Welcome to week seven of Mathematical
Methods for Quantitative Finance.
This week I'm going to cover two topics,
the first is
Lagrange's Method, which is a method for
solving constrained optimization problems.
And the second topic is Taylor Series, and
this is basically polynomial expansions of
nonlinear functions.
That are going to make that are going to
make it possible to compute
decimal representations of of those
functions.
So this week's menu, I'm going to start
off.
So the first five lectures will be on will
be on the subject of Lagrange method.
I'll start off with optimal investment
portfolios.
And so this is, a type of constrained
optimization
problem, that, that occurs quite often in
a financial setting.
So, if you've already have a, a set of
assets that you
want to buy, say you have ten stocks, and
you have a certain
amount of money to use to buy those
stocks.
Then you want to choose how much of each
one to buy with the constraint that the
sum of all of those has to be the amount
of money you're going to be using.
And then optimal means you want to have
some sort of optimal criteria.
So you might want to minimize the risk of
that portfolio
or you might want to maximize the expected
return of that portfolio.
So in the first section I'll just describe
the problem I'm trying to solve.
Then I'll talk a little bit about
relative extrema for functions of several
variables.
So we saw already that if I had a function
of one variable, if I wanted to
find the local minimum or a local maximum,
I had to use first and second order
conditions.
So the first order condition gave me the
critical points.
And then the second order condition
allowed me to classify those critical
points as either a maximum or a minimum.
And in section two, I'm going to go over
how
you can do that now for functions of
several variables.
Section three, I'll describe Lagrange's
Method.
Section four, I'll give an example; so
actually
solve a constrained optimization problem
using Lagrange's Method.
And then in section five, I'll go back to
one of the investment portfolios I talked
about in section
one, called the Minimum Variance
Portfolio.
And this turns out to be, have a very nice
properties.
So when you put this problem into the
Lagrange's Method, you end up
with a linear system, so ax equals b to
solve for the critical point.
So this builds very nicely on some of the
topics we just learned in linear algebra.
Then in the second half of this
week's lecture I'm going to talk about
Taylor's Formula.
So first I'll start off just by going
over Taylor's formula for a function of
one variable.
And then, a lot of what we're interested
in
with Taylor's Formula is, I'm using this
to make
approximations of difficult functions
using polynomials, so a polynomial
is generally considered a pretty easy
function to work with.
you know, highly non-linear functions,
like sin of x times z
to the x, or something like that, would
not be fun functions
to work with.
And what we'll be interested in is how
close
is this polynomial, my polynomial
approximation, to the actual function.
And one nice concept for sort of
understanding that is called the Big O
notation.
And so I'm going to go over that in
section seven.
Section eight, I'll go over the.
Taylor's formula, so the same thing I did
in section six, but now using functions of
several variables.
Then in section nine, I'll talk about a
Taylor series expansion.
So Taylor's formula is a polynomial, so
you could have a second order, Taylor
polynomial.
It's going to have a linear term and a
quadratic term.
If I had a third order Taylor polynomial,
it would
have a linear term, a quadratic term and a
cubic term.
Taylor series expansion is,
if I just take the limit.
So I take, you know, fourth order
polynomial, fifth order polynomial, n-th
order polynomial and then I take the limit
as n goes to infinity.
And so really what were interested in
there is does this Taylor series
expansion converge to the thing that I'm
trying to approximate or does it diverge?
And so we have a few results that are
going to answer that question.
And then I'm going to
finish up that section by talking about
Bond Convexity which is revisiting a
topic.
From the first set of lectures, where I
did bond duration, and
then used bond duration to make a linear
approximation of the yield curve.
With some of the, the tools we've learned
from Taylor's formula, we can
now extend that to have a, a second order,
so a quadratic term in the approximation.
And that quadratic term is called
convexity.