- 1999: Volume 8, No. 4 Puts & Calls: A Beast by One Name or Another

Colloquially, volatility is thought to be the characteristic of a financial asset to rise or fall sharply within a short period of time. While that definition may be sufficient to explain some market gyrations generally, it falls short in giving real shape to the beast.

The problem for options traders is that there's really two beasts commonly encountered: historic and implied volatility.

Historic Volatility

Statistically, historic volatility is the annualized standard deviation of daily returns over a period of time. It's not as foreboding as it may seem. Consider two stocks, ABC and XYZ, both of which gain 3-3/4 percent over the course of a month. ABC starts the month as a $200 stock, while XYZ is at $20. Look at the charts in Figures one and two and ask yourself, "Which is the more volatile?"

Figure 1: ABC Daily Closing Prices

Figure 2: XYZ Daily Closing Prices

It's deceptively easy to intuit ABC as the most volatile stock on the basis of just "eyeballing" the charts. But where volatility is concerned, a picture is not worth a thousand words-nor a thousand numbers. Look at the table of daily returns in Figure 3 below. Even though the absolute values of ABC's daily price changes are greater than XYZ's, XYZ's returns are more variable. And that's the true measure of volatility.

Figure 3

ABC				XYZ
Day	Close	Change	Return	Day	Close	Change	Return
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22	200 201 3/4 203 5/8 204 7/8 207 1/8 206 206 1/4 207 1/2 205 7/8 205 1/2 204 3/8 202 3/4 198 3/4 198 1/8 197 1/8 196 3/4 201 1/2 205 1/4 204 206 7/8 205 1/2 208 1/2 207 1/2	1 3/4 1 7/8 1 1/4 2 1/4 -1 1/8 1/4 1 1/4 -1 5/8 - 3/8 -1 1/8 -1 5/8 -4 - 5/8 -1 - 3/8 4 3/4 3 3/4 -1 1/4 2 7/8 -1 3/8 3 -1	0.9% 0.9% 0.6% 1.1% -0.5% 0.1% 0.6% -0.8% -0.2% -0.5% -0.8% -2.0% -0.3% -0.5% -0.2% 2.4% 1.9% -0.6% 1.4% -0.7% 1.5% -0.5%	0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22	20 19 3/4 20 1/8 20 1/4 20 1/8 20 1/8 20 1/4 20 1/2 20 20 1/4 20 1/2 21 20 5/8 20 1/2 20 5/8 19 7/8 20 3/8 20 5/8 20 1/8 20 3/8 20 7/8 20 5/8 20 3/4	-1/4 3/8 1/8 -1/8 0 1/8 1/4 -1/2 1/4 1/4 1/2 -3/8 -1/8 1/8 -3/4 1/2 1/4 -1/2 1/4 1/2 -1/4 1/8	-1.3% 1.9% 0.6% -0.6% 0.0% 0.6% 1.2% -2.4% 1.3% 1.2% 2.4% -1.8% -0.6% 0.6% -3.6% 2.5% 1.2% -2.4% 1.2% 2.5% -1.2% 0.6%
Mean	203 3/4	3/8	0.2%	Mean	20 1/4	0	0.2%
Standard Deviation		16.8%		Standard Deviation		27.3%

A common tendency is to look at an "average" as being definitive of an entire set. Saying that these two stocks have the same average daily returns, however, could lead to a dangerous assumption of similarity. The mean, or arithmetic average, daily return is the same for both stocks due to the additive properties of the signed numbers. But the median, or midpoint return, for ABC is -0.2 percent, quite a way off from XYZ's 0.6 percent median. This variance reflects the different distribution of returns for the two stocks. XYZ's returns ranged from -3.6 percent to 2.5 percent, while those for ABC varied between -2.0 percent and 2.4 percent only.

Volatility is merely a standard deviation adjusted for time. Standard deviation measures the dispersion of values in a sample around the mean. The wider the dispersion, the greater the standard deviation. Happily, you needn't be a statistical nerd to be able to calculate standard deviation nowadays. Spreadsheet programs like Excel have statistical functions that will do it for you with only the click of a button. But volatility with respect to financial assets is stated in annual terms, so you'll generally need to make an adjustment of the spreadsheet-generated standard deviation to "annualize" the value. For example, Excel would have calculated the standard deviation of ABC's returns below as 1.1 percent. But that's a daily reading representative of the 22 trading days in the sample. If we wish to extrapolate that volatility to an entire year, we need to multiply the standard deviation by the year's time factor, (252-a 252-day annualization factor takes out weekends and non-trading holidays). That's how ABC's 16.8 percent annualized standard deviation was derived.

