the at-the-money option is the most sensitive to changes involatility.
As is obvious, using options to hedge a position can be just as complex as usingoptions to construct trading strategies. Many factors go into the decision-makingprocess. When a potential hedger is confronted for the first time with the multitude of268 Chapter 13
possible strategies, he can understandably feel overwhelmed, to the point where hedecides to abandon options completely. When this happens the hedger, or someoneadvising the hedger, should lay out a limited number of strategies (perhaps four or five)which make sense, and compare the various risk/reward characteristics of the strate-gies. Given the hedger’s general market outlook, and his willingness or unwillingnessto accept certain risks, it should then be possible to make an intelligent decision. Asummary of common hedging strategies, with some of their advantages and disadvan-tages, is given in Figure 13-7.
PORTFOLIO INSURANCE
Suppose a hedger holds a long position in a certain asset, and he wants to protect hisholding against downward movement by purchasing a put option. Unfortunately, thehedger finds that no option market for this asset exists. Nor is any other source, such asan option dealer, available. Is there any way the hedger can create the put optionhimself?
If the hedger were really able to purchase a put, his position would be:
long underlying + long put
But we know from the synthetic relationships in Chapter 11 that a long underlyingposition together with a long put is equivalent to a long call. Therefore, the hedger reallywants to own a call with the same exercise price as the put.
What would be the characteristics of this call? We could analyze the call using atheoretical pricing model. In order to do that, we need to know what the underlyingasset is, and then we need to determine the inputs into the model. Suppose theunderlying asset is a futures contract, and that we plan to use the following inputs:
exercise price = 100
time to expiration = 10 weeksunderlying price = 101.35interest rate = 8%
volatility = 18.3%
Feeding these inputs into a theoretical pricing model, we find that the call option whichthe hedger would like to own has a delta of 57. This means that if the hedger actuallyowned the call (instead of owning the underlying asset together with a 100 put) intheory he would have a position equivalent to owning 57% of his asset. Therefore, if hewants to replicate the combination of the underlying asset and the 100 put, he must selloff 43% of his holdings in the asset. When he does that, he will have a positiontheoretically equivalent to owning a 100 call.
Now suppose a week passes and the value of his asset has risen to 102.26. Whatwould be the delta of the 100 call which the hedger desires to own? Again, feeding allthe inputs into a theoretical pricing model, we find that the call now has a delta of 62.If the hedger wants a position which is equivalent to a long call, he now needs to own 
269
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 270 Chapter 13
62% of his original holding in the underlying asset. He must therefore buy back 5% ofhis original holding in the asset.
Suppose another week passes and the value of the asset has fallen to 99.07. Usingthe new market conditions, the 100 call would now have a delta of 46. In order toreplicate the call position, the hedger must now sell off 16% of the asset, so that he ownsonly 46% of his original holding.
Notice what the hedger is doing. He is periodically rehedging his original holdingto create a position with the same delta characteristics as the call. If he continues to dothis over the ten-week period, he has, in effect, created a ten-week call at an exerciseprice of 100.
The reader may have already realized that we are simply repeating the example inFigure 5-1, but in a slightly different form. In that example, we purchased an under-priced call and replicated the sale of the same call through a continuous rehedgingprocess. In our current example we are also replicating the call through a continuousrehedging process. But here we want to purchase the call rather than sell it. All ouradjustments are therefore the opposite of those in Figure 5-1. Note also that at week ten(expiration), instead of liquidating the position, as we did in Figure 5-1, we plan to buyin the remainder of the asset. It was always our intention to retain our entire holding ofthe asset at expiration. The replicated position in our current example is shown in Figure13-8.
In order to replicate the call, the hedger will have to buy back some of the assetwhen the market rises and sell some of the asset when the market falls. The unfavorableadjustments (buying high, selling low) suggests that there will be a cost associated withthis rehedging process. What will the cost be? If the replication process offers the sameprotection as owning a 100 put, then one might expect the cost of the replicationProcess to be the same as the theoretical value of the ten week 100 put. This ts indeedthe case. Using the previously listed inputs, the Black-Scholes model yields a theoreticalvalue for the 100 put of 2.55. This is identical to the cost of the replication processdescribed in Figure 13-8.
