WEBVTT
Kind: captions
Language: en
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So if you go to the stock market and look
at the price of say a favourite company’s
00:00:24.460 --> 00:00:28.760
share. So what you would observe is the following.
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So if the horizontal axis is the time axis
and if this is the price axis it tells you
00:00:35.340 --> 00:00:48.600
what is the price then you will see starting
from a certain time price
00:00:48.600 --> 00:01:00.190
you will see some zigzagging motions like
this. So you observe it up to time T and you
00:01:00.190 --> 00:01:03.420
observe this zigzagging motion.
00:01:03.420 --> 00:01:08.390
This is of course random. Nobody knows what
is the next price is. So if a particular scenario
00:01:08.390 --> 00:01:19.760
evolves, you have a particular path. This
is called a sample path. So if another scenario
00:01:19.760 --> 00:01:29.610
evolves, if another scenario evolves there
would be another path, for example it could
00:01:29.610 --> 00:01:34.430
be like this. The stock price is going down
down down down and you are in a bad shape
00:01:34.430 --> 00:01:39.890
and then it again climbs up and again it falls
down down down and again then again climbs
00:01:39.890 --> 00:01:40.890
up.
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So under a different scenario it has a different
path. So it is sample path 1 it is sample
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path 2. So it is what type of scenario one
evolves. Now of course you can ask me what
00:01:52.761 --> 00:01:56.970
is this term scenario that you are talking
about, what is the meaning of this goddamn
00:01:56.970 --> 00:02:04.930
scenario? We will come to this very soon.
But how do I model such zigzagging paths,
00:02:04.930 --> 00:02:13.870
what way to model it. Is there any mathematical
way to say that us or can I construct the
00:02:13.870 --> 00:02:20.239
stochastic process whose sample paths are
represented in this form?
00:02:20.239 --> 00:02:24.389
Let us do, to do that we need to study what
is called Brownian motion. Brownian motion
00:02:24.389 --> 00:02:31.099
is a type of stochastic process which will
help us to model stock prices at the end.
00:02:31.099 --> 00:02:38.230
So the whole term Brownian motion comes from
the name of Robert Brown who first studied
00:02:38.230 --> 00:02:43.689
the movement of pollen grains in water and
he found that they were having a zigzag haphazard
00:02:43.689 --> 00:02:49.639
movements. But it is not so immediately apparent
that you can just start writing about this
00:02:49.639 --> 00:02:51.870
particular stochastic process.
00:02:51.870 --> 00:03:01.200
We need to have some more idea and built upon
some simpler stochastic process. So we will
00:03:01.200 --> 00:03:06.319
begin by introducing what is called symmetric
random works.
00:03:06.319 --> 00:03:14.669
Where there are only 2 possibilities you can
either go up or down that is like a coin toss
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head or tail and that here I can have infinite
such possibilities, infinite such sample paths
00:03:22.870 --> 00:03:27.809
and there are infinite possibilities also.
Here also we will have infinite possibilities
00:03:27.809 --> 00:03:39.569
but generated out of only 2 possibilities.
So symmetric random walk. So when you take
00:03:39.569 --> 00:03:46.389
a fair coin and then you keep on repeatedly
tossing it.
00:03:46.389 --> 00:03:56.249
So it is a so this is a stochastic process
which I will write in short form now as stochastic
00:03:56.249 --> 00:04:28.790
process generated by repeated tosses of a
fair coin okay and if you look at it very
00:04:28.790 --> 00:04:38.160
carefully what I mean by this? So you start
tossing the coin so repeatedly you are tossing
00:04:38.160 --> 00:04:45.870
omega 1 omega 2; so omega 1 is either head
or tail; omega 2, omega 3, omega 4, 5 and
00:04:45.870 --> 00:04:51.770
so and so forth. Suppose you have here head,
head, tail, tail, head, head, head, tail,
00:04:51.770 --> 00:04:55.030
tail, tail and it goes on.
