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Welcome to week seven of Mathematical
Methods for Quantitative Finance.

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This week I'm going to cover two topics,
the first is

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Lagrange's Method, which is a method for
solving constrained optimization problems.

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And the second topic is Taylor Series, and

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this is basically polynomial expansions of
nonlinear functions.

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That are going to make that are going to
make it possible to compute

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decimal representations of of those
functions.

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So this week's menu, I'm going to start
off.

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So the first five lectures will be on will
be on the subject of Lagrange method.

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I'll start off with optimal investment
portfolios.

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And so this is, a type of constrained
optimization

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problem, that, that occurs quite often in
a financial setting.

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So, if you've already have a, a set of
assets that you

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want to buy, say you have ten stocks, and
you have a certain

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amount of money to use to buy those
stocks.

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Then you want to choose how much of each
one to buy with the constraint that the

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sum of all of those has to be the amount
of money you're going to be using.

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And then optimal means you want to have
some sort of optimal criteria.

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So you might want to minimize the risk of
that portfolio

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or you might want to maximize the expected
return of that portfolio.

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So in the first section I'll just describe
the problem I'm trying to solve.

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Then I'll talk a little bit about

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relative extrema for functions of several
variables.

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So we saw already that if I had a function
of one variable, if I wanted to

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find the local minimum or a local maximum,

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I had to use first and second order
conditions.

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So the first order condition gave me the
critical points.

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And then the second order condition
allowed me to classify those critical

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points as either a maximum or a minimum.

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And in section two, I'm going to go over
how

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you can do that now for functions of
several variables.

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Section three, I'll describe Lagrange's
Method.

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Section four, I'll give an example; so
actually

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solve a constrained optimization problem
using Lagrange's Method.

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And then in section five, I'll go back to

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one of the investment portfolios I talked
about in section

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one, called the Minimum Variance
Portfolio.

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And this turns out to be, have a very nice
properties.

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So when you put this problem into the
Lagrange's Method, you end up

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with a linear system, so ax equals b to
solve for the critical point.

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So this builds very nicely on some of the
topics we just learned in linear algebra.

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Then in the second half of this

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week's lecture I'm going to talk about
Taylor's Formula.

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So first I'll start off just by going

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over Taylor's formula for a function of
one variable.

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And then, a lot of what we're interested
in

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with Taylor's Formula is, I'm using this
to make

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approximations of difficult functions
using polynomials, so a polynomial

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is generally considered a pretty easy
function to work with.

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you know, highly non-linear functions,
like sin of x times z

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to the x, or something like that, would
not be fun functions

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to work with.

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And what we'll be interested in is how
close

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is this polynomial, my polynomial
approximation, to the actual function.

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And one nice concept for sort of

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understanding that is called the Big O
notation.

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And so I'm going to go over that in
section seven.

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Section eight, I'll go over the.

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Taylor's formula, so the same thing I did

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in section six, but now using functions of
several variables.

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Then in section nine, I'll talk about a
Taylor series expansion.

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So Taylor's formula is a polynomial, so

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you could have a second order, Taylor
polynomial.

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It's going to have a linear term and a
quadratic term.

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If I had a third order Taylor polynomial,
it would

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have a linear term, a quadratic term and a
cubic term.

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Taylor series expansion is,

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if I just take the limit.

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So I take, you know, fourth order
polynomial, fifth order polynomial, n-th

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order polynomial and then I take the limit
as n goes to infinity.

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And so really what were interested in
there is does this Taylor series

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expansion converge to the thing that I'm
trying to approximate or does it diverge?

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And so we have a few results that are
going to answer that question.

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And then I'm going to

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finish up that section by talking about

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Bond Convexity which is revisiting a
topic.

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From the first set of lectures, where I
did bond duration, and

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then used bond duration to make a linear
approximation of the yield curve.

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With some of the, the tools we've learned
from Taylor's formula, we can

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now extend that to have a, a second order,
so a quadratic term in the approximation.

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And that quadratic term is called
convexity.