Welcome back. Today we are going to talk about how to generate meaningful, reasonable scenarios for asset returns using a random walk model, allows us to kind of the Monte-Carlo simulation in a fairly reasonable setting. So we are going to look at very simple case, very simple setting with two assets a risky asset. You may want to think about a risky asset as a stock index, for example. We're going to call the value of that risky asset S of t. That's the value of that stock index at time t, and then we're going to look at the risk-free asset. You might want to think about the risk-free asset as a one-year T-bill, say consistent with an investment horizon of one-year, that would be the risk-free asset and the value that risk-free asset we call B of t, before bond if you will this one-year bond. Now, the way we write this simple model, the simple random walk model in continuous time is to look on the left-hand side of the first equation are the return on the stock index. A return on the stock index is defined as dSt. DSt is the change in value of the stock index between time t and times t plus dt which is a time slightly later. So dSt is the change in dollar value and if you divide it by St, then it becomes a percentage change. So that's why dSt divided by St is return. The return over the infinitesimal extremely small period that we're going to call dt. Well interval of time. Now what we're going to see on the right-hand side of the equation is that this return on the risky asset has two components. The first component is like a trend, it's based on the expected return, that's how much the asset will go up on average because of this expected return, let's say 10 percent expected return. So it's going to go up by 10 percent on average. Now, it's only 10 percent on average, so around the average there will be lot of kind of up and down and that's the second component on the right-hand side of the equation which is the Stochastic component. Now, if you look at the trend it's about expected return. The expected return we typically call it mu.. Here the mu is being decomposed into two components. Mu is given by r plus lambda times sigma. So r is the risk-free rate, sigma is the volatility of the stock index, and lambda is the Sharpe ratio on that stock index. Now the reason why we've decided to decompose it not calling mu, which would have been very simple we call it mu as its expected return. The reason why we decided to call it not mu but r plus lambda sigma is to try and better disentangle all of those components that get into expected return. In this way we emphasize in particular that the 10 percent expected return mu is equal to risk-free rate, say two percent plus eight percent risk premium. Now where does the eight percent risk premium coming from? Well let's say the eight percent risk premium come from volatility which is stock index, it's typically around say 16 percent annualized, 16 percent volatility times the Sharpe Ratio, that's a.5 half would be a reasonable value for the Sharpe Ratio. So we get two percent risk-free rate plus half time. 16 percent, that makes it two plus eight that makes it 10 percent. So in this way we recognize that expected return on the risky asset is risk-free rate plus the risk premium and the risk premium is given by number of unit of risk which is volatility sigma times the unit price of the unity one per unit of risk which is lambda, the Sharpe Ratio. So that's for the 10 percent expected return on average over a year, and then it's multiplied by dt, because dt again is the interval of time that you're looking at. So if dt for example is let's say one month, then the 10 percent expected return over a year gets divided by 12, then one month is one divided by 12 in terms of fraction of a year. So that gives you how much return you get on average over a one-month period of time. So that's the first component on the right-hand side of the equation. The second component is volatilities of volatility sigma times these dWt where dWt is the Brownian Motion process that we are going to talk about in just a moment and the Brownian Motion process is essentially our best model for a random walk as we will explain. Before we do that let's look in contrast at what happens for the risk-free asset. Well, for the risk-free asset you get only the deterministic component there is no random walk type component, which suggests that the return between now and in a year from now is purely predictable. Well that's exactly the case by definition of this asset being the risk-free asset. So if you buy a pure discount bond at 100 and you get as face value 105, that you're going to get five percent over one year there is no uncertainty. If you invest $100 in the S and P 500 index you just don't know how much you're going to get in a year from now. Maybe you're not going to get 110 on average but there will be uncertainty around the average. Now let's go back to the Brownian motion process and let's try and explain what it looks like. Well the Brownian Motion process essentially looks like some kind of noise or random walk, something that goes up and down in a very kind of stochastic and predictable way. This process has been introduced in 1900 by a French mathematician goes by the name of Louis Bachelier, who actually back then, 1900, I know it's going to sound crazy but this guy was actually trying to do option pricing as part of his math PhD dissertation at La Sorbonne in 1900. Now, the process was eventually rediscovered independently in 1905 by Albert Einstein, who actually used the Brownian Motion model for different things that he discovered and different advances made in the field of physics in 1905 which is known as the miracle year. Albert Einstein published four papers in that year. One of those four papers was about the Brownian Motion. So what is the Brownian motion? Well, it's perhaps easier to understand what the Brownian Motion looks like if you think about the discrete time formulation trying to make it explicit. So what we're seeing is the return St plus dt minus St divided by St. That's the percentage return between time t and time t plus dt, is given by the mu which is r plus lambda sigma which is the expected return plus the sigma times the Brownian Motion. Now the size of the Brownian Motion is defined in terms of some kind of random variables. Let's call them psi of t. Those random variables are mean Z0 and unit variance, they are just like noise. So the way they are being constructed it's very easy to see that the expected return on that stock price is then given by mu times dt so that mu is the annualized expected return. In the same way what we see is that the variance of that return is given by sigma square time dt and therefore sigma squared happens to be identified as the annualized variance. So in other words the variance over a given time interval lets say a month and then divided by dt in this case divided by one divided by 12. In other words multiply by 12 that allows you to scale monthly volatility into annualized volatility. So essentially what the Brownian Motion is, it's a process, it's a stochastic process, and the change in that Brownian motions are modeled as a Gaussian variables with mean zero and volatility equal to delta t or dt which is the time interval. What's important about this Brownian Motion is the concept of independent increments. So whether you're coming from an up-move or whether you're coming from a down-move, at all points in time the probability to go up or to go down for these Brownian Motion is always half. There is no serial correlation. There has no momentum or there's no reversal. This is just a pure random walk. Well, this has been recognized as a fair approximation of the returns on securities. Of course if you look closely you could find some serial correlation but at least it's a reasonable model for reasonable situation when we have to simulate as written. In closing, today we've been introducing the Brownian Motion and we've seen how the Brownian Motion can be used as building blocks for generating scenarios for stock returns. For bond returns we don't necessarily need the Brownian Motion, at least if you look at Salton Bob returns because these are regarded as the safe asset and we don't have to worry about uncertainty. Next time we are going to look at extension surround the basic Brownian motion models and trying to make those models a little more realistic.