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. What's the implying meaning of "sentence" in "Home is the first sentence"? 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. 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. In fact, this was clear, from the previous slide where we had this result here. This will give you an entire set of statistics associated with portfolio performance from maximum drawdown to expected return. Okay so what are these three properties? There are uses for geometric Brownian motion in pricing derivatives as well. 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. Most economists prefer Geometric Brownian Motion as a simple model for market prices because it is everywhere positive (with probability 1), in contrast to Brownian Motion, even Brownian Motion with drift. A couple of observations about Geometric Brownian Motion. I knew some topics prior to the course and it helped me to strengthen my knowledge on derivative market systematically, particularly on interest rate derivatives. 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. Random walks in porous media or fractals are anomalous. 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. I would like to understand Geometric Brownian Motion. Geometric Brownian motion is a very important Stochastic process, a random process that's used everywhere in finance. Z_{t+\delta t} = Z_t + (\mu − \frac 1 2 \sigma^2) \delta t + \sigma \sqrt{\delta t} * \epsilon_t $\endgroup$ – Jan Stuller Nov 8 at 8:35 add a comment | Your Answer 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. Brownian motion is the random movement of particles in a fluid due to their collisions with other atoms or molecules. Shouldn't some stars behave as black hole? In my lecture it was introduced as follows: Let $\mathcal T = \{ 0 , \delta t, 2 \delta t, \dots, n \delta t = T\}$ be a discrete time grid. (independently and identically distributed) sequence. AGeometric Brownian Motion X(t) is the solution of an SDE with linear drift and difiusion coe–cients dX(t) =„X(t)dt+¾X(t)dW(t); with initial valueX(0) =x0. PS: We didn't talk about Wiener Process or Brownian Motion or SDE, so the definition above is the only thing I know. Geometric Brownian motion is useful in the modeling of stock prices over time when you feel that the percentage changes are independent and identically distributed. Z_{n \delta t} = Z_0 + n (\mu - \frac 1 2 \sigma^2) \delta t + \sigma \sqrt{\delta t} * \sum_{i = 0}^{n-1} \epsilon_i 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.

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