1 00:00:00,000 --> 00:00:01,940 The following content is provided 2 00:00:01,940 --> 00:00:05,110 by MIT OpenCourseWare under a Creative Commons license. 3 00:00:05,110 --> 00:00:07,759 Additional information about our license, 4 00:00:07,759 --> 00:00:10,468 and MIT OpenCourseWare in general 5 00:00:10,468 --> 00:00:12,541 is available at ocw.mit.edu. 6 00:00:15,340 --> 00:00:16,530 PROFESSOR: Vasily. 7 00:00:16,530 --> 00:00:21,970 Vasily Strela who works now for Morgan Stanley, 8 00:00:21,970 --> 00:00:27,000 did his PhD here in the math department, and kindly said he 9 00:00:27,000 --> 00:00:30,610 would tell us about financial mathematics. 10 00:00:30,610 --> 00:00:32,730 So, it's all yours. 11 00:00:36,450 --> 00:00:38,750 GUEST SPEAKER: Let me thank Professor Strang 12 00:00:38,750 --> 00:00:41,020 for giving this opportunity to talk here, 13 00:00:41,020 --> 00:00:46,930 and it feels very good to be back, be back to 18.086. 14 00:00:46,930 --> 00:00:50,420 So, a few more words about myself. 15 00:00:50,420 --> 00:00:54,100 I've been Professor Strang's student in mathematics 16 00:00:54,100 --> 00:00:56,060 about ten years ago. 17 00:00:56,060 --> 00:01:00,760 So after receiving my PhD, I taught mathematics 18 00:01:00,760 --> 00:01:02,870 for a few years. 19 00:01:02,870 --> 00:01:07,400 Then ended up working for a financial institution, 20 00:01:07,400 --> 00:01:11,660 for investment bank, Morgan Stanley in particular. 21 00:01:11,660 --> 00:01:17,950 I'm part of an analytic modeling group in fixed income division. 22 00:01:17,950 --> 00:01:22,740 What we are doing, we are doing math applications 23 00:01:22,740 --> 00:01:26,910 in finance and modeling derivatives, fixed income 24 00:01:26,910 --> 00:01:28,970 derivatives. 25 00:01:28,970 --> 00:01:31,530 That's actually what I'm going to talk about today. 26 00:01:31,530 --> 00:01:36,390 I want to show how 18.086, it's a wonderful class 27 00:01:36,390 --> 00:01:41,080 which I admire a lot, which applications 28 00:01:41,080 --> 00:01:44,720 it has in the real world, and in particular in finance 29 00:01:44,720 --> 00:01:47,650 and derivatives pricing. 30 00:01:47,650 --> 00:01:50,870 Let's start with a simple example, 31 00:01:50,870 --> 00:01:54,140 which actually comes not from finance, but rather 32 00:01:54,140 --> 00:01:57,050 from gambling. 33 00:01:57,050 --> 00:02:03,020 Well, let's look at horse racing or cockroach 34 00:02:03,020 --> 00:02:06,180 racing, if you prefer. 35 00:02:06,180 --> 00:02:09,520 Suppose there are two horses, and sure enough, 36 00:02:09,520 --> 00:02:14,190 people bet on them and bookie is a clever one, very 37 00:02:14,190 --> 00:02:16,700 scientific-minded guy and he made 38 00:02:16,700 --> 00:02:22,840 a very good research of previous history of these two horses. 39 00:02:22,840 --> 00:02:28,980 He found out that the first horse has 20% chance to win. 40 00:02:28,980 --> 00:02:32,660 The second horse has 80% chance to win. 41 00:02:32,660 --> 00:02:38,250 He is actually right about his knowledge about chances to win. 42 00:02:38,250 --> 00:02:41,250 On the other hand, general public, people who bet, 43 00:02:41,250 --> 00:02:44,500 they don't have access to all information, 44 00:02:44,500 --> 00:02:46,870 and the bets are split slightly differently. 45 00:02:46,870 --> 00:02:53,390 So 10,000 is placed on the first horse, 46 00:02:53,390 --> 00:02:57,970 and 50,000 is placed on the second horse. 47 00:02:57,970 --> 00:03:01,650 Bookie, sticking to his scientific knowledge, 48 00:03:01,650 --> 00:03:06,560 splits the odds 4 to 1, meaning that if the first horse wins, 49 00:03:06,560 --> 00:03:09,950 then whoever put on the horse gets his money back 50 00:03:09,950 --> 00:03:13,370 and four times his money back on top of it. 51 00:03:13,370 --> 00:03:17,950 Or if the second horse wins then whoever put money on this horse 52 00:03:17,950 --> 00:03:24,190 will get the money back and 1/4 on top of it. 53 00:03:24,190 --> 00:03:26,360 So, let's see. 54 00:03:26,360 --> 00:03:34,460 What are chances for bookie to win or lose in this situation? 55 00:03:34,460 --> 00:03:36,510 Well, if the first horse wins then he 56 00:03:36,510 --> 00:03:41,530 has to give back 10,000 plus 40,000, 50,000, 57 00:03:41,530 --> 00:03:46,500 and he got 60,000, so he gains 10,000. 58 00:03:46,500 --> 00:03:48,450 Well, good, good for him. 59 00:03:48,450 --> 00:03:51,080 On the other hand, if the second horse wins, 60 00:03:51,080 --> 00:03:56,400 then he has to give back 50,000 plus 1/4 of 50, 61 00:03:56,400 --> 00:04:05,360 which is 12,500, so 62.50 altogether, 62 00:04:05,360 --> 00:04:10,210 and he loses $2,500. 63 00:04:10,210 --> 00:04:15,320 After many runs, the expected win or loss of the bookmaker 64 00:04:15,320 --> 00:04:17,460 is the probability of the first horse 65 00:04:17,460 --> 00:04:20,300 to win times the expected win, plus the probability 66 00:04:20,300 --> 00:04:23,830 of second horse to win times the expected loss, which turns out 67 00:04:23,830 --> 00:04:25,770 to be exactly zero. 68 00:04:25,770 --> 00:04:31,450 So in each particular run, bookie may lose or win, 69 00:04:31,450 --> 00:04:38,390 but in the long run he expects to break even. 70 00:04:38,390 --> 00:04:41,640 On the other hand, if he would put the chances, 71 00:04:41,640 --> 00:04:47,230 he would set the odds according to the money bet, 5 to 1. 72 00:04:47,230 --> 00:04:48,790 what would be the outcome? 73 00:04:48,790 --> 00:04:54,960 Well, if the first horse wins he gives back 10,000 plus 50,000, 74 00:04:54,960 --> 00:05:00,820 60,000, exactly the amount he collected. 75 00:05:00,820 --> 00:05:04,680 Or if the second horse wins, again, he 76 00:05:04,680 --> 00:05:09,390 gives back 50 plus 1/5 of that, 60, he breaks even. 77 00:05:09,390 --> 00:05:14,010 So no matter which horse wins in this scenario, 78 00:05:14,010 --> 00:05:15,890 the bookie breaks even. 79 00:05:15,890 --> 00:05:18,560 How bookie operates, well he actually 80 00:05:18,560 --> 00:05:21,070 charges a fee for each bet, right. 81 00:05:21,070 --> 00:05:25,200 So the second situation is much more preferable for him. 82 00:05:25,200 --> 00:05:30,520 When he doesn't care which horse wins, he just collects the fee. 83 00:05:30,520 --> 00:05:33,830 Well here, he may lose or gain money. 84 00:05:33,830 --> 00:05:37,240 This is quite beautiful observation, 85 00:05:37,240 --> 00:05:43,360 which we will see how it works in derivatives. 86 00:05:43,360 --> 00:05:47,240 So now back to finance, back to derivatives. 87 00:05:47,240 --> 00:05:49,970 So we are actually interested in pricing 88 00:05:49,970 --> 00:05:52,880 a few financial derivatives, and what is a financial derivative? 89 00:05:52,880 --> 00:05:56,180 Well, a financial derivative is a contract, 90 00:05:56,180 --> 00:05:59,980 payoff of which at maturity, at some time T, 91 00:05:59,980 --> 00:06:04,920 depends on underlying security -- in our case, 92 00:06:04,920 --> 00:06:07,710 we always we will be talking about a stock as underlying 93 00:06:07,710 --> 00:06:13,250 security, and probably interest rates. 94 00:06:13,250 --> 00:06:17,500 What are the examples of financial derivatives? 95 00:06:17,500 --> 00:06:22,090 Well, the most simple example is probably a forward contract. 