4 – M1L5 06 Distribution Of Stock Prices Part 2 V1

Let’s talk about how a series of daily price values arises. It has to do with our earlier discussion of compounding. Let’s say, a stock starts at P sub zero, and each day the price changes by some small percent, the return. We saw earlier how the price at time T could be written as a product of all of these one plus little r sub i terms. Remember, one plus little r equals P sub t divided by P sub t minus one. If we take the log of both sides of this equation, we get log of P sub t on one side and a sum of log of one plus little r sub i terms on the other side. Here, for convenience, I’m going to move log of P sub zero back to the left-hand side. Remember, P sub zero is a constant, it’s not dependent on time. Now, we’re going to use the central limit theorem. You might remember the central limit theorem from a class on probability, but if you don’t I’ll just explain it briefly here. It says that, the sum of random variables that have the same distribution and are not dependent on each other approaches a normal distribution and the limit that the number of random variables and the sum goes to infinity. So, let’s make the assumption that the returns on each day are independent, but come from the same underlying distribution. This seems reasonable certainly more reasonable for returns and for prices. After all, the price of a stock on one day almost certainly depends on its price the day before. So, if we make that assumption, we can see that the right hand side, the wholesome is a random variable that follows a normal distribution. So, we have that log of P sub t minus log of P sub zero is distributed normally. But if P sub 0 is a constant, then log of P sub t is also distributed normally, because a normally distributed random variable minus the constant is still normally distributed. Now, I’m going to tell you about a new distribution called the log-normal distribution. Guess what, it looks kind of like that right-skewed distribution we talked about earlier. What you need to know about the log normal distribution is that if Y is distributed normally then e to the y is distributed log normally. I’ll say that again, if Y is distributed normally, then e to the y is distributed log normally. So, if log of P sub t is distributed normally, then P sub t which equals e to the log of P sub t is distributed log normally. In fact, prices are frequently assumed to arise from a log normal distribution. This seems reasonable from just eyeballing the distribution because we can see that it doesn’t go below zero and it’s skewed to the right. Those properties match our actual price distribution well. However, when working with real data, the assumption that prices are distributed log normally, may or may not be a good one. These distributions can be convenient models, but don’t confuse them with actual distributions of stock prices or returns.

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