1 00:00:01,420 --> 00:00:04,270 So, we're finally arriving at week number eight 2 00:00:05,430 --> 00:00:07,960 where I'm going to talk about numerical methods. 3 00:00:07,960 --> 00:00:11,730 So I think it's you know, when we looked at the Lagrange 4 00:00:11,730 --> 00:00:15,480 multipliers, we started to arrive at things where they were getting just. 5 00:00:15,480 --> 00:00:18,250 You could still solve them in special cases. 6 00:00:18,250 --> 00:00:22,780 for instance, when we looked at the minimum, minimum variance portfolio. 7 00:00:22,780 --> 00:00:26,570 It was still possible to, where we ended up like the linear system, 8 00:00:26,570 --> 00:00:28,790 we need to dissolve to do the Lagrange multipliers. 9 00:00:31,230 --> 00:00:32,700 here to find the critical point for the 10 00:00:32,700 --> 00:00:36,155 Lagrange multiplier, the critical point for the Lagrangian. 11 00:00:36,155 --> 00:00:37,850 [COUGH] 12 00:00:37,850 --> 00:00:42,870 But pretty much anything beyond that, any anything that is not sort of 13 00:00:42,870 --> 00:00:47,570 contrived to be simple is going to be far too difficult to solve. 14 00:00:47,570 --> 00:00:50,200 And so that's why we have something called the numerical methods. 15 00:00:50,200 --> 00:00:54,720 Which I sort of think of as just the logical extension 16 00:00:54,720 --> 00:00:58,430 of the guess and check method that I learned in elementary school. 17 00:00:58,430 --> 00:01:01,700 So you basically are just trying to make a guess. 18 00:01:01,700 --> 00:01:02,940 You'll measure how 19 00:01:02,940 --> 00:01:05,830 wrong you are, and that will hopefully suggest 20 00:01:05,830 --> 00:01:08,000 a way for you to improve your guess. 21 00:01:08,000 --> 00:01:10,160 And you want to do that in a way where you 22 00:01:10,160 --> 00:01:13,270 find the answer you're looking for with a minimum amount of work. 23 00:01:15,590 --> 00:01:17,360 So the motivation for this, I'm, I'm 24 00:01:17,360 --> 00:01:20,760 going to look at a problem called implied volatility. 25 00:01:20,760 --> 00:01:23,490 And then I'll talk about two methods for solving that. 26 00:01:23,490 --> 00:01:26,060 So one is bisection method, which is just a 27 00:01:26,060 --> 00:01:30,750 very easy approach to solving this implied volatility problem. 28 00:01:30,750 --> 00:01:36,090 Then Newton's method, which is not much more difficult, but theoretically 29 00:01:36,090 --> 00:01:40,950 it's a bit more, it, it has some theoretical, some nice theoretical 30 00:01:40,950 --> 00:01:43,290 results we can, we can look at. 31 00:01:43,290 --> 00:01:46,430 Now, also the nice thing about Newton's method 32 00:01:46,430 --> 00:01:49,409 is it can be extended for N dimensional problems. 33 00:01:51,210 --> 00:01:54,570 And then finally, and I don't know if we'll have time to get to this today. 34 00:01:56,200 --> 00:02:01,110 we'll look at solving a problem with Lagrange method, so it's actually 35 00:02:01,110 --> 00:02:05,990 one of the same problems I looked at in the Lagrange slides. 36 00:02:05,990 --> 00:02:09,960 But instead of just trying to do algebra once I need to find 37 00:02:09,960 --> 00:02:13,250 the critical point, I'll use Newton's method to try and find the critical point. 38 00:02:15,490 --> 00:02:18,130 So we get started by talking about implied volatility. 39 00:02:21,938 --> 00:02:25,820 So if we remember from the first half of the course, we were looking at the, 40 00:02:25,820 --> 00:02:28,090 the Black-Scholes formula for a while, and mostly 41 00:02:28,090 --> 00:02:30,259 when we were looking at computing partial derivatives. 42 00:02:31,670 --> 00:02:33,420 And so, this is just to remind you what it 43 00:02:33,420 --> 00:02:36,570 is, and hopefully I haven't made any, any typos in it. 44 00:02:39,390 --> 00:02:42,170 And if you look, well, I think I've got all of the, 45 00:02:44,730 --> 00:02:50,240 so the maturity, so the, the maturity date of an option, this is 46 00:02:50,240 --> 00:02:54,380 a, a contract that I go and buy and sell in a market. 