ility asmuch 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 onhistorical data, and that the implied volatility is currently 24%. If the trader decides togive the implied volatility 75% of the weight, his final forecast will be:
(75% x 24%) +. (25% x 20%) = 23%
On the other hand, if the trader decides to give the implied volatility 25% of the weight,his final forecast will be:
 
2. For further information an ARCH and GARCH models see:
Engle, R.F., “Autoregressive Conditional Heteroskedaticity with Estimates of the Variance of UnitedKingdom Inflation,” Econometrica, Vol. 50, Na. 4, 1982, pages 987-1000.
Bollerslev, T., “Generalized Autoregressive Conditional Heteraskedasticity,” Journal of Economics,No. 31, April 1986, pages 307-327.
Bollerslev, T., “A Conditional Heteroskedastic Time Serles Model for Speculative Prices and Rates ofReturn,” Review of Economics and Statistics, No. 69, August 1987, pages 542-547,
Nelson, David B., “Conditional Heteraskedasticity in Asset Returns: A New Approach,” Econometrica,No. 59, 1991, pages 347-370
Kuberek, Robert C., “Predicting Interest Rate Volatility: A Conditional Heteroskedastic Model ofInterest Rate Movements,” Journal of Fixed Income, Vol. 1, No. 4, March 1992, pages 21-27.282 Chapter 14
(25% x 24%) + (75% x 20%) = 21%
Finally, if the trader decides to give the implied volatility half the weight, his finalforecast Will be:
(50% x 24) + (50% x 20) = 22%
A PRACTICAL APPROACH
No matter how painstaking a trader’s method, he is likely to find that his volatilityforecasts 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 whatthe 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 willtry 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, Whatis the long-term mean volatility of the underlying contract?What has been the recent historical volatility in relation to the mean volatility?What is the trend in the recent historical volatility?
2
3
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) toexpiration and we are trying to decide on an appropriate volatility strategy. To make adecision, we might look at the 50-day historical volatility (the historical volatility whichcorresponds to the amount of time remaining to expiration), its relationship to thelong-term mean volatility, and of course the implied volatility. Having done this, supposewe 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 movingdownward. The recent 50-day historical volatility (approx. 20.6%} is still above thelong-term historical mean (approx. 18.7%), so there is reason to believe that thevolatility will continue to decline. The implied volatility is also declining, but stillappears to be about 1% percentage points higher (22.1%} than the 50-day historicalvolatility. Al! volatility conditions seem to be pointing in the same direction. Thehistorical volatility is above the mean but declining, and the implied volatility is abovethe historical volatility and also declining. A short volatility position (negativegamma/negative vega] is strongly indicated.
There are of course a variety of short volatility strategies available, and the beststrategy will still depend on a trader’s experience in the market and the amount of riskhe is willing to take. Suppose we have additional volatility data (perhaps a volatility coneof the type in Figure 14-3) showing that the 50-day volatility can easily vary by as muchas 10 percentage points. We still want to sell volatility, but given this instability and the 
 
 
 
           
 
 
 
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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 ariskier strategy is chosen, perhaps it should be done only in small size. On the otherhand, if implied volatility is currently 25% and we find that the volatility tends to bequite stable, varying only five or six percentage points for any 50-day period, a tradermay be willing to take ona 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, considerFigure 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 tradermay still choose a short volatility position, but he is unlikely to have the same degree ofconfidence. Indeed, the situation may be even more confused. Suppose the historicalvolatility is above the long-term mean but seems to be increasing (Figure 14-5C}. Orsuppose 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 positionin 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 involatility seems to be up, and there is no telling how long it will take for the volatilityto revert to its mean of 11.2%. The current implied volatility of 14.6% is also well abovethe 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 anupward trend in the implied volatility. With so many contradictory signals, a trader isunlikely to have a strong opinion about whether he should buy or sell volatility. He maychoose 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 Figure14-6B. While a trader may be hesitant to take a position in six-week options alone, whencombined with {9-week options, he may be able to construct a strategy with much moreacceptable risk characteristics. Here the 95-day historical volatility of 12.6% is above the11.2% mean volatility. Moreover the current implied volatility of 19-week options, at14.5%, is well above both the 95-day historical volatility and the long-term meanvolatility. There is a much better chance that the volatility will revert to its mean overa 19-week period than over a six-week period, so there are much stronger reasons fortaking a short volatility position in 19-week options than in six-week options. Still, theupward trend in both the historical and implied volatilities will cause some worry if wedo take a short volatility position. However, if we were to simultaneously take a longvolatility position in six-week options, we would at least be protected against acontinuing 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 takea position which should be profitable based on our knowledge of volatility charac-teristics, but which also has acceptable risk characteristics if we are wrong.
