Chapter 14 Volatility Revisited

While there are many ways in which traders interpret volatility, in theory the value of an option depends on only one volatility, the volatility of the underlying contract which will occur over the life of the option. Of course, this volatility is unknown to a trader since it will occur in the future. Nevertheless, If a trader wants to use a theoretical pricing model, he will be required to make some prediction about the volatility of the underlying contract over the life of the option.

Making an Intelligent volatility forecast can be a difficult and frustrating exercise, especially for a new option trader. The forecasting of directional price movements through technical analysis is a commonly studied area in trading, and there are many sources to which a trader can turn for information on this subject. Unfortunately, volatility is a much newer concept, and there is relatively little to guide a trader. In spite of this difficulty, an option trader must make some effort to come up with a reasonable volatility input if he is going to rely on a theoretical pricing model to make intelligent trading decisions.

SOME VOLATILITY CHARACTERISTICS

The first step in making a sensible volatility forecast is to understand some of the basic characteristics of volatility. First, let's compare two graphs. Figure 14-1 shows Deutschemark prices from 1982 to 1991. Figure 14-2 shows the 50-day volatility of Deutschemarks over the same period. Are there any generalizations one can make from these graphs? Obviously both prices and volatility sometimes rise and sometimes fall. But unlike the prices of an underlying instrument, which appear to move freely in either direction, there seems to be an equilibrium number to which the volatility always returns. Over a period of three years, from early 1985 to late 1987, the price of Deutschemarks rose from a low of 29 to a high of 63. While prices fluctuated after 1987, they never reached the lows of the early 1980s. Someday, economic forces may cause Deutschemarks to rise or fall dramatically, never again returning to prices in the 50s or 60s. In other words, prices of an underlying contract are open-ended. There is no reason why they have to return to some previous level.

Such does not appear to be the case with volatility. The 50-day volatility of Deutschemarks over the 10-year period in question fluctuated from a low of 5% to a high of 20%. Yet no matter how much it fluctuated, at some point volatility always reversed itself and retraced almost all of its previous rise or fall. Indeed, we might try to find an equilibrium volatility such that there are equal fluctuations above and below this number. In the case of Deutschemarks, this equilibrium volatility seems to be about 11% to 12%. Volatility may rise well above 12%, or fall well below 11%, but eventually it always seems to return to this area.

If we were to generalize about volatility characteristics from the Deutschemark volatility in Figure 14-2, we might surmise that an underlying contract is likely to have a typical long-term average, or mean volatility. Moreover, the volatility of the underlying contract appears to be mean reverting. When volatility rises above the mean, one can be fairly certain that it will eventually fall back to its mean; when volatility falls below the mean, one can be fairly certain that it will eventually rise to its mean. There is a constant gyration back and forth through this mean.

This mean reverting characteristic of volatility can also be seen in Figure 14-3, which shows the distribution of Deutschemark volatility from 1982 to 1991. Beginning at the extreme left on the x-axis (the amount of time remaining to expiration), we can see that over any two-week period during the 10 years in question, there was a 20% chance that volatility would be either less than 6.0% or greater than 17.3% (the 10th and 90th percentiles). There was a 50% chance that volatility would be either less than 7.6% or greater than 13.0% (the 25th and 75th percentiles). The mean volatility for any two week period was 9.7%. Moving to the extreme right on the x-axis, we can see that over any 50 week period there was a 20% chance that volatility would be either less than 9.8% or greater than 14%. There was a 50% chance that volatility would be either less than 10.6% and greater than 12.7%. The mean volatility for any 50-week period was about 11.5%.

Figure 14-3 has an easily identifiable structure. As one moves further out in time the percentile lines tend to converge towards the mean, and the mean becomes stable. This reinforces the assumption that volatility is indeed mean reverting. This type of volatility graph, sometimes referred to as a volatility cone, is an effective method of presenting the volatility characteristics of an underlying instrument. (Footnote 1: For a more detailed discussion of volatility cones, see:
Burghardt, Galen and Lane, Morton; "How to Tell if Options are Cheap"; The Journal of Portfolio Management, Winter 1990, pages 72-78.)