The rehedging process is an attempt to replicate the automatic rehedging charac-teristics of the option. As a hedge against an underlying position, an option has the verydesirable characteristic of offering greater protection as the market moves adversely, andless protection as the market moves favorably. In effect, the option automaticallyrehedges itself to fit the required amount of protection. This is the influence of thegamma on the option’s delta. When a hedger attempts to replicate the characteristics ofan option by continuously rehedging the underlying position himself, he is simply tryingto replicate the automatic rehedging (gamma) characteristics of the option. The cost ofthe option, or the cost of the rehedging process, is the price the hedger pays for theoption’s gamma characteristics.
Since we need to continuously compute the delta to replicate an option through arehedging process, this method of replication requires us to use a theoretical pricingmodel. We are therefore likely to encounter many of the same problems which weencounter anytime we use a theoretical pricing model to evaluate an option. Is themodel itself correct? Do we have the right inputs? Leaving aside the question of theHedging with Options 271
 
Figure 13-8% of AssetAsset Delta of Needed for RequiredWeek Value 100 call Replication Adjustment0 1014.35 57 57% sell 43%4 102.26 62 62% buy 5%2 99.07 46 46% sell 16%3 400.39 53 53% buy 7%4 400.76 56 56% buy 3%5 103.59 74 74% buy 18%6 99,26 45 45% sell 29%7 98.28 35 38% sell 10%8 99.98 50 50% buy 15%9 103,78 93 93% buy 43%10 102.54 100 100% buy 7%
 
model’s correctness, anyone wishing to use this method will have to come up with areasonable volatility estimate. If the volatility estimate turns out to be too high or toolow, the cost of replicating the option will be more or less than originally expected. Butnotice that the cost of replication will always be the right cost. If the market is morevolatile than expected, the cost will be greater in terms of required adjustments. But thevalue of the option would also have been greater. Higher volatility means a highertheoretical value. In the same way, if the market is less volatile than expected, the costwill be less in terms of required adjustments. The value of the option would also havebeen less. Lower volatility means a lower theoretical value.
This process of continuously rehed ging an underlying position to replicate an optionposition is often referred to as portfolio insurance. The method can be used to protectany long or short position against adverse movement, but it is most commonly used byfund managers who wish to insure the value of the securities in a portfolio against adrop in value. For example, suppose a manager has a portfolio of securities currentlyvalued at $100 million. If he wants to insure the value of the portfolio against a drop invalue below $90 million, he can either buy a $90 million put, or he can replicate thecharacteristics of a $90 million call, If he is unable to find someone willing to sell hima $90 million put, he can evaluate the characteristics of the $90 million call andcontinuously buy or sell the portion of his portfolio required to replicate the call position.In effect, he has created his own put.
Unfortunately, the costs of buying and selling a large number of securities in oddamounts may be quite high. While the concept of portfolio insurance may be attractivefrom a theoretical perspective, the transaction costs may make the process too expensiveto be of practical value. Is there any way a fund manager can use portfolio insurancewhile keeping transaction costs within acceptable bounds? One common methodinvolves using futures contracts as a substitute for the portfolio holding. If the mix ofSecurities in a portfolio approximates an index, and futures contracts are available onthat index, the manager can approximate the results of portfolio insurance by purchas-ing or selling futures contracts to increase or decrease the holdings in his portfolio. In272 Chapter 13
the United States several index futures are available, the most popular being the futurescontract on the S&P 500 index traded at the Chicago Mercantile Exchange. Many fundManagers use S&P 500 futures to insure their portfolios against adverse movement inthe stock market. This method is not without risk, since the S&P 500 index is unlikelyto exactly duplicate the holdings of any one portfolio. Still, the risk may be acceptablegiven the reduced transaction costs.
