --> 01:21:44,420 So next step would be hey, why don't we just add 1243 01:21:44,420 --> 01:21:48,418 another term, plus sigma dW? 1244 01:21:51,340 --> 01:21:54,760 So without the jump, this is just 1245 01:21:54,760 --> 01:22:01,440 a standard, log-normal process that you know how to do. 1246 01:22:01,440 --> 01:22:03,690 Now we add jump, essentially. 1247 01:22:03,690 --> 01:22:05,160 So you take a log-normal process. 1248 01:22:05,160 --> 01:22:07,050 You add a jump process to it. 1249 01:22:07,050 --> 01:22:10,170 And you repeat the same things we 1250 01:22:10,170 --> 01:22:13,190 were going through so far-- pricing 1251 01:22:13,190 --> 01:22:16,269 euro bonds, dollar bonds, and coming up 1252 01:22:16,269 --> 01:22:17,435 with a replication strategy. 1253 01:22:20,290 --> 01:22:26,510 This is, for example, a model that-- we're currently 1254 01:22:26,510 --> 01:22:32,590 working to implement a model like that at Morgan Stanley. 1255 01:22:32,590 --> 01:22:37,290 Our model has non-zero interest rates. 1256 01:22:37,290 --> 01:22:40,910 It has dynamic interest rates. 1257 01:22:40,910 --> 01:22:43,730 So that makes it a little bit more complex, 1258 01:22:43,730 --> 01:22:47,390 but overall, it doesn't make it too much more complex. 1259 01:22:47,390 --> 01:22:49,470 Having non-zero interest rates just kind of 1260 01:22:49,470 --> 01:22:51,280 has an extra drift term. 1261 01:22:51,280 --> 01:22:56,160 It doesn't really change that much the mathematics of it. 1262 01:22:56,160 --> 01:23:00,830 And the reason why we want to do that is because we want 1263 01:23:00,830 --> 01:23:03,100 to be able to price, essentially, 1264 01:23:03,100 --> 01:23:07,650 the contracts which are credit contingent, meaning the payoff 1265 01:23:07,650 --> 01:23:09,560 depends on whether something has survived 1266 01:23:09,560 --> 01:23:12,820 or not, whether credit default has occurred or not. 1267 01:23:12,820 --> 01:23:15,810 And the payoff is in units, anything, 1268 01:23:15,810 --> 01:23:17,210 like foreign currency. 1269 01:23:17,210 --> 01:23:20,780 A typical example would be a credit default swap 1270 01:23:20,780 --> 01:23:24,520 denominated in Brazilian reais. 1271 01:23:24,520 --> 01:23:27,160 Or that happens-- a credit default swap 1272 01:23:27,160 --> 01:23:29,140 on Brazil denominated in Brazilian reais. 1273 01:23:29,140 --> 01:23:33,050 Now, common sense is that when Brazil defaults, 1274 01:23:33,050 --> 01:23:35,234 Brazilian real is not going to cost very much. 1275 01:23:35,234 --> 01:23:37,150 It's not going to be very valuable, just as we 1276 01:23:37,150 --> 01:23:39,930 saw on the graph with the Argentinian peso, which totally 1277 01:23:39,930 --> 01:23:42,370 devalued, it would devalue as well. 1278 01:23:42,370 --> 01:23:45,540 Now Brazil is a very big economy, strong country. 1279 01:23:45,540 --> 01:23:48,590 So right now, people are buying a lot of their bonds. 1280 01:23:48,590 --> 01:23:51,650 People are investing in it. 1281 01:23:51,650 --> 01:23:53,640 Still, it has credit risk. 1282 01:23:53,640 --> 01:23:56,690 And you can buy-- you can trade the credit risk. 1283 01:23:56,690 --> 01:23:59,110 You can trade credit default swaps in dollars. 1284 01:23:59,110 --> 01:24:01,180 And you can also enter into contracts 1285 01:24:01,180 --> 01:24:04,590 that essentially quanto the credit risk 1286 01:24:04,590 --> 01:24:06,680 into Brazilian currency itself. 