1 00:00:11,500 --> 00:00:16,660 Okay. Good morning. Welcome to 2020. I guess this 2 00:00:16,660 --> 00:00:21,790 is the first lecture, certainly some of you. So we're here today 3 00:00:21,790 --> 00:00:27,160 for mathematical models of financial derivatives. I'm 4 00:00:27,160 --> 00:00:32,380 Sam Cohen. I'm one of the lecturers here in my finance. And what we're gonna do 5 00:00:32,380 --> 00:00:37,810 is in this course, we are going to try and understand some 6 00:00:37,810 --> 00:00:43,030 of the contracts that are commonly traded in finance. We're going to try and ask ourselves how we use 7 00:00:43,030 --> 00:00:48,070 mathematics to understand the better. We understand the risk. How do we manage it? How 8 00:00:48,070 --> 00:00:53,290 can we price it, et cetera. So before we begin, we really should 9 00:00:53,290 --> 00:00:59,080 be asking ourselves, why are we wanting to do this? 10 00:00:59,080 --> 00:01:04,480 And it's a it's a pertinent question because mathematical models of financial 11 00:01:04,480 --> 00:01:10,480 derivatives have been more than a little bit controversial. Now, 12 00:01:10,480 --> 00:01:15,540 you guys may not remember it as well as I do, but 13 00:01:15,540 --> 00:01:21,040 12, 13 years ago, there was a very large financial crisis, the GFC. 14 00:01:21,040 --> 00:01:26,170 And one of the things that are being blamed for this was an overreliance 15 00:01:26,170 --> 00:01:31,750 on mathematical models of financial derivatives. You've got the article, the formula 16 00:01:31,750 --> 00:01:37,060 that killed Wall Street. So is this a sensible 17 00:01:37,060 --> 00:01:43,540 thing for us to be studying at all? It's an open question. It's a serious one. 18 00:01:43,540 --> 00:01:48,680 I would argue it is for a variety of reasons. As 19 00:01:48,680 --> 00:01:53,690 we'll see, financial derivatives have always existed pretty 20 00:01:53,690 --> 00:01:59,030 much since agriculture. We have had financial derivatives 21 00:01:59,030 --> 00:02:04,250 and because of that, we we can't close our eyes to the fact that these things exist, 22 00:02:04,250 --> 00:02:09,260 that they are traded, that they are real and building models of 23 00:02:09,260 --> 00:02:14,610 them trying to understand their risk is a sensible thing to do. 24 00:02:14,610 --> 00:02:19,680 We also need to understand where those models are going to fail. We need to understand the restrictions 25 00:02:19,680 --> 00:02:24,840 of what we're doing. And so that's something I'm hoping that we'll touch on quite a lot in this 26 00:02:24,840 --> 00:02:30,960 course. Where is the model going to go wrong? Where are the restrictions? Where have we made 27 00:02:30,960 --> 00:02:36,600 abstractions or simplifications that are a little too brave, a little 28 00:02:36,600 --> 00:02:41,760 too far from reality? And I'm hoping that we'll cover some of those things as 29 00:02:41,760 --> 00:02:47,550 we go along. So what are we trying to actually do in this course? 30 00:02:47,550 --> 00:02:53,580 We are trying. As I say, we've got three broad goals. 31 00:02:53,580 --> 00:02:59,490 The first is to build 32 00:02:59,490 --> 00:03:07,020 models of contracts 33 00:03:07,020 --> 00:03:19,410 traded on financial markets. 34 00:03:19,410 --> 00:03:24,420 So that's the basic idea of what we're going to be doing. We're going to try and build models of some of 35 00:03:24,420 --> 00:03:29,460 these contracts. We're going to try and understand their payoffs. And then from that, we want to 36 00:03:29,460 --> 00:03:34,530 understand where they fail. We want to understand how to 37 00:03:34,530 --> 00:03:40,320 use them, 38 00:03:40,320 --> 00:03:46,700 where they fail. 39 00:03:46,700 --> 00:03:56,090 And that's where we need to take particular care. 40 00:03:56,090 --> 00:04:01,860 Well, we need to be careful. 41 00:04:01,860 --> 00:04:08,890 And is the third point, we want to develop 42 00:04:08,890 --> 00:04:18,160 the technical proficiency 43 00:04:18,160 --> 00:04:25,630 to build, understand 44 00:04:25,630 --> 00:04:34,470 and use better models. 45 00:04:34,470 --> 00:04:39,480 So this course is not the end point. We're not going to get to the advanced things 46 00:04:39,480 --> 00:04:44,610 that one can do in the world of mathematical finance, but we're going to try and get enough technical 47 00:04:44,610 --> 00:04:50,970 proficiency that we could at least move on to that step at the end of this course. 48 00:04:50,970 --> 00:04:56,310 Okay. So when we're building a model in finance, what 49 00:04:56,310 --> 00:05:01,410 are the key concerns? Now, the usual thing people think is that you're trying to build 50 00:05:01,410 --> 00:05:06,660 a model so that you make money. Sounds good. The truth 51 00:05:06,660 --> 00:05:12,840 of the matter is that usually making money is not our first concern. 52 00:05:12,840 --> 00:05:17,880 Usually our key concern in much 53 00:05:17,880 --> 00:05:25,730 financial modelling, a key concern. 54 00:05:25,730 --> 00:05:32,510 Is don't they exploited? 55 00:05:32,510 --> 00:05:37,730 We are more worried about if we are trying to give a price for something, that we will give a price 56 00:05:37,730 --> 00:05:43,820 that allows someone else to come and rip us off. That is actually our first concern. 57 00:05:43,820 --> 00:05:48,830 And this is going to come up again and again to the concept of arbitrage that if we are giving prices 58 00:05:48,830 --> 00:05:55,520 that allow us to be exploited, then we're not going to do very well. And so this is our first concern. 