| The
stock volatility is a crucial parameter for calculation
the option prices. In the Black-Scholes model one should
use annual volatility. There are many ideas how to calculate
this parameters. Some researchers suggest using shorter
period for calculating the volatility and then using a
simple equation to calculate the annual volatility.
V(annual)
= V(t) / SQRT (t)
(1)
where t is some period
of time expressed as a part of a year. SQRT is the square
root function. If, for example, one considers 3 month
period then t = 0.25.
In general, it is a good
idea. Let us show a plot of average volatilities calculated
for various values of t for Intel Co. (INTC). The averaging
has been performed for the period from 1986 to 2002.
In the log-log
scale this dependence is well described by a straight
line with the slope = 0.5. It confirms a validity of
equation 1.
In the real
life using this equation can be a dangerous game. For
the INTC stock price history we calculated 3 month and
preceding year volatilities.
| |<--------------------------------------
V(1 year) ------------------------------------------->| |
| |<-
V(1 quarter) ->| |
The ratios
of this numbers must be equal to 2 if equation 1 is
always valid. The next plot shows these ratios versus
time during studied 15 years of the INTC stock price
history.
One can see
that using equation 1 can provide 400% errors in calculated
annual volatility values! On the other hand, this picture
clearly shows that using the annual volatility does
not reflect recent stock price volatility and probably
using short-term volatility for calculation the annual
volatility presents the option prices better.
More detailed
study of this problem will be published soon.
|
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