96 00:06:22,090 --> 00:06:26,810 Forward contracts is a contract when you agree to purchase 97 00:06:26,810 --> 00:06:30,960 the security for a price which is set today -- 98 00:06:30,960 --> 00:06:33,740 you've agreed to purchase the security in the future 99 00:06:33,740 --> 00:06:35,890 for the price agreed today. 100 00:06:35,890 --> 00:06:42,230 Well, for example, if you needed 1,000 barrel of oil to heat 101 00:06:42,230 --> 00:06:47,165 your house, but not today, but rather for the next winter, 102 00:06:47,165 --> 00:06:48,540 on the other hand, you don't want 103 00:06:48,540 --> 00:06:52,700 to take the risks of waiting until the next winter 104 00:06:52,700 --> 00:06:57,670 and buying oil then, you would rather agree on the price 105 00:06:57,670 --> 00:07:03,880 now and pay it in the future and get the oil. 106 00:07:03,880 --> 00:07:05,300 What the price should be? 107 00:07:05,300 --> 00:07:10,590 What is the fair price for this contract? 108 00:07:10,590 --> 00:07:12,500 Well, we will see how to price it. 109 00:07:12,500 --> 00:07:17,820 Well, the few observation here is that this line represents 110 00:07:17,820 --> 00:07:21,440 the payout -- it's always useful to represent the payout 111 00:07:21,440 --> 00:07:22,402 graphically. 112 00:07:22,402 --> 00:07:24,360 This is just a straight line because the payout 113 00:07:24,360 --> 00:07:27,720 of our contract is S minus K at time T. 114 00:07:27,720 --> 00:07:32,520 This actually gives the current price of the contract for all 115 00:07:32,520 --> 00:07:35,570 different values of the underlying. 116 00:07:35,570 --> 00:07:38,920 Usually, the price of the forward contract 117 00:07:38,920 --> 00:07:42,860 is set such that for the current value of the underlying, 118 00:07:42,860 --> 00:07:45,010 the price of the contract is zero. 119 00:07:45,010 --> 00:07:47,490 It costs nothing to enter a forward contract, 120 00:07:47,490 --> 00:07:52,470 so that's why it intersects zero here. 121 00:07:52,470 --> 00:07:56,576 What are other common derivatives? 122 00:07:56,576 --> 00:07:58,450 Another common derivative are calls and puts. 123 00:07:58,450 --> 00:08:03,210 And I put European call and put here. 124 00:08:03,210 --> 00:08:05,300 Don't be confused by European or American. 125 00:08:05,300 --> 00:08:08,030 It has nothing to do with Europe or America, 126 00:08:08,030 --> 00:08:12,450 it has to do with the structure of the contract. 127 00:08:12,450 --> 00:08:16,230 European basically means that the contract 128 00:08:16,230 --> 00:08:19,280 expires at certain time T. American means 129 00:08:19,280 --> 00:08:22,240 that's you can exercise this contract at any time 130 00:08:22,240 --> 00:08:24,690 between now and future. 131 00:08:24,690 --> 00:08:27,270 We'll be talking only about European contracts. 132 00:08:27,270 --> 00:08:30,860 So European call option is a contract 133 00:08:30,860 --> 00:08:35,650 which gives you the right, but not obligation, 134 00:08:35,650 --> 00:08:40,530 to purchase the underlying security at set price K, which 135 00:08:40,530 --> 00:08:43,640 is called strike price, at a future time 136 00:08:43,640 --> 00:08:46,020 T, which is expiration time. 137 00:08:46,020 --> 00:08:52,270 So if your security at time T ends up below K, 138 00:08:52,270 --> 00:08:55,170 below the strike, then sure enough 139 00:08:55,170 --> 00:08:59,330 there is no point of buying the security 140 00:08:59,330 --> 00:09:01,390 for a more expensive price. 141 00:09:01,390 --> 00:09:04,170 So the contract expires worthless. 142 00:09:04,170 --> 00:09:08,980 On the other hand, if your stock ends up being greater than K 143 00:09:08,980 --> 00:09:12,550 at expiration time C, then you would 144 00:09:12,550 --> 00:09:16,340 make money but my purchasing this stock for K 145 00:09:16,340 --> 00:09:19,930 dollars and your payout will be S minus K, 146 00:09:19,930 --> 00:09:22,100 and this is a graph of your payout. 147 00:09:22,100 --> 00:09:28,110 This line here, as we will see, is the current price 148 00:09:28,110 --> 00:09:30,260 of the contract, and we'll see how to obtain 149 00:09:30,260 --> 00:09:33,880 this line in a few minutes. 150 00:09:33,880 --> 00:09:38,250 Another common contract is a put. 151 00:09:38,250 --> 00:09:43,750 While call was basically a bet that your stock will grow, 152 00:09:43,750 --> 00:09:48,640 right, the put is the bet that your stock will not grow. 153 00:09:48,640 --> 00:09:54,090 So, in this case, the put is the right, but not the obligation 154 00:09:54,090 --> 00:09:58,260 to sell the stock for a certain price K. 155 00:09:58,260 --> 00:10:03,640 Here is the payout, which is similar to the put, 156 00:10:03,640 --> 00:10:06,830 but just flipped. 157 00:10:06,830 --> 00:10:10,250 This is the current price of a put option. 158 00:10:10,250 --> 00:10:13,180 Calls and puts, being very common contracts, 159 00:10:13,180 --> 00:10:21,230 are traded on exchanges -- Chicago exchange is probably 160 00:10:21,230 --> 00:10:25,790 the most common place for the calls and puts on stocks 161 00:10:25,790 --> 00:10:27,040 to trade. 162 00:10:27,040 --> 00:10:29,510 I just printed out a Bloomberg screen, 163 00:10:29,510 --> 00:10:33,880 which gives the information about a few calls 164 00:10:33,880 --> 00:10:36,370 and puts on IBM stock. 165 00:10:36,370 --> 00:10:41,450 So I did it on March 8, and the IBM stock 166 00:10:41,450 --> 00:10:44,410 was trading at this time at $81.14, 167 00:10:44,410 --> 00:10:46,900 and here are descriptions of the contract, 168 00:10:46,900 --> 00:10:50,260 they expire on 22nd of April, so it's 169 00:10:50,260 --> 00:10:52,520 pretty short-dated contract. 170 00:10:52,520 --> 00:10:56,420 They can go as far as two years from now, usually. 171 00:10:56,420 --> 00:11:01,990 Here is a set of strikes, and here are a set of prices. 172 00:11:01,990 --> 00:11:05,150 As you can see, there is no single price, 173 00:11:05,150 --> 00:11:07,830 there is always a bid and ask, and that's how dealers 174 00:11:07,830 --> 00:11:10,930 and brokers make their money -- like a bookie, 175 00:11:10,930 --> 00:11:15,520 they basically charge you a fee for selling or buying 176 00:11:15,520 --> 00:11:16,390 the contract. 177 00:11:21,420 --> 00:11:24,740 That's how the money are made -- they are made on this spread, 178 00:11:24,740 --> 00:11:27,390 but not on the price of the contract itself, 179 00:11:27,390 --> 00:11:29,630 because as we will see in a second, 180 00:11:29,630 --> 00:11:33,430 we actually can price the contract exactly, 181 00:11:33,430 --> 00:11:37,340 and there is no uncertainty once the price of the stock is set. 182 00:11:41,680 --> 00:11:44,460 There are plenty of other options. 183 00:11:44,460 --> 00:11:47,230 Slightly more exotics contracts, either digital 184 00:11:47,230 --> 00:11:50,060 which pays either zero or one depending 185 00:11:50,060 --> 00:11:52,770 on where your stock ends up. 186 00:11:52,770 --> 00:11:54,370 It probably is not exchange-traded, 187 00:11:54,370 --> 00:11:56,840 also I'm not sure. 188 00:11:56,840 --> 00:12:00,040 There are hundreds, if not thousands, 189 00:12:00,040 --> 00:12:04,790 of exotic options where you can say that, well, 190 00:12:04,790 --> 00:12:06,980 how much would be the right to purchase 191 00:12:06,980 --> 00:12:12,040 a stock for the maximum price between today and two years 192 00:12:12,040 --> 00:12:13,122 from now. 193 00:12:13,122 --> 00:12:14,330 So it will be past-dependent. 