Regarding index options. They are always less volatile then their
constituents. The best case for this statement is when the under-
lying securities are independent (this is not the case when market
risk is present. Market risk is risk that the market as a whole will
change, non-market risk is risk that the particular stock will change
relative to the market.
Example calculation:
Four stocks, A, B, C, and D make up index X, X = (A+B+C+D)/4.
Suppose A, B, C, and D will be each be worth either $4 or $6
three weeks from now, with equal probability. (I.e. I am using
simple probability examples that can be worked without calculus.)
Further, suppose they each currently sell for $5.00.
Easy to calculate their current variances:
0.5 * (5-4)^2 + 0.5 * (5-6)^2 = 1.
So standard deviation = sqrt of variance = 1.
Therefore their options should be priced with an implied volatility
of 1.
Now look at the index. Here are the possible outcomes, with
probabilities:
(4+4+4+4)/4 = 4.00 Prob = 1/16 = 0.0625
(4+4+4+6)/4 = 4.50 Prob = 4/16 = 0.2500
(4+4+6+6)/4 = 5.00 Prob = 6/16 = 0.3750
(4+6+6+6)/4 = 5.50 Prob = 4/16 = 0.2500
(6+6+6+6)/4 = 6.00 Prob = 1/16 = 0.0625
So variance is:
0.0625*(5-4.0)^2 + 0.2500*(5-4.5)^2 + 0.3750*(5-5.0)^2
+0.2500*(5-5.5)^2 + 0.0625*(5-6.0)^2 = 0.2500
So standard deviation is sqrt(0.25) = 0.5
So the index options should (in this case of 4 independent
stocks) be priced with an implied volatility just half that of
the constituent stocks. This is a theoretical calculation,
and not an indication that one or the other is a better deal.
These calculations compute the "fair" deal.
This shows why it is that indexes are supposed to carry lower
implied volatility than the constituent stocks that make them up.
Similarly, the SPX should have lower implied volatility than the
SOX or XCI, for example.
The point is that just because INTC has higher implied volatility,
doesn't mean that its options are a worse deal. If that were the
case, then options buyers would only buy index options. In fact,
options are more or less fairly priced. The problem with purchasing
them is the spreads, the commissions, and the decay of time
premium... :)
To do this comparison on real-life stocks and indexs, look up
time premiums for same expiration calls/puts struck at the money.
-- Carl |