1287 01:24:06,680 --> 01:24:11,590 And to be able to really price this, you can do it. 1288 01:24:11,590 --> 01:24:14,650 We've done it for many years without having a jump model. 1289 01:24:14,650 --> 01:24:17,660 But then your hedge ratios are not very good. 1290 01:24:17,660 --> 01:24:20,720 And you cannot really explain the prices you see 1291 01:24:20,720 --> 01:24:22,370 in the market. 1292 01:24:22,370 --> 01:24:26,120 So we are essentially implementing infrastructure 1293 01:24:26,120 --> 01:24:29,390 to-- we've already implemented this model 1294 01:24:29,390 --> 01:24:31,280 or a version of this model, but we're 1295 01:24:31,280 --> 01:24:33,670 implementing infrastructure to kind of really 1296 01:24:33,670 --> 01:24:35,450 put it in production. 1297 01:24:35,450 --> 01:24:37,810 As you can see in this model now, 1298 01:24:37,810 --> 01:24:39,540 your FX process depends on credit. 1299 01:24:39,540 --> 01:24:42,830 So it actually-- calibration and all these things 1300 01:24:42,830 --> 01:24:44,900 become a little bit more tricky. 1301 01:24:44,900 --> 01:24:47,950 Which I don't want to worry about for your final project, 1302 01:24:47,950 --> 01:24:52,250 but I think it would be a very interesting exercise 1303 01:24:52,250 --> 01:24:59,780 to take something like that, and basically 1304 01:24:59,780 --> 01:25:02,360 work out all the steps. 1305 01:25:02,360 --> 01:25:04,360 It does get a little bit more complicated, 1306 01:25:04,360 --> 01:25:06,830 because now you have to-- if you're doing Ito's lemma, 1307 01:25:06,830 --> 01:25:08,970 you've got to do it both for diffusive processes 1308 01:25:08,970 --> 01:25:11,173 and for a jump process, so you're 1309 01:25:11,173 --> 01:25:15,140 going to have two terms in your Ito's lemma. 1310 01:25:15,140 --> 01:25:17,340 But you've seen them both. 1311 01:25:17,340 --> 01:25:19,629 They're in your class notes. 1312 01:25:19,629 --> 01:25:21,170 If you're so inclined, you can do it. 1313 01:25:23,950 --> 01:25:29,670 And you can-- once you solve the model, 1314 01:25:29,670 --> 01:25:33,810 then you can kind of check your results. 1315 01:25:33,810 --> 01:25:35,970 You can actually build a Monte Carlo simulation, 1316 01:25:35,970 --> 01:25:39,440 or actually run a bunch of paths where 1317 01:25:39,440 --> 01:25:43,850 you simulate both the default and the diffusive part, 1318 01:25:43,850 --> 01:25:47,200 and see if your prices arrived at analytically match 1319 01:25:47,200 --> 01:25:51,410 with your expectations computed by Monte Carlo. 1320 01:25:51,410 --> 01:25:54,870 This will be a good-- it's always a very good check 1321 01:25:54,870 --> 01:25:59,300 to see if-- usually, we do this exercise to check if our Monte 1322 01:25:59,300 --> 01:26:01,650 Carlo simulations is correct, because we 1323 01:26:01,650 --> 01:26:04,960 know that our math is right. 1324 01:26:04,960 --> 01:26:08,350 But you can also do it to check the other way around. 1325 01:26:10,880 --> 01:26:17,630 OK, so in real life, as we went over-- 1326 01:26:17,630 --> 01:26:20,550 I mentioned a couple of times during the lecture-- 1327 01:26:20,550 --> 01:26:22,550 our models are more complicated. 1328 01:26:22,550 --> 01:26:26,610 We have stochastic interest rates, stochastic hazard rates. 1329 01:26:26,610 --> 01:26:29,030 So currently, we assumed that our hazard rate, h, 1330 01:26:29,030 --> 01:26:31,440 is a fixed number. 1331 01:26:31,440 --> 01:26:33,445 h can be stochastic as well. 1332 01:26:33,445 --> 01:26:35,070 It can have its own distribution, 1333 01:26:35,070 --> 01:26:40,790 and typically that's what we use in our models-- stochastic 1334 01:26:40,790 --> 01:26:42,250 effects. 