| In
this short educational paper we consider calculations
of the implied volatility of call and put stock options.
Using the Black-Sholes (BS) theoretical equation for calculation
of the option prices can give a wrong price values. It
is related to complicated calculation of the stock volatility.
It seems that the best way to calculate volatility if
using a standard deviation of the daily log returns:
 X(i)
= lnP(i) - ln P(i-1)
where P(i)
= closing price on day i; ln = natural logarithm.
Usually this equation provides
smaller values of volatility than one needs to describe
the option prices. This is particularly important for
volatile stocks. The implied volatility is calculated
from the BS equation while letting volatility be the
unknown parameter. If one knows option striking prices,
time remaining until expiration, option and stock current
prices you can use computer to calculate the volatility.
"This method of computing volatility is quite accurate
and proves to be sensitive to changes in the volatility
of a stock". (L. McMillan, Options as a Strategic
Investment).
Here we will show that
the implied volatility is different for call and put
options. The first figure shows the implied volatilities
of call and put option of DELL for the period from July
to December 1998.
One can see that for all
periods implied volatilities of the put options is larger
than the volatililties of the call options. It is true
for all stock trends as one can see from the lower panel
of the figure. The average implied volatility becomes
larger when a stock price moves significantly but the
put options always cost more than the call options.
 The
next figure shows relationship between volatililtires
of put and call options of various stocks. In average
the volatility of put option is 9% larger than the volatility
of call options.
We did not find any significant
correlation between these volatility differences
Dv =- v(put)
- v(call)
and option trading volumes,
option prices.The only correlation was found between
Dv and stock prices: the larger stocks price, the larger
values of Dv (plot is not shown). However the correlation
coefficient is small: 0.18.
Conclusion.
Volatilities of the put stock options are about 9% larger
than the volatilities of the call options. This fact
can be used for quick estimation of profitability of
some option trading strategy using puts and calls. It
is important for short-term option traders: because
of larger volatility the put options cost more than
the call options. It means that trading puts is safer
then trading calls. The probability of very large drops
of puts is less than that one for calls.
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