The Percentage Change Model
Box Box Models VIII – Random Prices
Today’s price is 100, and in our simple model the box that represents the price change under Model A between now and expiration is:
If you draw from this box many times and add up your draws, you can get a price change more negative than -100, which implies a price at expiration less than 0, so this is not a good model for actual price change behavior. A simple change will do the trick, however. Instead of letting the tickets stand for dollar price change, let them stand for percentage price change:
Using this model, we never get a price below 0. In two draws, for instance, if we get -10% each time, we apply the -10% to the original price of 100 twice: after the first draw the price falls to 100-10% of 100 = 90, and after the second draw the price falls to 90-10% of 90 = 81. Every time you get a -10%, the price becomes 90% of its previous value, and no matter how many times you multiply by 90% you will never get to 0.
This is for most practical purposes the same as the model commonly used by mathematicians and economists for describing stock price behavior. Their model is called the “log normal model”, for reasons we will discuss later, and it uses logarithms instead of percentages, but the results are practically the same. I will call the above model the “percentage change model”.
If each draw is supposed to represent the percentage change in price for a day, the +10% and -10% values are probably too extreme. Most stocks do not fluctuate this much in a day. How do you choose the percentages to put in the box? The usual way is to look at past behavior of the stock and measure the typical percentage change in a day, and the usual measurement is the standard deviation of the daily percentage changes in the last 50 or so days (we use 100 days on the web site).
And even in only one day there are many more possible percentage changes than just two. But to a first approximation, especially if the standard deviation of percentage changes is not large, two are sufficient to give a pretty good idea of option values.