47 00:02:54,380 --> 00:02:57,730 And when it's written, it's going to have a maturity date. 48 00:02:57,730 --> 00:03:00,210 So, it's good for 3 months or it's good for 6 months. 49 00:03:00,210 --> 00:03:00,940 Something like that. 50 00:03:00,940 --> 00:03:04,510 It has something called the strike price, which is, this 51 00:03:04,510 --> 00:03:07,220 is also something going to be written on the contract. 52 00:03:07,220 --> 00:03:10,030 So basically, once this contract comes 53 00:03:10,030 --> 00:03:17,150 into existence, the T Oops, and the K are fixed, so they're never going to change. 54 00:03:20,720 --> 00:03:23,330 And then there's a secondary market for these contracts. 55 00:03:23,330 --> 00:03:28,150 So once I have the contract, if it's between the time that it was written 56 00:03:28,150 --> 00:03:33,050 and the maturity, I can go to the secondary market and I can sell my option. 57 00:03:33,050 --> 00:03:37,410 So I don't actually have to wait for the condition in the auction to come to pass. 58 00:03:37,410 --> 00:03:41,210 I can go and find somebody else who's interested in having that 59 00:03:41,210 --> 00:03:45,140 same type of protection and I can sell my option to him. 60 00:03:45,140 --> 00:03:46,380 And so if I 61 00:03:46,380 --> 00:03:53,780 sell the option, at the time that I sell the option the price that I sell it for 62 00:03:53,780 --> 00:03:55,670 is going to be known, because you know, I 63 00:03:55,670 --> 00:03:57,770 give him the contract, he gives me some money. 64 00:03:57,770 --> 00:03:59,900 So I can look at the amount of money he gave me 65 00:03:59,900 --> 00:04:02,840 and, and I know exactly what the price of that option is. 66 00:04:03,970 --> 00:04:08,660 The option is based on some underlying asset that has a, 67 00:04:08,660 --> 00:04:11,680 a price S, and so I can also look in the market 68 00:04:11,680 --> 00:04:17,160 at the same time that I sold my option for this price C. 69 00:04:18,350 --> 00:04:20,240 I can look at the most recent trade for 70 00:04:20,240 --> 00:04:22,870 that underlying assett and see what the asset price is. 71 00:04:22,870 --> 00:04:26,480 So, I can also assume that when these transactions take 72 00:04:26,480 --> 00:04:29,280 place, the C and the S are going to be known. 73 00:04:33,440 --> 00:04:35,760 And then I'm also going to assume that the 74 00:04:35,760 --> 00:04:39,600 risk free rate is constant, and that the dividend rate. 75 00:04:39,600 --> 00:04:42,980 So, because options exist on kind of a short time scale 76 00:04:42,980 --> 00:04:47,500 most of the time, you know either 3 months or 6 months. 77 00:04:47,500 --> 00:04:49,900 You generally know what the, the company's already 78 00:04:49,900 --> 00:04:51,930 announced when it's going to be paying dividends. 79 00:04:51,930 --> 00:04:53,290 So you know if you're going to get 80 00:04:53,290 --> 00:04:56,030 dividends during the life of the option or not. 81 00:04:56,030 --> 00:04:59,110 And the risk free risk, well, it's definitely changing. 82 00:04:59,110 --> 00:05:01,230 It changes on a very slow time scale. 83 00:05:01,230 --> 00:05:06,130 So on the, during the life of the option, you can assume that that's fixed. 84 00:05:06,130 --> 00:05:12,641 And so it turns that if you look in this formula, so it depends on this d1 and d2 85 00:05:12,641 --> 00:05:18,700 as well, you know everything except this variable sigma. 86 00:05:23,860 --> 00:05:27,160 So the, the only quantity that's not known is the volatility, and 87 00:05:27,160 --> 00:05:30,110 this is the one thing that it's actually quite difficult to know, 88 00:05:30,110 --> 00:05:32,970 that there's not any one thing in the market where you can 89 00:05:32,970 --> 00:05:36,585 just say oh, this is, this is definitely the rate of sigma. 