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Language: en
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So, I am going to now speak about the use
of Ito calculus, to understand interest rate
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models. So, do not get into this issue of
why this model was built like this, and why
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it was built like that. So, we are going to
talk about the Vasicek’s model. The Vasicek’s
00:01:04.760 --> 00:01:19.520
model of, so this is, we are what, we are
so this is what we are going to now discuss.
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So, this is again the second version of application
of Ito’s calculus.
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So here we will see two aspects of Ito’s
calculus. The first Vasicek define, the Rt
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to be an Ito process, which is governed by
the stochastic differential equation. Of course,
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you know that these are only shorthand’s
but okay these are helpful to a certain extent.
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So, you observe that if I put Rt equal to
alpha by beta, there will be no drift term.
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It will be completely purely a random term.
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Now interestingly enough, if you assume that,
we will assume naturally, that alpha beta
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gamma are positive. Let us note that, every
stochastic differential equation, cannot be
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given a closed form solution. We will not
go in to show how to get a closed form solution,
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but it cannot be given a closed form solution
every time. But luckily in the Vasicek’s
00:03:06.569 --> 00:03:11.299
case, you can give it a closed form solution
and the closed form solution is given in the
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following way.
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So, you will immediately see that it will
tell you that this is an Ito process. So,
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do not think that okay this is something why
this is outside the integral. This was anyway
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outside the integral always. The integral
was this was into this integral. So, this
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is actually a, closed form solution. Our first
step of the use of Ito calculus here would
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be to show that this is indeed this Ito process
is indeed a solution of this.
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The question would be that is, Ito calculus
only helpful to show that something is a solution
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of this, but and I do not know how to solve
it, but I am telling you that, many of these
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equations do not have a closed form solution
and when we will go to the Cox-Rand-Ingersoll
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model, another model of interest rate. We
will see that it does not have a solution,
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but Ito calculus would help us, even if it
does not have a solution can compute its mean
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and variance.
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Now the interesting part or you will very
soon see that we can show that Rt will satisfy
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a normal distribution and hence by doing so
we will come to the following conclusion that
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with a mean 0 in fact this Rt is a random
variable which can also take negative values
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because this is a normal distribution with
mean 0. So, that something is not favorable
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you do not have negative interest rates. The
interest rates have to be positive or 0.
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So, but though it is a initially good model
to understand the behavior of interest rate.
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Of course, I am not going to the detail of
how this model was built and why it was built.
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That will take us too much of a field into
finance. So, when we are going to talk about
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finance, in our next course, then we are really
going to talk about these sorts of models
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right. That is something that we will build
up. But anybody, who would like to do this
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course in finance, will have an access to
these lectures which I gave you in the very
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basics.
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So, here we are trying to see how Ito calculus
can be applied, what are the techniques. For
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example, you know when we do Ito calculus,
our first idea would be to know how to choose
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my ft a for f tx. So, that choosing the function
f is of fundamental importance to know how
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to apply Ito calculus. Here of course, you
want to use this as some sort of your ft,
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,f tx.
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So, how will you do it. So here, we define
f tx as, basically here, we are bringing a
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process within a process. This is my, so I
am getting a smaller Ito process. So, I am
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having Xt, so your shorthand d xt, this is
the shorthand right. So, shorthand is very
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useful. It is like game playing so you can
have some fun with it always.
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Now on this, I would start applying the Ito’s
calculus. To do this, we must first know,
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so you know that, I need to have the notion
of three derivatives. The derivative of f
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with respect to x with respect to t and the
second derivative with respect to x both the
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times. So, partial derivative second derivative
with respect to x del 2 x, del x 2.
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So, what is my f t x. So, f tx would have
a larger say here, because we have terms in
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t, so this will become minus beta e to the
power minus beta t plus alpha into e to the
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power minus beta t minus minus will cancel
and give you a plus. Here you will have minus
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sigma e to the power minus sorry, beta e to
the power minus beta t into x.
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Now if you do f of x the derivative, partial
derivative with respect to x all these parts
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would be 0 and we will simply have, sigma
e to the power minus beta t and if you want
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to do it, f xx of course, it is 0 because
there is no x term anymore here. So, you know
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one of the terms of the Ito’s calculus goes
away.
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So, all these two classes, we are trying to
see how Ito’s calculus is useful. So, do
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not bother much about the financial implications
of the model. That only would come in the
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next course, as why this model is important
what is the issue etc., etc., etc.
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So, the finance game, is a very different
game, and here we are trying to learn the
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math associated with it, with examples coming
from finance of course. So, here let me start
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writing the Ito’s calculus. So hence by
Ito’s formula we have let us see what we
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have by Ito’s formula.
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So, d of f t xt is equal to ft t xt dt plus
that is now I am just writing the Ito’s
00:10:38.720 --> 00:10:48.290
formula. f x t xt d Wt because you know only
with this variable, this partial derivative
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where you have the d Wt term, plus half f
x xt xt d xt d xt which is the quadratic variation
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term and you that this is 0. so it just goes
there is nothing. So, the interesting part.
