39 lines
2.9 KiB
Plaintext
39 lines
2.9 KiB
Plaintext
Chapter 28: Mathematical Applications 463
|
||
This is then the proper way to calculate historical volatility. Obviously, the
|
||
strategist can calculate 10-, 20-, and 50-day and annual volatilities if he wishes - or
|
||
any other number for that matter. In certain cases, one can discern valuable infor
|
||
mation about a stock or future and its options by seeing how the various volatilities
|
||
compare with one another.
|
||
There is, in fact, a way in which the strategist can let the market compute the
|
||
volatility for him. This is called using the implied volatility; that is, the volatility that
|
||
the market itself is implying. This concept makes the assumption that, for options
|
||
with striking prices close to the current stock price and for options with relatively
|
||
large trading volume, the market is fairly priced. This is something like an efficient
|
||
market hypothesis. If there is enough trading interest in an option that is close to the
|
||
money, that option will generally be fairly priced. Once this assumption has been
|
||
made, a corollary arises: If the actual price of an option is the fair price, it can be fixed
|
||
in the Black-Scholes equation while letting volatility be the unknown variable. The
|
||
volatility can be determined by iteration. In fact, this process of iterating to compute
|
||
the volatility can be done for each option on a particular underlying stock This might
|
||
result in several different volatilities for the stock If one weights these various results
|
||
by volume of trading and by distance in- or out-of-the-money, a single volatility can
|
||
be derived for the underlying stock This volatility is based on the closing price of all
|
||
the options on the underlying stock for that given day.
|
||
Example: XYZ is at 33 and the closing prices are given in Table 28-1. Each option
|
||
has a different implied volatility, as computed by determining what volatility in the
|
||
Black-Scholes model would result in the closing price for each option: That is, if .34
|
||
were used as the volatility, the model would give 4¼ as the price of the January 30
|
||
call. In order to rationally combine these volatilities, weighting factors must be
|
||
applied before a volatility for XYZ stock itself can be arrived at.
|
||
The weighting factors for volume are easy to compute. The factor for each
|
||
option is merely that option's daily volume divided by the total option volume on all
|
||
XYZ options (Table 28-2). The weighting functions for distance from the striking
|
||
price should probably not be linear. For example, if one option is 2 points out-of-the
|
||
money and another is 4 points out-of-the-money, the former option should not nec
|
||
essarily get twice as much weight as the latter. Once an option is too far in- or out-of
|
||
the-money, it should not be given much or any weight at all, regardless of its trading
|
||
volume. Any parabolic function of the following form should suffice:
|
||
{
|
||
(x - a)2 if xis less than a
|
||
Weighting factor = -;;,r-
|
||
= 0 if x is greater than a |