90 00:05:36,585 --> 00:05:39,850 So going in the, in the forward sense when you want to use sigma as 91 00:05:39,850 --> 00:05:44,050 an input to this formula, it's something you have to actually estimate from data. 92 00:05:44,050 --> 00:05:49,040 So you have to look at, it's, it's trying to describe the, how how 93 00:05:49,040 --> 00:05:54,680 wide the fluctuations are of this underlying asset price S. 94 00:05:54,680 --> 00:05:57,595 And so it's, you can estimate it, but you can never actually know it. 95 00:05:57,595 --> 00:06:02,640 Whereas if you go to the market you know, on a particular day, I can see what was 96 00:06:02,640 --> 00:06:05,630 you know, what when that asset changed hands, what 97 00:06:05,630 --> 00:06:09,030 were those prices so I can actually observe S. 98 00:06:09,030 --> 00:06:11,070 Sigma, I'm not able to do that in any way. 99 00:06:15,520 --> 00:06:17,240 So the implied volatility problem. 100 00:06:17,240 --> 00:06:23,250 If I know every variable and the answer to the Black-Scholes pricing formula except 101 00:06:23,250 --> 00:06:28,950 sigma, then what I want to do is just sort of solve this equation backwards. 102 00:06:28,950 --> 00:06:32,320 So, essentially, if I could, I'd like to isolate sigma and 103 00:06:32,320 --> 00:06:36,440 have a new functions that's sigma as a function of everything else. 104 00:06:36,440 --> 00:06:41,010 But it turns out, if, if we look at this guy, it's not exactly easy to do that. 105 00:06:41,010 --> 00:06:44,010 So extra credit if anybody can come up with the formula. 106 00:06:49,270 --> 00:06:51,230 And so what I want to do, as this is 107 00:06:51,230 --> 00:06:54,530 where I was getting at this trial and error thing. 108 00:06:54,530 --> 00:06:57,425 I want to just plug in values of sigma into my 109 00:06:57,425 --> 00:07:01,410 Black-Scholes formula, and I know what the call price should be. 110 00:07:01,410 --> 00:07:03,260 And for every different value of sigma that 111 00:07:03,260 --> 00:07:06,040 I plug in, I'm going to get some price. 112 00:07:06,040 --> 00:07:09,690 And eventually, if I just keep guessing long enough, I'm going to get a value of 113 00:07:09,690 --> 00:07:13,820 sigma that gets me very close to the price that I know it should, should be. 114 00:07:13,820 --> 00:07:14,150 So I, 115 00:07:14,150 --> 00:07:17,770 if I can get close to that market price C, then that's 116 00:07:17,770 --> 00:07:22,040 probably, well, I mean, that's the, the value of sigma that would 117 00:07:22,040 --> 00:07:25,858 have been in the market. And so even though I can't observe sigma, 118 00:07:25,858 --> 00:07:30,560 all the other things that I can observe imply a certain level for sigma. 119 00:07:30,560 --> 00:07:39,200 They imply a certain amount sigma, and so that's going to be called the 120 00:07:39,200 --> 00:07:41,992 implied volatility. 121 00:07:41,992 --> 00:07:47,262 So what I want to look at is, is if, if S, K, Q, R, T and little t. 122 00:07:47,262 --> 00:07:50,034 So this is basically all of the inputs to the 123 00:07:50,034 --> 00:07:54,885 Black-Scholes pricing formula for our European call option are known and 124 00:07:54,885 --> 00:07:57,734 I also know what the price of that option is, 125 00:07:57,734 --> 00:08:00,799 then I can think of this as a function of sigma. 126 00:08:00,799 --> 00:08:04,735 So there's only one unknown in this highlighted 127 00:08:04,735 --> 00:08:09,816 side of the equation and that's sigma. And the way I've set 128 00:08:09,816 --> 00:08:14,713 this up, I've, I've, put the theoretical price for European call 129 00:08:14,713 --> 00:08:19,981 option minus the price for that option that I've observed in the market. 130 00:08:19,981 --> 00:08:25,129 So when I have the correct value of sigma, this 131 00:08:25,129 --> 00:08:30,014 whole thing f of sigma should be equal to 0. 