00:11:21.230 --> 00:11:26.920
which we have not told is that. all these
equations that you write here. is all are
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in the form of almost everywhere.
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These terms of Ito’s inequality, what we
are writing and we are not bothering to say,
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these are all almost everywhere type inequalities.
When I am writing equality in a stochastic
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process setup, it is always almost everywhere
type. There could be some omegas for which
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this does not hold because, we are taking
limits and we are taking quadratic variations
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and all these things are all in almost in
everywhere sense. So, all these inequalities
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it is very important.
00:11:57.670 --> 00:12:03.520
So, all inequalities are in almost everywhere
sense sorry, equalities. The equalities are
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in almost everywhere sense. Please note this
though it is very nice, so because of it only
00:12:10.790 --> 00:12:15.190
fails on a null event, we do not bother about
it, we just write equality, so while doing
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just as a working tool, I can just write equality,
but all equalities are in almost everywhere
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sense or it is same as they probably say almost
surely sense, and this occurs with probability
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one and equality occurs with probability one.
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It is a random phenomenon. Do not think that
it is a nonrandom phenomenon. So, then if
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I go and write down this and do the calculations
here, what you will get is, alpha minus beta
00:12:55.230 --> 00:13:12.070
f this is 0 that part f t x t and you know
what is this f x. So, once you write everything
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the whole thing once you put everything and
do the simplifications you will get this formula.
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I leave it for you to do the simplifications.
I am not writing the simplifications down
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because that will take too much of time.
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So, you see here you see this is exactly the
same thing. So, this we have verified by Ito’s
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calculus rule what is the that this is actually
the closed form solution of this stochastic
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differential equation. So, let us put f0 x0
and f0 x0 is equal to R0. So, f0 x0 is r0.
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So, the initial condition of this process
and f t xt. xt is now this one so if you put
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f t xt this is Rt.
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Actually, what happens ft xt has the same
behavior. Essentially if you look at it if
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I put xt here, it is exactly your Rt. So essentially,
we are if you once we are trying to apply
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Ito’s formula, we have to extract from the
given Ito process, the form of the function.
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That is the key thing that you have to observe
when we do the job.
00:15:08.670 --> 00:15:16.310
Of course, there are certain issues. So, what
about, of course you know that, what about
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this xt, what is the nature of this xt? xt
is also Ito process. So of course, xt is a
00:15:24.390 --> 00:15:50.630
process where x0 is 0. So, xt is a process
where x0 is 0 naturally. So, expectation of
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xt, what is expectation of xt is 0. So, what
is expectation of Rt? Expectation of Rt, is
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nothing but sigma e to the power minus beta
t expectation of x t right. So, over these
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constants. So, expectation of this plus expectation
of this. This part is it will come out.
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So, let us see what is the expectation of
Rt. Now if you look at it very carefully.
00:16:43.840 --> 00:16:50.040
You can find also the variance of this. So,
what would be the variance of this? Again,
00:16:50.040 --> 00:17:08.130
by the Ito isometry, the variance of this
would be, which is equal to 1 by 2 beta e
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to the power 2 beta t minus 1.
00:17:12.640 --> 00:17:24.569
So, what we have done, what we had shown,
that this x t so W is following normal distribution
00:17:24.569 --> 00:17:36.710
right. So, xt, so scaling of a normal distribution
also gives me a normal distribution. So, xt
00:17:36.710 --> 00:17:44.690
also follows a normal distribution with mean
0 and variance this. So, xt follows normal
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with mean 0 and variance. So, that shows from
here this is just a constant so this shows
00:18:00.779 --> 00:18:05.929
that expectation of Rt is.
00:18:05.929 --> 00:18:22.840
So, Rt is also normal distribution, with mean
this and variance which I am not writing in
00:18:22.840 --> 00:18:34.230
detail. You can calculate out this thing.
So basically, when you take the variance,
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the variance of this constant part is 0. You
do not have variance of a constant, where
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the constant term does not vary.
So, your Rt also follows normal distribution
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with mean this and okay.
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So, what is the lesson here. The lesson is
that if this is following normal distribution,
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and this might give me negative values, so
what it cannot have negative values. So how
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do I modify this? The question is how do I
modify this.
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This modification was done by Cox-Ross-Ingersoll
model. To study the Cox-Ross-Ingersoll model,
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we will see they have just changed the drift
term slightly, sorry, not Cox-Ross-Ingersoll
00:19:34.470 --> 00:19:42.830
but Cox-Ingersoll-Rand model. I will just
CIR, this is called the CIR model, or the
00:19:42.830 --> 00:19:54.840
Cox-Ingersoll-Rand Ross model CIR. These are
all big names in financial economics. Cox-Ingersoll-Ross
00:19:54.840 --> 00:20:02.990
CIR model.