What else can we say about volatility? Looking at the more detailed Deutschemark volatility chart in Figure 14-4, we might surmise that volatility has some trending characteristics. From July 1989 through June 1990 there was a downward trend in volatility. From July 1990 to April 1991 there was an upward trend. And from April 1991 to October 1991 there was again a downward trend. Moreover, within these major trends there were minor trends as volatility rose and fell for short periods of time.

In this respect volatility charts seem to display some of the same characteristics as price charts, and it would not be unreasonable to apply some of the same principles used in technical analysis to volatility analysis. It is important to remember, however, that while price changes and volatility are related, they are not the same thing. If a trader tries to apply exactly the same rules of technical analysis to volatility analysis, he is likely to find that in some cases the rules have no relevance, and that in other cases the rules must be modified to take into account the unique characteristics of volatility. Since the author claims no particular expertise in the area of technical analysis, the reader is left to his own devices in this regard.

VOLATILITY FORECASTING

Given the volatility characteristics that we have identified, how might we go about making a volatility forecast? First we need some volatility data. Suppose we have the following historical volatility data on a certain underlying instrument:

last 30 days	24%
last 60 days	20%
last 120 days	18%
last 250 days	18%

Certainly, we would like as much volatility data as possible. But if this is the only data available, how might we use it to make a forecast? One method might be to simply take the average volatility over the periods which we have:

(24% + 20% + 18% + 18%) / 4 = 20%

Using this method, each piece of data is given identical weight. Might it not be reasonable to assume that some data is more important than other data? A trader might assume, for example, that the more current the data, the greater its importance. Since the 24% volatility over the last 30 days is clearly more current than the other volatility data, perhaps 24% should play a greater role in our volatility forecast. We might, for example, give twice as much weight to the 30-day data as to the other data:

(40% x 24%) + (20% x 20%) + (20% x 18%) + (20% x 18%) = 20.8%

Our forecast has gone up slightly because of the extra weight given to the more recent data.

Of course, if it is true that the more recent volatility over the last 30 days is more important than the other data, it follows that the volatility over the last 60 days ought to be more important than the volatility over the last 120 days and 250 days. It also follows that the volatility over the last 120 days must be more important than the volatility over the last 250 days. We can factor this into our forecast by using a regressive weighting, giving more distant volatility data progressively less weight in our forecast.

For example, we might calculate:

(40% x 24%) + (30% x 20%) + (20% × 18%) + (10% × 18%) = 21.0%

Here we have given the 30-day volatility 40% of the weight, the 60-day volatility 30% of the weight, the 120-day volatility 20% of the weight, and the 250-day volatility 10% of the weight.

We have made the assumption that the more recent the data, the greater its importance. Is this always true? If we are interested in evaluating short term options, it may be true that data which covers short periods of time is the most important. But suppose we are interested in evaluating very long-term options. Over long periods of time the mean reverting characteristic of volatility is likely to reduce the importance of any short-term fluctuations in volatility. In fact, over very long periods of time the most reasonable volatility forecast is simply the long-term mean volatility of the instrument. Therefore the relative weight we give to the different volatility data will depend on the amount of time remaining to expiration for the options in which we are interested.

In a sense, all the historical volatility data we have at our disposal are current data; they simply cover different periods of time. How do we know which data is the most important? In addition to the mean reverting characteristic, volatility also tends to exhibit serial correlation. The volatility over any given period is likely to depend on, or correlate with, the volatility over the previous period, assuming that both periods cover the same amount of time. If the volatility of a contract over the last four weeks was 15%, the volatility over the next four weeks is more likely to be close to 15% than far away from 15%. We can again use the weather analogy from Chapter 4. If the high temperature yesterday was 25°, and we had to guess what the high temperature today would be, a guess of 30° would make more sense than a guess of 50°. Once we realize this, we might logically choose to give the greatest weight to the volatility data covering a time period closest to the life of the options in which we are interested. That is, if we are trading very long-term options, the long-term data should get the most weight. If we are trading very short term options, the short term data should get the most weight. And if we are trading intermediate-term options, the intermediate-term data should get the most weight.