194 00:12:14,330 --> 00:12:17,310 Depending on how the stock will go, 195 00:12:17,310 --> 00:12:20,300 the payout will be defined by this path. 196 00:12:20,300 --> 00:12:22,540 There are American options where you 197 00:12:22,540 --> 00:12:26,120 can exercise your option any time between now and maturity, 198 00:12:26,120 --> 00:12:27,110 and so on and so forth. 199 00:12:27,110 --> 00:12:32,710 So, just before we go into mathematics 200 00:12:32,710 --> 00:12:40,600 of pricing, just a few observations and statements. 201 00:12:40,600 --> 00:12:43,550 First of all, it turns out that thanks 202 00:12:43,550 --> 00:12:47,760 to developed mathematics, mathematical theory, 203 00:12:47,760 --> 00:12:51,480 if you make certain assumptions on the dynamics of the stock, 204 00:12:51,480 --> 00:12:56,390 then there is no uncertainty in the price of the option. 205 00:12:56,390 --> 00:13:01,360 You can say exactly how much the option costs now, 206 00:13:01,360 --> 00:13:06,180 and that's what provides, and this 207 00:13:06,180 --> 00:13:08,020 is a big driver for the market. 208 00:13:08,020 --> 00:13:13,670 So dealers quote these contracts and there is a great agreement 209 00:13:13,670 --> 00:13:17,380 on the prices. 210 00:13:17,380 --> 00:13:22,250 The price of the derivative contract 211 00:13:22,250 --> 00:13:25,470 is defined completely by the stock price 212 00:13:25,470 --> 00:13:29,820 and not by risk preferences of the market participant. 213 00:13:29,820 --> 00:13:35,060 So it doesn't matter what are your views on the growth 214 00:13:35,060 --> 00:13:38,080 prospects of the stock. 215 00:13:38,080 --> 00:13:44,150 It will not affect the price of the derivative contract. 216 00:13:44,150 --> 00:13:48,210 As I said, so the mathematical part of it 217 00:13:48,210 --> 00:13:52,190 comes into giving the exact price without any uncertainty. 218 00:13:55,080 --> 00:13:59,880 So let's consider a simple example now. 219 00:13:59,880 --> 00:14:02,690 Let's assume that we are in a very simple world. 220 00:14:02,690 --> 00:14:07,180 Well, first of all, in our world there are only three objects -- 221 00:14:07,180 --> 00:14:11,690 the stock itself, the riskless money market account, 222 00:14:11,690 --> 00:14:15,900 meaning that it is an account where we can either borrow 223 00:14:15,900 --> 00:14:20,560 money or invest money at the riskless rate r, 224 00:14:20,560 --> 00:14:22,800 and finally our derivative contract. 225 00:14:22,800 --> 00:14:25,620 Here we are not making any assumptions of what kind 226 00:14:25,620 --> 00:14:29,200 of derivative contract it is -- it could be forward, 227 00:14:29,200 --> 00:14:33,280 it could be call, it could be put, it can be anything. 228 00:14:33,280 --> 00:14:35,830 Moreover, our world is so simple, that first of all, 229 00:14:35,830 --> 00:14:38,620 it's discrete, and second of all, 230 00:14:38,620 --> 00:14:42,396 there is only one time step to the expiration of power 231 00:14:42,396 --> 00:14:44,360 of contract, dt. 232 00:14:44,360 --> 00:14:46,570 Not only there is only one step left, 233 00:14:46,570 --> 00:14:51,690 we actually know exactly what our transition probabilities. 234 00:14:51,690 --> 00:14:53,430 There are only two states at the end, 235 00:14:53,430 --> 00:14:55,170 and we know the transition probability. 236 00:14:55,170 --> 00:14:58,420 So with probability p, we move from the state zero 237 00:14:58,420 --> 00:15:01,890 to the state one, and with probability of one minus p, 238 00:15:01,890 --> 00:15:04,750 we move to the state two. 239 00:15:04,750 --> 00:15:08,100 And just notice, because this is riskless money market account, 240 00:15:08,100 --> 00:15:10,670 it's the same in both cases. 241 00:15:10,670 --> 00:15:15,892 You just invest money and it grows with risk-free interest 242 00:15:15,892 --> 00:15:18,420 rate. 243 00:15:18,420 --> 00:15:23,830 So, what can we say about the price of our derivative f? 244 00:15:23,830 --> 00:15:28,110 Well a simple-minded -- well, let's start with the forward 245 00:15:28,110 --> 00:15:29,310 contract. 246 00:15:29,310 --> 00:15:32,490 We know what the payout in delta t of our forward contract 247 00:15:32,490 --> 00:15:35,200 will be, it will be just the difference between the stock 248 00:15:35,200 --> 00:15:39,390 price and our strike. 249 00:15:39,390 --> 00:15:42,940 Well, a simple-minded approach would be -- well, 250 00:15:42,940 --> 00:15:44,990 we know the transition probabilities, 251 00:15:44,990 --> 00:15:48,980 let's just compute the expected value of our contract, 252 00:15:48,980 --> 00:15:54,510 and that's what we would expect to get if there were many such 253 00:15:54,510 --> 00:15:55,820 experiments. 254 00:15:55,820 --> 00:15:58,470 Well, you take the probability of going to state one, 255 00:15:58,470 --> 00:16:01,254 you multiply by the payoff at stage one. 256 00:16:01,254 --> 00:16:03,420 Take, minus p for probability of going to state two, 257 00:16:03,420 --> 00:16:08,590 multiply by the payout in that state two. 258 00:16:08,590 --> 00:16:12,600 Sum them up and you get the expression. 259 00:16:12,600 --> 00:16:16,720 As I said, the common thing to choose 260 00:16:16,720 --> 00:16:19,280 the strike such that the contract 261 00:16:19,280 --> 00:16:22,796 has zero value now, so you get your strike. 262 00:16:22,796 --> 00:16:24,170 Well, in particular you could say 263 00:16:24,170 --> 00:16:26,440 that if you research the market well 264 00:16:26,440 --> 00:16:29,350 and you know that the stock has equal probability of going 265 00:16:29,350 --> 00:16:33,552 up and down, then actually you expect your strike 266 00:16:33,552 --> 00:16:40,400 to be an average of end values of the stock. 267 00:16:40,400 --> 00:16:44,060 But as we can imagine, following our bookie example, 268 00:16:44,060 --> 00:16:46,270 this is not the right price. 269 00:16:46,270 --> 00:16:48,820 There is actually a definite price 270 00:16:48,820 --> 00:16:52,580 which doesn't depend on transition probability. 271 00:16:52,580 --> 00:16:57,220 Here is the reason why there is a definite price. 272 00:16:57,220 --> 00:17:02,400 Well let's just consider a very simple strategy. 273 00:17:02,400 --> 00:17:05,960 Let's borrow just enough to purchase a stock. 274 00:17:05,960 --> 00:17:09,520 So let's borrow S_0 dollars right now and buy 275 00:17:09,520 --> 00:17:12,110 the stock for this money. 276 00:17:12,110 --> 00:17:13,920 And let's enter the forward contract. 277 00:17:13,920 --> 00:17:17,410 Well, by definition forward contract has price zero now, 278 00:17:17,410 --> 00:17:20,440 so we enter the forward contract. 279 00:17:20,440 --> 00:17:25,530 Now, at the time dt when our contract expires, what happens? 280 00:17:25,530 --> 00:17:27,950 Well, we deliver our stock, which we already 281 00:17:27,950 --> 00:17:32,970 have in our hand in exchange of K dollars. 282 00:17:32,970 --> 00:17:35,220 That's our forward contract. 283 00:17:35,220 --> 00:17:38,160 On the other hand, we have to repay our loan, 284 00:17:38,160 --> 00:17:40,580 and because it was a loan, it grew. 285 00:17:40,580 --> 00:17:46,520 It grew to S_0 times e to the r*dt. 286 00:17:46,520 --> 00:17:47,870 Now, let's see. 287 00:17:47,870 --> 00:17:53,460 What would happen if K was greater than S times e 288 00:17:53,460 --> 00:17:54,460 to the r*dt? 289 00:17:54,460 --> 00:17:58,210 Then we know for sure, we know now for sure, 290 00:17:58,210 --> 00:18:02,460 that we would make money. 