59 00:05:55,520 --> 00:06:01,340 Our second concern is we need to understand 60 00:06:01,340 --> 00:06:08,910 and manage our risk. 61 00:06:08,910 --> 00:06:14,040 It's not enough to have something which pays off. Well, if it's so risky that we're going 62 00:06:14,040 --> 00:06:19,110 to lose before we win, and so we need to understand to manage our 63 00:06:19,110 --> 00:06:28,230 risk. And then a third goal is. 64 00:06:28,230 --> 00:06:33,420 Make a profit in what we do in this course. This is actually not 65 00:06:33,420 --> 00:06:38,490 going to be a key aim. We're going to be trying to understand these steps because only 66 00:06:38,490 --> 00:06:43,530 once we've done these, that this is even possible to even talk about making profit, unless 67 00:06:43,530 --> 00:06:50,180 you can understand your risk and you're sure you're not going to be exploited. There's no way you can turn a profit. 68 00:06:50,180 --> 00:06:55,530 Okay. So what are the economic assumptions that underlie 69 00:06:55,530 --> 00:07:00,690 most our modelling? Let's try and start moving towards actually building 70 00:07:00,690 --> 00:07:06,540 a model of something. So will begin with 71 00:07:06,540 --> 00:07:12,010 a fairly simple set of assumptions about how financial markets work. 72 00:07:12,010 --> 00:07:17,010 And these are deliberately simplified, but they allow us to 73 00:07:17,010 --> 00:07:22,260 start making some conclusions. So the first assumption that we're going to use in 74 00:07:22,260 --> 00:07:30,740 a lot of the models is that there is a riskless 75 00:07:30,740 --> 00:07:35,790 investment. 76 00:07:35,790 --> 00:07:41,100 Now, what do I mean by riskless? I mean that if I put money into it now, I know how much money 77 00:07:41,100 --> 00:07:46,320 I'm going to have in a year's time. So this is a fairly common 78 00:07:46,320 --> 00:07:51,390 assumption. You can think of this as being cash or more usually as 79 00:07:51,390 --> 00:07:56,580 being a bond or a bank account with a fixed interest rate. I can invest in it today and I know 80 00:07:56,580 --> 00:08:01,860 how much money I'm going to have in the future. So we'll often refer to this either as 81 00:08:01,860 --> 00:08:07,410 a bank account or 82 00:08:07,410 --> 00:08:12,450 a bond. If you're 83 00:08:12,450 --> 00:08:17,460 not sure a bond is just a an agreement to pay in the future. I thought of it. 84 00:08:17,460 --> 00:08:23,190 I would have. I have one in my office and old physical bond. It's a contract 85 00:08:23,190 --> 00:08:28,320 where what you do is a company or government will issue 86 00:08:28,320 --> 00:08:33,450 bonds. They sell them in the market and they give the holder of the bonds the right to certain cash 87 00:08:33,450 --> 00:08:38,490 flows. So if you buy a bond, you then can go to the company. And every month or however often the 88 00:08:38,490 --> 00:08:43,620 payment is they will give you money. Now, to make 89 00:08:43,620 --> 00:08:50,730 things simple. This grows 90 00:08:50,730 --> 00:08:56,280 at a constant, continuously 91 00:08:56,280 --> 00:09:05,640 compounded 92 00:09:05,640 --> 00:09:11,820 interest rate, 93 00:09:11,820 --> 00:09:17,130 which will tonight buy a lower case. Now, 94 00:09:17,130 --> 00:09:22,290 the fact it's constant just simplifies things a lot. We'll come back to whether this is a good assumption at various 95 00:09:22,290 --> 00:09:27,390 points. The continuous compounding, that's more of a just a technical 96 00:09:27,390 --> 00:09:32,460 simplification. The fact that something is compounding continuously means that I can figure out the value 97 00:09:32,460 --> 00:09:37,830 of my riskless investment at any point in time without worrying about when exactly 98 00:09:37,830 --> 00:09:42,930 the interest is being determined. Now, in many cases, interest is calculated on a day 99 00:09:42,930 --> 00:09:47,970 to day basis. So at the end of the day. And what that means is that this is 100 00:09:47,970 --> 00:09:54,000 only an approximation. It's the first point where we've already started making bad assumptions, 101 00:09:54,000 --> 00:09:59,190 assumptions that are not technically true, but they are good enough. If I'm pricing something which is a long 102 00:09:59,190 --> 00:10:04,500 way in the future, say a few months, the fact that interest is being computed daily rather 103 00:10:04,500 --> 00:10:11,200 than continuously makes very, very little difference. In most cases. 104 00:10:11,200 --> 00:10:16,230 Okay. So what does this practically mean mathematically? Well, 105 00:10:16,230 --> 00:10:23,380 it means that if a quantity MDT is invested 106 00:10:23,380 --> 00:10:28,950 at time t 107 00:10:28,950 --> 00:10:34,200 then it grows 108 00:10:34,200 --> 00:10:39,210 to well how much a big T, which is E to 109 00:10:39,210 --> 00:10:46,050 the R. Big T minus little T. Empty 110 00:10:46,050 --> 00:10:51,210 time. And so this is this continuous 111 00:10:51,210 --> 00:10:58,630 compounding, you get an exponential coming in. Okay. 112 00:10:58,630 --> 00:11:04,270 A guaranteed 113 00:11:04,270 --> 00:11:09,400 amount. B t to be 114 00:11:09,400 --> 00:11:17,570 paid at time t. 115 00:11:17,570 --> 00:11:24,350 Is therefore worth 116 00:11:24,350 --> 00:11:30,510 little to which is E to the minus R t 117 00:11:30,510 --> 00:11:36,400 tape. So if this is true, if I can invest 118 00:11:36,400 --> 00:11:41,870 a time little T so today I invest into the future and my amount that I've invested 119 00:11:41,870 --> 00:11:47,360 grows by multiplying by this call coefficient eight of the R big T minus subtlety, 120 00:11:47,360 --> 00:11:52,430 the time duration. Well that means if I want to get a contract which will pay 121 00:11:52,430 --> 00:11:57,680 me in the future a fixed amount B big T. How much do I have to invest today. 