Suppose we are interested in evaluating six-month options. How should we weight our data? Since 120 (trading) days is closest to six months, we can give the 120-day data the greatest weight, and give other data correspondingly lesser weight:

(15% × 24%) + (25% x 20%) + (35% x 18%) + (25% × 18%) = 19.4%

Alternatively, if we are interested in 10-week options, we can give the greatest weight to the 60-day volatility data:

(25% x 24%) + (35% x 20%) + (25% x 18%) + (15% × 18%) = 20.2%

In the foregoing examples we used only four historical volatilities, but the more volatility data which is available, the more accurate any volatility forecast is likely to be. Not only will more data, covering different periods of time, give a better overview of the volatility characteristics of an underlying instrument, but it will enable a trader to more closely match historical volatilities to options with different amounts of time to expiration. In our example we used historical volatilities over the last 60 days and 120 days as approximations to forecast volatilities for six-month and 10-week options. Ideally, we would like historical data covering exactly a six-month period and exactly a 10-week period.

The method we have described is one which many traders intuitively use to forecast volatility. It depends on identifying the typical characteristics of volatility, and then projecting a volatility over the forecasting period. Theoreticians have recently tried to take essentially the same approach to volatility forecasting, and this has led to the development of autoregressive conditional heteroskedasticity (ARCH) and generalized autoregressive conditional heterosexuality (GARCH) volatility models. A detailed discussion of such models is beyond the scope of this text, since they can be mathematically complex and are not widely used among traders. Nevertheless, an option trader should be aware that these models do exist, and that they are simply an attempt to apply the mean reverting and serial correlation characteristics of volatility to volatility forecasting. (Footnote 2: For further information on ARCH and GARCH models see:
Engie, R.F., "Autoregressive Conditional Heteroskedaticity with Estimates of the Varlance of United Kingdom Inflation," Econometrica, Vol. 50, No. 4, 982, pages 987-1000.
Bollerslev, T., "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Economics, No. 31, April 1986, pages 307-327.
Bollerslev, T., "A Conditional Heteroskedastic Time Serles Model for Speculative Prices and Rates of Return," Revtew of Economics and Statistics, No. 69, August 1987, pages 542-547.
Nelson, David B., "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, No. 59, 1991, pages 347-370
Kuberek, Robert C., "Predicting Interest Rate Volatility: A Conditional Heteroskedastic Model of Interest Rate Movements," Journal of Fixed Income, Vol. 1, No. 4, March 1992, pages 21-27.)

Thus far we have focused only on the historical volatility characteristics of the underlying instrument in trying to forecast a volatility. Is there any other information which might be useful? No individual trader can hope to know everything affecting price changes in an underlying contract. Perhaps there are factors which could affect the future volatility, but about which the trader is unaware. If one believes that such information is likely to be reflected in the prices of the contracts being traded, one way to ferret out additional volatility information is to look at the prices of options. In other words, a trader will want to look at the implied volatility in the marketplace to find the consensus volatility. Once he has done this, he will want to use this information in any volatility forecast he intends to make.

How much weight should a trader give the implied volatility? Some traders subscribe to the efficient market principle, and believe that the implied volatility is always the best volatility forecast since it reflects all available information. Most traders believe, however, that while the implied volatility is important, it isn't the whole story. Typically, a trader might give the implied volatility a weighting somewhere between 25% and 75% in making a volatility forecast. How much depends on the trader's confidence in forecasting a volatility based on historical volatility data? If a trader feels very confident about his forecast, he might give the implied volatility as little as a 25% weight; if he feels uncertain about his forecast, he might give the implied volatility as much as a 75% weight. Of course, his confidence level will depend on his experience, as well as how conclusive the historical data is.

