>> In this module we're going to discuss Geometric Brownian Motion. Geometric Brownian motion is a very important Stochastic process, a random process that's used everywhere in finance. We have the following definition, we say that a random process, Xt, is a Geometric Brownian Motion if for all t, Xt is equal to e to the mu minus sigma squared over 2 times t plus sigma Wt, where Wt is the standard Brownian motion. So we've discussed Brownian Motion, in a separate module, so you can look at that module, if you'd like, to remind yourself what a Brownian Motion is. But one thing to keep in mind with the Brownian Motion, is that Wt, has got a normal distribution, with mean 0, and variance t, This is one of the properties of a Brownian Motion. Recall mu the drift, sigma the volatility, and write Xt till GBM mu sigma. An interesting observation to make is the following, let's take a look at this expression, but let's replace t with t plus s, if we do that, we'll see that Xt plus s equals X0, and in fact I should have had an X0 here. So Xt plus s equals X0, e to the mu, minus sigma squared over 2 times t plus s, plus sigma plus Wt plus s. And now what we can do, is we can rewrite this expression up here in the exponential. We can subtract a minus Wt, and add a Wt here, and we can break this summation up into t times this plus s times this. If we do that, we get this term here in the right hand side, but what's interesting is that this quantity here, is actually equal to Xt. So we can write Xt plus s equals Xt times the exponential of mu minus sigma squared over 2 times s plus sigma times Wt plus s, and this representation is very useful, it's in fact very useful for simulating security prices, when those security prices follow a Geometric Brownian Motion. This quantity here, Wt plus s minus Wt, well that's just a normal random variable with mean 0, and variance s. Moreover, it is actually independent of Xt and this follows from the independent increment property of Brownian motion that we discussed in that other module on Brownian motion. So that means for example, suppose that we wanted to generate values of a Geometric Brownian Motion at time 0 and at time t, bit also may be at these intermediate times may be delta, 2 delta, 3 delta, and so on. Well, what we can do, so we want to generate x delta, x2 delta, x3 delta, and so on. Well, what we can do is we can actually simulate the Geometric Brownian Motion at these time periods by just simulating, and zero delta random variables, that's very easy to do in standard software, you can even do it easily in Excel. So you could generate a sample path of your Geometric Brownian Motion or a sample path of your stock. You start at time zero with x zero which you know, and then you get x delta, using this formula here, with t equal to 0 and Wt plus s minus Wt, well that's just equal to a normal mean 0 variance delta random variable, so that would give you x delta. You could then get x2 delta by taking t equal to delta and s equal to delta, so you will get x delta plus delta is x2 delta, that's equal to x delta times this quantity again here. And again, to generate this term you could just generate a standard normal random variable, with mean zero and variance delta. So it's actually very useful simulating a Geometric Brownian Motion and we may return to this again, later in the course. Here's a question, suppose Xt is a Geometric Brownian Motion with parameters mu and sigma, what is the expected value of Xt plus s given little t? Well from equation 10 of the previous slide, we know that Xt plus s is equal to Xt times the exponential of this term here. Well, at time t, all of this is known to us, so we can take this outside the expectation, and we're left with this times the expected value of each of the sigma Wt plus s minus Wt. Well this term here, as I've already said, is normal with mean 0 and variance s, so all you're trying to do when you compute this expectation, is actually compute the moment generating function of a normal rounding variable. How many have seen that before? Suppose Zed is normal with mean a and variance b squared. Then it implies that the expected value of e to the s times Zed is equal to e to the a s plus a half, b squared times s squared, so this the moment generating function of a normal rounding variable. And we can just use the standard result up here, to recognize that this must be equal to e to the sigma squared over 2 times s. This term, cancels with this term, and we get this expectation equals e to the mu s times Xt, so that the expected growth rate of Xt, is in fact, mu. Here are some sample paths of Geometric Brownian Motion. The important thing to notice with these paths, is that they are continuous, they are very jagged. If I was to zoom in, I would still see that they are very jagged and they are continuous as I said, so they do not jump. I can draw any one of these paths, by keeping my pen on the page. The following properties of Geometric Brownian Motion, follow immediately from the definition of Brownian Motion. Recall that we saw the following, so we know that Xt plus s, is equal to Xt, e to the mu minus sigma squared, over 2 times s, plus sigma times Wt plus s minus Wt. Okay so what are these three properties? Well the first property states, that these ratios xt2 over xt1, xt3 over xt2 and so on, they're mutually independent. And that follows, because if I divide across here by Xt, I can see I've got the only random variable here is this increment, and the independent property, independent increments property of Brownian Motion will actually imply this first property here. The second property is, the property I mentioned on the previous slide that is that the paths of Xt our continuous as a function of t, they do not jump. The third property states, that the log of Xt plus s over Xt has got a normal distribution as follows, and that also follows from equation 10, which I've rewritten here. So I can easily see that the log of Xt plus s divided by Xt is equal to, well it's just this term up here in the exponent, it's equal to mu, minus sigma squared over 2 times s, plus sigma times Wt plus s, minus Wt. This guy is normal with mean 0 and variance s, and so this quantity is normal with mean mu minus sigma squared over 2s, and variance sigma squared s, which is exactly what we have here. A couple of observations about Geometric Brownian Motion. It is clear #1, that if Xt is greater than 0, than Xt plus s is always positive for any value of s greater than 0. And again, let's write out equation 10 here just to see this more clearly. It's, so this is our equation 10 from an earlier slide. We can see that if Xt is greater than 0, then of course the exponential of this would be greater than 0, and so then Xt plus s would be greater than 0. So if we are using a Geometric Brownian Motion to model stock prices, then we can see that the limited liability of a stock price, i.e., the fact that the stock price cannot go negative, is not violated. Another observation, is that the distribution of Xt plus s divided by Xt, only depends on s and not on Xt. In fact, this was clear, from the previous slide where we had this result here. The log of Xt plus s is a normal distribution, and this normal distribution does not depend on Xt, it only depends on s and the parameters mu and sigma. And this is nice, because we wouldn't expect returns to depend on Xt, so we can view this as being the return on a stock the return between times t and t plus s and we don't expect in general that this return should depend n the current value of the stock. So again, this is another nice property that Geometric Brownian Motion has, that is generally reflected in stock prices as well. So these two properties suggest that Geometric Brownian Motion might be a reasonable model for stock prices. And indeed, Geometric Brownian Motion is the underlying model for the famous Black-Scholes option formula that we will also see in this course.