<<Perhaps one of the wunderkind (Ethan or Mathmage) will be kind enough to offer a mathametical explanation of the difference between the two terms in practical applications.>>
First a disclaimer: I have known some wunderkinds and I am no wunderkind.
For option writers volatility is important in that it is one factor that determines the premiums. Beta does not and I don't think there is necessarily any correlation between the two. A high beta stock could have a low volatility and vice versa. Volatility is a measurement of movement based on shorter time intervals. Beta is a measurement not only of movement but also direction (a vector?) over longer time intervals. Mathematically perhaps beta is something like the integral of volatility. But now I am out of my depth. I have probably said enough wrong that it will stimulate the mathematicians lurking here to step forward and tell how these terms really relate.
Your question caused me to read further on the subject (always dangerous since it can lead to facts getting in the way of my opinions) and I came across the following from the CBOE site:
cboe.com
Volatility
With regard to stock prices and stock index levels, volatility is a measure of changes in price
expressed in percentage terms without regard to direction. This means that a rise from 200 to 202
in one index is equal in volatility terms to a rise from 100 to 101 in another index, because both
changes are 1 percent. Also, a 1 percent price rise is equal in volatility terms to a 1 percent price
decline. While volatility simply means movement, there are four ways to describe this movement:
1. Historic volatility is a measure of actual price changes during a specific time period in the past.
Mathematically, historic volatility is the annualized standard deviation of daily returns during a
specific period. CBOE provides 30 day historical volatility data for obtainable stocks in the Trader's
Tools section of this Web site.
2. Future volatility means the annualized standard deviation of daily returns during some future
period, typically between now and an option expiration. And it is future volatility that option
pricing formulas need as an input in order to calculate the theoretical value of an option.
Unfortunately, future volatility is only known when it has become historic volatility. Consequently,
the volatility numbers used in option pricing formulas are only estimates of future volatility. This
might be a shock to those who place their faith in theoretical values, because it raises a question
about those values. Theoretical values are only estimates, and as with any estimate, they must be
interpreted carefully.
3. Expected volatility is a trader's forecast of volatility used in an option pricing formula to
estimate the theoretical value of an option. Many option traders study market conditions and
historical price action to forecast volatility. Since forecasts vary, there is no specific number that
everyone can agree on for expected volatility.
4. Implied volatility is the volatility percentage that explains the current market price of an
option; it is the common denominator of option prices. Just as p/e ratios allow comparisons of stock
prices over a range of variables such as total earnings and number of shares outstanding, implied
volatility enables comparison of options on different underlying instruments and comparison of the
same option at different times. Theoretical value of an option is a statistical concept, and traders
should focus on relative value, not absolute value. The terms "overvalued" and "undervalued"
describe a relationship between implied volatility and expected volatility. Two traders could differ
in their opinion of the relative value of the same option if they have different market forecasts and
trading styles.
CBOE Volatility Index - VIX™
One measure of the level of implied volatility in index options is CBOE's Volatility Index, known by
its ticker symbol VIX. VIX, introduced by CBOE in 1993, measures the volatility of the U.S. equity
market. It provides investors with up-to-the-minute market estimates of expected volatility by
using real-time OEX index option bid/ask quotes. This index is calculated by taking a weighted
average of the implied volatilities of eight OEX calls and puts. The chosen options have an average
time to maturity of 30 days. Consequently, the VIX is intended to indicate the implied volatility of
30-day index options. It is used by some traders as a general indication of index option implied
volatility. Implied volatility levels in index options change frequently and substantially.
Consequently, when trading short-term index options, traders should forecast the index level, the
time period, and the volatility level. Traders of long-term index options should also include a
forecast of interest rates. (The volatility discussions above are excerpts from the book
Trading Index Options by James B. Bittman. This book is available through our
online bookstore.) |