1335 01:26:42,250 --> 01:26:45,970 So when I say stochastic, both jump and diffusion processes. 1336 01:26:45,970 --> 01:26:49,850 And then if you get really fancy, 1337 01:26:49,850 --> 01:26:52,470 then you can start putting correlations between interest 1338 01:26:52,470 --> 01:26:56,350 rates, FX and hazard rates. 1339 01:26:56,350 --> 01:27:01,900 So in particular, having a jump of FX on default 1340 01:27:01,900 --> 01:27:03,720 naturally introduces a correlation 1341 01:27:03,720 --> 01:27:07,710 between credit and FX. 1342 01:27:07,710 --> 01:27:10,380 When credit occurs, FX devalues. 1343 01:27:10,380 --> 01:27:13,420 So clearly, there's going to be a correlation. 1344 01:27:13,420 --> 01:27:16,110 But there also could be a correlation between the hazard 1345 01:27:16,110 --> 01:27:18,530 rates themselves and FX. 1346 01:27:18,530 --> 01:27:20,940 So it's another source of correlation. 1347 01:27:20,940 --> 01:27:23,220 And these correlations would produce different effects 1348 01:27:23,220 --> 01:27:24,010 in the market. 1349 01:27:24,010 --> 01:27:31,550 So basically, you can, if you have enough data points, 1350 01:27:31,550 --> 01:27:34,260 you'd be able to say, well, this model 1351 01:27:34,260 --> 01:27:37,270 seems like it describes the market better than that model. 1352 01:27:37,270 --> 01:27:39,460 Both of them produce quanto effects, though. 1353 01:27:42,040 --> 01:27:49,930 And whether we use analytic solutions or Monte Carlo, 1354 01:27:49,930 --> 01:27:52,680 they're different approaches to price derivatives and compute 1355 01:27:52,680 --> 01:27:54,860 risk. 1356 01:27:54,860 --> 01:27:57,900 It depends, really, on how complex your model is. 1357 01:27:57,900 --> 01:27:59,760 And for certain markets, you'd rather 1358 01:27:59,760 --> 01:28:02,300 have a more complex model that is slower 1359 01:28:02,300 --> 01:28:04,670 and requires Monte Carlo. 1360 01:28:04,670 --> 01:28:07,780 And in other places, you want to have faster, more 1361 01:28:07,780 --> 01:28:13,040 tractable models that can price your derivatives analytically. 1362 01:28:13,040 --> 01:28:15,280 But maybe your models, they don't 1363 01:28:15,280 --> 01:28:17,780 have as many features in them. 1364 01:28:17,780 --> 01:28:19,980 So there's a whole range of models 1365 01:28:19,980 --> 01:28:22,980 implemented for various markets in Morgan Stanley. 1366 01:28:22,980 --> 01:28:27,860 It's a very big area of expertise for us. 1367 01:28:27,860 --> 01:28:30,520 So I think that's it. 1368 01:28:30,520 --> 01:28:32,070 I think I ran a little bit over time. 1369 01:28:32,070 --> 01:28:35,150 I apologize-- five minutes. 1370 01:28:35,150 --> 01:28:37,947 PROFESSOR: Thank you very, very much. 1371 01:28:37,947 --> 01:28:39,738 And we'll thank our speaker first, I guess. 1372 01:28:39,738 --> 01:28:42,096 [APPLAUSE] 1373 01:28:42,096 --> 01:28:43,845 I think there's probably a question or two 1374 01:28:43,845 --> 01:28:47,544 that people might have. 1375 01:28:47,544 --> 01:28:49,793 AUDIENCE: I was wondering if we could now answer which 1376 01:28:49,793 --> 01:28:52,055 of the Italian bets was better. 1377 01:28:52,055 --> 01:28:53,180 STEFAN ANDREEV: Which what? 1378 01:28:53,180 --> 01:28:55,220 AUDIENCE: Which of the bets that we initially 1379 01:28:55,220 --> 01:28:57,410 were considering on the Italian bonds was better? 