1 00:00:01,060 --> 00:00:05,070 Now we get to the, the fun part which is actually taking derivitaives. 2 00:00:05,070 --> 00:00:08,420 So we'll start off by calculating delta. 3 00:00:09,860 --> 00:00:15,550 So the delta of a European call option is the rate of change, oops. 4 00:00:15,550 --> 00:00:20,330 Of c of s t, so again remember this is the actual Black-Scholes pricing formula. 5 00:00:20,330 --> 00:00:22,850 So, even though I've written it here as c of s 6 00:00:22,850 --> 00:00:26,350 comma t it's really a function of all of the seven variables 7 00:00:26,350 --> 00:00:28,900 that I had on the, the first set of slides. 8 00:00:31,190 --> 00:00:32,610 And I'm going to take that, it says the rate 9 00:00:32,610 --> 00:00:35,130 of change with respect to the asset price s. 10 00:00:35,130 --> 00:00:39,270 So, how how much does the value of the call option change? 11 00:00:39,270 --> 00:00:41,760 As the price of the underlying asset changes. 12 00:00:45,660 --> 00:00:50,080 And so mathematically I can write that just as, delta is equal to the partial 13 00:00:50,080 --> 00:00:52,484 derivative with respect to s of the 14 00:00:52,484 --> 00:00:56,560 Black-Scholes pricing formula for a European call option. 15 00:01:02,630 --> 00:01:05,980 And so this is the pricing formula for a European 16 00:01:05,980 --> 00:01:08,210 call option that I had in the first set of slides. 17 00:01:11,000 --> 00:01:14,560 And, I'm just going to go ahead and start using the rules we've 18 00:01:14,560 --> 00:01:18,360 been developing er, earlier in this course to start taking the dcerivative here. 19 00:01:18,360 --> 00:01:19,799 So the first thing I'll, I'll notice. 20 00:01:21,185 --> 00:01:24,320 Is that the pricing formula has two terms. 21 00:01:24,320 --> 00:01:26,740 So I can use the linearity property of the derivative to 22 00:01:26,740 --> 00:01:31,030 write this as the, so this is a, a sum essentially. 23 00:01:31,030 --> 00:01:33,780 And so the derivative of a sum is the sum of the derivatives. 24 00:01:37,460 --> 00:01:41,570 And then I'm also in, in this one step going to combine the. 25 00:01:41,570 --> 00:01:46,900 Scaling properties, so when I'm taking the derivative with respect to s. 26 00:01:46,900 --> 00:01:50,140 So, I am taking a partial derivative with respect to s. 27 00:01:50,140 --> 00:01:53,250 Anything that isn't a function of s, I can treat as 28 00:01:53,250 --> 00:01:57,218 a constant, and any constant I can move across the derivative operator. 29 00:02:00,440 --> 00:02:06,480 So, in the first case here. This part the e to the minus qt minus t. 30 00:02:06,480 --> 00:02:08,660 That doesn't depend on s so I'm going to take 31 00:02:08,660 --> 00:02:12,180 that out and I end up with e to the minus 32 00:02:12,180 --> 00:02:16,140 qt minus t times the partial derivative with respect to 33 00:02:16,140 --> 00:02:20,699 s of the product s times capital phi of d plus. 34 00:02:22,580 --> 00:02:25,590 And on, on the right hand side I'm going to do the same thing, except here 35 00:02:25,590 --> 00:02:28,480 I have two parts that don't depend on s, 36 00:02:28,480 --> 00:02:30,860 so I can take those across the derivative operator. 37 00:02:30,860 --> 00:02:33,760 If I have minus ke to the r. 38 00:02:33,760 --> 00:02:34,120 Sorry. 39 00:02:34,120 --> 00:02:37,190 Minus ke to the minus rt minus t, times the 40 00:02:37,190 --> 00:02:40,620 partial derivative with respect to s of phi of d minus. 41 00:02:43,530 --> 00:02:47,940 And so, the next step here is both phi of d plus. 42 00:02:47,940 --> 00:02:50,180 D plus depends on all of the arguments 43 00:02:50,180 --> 00:02:53,310 of the Black Scholes pricing formula including s. 44 00:02:53,310 --> 00:02:55,380 And so does d minus. 