122 00:11:57,680 --> 00:12:08,450 Will you rearrange this formula and you get it. But with a negative exponent, 123 00:12:08,450 --> 00:12:14,960 we assume borrowing 124 00:12:14,960 --> 00:12:20,420 and lending rates 125 00:12:20,420 --> 00:12:25,580 are the same. Well, there's an assumption that's clearly 126 00:12:25,580 --> 00:12:31,010 not true. But if you're a big financial institution. It gets 127 00:12:31,010 --> 00:12:36,140 close to being true that the the amount that you're paying to borrow 128 00:12:36,140 --> 00:12:41,930 and to lend is very similar. And so we're just going to assume that they're the same. 129 00:12:41,930 --> 00:12:49,080 But we are going to remember that's an assumption which we know is sketchy. 130 00:12:49,080 --> 00:12:54,320 OK. So this is the first thing that we've got. We've got this riskless investment 131 00:12:54,320 --> 00:12:59,420 in the world where we can invest now for the future and 132 00:12:59,420 --> 00:13:04,630 it's going to grow at a fixed rate that we know beforehand. 133 00:13:04,630 --> 00:13:09,770 Okay. What are the assumptions about the market? Because so far we've said nothing about 134 00:13:09,770 --> 00:13:15,350 a market. We've just said there is an investment. So we're going to make two assumptions, 135 00:13:15,350 --> 00:13:20,480 which are sort of our basic modelling assumptions about financial markets. 136 00:13:20,480 --> 00:13:25,850 So there are no 137 00:13:25,850 --> 00:13:33,320 transaction costs. 138 00:13:33,320 --> 00:13:38,420 So what does that mean? It means that if you want to buy something or if you want 139 00:13:38,420 --> 00:13:43,610 to sell it, you're going to pay the same price to make either side of the trade. 140 00:13:43,610 --> 00:13:52,970 So you pay the same price. You pay the same price 141 00:13:52,970 --> 00:14:01,760 to buy or sell. 142 00:14:01,760 --> 00:14:09,560 And for any quantity. 143 00:14:09,560 --> 00:14:14,990 So the price that you would pay for buying one unit of an asset is the same as what you'd get for selling 144 00:14:14,990 --> 00:14:20,660 one unit of an asset, which is one hundredth of what you'd get for buying or selling 145 00:14:20,660 --> 00:14:26,060 100 units of the asset. Now, again. Is this assumption 146 00:14:26,060 --> 00:14:31,280 reasonable? Well, it's a reasonable first approximation, but it's not 147 00:14:31,280 --> 00:14:36,440 technically true. We all know that if I came to you and said I'd like to sell you a TV, maybe 148 00:14:36,440 --> 00:14:41,480 you'd pay me one price if I said I'd like to buy a TV. You'd give me a different price. And if I said I'd like to sell 149 00:14:41,480 --> 00:14:46,580 you a thousand TV shows, you'd look at me strangely and ask, where have I gotten a thousand TV's 150 00:14:46,580 --> 00:14:52,370 from and why am I trying to sell them? So the quantity effects 151 00:14:52,370 --> 00:14:58,670 are important, but we're just going to ignore them as a first approximation. 152 00:14:58,670 --> 00:15:04,130 Okay. A related assumption is that assets 153 00:15:04,130 --> 00:15:13,590 are infinitely divisible. 154 00:15:13,590 --> 00:15:20,960 OK. You can own 155 00:15:20,960 --> 00:15:26,480 zero point two shares. This is more technical 156 00:15:26,480 --> 00:15:31,520 assumption than anything else. It means that we don't have to worry about discrete numbers of shares, that you've got to 157 00:15:31,520 --> 00:15:37,130 buy one share or two shares. You can't buy one and a half shares 158 00:15:37,130 --> 00:15:42,140 practically. It doesn't matter because once you're working on a large scale, if you're working 159 00:15:42,140 --> 00:15:48,970 on a deal for ten thousand shares, the difference between buying. 160 00:15:48,970 --> 00:15:54,370 Eight thousand and eight thousand and one shares is pretty small. It's a ization area, 161 00:15:54,370 --> 00:15:59,880 but it's small enough that we don't need to worry about that as a first approximation. 162 00:15:59,880 --> 00:16:05,260 Okay. Something a bit more controversial than those is that we're usually 163 00:16:05,260 --> 00:16:12,040 going to build models where short selling 164 00:16:12,040 --> 00:16:18,040 is allowed. OK, a little bit of terminology. 165 00:16:18,040 --> 00:16:23,710 Whenever I say short, I mean, selling something. Whenever I say long. I mean owning something. 166 00:16:23,710 --> 00:16:29,350 So what's a short sale? That's where I own a negative quantity 167 00:16:29,350 --> 00:16:34,810 of an asset. Now, is this possible? 168 00:16:34,810 --> 00:16:40,150 Well, it depends on what you mean. It's certainly true that in many markets, 169 00:16:40,150 --> 00:16:45,160 what you can do, say, in the equity market for big stocks, if I want 170 00:16:45,160 --> 00:16:50,170 to hold a negative amount of some big stock, what can I do? I can go 171 00:16:50,170 --> 00:16:56,110 and I can borrow that stock from someone else. I pay a fee. We'll ignore the fee, 172 00:16:56,110 --> 00:17:02,160 but I pay a fee to borrow the stock. I then hold the stock. I can sell the stock. 