00:04:55.030 --> 00:05:02.180
So this is one particular scenario that has
evolved. You could have another scenario say
00:05:02.180 --> 00:05:10.890
omega bar which is consisting of say omega
1 bar, omega 2 bar, omega 3 bar, omega 4 bar,
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omega 5 bar, it could be something like this;
tail, tail, head, head, head, tail, tail,
00:05:17.420 --> 00:05:23.030
tail, head, head, head and so on. So these
2 are different scenarios and these 2 each
00:05:23.030 --> 00:05:29.650
would generate 2 different sample paths. So
how do we generate this symmetric random walk?
00:05:29.650 --> 00:05:49.480
So these are 2 different scenarios, 2 different
scenarios. Now construct a random variable
00:05:49.480 --> 00:06:21.370
Xj which takes the value 1 if j is equal sorry
if omega j is equal to head and takes the
00:06:21.370 --> 00:06:31.060
value -1 if omega j is equal to tail if tail
appears and 1 if head appears. So now you
00:06:31.060 --> 00:06:52.220
define a stochastic process, define a new
stochastic process Mk, k=0 to infinity or
00:06:52.220 --> 00:06:59.830
let us we can need not bother we can also
fix it after some time.
00:06:59.830 --> 00:07:07.240
It could be some time capital say K is say
25 something here 25 or 30 whatever. But in
00:07:07.240 --> 00:07:15.500
general it is alright to take plus infinity
just a sequence where M k is given as follows.
00:07:15.500 --> 00:07:28.150
Each of these M ks are calculated by starting
from M 0 equal to 0. M k is equal to j is
00:07:28.150 --> 00:07:37.050
the sum from j from 1 to k to X j. So let
us see what would happen if one particular
00:07:37.050 --> 00:07:43.240
scenario like this evolves. Let us see then
what is the sample path of this. This symmetric
00:07:43.240 --> 00:07:50.780
random walk is also called a drunkards walk.
00:07:50.780 --> 00:08:00.760
So somebody has had a good drink and he has
become drunk and if you look at his walk so
00:08:00.760 --> 00:08:06.110
a drunkard would walk like this if I am here
so I start from here then I can just go like
00:08:06.110 --> 00:08:11.740
this and he goes like this just it is just
or like this you know I am coming here and
00:08:11.740 --> 00:08:16.620
then I am going there something like this.
So this sort of thing you will immediately
00:08:16.620 --> 00:08:26.810
observe as I start say checking out with this
scenario.
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So here is my k and here is my M k value.
Now the first one here has turned out to be
00:08:39.880 --> 00:09:01.149
head. So M0 is 0. Let me write -1, -2, -3
and so on -4 here 1 2 3 4 and so on and of
00:09:01.149 --> 00:09:14.620
course here also you have to have k values
which is 1 or maybe 1 2 3 4 5 6 7 8 9 10 and
00:09:14.620 --> 00:09:25.240
so and so forth. So M0 is 0 this is 0 is 0.
Now you toss a coin and you have head.
00:09:25.240 --> 00:09:34.050
Omega 1 is head so you go up by +1 because
X j will take +1 because m1 is just X 1 so
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here is the value of M1 this is your M1 so
you join the M0 and 1 by line so 0 is M0 and
00:09:44.260 --> 00:09:51.509
then you again had head so M2 is again 1.
I am looking at this scenario M2 is you go
00:09:51.509 --> 00:10:01.950
by 1 so it is 1+1 now 2. So M2 is 2. So you
join again by this line. But M 3 is tail so
00:10:01.950 --> 00:10:06.839
you will drop by 1 so it will be -1 so it
will again drop back to the point 1.
00:10:06.839 --> 00:10:17.769
So this is your M2 and this is M3 and then
omega 4 is again tail so it drops back to
00:10:17.769 --> 00:10:30.499
0 again you -1 subtract. So this is your M4
0. Again then you have head for omega 5 say
00:10:30.499 --> 00:10:41.640
so here you have omega 6, omega 7, omega 8,
omega 9, so for M4 you have tail you have
00:10:41.640 --> 00:10:46.279
come to 0 again then it goes up again for
M it goes to plus 1 again.
