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>> We're now going to introduce Brownian Motion.

Â Brownian Motion is a very commonly used stercastic process in finance.

Â It is the process that underlies the Black-Scholes methodology and we're going

Â to discuss it now. So, let's define our Brownian Motion

Â first. We say that a random process or stercastic

Â process xt where t greater than or equal to 0 is a Brownian motion with parameters

Â mu and sigma if, for the following fixed times: t1 less than t2 up to tn.

Â The following increments: xt2 minus xt1, xt3 minus xt2, up to xtn minus xtn minus

Â 1, if they are mutually independent. For s greater than 0, xt plus s minus xt,

Â must have a normal distribution with mean mu-s, and variance sigma squared s.

Â So notice mu and sigma are the parameters of the Brownian motion, and the increment,

Â so this is an increment of length s, it's xt plus s minus xt, that increment must

Â have mean, mu-s variance sigma squared s and be normally distributed.

Â And the third condition that must be satisfied is that xt is a continuous

Â function of t. In other word, if I was to plot and we'll

Â see this in a moment, Brownian Motion, okay through time, in fact it's actually a

Â lot more jagged than I've shown you here, but it actually never jump.

Â So I can draw a path of Brownian Motion with my pen never leaving the page.

Â And that's what I mean when I say xt is a continuous function of t.

Â We say that xt is a b-mu sigma Brownian motion.

Â Mu is the drift, okay and sigma is the volatility.

Â Property number one, is often called the Independent Increments Property.

Â So they're among the first people to introduce Brownian Motion from a

Â mathematical viewpoint as we've defined here.

Â Were Bachelier in 1900 and Einstein in 1905, it's interesting that Bachelier,

Â very little is known about him, he was a French mathematician and in fact, it turns

Â out he, he has had a great role to play in, in, in finance.

Â He was trying to model stock prices on the Paris stock exchange way back in 1900 and

Â he tried to introduce the idea of a Brownian motion to do that.

Â So it's very interesting to see, that a concept as important as Brownian motion,

Â which is used throughout the physical sciences and engineering was actually

Â introduced by Bachelier in a financial context.

Â Wiener, in the 1920s, was the first to show that it actually exists as a well

Â defined mathematical entity. So Brownian motion, it's a hugely

Â important stochastic process, and it plays a very big role in, in finance as well.

Â Some other pieces of information when mu equals 0 and sigma equals 1, we have

Â what's called a standard Brownian motion. We will use wt to denote a standard

Â Brownian motion, and, we also assume that it begins at 0.

Â So w0 is equal to 0. Note that if Xt is a b-mu sigma Brownian

Â motion and X0 equals little x then we can write Xt equals little x plus mu-t plus

Â sigma Wt, where Wt is our standard Brownian motion.

Â We therefore see that Xt, is normally distributed with mean x plus mu-t and

Â variance sigma squared plus t. Because of course, if Xt equals this, then

Â the expected value of Xt is equal to the constants X plus mu-t plus sigma times the

Â expected value of Wt. Wt is a standard Brownian motion, so has

Â mean mu equals 0 times t, so this is equal to, X plus mu-t and the variance of Xt.

Â Well the constants don't matter, they don't factor into the variance, so the

Â variance of Xt is equal to sigma squared times the variance of Wt.

Â And the variance of a standard Brownian motion has sigma equals 1, So it's equal

Â to sigma squared times t, and that's where we get this calculation from here.

Â So here's a sample path for Brownian motion.

Â I've been simulating this Brownian motion by simulating these increments, which are

Â normally distributed between t equals 0, and t equals 2 years.

Â So this axis represents, a time period of 2 years and I've been simulating a

Â Brownian motion. I've-, assuming it's-, I've been assuming

Â it starts at a hundred, so I'm thinking maybe of a security price, although you

Â wouldn't model a security price as a Brownian motion typically, but for the

Â purposes of this demonstration you can think of doing so.

