Box Models IV – Volatility/Puts
Box Model A is:
The number drawn from this box represents the value of the underlying at expiration.
Today’s price is 100, so you could also put tickets into the box to represent the price change under Model A:
The average price change is just the average of the tickets in this box, and it is obviously zero. The standard deviation of a group of numbers measures the ‘typical’ deviation from their average. You could use the formula to compute the standard deviation of the numbers in the Model A Price Change box above, but you don’t have to if you merely observe that every deviation from the average of 0 is equal exactly to 20; so the standard deviation must be 20.
Suppose you used Model A twice, drawing a ticket and then replacing it and then drawing a ticket again, adding the two price changes to get a net price change. Since the price change version of Model A represents the price change after a fixed time period (say it is one month), the sum of the two draws would represent the price change after two months. What values are possible?
You could get two -10’s, or two +10’s, or one -10 and one +10. But these are not all equally likely. It is more likely, twice as likely to be exact, that you would get one -10 and one +10 than two -10’s or two +10’s. You can see this clearly if you just list the possibilities, keeping the order of the draws intact: 1st draw 2nd draw Net Price Change -10 -10 -20 -10 +10 0 +10 -10 0 +10 +10 +20
Thus the sum of two draws from the Model A Price Change box is like one draw from the above box containing four tickets. There is one-half (or 2/4) a chance that the price change will be 0; one fourth a chance that it will be -20; and one fourth a chance it will be +20.
Again, you could use the formula to compute the standard deviation of the tickets in the above box. If you did all the arithmetic correctly, you would find it to be 10 times the square root of 2, which is about 10 x 1.414 = 14.14. This is just like the square root law I have spoken of in the past, applied to the sum of two draws from the box instead of to the volatility. I will continue this next time, but you might try and see if you can understand why the two square root laws are really the same.