00:10:46.279 --> 00:10:57.869
So this is your M5. Again, it goes up to 2
M6 but then you have tail again so M7 comes
00:10:57.869 --> 00:11:07.439
down to +1. Again, you have tail so M8 comes
down to 0 because you are adding up everything.
00:11:07.439 --> 00:11:15.420
At every step you are going up or down. So
you add up in this fashion and you move like
00:11:15.420 --> 00:11:24.199
this. So M8 I have here omega 8 it is again
omega 9 is tail so again I have to go down
00:11:24.199 --> 00:11:30.100
so I go down by so from 0, I will have to
go down by 1 so I will come to -1.
00:11:30.100 --> 00:11:39.559
So this will be your M9 and if suppose M10
omega 10 is head then it will again go up
00:11:39.559 --> 00:11:46.960
to 0 at the 10th place because you will again
add 1 so it will become your M10. So what
00:11:46.960 --> 00:11:57.589
you see that the symmetric random walk is
providing me some zigzag looking curve which
00:11:57.589 --> 00:12:04.809
might tempt you to think that possibly these
2 have some relationships. They are they do
00:12:04.809 --> 00:12:10.980
have some relationships and we will talk about
that slightly down the talk.
00:12:10.980 --> 00:12:18.519
But let me tell you some more properties of
this symmetric random walk M k. So here is
00:12:18.519 --> 00:12:36.529
my stochastic process and this stochastic
process is called the symmetric random walk.
00:12:36.529 --> 00:12:41.389
Of course we are not mentioning but underlying
we are always taking some probability space
00:12:41.389 --> 00:12:43.730
and all of those things.
00:12:43.730 --> 00:12:51.800
So now we will these random this particular
random process of stochastic process has independent
00:12:51.800 --> 00:13:08.800
increments. What do I mean by the fact that
they have independent increments? What I mean
00:13:08.800 --> 00:13:22.779
is the following. So if you have you take
certain numbers sometime say some K m then
00:13:22.779 --> 00:13:24.089
you have the following.
00:13:24.089 --> 00:13:37.230
You have M k1 that is once you consider non-overlapping
intervals then this difference is independent
00:13:37.230 --> 00:13:43.990
because they depend on independent coin tosses
because coin tosses are independent when the
00:13:43.990 --> 00:13:48.360
coin tossed at the second level really does
not the second outcome does not really depend
00:13:48.360 --> 00:14:00.970
on the first outcome right when you do a repeated
coin toss sorry M k2 - M k1,…, M k m- M
00:14:00.970 --> 00:14:08.000
k m-1.
00:14:08.000 --> 00:14:15.829
So these random variables are independent.
All of these random variables these form a
00:14:15.829 --> 00:14:24.189
set of independent random variables. So that
is once this happens this is when we say that
00:14:24.189 --> 00:14:29.279
it has independent increments and this actually
has independent increment. These are independent
00:14:29.279 --> 00:14:37.629
because their difference which really does
not depend on the coin tosses here does not
00:14:37.629 --> 00:14:43.879
depend on the coin tosses here and here and
so you have independent increments.
00:14:43.879 --> 00:14:48.110
So this change that you see here does not
depend on the change that you see in this
00:14:48.110 --> 00:14:55.839
interval or the change that you see in this
interval right. So you can still observe that
00:14:55.839 --> 00:15:02.070
it is like a drunkard walk so drunkard walks
like this goes down goes up. In George Gamow’s
00:15:02.070 --> 00:15:08.860
famous book One, Two, Three Infinity this
has been described in a very very nice way.