For example, suppose a trader has made a volatility forecast of 20% based on historical data, and that the implied volatility is currently 24%. If the trader decides to give the implied volatility 75% of the weight, his final forecast will be:

(75% x 24%) + (25% × 20%) = 23%

On the other hand, if the trader decides to give the implied volatility 25% of the weight, his final forecast will be:

(25% x 24%) + (75% x 20%) = 21%

Finally, If the trader decides to give the implied volatility half the weight, his final forecast will be:

(50% × 24) + (50% × 20) = 22%

A PRACTICAL APPROACH

No matter how painstaking a trader's method, he is likely to find that his volatility forecasts are often incorrect, and sometimes to a large degree. Given this difficulty, many traders find it easier to take a more general approach. Rather than asking what the correct volatility is, a trader might instead ask, given the current volatility climate, what's the right strategy? Rather than trying to forecast an exact volatility, a trader will try to pick a strategy that best fits the volatility conditions in the marketplace. To do this, a trader will want to consider several factors:

1. What is the long-term mean volatility of the underlying contract?
2. What has been the recent historical volatility in relation to the mean volatility?
3. What is the trend in the recent historical volatility?
4. Where is implied volatility and what is its trend?
5. Are we dealing with options of shorter or longer duration?
6. How stable does the volatility tend to be?

For example, suppose there are ten weeks (approximately 50 trading days) to expiration and we are trying to decide on an appropriate volatility strategy. To make a decision, we might look at the 50-day historical volatility (the historical volatility which corresponds to the amount of time remaining to expiration), its relationship to the long-term mean volatility, and of course the implied volatility. Having done this, suppose we find the conditions shown in Figure 14-5A. What are our conclusions?

Clearly we are coming off a period of high volatility and seem to be moving downward. The recent 50-day historical volatility (approx. 20.6%) is still above the long-term historical mean (approx. 18.7%), so there is reason to believe that the volatility will continue to decline. The implied volatility is also declining, but still appears to be about 1½ percentage points higher (22.1%) than the 50-day historical volatility. All volatility conditions seem to be pointing in the same direction. The historical volatility is above the mean but declining, and the implied volatility is above the historical volatility and also declining. A short volatility position (negative gamma/negative vega) is strongly indicated.

There are of course a variety of short volatility strategies available, and the best strategy will still depend on a trader's experience in the market and the amount of risk he is willing to take. Suppose we have additional volatility data (perhaps a volatility cone of the type in Figure 14-3) showing that the 50-day volatility can easily vary by as much as 10 percentage points. We still want to sell volatility, but given this instability and the fact that implied volatility at 22.2% is only slightly above the 50-day historical volatility, perhaps the wisest choice is a less risky strategy such as the purchase of butterflies. If a riskier strategy is chosen, perhaps it should be done only in small size. On the other hand, if implied volatility is currently 25% and we find that the volatility tends to be quite stable, varying only five or six percentage points for any 50-day period, a trader may be willing to take on a much riskier position, perhaps selling a straddles or strangles.

In real life the situation is rarely as simple as in Figure 14-5A. For example, consider Figure 14-5B. Now the historical volatility is above the historical mean and declining, but the implied volatility has already moved ahead of the historical volatility. A trader may still choose a short volatility position, but he is unlikely to have the same degree of confidence. Indeed, the situation may be even more confused. Suppose the historical volatility is above the long-term mean but seems to be increasing (Figure 14-5C). Or suppose implied volatility is moving in the opposite direction of the historical volatility (Figure 14-5D). In each of these situations some factors dictate one type of position, while other factors dictate a different position.

Consider the situation in Figure 14-6A where we are thinking of taking a position in six-week options (approximately 30 trading days). The 30-day historical volatility, currently at 15.8%, is well above the long-term mean of 11.2%. However, the trend in volatility seems to be up, and there is no telling how long it will take for the volatility to revert to its mean of 11.2%. The current Implied volatility of 14.6% is also well above the long term mean volatility, but is lower than the 30-day historical volatility. Corresponding to the upward trend in historical volatility, there also seems to be an upward trend in the implied volatility. With so many contradictory signals, a trader is unlikely to have a strong opinion about whether he should buy or sell volatility. He may choose to take no position at all, preferring to wait for clearer indications.