132 00:08:30,014 --> 00:08:34,118 So finding the implied volatility really just boils down to solving 133 00:08:34,118 --> 00:08:38,392 this thing, it, it's called a non-linear optimization problem. 134 00:08:38,392 --> 00:08:46,549 And I just want to find the value of sigma so that f of sigma is equal to zero. 135 00:08:46,549 --> 00:08:50,749 And this is something like I said you're not going to be able to just 136 00:08:50,749 --> 00:08:55,024 isolate sigma and have a nice f of sigma function you can just plug in a 137 00:08:55,024 --> 00:08:58,538 zero and find out what should it have been. 138 00:08:58,538 --> 00:09:02,350 so you have to solve this numerically. 139 00:09:02,350 --> 00:09:06,292 And so when I say, solve numerically, there, there are lots of 140 00:09:06,292 --> 00:09:10,864 different prices, lots of different ways I can try and do this. 141 00:09:10,864 --> 00:09:15,679 So, the, the simplest one would just be to make a plot 142 00:09:15,679 --> 00:09:19,961 of f of sigma and see where it crosses the x axis. 143 00:09:19,961 --> 00:09:23,174 And so, where it crosses the x axis is where f of sigma is going 144 00:09:23,174 --> 00:09:27,171 to be equal to 0, and that's what I'm trying to find in the, the value. 145 00:09:27,171 --> 00:09:30,755 So, the x axis in this case is going to be the sigma axis. 146 00:09:30,755 --> 00:09:36,663 The value of sigma where it crosses the axis is this f of sigma equals 0. 147 00:09:38,900 --> 00:09:43,785 And so, to do that in R, the first thing I need is a function to compute the 148 00:09:43,785 --> 00:09:48,940 Black-Scholes call price. So, this is just our implementation of 149 00:09:48,940 --> 00:09:54,840 what was on the slide, a couple of slides ago, so I've made an expression for d1. 150 00:09:57,760 --> 00:10:02,990 Then the expression for d2, and then finally the expression 151 00:10:02,990 --> 00:10:08,060 for the Black-Scholes call price. 152 00:10:08,060 --> 00:10:12,800 And I, I don't know if you guys are getting used to looking at r 153 00:10:12,800 --> 00:10:17,400 yet, but basically what I'm doing here is I've I'm taking this expression, 154 00:10:19,250 --> 00:10:24,020 I'm assigning it to a piece of memory, a variable called d1. 155 00:10:24,020 --> 00:10:27,420 Then I'm taking this expression, assigning it to 156 00:10:27,420 --> 00:10:31,290 a piece of memory, a variable called d2. 157 00:10:31,290 --> 00:10:35,440 But then in R, if the last expression in a function. 158 00:10:35,440 --> 00:10:38,330 So, because this has this keyword function here, this thing 159 00:10:38,330 --> 00:10:43,780 is a function, the last expression in a function gets returned. 160 00:10:44,970 --> 00:10:49,140 So, when I evaluate this, I don't have to, you can if you want to 161 00:10:49,140 --> 00:10:53,650 say you know, assign this to say Black-Scholes call price and 162 00:10:53,650 --> 00:10:57,560 then have one more line that says return Black-Scholes call price. 163 00:10:57,560 --> 00:11:00,850 Or you can just put an expression for the thing 164 00:11:00,850 --> 00:11:04,110 that you want your function to return as the last line. 165 00:11:04,110 --> 00:11:05,770 So that's why this is going to work. 166 00:11:08,340 --> 00:11:12,570 And then one of the nice features R has is it's treating every 167 00:11:12,570 --> 00:11:16,840 variable like a vector. So, instead of having to do this in a for 168 00:11:16,840 --> 00:11:22,430 loop, or evaluate the function by hand you know, 50 or 60 times to get a set of 169 00:11:22,430 --> 00:11:30,050 points to, to plot, I can just give I can give a vector argument. 170 00:11:30,050 --> 00:11:33,358 So right now I'm giving every argument that's just a 171 00:11:33,358 --> 00:11:36,860 length one vector. And so I get a length one answer. 