00:15:08.860 --> 00:15:17.649
How do you, see here, my success probability
this occurs with probability half and this
00:15:17.649 --> 00:15:25.399
also occurs with probability half so if you
observe that exponential of X j sorry expectation
00:15:25.399 --> 00:15:48.100
of X j is 0 because this is one into half
plus minus one into half. Variance of X j
00:15:48.100 --> 00:15:58.189
so what is variance of X j exponential X minus
X j whole square which is 0 so it is 1 into
00:15:58.189 --> 00:16:05.970
half plus minus 1 minus 0 whole square plus
1 into half which is 1.
00:16:05.970 --> 00:16:16.579
Once you have this information this is true
for all j. It is immediate that exponential
00:16:16.579 --> 00:16:50.389
M of k i plus 1 minus M of k i is 0 and the
variance of M of k i plus 1 minus M of k i
00:16:50.389 --> 00:17:03.699
is equal to I leave it to you to calculate
these stuffs. Our second property about this
00:17:03.699 --> 00:17:11.860
random walk is to show that this is also a
Martingale. You see Martingale thing comes
00:17:11.860 --> 00:17:46.120
up. So symmetric random walk is a discrete
Martingale.
00:17:46.120 --> 00:17:52.480
So you can easily prove that it is a Martingale.
So you take any k strictly less than l and
00:17:52.480 --> 00:18:04.420
look at the expectation of M l conditioned
on the filtration, the Sigma-algebra F k which
00:18:04.420 --> 00:18:10.420
is the part of the filtration okay.
00:18:10.420 --> 00:18:36.070
So you can write this as
M l minus M k plus M k. So these can be summed
00:18:36.070 --> 00:18:40.520
up just like expectation can be some conditional
expectation this random variable can be decomposed
00:18:40.520 --> 00:18:44.400
into 2 parts which you can actually prove
which will be a part of your exercise but
00:18:44.400 --> 00:19:05.880
we are just using this fact here so, anyway
I should rub the board a bit.
00:19:05.880 --> 00:19:24.050
Now let us look at the first part. Since l
is strictly bigger than k this increment M
00:19:24.050 --> 00:19:31.860
l minus M k is independent of F k. F k does
not have the information of anything which
00:19:31.860 --> 00:19:39.730
is beyond the time k. So here by one of our
rules for conditional expectation this is
00:19:39.730 --> 00:19:55.120
nothing but M l minus M k and here at time
k everything about M k is known. So F k contains
00:19:55.120 --> 00:20:02.490
all information about M k. So the first law
was taking out what is known, I can write
00:20:02.490 --> 00:20:09.260
this M k as M k dot 1 where 1 is the constant
random variable 1. So whatever be the scenario
00:20:09.260 --> 00:20:11.940
it will just give you the value 1.
00:20:11.940 --> 00:20:18.640
So I can write this as M k so I will write
this as M k dot 1 so I can take out what is
00:20:18.640 --> 00:20:26.250
known 1 dot F k. Of course, 1 is a constant
random variable. It does not really depend
00:20:26.250 --> 00:20:33.550
on is independent of F k so it will be E of
1 which is a constant which will be just 1.
00:20:33.550 --> 00:20:38.850
So everything will be 1 so the sum of the
probabilities will sum up to 1. So the expectation
00:20:38.850 --> 00:20:41.260
will be just the number.
00:20:41.260 --> 00:20:52.940
So this again is 0 which we already know plus
M k into 1 which is M k and so this shows
00:20:52.940 --> 00:21:02.590
that M l is a this symmetric random walk this
thing forms a discrete Martingale. Of course,
00:21:02.590 --> 00:21:09.040
F k is M k has to be adapted to this filtration
that is the basic definition of Martingale.
00:21:09.040 --> 00:21:16.210
There is another notion which crops up in
the study of these sort of processes is called
00:21:16.210 --> 00:21:21.360
the quadratic variation. So you essentially
look at path by path.