291 00:18:02,460 --> 00:18:06,060 There is no uncertainty about it now. 292 00:18:06,060 --> 00:18:09,190 Similarly, if K is less than this value, 293 00:18:09,190 --> 00:18:12,550 then we know that we will lose money. 294 00:18:12,550 --> 00:18:14,670 That's not how the rational market works. 295 00:18:14,670 --> 00:18:18,170 If everybody knew that by setting this price 296 00:18:18,170 --> 00:18:21,430 you would make money, people would do it all day long 297 00:18:21,430 --> 00:18:23,760 and make infinite money. 298 00:18:23,760 --> 00:18:26,720 So there will be no other side of the market. 299 00:18:26,720 --> 00:18:28,280 So the price has to go down. 300 00:18:28,280 --> 00:18:35,280 So the only choice for K, the only market-implied choice, 301 00:18:35,280 --> 00:18:42,622 is that K has to be equal to S times e to the r*dt. 302 00:18:42,622 --> 00:18:44,580 As you can see, it doesn't depend on transition 303 00:18:44,580 --> 00:18:46,200 probabilities at all. 304 00:18:46,200 --> 00:18:49,560 That's what market implies us. 305 00:18:49,560 --> 00:18:51,490 That's the price of forward contract, 306 00:18:51,490 --> 00:18:56,190 and that actually explains why, when 307 00:18:56,190 --> 00:18:59,130 I was plotting the forward contract, 308 00:18:59,130 --> 00:19:02,010 current price was just the straight line, 309 00:19:02,010 --> 00:19:06,890 it's just discounted payoff. 310 00:19:06,890 --> 00:19:15,630 The payout is linear, so just the parallel to the payoff. 311 00:19:15,630 --> 00:19:17,690 That's the idea, basically. 312 00:19:17,690 --> 00:19:25,280 The idea is to try to find such a portfolio of stock 313 00:19:25,280 --> 00:19:30,290 and the money market account with such a payout , 314 00:19:30,290 --> 00:19:35,620 which will exactly replicate the payoff of our derivative. 315 00:19:35,620 --> 00:19:37,896 If we found such of a portfolio, than we know for sure 316 00:19:37,896 --> 00:19:39,270 that the value of this portfolio, 317 00:19:39,270 --> 00:19:41,520 the replicating portfolio today is 318 00:19:41,520 --> 00:19:44,590 equal to the value of the derivative, because otherwise, 319 00:19:44,590 --> 00:19:48,110 you would make or lose money risklessly. 320 00:19:48,110 --> 00:19:51,550 That's no-arbitrage condition. 321 00:19:51,550 --> 00:19:56,300 So, can we apply it to our general one-step world? 322 00:19:56,300 --> 00:20:01,910 Well, if we have a general payout f, what we want to do, 323 00:20:01,910 --> 00:20:05,140 we want to form a replicating portfolio such 324 00:20:05,140 --> 00:20:09,970 that at expiration time, it will replicate our payouts. 325 00:20:09,970 --> 00:20:15,540 So we want to choose such constants a and b that such 326 00:20:15,540 --> 00:20:17,550 that the combination of stock and money market 327 00:20:17,550 --> 00:20:21,430 account in both states will replicate 328 00:20:21,430 --> 00:20:23,570 the payout of our option. 329 00:20:23,570 --> 00:20:27,940 Then, if we are able to find such constants a and b, 330 00:20:27,940 --> 00:20:33,900 then we just look at the current price of the contract 331 00:20:33,900 --> 00:20:36,620 and it has to be equal to the current price 332 00:20:36,620 --> 00:20:38,930 of our derivative. 333 00:20:38,930 --> 00:20:41,360 Well, but in our particular case, this is easy. 334 00:20:41,360 --> 00:20:45,770 It's just two linear equations with two unknowns, easily 335 00:20:45,770 --> 00:20:52,020 solved, and here is current price of our derivative. 336 00:20:52,020 --> 00:20:54,500 No matter what payout is -- I mean you just substitute 337 00:20:54,500 --> 00:21:02,980 the payout here, and if you know S_1 and S_2, that's it. 338 00:21:02,980 --> 00:21:06,320 A useful way to look at this, just to re-write this equation, 339 00:21:06,320 --> 00:21:13,350 is in this form, and then notice that actually the current price 340 00:21:13,350 --> 00:21:20,510 of our derivative can be viewed as a discounted expected 341 00:21:20,510 --> 00:21:25,350 payout of the derivative, but with very certain probability. 342 00:21:25,350 --> 00:21:28,460 This probability, it doesn't come 343 00:21:28,460 --> 00:21:32,020 from statistical properties of the stock or from any research, 344 00:21:32,020 --> 00:21:34,720 it actually is defined by the market. 345 00:21:34,720 --> 00:21:37,080 So it's called a risk-neutral probability. 346 00:21:37,080 --> 00:21:42,050 So this probability doesn't depend 347 00:21:42,050 --> 00:21:48,660 on the views on the market by the market participants. 348 00:21:48,660 --> 00:21:51,860 An interesting observation is that actually, 349 00:21:51,860 --> 00:21:57,435 the value of this stock, the discounted value of the stock 350 00:21:57,435 --> 00:22:02,860 is actually also is expected value of our outcomes 351 00:22:02,860 --> 00:22:06,850 under this risk-neutral probability. 352 00:22:06,850 --> 00:22:09,450 That's basically general idea. 353 00:22:09,450 --> 00:22:14,330 Now let's move one notch up and try to apply 354 00:22:14,330 --> 00:22:17,760 these idea to continuous case. 355 00:22:17,760 --> 00:22:20,600 Well, if you live in continuous world now, 356 00:22:20,600 --> 00:22:27,270 we need to make some assumptions on the behavior of the stock. 357 00:22:27,270 --> 00:22:31,225 The very common assumption is that the dynamics of the stock 358 00:22:31,225 --> 00:22:32,770 is log-normal. 359 00:22:32,770 --> 00:22:35,870 Log-normal meaning that the logarithm of the stock 360 00:22:35,870 --> 00:22:38,300 is actually normally distributed. 361 00:22:38,300 --> 00:22:43,520 So, here mu is some drift, sigma is the volatility of our stock, 362 00:22:43,520 --> 00:22:48,530 and dW is a Wiener process, W is a Wiener process 363 00:22:48,530 --> 00:22:50,970 such that dW is normally distributed 364 00:22:50,970 --> 00:22:56,320 with mean zero and variance square root dt. 365 00:22:56,320 --> 00:23:00,530 Our approach would be to find the replicating portfolio. 366 00:23:00,530 --> 00:23:01,550 And what does it mean? 367 00:23:01,550 --> 00:23:07,020 It means that we want to find such constants, over time dt -- 368 00:23:07,020 --> 00:23:10,900 so we assume that a and b are constant over the next step, 369 00:23:10,900 --> 00:23:15,730 dt -- such that the change in our derivative is a linear 370 00:23:15,730 --> 00:23:20,600 combination with this constant of the change of our underlying 371 00:23:20,600 --> 00:23:25,740 security and the change of money market account. 372 00:23:25,740 --> 00:23:31,800 Now we just need to look more closely at this equation. 373 00:23:31,800 --> 00:23:34,520 First of all, let's concentrate on df. 374 00:23:34,520 --> 00:23:44,910 So, f, our derivative, is a function 375 00:23:44,910 --> 00:23:48,660 of stock value and time. 376 00:23:48,660 --> 00:23:52,060 But unfortunately, our stock value is stochastic, 377 00:23:52,060 --> 00:24:00,110 so df is not that simple, and to write df out we have to use 378 00:24:00,110 --> 00:24:03,180 a famous -- Ito's formula from stochastic calculus, 379 00:24:03,180 --> 00:24:10,910 which actually is analogous of Taylor's formula for stochastic 380 00:24:10,910 --> 00:24:12,080 variables. 381 00:24:12,080 --> 00:24:12,710 Let's see. 382 00:24:12,710 --> 00:24:15,240 If our S will not be stochastic, if it 383 00:24:15,240 --> 00:24:17,110 would be completely deterministic 384 00:24:17,110 --> 00:24:19,640 and depend only on dt, then there 385 00:24:19,640 --> 00:24:24,670 would be no term and differential f is just 386 00:24:24,670 --> 00:24:26,800 the standard expression. 