A short time spread will not eliminate every risk. The market might suddenlybecome very quiet, with historical volatility quickly dropping to, or even below, its28522=rs
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long-term mean. At the same time, implied volatility may remain relatively high. Suchconditions will do the most damage to a short time spread. Still, if we believe that theimplied volatility tends to follow the historical volatility, we may conclude that it isunlikely 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 ofmarket conditions, whether a directional opinion or a volatility opinion. Given the factthat there is such a wide variety of market conditions, a trader who is familiar with thegreatest number of strategies will have the best chance of surviving and prospering. Hewill be in a position to pick strategies with the best risk/reward characteristics, strategieswhich will be profitable when things go right, but which won’t give back all the profitswhen things go wrong. This skill comes not only from a technical knowledge of optionevaluation and theory, but also from a practical knowledge of what really happens inthe marketplace.
SOME THOUGHTS ON IMPLIED VOLATILITY
Since many option strategies are sensitive to changes in implied volatility, and sinceimplied volatility will often play a role in forecasting the volatility of the underlyingcontract, it may be worthwhile to consider some of the characteristics of impliedvolatility.
Implied versus Historical Volatility
Implied volatility can be thought of as a consensus volatility among all market partici-pants with respect to the expected amount of underlying price fluctuation over theremaining life of an option. In the same way that an individual trader is likely to changehis volatility forecast in response to changing historical volatility, it is logical to assumethat the marketplace as a whole will also change its consensus volatility in response tochanging historical volatility. As the market becomes more volatile, implied volatilitycan be expected to rise; as the market becomes less volatile, implied volatility can beexpected to fall. Market participants are making the logical assumption that what hashappened 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 ChicagoBoard of Trade from 1989 to 1991. In late 1989, and again in mid-1991, there weredeclines in the volatility of Treasury Bond futures, and these were accompanied bycorresponding declines in implied volatility. From August 1990 through January 1991(the period of the Iraqi invasion of Kuwait), there were several sharp increases in thevolatility of the futures, and these were accompanied by similar increases in impliedvolatility. Clearly, the marketplace, in the form of changing implied volatility, wasresponding to the changing historical volatility of the underlying contract.
Notice, however, that the fluctuations in implied volatility were usually less thanthe fluctuations in historical volatility. When the historical volatility declined, theimplied volatility rarely declined by an equal amount. And when historical volatilityincreased, the implied volatility rarely increased by an equal amount. Because volatility291
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tends to be mean reverting, when historical volatility is above its mean there is a greaterlikelihood that it will decline, and when historical volatility is below its mean there is agreater likelihood that it will increase.
Moreover, the further out in time we go, the greater the likelihood that thevolatility 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 themean volatility of an underlying contract than the implied volatility of short-termoptions. 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 oftime, the implied volatility of long-term options will tend to rise less than the impliedvolatility of short-term options. As historical volatility falls, the implied volatility of alloptions is likely to fall. But the implied volatility of long-term options will tend to fallless 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 fromSeptember 1990 to May 1991. Note the increase in implied volatilities during January1991. But the increase in the implied volatility of the short-term contract (March) wasmuch 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 theroles were reversed. The March contract declined the most rapidly, followed by the Junecontract, and the September contract. This is typical of the way in which impliedvolatility tends to change in response to changing volatility environments.