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So, the CIR model, they assume that it should
be of this form. The interesting part is that
00:20:34.529 --> 00:20:42.360
this model does not have this drawback of
this becoming nonnegative. Rt would never
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become nonnegative, but this is the example
of a stochastic differential equation, which
00:20:46.779 --> 00:20:53.929
does not have a closed form solution. Even
if does not have a closed form solution, you
00:20:53.929 --> 00:20:59.259
can still get some idea of its mean and variance.
00:20:59.259 --> 00:21:04.150
If you want to compute that okay what sort
of distribution Rt satisfies. Then it will
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be quite a difficult thing to compute the
distribution, but you can at least compute
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its important parameters, mean and variance.
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So, let us just give a hint, as how to compute
mean of Rt and you will see how the Ito’s
00:21:35.690 --> 00:21:39.830
calculus would be helpful here. See Ito’s
calculus is not just to verify some given
00:21:39.830 --> 00:21:51.330
solution our SDE. It is helpful in many-many
states. So, to do this, you have to choose.
00:21:51.330 --> 00:21:57.960
Now this choice of f tx it must look very
arbitrary to you, but it is done because you
00:21:57.960 --> 00:22:03.620
have done some work. It is done from guess
and test. So, if you always bring in exponential
00:22:03.620 --> 00:22:11.280
things because, you want to have things nonnegative.
So, exponential thing always comes up usually.
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So, your xt is your Rt. So, your so d of e
to the power beta t Rt. So, that is again
00:22:25.460 --> 00:22:53.809
ft t rt dt plus fx x Rt dt plus half f x x
t Rt d Rt d Rt. So, I will ask you to calculate
00:22:53.809 --> 00:22:58.950
this thing through okay. If you calculate
this thing, as you can have some fun computing
00:22:58.950 --> 00:23:06.720
d Rt d Rt because, it will become alpha square
R t d t. So, if you do this calculation this
00:23:06.720 --> 00:23:15.789
will turn out to be the final form would be
of this form. Alpha e to the power beta t
00:23:15.789 --> 00:23:38.550
dt plus sigma e to the power beta t root Rt
d Wt sorry here I will have d Wt.
00:23:38.550 --> 00:23:49.289
So, when you write put in all the values,
you know what is e tx ft is beta e to the
00:23:49.289 --> 00:23:56.450
power beta tx. So, you know what it is and
what is x. It is e to the power beta t. What
00:23:56.450 --> 00:24:14.389
is f xx. f xx is, what would be f xx here.
Here f xx would be 0. So, once you know that
00:24:14.389 --> 00:24:18.820
fact, so you do not have to bother about the
quadratic variation, and then you write down
00:24:18.820 --> 00:24:27.100
everything, alpha e to the power beta t for
example here ft would be beta e to the power
00:24:27.100 --> 00:24:28.730
beta t x which is Rt.
00:24:28.730 --> 00:24:34.379
So, it will be alpha beta e to the power beta
t R t into d t. So, write the full form Rt
00:24:34.379 --> 00:24:41.320
again this one. So, once you write, sorry,
you do not have to write the full form R t.
00:24:41.320 --> 00:24:52.269
So here you have f x so x e to the power beta
t x e to the power beta t x means 1 so f t
00:24:52.269 --> 00:24:58.690
R t e to the power beta t into d omega t.
So, once you write this you will get back
00:24:58.690 --> 00:25:05.919
this formula. You see we are just essentially
getting that.
00:25:05.919 --> 00:25:18.639
So, if you now know this formula, I would
similarly ask you. You can now integrate both
00:25:18.639 --> 00:25:23.539
sides. So, you can integrate both sides take
off the differential and you can write e to
00:25:23.539 --> 00:25:39.080
the power beta t R t is, what you have alpha
e to the power beta t dt 0 to t plus 0 to
00:25:39.080 --> 00:25:43.610
t sigma e to the power bt root R t d Wt.
00:25:43.610 --> 00:25:51.350
So, once you know this you can immediately
use your other ideas about Ito’s integral
00:25:51.350 --> 00:25:59.039
and this is a simple case. To really figure
out, what is the mean, you just take the beta
00:25:59.039 --> 00:26:04.440
t on the other side compute this out and then
take them in. So, I will leave that to you
00:26:04.440 --> 00:26:10.200
as homework. You should try it out something
yourself. If you do not try it out yourself
00:26:10.200 --> 00:26:12.269
things might not always be fine.
00:26:12.269 --> 00:26:19.320
So, with this I end my talk to give you some
idea about how things can be used. Please
00:26:19.320 --> 00:26:23.570
check up these calculations. Do not completely
rely on what I have written at the end. Just
00:26:23.570 --> 00:26:28.220
checkup that you come to that conclusion.
00:26:28.220 --> 00:26:34.720
So, thank you very much.