45 00:02:55,380 --> 00:02:56,680 So, this is, I'm going to look at it 46 00:02:56,680 --> 00:02:59,040 and say, that's the product of two functions of s. 47 00:02:59,040 --> 00:03:02,440 That's s, which is a very simple function of s. 48 00:03:02,440 --> 00:03:05,450 And phi of capitol d plus. 49 00:03:05,450 --> 00:03:07,510 So I'll have to use the product rule on the first term. 50 00:03:12,340 --> 00:03:15,030 And so the product rule says if I have the product of two 51 00:03:15,030 --> 00:03:19,270 functions, I take the derivative of the first function times the second function. 52 00:03:19,270 --> 00:03:25,800 So that gives me just a capital phi of d plus Plus the first function, s. 53 00:03:25,800 --> 00:03:28,240 So this s comes from right here. 54 00:03:28,240 --> 00:03:30,560 Times the derivative of the second function. 55 00:03:30,560 --> 00:03:35,289 So times the derivative with respect to s of capital phi of d plus. 56 00:03:43,470 --> 00:03:46,070 So now I have a pretty complicated formula so what 57 00:03:46,070 --> 00:03:48,950 I'm going to do is kind of attack pieces of it 58 00:03:48,950 --> 00:03:50,660 and then we'll be able to put all of those 59 00:03:50,660 --> 00:03:54,110 together to get an expression for delta in the end. 60 00:03:54,110 --> 00:03:56,761 So the first thing I'm going to look at. 61 00:03:56,761 --> 00:04:02,250 Are we have these partial derivatives with respect to s of phi of, and 62 00:04:02,250 --> 00:04:03,920 so I've used this plus, minus sign 63 00:04:03,920 --> 00:04:06,720 here just because the functions are very similar. 64 00:04:06,720 --> 00:04:09,170 So quite often when 65 00:04:09,170 --> 00:04:11,740 I take derivatives of that I'm going to get the same answer. 66 00:04:13,040 --> 00:04:15,040 if not, I'll put a plus, minus. 67 00:04:15,040 --> 00:04:17,290 And where it's important, and then if I'm just talking 68 00:04:17,290 --> 00:04:20,480 about d plus, you'd think of this as a plus. 69 00:04:20,480 --> 00:04:23,450 If I'm talking about d minus, you think about this as a minus. 70 00:04:26,740 --> 00:04:32,970 So the chain rule says that if I want to take the derivative of capital phi of d 71 00:04:32,970 --> 00:04:35,310 plus with respect to s, that's going to be 72 00:04:35,310 --> 00:04:40,080 equal to the derivative with respect to its argument. 73 00:04:40,080 --> 00:04:46,720 So with respect to this d plus or minus, of fee of d plus or minus, 74 00:04:46,720 --> 00:04:51,350 times the partial derivative of d with respect to s. 75 00:04:53,920 --> 00:04:54,420 And 76 00:04:56,970 --> 00:04:59,250 now remember the definition of capital fee. 77 00:05:00,270 --> 00:05:03,010 As just we were integrating from minus infinity up to 78 00:05:03,010 --> 00:05:07,360 d of this function, lower case phi of x, dx, 79 00:05:09,550 --> 00:05:11,770 and so, I, I like to use the shorthand 80 00:05:11,770 --> 00:05:14,930 notation little phi of x for this function here. 81 00:05:14,930 --> 00:05:16,880 Just it saves me writing some stuff. 82 00:05:16,880 --> 00:05:19,270 Also, it's less likely that I'm going to make a 83 00:05:19,270 --> 00:05:21,810 mistake if I just have to write that down rather than. 84 00:05:21,810 --> 00:05:22,620 This whole expression. 85 00:05:25,060 --> 00:05:27,810 And we saw in the, in the second week how 86 00:05:27,810 --> 00:05:30,280 I could take the derivative of, of something like that. 87 00:05:30,280 --> 00:05:33,790 So it's just the derivative of an indefinite integral. 88 00:05:40,840 --> 00:05:44,170 And so, again, I'm just, I'm doing the same calculation two times. 89 00:05:44,170 --> 00:05:46,540 So it's either, I'm treating it as phi of x. 90 00:05:46,540 --> 00:05:48,070 Or I'm treating it as. 91 00:05:48,070 --> 00:05:50,000 The actual expression for that function, and 92 00:05:51,350 --> 00:05:53,280 then when I evaluate these it turns out 93 00:05:55,450 --> 00:05:57,900 that the derivative is either, is just lower 94 00:05:57,900 --> 00:06:00,480 case phi of either d plus or d minus. 