Over long periods of time the historical volatility of the underlying contract will bethe 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 market-place foresees events which could cause the underlying contract to become morevolatile, anticipation of these events might cause implied volatility to change in waysthat are not necessarily consistent with historical volatility. For example, governmentreports on economic conditions are issued periodically, and these reports have beenknown to contain surprises for the interest rate and foreign exchange market. Thispotential for surprise can cause uncertainty in the marketplace, and this uncertainty isoften reflected in an increase in implied volatility. Going into government reports, thereis a strong tendency for implied volatility to rise, even in the face of low historicalvolatility 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 onimplied volatility. In the currency markets, upcoming meetings of finance ministers—orin 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 newproducts, or (most dramatically) the possibility of a takeover, can all cause increases inimplied volatility, regardless of the historical volatility of the stock.
In a similar way, if the marketplace believes that no significant events are likely tooccur in the foreseeable future, uncertainty is removed from the market. In such a casethe implied volatility may start to fall, even if the actual historical volatility has been293
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relatively high. This is why implied volatility sometimes drops right after large moves inthe underlying contract. Once the big event has occurred, there may be a perceptionthat all the uncertainty has been removed from the market.
Regardless of short-term changes in implied volatility, it is still important fora traderto remember that the volatility of the underlying contract will eventually overwhelmany considerations of implied volatility. As an example, consider the following situation:
futures price = 97.73
time to expiration = 60 daysinterest rate = 6%
implied volatility = 20%
Given these conditions, the 100 call would be trading for 2.17, with an implied deltaof 40. Suppose we create a delta neutral position by purchasing ten 100 calls for 2.17each, and selling four futures contracts at 97.73. What will happen to our position ifimplied volatility rises to 22%?
If implied volatility immediately goes to 22%, the new price of the 100 call wilt be2.47, and we will show a profit of
10 x (2.47 -— 2.17) = +3.00
Suppose, however, that the increase in implied volatility occurs very slowly, over aperiod of 20 days, and during this period the price of the underlying futures contractremains at 97.73. Under these conditions, even if implied volatility rises from 20% to22%, the 100 call will now be worth only 1.87. Our position will then show a loss of
10 x (1.87 ~ 2.17) = -3.00
Even though the implied volatility increased, the fact that the underlying futurescontract 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 futurescontracts) 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 willbe 1.86, and we will show a loss of
10x (1.86 - 2.17) =-3.10
Suppose, however, that the decline in implied volatility is accompanied by a swift movein the underlying contract. If the underlying futures contract falls to 93.00, and impliedvolatility falls to 18%, the price of the 100 call will be .59, and we will show a profit of
4 x (97.73 — 93.00) — 10 x (2.17 — 59) = +3.12
On the other hand, if the underlying futures contract rises to 102.50 while impliedvolatility falls to 18%, the price of the 100 call will be 4.32, and we will again show aprofit
4 x (97.73 - 102.50) + 10 x (4.32 - 2.17) = +2.42Volatility Revistted 295
In both cases the movement in the underlying contract has more than offset any declinein the option’s price due to a decline in implied volatility.
The foregoing examples are of course simplified. As market conditions change-anactive trader may very well make adjustments to his position in order to remain deltaneutral. If so, the actual profit or loss will be affected by the cash flow from thisadjustment process. The important point is that the volatility of the underlying contract,whether the contract moves or sits still over time, will eventually overwhelm anychanges 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 ofan option is determined by the volatility of the underlying contract over the life of theoption.
Implied versus Future Volatility
If, as many traders believe, prices in the marketplace reflect all available informationaffecting the value of a contract, the best predictor of the future volatility ought to bethe 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 wouldrequire a detailed study of many markets over long periods of time, we might still gainsome insight by looking at a limited number of examples.
Clearly, no one knows the future volatility. We can, however, record the impliedvolatility at any moment in time and then, when expiration arrives, look back andcalculate the actual volatility of the underlying contract between the time we recordedthe 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 volatilitythat occurred over this period. This has been done in Figures 1 4-9A, 9B, and 9C for theJune 1992, March 1993, and December 1993 options on Treasury Bond futures. It isadmittedly dangerous to generalize from such limited data, but are there any conclu-sions we might draw from these graphs?