Suppose, in addition to six-week options, we also have available 19-week options (approximately 95 trading days). Volatility data for this contract is shown in Figure 14-6B. While a trader may be hesitant to take a position in six-week options alone, when combined with 19-week options, he may be able to construct a strategy with much more acceptable risk characteristics. Here the 95-day historical volatility of 12.6% is above the 11.2% mean volatility. Moreover the current implied volatility of 19-week options, at 14.5%, is well above both the 95-day historical volatility and the long-term mean volatility. There is a much better chance that the volatility will revert to its mean over a 19-week period than over a six-week period, so there are much stronger reasons for taking a short volatility position in 19 week options than in six-week options. Still, the upward trend in both the historical and implied volatilities will cause some worry if we do take a short volatility position. However, If we were to simultaneously take a long volatility position in six-week options, we would at least be protected against a continuing increase in the volatility of the underlying contract over the next six weeks. By creating a short time spread (buy six-week options/ sell 19-week options) we can take a position which should be profitable based on our knowledge of volatility characteristics, but which also has acceptable risk characteristics if we are wrong.

A short time spread will not eliminate every risk. The market might suddenly become very quiet, with historical volatility quickly dropping to, or even below, its long-term mean. At the same time, implied volatility may remain relatively high. Such conditions will do the most damage to a short time spread. Still, if we believe that the implied volatility tends to follow the historical volatility, we may conclude that it is unlikely that the implied volatility will stay high if the historical volatility drops.

A trader will always attempt to pick the strategy which best fits his opinion of market conditions, whether a directional opinion or a volatility opinion. Given the fact that there is such a wide variety of market conditions, a trader who is familiar with the greatest number of strategies will have the best chance of surviving and prospering. He will be in a position to pick strategies with the best risk/ reward characteristics, strategies which will be profitable when things go right, but which won't give back all the profits when things go wrong. This skill comes not only from a technical knowledge of option evaluation and theory, but also from a practical knowledge of what really happens in the marketplace.

SOME THOUGHTS ON IMPLIED VOLATILITY

Since many option strategies are sensitive to changes in implied volatility, and since implied volatility will often play a role in forecasting the volatility of the underlying contract, it may be worthwhile to consider some of the characteristics of implied volatility.

Implied versus Historical Volatility

Implied volatility can be thought of as a consensus volatility among all market participants with respect to the expected amount of underlying price fluctuation over the remaining life of an option. In the same way that an individual trader is likely to change his volatility forecast in response to changing historical volatility, it is logical to assume that the marketplace as a whole will also change its consensus volatility in response to changing historical volatility. As the market becomes more volatile, implied volatility can be expected to rise; as the market becomes less volatile, implied volatility can be expected to fall, Market participants are making the logical assumption that what has happened in the past is a good indicator of what will happen in the future.

The influence of historical volatility on implied volatility can be seen in Figure 14-7, the historical and implied volatility of U.S. Treasury Bond futures traded on the Chicago Board of Trade from 1989 to 1991. In late 1989, and again in mid-1991, there were declines in the volatility of Treasury Bond futures, and these were accompanied by corresponding declines in implied volatility. From August 1990 through January 1991 (the period of the Iraqi invasion of Kuwait), there were several sharp increases in the volatility of the futures, and these were accompanied by similar increases in implied volatility. Clearly, the marketplace, in the form of changing implied volatility, was responding to the changing historical volatility of the underlying contract.

Notice, however, that the fluctuations in implied volatility were usually less than the fluctuations in historical volatility. When the historical volatility declined, the implied volatility rarely declined by an equal amount. And when historical volatility increased, the implied volatility rarely increased by an equal amount. Because volatility tends to be mean reverting, when historical volatility is above its mean there is a greater likelihood that it will decline, and when historical volatility is below its mean there is a greater likelihood that it will increase.