95 00:06:01,660 --> 00:06:04,550 And then it has this functional form on the right hand side here. 96 00:06:05,860 --> 00:06:08,230 So that gets me this part the derivative. 97 00:06:08,230 --> 00:06:11,670 The partial derivative of capital phi with respect to s. 98 00:06:15,120 --> 00:06:18,430 So, what I have so far, I have this expression for delta, 99 00:06:22,760 --> 00:06:27,090 and then we just evaluated this thing here and this thing here. 100 00:06:29,560 --> 00:06:33,020 So I have this expression for the derivative of capital fee, 101 00:06:36,560 --> 00:06:39,800 and I still need this part here so the derivative of, 102 00:06:39,800 --> 00:06:43,340 the partial derivative of d plus or minus with respect to s. 103 00:06:43,340 --> 00:06:43,840 So 104 00:06:49,890 --> 00:06:53,020 I have the first term here, is this guy. 105 00:06:53,020 --> 00:06:57,106 And now to get the full derivative I still need to evaluate the 106 00:06:57,106 --> 00:07:00,030 derivative of phi plus, that kind of came out by the chain rule. 107 00:07:02,300 --> 00:07:06,590 So, I'll move one step ahead with our formula. 108 00:07:06,590 --> 00:07:09,900 So I'm going to substitute this result here. 109 00:07:09,900 --> 00:07:10,520 Oops. 110 00:07:10,520 --> 00:07:12,780 This result in the second line, into the 111 00:07:12,780 --> 00:07:16,400 first line, to update my expression for delta. 112 00:07:16,400 --> 00:07:19,800 But I still have this partial of d plus with respect 113 00:07:19,800 --> 00:07:23,410 to s, and partial of d minus with respect to s. 114 00:07:23,410 --> 00:07:25,620 So that's what I'm going to do in the next slide. 115 00:07:25,620 --> 00:07:27,580 So I need to compute the partial derivatives 116 00:07:27,580 --> 00:07:29,010 of d plus and d minus. 117 00:07:32,320 --> 00:07:34,510 So remember from the, from the first set of slides 118 00:07:34,510 --> 00:07:37,010 I said, I gave definitions for what d plus and 119 00:07:37,010 --> 00:07:38,840 minus look like and so now I can write them 120 00:07:38,840 --> 00:07:44,270 as, just one, formula, using this plus minus notation, so. 121 00:07:44,270 --> 00:07:47,870 The only difference between d plus and d minus was this plus 122 00:07:47,870 --> 00:07:51,240 or minus sign here, so I've just replaced it with this little placeholder. 123 00:07:51,240 --> 00:07:56,080 And now I want to take the derivative of this with respect to s. 124 00:08:01,660 --> 00:08:06,560 And so I'm going to make a little simplification here, so If I have a sum in 125 00:08:06,560 --> 00:08:12,860 the numerator, I can write this as log of s over k divided by 126 00:08:12,860 --> 00:08:18,589 sigma times the square root of t minus t, so that'll be one fraction, and then plus. 127 00:08:19,870 --> 00:08:23,560 R minus q plus or minus sigma squared over 2 times 128 00:08:23,560 --> 00:08:26,860 t minus t divided by sigma, square root of t minus t. 129 00:08:26,860 --> 00:08:30,060 So that's a second fraction, I'm just adding those two fractions together. 130 00:08:32,330 --> 00:08:36,750 And so the reason I'm going to want to do that is because after, this 131 00:08:36,750 --> 00:08:40,070 s here only appears in the, in the first one of those fractions. 132 00:08:40,070 --> 00:08:42,030 So what I'm going to end up with is. 133 00:08:42,030 --> 00:08:48,330 I'll want to take the partial derivative of the leftover part with respect to s but 134 00:08:48,330 --> 00:08:51,500 there is no s's in this part so I can treat that as a constant. 