1380 01:28:57,410 --> 01:29:00,056 Could we answer that now? 1381 01:29:00,056 --> 01:29:01,965 Because we haven't, I think. 1382 01:29:01,965 --> 01:29:02,840 STEFAN ANDREEV: Yeah. 1383 01:29:02,840 --> 01:29:05,920 Yeah, let's go back. 1384 01:29:05,920 --> 01:29:08,565 Which Italian bonds was better? 1385 01:29:11,170 --> 01:29:13,730 What was that? 1386 01:29:13,730 --> 01:29:16,090 OK, so let's try to answer that together. 1387 01:29:19,640 --> 01:29:23,040 And we can answer it within our model, right? 1388 01:29:23,040 --> 01:29:25,680 So in reality, there's all kinds of factors 1389 01:29:25,680 --> 01:29:28,120 going into the price. 1390 01:29:28,120 --> 01:29:30,680 So there's supply and demand, liquidity 1391 01:29:30,680 --> 01:29:32,740 in euros, liquidity in dollars. 1392 01:29:32,740 --> 01:29:36,890 Well, let's say if you're trying to invest in euros, 1393 01:29:36,890 --> 01:29:42,010 or trying to invest in dollars, if I invest 1394 01:29:42,010 --> 01:29:46,260 in dollars, if a default happens, 1395 01:29:46,260 --> 01:29:50,480 I lose essentially-- let's say the recovery was zero. 1396 01:29:50,480 --> 01:29:52,261 So I lose all my money in dollars. 1397 01:29:52,261 --> 01:29:54,260 I thought I had some amount of money in dollars. 1398 01:29:54,260 --> 01:29:56,170 Default occurs. 1399 01:29:56,170 --> 01:29:59,607 I lost my dollars. 1400 01:29:59,607 --> 01:30:00,440 Same thing in euros. 1401 01:30:00,440 --> 01:30:03,260 If I invest in euros, if default occurs, I lose my euros. 1402 01:30:03,260 --> 01:30:06,790 So how much do I lose in a case of euros 1403 01:30:06,790 --> 01:30:08,740 and in the case of dollars? 1404 01:30:08,740 --> 01:30:13,980 So if I invested euros, you say, well, if a default happens, 1405 01:30:13,980 --> 01:30:19,250 my euros are maybe not as valuable. 1406 01:30:19,250 --> 01:30:25,010 So euros are not as valuable, so I lost my euros, 1407 01:30:25,010 --> 01:30:27,370 but what I lost was not as much, because already it's 1408 01:30:27,370 --> 01:30:28,910 also the value of the lot. 1409 01:30:28,910 --> 01:30:34,570 Conversely, is we saw, because of the compensator drift-- 1410 01:30:34,570 --> 01:30:39,840 remember, if you have a jump that makes the currency 1411 01:30:39,840 --> 01:30:42,300 devalue upon default, the currency 1412 01:30:42,300 --> 01:30:45,760 will tend to appreciate if default doesn't happen. 1413 01:30:45,760 --> 01:30:48,590 Because we want the-- the expected value of the currency 1414 01:30:48,590 --> 01:30:52,110 has to be-- that's determined by interest rates parity, 1415 01:30:52,110 --> 01:30:55,310 the first thing we talked about-- the interest rate 1416 01:30:55,310 --> 01:30:56,760 differential. 1417 01:30:56,760 --> 01:31:01,422 So that is kind of an ironclad arbitrage condition 1418 01:31:01,422 --> 01:31:02,380 that we have to follow. 1419 01:31:02,380 --> 01:31:06,240 So if you want your FX forward to really-- the expected value 1420 01:31:06,240 --> 01:31:10,400 of your FX to remain fixed by the interest rate differential, 1421 01:31:10,400 --> 01:31:13,610 and you know that upon default, your currency would devalue, 1422 01:31:13,610 --> 01:31:16,150 that means if the currency does not devalue, 1423 01:31:16,150 --> 01:31:17,539 it's going to appreciate. 1424 01:31:17,539 --> 01:31:19,080 Because if a default does not happen, 1425 01:31:19,080 --> 01:31:22,220 the currency would appreciate relatively speaking. 