So if you go to the stock market and look
at the price of say a favourite company’s share. So what you would observe is the following. So if the horizontal axis is the time axis
and if this is the price axis it tells you what is the price then you will see starting
from a certain time price you will see some zigzagging motions like
this. So you observe it up to time T and you observe this zigzagging motion. This is of course random. Nobody knows what
is the next price is. So if a particular scenario evolves, you have a particular path. This
is called a sample path. So if another scenario evolves, if another scenario evolves there
would be another path, for example it could be like this. The stock price is going down
down down down and you are in a bad shape and then it again climbs up and again it falls
down down down and again then again climbs up. So under a different scenario it has a different
path. So it is sample path 1 it is sample path 2. So it is what type of scenario one
evolves. Now of course you can ask me what is this term scenario that you are talking
about, what is the meaning of this goddamn scenario? We will come to this very soon.
But how do I model such zigzagging paths, what way to model it. Is there any mathematical
way to say that us or can I construct the stochastic process whose sample paths are
represented in this form? Let us do, to do that we need to study what
is called Brownian motion. Brownian motion is a type of stochastic process which will
help us to model stock prices at the end. So the whole term Brownian motion comes from
the name of Robert Brown who first studied the movement of pollen grains in water and
he found that they were having a zigzag haphazard movements. But it is not so immediately apparent
that you can just start writing about this particular stochastic process. We need to have some more idea and built upon
some simpler stochastic process. So we will begin by introducing what is called symmetric
random works. Where there are only 2 possibilities you can
either go up or down that is like a coin toss head or tail and that here I can have infinite
such possibilities, infinite such sample paths and there are infinite possibilities also.
Here also we will have infinite possibilities but generated out of only 2 possibilities.
So symmetric random walk. So when you take a fair coin and then you keep on repeatedly
tossing it. So it is a so this is a stochastic process
which I will write in short form now as stochastic process generated by repeated tosses of a
fair coin okay and if you look at it very carefully what I mean by this? So you start
tossing the coin so repeatedly you are tossing omega 1 omega 2; so omega 1 is either head
or tail; omega 2, omega 3, omega 4, 5 and so and so forth. Suppose you have here head,
head, tail, tail, head, head, head, tail, tail, tail and it goes on. So this is one particular scenario that has
evolved. You could have another scenario say omega bar which is consisting of say omega
1 bar, omega 2 bar, omega 3 bar, omega 4 bar, omega 5 bar, it could be something like this;
tail, tail, head, head, head, tail, tail, tail, head, head, head and so on. So these
2 are different scenarios and these 2 each would generate 2 different sample paths. So
how do we generate this symmetric random walk? So these are 2 different scenarios, 2 different
scenarios. Now construct a random variable Xj which takes the value 1 if j is equal sorry
if omega j is equal to head and takes the value -1 if omega j is equal to tail if tail
appears and 1 if head appears. So now you define a stochastic process, define a new
stochastic process Mk, k=0 to infinity or let us we can need not bother we can also
fix it after some time. It could be some time capital say K is say
25 something here 25 or 30 whatever. But in general it is alright to take plus infinity
just a sequence where M k is given as follows. Each of these M ks are calculated by starting
from M 0 equal to 0. M k is equal to j is the sum from j from 1 to k to X j. So let
us see what would happen if one particular scenario like this evolves. Let us see then
what is the sample path of this. This symmetric random walk is also called a drunkards walk. So somebody has had a good drink and he has
become drunk and if you look at his walk so a drunkard would walk like this if I am here
so I start from here then I can just go like this and he goes like this just it is just
or like this you know I am coming here and then I am going there something like this.
So this sort of thing you will immediately observe as I start say checking out with this
scenario. So here is my k and here is my M k value.
Now the first one here has turned out to be 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 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 so and so forth. So M0 is 0 this is 0 is 0.
Now you toss a coin and you have head. Omega 1 is head so you go up by +1 because
X j will take +1 because m1 is just X 1 so here is the value of M1 this is your M1 so
you join the M0 and 1 by line so 0 is M0 and then you again had head so M2 is again 1.