172 00:11:39,680 --> 00:11:45,540 On the other hand, suppose I take, so the second to last thing, this little s so. 173 00:11:46,800 --> 00:11:49,000 I, I guess I should also mention this is sort of 174 00:11:49,000 --> 00:11:53,969 bad bad programming, because none of the variables have very meaningful names. 175 00:11:55,510 --> 00:11:57,670 I just had to do that to make it fit on the slide, the 176 00:11:57,670 --> 00:12:01,170 first way, I wrote this only about the first half of it on the slide. 177 00:12:01,170 --> 00:12:02,970 I also think it's a lot easier to read 178 00:12:02,970 --> 00:12:05,044 if you know, when I say something like R minus 179 00:12:05,044 --> 00:12:08,080 Q, that should be R space minus space Q. 180 00:12:09,150 --> 00:12:13,094 you want to group things, so it's a little bit easier to see what the function is. 181 00:12:13,094 --> 00:12:17,940 But in this particular case I had a another constraint which it needed to 182 00:12:17,940 --> 00:12:20,900 fit up here for the, for the lecture, so I had to skip that. 183 00:12:22,070 --> 00:12:25,360 But if we look at the second to last argument, this lower 184 00:12:25,360 --> 00:12:28,715 case s, this is where I'm putting in my value of sigma. 185 00:12:28,715 --> 00:12:30,613 And so, when I call my function 186 00:12:30,613 --> 00:12:33,533 this time, instead of putting in a single value 187 00:12:33,533 --> 00:12:36,798 for sigma, I'm putting in three values for sigma. 188 00:12:36,798 --> 00:12:42,512 I want to end that getting out are three values for this Black-Scholes call price. 189 00:12:42,512 --> 00:12:48,947 So all of the other variables stay the same, but the first one here 190 00:12:48,947 --> 00:12:55,616 corresponds to sigma of 0.15, the second one here corresponds 191 00:12:55,616 --> 00:13:02,798 to sigma of 0.2, and the third one corresponds to sigma of 0.25. 192 00:13:02,798 --> 00:13:08,334 And now, let's suppose that the option sold for $7, and I want to find sigma. 193 00:13:08,334 --> 00:13:10,746 So you can think of this is, I want to find the 194 00:13:10,746 --> 00:13:14,761 sigma that's going to make my Black-Scholes call price equal to 7. 195 00:13:14,761 --> 00:13:18,067 But most nonlinear solvers are going to be set 196 00:13:18,067 --> 00:13:21,445 up just to solve this f of whatever equals 0. 197 00:13:21,445 --> 00:13:23,545 So you want to make a new objective 198 00:13:23,545 --> 00:13:26,900 function where the point you're looking for is 0. 199 00:13:26,900 --> 00:13:27,400 And 200 00:13:30,140 --> 00:13:32,140 I'm going to call that function f of sigma. 201 00:13:32,140 --> 00:13:35,343 And that's just going to be my Black-Scholes call price minus 7. 202 00:13:36,550 --> 00:13:38,620 And I'm going to plot that over a range of values 203 00:13:38,620 --> 00:13:41,790 of sigma and see where it crosses the x axis. 204 00:13:43,090 --> 00:13:45,850 So the first thing I'll do is make my range of sigmas. 205 00:13:45,850 --> 00:13:49,870 So seq is a function that just makes a sequence. 206 00:13:49,870 --> 00:13:51,954 It starts at 0.05. 207 00:13:51,954 --> 00:13:54,427 It's going to end at 0.5. 208 00:13:55,450 --> 00:14:01,926 And it's going to go in steps of 0.01. 209 00:14:01,926 --> 00:14:06,510 Then f sig, I'm going 210 00:14:06,510 --> 00:14:11,010 to evaluate my Black-Scholes call price, add all of 211 00:14:11,010 --> 00:14:15,870 the inputs that were fixed, and then at this entire vector of sigmas. 212 00:14:15,870 --> 00:14:21,580 So this is, I think this will end up being maybe 46 different values of sigma. 