Moreover, the further out in time we go, the greater the likelihood that the volatility of the underlying contract will return to its mean. (Look again at Figure 14-3.) Consequently, the implied volatility of long-term options tends to remain closer to the mean volatility of an underlying contract than the implied volatility of short-term options. As historical volatility rises, the implied volatility of all options is likely to rise. But given the stronger mean reverting characteristics of volatility over long periods of time, the implied volatility of long-term options will tend to rise less than the implied volatility of short-term options. As historical volatility falls, the implied volatility of all options is likely to fall. But the implied volatility of long-term options will tend to fall less than the implied volatility of short-term options. This is born out by Figure 14-8, the implied volatility of Treasury Bond options for various expiration months from September 1990 to May 1991. Note the increase in implied volatilities during January 1991. But the increase in the implied volatility of the short term contract (March) was much greater than the increase in the implied volatility of the mid-term (June) contract, which was in turn greater than the increase in the implied volatility of the long-term (September) contract. When implied volatility began to decline in late January 1991 the roles were reversed. The March contract declined the most rapidly, followed by the June contract, and the September contract. This is typical of the way in which implied volatility tends to change in response to changing volatility environments.

Over long periods of time the historical volatility of the underlying contract will be the dominant factor affecting implied volatility. Over short periods of time, however, other factors can also play a significant, perhaps even a dominant, role. If the marketplace foresees events which could cause the underlying contract to become more volatile, anticipation of these events might cause implied volatility to change in ways that are not necessarily consistent with historical volatility. For example, government reports on economic conditions are issued periodically, and these reports have been known to contain surprises for the interest rate and foreign exchange market. This potential for surprise can cause uncertainty in the marketplace, and this uncertainty is often reflected in an increase in implied volatility. Going into government reports, there is a strong tendency for implied volatility to rise, even in the face of low historical volatility in the underlying instrument.

Government reports are not the only factors which add uncertainty to the market. Any future events which could have unexpected consequences can have an effect on Implied volatility. In the currency markets, upcoming meetings of finance ministers-or in the energy markets upcoming OPEC meetings— often cause implied volatility to rise. In the stock option market, earnings news, the potential success or failure of new products, or (most dramatically) the possibility of a takeover, can all cause increases in implied volatility, regardless of the historical volatility of the stock.

In a similar way, if the marketplace believes that no significant events are likely to occur in the foreseeable future, uncertainty is removed from the market. In such a case the implied volatility may start to fall, even if the actual historical volatility has been relatively high. This is why implied volatility sometimes drops right after large moves in the underlying contract. Once the big event has occurred, there may be a perception that all the uncertainty has been removed from the market.

Regardless of short-term changes in implied volatility, it is still important for a trader to remember that the volatility of the underlying contract will eventually overwhelm any considerations of implied volatility. As an example, consider the following situation:

futures price = 97.73
time to expiration = 60 days
interest rate = 6%
implied volatility = 20%

Given these conditions, the 100 call would be trading for 2.17, with an implied delta of 40. Suppose we create a delta neutral position by purchasing ten 100 calls for 2.17 each, and selling four futures contracts at 97.73. What will happen to our position if implied volatility rises to 22%?

If implied volatility immediately goes to 22%, the new price of the 100 call will be 2.47, and we will show a profit of

10 × (2.47 - 2.17) = +3.00

Suppose, however, that the increase in implied volatility occurs very slowly, over a period of 20 days, and during this period the price of the underlying futures contract remains at 97.73. Under these conditions, even if implied volatility rises from 20% to 22%, the 100 call will now be worth only 1.87. Our position will then show a loss of

10 × (1.87 - 2.17) = -3.00

Even though the implied volatility increased, the fact that the underlying futures contract failed to make any significant move resulted in the option's price declining.

Now suppose we have the same position (long ten 100 calls, short four futures contracts) but this time instead of rising to 22%, the implied volatility drops to 18%. What will be the effect on our position?

If implied volatility immediately drops to 18%, the new price of the 100 call will be 1.86, and we will show a loss of

10 × (1.86 - 2.17) = -3.10

Suppose, however, that the decline in implied volatility is accompanied by a swift move in the underlying contract. If the underlying futures contract falls to 93.00, and implied volatility falls to 18%, the price of the 100 call will be .59, and we will show a profit of 4 × (97.73 - 93.00) - 10 × (2.17 - 59) = +3.12

On the other hand, if the underlying futures contract rises to 102.50 while implied volatility falls to 18%, the price of the 100 call will be 4.32, and we will again show a profit

4 × (97.73 - 102.50) + 10 × (4.32 - 2.17) = +2.42

In both cases the movement in the underlying contract has more than offset any decline in the option's price due to a decline in implied volatility.