So what exactly does a 16.8 percent volatility mean? If you believe the past is predictive of the future (and that's a big if), historic volatility can give you an idea of the future price potential for the stock. Statisically, if ABC is in a "normal" market, there's a 68 percent probability the stock will be within 16.8 percent-one standard deviation-of the sample's mean price at the end of a year. Simply put, having confidence in the constancy of this stock's price action, you'd be laying 2-to-3 odds of ABC ending up a year from now somewhere between 169-1/2 and 238.

Implied Volatility

Historic volatility looks backward to describe price or return tendencies. But there's no guarantee that history will repeat itself. Relying on the continuation of historical trends could lead you, in fact, to take wholly inappropriate risk positions at market turning points. You needn't rely solely on the past, however. You can find a market consensus for a stock's future volatility embedded in its option prices. This implied volatility estimate is the assumption used in option pricing models. Algorithms like Black-Scholes or Cox-Ross-Rubinstein require a volatility input, along with other quantities, to calculate an option's theoretical value. If the option's premium is already known, however, a model can be used "backwards" to estimate the expected price volatility of the underlying asset. Essentially this requires a series of calculations to be conducted until the volatility of the model satisfies the option's actual market price. Luckily, these calculations needn't be hand-cranked since many model-based option calculators can now be found on the Internet.

Let's suppose you obtained three offers for a three-month XYZ 25 call. From research, you know that XYZ pays no dividend and that three-month T-Bills are presently offered at a five percent yield. You're also wired into an online option pricing calculator. Running the numbers through the calculator, you derive the implied volatilities found in Figure 4.

Figure 4

XYZ = 20 3/4 XYZ 22 1/2 Call
Premium: Expiration: Dividends: Interest: Implied Volatility:	1 3/4 92 days None 5% 31.5%	2 7/8 92 days None 5% 34.5%	3 1 92 days None 5% 38.0%

The first thing you note is that implied volatility is higher than the historic volatility previously calculated for XYZ. That's the market's way of telling you that the future is expected to be more volatile than the recent past. Keep in mind that's only an expectation, not a guarantee.

Notable, too is the relationship between the option premium and the volatility assumption. It's simple and direct: the higher the volatility assumption, the higher the premium. Risk has a price, exactly as in the computation of insurance premiums. Consider two drivers attempting to insure identical vehicles with the same company. Why would one be charged a higher premium than the other for identical coverage (see Figure 5)?

Figure 5

	Driver A	Driver B
Value of vehicle Deductible Contract term Insururer's investment return Premium	$20,000 $500 1 year 5% $500	$20,000 $500 1 year 5% $800

The only factor that can explain the different premiums is risk. Driver B, for whatever reason, is perceived by the insurer as more risky. The insurer sees Driver B as being more likely to make a claim against the company assets. The company actuarial models quantified the risk, and derived the premium, in precisely the same manner that option pricing models relate implied volatility to contract prices (see Figure 6).

Figure 6

	Auto insurance	Stock insurance
Asset Value Deductible Contract term Investment return Risk Premium	Value of vehicle Deductible Term Net cost of money Insured's risk profile Premium	Stock price Strike price Term Net cost of money Volatility forecast Premium

The risk undertaken by the seller of the option is analogous to the insurer's position. A call seller perceives a higher likelihood of being assigned as expected volatility increases. Compensation for that risk is demanded in the form of additional premium.

In the case of your XYZ calls, the 3/4 offer is obviously the most attractive, being "cheapest" in dollars and in terms of volatility. Cheapness should be seen in relation to expectations of future volatility, too. Your expectations may vary radically from the market consensus. At times the market may underestimate the potential volatility of the stock, making certain options attractive buying candidates. Conversely, when implied volatilities are higher than your expectation for future volatility, you may consider being a net seller of options instead.

Online Volatility Resources

There are plenty of online resources available to help you tame the volatility beast. Here are just a few: Download free option educational software from the Options Industry Council at http://www.optionscentral.com/toolbox.html Obtain free historical and implied volatility data from McMillan Analysis Corp. at http://www.optionstrategist.com/data/volatility.htm Use a variety of option and volatility calculators provided by CIBC at: http://www.schoolfp.cibc.com/calc/index.html

Brad Zigler is Managing Director, Options Marketing, Research & Education at the Pacific Exchange in San Francisco. He can be reached through the Exchange's web site at www.pacificex.com. Send e-mail to: bzigler@pacificex.com. Any strategies discussed, including examples using actual securities and price data, are strictly for illustrative and educational purposes and are not to be construed as an endorsement, recommendation, or solicitation to buy or sell securities. The examples presented do not take into consideration commissions, tax considerations or other transaction costs, which may significantly affect the economic consequences of a given strategy. Options involve risk and are not for everyone.