213 00:14:21,580 --> 00:14:27,620 So, this is going to give me 46 different Black-Scholes call prices corresponding to 214 00:14:27,620 --> 00:14:32,460 each one of those input sigmas, and then I'm going to subtract seven from that. 215 00:14:32,460 --> 00:14:34,770 And hopefully, one of these will be pretty close to zero. 216 00:14:36,080 --> 00:14:41,670 And then I'm going to plot. So, I'll put sigma on the x axis, and 217 00:14:41,670 --> 00:14:47,190 then this fsig that I've just calculated here on the y axis, and then this optional 218 00:14:47,190 --> 00:14:49,010 arguments type equals l. 219 00:14:49,010 --> 00:14:50,800 It's just going to plot it with a line rather than 220 00:14:50,800 --> 00:14:54,589 with points, and I get a plot that looks like this. 221 00:14:56,720 --> 00:15:00,420 And so, on the, on the left here, I'm below 0, on the right I'm about 0. 222 00:15:00,420 --> 00:15:07,520 And it looks like about sigma equals 0.25 is about where I hit 0. 223 00:15:07,520 --> 00:15:11,470 So that gives me an implied volatility, 224 00:15:11,470 --> 00:15:15,560 so I denoted that. Sigma sub implied, think the book is 225 00:15:15,560 --> 00:15:23,050 called Sigma sub imp of 0.25. 226 00:15:23,050 --> 00:15:29,490 And so, this is a valid way of coming up with an implied volatility. 227 00:15:29,490 --> 00:15:33,500 But to do that, I had to evaluate the Black-Scholes formula 46 times. 228 00:15:33,500 --> 00:15:36,450 And now if you look at this plot, you know, I evaluated 229 00:15:36,450 --> 00:15:39,520 a whole lot over here where it wasn't telling me very much. 230 00:15:39,520 --> 00:15:41,460 And I did the same thing over here. 231 00:15:44,410 --> 00:15:47,910 And still the computer answer was not very precise. 232 00:15:47,910 --> 00:15:53,312 So I, I only plotted it in these steps of 0.01. 233 00:15:53,312 --> 00:16:00,620 And so it turned out that 0.25 was slightly less than 0. 234 00:16:00,620 --> 00:16:04,910 And when I put in 0.26, it was slightly more, but it 235 00:16:04,910 --> 00:16:08,750 was more slightly more so that 0.25 was the closest one to 0. 236 00:16:08,750 --> 00:16:09,750 So that's 237 00:16:09,750 --> 00:16:16,580 why I picked this one. And so I was off by, so this, 238 00:16:16,580 --> 00:16:20,970 this is on the scale of dollars here, so I was off by about one and a half cents. 239 00:16:22,280 --> 00:16:24,210 So it's, doesn't sound too bad, except that 240 00:16:24,210 --> 00:16:26,370 you might be trading, you know, thousands or tens 241 00:16:26,370 --> 00:16:28,230 of thousands of these things, and so that, that 242 00:16:28,230 --> 00:16:32,490 ends up being more worrisome to be off by. 243 00:16:34,940 --> 00:16:36,700 So what I'd really like to be able to do 244 00:16:37,710 --> 00:16:44,720 is compute this implied volatility to within a pre-specified tolerance. 245 00:16:44,720 --> 00:16:49,140 So right now, I've, I've just sort of done these 46 things, and I've ended 246 00:16:49,140 --> 00:16:53,780 up finding out how close I was just by going back and checking my solution. 247 00:16:53,780 --> 00:16:59,349 What I would like to have is an algorithm that's going to keep going until. 248 00:16:59,349 --> 00:17:03,210 So for instance, I was just going to buy one option, I 249 00:17:03,210 --> 00:17:06,890 would sort of want this to keep going until maybe the first significant 250 00:17:06,890 --> 00:17:10,010 digit was somewhere over here, and then I would be certain that 251 00:17:10,010 --> 00:17:14,720 I'm within less than a penny of the price that I'm aiming for. 252 00:17:14,720 --> 00:17:16,340 And so, if I only wanted to buy one. 253 00:17:16,340 --> 00:17:19,630 Well, if that's the smallest denomination of currency bill 254 00:17:19,630 --> 00:17:21,750 allowed me to use, then that will be good enough. 255 00:17:23,880 --> 00:17:27,650 So I want to get inside this pre-specified color, and at the same time 256 00:17:27,650 --> 00:17:32,330 to do that, I want to use the minimum number of function evaluations required. 257 00:17:34,920 --> 00:17:37,020 And so the methods, the, the general methods for 258 00:17:37,020 --> 00:17:41,186 solving this type of problem are called nonlinear solvers.