1426 01:31:22,220 --> 01:31:26,070 So in our case, when we're buying bonds, 1427 01:31:26,070 --> 01:31:29,690 we only get paid if default does not occur. 1428 01:31:29,690 --> 01:31:34,090 So you would rather, essentially, 1429 01:31:34,090 --> 01:31:37,740 buy the bonds in the currency that's 1430 01:31:37,740 --> 01:31:40,140 going to relatively appreciate, essentially. 1431 01:31:40,140 --> 01:31:43,200 Suppose interest rates were zero in both cases. 1432 01:31:43,200 --> 01:31:46,480 You would rather buy the bond where FX would appreciate it 1433 01:31:46,480 --> 01:31:47,710 default does not occur. 1434 01:31:47,710 --> 01:31:51,140 Because if it occurs, you get nothing in any case, right? 1435 01:31:51,140 --> 01:31:53,120 But if it doesn't occur, when you get paid, 1436 01:31:53,120 --> 01:31:55,525 you want something that would appreciate versus something 1437 01:31:55,525 --> 01:31:56,150 that would not. 1438 01:31:56,150 --> 01:31:57,590 So the dollar, for example, let's 1439 01:31:57,590 --> 01:32:01,320 say the dollar doesn't move versus other currencies 1440 01:32:01,320 --> 01:32:04,010 when the euro default happens. 1441 01:32:04,010 --> 01:32:08,550 So you'd rather get the euro bonds. 1442 01:32:08,550 --> 01:32:11,750 AUDIENCE: If you want to estimate recovery, 1443 01:32:11,750 --> 01:32:13,382 can you use a bunch-- I mean, not 1444 01:32:13,382 --> 01:32:15,090 necessarily factors already in the model, 1445 01:32:15,090 --> 01:32:19,180 but outside factors like macroeconomic factors 1446 01:32:19,180 --> 01:32:21,865 to predict the expected value of recovery? 1447 01:32:21,865 --> 01:32:23,240 STEFAN ANDREEV: Absolutely, yeah. 1448 01:32:23,240 --> 01:32:25,810 Recovery is something that we cannot really price, 1449 01:32:25,810 --> 01:32:30,550 necessarily, because usually we have bonds. 1450 01:32:30,550 --> 01:32:35,830 And the bond price-- you can say we model default, probability 1451 01:32:35,830 --> 01:32:38,019 of default versus probability of non-default. 1452 01:32:38,019 --> 01:32:40,060 But now if you introduce a second variable, which 1453 01:32:40,060 --> 01:32:43,500 is the recovery, now you have essentially both probability 1454 01:32:43,500 --> 01:32:46,760 of defaults and recovery amount as variables. 1455 01:32:46,760 --> 01:32:49,980 And you have only price as your data point. 1456 01:32:49,980 --> 01:32:53,250 And you can have infinitely many solutions. 1457 01:32:53,250 --> 01:32:55,870 So typically, what happens is you 1458 01:32:55,870 --> 01:32:57,710 fix the recovery at something. 1459 01:32:57,710 --> 01:33:00,080 Now what do we use to fix the recovery? 1460 01:33:00,080 --> 01:33:03,920 Well, for sovereign countries we use 25% 1461 01:33:03,920 --> 01:33:06,682 and for corporates we use 40%. 1462 01:33:06,682 --> 01:33:08,140 But these numbers-- everybody knows 1463 01:33:08,140 --> 01:33:10,650 that they're kind of just conventions, 1464 01:33:10,650 --> 01:33:12,190 really, more than anything. 1465 01:33:12,190 --> 01:33:17,270 We don't really believe that recovery is really 40% or 25%. 1466 01:33:17,270 --> 01:33:19,264 It varies a lot by corporation. 1467 01:33:19,264 --> 01:33:20,930 And there are studies by credit agencies 1468 01:33:20,930 --> 01:33:25,270 about how much recoveries-- what are the recoveries 1469 01:33:25,270 --> 01:33:28,520 for various bonds. 