Black-Scoles Merton-Hull reference

[k_gopala]

Dear David,

Ref: Hull 6th ed. page 283 and page 284.

In the example 13.3 , the annual volatlity is said to be 20% and the formula divides the volatility by square root of 3 to calcuate the probability distribution for the continuously compounded average return over the three year period. Can you kindly explain why we are not using the square root rule of standard deviation for three year period.(multiply by square root of three)

Thanks and regards

 

[David Harper CFA FRM]

Hi risklearner,

yes, these pages are harder than they look.

0.2 per annum*SQRT(3) = about 35% for the three-year volatility, scaled by square root rule.
To your point, this above is scaling by the square root.

This is just about the interval. He is saying, what is *average* volatility over the period, so divide by three
(and let 3 = sqrt(3)^2):
0.2*SQRT(3)/3 = 0.2*SQRT(3)/[SQRT(3)^2] = 0.2/SQRT(3)

So, the 35% you are thinking about is just three times the 11.55%. In 13.3, he is getting the annual *average* over the period instead of a three-year measure

IMO, that is one of the two hard parts about this. The other hard part here is: expected return has the two meanings; e.g., 17% and 15% where 15% corresponds to the geometric average that is used in the Black-Scholes and reflects that "volatility erodes returns"


...David