So, we're finally arriving at week number
eight
where I'm going to talk about numerical
methods.
So I think it's you know, when we looked
at the Lagrange
multipliers, we started to arrive at
things where they were getting just.
You could still solve them in special
cases.
for instance, when we looked at the
minimum, minimum variance portfolio.
It was still possible to, where we ended
up like the linear system,
we need to dissolve to do the Lagrange
multipliers.
here to find the critical point for the
Lagrange multiplier, the critical point
for the Lagrangian.
[COUGH]
But pretty much anything beyond that, any
anything that is not sort of
contrived to be simple is going to be far
too difficult to solve.
And so that's why we have something called
the numerical methods.
Which I sort of think of as just the
logical extension
of the guess and check method that I
learned in elementary school.
So you basically are just trying to make a
guess.
You'll measure how
wrong you are, and that will hopefully
suggest
a way for you to improve your guess.
And you want to do that in a way where you
find the answer you're looking for with a
minimum amount of work.
So the motivation for this, I'm, I'm
going to look at a problem called implied
volatility.
And then I'll talk about two methods for
solving that.
So one is bisection method, which is just
a
very easy approach to solving this implied
volatility problem.
Then Newton's method, which is not much
more difficult, but theoretically
it's a bit more, it, it has some
theoretical, some nice theoretical
results we can, we can look at.
Now, also the nice thing about Newton's
method
is it can be extended for N dimensional
problems.
And then finally, and I don't know if
we'll have time to get to this today.
we'll look at solving a problem with
Lagrange method, so it's actually
one of the same problems I looked at in
the Lagrange slides.
But instead of just trying to do algebra
once I need to find
the critical point, I'll use Newton's
method to try and find the critical point.
So we get started by talking about implied
volatility.
So if we remember from the first half of
the course, we were looking at the,
the Black-Scholes formula for a while, and
mostly
when we were looking at computing partial
derivatives.
And so, this is just to remind you what it
is, and hopefully I haven't made any, any
typos in it.
And if you look, well, I think I've got
all of the,
so the maturity, so the, the maturity date
of an option, this is
a, a contract that I go and buy and sell
in a market.
And when it's written, it's going to have
a maturity date.
So, it's good for 3 months or it's good
for 6 months.
Something like that.
It has something called the strike price,
which is, this
is also something going to be written on
the contract.
So basically, once this contract comes
into existence, the T Oops, and the K are
fixed, so they're never going to change.
And then there's a secondary market for
these contracts.
So once I have the contract, if it's
between the time that it was written
and the maturity, I can go to the
secondary market and I can sell my option.
So I don't actually have to wait for the
condition in the auction to come to pass.
I can go and find somebody else who's
interested in having that
same type of protection and I can sell my
option to him.
And so if I
sell the option, at the time that I sell
the option the price that I sell it for
is going to be known, because you know, I
give him the contract, he gives me some
money.
So I can look at the amount of money he
gave me
and, and I know exactly what the price of
that option is.
The option is based on some underlying
asset that has a,
a price S, and so I can also look in the
market
at the same time that I sold my option for
this price C.
I can look at the most recent trade for
that underlying assett and see what the
asset price is.
So, I can also assume that when these
transactions take
place, the C and the S are going to be
known.
And then I'm also going to assume that the
risk free rate is constant, and that the
dividend rate.
So, because options exist on kind of a
short time scale
most of the time, you know either 3 months
or 6 months.
You generally know what the, the company's
already
announced when it's going to be paying
dividends.
So you know if you're going to get
dividends during the life of the option or
not.
And the risk free risk, well, it's
definitely changing.
It changes on a very slow time scale.
So on the, during the life of the option,
you can assume that that's fixed.
And so it turns that if you look in this
formula, so it depends on this d1 and d2
as well, you know everything except this
variable sigma.
So the, the only quantity that's not known
is the volatility, and
this is the one thing that it's actually
quite difficult to know,
that there's not any one thing in the
market where you can
just say oh, this is, this is definitely
the rate of sigma.
So going in the, in the forward sense when
you want to use sigma as
an input to this formula, it's something
you have to actually estimate from data.