00:21:21.360 --> 00:21:27.500
You look at how much the random variable values
are varying between one end of the path to
00:21:27.500 --> 00:21:32.820
other end of the path that is between k1 and
k2 say how much it is varying but do not take
00:21:32.820 --> 00:21:37.790
just the sum of those variations they might
just be 0 so you have would not get any information
00:21:37.790 --> 00:21:45.030
but take the square of the variation. It is
like a mean square error type thing so we
00:21:45.030 --> 00:22:03.120
again take here and introduce the notion of
a quadratic variation.
00:22:03.120 --> 00:22:26.190
So the quadratic variation is expressed in
the following way. M, M k is defined as summation
00:22:26.190 --> 00:22:38.560
j equal to 1 to k, M j minus M j minus 1 whole
square is equal to and this if you look M
00:22:38.560 --> 00:22:48.780
j minus M j minus 1 whole square this value
is always 1. If you sum them up what will
00:22:48.780 --> 00:22:53.240
be left here, X j would be left here, the
X j. If you take the difference between M
00:22:53.240 --> 00:22:58.730
j and M j minus 1 you will have the value
X j left. X j is either plus 1 or minus 1
00:22:58.730 --> 00:23:01.310
so the squaring will always give you 1.
00:23:01.310 --> 00:23:11.060
So this expression of the square errors basically
or the square changes around every path of
00:23:11.060 --> 00:23:19.770
a given sample I can take the changes but
whatever be the path independent of the path
00:23:19.770 --> 00:23:25.630
it turns out to be k. If you do up to the
kth level it turns out to be the k independent
00:23:25.630 --> 00:23:29.470
of the path that you have taken which is very
very interesting.
00:23:29.470 --> 00:23:40.980
It does not happen for suppose you want to
compute the variance
00:23:40.980 --> 00:23:51.520
so M, M k is actually variance of M of k think
about it how it is possible. But you see to
00:23:51.520 --> 00:23:55.430
compute this I really do not need to bother
about the path but to take variance of M k
00:23:55.430 --> 00:24:00.820
we are essentially averaging over all the
paths. So this is a difference.
00:24:00.820 --> 00:24:12.560
Now how do I can I do something with this
process. Can I increase the jiggling of this
00:24:12.560 --> 00:24:18.290
process a bit this symmetric random walk a
bit and generate some sort of an approximation
00:24:18.290 --> 00:24:24.710
of a Brownian motion. Generate this sort of
zigzagging that we had just seen in the beginning
00:24:24.710 --> 00:24:28.620
when I had drawn the picture of the stock
price that this sort of zigzagging can we
00:24:28.620 --> 00:24:35.650
generate this sort of zigzagging this sort
of zigzagging can be generated by using the
00:24:35.650 --> 00:24:41.540
symmetric random walk and that leads to what
is called a scaled symmetric random walk.
00:24:41.540 --> 00:24:47.780
We will not go too much of details into it
because that might you know take you off track
00:24:47.780 --> 00:24:54.820
and you might feel a little bit of discomfort
for those who are not so very comfortable
00:24:54.820 --> 00:25:01.400
with very complicated analysis. So what we
are going to now show by this scaled random
00:25:01.400 --> 00:25:16.340
walk is that
00:25:16.340 --> 00:25:21.020
what we are going to show by scaled random
walk is that we can construct an nth level
00:25:21.020 --> 00:25:23.040
approximation for the Brownian motion.
00:25:23.040 --> 00:25:40.930
Let us construct an nth level approximation.
So if you are zigzagging by say +1 and -1,
00:25:40.930 --> 00:25:49.200
I might zigzag by 1 by 10th and minus 1 by
10th. So I will decrease my zigzagging steps
00:25:49.200 --> 00:25:57.190
but increases my time size right so we construct
the scaled symmetric random walk which is
00:25:57.190 --> 00:26:06.520
the nth level approximation of a Brownian
motion. So all these are stochastic processes
00:26:06.520 --> 00:26:11.450
so this is a discrete stochastic process from
which I am trying to go to a continuous stochastic
00:26:11.450 --> 00:26:35.480
process. I define it like this.