1470 01:33:28,520 --> 01:33:32,040 And this 25% for sovereign is based 1471 01:33:32,040 --> 01:33:34,650 on some study like that that went over the last 50 1472 01:33:34,650 --> 01:33:37,400 years, looked at the recoveries of sovereigns, 1473 01:33:37,400 --> 01:33:39,940 of which there are not that many every year. 1474 01:33:39,940 --> 01:33:42,250 But if you look at 50 years, there's quite a few. 1475 01:33:42,250 --> 01:33:44,664 And then they made some statements-- 1476 01:33:44,664 --> 01:33:46,330 some recover higher, some recover lower, 1477 01:33:46,330 --> 01:33:49,180 but on average, they recover 25%. 1478 01:33:49,180 --> 01:33:52,190 If you remember in Greece, what happened in Greece, 1479 01:33:52,190 --> 01:33:56,230 how much did bondholders in Greece get for their bonds? 1480 01:33:56,230 --> 01:33:58,270 Now, they didn't really default, technically. 1481 01:33:58,270 --> 01:33:59,728 Well, they did default technically, 1482 01:33:59,728 --> 01:34:01,510 but it was a very managed process. 1483 01:34:01,510 --> 01:34:05,541 But they got definitely less than 25%. 1484 01:34:05,541 --> 01:34:07,540 I think they got something on the order of $0.15 1485 01:34:07,540 --> 01:34:09,010 on the dollar. 1486 01:34:09,010 --> 01:34:14,420 So recovery there was, like I say, was less than 25%. 1487 01:34:17,000 --> 01:34:19,330 Same for this Argentinian default I'm talking about, 1488 01:34:19,330 --> 01:34:22,820 the 2001-- Argentina is still being 1489 01:34:22,820 --> 01:34:27,000 sued by creditors trying to get money back from this. 1490 01:34:27,000 --> 01:34:28,522 And it's a big thing in the news. 1491 01:34:28,522 --> 01:34:30,855 AUDIENCE: [INAUDIBLE] if you have a claim from Argentina 1492 01:34:30,855 --> 01:34:34,480 and they fly over, it can be seized by [INAUDIBLE] funds. 1493 01:34:34,480 --> 01:34:35,480 STEFAN ANDREEV: Exactly. 1494 01:34:35,480 --> 01:34:37,260 They tried to do some settlements. 1495 01:34:37,260 --> 01:34:39,302 So how much did people recover? 1496 01:34:39,302 --> 01:34:40,510 Well, it depends who you are. 1497 01:34:40,510 --> 01:34:44,300 If you took the original deal, maybe you got $0.20, $0.25, 1498 01:34:44,300 --> 01:34:45,634 $0.30 on the dollar. 1499 01:34:45,634 --> 01:34:47,050 Maybe you got $0.20 on the dollar. 1500 01:34:47,050 --> 01:34:48,800 But now if you hold out-- if you held out, 1501 01:34:48,800 --> 01:34:51,020 apparently you got a little bit more eventually. 1502 01:34:51,020 --> 01:34:53,992 So it's a little bit of a fuzzy concept. 1503 01:34:53,992 --> 01:34:55,700 But it's not something-- you usually make 1504 01:34:55,700 --> 01:34:59,170 an assumption of what it is. 1505 01:34:59,170 --> 01:35:00,670 AUDIENCE: And in a related question, 1506 01:35:00,670 --> 01:35:04,950 so how would we also estimate the other constants 1507 01:35:04,950 --> 01:35:08,190 like the hazard rate and the J? 1508 01:35:08,190 --> 01:35:11,240 STEFAN ANDREEV: So once you fix the recovery rate, 1509 01:35:11,240 --> 01:35:13,240 then you can take the bond price. 1510 01:35:13,240 --> 01:35:14,935 And because bond price theoretically 1511 01:35:14,935 --> 01:35:19,260 is e to the minus hT, you can estimate h from the bond price. 1512 01:35:19,260 --> 01:35:21,340 So if you observe a bond price in the market, 1513 01:35:21,340 --> 01:35:23,206 you can say, I'm going to estimate h. 1514 01:35:23,206 --> 01:35:25,580 So let's say I'm going to take some benchmark bonds which 1515 01:35:25,580 --> 01:35:28,490 I know the price of, and I'm going to estimate h 1516 01:35:28,490 --> 01:35:29,760 for each of these bond prices. 1517 01:35:29,760 --> 01:35:32,690 And I'm going to create a curve, which 1518 01:35:32,690 --> 01:35:34,510 is going to be my hazard curve. 