So you have to look at, it's, it's trying
to describe the, how how
wide the fluctuations are of this
underlying asset price S.
And so it's, you can estimate it, but you
can never actually know it.
Whereas if you go to the market you know,
on a particular day, I can see what was
you know, what when that asset changed
hands, what
were those prices so I can actually
observe S.
Sigma, I'm not able to do that in any way.
So the implied volatility problem.
If I know every variable and the answer to
the Black-Scholes pricing formula except
sigma, then what I want to do is just sort
of solve this equation backwards.
So, essentially, if I could, I'd like to
isolate sigma and
have a new functions that's sigma as a
function of everything else.
But it turns out, if, if we look at this
guy, it's not exactly easy to do that.
So extra credit if anybody can come up
with the formula.
And so what I want to do, as this is
where I was getting at this trial and
error thing.
I want to just plug in values of sigma
into my
Black-Scholes formula, and I know what the
call price should be.
And for every different value of sigma
that
I plug in, I'm going to get some price.
And eventually, if I just keep guessing
long enough, I'm going to get a value of
sigma that gets me very close to the price
that I know it should, should be.
So I,
if I can get close to that market price C,
then that's
probably, well, I mean, that's the, the
value of sigma that would
have been in the market.
And so even though I can't observe sigma,
all the other things that I can observe
imply a certain level for sigma.
They imply a certain amount sigma, and so
that's going to be called the
implied volatility.
So what I want to look at is, is if, if S,
K, Q, R, T and little t.
So this is basically all of the inputs to
the
Black-Scholes pricing formula for our
European call option are known and
I also know what the price of that option
is,
then I can think of this as a function of
sigma.
So there's only one unknown in this
highlighted
side of the equation and that's sigma.
And the way I've set
this up, I've, I've, put the theoretical
price for European call
option minus the price for that option
that I've observed in the market.
So when I have the correct value of sigma,
this
whole thing f of sigma should be equal to
0.
So finding the implied volatility really
just boils down to solving
this thing, it, it's called a non-linear
optimization problem.
And I just want to find the value of sigma
so that f of sigma is equal to zero.
And this is something like I said you're
not going to be able to just
isolate sigma and have a nice f of sigma
function you can just plug in a
zero and find out what should it have
been.
so you have to solve this numerically.
And so when I say, solve numerically,
there, there are lots of
different prices, lots of different ways I
can try and do this.
So, the, the simplest one would just be to
make a plot
of f of sigma and see where it crosses the
x axis.
And so, where it crosses the x axis is
where f of sigma is going
to be equal to 0, and that's what I'm
trying to find in the, the value.
So, the x axis in this case is going to be
the sigma axis.
The value of sigma where it crosses the
axis is this f of sigma equals 0.
And so, to do that in R, the first thing I
need is a function to compute the
Black-Scholes call price.
So, this is just our implementation of
what was on the slide, a couple of slides
ago, so I've made an expression for d1.
Then the expression for d2, and then
finally the expression
for the Black-Scholes call price.
And I, I don't know if you guys are
getting used to looking at r
yet, but basically what I'm doing here is
I've I'm taking this expression,
I'm assigning it to a piece of memory, a
variable called d1.
Then I'm taking this expression, assigning
it to
a piece of memory, a variable called d2.
But then in R, if the last expression in a
function.
So, because this has this keyword function
here, this thing
is a function, the last expression in a
function gets returned.
So, when I evaluate this, I don't have to,
you can if you want to
say you know, assign this to say
Black-Scholes call price and
then have one more line that says return
Black-Scholes call price.
Or you can just put an expression for the
thing
that you want your function to return as
the last line.
So that's why this is going to work.
And then one of the nice features R has is
it's treating every
variable like a vector.
So, instead of having to do this in a for
loop, or evaluate the function by hand you
know, 50 or 60 times to get a set of
points to, to plot, I can just give I can
give a vector argument.
So right now I'm giving every argument
that's just a
length one vector.
And so I get a length one answer.
On the other hand, suppose I take, so the
second to last thing, this little s so.
I, I guess I should also mention this is
sort of
bad bad programming, because none of the
variables have very meaningful names.