173 00:17:02,160 --> 00:17:07,380 Now, what that means is I have an obligation in the future to buy the stock again and return it to the person 174 00:17:07,380 --> 00:17:13,090 I borrowed it from. But in the meantime, I hold a negative quantity 175 00:17:13,090 --> 00:17:18,460 of stock. Why is that? If the stock goes up in price? I lose. The stock goes down. 176 00:17:18,460 --> 00:17:23,570 I win because I'm gonna have to buy it in the future. 177 00:17:23,570 --> 00:17:28,640 Now, that is what's called a covered short sale. I have gone. I have 178 00:17:28,640 --> 00:17:33,890 borrowed the stock from someone. So the stock is physically there and I'm able to sell 179 00:17:33,890 --> 00:17:39,200 it on. When I say physically, I don't mean physically. Pretty much everything's 180 00:17:39,200 --> 00:17:44,570 done electronically these days, but don't let that worry you. There is also 181 00:17:44,570 --> 00:17:49,580 what's called a naked short sale. And naked short sale is basically 182 00:17:49,580 --> 00:17:54,640 where I go into the market and say, I'll send you some stocks. And you say, 183 00:17:54,640 --> 00:18:00,080 all right, I'll buy them. And I go, great. And I know that the contract 184 00:18:00,080 --> 00:18:05,210 gives me two days to make good on the deal. So if you've bought stock from me, 185 00:18:05,210 --> 00:18:10,250 I know that I have two days in which I. In two days, time will buy 186 00:18:10,250 --> 00:18:15,770 two days time. I have to deliver the stock. So I am now currently 187 00:18:15,770 --> 00:18:20,780 short the stock, because the stock goes up in value, I'm in trouble. But 188 00:18:20,780 --> 00:18:25,850 there's the idea that I could go and if I do this, if I make a naked short sale, I say I'm 189 00:18:25,850 --> 00:18:31,760 gonna sell it. I could then go and buy it in the market or borrow it from someone after I've made the deal. 190 00:18:31,760 --> 00:18:36,830 This is a naked short sale. This has been very controversial in the past few years. 191 00:18:36,830 --> 00:18:41,900 Why is it? Because if I haven't already identified the stock that I'm 192 00:18:41,900 --> 00:18:47,300 going to be borrowing or if I haven't already borrowed it, it's very easy for me to start 193 00:18:47,300 --> 00:18:52,790 selling things I don't own. Very easy if I'm allowed to do that for me to go and say 194 00:18:52,790 --> 00:18:58,640 I'm going to sell 10000, 10 million, however many stocks I want. 195 00:18:58,640 --> 00:19:03,640 And that looks like lots of people are trying to sell a lot of stock. Even though I 196 00:19:03,640 --> 00:19:08,920 don't own it. And so that's been linked to the idea that it's going to push prices down. 197 00:19:08,920 --> 00:19:14,200 And so it's been quite controversial. We're going to assume that it's allowed. 198 00:19:14,200 --> 00:19:20,850 We're not going to go into the details of how short selling is actually manipulated. 199 00:19:20,850 --> 00:19:26,580 But we're just going to say you can hold 200 00:19:26,580 --> 00:19:35,290 a negative quantity. 201 00:19:35,290 --> 00:19:44,160 Of an asset. 202 00:19:44,160 --> 00:19:49,200 Okay, so these are the basic assumptions of how we are going to 203 00:19:49,200 --> 00:19:54,360 model markets, unless I say otherwise, we're going to take these is given, even though we know that most 204 00:19:54,360 --> 00:19:59,400 of them sort of. Not true. But this is true in most mathematical modelling. We have to 205 00:19:59,400 --> 00:20:04,530 abstract. We approximate. And we hope that we've captured the key features of what we're 206 00:20:04,530 --> 00:20:09,610 trying to understand. OK. So how do we proceed? 207 00:20:09,610 --> 00:20:14,680 Well, we haven't yet given any principles which will allow us to find a price 208 00:20:14,680 --> 00:20:21,010 of anything. But we have said one thing earlier here. 209 00:20:21,010 --> 00:20:27,460 That we're going to avoid being exploited. And the most basic form of exploitation 210 00:20:27,460 --> 00:20:34,010 is arbitration. 211 00:20:34,010 --> 00:20:39,200 So what's an arbitrage, an arbitrage is a deal that is too 212 00:20:39,200 --> 00:20:49,740 good to be true. So an arbitrage. 213 00:20:49,740 --> 00:20:58,760 Let's get the definition right. So is an investment. 214 00:20:58,760 --> 00:21:06,480 Which costs? Nothing. 215 00:21:06,480 --> 00:21:12,140 To set up. 216 00:21:12,140 --> 00:21:18,740 At time t. 217 00:21:18,740 --> 00:21:23,740 I. So if I think of the 218 00:21:23,740 --> 00:21:31,050 cost or value of this investment, it starts off being negative or most Zira. 219 00:21:31,050 --> 00:21:40,710 But. At a later time, 220 00:21:40,710 --> 00:21:46,660 later time. Big teeth. We have. 221 00:21:46,660 --> 00:21:53,530 A zero probability 222 00:21:53,530 --> 00:22:00,270 of it being negative. 