387 00:24:26,800 --> 00:24:31,070 On the other hand, if we have dependence 388 00:24:31,070 --> 00:24:35,040 on stochastic variables, then we have 389 00:24:35,040 --> 00:24:38,130 to have more terms, and why this happens? 390 00:24:38,130 --> 00:24:41,330 Well, in very rough words is that because the order 391 00:24:41,330 --> 00:24:45,510 of magnitude of dW is higher than dt's -- 392 00:24:45,510 --> 00:24:47,930 it's square root of dt. 393 00:24:47,930 --> 00:24:51,340 So we have to make into account more terms, 394 00:24:51,340 --> 00:24:56,970 and in particular, we have to take into account next order 395 00:24:56,970 --> 00:24:59,070 of dS squared. 396 00:24:59,070 --> 00:25:01,970 Formally, dS square can be written this way, 397 00:25:01,970 --> 00:25:04,900 and again, very rough explanation is as follows. 398 00:25:04,900 --> 00:25:07,970 If we would square this equation there 399 00:25:07,970 --> 00:25:09,490 will be three terms there. 400 00:25:09,490 --> 00:25:11,900 One would come from the square of this term, 401 00:25:11,900 --> 00:25:16,530 and this would be of the order of dt squared, 402 00:25:16,530 --> 00:25:20,240 next order of magnitude -- much smaller than dt. 403 00:25:20,240 --> 00:25:23,060 The second term will be cross-product of dW*dt. 404 00:25:23,060 --> 00:25:26,200 What order of magnitude we are talking about, 405 00:25:26,200 --> 00:25:32,990 it is dt to the power 3/2, again, much smaller than dt. 406 00:25:32,990 --> 00:25:37,090 On the other hand, the third term will be the square of dW, 407 00:25:37,090 --> 00:25:40,520 this is of order of magnitude of dt, 408 00:25:40,520 --> 00:25:43,500 so that's what we have to keep. 409 00:25:43,500 --> 00:25:48,990 And that's what Ito's formula is about. 410 00:25:48,990 --> 00:25:56,210 Now, we are basically, we know all terms here, 411 00:25:56,210 --> 00:26:01,560 and let me stress out that this term, dB, it is not stochastic, 412 00:26:01,560 --> 00:26:04,860 it's completely deterministic because we 413 00:26:04,860 --> 00:26:09,350 know that B grows with the rate r, that's what it is. 414 00:26:09,350 --> 00:26:14,510 So we substitute all those terms into our replicating equation. 415 00:26:14,510 --> 00:26:16,000 We collect the terms. 416 00:26:16,000 --> 00:26:17,120 We get this equation. 417 00:26:17,120 --> 00:26:19,580 And again, there is the deterministic part, 418 00:26:19,580 --> 00:26:20,850 there is stochastic part. 419 00:26:20,850 --> 00:26:23,710 So the only way for this equation to hold 420 00:26:23,710 --> 00:26:27,543 is this term to be equal to this term, and this term 421 00:26:27,543 --> 00:26:32,170 to be equal to this term, and that's what's written out here. 422 00:26:32,170 --> 00:26:34,780 So again, two equations, these two unknowns, and here 423 00:26:34,780 --> 00:26:35,660 is answer. 424 00:26:38,210 --> 00:26:47,800 Finally, let's take a*S to another part. 425 00:26:47,800 --> 00:26:50,720 Notice that this part of our equation 426 00:26:50,720 --> 00:26:53,880 is completely deterministic. 427 00:26:53,880 --> 00:26:55,810 So we know how it will grow. 428 00:26:55,810 --> 00:27:01,020 So basically, d of f minus a*S, which is b times dB, 429 00:27:01,020 --> 00:27:05,890 is r times b times dt. 430 00:27:05,890 --> 00:27:08,740 And we know all other terms, we substitute them 431 00:27:08,740 --> 00:27:12,670 here, take something to the left-hand side 432 00:27:12,670 --> 00:27:14,220 and get this equation. 433 00:27:14,220 --> 00:27:21,680 So this is partial differential equation for our derivative f, 434 00:27:21,680 --> 00:27:26,670 as a function of S and t, of second order, 435 00:27:26,670 --> 00:27:30,310 and this equation is the famous Black-Scholes equation. 436 00:27:30,310 --> 00:27:34,200 It was derived by Fischer Black and Myron Scholes 437 00:27:34,200 --> 00:27:38,940 in their famous paper published in 1973. 438 00:27:38,940 --> 00:27:41,240 Myron Scholes and Robert Merton actually 439 00:27:41,240 --> 00:27:44,640 received Nobel Prize for deriving and solving 440 00:27:44,640 --> 00:27:47,560 this equation in '97. 441 00:27:47,560 --> 00:27:51,250 Black was already dead by the time. 442 00:27:51,250 --> 00:27:55,320 This is really the cornerstone of math finance. 443 00:27:59,290 --> 00:28:05,780 The cornerstone is because using the replicating portfolio, 444 00:28:05,780 --> 00:28:12,080 using this reasoning, we were able to find an exact equation 445 00:28:12,080 --> 00:28:14,040 for our derivative. 446 00:28:14,040 --> 00:28:16,980 So a few remarks on Black-Scholes. 447 00:28:16,980 --> 00:28:20,930 So first of all, we made some assumptions 448 00:28:20,930 --> 00:28:24,600 on the dynamic of the stock, but we never made any assumptions 449 00:28:24,600 --> 00:28:27,150 on our derivative. 450 00:28:27,150 --> 00:28:32,610 Which means that any derivative has to satisfy this equation, 451 00:28:32,610 --> 00:28:35,040 and that's very strong result. So if you 452 00:28:35,040 --> 00:28:36,740 assume that our stock is lognormal, 453 00:28:36,740 --> 00:28:39,780 which is not a bad assumption and agrees quite well 454 00:28:39,780 --> 00:28:42,810 with the market, then we basically, in principle, 455 00:28:42,810 --> 00:28:44,470 can price any derivative. 456 00:28:44,470 --> 00:28:48,400 We know the equation for any derivative. 457 00:28:48,400 --> 00:28:51,800 The other thing is that our Black-Scholes equation 458 00:28:51,800 --> 00:28:55,530 doesn't depend on the actual drift mu 459 00:28:55,530 --> 00:28:57,180 in the dynamics of our stock. 460 00:28:57,180 --> 00:29:06,320 So again, it is the manifest of risk-neutral dynamic. 461 00:29:06,320 --> 00:29:12,150 Not only we wrote down the equation for our derivative, 462 00:29:12,150 --> 00:29:15,030 we also found a replicating portfolio. 463 00:29:15,030 --> 00:29:18,280 So in other words, we found a hedging strategy, 464 00:29:18,280 --> 00:29:21,795 meaning that at any given time we 465 00:29:21,795 --> 00:29:28,360 can form this portfolio with rates a and b. 466 00:29:28,360 --> 00:29:33,460 If we hold both the derivative and both replicating portfolio, 467 00:29:33,460 --> 00:29:38,490 altogether, this is zero sum gain. 468 00:29:38,490 --> 00:29:41,950 We know that no matter where stock moves, 469 00:29:41,950 --> 00:29:44,710 we will not lose money or gain money. 470 00:29:44,710 --> 00:29:51,110 So if we just charge bid-offer on the derivative, 471 00:29:51,110 --> 00:29:53,330 if we charge a fee on the contract, 472 00:29:53,330 --> 00:29:56,420 we can hedge ourself perfectly, buy the contract 473 00:29:56,420 --> 00:29:58,970 or sell the contract, hedge perfectly ourself 474 00:29:58,970 --> 00:30:01,460 and just make money on the fee, that's it. 475 00:30:06,180 --> 00:30:09,060 Finally, more mathematical remark 476 00:30:09,060 --> 00:30:13,250 is that actually after a few manipulations, a few change 477 00:30:13,250 --> 00:30:16,710 of variables, the Black-Scholes equation comes out 478 00:30:16,710 --> 00:30:19,980 to be just a heat question, which you already 479 00:30:19,980 --> 00:30:22,190 saw in this class. 480 00:30:22,190 --> 00:30:23,650 This is very good news. 481 00:30:23,650 --> 00:30:25,000 Why is this good news? 482 00:30:25,000 --> 00:30:27,250 Well, because heat equation is very well studied. 