I just had to do that to make it fit on
the slide, the
first way, I wrote this only about the
first half of it on the slide.
I also think it's a lot easier to read
if you know, when I say something like R
minus
Q, that should be R space minus space Q.
you want to group things, so it's a little
bit easier to see what the function is.
But in this particular case I had a
another constraint which it needed to
fit up here for the, for the lecture, so I
had to skip that.
But if we look at the second to last
argument, this lower
case s, this is where I'm putting in my
value of sigma.
And so, when I call my function
this time, instead of putting in a single
value
for sigma, I'm putting in three values for
sigma.
I want to end that getting out are three
values for this Black-Scholes call price.
So all of the other variables stay the
same, but the first one here
corresponds to sigma of 0.15, the second
one here corresponds
to sigma of 0.2, and the third one
corresponds to sigma of 0.25.
And now, let's suppose that the option
sold for $7, and I want to find sigma.
So you can think of this is, I want to
find the
sigma that's going to make my
Black-Scholes call price equal to 7.
But most nonlinear solvers are going to be
set
up just to solve this f of whatever equals
0.
So you want to make a new objective
function where the point you're looking
for is 0.
And
I'm going to call that function f of
sigma.
And that's just going to be my
Black-Scholes call price minus 7.
And I'm going to plot that over a range of
values
of sigma and see where it crosses the x
axis.
So the first thing I'll do is make my
range of sigmas.
So seq is a function that just makes a
sequence.
It starts at 0.05.
It's going to end at 0.5.
And it's going to go in steps of 0.01.
Then f sig, I'm going
to evaluate my Black-Scholes call price,
add all of
the inputs that were fixed, and then at
this entire vector of sigmas.
So this is, I think this will end up being
maybe 46 different values of sigma.
So, this is going to give me 46 different
Black-Scholes call prices corresponding to
each one of those input sigmas, and then
I'm going to subtract seven from that.
And hopefully, one of these will be pretty
close to zero.
And then I'm going to plot.
So, I'll put sigma on the x axis, and
then this fsig that I've just calculated
here on the y axis, and then this optional
arguments type equals l.
It's just going to plot it with a line
rather than
with points, and I get a plot that looks
like this.
And so, on the, on the left here, I'm
below 0, on the right I'm about 0.
And it looks like about sigma equals 0.25
is about where I hit 0.
So that gives me an implied volatility,
so I denoted that.
Sigma sub implied, think the book is
called Sigma sub imp of 0.25.
And so, this is a valid way of coming up
with an implied volatility.
But to do that, I had to evaluate the
Black-Scholes formula 46 times.
And now if you look at this plot, you
know, I evaluated
a whole lot over here where it wasn't
telling me very much.
And I did the same thing over here.
And still the computer answer was not very
precise.
So I, I only plotted it in these steps of
0.01.
And so it turned out that 0.25 was
slightly less than 0.
And when I put in 0.26, it was slightly
more, but it
was more slightly more so that 0.25 was
the closest one to 0.
So that's
why I picked this one.
And so I was off by, so this,
this is on the scale of dollars here, so I
was off by about one and a half cents.
So it's, doesn't sound too bad, except
that
you might be trading, you know, thousands
or tens
of thousands of these things, and so that,
that
ends up being more worrisome to be off by.
So what I'd really like to be able to do
is compute this implied volatility to
within a pre-specified tolerance.
So right now, I've, I've just sort of done
these 46 things, and I've ended
up finding out how close I was just by
going back and checking my solution.
What I would like to have is an algorithm
that's going to keep going until.
So for instance, I was just going to buy
one option, I
would sort of want this to keep going
until maybe the first significant
digit was somewhere over here, and then I
would be certain that
I'm within less than a penny of the price
that I'm aiming for.
And so, if I only wanted to buy one.
Well, if that's the smallest denomination
of currency bill
allowed me to use, then that will be good
enough.
So I want to get inside this pre-specified
color, and at the same time
to do that, I want to use the minimum
number of function evaluations required.
And so the methods, the, the general
methods for
solving this type of problem are called
nonlinear solvers.