Â So, that's one sample path, here's another one.

Â We can see there's lots of different behavior, very jagged, In fact if I was to

Â zoom in here, you would see that jaggedness still up here.

Â Key thing to note is that these paths they're continuous, even though they're

Â very jagged none of them jump, okay so again, I could draw one of these paths,

Â and make sure that my pen never leaves the page.

Â It doesn't suddenly jump from here down to here, okay?

Â So Brownian motion is continuous, the paths of it are continuous.

Â On this slide, I want to introduce an important fact about Brownian motion, but

Â before I do so, let us review by what we mean by an information filtration.

Â For any random process, we will us Ft to denote the information available at time

Â t. And then Ft for all values of t greater

Â than or equal to 0 is called the information filtration.

Â And we actually discussed this in a previous module when we spoke and

Â introduced, when we spoke about and introduced Martingales.

Â This quantity here, this expectation, conditional on Ft, then denotes an

Â expectation conditional on the time t information that's available to us.

Â And usually, it would be very clear what that information is.

Â So really, this, this information filtration ss just a mathematical way of

Â describing what is intuitively obvious to us anyway.

Â The important fact I want to introduce is the following.

Â The independent increments property of Brownian motion implies that any function

Â of Wt plus s, minus Wt is independent of Ft.

Â In other words, knowing all of the information available at time level t,

Â that tells us nothing about the increment Wt plus s minus Wt.

Â So that in the predictor means that Wt plus s minus Wt is normal with mean 0 and

Â variant, variance s, and that's in-, and that's even conditional on time Ft

Â information. So let's do a calculation with Brownian

Â Motion. We probably won't use this calculation

Â during the course, but there's no problem in doing such a calculation, and it helps

Â improve our intuition of what's going on with the Brownian Motion.

Â So let's compute the expected value at time 0, conditional times 0, information

Â of Wt plus s times Ws. Well we can use a version of the

Â conditional expectation identity to obtain the following.

Â So Wt plus s, I can rewrite this as Wt plus s minus Ws plus Ws.

Â Okay, and then i'm multiplying by Ws outside, so that's this second Ws out

Â here. I can multiply through this Ws through

Â the-, this term here and break it down into two terms.

Â I get Ws times this term, so that's what comes into the first term here, and then I

Â get Ws times Ws is Ws squared, and that goes to that term, so this goes here.

Â So now I've got two terms. Well the first thing is, let's deal with

Â this guy first. I claim that the expected value of Ws

Â squared is equal to s, how do I know that? Well I know that because of the following.

Â I know, that s is equal to the variance of Ws, but the variance of Ws is of course

Â equal to the expected value of Ws squared minus the expected value of Ws all to b

Â squared. But the expected value of, Ws is equal to

Â 0, because it's a standard Brownian Motion, so therefore the variance at w s

Â is just the expected value of Ws squared, and that's equal to s as we've seen over

Â here.So that handles this second term on the right hand side of 9.

Â How about the first term? Well a version of the conditional

Â expectation identity implies the following.

Â So I want to compute the expected value of Wt plus s minus Ws times Ws, so what I'm

Â going to do is first condition on time s information.

Â So I can actually rewrite this expectation by conditioning first of all in time s

Â information, I get dou-, Wt plus s minus Ws times Ws, conditional on time s

Â information this Ws term can actually come outside this inner expectation, which is

Â where it is over here. And I'm left with, inside the inner

Â expectation with Wt plus s minus Ws. But that important fact over here.

Â So let's call this star. That important fact tells me, that this

Â guy is normal with mean 0 and variance, in this case t.

Â So therefore, the expect-, and it's independent of Fs.

Â So therefore, this quantity here, this inner expectation, has expected value 0,

Â and that's where the 0 comes from, and so I get 0 here.

Â And so, therefore what we've shown, is that the expected value of Wt plus s,

Â times Ws, is equal to s.

Â