1519 01:35:34,510 --> 01:35:37,650 And then I take another derivative or bond 1520 01:35:37,650 --> 01:35:41,290 that I don't know the price of, and I can use the same curve 1521 01:35:41,290 --> 01:35:42,730 to price it. 1522 01:35:42,730 --> 01:35:46,270 So essentially by doing this, what I'm saying is, 1523 01:35:46,270 --> 01:35:47,890 I'm going to replicate my derivative 1524 01:35:47,890 --> 01:35:50,030 using these benchmark bonds as much as I can. 1525 01:35:52,630 --> 01:35:54,997 That's the assumption that I'm making. 1526 01:35:57,931 --> 01:36:01,360 AUDIENCE: And how long [INAUDIBLE] 1527 01:36:01,360 --> 01:36:03,710 if multiple currencies are involved, 1528 01:36:03,710 --> 01:36:06,700 if we are trying to trade with multiple different currencies, 1529 01:36:06,700 --> 01:36:09,144 how does the whole model differ? 1530 01:36:09,144 --> 01:36:11,310 STEFAN ANDREEV: If multiple currencies are involved, 1531 01:36:11,310 --> 01:36:17,310 you can-- first you can-- it becomes tricky. 1532 01:36:17,310 --> 01:36:20,429 You can say each currency can devalue X amount. 1533 01:36:20,429 --> 01:36:22,720 If default happens, you can have more than one currency 1534 01:36:22,720 --> 01:36:25,090 being devalued. 1535 01:36:25,090 --> 01:36:26,640 If you have more than one currency, 1536 01:36:26,640 --> 01:36:28,230 if you have more than two currencies, 1537 01:36:28,230 --> 01:36:30,550 like three currencies, there's other identities 1538 01:36:30,550 --> 01:36:33,020 you have to take care. 1539 01:36:33,020 --> 01:36:35,770 You really simulate-- if you have three currencies, 1540 01:36:35,770 --> 01:36:38,140 there is a triangle identity that, say, 1541 01:36:38,140 --> 01:36:40,260 dollar-euros times euro-yen exchange rate 1542 01:36:40,260 --> 01:36:42,230 has to equal to dollar-yen exchange rate. 1543 01:36:42,230 --> 01:36:45,440 That's kind of an arbitrage condition. 1544 01:36:45,440 --> 01:36:48,390 Just like interest rate FX forward parity-- 1545 01:36:48,390 --> 01:36:50,420 even stronger in some sense. 1546 01:36:50,420 --> 01:36:55,820 And so you can basically, you can write down 1547 01:36:55,820 --> 01:36:57,680 multiple processes and price stuff. 1548 01:37:00,479 --> 01:37:02,020 AUDIENCE: How much do these equations 1549 01:37:02,020 --> 01:37:04,900 change when you add in bonds that are paying coupons? 1550 01:37:04,900 --> 01:37:07,340 And how do you factor in duration and all that? 1551 01:37:07,340 --> 01:37:09,975 STEFAN ANDREEV: Well, you just-- it's not hard, really. 1552 01:37:09,975 --> 01:37:12,630 You just, instead of having this, you just 1553 01:37:12,630 --> 01:37:15,560 write down all the coupon payments, when you pay them. 1554 01:37:15,560 --> 01:37:18,510 And then you just take an expectation 1555 01:37:18,510 --> 01:37:20,220 of all the coupon payments. 1556 01:37:20,220 --> 01:37:21,940 So it's really the same process. 1557 01:37:21,940 --> 01:37:23,440 You just repeat it for every coupon. 1558 01:37:26,020 --> 01:37:28,380 PROFESSOR: Why don't we shut the formal class over now. 1559 01:37:28,380 --> 01:37:30,839 But if people have questions afterwards, we'll [INAUDIBLE]. 1560 01:37:30,839 --> 01:37:32,296 STEFAN ANDREEV: Yeah, I'm certainly 1561 01:37:32,296 --> 01:37:34,270 around to answer questions, if anybody wants. 1562 01:37:34,270 --> 01:37:34,570 PROFESSOR: Thank you very much. 1563 01:37:34,570 --> 01:37:36,420 STEFAN ANDREEV: Thank you.