Box Models VII – Random Prices
Box Model A is:
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 shown above.
If you draw from this box 9 times and add the numbers you draw, the sum of all these 9 draws will be at the largest, 90 (if every ticket drawn is a +10, most unlikely) and at the smallest -90 (if you get all -10’s, equally unlikely). The question I closed with last time was: what will be the number in the composite 1-draw box with the most tickets?
The answer is that there will be two numbers with the most tickets, -10 and +10. Notice that 0 is impossible — you cannot add up 9 +10’s and -10’s and get 0, since for that to happen you would have to get the same number of +10’s and -10’s, and 9 is an odd number, so the closest you can get is 4 +10’s and 5 -10’s (giving a sum of -10) or 5 +10’s and 4 -10’s (giving a sum of +10).
This is an example of the little puzzles that crop up when you analyze price behavior. These problems may be interesting in themselves as little mathematical exercises, but they don’t contribute anything to the intuition you need about options and price behavior, so I will circumvent them as much as possible.
This kind of problem arises when thinking about short term behavior. Fortunately, we are concerned mostly with longer term price behavior. The stock we are considering has a price today of 100. Could the sum of 9 draws from the box model above reasonably represent the price of our stock 9 days from now?
The price changes that arise out of the sum of 9 draws have a lowest value of -90. This has, of course, a very low probability, but it leads to an expiration price of 10 for the stock. Is this reasonable? You might imagine it to be if the volatility of the stock is high enough. But suppose expiration is 11 days off. Would the model then produce reasonable results?
11 draws from the box produce a lowest possible sum of -110, supposedly representing the price change in 11 days. But this leads to a price of -10, clearly nonsensical. So this box model, aside from the simplicity of the numbers in the box, cannot be a good representation of the price of the stock 11 days from now, and certainly not for any larger number of days, and thus even a number of days between 1 and 10 is suspect.
How can we alter the box model so that it is a more reasonable portrait of the random character of the future stock price? If you have absorbed the previous commentaries, you should be able to guess the answer, which I’ll discuss next time.