223 00:22:00,270 --> 00:22:05,440 So the probability that X a big T is less than 224 00:22:05,440 --> 00:22:10,750 zero zero. Because if it was worth a negative 225 00:22:10,750 --> 00:22:16,630 amount. That would be costly for me in the future. I'd have to pay it. 226 00:22:16,630 --> 00:22:21,670 So this is a deal which costs me nothing to set up in the future. I can't 227 00:22:21,670 --> 00:22:26,920 lose the probability of me losing is zero and 228 00:22:26,920 --> 00:22:33,940 it has a strictly positive 229 00:22:33,940 --> 00:22:40,330 probability of a 230 00:22:40,330 --> 00:22:45,970 strictly positive 231 00:22:45,970 --> 00:22:51,040 payoff. So, I mean that the 232 00:22:51,040 --> 00:22:56,080 probability that X T is greater than zero should 233 00:22:56,080 --> 00:23:01,090 be positive. So an arbitrage, if we think of simple 234 00:23:01,090 --> 00:23:06,100 games, is something like the game where we flip a coin. If it's heads, I'll give you 235 00:23:06,100 --> 00:23:11,340 a pound. If it's a tails, nothing happens. Now, if that doesn't 236 00:23:11,340 --> 00:23:16,590 cost you anything to enter into, it's a very good game to play because, well, maybe 237 00:23:16,590 --> 00:23:23,730 you win something, maybe you don't. But you're never going to lose. So that is an arbitrage. 238 00:23:23,730 --> 00:23:30,120 And we are going to basically assume that arbitragers shouldn't exist. 239 00:23:30,120 --> 00:23:36,270 So, Al, the way we're gonna build models is based on the principle of no arbitrage. 240 00:23:36,270 --> 00:23:41,430 Now, does that mean arbitrages actually don't exist? Well, no, they do exist, but 241 00:23:41,430 --> 00:23:46,500 on well functioning markets with lots of players. Whenever 242 00:23:46,500 --> 00:23:52,950 an arbitrage exists, someone will exploit it and some will exploit it quickly. 243 00:23:52,950 --> 00:23:58,080 And if someone exploits it, they start changing prices. And when 244 00:23:58,080 --> 00:24:03,150 they change prices, they do so in a way which gets rid of the arbitrage, sort of 245 00:24:03,150 --> 00:24:08,550 eat up all of the money that can be made out of the arbitrage. By changing the price. 246 00:24:08,550 --> 00:24:14,100 So we're going to assume there are no arbitragers when we build models. What does that mean? 247 00:24:14,100 --> 00:24:20,160 It means we're not building a model to try and detect and exploit arbitrage. 248 00:24:20,160 --> 00:24:25,860 We're going to be building models which say, let's assume we're not fast enough. We're not aiming 249 00:24:25,860 --> 00:24:31,890 to exploit this. What can we then say about the market? 250 00:24:31,890 --> 00:24:36,990 OK, so we 251 00:24:36,990 --> 00:24:42,610 assume. No. 252 00:24:42,610 --> 00:24:48,310 Arbitragers, no arbitragers 253 00:24:48,310 --> 00:24:53,770 exist. 254 00:24:53,770 --> 00:24:59,260 So we've got some basic assumptions about markets and how they work, 255 00:24:59,260 --> 00:25:04,780 but we haven't actually considered any contracts yet. So where we'll begin 256 00:25:04,780 --> 00:25:11,770 is with probably the oldest form of financial contract. This is what's called a forward 257 00:25:11,770 --> 00:25:17,110 and what it is. I'll give you a simple example. Imagine for a moment that you are 258 00:25:17,110 --> 00:25:22,130 an exporter. So you are based here 259 00:25:22,130 --> 00:25:27,380 in the UK. You are selling things in the US, 260 00:25:27,380 --> 00:25:32,580 for example. So in one year's time. You will make 261 00:25:32,580 --> 00:25:37,650 a sale. You've already signed the deal today to make a sale in the US 262 00:25:37,650 --> 00:25:43,650 and you will receive some quantity of US dollars. You know 263 00:25:43,650 --> 00:25:49,530 that you're going to receive them. On that day, you will have to pay 264 00:25:49,530 --> 00:25:54,910 all your costs in the UK. All those costs have to be paid 265 00:25:54,910 --> 00:26:00,250 in pound sterling. OK. You have calculated 266 00:26:00,250 --> 00:26:05,680 that if the exchange rate is as it is today, then this is a good deal. 267 00:26:05,680 --> 00:26:10,780 You will make money, but you're worried. You've looked at the 268 00:26:10,780 --> 00:26:16,120 last couple of years and the price of pound sterling versus the US dollar has fluctuated 269 00:26:16,120 --> 00:26:21,170 quite a lot. And your worries that in one year's time when 270 00:26:21,170 --> 00:26:26,270 you receive a whole lot of U.S. dollars but have a whole lot of expenses in pounds, that you 271 00:26:26,270 --> 00:26:31,580 won't have the money to cover it and you might lose. Even though you think today this is a good deal. 272 00:26:31,580 --> 00:26:37,130 So what could you do? Well, what you do is you can go to a bank 273 00:26:37,130 --> 00:26:42,230 and you say to them, look, this is my situation. I would like an agreement with 274 00:26:42,230 --> 00:26:47,240 you to sell us dollars into pound 275 00:26:47,240 --> 00:26:52,430 sterling in one year's time. Now, there's nothing strange about this contract. All 276 00:26:52,430 --> 00:26:57,620 it is, is in one year I will be giving you say one million U.S. 277 00:26:57,620 --> 00:27:03,230 dollars. And I would like you to give me however many it is pound sterling. 278 00:27:03,230 --> 00:27:08,840 And what you try and do is you go to the bank and say, could we please agree today 279 00:27:08,840 --> 00:27:14,120 on the price at which we will do this deal? OK. So we're going to agree today 280 00:27:14,120 --> 00:27:19,520 on the exchange rate at which we will trade in one year's time. And that's the basic idea 281 00:27:19,520 --> 00:27:25,370 of a forward. It's about taking a price today 282 00:27:25,370 --> 00:27:30,540 at which we will trade in the future. The question is, 283 00:27:30,540 --> 00:27:36,910 what price should we trade at? So how do we determine the price today? 284 00:27:36,910 --> 00:27:47,070 Okay, so let's try and write this out. Okay. 285 00:27:47,070 --> 00:27:52,590 So a forward is an 286 00:27:52,590 --> 00:27:57,630 agreement to 287 00:27:57,630 --> 00:28:03,390 trade. 