483 00:30:27,250 --> 00:30:32,986 So the solutions are well-known, and numerical methods, the ways 484 00:30:32,986 --> 00:30:34,860 to solve it, in particular the numerical ways 485 00:30:34,860 --> 00:30:36,760 to solve it are well-known. 486 00:30:36,760 --> 00:30:38,220 So we are in business. 487 00:30:38,220 --> 00:30:45,130 But as any partial differential equation, 488 00:30:45,130 --> 00:30:49,450 the equation itself doesn't make much sense because to find 489 00:30:49,450 --> 00:30:55,610 a particular solution we need boundary and initial condition. 490 00:30:58,520 --> 00:31:01,940 And although any derivative satisfies Black-Scholes 491 00:31:01,940 --> 00:31:08,310 equation, the final and boundary conditions 492 00:31:08,310 --> 00:31:10,720 will vary from contract to contract. 493 00:31:10,720 --> 00:31:15,510 Here are a few examples of the final and boundary conditions. 494 00:31:15,510 --> 00:31:18,680 Here, an interesting remark that if, usually, 495 00:31:18,680 --> 00:31:22,270 we would talk about initial condition, here 496 00:31:22,270 --> 00:31:24,650 we are talking about final condition. 497 00:31:24,650 --> 00:31:26,340 The time goes in reverse; we know 498 00:31:26,340 --> 00:31:30,150 the state of the world at the end, at expiration, not today. 499 00:31:30,150 --> 00:31:36,810 So, here are final and boundary conditions for call and put, 500 00:31:36,810 --> 00:31:40,640 and let's look a little bit at the pictures for our call 501 00:31:40,640 --> 00:31:44,250 and put to see where they come from. 502 00:31:44,250 --> 00:31:48,950 So for example, for calls, well, this is our final condition, 503 00:31:48,950 --> 00:31:50,930 right, this is defined by the payout. 504 00:31:50,930 --> 00:31:54,690 On the other hand, the boundary condition, well, what happens, 505 00:31:54,690 --> 00:31:57,646 we put them at zero at an infinity, we [INAUDIBLE PHRASE] 506 00:31:57,646 --> 00:31:59,020 to put them at zero and infinity. 507 00:31:59,020 --> 00:31:59,519 And why? 508 00:31:59,519 --> 00:32:03,340 Well, because if stock hits zero then it stays at zero. 509 00:32:03,340 --> 00:32:06,330 That's what our dynamics show. 510 00:32:06,330 --> 00:32:11,400 So the value of our contract at maturity will become zero. 511 00:32:11,400 --> 00:32:14,990 On the other hand, if the stock grows, grows to infinity, 512 00:32:14,990 --> 00:32:17,360 a good assumption to make is that actually it 513 00:32:17,360 --> 00:32:19,830 becomes similar to stock itself, so it would just 514 00:32:19,830 --> 00:32:26,160 become parallel to the stock, and that's the conditions 515 00:32:26,160 --> 00:32:27,150 which we impose here. 516 00:32:30,270 --> 00:32:36,290 Similarly for the put, you can derive these conditions. 517 00:32:36,290 --> 00:32:41,740 And again, just because it is a heat equation, 518 00:32:41,740 --> 00:32:44,910 it turns out that for a simple derivative such as the calls 519 00:32:44,910 --> 00:32:50,070 and puts, it is possible to find an exact analytic solution. 520 00:32:50,070 --> 00:32:54,350 Here are exact analytic set of solutions for a call, put 521 00:32:54,350 --> 00:32:57,090 and the digital contracts. 522 00:32:57,090 --> 00:33:01,310 Well, not surprising again, I mean 523 00:33:01,310 --> 00:33:04,310 they're all connected to the error function, 524 00:33:04,310 --> 00:33:07,760 so to the normal distribution, basically, 525 00:33:07,760 --> 00:33:12,400 as the solutions of heat equation ought to be. 526 00:33:12,400 --> 00:33:14,580 Why do they look exactly the same? 527 00:33:14,580 --> 00:33:17,020 If we have five minutes at the end, 528 00:33:17,020 --> 00:33:21,690 we'll probably shed some light on the specific form 529 00:33:21,690 --> 00:33:22,750 of equations. 530 00:33:22,750 --> 00:33:28,090 But let me just stress that we can see that it's discounted, 531 00:33:28,090 --> 00:33:32,790 and what I'm claiming, it's expected value of our payout 532 00:33:32,790 --> 00:33:36,810 under risk-neutral measure. 533 00:33:36,810 --> 00:33:42,290 Here is an example -- it's of a particular call option 534 00:33:42,290 --> 00:33:44,160 on the same IBM stock. 535 00:33:44,160 --> 00:33:46,580 So I chose the short-dated contract, 536 00:33:46,580 --> 00:33:50,300 just to avoid the dividend payment. 537 00:33:50,300 --> 00:33:53,820 So it's a contract expiring on March 18. 538 00:33:53,820 --> 00:33:56,750 So there is 10 days to expiration. 539 00:33:56,750 --> 00:34:02,170 The stock, as we saw, the expirations of stock, 540 00:34:02,170 --> 00:34:06,750 as we saw, is what's trading at 81.14. 541 00:34:06,750 --> 00:34:11,800 The volatility is somewhere around 14%, 542 00:34:11,800 --> 00:34:17,110 estimated either from other options or historically. 543 00:34:17,110 --> 00:34:20,210 Here is the price of our contract. 544 00:34:20,210 --> 00:34:23,750 I also have a simple Black-Scholes calculator here, 545 00:34:23,750 --> 00:34:28,800 and let's see if we can match this price. 546 00:34:28,800 --> 00:34:39,280 So let's see, I believe the volatility was 13%, right, 547 00:34:39,280 --> 00:34:42,240 13.47. 548 00:34:42,240 --> 00:34:45,220 The interest rate, it's already here. 549 00:34:45,220 --> 00:34:48,540 As we all know, Fed just bumped the interest rate, 550 00:34:48,540 --> 00:34:51,980 so they are at 4.75% right now. 551 00:34:51,980 --> 00:34:56,360 The strike of our option was 80. 552 00:34:56,360 --> 00:34:58,840 Time to expiration was actually 10 days, 553 00:34:58,840 --> 00:35:02,390 and this should be measured at a fraction of year. 554 00:35:02,390 --> 00:35:08,440 So we divide 10 by 365. 555 00:35:08,440 --> 00:35:15,350 This stock was trading at 81.14, if I'm not mistaken. 556 00:35:15,350 --> 00:35:21,210 Here is the price of our call options contract, which is 150. 557 00:35:21,210 --> 00:35:26,630 Well, it's within the offer. 558 00:35:26,630 --> 00:35:29,070 So maybe our volatility's slightly off 559 00:35:29,070 --> 00:35:35,140 and if we increase [UNINTELLIGIBLE] to say, 560 00:35:35,140 --> 00:35:42,330 increase it to 14%, it will go slightly up. 561 00:35:42,330 --> 00:35:44,800 152. 562 00:35:44,800 --> 00:35:48,580 Well, in general, let's play a little bit with it. 563 00:35:48,580 --> 00:35:50,480 Well, it is very short-dated option, 564 00:35:50,480 --> 00:35:55,990 so the value of our option is very close to the payout. 565 00:35:55,990 --> 00:35:58,870 So if we increase the time to maturity, 566 00:35:58,870 --> 00:36:02,280 let's make it two years just to see where -- 567 00:36:02,280 --> 00:36:08,980 so now value of our option is -- well, that's what it is. 568 00:36:08,980 --> 00:36:15,040 If increase volatility, sure enough, let's make it 30%. 569 00:36:15,040 --> 00:36:15,970 So what do we expect? 570 00:36:15,970 --> 00:36:19,340 We expect if volatility is higher, the uncertainty higher, 571 00:36:19,340 --> 00:36:25,300 so the value of our contract should go up, and it sure does. 572 00:36:29,120 --> 00:36:31,520 So basically that's how Black-Scholes works. 573 00:36:39,690 --> 00:36:41,520 And plenty of those contracts trade 574 00:36:41,520 --> 00:36:47,670 on the market, but unfortunately not all of these contracts 575 00:36:47,670 --> 00:36:50,600 are so simple as calls and puts. 576 00:36:50,600 --> 00:36:55,610 First of all, there are many more complicated products 577 00:36:55,610 --> 00:37:03,390 with more difficult payout, which will constitute different 578 00:37:03,390 --> 00:37:08,800 and probably discontinuous final conditions on our Black-Scholes 579 00:37:08,800 --> 00:37:10,930 equation. 