The foregoing examples are of course simplified. As market conditions change-an active trader may very well make adjustments to his position in order to remain delta neutral. If so, the actual profit or loss will be affected by the cash flow from this adjustment process. The important point is that the volatility of the underlying contract, whether the contract moves or sits still over time, will eventually overwhelm any changes in implied volatility. This is not to say that implied volatility is unimportant. The price of a contract is always an important consideration in making trading decisions. But in order to trade intelligently, we need to know value as well as price. The value of an option is determined by the volatility of the underlying contract over the life of the option.

Implied versus Future Volatility

If, as many traders believe, prices in the marketplace reflect all available information affecting the value of a contract, the best predictor of the future volatility ought to be the implied volatility. Just how good a predictor of future volatility is implied volatility? While it would be impossible to answer this question definitively, since that would require a detailed study of many markets over long periods of time, we might still gain some insight by looking at a limited number of examples.

Clearly, no one knows the future volatility. We can, however, record the implied volatility at any moment in time and then, when expiration arrives, look back and calculate the actual volatility of the underlying contract between the time we recorded the implied volatility and expiration. We can do this every day during an option's life, recording the implied volatility and then at expiration calculating the actual volatility that occurred over this period. This has been done in Figures 14-9A, 9B, and 9C for the June 1992, March 1993, and December 1993 options on Treasury Bond futures. It is admittedly dangerous to generalize from such limited data, but are there any conclusions we might draw from these graphs?

We can see that with a great deal of time remaining to expiration, the future volatility of the underlying contract (the solid line) is relatively stable. But as we get closer to expiration the future volatility can become much less stable. This is logical if we again recall that the mean reverting characteristics of volatility are much less certain over short periods of time than over long periods. One large move in the underlying contract with only several days remaining to expiration will result in a sharp increase in the volatility to expiration (see Figure 14-9C). On the other hand, if the underlying contract is relatively quiet over the last several days in an option's life, the volatility to expiration will collapse (see Figure 14-9B).

How does the marketplace react to these volatility characteristics? With long periods of time remaining to expiration, the volatility to expiration is relatively stable. One would therefore expect the implied volatility to also be relatively stable. Conversely, with short periods of time remaining to expiration, the volatility to expiration can be very unstable, and one would therefore expect the implied volatility to be unstable. These conclusions are born out by the implied volatility (the broken line) in Figures 14-9A, 9B, and 9C. Over long periods of time, the marketplace is reacting to many events. This is easier than reacting to a limited number of events, which is what the marketplace is faced with over a short period of time. The marketplace knows that the laws of probability are more likely to balance out over many occurrences than over only a few occurrences.

Note also that there is no guarantee that the marketplace will have the correct implied volatility. The implied volatility is a guess, and guesses carry with them the possibility of error, sometimes very large error. In Figure 14-9B the implied volatility turned out to be much too high over almost all of the option's life. Had a trader sold premium at any time, he would have shown a profit. At its most extreme, during October 1992, there was almost a four percentage-point difference between the implied volatility of the March 1993 options and the future volatility of the March 1993 futures contract. In Figure 14-9C the implied volatility was too high during the early part of the option's life, but too low during the latter part. During the few weeks prior to expiration, the implied volatility was too low by as much as three percentage points. Finally, in Figure 14-9A the implied volatility was relatively accurate over the early part of the option's life, but too high during the latter part.

It should be clear by now that dealing with volatility is a difficult task. To facilitate the decision-making process we have attempted to make some generalizations about volatility characteristics. Even then, if one decides to become involved in a market it may not be at all clear what the right strategy is. Moreover, we have looked at a limited number of examples, making the generalizations even less reliable. Every market has its own characteristics, and knowing the volatility characteristics of a particular market, whether interest rates, foreign currencies, stocks, or a physical commodity, is at least as important as knowing the technical characteristics of volatility. And this knowledge can only come from careful study of a market combined with actual trading experience.