Welcome to week 8, the 8th and
final week of Mathematical Methods for
Quantitative Finance.
This week, I'm going to talk about
numerical methods.
So, so far in the course, I've introduced
a
lot of topics for solving different types
of the problems.
But a lot of the examples in the homework
have been
what, what I sort of think of as sort of
toy examples.
Because they need to be complicated enough
that you understand what's going on.
At the same time they need to be simple
enough that you
can actually solve them with a, pen and a
piece of paper.
In practice though, we're going to want to
solve much more complicated problems.
And so to do that we, we're going to want
to do that with a computer.
And so that brings us to this topic of
numerical methods.
So the outline for this week, I'll talk
about a problem called Implied Volatility.
So this is basically going back to the
Black-Sholes pricing formula.
But instead of thinking of the, the value
that I
want to calculate as the call price or the
put price.
I'm going to think that I could actually
go,
or I'm going to approach the problem in
this sense.
I can actually go to the
market and I can observe all of the inputs
for the Black-Sholes formula.
And I can see what that put or call price
actually is.
And the one thing that's going to be the
most
difficult to observe is the volatility
parameter, the sigma.
And so, given that I have the price, I can
actually use Black-sholes kind of
backwards to figure out what the sigma
parameter had to be to justify that price.
And that turns out to involve solving
something called a non linear equation.
And so then in sections two and three, I'm
going to
consider two methods for solving non
linear equations, the Bisection method.
So essentially what I'm trying to do with
the Bisection method.
So I want to find two endpoints so that I
know I'm bigger than the value I
want on one side, and I'm smaller than the
value I want on the other side.
And then, I'm going to just start chopping
that interval in half.
By preserving this property that the, the
right end point and
the left end point bracket the point that
I'm looking for.
And so, as I chop this in half, you know,
the end points
are getting closer and closer to the
number that I want to compute.
Another approach for solving this is
called Newton's method.
So this has a little bit stronger
assumptions that I need
the function I'm trying to find the value
of to be differentiable.
And then the other nice property that
Newton's Method
has is there's a n dimensional version of
Newton's Method.
So the, the bisection method is only
going to work if there's a single
variable.
It get, it starts to break down if I'm, if
there's
a variable going this way and a variable
going this way.
And so once I've gone over how I can
solve these nonlinear problems using
Newton's method and bisection.
Then I'm going to look at some of the
problems we had from last week's lecture.
So in particular, the problems that came
up using Lagrange's method.
We're going to have generally a non linear
system that we
need to solve to find the critical point
of the Lagrange
[UNKNOWN].
And so we can use Newton's method do that.
So I'll talk about about how you can set
that
problem up in section 5 and go through a
small example.
And then in section 6, I'm going to go
through another example of that.
And then finally in section 7, I'm going
to look at the other optimal
investment portfolio that I talked about
in
week 7 called the maximum expected returns
portfolio.
And so this ends up being a
bit more complicated than the minimum
variance portfolio.
But it's something that can be solved
using the n dimensional version of
Newton's method.
And so I have a example of how that can be
done.