580 00:37:10,930 --> 00:37:14,590 Moreover, we made an assumption that the volatility 581 00:37:14,590 --> 00:37:16,650 is constant with time, and interest rate 582 00:37:16,650 --> 00:37:18,210 is constant with time. 583 00:37:18,210 --> 00:37:21,860 It is certainly not true for the real world. 584 00:37:21,860 --> 00:37:24,790 Volatility probably should be time-dependent, 585 00:37:24,790 --> 00:37:26,540 and this would make the coefficients 586 00:37:26,540 --> 00:37:30,090 in our Black-Scholes equation time-dependent. 587 00:37:30,090 --> 00:37:33,250 Unfortunately, these cannot be solve analytically. 588 00:37:33,250 --> 00:37:36,420 So in most of the cases in practice, 589 00:37:36,420 --> 00:37:42,530 we will have to use some kind of numerical solution. 590 00:37:42,530 --> 00:37:45,870 Finite difference methods is the typical approach 591 00:37:45,870 --> 00:37:47,200 for the heat equation. 592 00:37:47,200 --> 00:37:50,330 As you know, both explicit and implicit schemes, 593 00:37:50,330 --> 00:37:57,910 and you will discuss some of those in 18.086. 594 00:37:57,910 --> 00:37:58,630 Tree methods. 595 00:37:58,630 --> 00:38:05,090 Tree methods meaning that we go back to our one-step tree, 596 00:38:05,090 --> 00:38:08,380 and basically assume that our time to expiration 597 00:38:08,380 --> 00:38:12,000 is many time steps away and we'll grow the tree further, 598 00:38:12,000 --> 00:38:14,270 so from this node we have two more nodes, 599 00:38:14,270 --> 00:38:15,550 and so and so forth. 600 00:38:15,550 --> 00:38:19,670 That would imply the final condition at the end, 601 00:38:19,670 --> 00:38:23,640 and discount back using our risk-neutral probabilities, 602 00:38:23,640 --> 00:38:25,170 and get the price now. 603 00:38:25,170 --> 00:38:27,240 So those are called tree methods. 604 00:38:27,240 --> 00:38:30,330 One can show that actually those tree methods are equivalent 605 00:38:30,330 --> 00:38:36,100 to finite difference -- explicit finite difference schemes. 606 00:38:36,100 --> 00:38:38,110 Those are very popular. 607 00:38:38,110 --> 00:38:41,830 But again, in tree methods, what is very important 608 00:38:41,830 --> 00:38:44,746 is to set the probabilities from your tree, the transition 609 00:38:44,746 --> 00:38:46,120 probabilities, to the right ones, 610 00:38:46,120 --> 00:38:48,960 and the right ones are risk-neutral probabilities. 611 00:38:48,960 --> 00:38:53,170 Probabilities implied by the market, actually. 612 00:38:53,170 --> 00:38:58,280 Another important numerical method is Monte Carlo 613 00:38:58,280 --> 00:39:02,910 simulation where you would simulate many different 614 00:39:02,910 --> 00:39:05,650 scenarios of the development of your stock up to the maturity, 615 00:39:05,650 --> 00:39:09,840 and then, basically find -- using this path, 616 00:39:09,840 --> 00:39:15,990 you will find the expected value of your payout. 617 00:39:15,990 --> 00:39:20,210 But again, in order for this expected value 618 00:39:20,210 --> 00:39:24,810 to be the same as the risk-neutral value, 619 00:39:24,810 --> 00:39:27,050 as the arbitrage-free value, you have 620 00:39:27,050 --> 00:39:29,910 to develop your Monte Carlo simulations 621 00:39:29,910 --> 00:39:31,800 with risk-neutral probabilities. 622 00:39:31,800 --> 00:39:37,360 So, risk-neutral valuation is extremely important. 623 00:39:40,780 --> 00:39:45,810 Here is actually the general risk-neutral statement, which 624 00:39:45,810 --> 00:39:51,580 one can prove, is that actually, the value of any derivative 625 00:39:51,580 --> 00:39:56,140 is just discounted expected value of the payout 626 00:39:56,140 --> 00:39:59,280 of this derivative at maturity, but you 627 00:39:59,280 --> 00:40:02,140 have to take this expectation at the right measure. 628 00:40:02,140 --> 00:40:05,230 Using the right measure, meaning that you have to set correctly 629 00:40:05,230 --> 00:40:08,705 the transition probability -- you have to make them 630 00:40:08,705 --> 00:40:09,330 market-neutral. 631 00:40:12,710 --> 00:40:19,950 Under this measure, actually the dynamics of our stocks 632 00:40:19,950 --> 00:40:22,660 looks slightly different, and as you can see, 633 00:40:22,660 --> 00:40:25,240 our drift becomes the interest rate. 634 00:40:25,240 --> 00:40:27,810 So under risk-neutral measure, everything 635 00:40:27,810 --> 00:40:32,430 grows with our risk-free interest rate. 636 00:40:32,430 --> 00:40:38,720 Just to shed a little bit of light on how we go 637 00:40:38,720 --> 00:40:42,630 at the solutions for calls and puts, 638 00:40:42,630 --> 00:40:44,920 Black-Scholes solutions for call and put, 639 00:40:44,920 --> 00:40:47,920 well this is the distribution of our stock, 640 00:40:47,920 --> 00:40:51,790 log-normal distribution of our stock at time T, 641 00:40:51,790 --> 00:40:57,660 and if we take this distribution and integrate our payout 642 00:40:57,660 --> 00:41:01,950 of our call option against this distribution -- in other words, 643 00:41:01,950 --> 00:41:10,260 find the expected value of payout of our call option under 644 00:41:10,260 --> 00:41:17,810 risk-neutral measure, then, sure enough, 645 00:41:17,810 --> 00:41:21,200 you will get [UNINTELLIGIBLE PHRASE]. 646 00:41:21,200 --> 00:41:24,000 This illustrates the best -- because what is digital? 647 00:41:24,000 --> 00:41:26,750 Digital is just the probability to end up 648 00:41:26,750 --> 00:41:29,590 at above the strike at time T, right? 649 00:41:29,590 --> 00:41:34,110 So if you integrate this log-normal pdf 650 00:41:34,110 --> 00:41:42,310 from the strike K to infinity, that will be your answer. 651 00:41:42,310 --> 00:41:44,930 This is a good exercise in integration, 652 00:41:44,930 --> 00:41:47,760 to make sure that it's correct. 653 00:41:47,760 --> 00:41:49,270 So let's see. 654 00:41:49,270 --> 00:41:52,170 To conclude, what we've seen. 655 00:41:52,170 --> 00:42:02,180 So, we have seen that modern derivatives business makes 656 00:42:02,180 --> 00:42:04,600 use of quite advanced mathematics, 657 00:42:04,600 --> 00:42:08,050 and what kinds of mathematics is used there? 658 00:42:08,050 --> 00:42:12,630 Well, partial differential equations are used heavily. 659 00:42:12,630 --> 00:42:14,530 Numerical methods for the solution 660 00:42:14,530 --> 00:42:20,470 of this partial differential equations are naturally used. 661 00:42:20,470 --> 00:42:23,890 In order to get these equations, we actually 662 00:42:23,890 --> 00:42:26,870 need to operate in terms of stochastic calculus, 663 00:42:26,870 --> 00:42:31,180 meaning that we need to know how to deal with Ito calculus, 664 00:42:31,180 --> 00:42:36,580 Ito formula, Girsanov theorem, and so on and so forth. 665 00:42:36,580 --> 00:42:41,330 The other thing is to be able to build simulations 666 00:42:41,330 --> 00:42:45,380 to solve the heat equation and all other equations 667 00:42:45,380 --> 00:42:47,780 that you might encounter. 668 00:42:47,780 --> 00:42:51,230 The topic which we didn't touch upon 669 00:42:51,230 --> 00:42:54,660 is statistics because, of course, very advanced 670 00:42:54,660 --> 00:42:58,690 statistics is used for many, many things, 671 00:42:58,690 --> 00:43:02,320 for analyzing historical data, which 672 00:43:02,320 --> 00:43:04,040 can be quite beautiful for trading 673 00:43:04,040 --> 00:43:07,660 strategies and many others. 