We can see that with a great deal of time remaining to expiration, the futurevolatility of the underlying contract (the solid line) is relatively stable. But as we getCloser to expiration the future volatility can become much less stable. This is logical ifWe again recall that the mean reverting characteristics of volatility are much less certainover short periods of time than over long periods. One large move in the underlyingcontract with only several days remaining to expiration will result in a sharp increase inthe volatility to expiration (see Figure 14-9C). On the other hand, if the underlyingcontract is relatively quiet over the last several days in an option’s life, the volatility toexpiration will collapse (see Figure 14-9B).
How does the marketplace react to these volatility characteristics? With longPeriods of time remaining to expiration, the volatility to expiration is relatively stable.One would therefore expect the implied volatility to also be relatively stable. Con-versely, with short periods of time remaining to expiration, the volatility to expirationCan be very unstable, and one would therefore expect the implied volatility to beUnstable. These conclusions are born out by the implied volatility (the broken line) in296
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Figures 14-9A, 9B, and 9C. Over long periods of time, the marketplace is reacting tomany events. This is easier than reacting to a limited number of events, which is whatthe marketplace is faced with over a short period of time. The marketplace knows thatthe laws of probability are more likely to balance out over many occurrences than overonly a few occurrences.
Note also that there is no guarantee that the marketplace will have the correctimplied volatility. The implied volatility is a guess, and guesses carry with them thepossibility of error, sometimes very large error. In Figure 14-9B the implied volatilityturned out to be much too high over almost all of the option’s life. Had a trader soldpremium at any time, he would have shown a profit. At its most extreme, duringOctober 1992, there was almost a four-percentage-point difference between the impliedvolatility of the March 1993 options and the future volatility of the March 1993 futurescontract. In Figure 1 4-9C the implied volatility was too high during the early part of theoption’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, inFigure 14-9A the implied volatility was relatively accurate over the early part of theoption’s life, but too high during the latter part.
It should be clear by now that dealing with volatility is a difficult task. To facilitatethe decision-making process we have attempted to make some generalizations aboutvolatility characteristics. Even then, if one decides to become involved in a market itmay not be at all clear what the right strategy is. Moreover, we have looked at a limitednumber of examples, making the generalizations even less reliable. Every market has itsown characteristics, and knowing the volatility characteristics of a particular market,whether interest rates, foreign currencies, stocks, or a physical commodity, is at least asimportant as knowing the technical characteristics of volatility. And this knowledge canonly come from careful study of a market combined with actual trading experience.Volatility Revisited 299
Figures 14-9A, 9B, and 9C. Over long periods of time, the marketplace is reacting tomany events. This is easier than reacting to a limited number of events, which is whatthe marketplace is faced with over a short period of time. The marketplace knows thatthe laws of probability are more likely to balance out over many occurrences than overonly a few occurrences.
Note also that there is no guarantee that the marketplace will have the correctimplied volatility. The implied volatility is a guess, and guesses carry with them thepossibility of error, sometimes very large error. In Figure 14-9B the implied volatilityturned out to be much too high over almost all of the option’s life. Had a trader soldpremium at any time, he would have shown a profit. At its most extreme, duringOctober 1992, there was almost a four-percentage-point difference between the impliedvolatility of the March 1993 options and the future volatility of the March 1993 futurescontract. In Figure 1 4-9C the implied volatility was too high during the early part of theoption’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, inFigure 14-9A the implied volatility was relatively accurate over the early part of theoption’s life, but too high during the latter part.
It should be clear by now that dealing with volatility is a difficult task. To facilitatethe decision-making process we have attempted to make some generalizations aboutvolatility characteristics. Even then, if one decides to become involved in a market itmay not be at all clear what the right strategy is. Moreover, we have looked at a limitednumber of examples, making the generalizations even less reliable. Every market has itsown characteristics, and knowing the volatility characteristics of a particular market,whether interest rates, foreign currencies, stocks, or a physical commodity, is at least asimportant as knowing the technical characteristics of volatility. And this knowledge canonly come from careful study of a market combined with actual trading experience.a 15 &
Stock Index Futuresand Options
Since their introduction on U.S. exchanges in the early 1980s, stock index futures andoptions have experienced phenomenal growth, with increasing participation by allsegments of the trading community.