00:26:35.480 --> 00:26:43.610
You see if I do not have nt to be an integer
I cannot define this. So here my t is a t
00:26:43.610 --> 00:26:49.960
say t between t starts from 0 and say it is
up to t or even t goes to infinity so basically
00:26:49.960 --> 00:27:01.730
for me here this t is just greater than equal
to 0. So using the discrete thing I am trying
00:27:01.730 --> 00:27:07.900
to construct a continuous stochastic process
but I have to be aware that if I really want
00:27:07.900 --> 00:27:16.020
to use it so at the nth level approximation
this m and t this nt has to be integer if
00:27:16.020 --> 00:27:18.490
I want to actually compute this.
00:27:18.490 --> 00:27:23.270
Otherwise m is a discrete thing it is computed
only at integer points you cannot compute
00:27:23.270 --> 00:27:31.091
it at non integer points. So what happens
if it is not computed, if nt does not turn
00:27:31.091 --> 00:27:39.250
out to be an integer? So basically what you
are considering for n very large at various
00:27:39.250 --> 00:27:48.330
time points nt would be an integer and you
are actually computing out of nt. So if nt
00:27:48.330 --> 00:27:55.070
is not an integer take the t for which is
not an integer then take some u and take some
00:27:55.070 --> 00:28:03.350
s which is nearest to t such that ns and nu.
00:28:03.350 --> 00:28:14.880
These are integers and then compute the value
of Wnu and Wns and then make an interpolation
00:28:14.880 --> 00:28:20.810
linear interpolation to approximate the value
of Wnt and that is how you can actually do
00:28:20.810 --> 00:28:32.090
you can generate it in a machine by taking
a sample so you can take a sample of say so
00:28:32.090 --> 00:28:43.620
you can do the coin tossing 400 times with
one by tenth
00:28:43.620 --> 00:28:49.700
you can toss the coin 400 times with probability
of half of going you go one by tenth if it
00:28:49.700 --> 00:28:53.610
is h and you go minus one by tenth if it is
tail.
00:28:53.610 --> 00:29:00.220
So you decrease your movement so you actually
increase the zigzag by decreasing the movement
00:29:00.220 --> 00:29:08.640
and at every time you have to observe that
your Mnt nt has to be an integer. Once you
00:29:08.640 --> 00:29:15.230
do that you will find all the properties that
you had for here is in here provided that
00:29:15.230 --> 00:29:20.132
this M n into t is an integer. So you first
do it only for n into t is a integer. Whatever
00:29:20.132 --> 00:29:24.970
is left you do the interpolation and you will
see you will start getting a zigzagging curve
00:29:24.970 --> 00:29:29.870
much zigzagged than the symmetric random walk
itself.
00:29:29.870 --> 00:29:43.220
Actually it can be shown that as n tends to
infinity as n tends to infinity this W nt
00:29:43.220 --> 00:30:02.510
converges this random variable converges almost
surely sorry not almost surely I made a mistake
00:30:02.510 --> 00:30:08.270
converges in distribution rather converges
in distribution.
00:30:08.270 --> 00:30:14.170
Okay these are terms which I have not mentioned.
Just forget them for a while, converges. In
00:30:14.170 --> 00:30:22.330
some sense W nt as n becomes infinity. This
this stochastic processes gets changed into
00:30:22.330 --> 00:30:29.370
what is called a Brownian motion. So this
is what we are going to talk about in the
00:30:29.370 --> 00:30:37.840
next class. So tomorrow we are going to study
the properties of Brownian motion for the
00:30:37.840 --> 00:30:39.430
next 2 classes.
00:30:39.430 --> 00:30:44.930
So tomorrow’s class would be the last for
the second week of the course. In the third
00:30:44.930 --> 00:30:49.970
week we continue our discussion on Brownian
motion and then go to understand stochastic
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integrals or Ito integrals and doing Ito calculus
which is the foundation of any financial mathematics
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that you do. Thank you very much.