288 00:28:03,390 --> 00:28:15,440 On a fixed date in the future. 289 00:28:15,440 --> 00:28:25,050 For a price determined. 290 00:28:25,050 --> 00:28:30,690 Tonight. Now, one thing to notice, this is an agreement to trade. 291 00:28:30,690 --> 00:28:35,820 All parties must trade on that date. There's no way 292 00:28:35,820 --> 00:28:41,040 I can get out of this once I've signed the contract with the bank. I am legally bound to deliver 293 00:28:41,040 --> 00:28:46,650 US dollars in one year's time. And they are legally bound to deliver. Pound sterling 294 00:28:46,650 --> 00:28:52,490 in one year's time. No one is allowed to back out of the deal. 295 00:28:52,490 --> 00:28:57,640 So both parties 296 00:28:57,640 --> 00:29:03,160 to the contract have 297 00:29:03,160 --> 00:29:09,070 the obligation 298 00:29:09,070 --> 00:29:16,090 to try and. So 299 00:29:16,090 --> 00:29:21,190 how are we going to find a price for this? Okay, let's 300 00:29:21,190 --> 00:29:26,470 think about this. How could I? What other options 301 00:29:26,470 --> 00:29:32,260 could I do? Well, I know that I'm going to be. 302 00:29:32,260 --> 00:29:38,410 I'm going to have to sell us dollars in one year to get pound sterling. 303 00:29:38,410 --> 00:29:43,540 So what I could do. This is called the short 304 00:29:43,540 --> 00:29:48,730 position from the U.K. perspective, because I am selling the asset, 305 00:29:48,730 --> 00:29:55,020 the asset is US dollars. OK. So, for 306 00:29:55,020 --> 00:30:02,250 example, suppose 307 00:30:02,250 --> 00:30:07,890 we wish to sell 308 00:30:07,890 --> 00:30:13,350 one U. S. T in one 309 00:30:13,350 --> 00:30:24,000 year. For simplicity, 310 00:30:24,000 --> 00:30:30,990 we assume no interest 311 00:30:30,990 --> 00:30:40,800 is paid for holding US dollars, 312 00:30:40,800 --> 00:30:46,200 although we'll come back to that assumption. But let's begin. As soon as you hold any US dollars, 313 00:30:46,200 --> 00:30:51,920 there's no interest being paid on that. So what could I do? 314 00:30:51,920 --> 00:31:02,250 Well, one thing I could say is I could. 315 00:31:02,250 --> 00:31:10,770 Let's suppose 316 00:31:10,770 --> 00:31:15,990 the price of one U.S. 317 00:31:15,990 --> 00:31:22,170 dollar today is 318 00:31:22,170 --> 00:31:27,220 s little take. So that's the price at which I can buy one U.S. dollar 319 00:31:27,220 --> 00:31:32,740 in pounds right now. And we've assumed no transaction costs so I could buy or sell 320 00:31:32,740 --> 00:31:37,780 at the same price. So what could I do? Well, 321 00:31:37,780 --> 00:31:43,190 I could. So consider 322 00:31:43,190 --> 00:31:51,750 borrowing S-T. 323 00:31:51,750 --> 00:31:58,270 Hey. Buying 324 00:31:58,270 --> 00:32:07,460 one new installer and holding it. 325 00:32:07,460 --> 00:32:18,530 For a year. 326 00:32:18,530 --> 00:32:27,860 The payoff for doing this. 327 00:32:27,860 --> 00:32:33,380 Is well. What's the value of my US dollar? That I've bought in one years time? 328 00:32:33,380 --> 00:32:39,320 S. S t. That's the future value of one U.S. dollar. 329 00:32:39,320 --> 00:32:44,540 I don't know its value now. I'll only know that at time, big T. But I've also had to borrow 330 00:32:44,540 --> 00:32:50,030 money. And we remember by borrowing S-T today, I've had to 331 00:32:50,030 --> 00:32:55,160 pay interest on that. So I've got E to the R team 332 00:32:55,160 --> 00:33:00,170 honesty. This is my interest rate on the money that I've borrowed in Pownce times. The amount that I 333 00:33:00,170 --> 00:33:06,150 borrowed, which is as little T. 334 00:33:06,150 --> 00:33:12,240 OK, if I hold 335 00:33:12,240 --> 00:33:20,910 this and move forward, 336 00:33:20,910 --> 00:33:26,010 which has pay off, let's think about 337 00:33:26,010 --> 00:33:32,950 this, what's the forward contract worth? Five agreed a price today. 338 00:33:32,950 --> 00:33:39,420 De. Then I've got. Oh. 339 00:33:39,420 --> 00:33:47,250 My US dollar in one year's time, I can sell it for some amount. 340 00:33:47,250 --> 00:33:53,130 Which is the forward price. 341 00:33:53,130 --> 00:33:59,220 De. So I 342 00:33:59,220 --> 00:34:06,070 agreed. Agreed price 343 00:34:06,070 --> 00:34:11,140 F.T. So if I've agreed that price. What's the value of this contract? 344 00:34:11,140 --> 00:34:16,460 Well, in one year I have agreed to traded s little T. The asset, 345 00:34:16,460 --> 00:34:22,240 the U.S. dollar is worth. S big T. 346 00:34:22,240 --> 00:34:28,650 So I have a pay off, which is evidently 347 00:34:28,650 --> 00:34:34,760 which is what I'm actually getting minus 348 00:34:34,760 --> 00:34:40,280 S Big T, which is what the price would be if I didn't have the contract. 349 00:34:40,280 --> 00:34:45,370 I didn't have the forward. OK. So if I 350 00:34:45,370 --> 00:34:52,090 hold this and the forward my total 351 00:34:52,090 --> 00:34:57,460 cash flow is 352 00:34:57,460 --> 00:35:02,980 o f t minus eight to the R. 353 00:35:02,980 --> 00:35:08,030 Hey, let's take. So it's my cash 354 00:35:08,030 --> 00:35:17,800 flow story titled Profit. 