I am looking at this scenario M2 is you go 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 you will drop by 1 so it will be -1 so it
will again drop back to the point 1. So this is your M2 and this is M3 and then
omega 4 is again tail so it drops back to 0 again you -1 subtract. So this is your M4
0. Again then you have head for omega 5 say so here you have omega 6, omega 7, omega 8,
omega 9, so for M4 you have tail you have come to 0 again then it goes up again for
M it goes to plus 1 again. So this is your M5. Again, it goes up to 2
M6 but then you have tail again so M7 comes down to +1. Again, you have tail so M8 comes
down to 0 because you are adding up everything. At every step you are going up or down. So
you add up in this fashion and you move like this. So M8 I have here omega 8 it is again
omega 9 is tail so again I have to go down so I go down by so from 0, I will have to
go down by 1 so I will come to -1. So this will be your M9 and if suppose M10
omega 10 is head then it will again go up to 0 at the 10th place because you will again
add 1 so it will become your M10. So what you see that the symmetric random walk is
providing me some zigzag looking curve which might tempt you to think that possibly these
2 have some relationships. They are they do have some relationships and we will talk about
that slightly down the talk. But let me tell you some more properties of
this symmetric random walk M k. So here is my stochastic process and this stochastic
process is called the symmetric random walk. Of course we are not mentioning but underlying
we are always taking some probability space and all of those things. So now we will these random this particular
random process of stochastic process has independent increments. What do I mean by the fact that
they have independent increments? What I mean is the following. So if you have you take
certain numbers sometime say some K m then you have the following. You have M k1 that is once you consider non-overlapping
intervals then this difference is independent because they depend on independent coin tosses
because coin tosses are independent when the coin tossed at the second level really does
not the second outcome does not really depend on the first outcome right when you do a repeated
coin toss sorry M k2 - M k1,…, M k m- M k m-1. So these random variables are independent.
All of these random variables these form a set of independent random variables. So that
is once this happens this is when we say that it has independent increments and this actually
has independent increment. These are independent because their difference which really does
not depend on the coin tosses here does not depend on the coin tosses here and here and
so you have independent increments. So this change that you see here does not
depend on the change that you see in this interval or the change that you see in this
interval right. So you can still observe that it is like a drunkard walk so drunkard walks
like this goes down goes up. In George Gamow’s famous book One, Two, Three Infinity this
has been described in a very very nice way. How do you, see here, my success probability
this occurs with probability half and this also occurs with probability half so if you
observe that exponential of X j sorry expectation of X j is 0 because this is one into half
plus minus one into half. Variance of X j so what is variance of X j exponential X minus
X j whole square which is 0 so it is 1 into half plus minus 1 minus 0 whole square plus
1 into half which is 1. Once you have this information this is true
for all j. It is immediate that exponential 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 is equal to I leave it to you to calculate
these stuffs. Our second property about this random walk is to show that this is also a
Martingale. You see Martingale thing comes up. So symmetric random walk is a discrete
Martingale. So you can easily prove that it is a Martingale.
So you take any k strictly less than l and look at the expectation of M l conditioned
on the filtration, the Sigma-algebra F k which is the part of the filtration okay. So you can write this as
M l minus M k plus M k. So these can be summed up just like expectation can be some conditional
expectation this random variable can be decomposed into 2 parts which you can actually prove
which will be a part of your exercise but we are just using this fact here so, anyway
I should rub the board a bit. Now let us look at the first part. Since l
is strictly bigger than k this increment M l minus M k is independent of F k. F k does
not have the information of anything which is beyond the time k. So here by one of our
rules for conditional expectation this is nothing but M l minus M k and here at time
k everything about M k is known. So F k contains all information about M k. So the first law
was taking out what is known, I can write this M k as M k dot 1 where 1 is the constant
random variable 1. So whatever be the scenario it will just give you the value 1. 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 known 1 dot F k. Of course, 1 is a constant
random variable. It does not really depend on is independent of F k so it will be E of
1 which is a constant which will be just 1. So everything will be 1 so the sum of the
probabilities will sum up to 1. So the expectation will be just the number. So this again is 0 which we already know plus
M k into 1 which is M k and so this shows that M l is a this symmetric random walk this
thing forms a discrete Martingale. Of course, F k is M k has to be adapted to this filtration
that is the basic definition of Martingale. There is another notion which crops up in
the study of these sort of processes is called the quadratic variation. So you essentially