674 00:43:07,660 --> 00:43:12,450 Besides these five topics, there is much, much more 675 00:43:12,450 --> 00:43:17,400 to mathematical finance, which makes it a very, very 676 00:43:17,400 --> 00:43:19,760 exciting field to work in. 677 00:43:19,760 --> 00:43:22,126 That's what I wanted to talk about. 678 00:43:22,126 --> 00:43:23,750 Thank you very much for your attention. 679 00:43:30,890 --> 00:43:35,430 PROFESSOR: Maybe I'll ask a firm question 680 00:43:35,430 --> 00:43:38,100 about boundary conditions, because you 681 00:43:38,100 --> 00:43:40,060 had said that those are different 682 00:43:40,060 --> 00:43:42,860 for different contracts. 683 00:43:42,860 --> 00:43:47,290 And how do you deal with them in the finite differences 684 00:43:47,290 --> 00:43:52,140 or the tree model or whatever? 685 00:43:52,140 --> 00:43:54,750 What would be a typical one? 686 00:43:54,750 --> 00:43:57,560 GUEST SPEAKER: Well, typical one -- two very typical ones. 687 00:43:57,560 --> 00:44:05,500 So those, you basically make a grid of your problem, 688 00:44:05,500 --> 00:44:08,100 in particular, you build a tree, which 689 00:44:08,100 --> 00:44:11,650 is actually a grid of all possible outcomes. 690 00:44:11,650 --> 00:44:28,490 You set them up at the end, so your tree grows -- 691 00:44:28,490 --> 00:44:36,310 so you set your boundary here at the end, and well, you set, 692 00:44:36,310 --> 00:44:45,480 probably, some initial -- this is final condition, 693 00:44:45,480 --> 00:44:48,690 so you set some boundary conditions here. 694 00:44:48,690 --> 00:44:55,710 So this is your time T. This is t = 0, time t -- this is 0, 695 00:44:55,710 --> 00:45:04,030 this is 1, this is 2, this is T. So you set your payout here, 696 00:45:04,030 --> 00:45:11,340 so it will be maximum of S minus K and 0. 697 00:45:11,340 --> 00:45:14,940 PROFESSOR: How many time steps might you take in this? 698 00:45:14,940 --> 00:45:20,315 GUEST SPEAKER: Well, you would do like daily for three months 699 00:45:20,315 --> 00:45:22,476 -- if you three-month options. 700 00:45:22,476 --> 00:45:23,600 PROFESSOR: Maybe 100 steps. 701 00:45:23,600 --> 00:45:25,308 GUEST SPEAKER: Yeah, something like that. 702 00:45:25,308 --> 00:45:28,470 Well, if it's two-year option, that you probably would do it 703 00:45:28,470 --> 00:45:30,820 weekly or something like that. 704 00:45:30,820 --> 00:45:32,350 PROFESSOR: You don't get into large, 705 00:45:32,350 --> 00:45:35,490 what would be scientifically, large-scale calculating. 706 00:45:35,490 --> 00:45:37,490 PROFESSOR: No, in finance we usually 707 00:45:37,490 --> 00:45:39,040 don't keep this problem--. 708 00:45:39,040 --> 00:45:40,810 PROFESSOR: In finite differences, 709 00:45:40,810 --> 00:45:44,670 do you use like higher-order -- suppose, well, 710 00:45:44,670 --> 00:45:47,230 you had second derivatives, would you always use second 711 00:45:47,230 --> 00:45:51,449 differences or second-order accuracy? 712 00:45:51,449 --> 00:45:52,740 GUEST SPEAKER: In general, yes. 713 00:45:52,740 --> 00:45:55,790 In general, second-order accuracy. 714 00:45:55,790 --> 00:45:58,210 In general you don't go higher. 715 00:45:58,210 --> 00:46:01,920 I mean the precision -- well, it's within cents, right. 716 00:46:01,920 --> 00:46:06,010 So you cannot do better than that. 717 00:46:06,010 --> 00:46:11,650 So it depends -- well, it depends what kind of amount you 718 00:46:11,650 --> 00:46:12,830 are dealing with. 719 00:46:12,830 --> 00:46:16,775 If you're actually selling and buying units of stock, 720 00:46:16,775 --> 00:46:21,080 you might consider something more precise. 721 00:46:21,080 --> 00:46:25,220 But it's very problem-defined. 722 00:46:30,280 --> 00:46:33,370 So that's how we deal with it. 723 00:46:33,370 --> 00:46:35,770 PROFESSOR: Any questions? 724 00:46:35,770 --> 00:46:38,440 You can put the mic on if you have a question. 725 00:46:38,440 --> 00:46:45,080 STUDENT: [UNINTELLIGIBLE PHRASE]. 726 00:46:55,800 --> 00:46:58,000 GUEST SPEAKER: Well, it is Markov process. 727 00:46:58,000 --> 00:46:58,500 Yes. 728 00:46:58,500 --> 00:47:01,480 I mean, this is just a numerical solution. 729 00:47:01,480 --> 00:47:02,910 So yeah, it is Markov process. 730 00:47:02,910 --> 00:47:04,990 and basically all stochastic calculus 731 00:47:04,990 --> 00:47:07,590 is about Markov process, continuous Markov process. 732 00:47:15,030 --> 00:47:17,010 PROFESSOR: Is the mathematics that you 733 00:47:17,010 --> 00:47:22,600 get involved with pretty well set now or is there 734 00:47:22,600 --> 00:47:27,000 a need for more mathematics, if I 735 00:47:27,000 --> 00:47:28,920 can ask the question that way? 736 00:47:28,920 --> 00:47:31,140 PROFESSOR: Yeah, well, in this field 737 00:47:31,140 --> 00:47:33,610 it is probably quite well set. 738 00:47:33,610 --> 00:47:37,410 But if you get into more complicated fields, 739 00:47:37,410 --> 00:47:44,690 especially into credit modeling, the model for the credits 740 00:47:44,690 --> 00:47:49,810 of certain companies, then mathematics is not quite set, 741 00:47:49,810 --> 00:47:53,730 because there, you start talking about jump processes 742 00:47:53,730 --> 00:47:58,860 and not Wiener processes, not just log-normal processes. 743 00:47:58,860 --> 00:48:03,070 This stochastic differential equation become very hard, 744 00:48:03,070 --> 00:48:06,150 but maybe still analytically tractable. 745 00:48:06,150 --> 00:48:09,330 So from this point of view there is need -- 746 00:48:09,330 --> 00:48:13,281 but it's not a fundamental mathematics, 747 00:48:13,281 --> 00:48:15,030 it's not that you are opening a new field, 748 00:48:15,030 --> 00:48:21,170 but definitely trying to solve a stochastic differential 749 00:48:21,170 --> 00:48:23,840 equation -- which usually boils down to solving a partial 750 00:48:23,840 --> 00:48:26,780 differential equation analytically -- 751 00:48:26,780 --> 00:48:30,280 can be pretty hard a mathematical problem, 752 00:48:30,280 --> 00:48:32,280 viewed as a mathematical problem. 753 00:48:32,280 --> 00:48:33,790 PROFESSOR: So you showed the example 754 00:48:33,790 --> 00:48:37,700 of Black-Scholes solver. 755 00:48:37,700 --> 00:48:39,872 Everybody has that available all the time? 756 00:48:39,872 --> 00:48:40,830 GUEST SPEAKER: Oh yeah. 757 00:48:40,830 --> 00:48:44,550 On Chicago trading floor, the traders 758 00:48:44,550 --> 00:48:47,520 have calculators where they just press a button 759 00:48:47,520 --> 00:48:49,104 and it's just hard-wired there. 760 00:48:49,104 --> 00:48:51,270 PROFESSOR: And they're printing out error functions, 761 00:48:51,270 --> 00:48:55,060 basically -- a combination of error function, yeah. 762 00:48:55,060 --> 00:49:01,110 GUEST SPEAKER: Well, sure enough, nobody uses just -- 763 00:49:01,110 --> 00:49:04,820 I mean this was very approximate example and that's why I chose 764 00:49:04,820 --> 00:49:09,390 such short-dated stock, that before it pays any dividends, 765 00:49:09,390 --> 00:49:12,920 and where we can assume the volatility is constant, 766 00:49:12,920 --> 00:49:15,400 and so on and so forth, to match the prices. 767 00:49:15,400 --> 00:49:28,430 Otherwise, the prices wouldn't match. 768 00:49:28,430 --> 00:49:29,930 PROFESSOR: Thank you.