355 00:35:17,800 --> 00:35:22,810 Everyone see where I've got that from. So I've got my terminal pay off. This is what I paid 356 00:35:22,810 --> 00:35:28,630 for, the amount that I borrowed. I've got my forward contract, which I agree a price on today, 357 00:35:28,630 --> 00:35:33,670 but I'm effectively losing out because I'm not getting the value of the stock in 358 00:35:33,670 --> 00:35:39,630 the future. But if I add these two together, I just get this payoff here. 359 00:35:39,630 --> 00:35:44,730 The interesting thing is, while the problem of figuring out the value of this was that I don't 360 00:35:44,730 --> 00:35:49,800 know the value of a yet, that I'll only know in one year's time at the terminal point 361 00:35:49,800 --> 00:35:55,970 of my contracts. But this depends on things that I know today. 362 00:35:55,970 --> 00:36:01,040 There's nothing risky in here. I've got. This is the amount that I borrowed. I know what I'm going to pay for that. This is 363 00:36:01,040 --> 00:36:08,430 the price I'm agreeing on today. So there's my total profit, which I know today 364 00:36:08,430 --> 00:36:16,100 by no. Of a trash. 365 00:36:16,100 --> 00:36:22,450 This cannot. Uh, to have 366 00:36:22,450 --> 00:36:42,640 a positive value. 367 00:36:42,640 --> 00:36:48,130 That has to be negative because if that was positive, this would be free money. 368 00:36:48,130 --> 00:36:53,260 I'd go and do this deal, I'd enter into the forward contract. I would borrow money out by U.S. 369 00:36:53,260 --> 00:36:58,300 dollars. And in one year's time, I would have money. It costs me 370 00:36:58,300 --> 00:37:04,080 nothing today. Forward contracts. I'm not paying you anything today, we're just agreeing a price. 371 00:37:04,080 --> 00:37:10,420 No cash exchange tonight. And so if this were a positive amount. 372 00:37:10,420 --> 00:37:15,640 Then I'll be able to make money for nothing. So that's an arbitrage which we've excluded from our models. 373 00:37:15,640 --> 00:37:21,850 So we certainly know that this quantity must be negative. 374 00:37:21,850 --> 00:37:27,200 Well, what about the other side? So this was if I was 375 00:37:27,200 --> 00:37:32,310 if I was thinking as myself. But what I could do is I could 376 00:37:32,310 --> 00:37:37,360 instead think, what if I was my bank? So what's my bank's position? They're doing 377 00:37:37,360 --> 00:37:43,210 everything opposite to me. So they're selling me. They're agreeing the price to do the trade the other way. 378 00:37:43,210 --> 00:37:49,820 And they could borrow money. They could effectively 379 00:37:49,820 --> 00:37:55,230 take money invested and sell us dollars. OK, so instead of buying 380 00:37:55,230 --> 00:38:00,720 a US dollar, they short selling US dollar, instead of borrowing money, they invest money. 381 00:38:00,720 --> 00:38:10,480 So they do everything with the opposite sign to me. So considering 382 00:38:10,480 --> 00:38:17,050 the long forward 383 00:38:17,050 --> 00:38:23,500 with the direction 384 00:38:23,500 --> 00:38:42,030 of trades. First. 385 00:38:42,030 --> 00:38:47,180 We get that. Because if it was possible for my bank to just go and make 386 00:38:47,180 --> 00:38:52,240 money. Then that's an albatross for them. In fact, if 387 00:38:52,240 --> 00:38:57,310 they're offering me a deal, which OK. There's money on the table, I can make 388 00:38:57,310 --> 00:39:02,500 money for doing nothing. Well, we do this. That's an arbitrage. So we're going to assume this can't 389 00:39:02,500 --> 00:39:10,610 happen either. But now we put those two together and we find 390 00:39:10,610 --> 00:39:16,620 that the forward price is equal to the R 391 00:39:16,620 --> 00:39:21,660 t take t times. Let's take. 392 00:39:21,660 --> 00:39:26,680 That is the current price of one U.S. dollar. So the price at 393 00:39:26,680 --> 00:39:32,120 which we agree to trade in the future. Depends on the current exchange 394 00:39:32,120 --> 00:39:39,930 rate. Well, the current exchange rate and the interest rate payments. 395 00:39:39,930 --> 00:39:45,070 Okay, so just before we continue, we should just think 396 00:39:45,070 --> 00:39:50,160 a little bit about this this pay off here. 397 00:39:50,160 --> 00:39:55,820 OK, so the 398 00:39:55,820 --> 00:40:02,950 pay off of a forward 399 00:40:02,950 --> 00:40:08,240 is. So if this is the 400 00:40:08,240 --> 00:40:14,250 stock price. OK. 401 00:40:14,250 --> 00:40:19,260 This is the payoff. Well, if I go for the 402 00:40:19,260 --> 00:40:24,780 short forward, which is what I've described here. So that's an agreement to sell the asset. 403 00:40:24,780 --> 00:40:38,920 Then the payoff looks like this. 404 00:40:38,920 --> 00:40:44,230 On the other hand, if I look at the other side of the deal, an agreement to buy the asset 405 00:40:44,230 --> 00:40:49,550 in one year's time, then I get the opposite. 406 00:40:49,550 --> 00:41:06,350 This is the long forward. 407 00:41:06,350 --> 00:41:13,030 OK. So what are some peculiar things that we notice 408 00:41:13,030 --> 00:41:22,650 when we're doing this? 409 00:41:22,650 --> 00:41:27,960 Observations of 410 00:41:27,960 --> 00:41:35,010 the forward value forward price 411 00:41:35,010 --> 00:41:41,980 is based on current. 412 00:41:41,980 --> 00:41:55,640 Pryce's. And interest payments. 413 00:41:55,640 --> 00:42:00,800 It does not 414 00:42:00,800 --> 00:42:08,330 depend on whether. 415 00:42:08,330 --> 00:42:16,430 S t is a fair price 416 00:42:16,430 --> 00:42:21,520 for the asset. 