1 00:00:00,960 --> 00:00:02,607 Welcome to week 8, the 8th and 2 00:00:02,607 --> 00:00:06,950 final week of Mathematical Methods for Quantitative Finance. 3 00:00:06,950 --> 00:00:09,660 This week, I'm going to talk about numerical methods. 4 00:00:09,660 --> 00:00:12,054 So, so far in the course, I've introduced a 5 00:00:12,054 --> 00:00:15,390 lot of topics for solving different types of the problems. 6 00:00:15,390 --> 00:00:18,424 But a lot of the examples in the homework have been 7 00:00:18,424 --> 00:00:21,820 what, what I sort of think of as sort of toy examples. 8 00:00:21,820 --> 00:00:26,550 Because they need to be complicated enough that you understand what's going on. 9 00:00:26,550 --> 00:00:29,014 At the same time they need to be simple enough that you 10 00:00:29,014 --> 00:00:31,720 can actually solve them with a, pen and a piece of paper. 11 00:00:33,100 --> 00:00:37,629 In practice though, we're going to want to solve much more complicated problems. 12 00:00:38,630 --> 00:00:43,280 And so to do that we, we're going to want to do that with a computer. 13 00:00:43,280 --> 00:00:46,070 And so that brings us to this topic of numerical methods. 14 00:00:48,830 --> 00:00:54,370 So the outline for this week, I'll talk about a problem called Implied Volatility. 15 00:00:54,370 --> 00:00:59,122 So this is basically going back to the Black-Sholes pricing formula. 16 00:00:59,122 --> 00:01:02,482 But instead of thinking of the, the value that I 17 00:01:02,482 --> 00:01:06,467 want to calculate as the call price or the put price. 18 00:01:06,467 --> 00:01:09,347 I'm going to think that I could actually go, 19 00:01:09,347 --> 00:01:12,810 or I'm going to approach the problem in this sense. 20 00:01:12,810 --> 00:01:14,007 I can actually go to the 21 00:01:14,007 --> 00:01:18,560 market and I can observe all of the inputs for the Black-Sholes formula. 22 00:01:18,560 --> 00:01:23,470 And I can see what that put or call price actually is. 23 00:01:23,470 --> 00:01:25,255 And the one thing that's going to be the most 24 00:01:25,255 --> 00:01:28,780 difficult to observe is the volatility parameter, the sigma. 25 00:01:28,780 --> 00:01:33,525 And so, given that I have the price, I can actually use Black-sholes kind of 26 00:01:33,525 --> 00:01:39,450 backwards to figure out what the sigma parameter had to be to justify that price. 27 00:01:39,450 --> 00:01:43,380 And that turns out to involve solving something called a non linear equation. 28 00:01:45,420 --> 00:01:47,839 And so then in sections two and three, I'm going to 29 00:01:47,839 --> 00:01:51,947 consider two methods for solving non linear equations, the Bisection method. 30 00:01:51,947 --> 00:01:55,242 So essentially what I'm trying to do with the Bisection method. 31 00:01:55,242 --> 00:01:58,954 So I want to find two endpoints so that I know I'm bigger than the value I 32 00:01:58,954 --> 00:02:03,560 want on one side, and I'm smaller than the value I want on the other side. 33 00:02:03,560 --> 00:02:07,290 And then, I'm going to just start chopping that interval in half. 34 00:02:07,290 --> 00:02:10,930 By preserving this property that the, the right end point and 35 00:02:10,930 --> 00:02:15,360 the left end point bracket the point that I'm looking for. 36 00:02:15,360 --> 00:02:17,757 And so, as I chop this in half, you know, the end points 37 00:02:17,757 --> 00:02:20,629 are getting closer and closer to the number that I want to compute. 38 00:02:22,190 --> 00:02:25,000 Another approach for solving this is called Newton's method. 39 00:02:25,000 --> 00:02:29,488 So this has a little bit stronger assumptions that I need 40 00:02:29,488 --> 00:02:34,600 the function I'm trying to find the value of to be differentiable. 41 00:02:36,330 --> 00:02:38,906 And then the other nice property that Newton's Method 42 00:02:38,906 --> 00:02:42,150 has is there's a n dimensional version of Newton's Method. 43 00:02:42,150 --> 00:02:43,762 So the, the bisection method is only 44 00:02:43,762 --> 00:02:45,650 going to work if there's a single variable. 45 00:02:45,650 --> 00:02:48,246 It get, it starts to break down if I'm, if there's 46 00:02:48,246 --> 00:02:51,030 a variable going this way and a variable going this way. 47 00:02:54,280 --> 00:02:56,165 And so once I've gone over how I can 48 00:02:56,165 --> 00:03:00,190 solve these nonlinear problems using Newton's method and bisection. 49 00:03:00,190 --> 00:03:04,680 Then I'm going to look at some of the problems we had from last week's lecture. 50 00:03:04,680 --> 00:03:08,340 So in particular, the problems that came up using Lagrange's method. 51 00:03:09,380 --> 00:03:13,176 We're going to have generally a non linear system that we 52 00:03:13,176 --> 00:03:16,826 need to solve to find the critical point of the Lagrange 53 00:03:16,826 --> 00:03:17,585 [UNKNOWN]. 54 00:03:17,585 --> 00:03:20,196 And so we can use Newton's method do that. 55 00:03:20,196 --> 00:03:23,008 So I'll talk about about how you can set that 56 00:03:23,008 --> 00:03:26,780 problem up in section 5 and go through a small example. 57 00:03:26,780 --> 00:03:29,680 And then in section 6, I'm going to go through another example of that. 58 00:03:31,410 --> 00:03:35,599 And then finally in section 7, I'm going to look at the other optimal 59 00:03:35,599 --> 00:03:38,226 investment portfolio that I talked about in 60 00:03:38,226 --> 00:03:42,470 week 7 called the maximum expected returns portfolio. 61 00:03:42,470 --> 00:03:43,730 And so this ends up being a 62 00:03:43,730 --> 00:03:46,823 bit more complicated than the minimum variance portfolio. 63 00:03:46,823 --> 00:03:48,679 But it's something that can be solved 64 00:03:48,679 --> 00:03:51,800 using the n dimensional version of Newton's method. 65 00:03:51,800 --> 00:03:53,990 And so I have a example of how that can be done.