It is time for us to start going at a much higher mathematical level. It's been very simple mathematically so far. Now we are going to have to introduce some sophisticated mathematics, trying not to go too much into technical details, but still to explain the intuition behind the model. Time to introduce the Brownian motion process, which is going to be the main stochastic factor behind the model. Just a little bit of history to give you a perspective. The name comes from a biologist, Brown, which in 1800s was studying movement of molecules in liquid, which was very irregular movement, and the stochastic process, which is later developed, which is called Brownian motion, will also have very irregular movements. But really, Brown didn't have any mathematical model, he was just considering this movement of molecules. For a long time, it was thought that it was Einstein, basically the first one who used Brownian motion in physics. But then it was discovered a couple of decades ago that it was really Louis Bachelier who had a PhD thesis in Paris in France in 1900s, which did two things, it introduced Brownian motion mathematically and it used it as a model for stock prices on Paris Stock Exchange. In terms of economics, he was way before his time, and his work was forgotten for a long time and then it was discovered later. There're two reasons why it was forgotten. One is that mathematically, that time in France, the Bachelier school was popular, very abstract and rigorous math, and they just didn't consider him or they didn't even pay attention to him that much. In terms of economics, it was just too early for economists to consider such sophisticated mathematical models. Then it was mathematicians Wiener and Levy who developed the mathematical theory, and in particular Ito in the '40s, and we are going to see his name, he really brought it into a form that can be used in finance in perfect way. Ito died a few years ago at an old age. He had received a number of prizes from financial institutions, even though he never himself really applied his theory to finance. He was a pure mathematician, but the theory was custom-made or tailor-made for financial applications really. It was popularized in economics by a later Nobel Prize winner Paul Samuelson, a great American economist in the '60s, and then the Option Pricing theory using that model was done by Merton, and Black, and Scholes in the '70s, and Scholes and Merton got a Nobel Prize for it. Black had died before that, otherwise he would have gotten the Nobel Prize together with Merton and Scholes. These are the main names at least up to the '70s behind the model. Let me give you a one-slide introduction of what the model is without going into mathematical details, and then it'll be more precise about the mathematics behind. There will be one risk-free asset, the bank account with a continuously compounded and constant interest rate r, and B of t is e^rt, and there will be a stock which has so-called log-normal distribution, meaning that the logarithm of the stock prices will have a normal distribution. The log of S of t is log of S of 0 plus some constant time t, linear function of time. There is a reason why this constant is written in this way, Mu minus 1/2 Sigma squared t. We'll get to that later. Then the same Sigma is here, square root of t, which is the time scaling, which happens to be appropriate here, and times z of t or z or t is a standard normal random variable. Gaussian random variable mean 0, variance 1. This looks simple. It's a linear combination of time and the standard normal random variable, and times square root of t. This is simple. For fixed t, the mathematical problem arises when you try to move this t and have this as a stochastic process, the question is, does this make sense as a stochastic process in time and how do you construct it? That's where the mathematics behind it becomes harder. But conceptually, this is, as a random variable, what the stock price looks like in the Black-Scholes-Merton model. I'm calling it Black-Scholes-Merton or Merton-Black-Scholes smaller, even though the model goes historically before them, they use it for option pricing, but they didn't invent the model. Still I'm going to keep that name. That's in terms of a fixed fee, what the stock price is as a random variable. If we get rid of the logarithm, it's going to look like this. S of t is initial price times the exponential of our linear combination is something Nu minus 1/2 Sigma square t plus Sigma square root of t, z of t. Just to give you a little bit of the feel for what Mu and Sigma stand for, using standard probability, it can be shown that the expected value of S of t is S of 0 e^Mu t. Mu is basically expected return rate of the stock. On average the stock behaves as a risk-free or as a bank account with a continuously compounded interest rate Mu. Mu is called the return rate or expected return rate of the stock. It's also not hard to compute the variance of the log of S of t. Namely, if you look at the variance of the log of S of t over S of 0, this is deterministic, doesn't have a variance. Contribution to the variance is going to be zero. The only term that contributes to the variance is this term. How do you compute the variance, but it doesn't mean zero, so the variance is just the expected value of the squared term. So it's going to be Sigma squared, square root of t squared, which is t, and then the variance of z of t is 1. Then if I divide by that t, I'm going to get that variance of the log of S of t over S of 0 divided by t Sigma squared. Sigma squared is the variance of the log stock returns if you want log stock, S of t over S of 0 per unit time, variance of log of S of 0 per unit time. That's what sigma squared stands for, for the variance of this log. Log difference is log increments in the stock price. This is to keep big picture in mind, global motivation for why we need to develop a Brownian motion. Brownian motion process is going to be really this z of t thing here, but I have to be more careful how I define it.