417 00:42:21,520 --> 00:42:26,830 I have made no assumption anywhere that the current exchange 418 00:42:26,830 --> 00:42:32,050 rate is a sensible exchange rate. And there's something 419 00:42:32,050 --> 00:42:37,360 a little bit weird going on here that I haven't even assumed anything about the future 420 00:42:37,360 --> 00:42:42,430 evolution of the exchange rate. We're making an agreement for what's 421 00:42:42,430 --> 00:42:47,920 going to happen in one year's time. And I have told you nothing about what I think the exchange 422 00:42:47,920 --> 00:42:52,960 rate will be in one year's time. That has not entered 423 00:42:52,960 --> 00:42:58,030 into our calculations at all. So even though we've got a contract, the payoff of 424 00:42:58,030 --> 00:43:03,160 which depends on the exchange rate in one year's time. 425 00:43:03,160 --> 00:43:08,500 Whether it's profitable or whether I make a profit or loss depends 426 00:43:08,500 --> 00:43:13,780 on what the exchange rate one year time is. We have not used any statements 427 00:43:13,780 --> 00:43:20,690 about what I think this is what I think the exchange rate will be. 428 00:43:20,690 --> 00:43:26,810 Now, if you've got an efficient market, then this should be a fair price. 429 00:43:26,810 --> 00:43:32,060 And there is a secret assumption. It's not so secret. We made 430 00:43:32,060 --> 00:43:37,250 it up here. There are no transaction costs. And I can buy and sell any 431 00:43:37,250 --> 00:43:43,220 amount for the same price. So because of that, if there was 432 00:43:43,220 --> 00:43:48,270 money on the table, if this was a bad price. This would be a very strange assumption 433 00:43:48,270 --> 00:43:53,340 to be making, because what it's saying is I can start just buying as much as 434 00:43:53,340 --> 00:43:58,490 I like without changing the price 435 00:43:58,490 --> 00:44:03,660 anyway. But our calculation for the forward price actually 436 00:44:03,660 --> 00:44:11,160 doesn't depend on whether we have a fair price for the asset at all. 437 00:44:11,160 --> 00:44:20,250 We did not need to model. 438 00:44:20,250 --> 00:44:26,360 Didn't need a model for what the future exchange rate will be. So if 439 00:44:26,360 --> 00:44:31,490 interest rates are 440 00:44:31,490 --> 00:44:38,380 positive, she's usually a good assumption. 441 00:44:38,380 --> 00:44:43,850 OK? And 442 00:44:43,850 --> 00:44:52,030 there is no cost or benefit. 443 00:44:52,030 --> 00:45:00,930 Holding the asset. 444 00:45:00,930 --> 00:45:06,080 Then f t you look at this and you say, well, this is a positive. 445 00:45:06,080 --> 00:45:11,090 This is a number. Well, if are is 446 00:45:11,090 --> 00:45:18,470 positive. This quantity is going to be bigger than one. 447 00:45:18,470 --> 00:45:26,270 If I made a silly mistake, someone. 448 00:45:26,270 --> 00:45:35,840 And F.T. is. 449 00:45:35,840 --> 00:45:43,780 Above the current. 450 00:45:43,780 --> 00:45:52,720 Prior asset price. 451 00:45:52,720 --> 00:45:57,970 OK. But we've made an assumption in there, so 452 00:45:57,970 --> 00:46:07,240 question. 453 00:46:07,240 --> 00:46:12,260 What if. Interest 454 00:46:12,260 --> 00:46:18,930 is earned on holding. 455 00:46:18,930 --> 00:46:25,800 That's right. 456 00:46:25,800 --> 00:46:30,800 So let's quickly. In the last minute. So what happens if interest isn't on holding you 457 00:46:30,800 --> 00:46:35,820 dollar solicitor, right? Aha. Well. 458 00:46:35,820 --> 00:46:41,310 Okay. The thing is here, we made our contract 459 00:46:41,310 --> 00:46:46,690 about one U.S. dollar. But if I'm gonna earn interest on holding U.S. dollars 460 00:46:46,690 --> 00:46:51,740 in this deal, I'd buy one U.S. dollar and I hold it to the end. 461 00:46:51,740 --> 00:46:58,150 So how about I change this to holding E to the minus hat? 462 00:46:58,150 --> 00:47:03,290 Do you want to see yourselves? Why is that amount? Well, 463 00:47:03,290 --> 00:47:08,420 this is the amount of U.S. dollars I need to buy today to have one U.S. dollar in 464 00:47:08,420 --> 00:47:13,970 a year's time or in the future. So if I hope if I do this, 465 00:47:13,970 --> 00:47:19,600 I get the same payoff. OK, 466 00:47:19,600 --> 00:47:25,690 but what's it going to cost me? Well, I have to correct this because I no longer. 467 00:47:25,690 --> 00:47:33,020 How much do I borrow? Well, I get eight to the minus our hat. 468 00:47:33,020 --> 00:47:39,050 Got to change the amount I buy. These combine. 469 00:47:39,050 --> 00:47:44,060 OK, so you get now this is E to the minus, 470 00:47:44,060 --> 00:47:50,280 I had to take T think. 471 00:47:50,280 --> 00:47:55,370 So this changes and you end up. 472 00:47:55,370 --> 00:48:00,410 With each of the R minus R hat to take 473 00:48:00,410 --> 00:48:05,480 T. S t. And this is the sort 474 00:48:05,480 --> 00:48:10,550 of formula you get whenever there is a cost or benefit for holding the asset. Now, 475 00:48:10,550 --> 00:48:15,680 forwards are traded on many things, including commodities. That was the 476 00:48:15,680 --> 00:48:20,720 classic example, is buying agricultural products. They've been traded since 477 00:48:20,720 --> 00:48:25,820 antiquity. So there's evidence for them in the Code of Hammurabi, which is 18th century 478 00:48:25,820 --> 00:48:33,020 B.C. There's examples of them in Aristotle, in the politics. 479 00:48:33,020 --> 00:48:38,270 On the other hand, trading on financial products like interest rates, exchange 480 00:48:38,270 --> 00:48:43,370 rates is much more recent. That's really since the 1970s. So they were 481 00:48:43,370 --> 00:48:48,410 out of time. We'll continue talking about this tomorrow morning where we 482 00:48:48,410 --> 00:48:53,510 will also discuss what are the criticisms of what we've done, whereas what we've 483 00:48:53,510 --> 00:49:13,400 done gone wrong.