look at path by path. You look at how much the random variable values
are varying between one end of the path to other end of the path that is between k1 and
k2 say how much it is varying but do not take just the sum of those variations they might
just be 0 so you have would not get any information but take the square of the variation. It is
like a mean square error type thing so we again take here and introduce the notion of
a quadratic variation. So the quadratic variation is expressed in
the following way. M, M k is defined as summation j equal to 1 to k, M j minus M j minus 1 whole
square is equal to and this if you look M j minus M j minus 1 whole square this value
is always 1. If you sum them up what will be left here, X j would be left here, the
X j. If you take the difference between M j and M j minus 1 you will have the value
X j left. X j is either plus 1 or minus 1 so the squaring will always give you 1. So this expression of the square errors basically
or the square changes around every path of a given sample I can take the changes but
whatever be the path independent of the path it turns out to be k. If you do up to the
kth level it turns out to be the k independent of the path that you have taken which is very
very interesting. It does not happen for suppose you want to
compute the variance so M, M k is actually variance of M of k think
about it how it is possible. But you see to compute this I really do not need to bother
about the path but to take variance of M k we are essentially averaging over all the
paths. So this is a difference. Now how do I can I do something with this
process. Can I increase the jiggling of this process a bit this symmetric random walk a
bit and generate some sort of an approximation of a Brownian motion. Generate this sort of
zigzagging that we had just seen in the beginning when I had drawn the picture of the stock
price that this sort of zigzagging can we generate this sort of zigzagging this sort
of zigzagging can be generated by using the symmetric random walk and that leads to what
is called a scaled symmetric random walk. We will not go too much of details into it
because that might you know take you off track and you might feel a little bit of discomfort
for those who are not so very comfortable with very complicated analysis. So what we
are going to now show by this scaled random walk is that what we are going to show by scaled random
walk is that we can construct an nth level approximation for the Brownian motion. Let us construct an nth level approximation.
So if you are zigzagging by say +1 and -1, I might zigzag by 1 by 10th and minus 1 by
10th. So I will decrease my zigzagging steps but increases my time size right so we construct
the scaled symmetric random walk which is the nth level approximation of a Brownian
motion. So all these are stochastic processes so this is a discrete stochastic process from
which I am trying to go to a continuous stochastic process. I define it like this. You see if I do not have nt to be an integer
I cannot define this. So here my t is a t say t between t starts from 0 and say it is
up to t or even t goes to infinity so basically for me here this t is just greater than equal
to 0. So using the discrete thing I am trying to construct a continuous stochastic process
but I have to be aware that if I really want to use it so at the nth level approximation
this m and t this nt has to be integer if I want to actually compute this. Otherwise m is a discrete thing it is computed
only at integer points you cannot compute it at non integer points. So what happens
if it is not computed, if nt does not turn out to be an integer? So basically what you
are considering for n very large at various time points nt would be an integer and you
are actually computing out of nt. So if nt is not an integer take the t for which is
not an integer then take some u and take some s which is nearest to t such that ns and nu. These are integers and then compute the value
of Wnu and Wns and then make an interpolation linear interpolation to approximate the value
of Wnt and that is how you can actually do you can generate it in a machine by taking
a sample so you can take a sample of say so you can do the coin tossing 400 times with
one by tenth you can toss the coin 400 times with probability
of half of going you go one by tenth if it is h and you go minus one by tenth if it is
tail. So you decrease your movement so you actually
increase the zigzag by decreasing the movement and at every time you have to observe that
your Mnt nt has to be an integer. Once you do that you will find all the properties that
you had for here is in here provided that this M n into t is an integer. So you first
do it only for n into t is a integer. Whatever is left you do the interpolation and you will
see you will start getting a zigzagging curve much zigzagged than the symmetric random walk
itself. Actually it can be shown that as n tends to
infinity as n tends to infinity this W nt converges this random variable converges almost
surely sorry not almost surely I made a mistake converges in distribution rather converges
in distribution. Okay these are terms which I have not mentioned.
Just forget them for a while, converges. In some sense W nt as n becomes infinity. This
this stochastic processes gets changed into what is called a Brownian motion. So this
is what we are going to talk about in the next class. So tomorrow we are going to study
the properties of Brownian motion for the next 2 classes. So tomorrow’s class would be the last for
the second week of the course. In the third week we continue our discussion on Brownian
motion and then go to understand stochastic integrals or Ito integrals and doing Ito calculus
which is the foundation of any financial mathematics that you do. Thank you very much.