135 00:08:52,590 --> 00:08:55,680 So this term in the right hand side that's just 136 00:08:55,680 --> 00:08:58,500 going to go away that will be equal to zero. 137 00:08:58,500 --> 00:09:03,400 And then I'm left with 1 over sigma times the square root of t minus 138 00:09:03,400 --> 00:09:07,610 t, times the partial derivative with respect to s of the log of s over k. 139 00:09:10,350 --> 00:09:14,880 So I'll do a little substitution here. So I'll let u equal s over k. 140 00:09:16,540 --> 00:09:20,050 Then I can rewrite this Here, as the partial 141 00:09:20,050 --> 00:09:24,500 derivative of, with respect to s, of log of u. 142 00:09:25,820 --> 00:09:28,010 And so by the chain rule, that's going to be the derivative 143 00:09:28,010 --> 00:09:31,900 of the log times the derivative of u with respect to s. 144 00:09:34,090 --> 00:09:38,290 So, the derivative of log of u is just 1 over u. 145 00:09:38,290 --> 00:09:40,610 And then I have this partial u, partial s. 146 00:09:41,980 --> 00:09:44,620 And I'm going to calculate that just by taking the 147 00:09:44,620 --> 00:09:48,800 partial derivative with respect to s of u here. 148 00:09:48,800 --> 00:09:50,780 So, that's going to be 1 over k. 149 00:09:53,700 --> 00:09:58,700 So what I'm left with is 1 over sigma, square root of t minus t. 150 00:09:58,700 --> 00:10:00,440 K over s. 151 00:10:00,440 --> 00:10:01,980 So, u was s over k. 152 00:10:01,980 --> 00:10:05,800 If I take one over that, I get k over s. 153 00:10:05,800 --> 00:10:09,790 And then times 1 over k, which is just this partial derivative, partial 154 00:10:09,790 --> 00:10:13,300 u, partial s, which I, which popped out because of the chain rule. 155 00:10:15,160 --> 00:10:17,760 So, we'll see that the k's are going to cancel here. 156 00:10:20,000 --> 00:10:25,360 And I end up with partial derivative with respect to s of either d plus or d 157 00:10:25,360 --> 00:10:31,070 minus is just 1 over s times sigma times the square root of t minus t. 158 00:10:31,070 --> 00:10:33,819 So in this case I was able to get both of those calculations. 159 00:10:34,860 --> 00:10:38,070 Because the, the plus or minus bit didn't depend on 160 00:10:38,070 --> 00:10:40,360 s, so that ended up being zero, so it wouldn't 161 00:10:40,360 --> 00:10:42,780 matter if I was taking the derivative of d plus 162 00:10:42,780 --> 00:10:44,740 or d minus, I'm going to get the same expression here. 163 00:10:47,280 --> 00:10:53,660 So, putting everything together, all I'm , all I need to do now is, I have this. 164 00:10:53,660 --> 00:10:56,960 This is my most up to date expression for delta, but it still 165 00:10:56,960 --> 00:11:01,080 has this partial derivative of d plus and d minus with respect to s. 166 00:11:02,350 --> 00:11:05,720 But that's exactly what we calculated on the previous slide here, so I just 167 00:11:05,720 --> 00:11:10,020 have to put a 1 over S sigma squared to t minus t into here. 168 00:11:12,770 --> 00:11:14,400 Oops, I guess I had it on the next line. 169 00:11:15,860 --> 00:11:18,640 And so, that's going to give the following expression for delta. 170 00:11:18,640 --> 00:11:19,140 So, 171 00:11:20,810 --> 00:11:26,090 I'm going to get e to the minus q, t minus t, times capital phi of d plus. 172 00:11:26,090 --> 00:11:28,760 And I have the rest of this in light gray, because 173 00:11:28,760 --> 00:11:32,320 if you look in the text book, this is the entire. 174 00:11:32,320 --> 00:11:34,620 Oops, maybe I got it backwards. Nope. 175 00:11:36,550 --> 00:11:40,070 just the part in the dark black is the entire expression for delta. 176 00:11:40,070 --> 00:11:42,890 But we still have these other terms left over. 177 00:11:44,560 --> 00:11:46,130 And so in the next set of slides 178 00:11:46,130 --> 00:11:49,770 I'm going to go through an argument about why that's going to be equal to zero.