Even if options are available on an underlying asset, a hedger may still choose toeffect a portfolio insurance strategy himself rather then purchasing the option in themarketplace. For one thing, he may consider the option too expensive. If he believesthe option is theoretically overpriced, in the long run it will be cheaper to continuouslyrehedge the portfolio. Or he may find insufficient liquidity in the option market to absorbthe number of option contracts he needs to hedge his position. Finally, the expirationof options which are available may not exactly correspond to the period over which hewants to protect his position. If an option is available, but expires earlier than desired,the hedger might still choose to purchase options in the marketplace, and then pursuea portfolio insurance strategy over the period following the option’s expiration. For allthese reasons, portfolio insurance has become an increasingly popular method ofprotecting a position in an underlying asset.a 14 %
Volatility Revisited
While there are many ways in which traders interpret volatility, in theory the value ofan option depends on only one volatility, the volatility of the underlying contract whichwill occur over the life of the option. Of course, this volatility is unknown to a tradersince it will occur in the future. Nevertheless, if a trader wants to use a theoreticalpricing model, he will be required to make some prediction about the volatility of theunderlying 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 movementsthrough technical analysis is a commonly studied area in trading, and there are manysources 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 spiteof this difficulty, an option trader must make some effort to come up with a reasonablevolatility input if he is going to rely on a theoretical pricing model to make intelligenttrading decisions.
SOME VOLATILITY CHARACTERISTICS
The first step in making a sensible volatility forecast is to understand some of the basiccharacteristics of volatility. First, let’s compare two graphs. Figure 14-1 shows Deutsche-mark prices from 1982 to 1991. Figure 14-2 shows the 50-day volatility of Deutsche-marks over the same period. Are there any generalizations one can make from thesegraphs? Obviously both prices and volatility sometimes rise and sometimes fall. Butunlike the prices of an underlying instrument, which appear to move freely in eitherdirection, there seems to be an equilibrium number to which the volatility alwaysreturns. Over a period of three years, from early 1985 to late 1987, the price ofDeutschemarks 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 causeDeutschemarks to rise or fall dramatically, never again returning to prices in the 50s or60s. In other words, prices of an underlying contract are open-ended. There is no reasonWhy they have to return to some previous level.
Such does not appear to be the case with volatility. The 50-day volatility ofDeutschemarks over the 10-year period in question fluctuated from a low of 5% to ahigh of 20%. Yet no matter how much it fluctuated, at some point volatility alwaysTeversed itself and retraced almost all of its previous rise or fall. Indeed, we might try tofind an equilibrium volatility such that there are equal fluctuations above and below this
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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 italways seems to return to this area.
If we were to generalize about volatility characteristics from the Deutschemarkvolatility in Figure 14-2, we might surmise that an underlying contract is likely to havea typical long-term average, or mean volatility. Moreover, the volatility of the underlyingcontract appears to be mean reverting. When volatility rises above the mean, one canbe fairly certain that it will eventually fall back to its mean; when volatility falls belowthe mean, one can be fairly certain that it will eventually rise to its mean. There is aconstant gyration back and forth through this mean.
This mean reverting characteristic of volatility can also be seen in Figure 143,which shows the distribution of Deutschemark volatility from 1982 to 1991. Beginningat the extreme left on the x-axis (the amount of time remaining to expiration), we cansee 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 10thand 90th percentiles}. There was a 50% chance that volatility would be either less than7.6% or greater than 13.0% (the 25th and 75th percentiles]. The mean volatility for anytwo week period was 9.7%. Moving to the extreme right on the x-axis, we can see thatover any 50 week period there was a 20% chance that volatility would be either lessthan 9.8% or greater than 14%. There was a 50% chance that volatility would be eitherless than 10.6% and greater than 12.7%. The mean volatility for any 50-week periodwas about 11.5%.
Figure 14-3 has an easily identifiable structure. As one moves further out in timethe 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 ofvolatility graph, sometimes referred to as a volatility cone, is an effective method ofpresenting the volatility characteristics of an underlying instrument.!
What else can we say about volatility? Looking at the more detailed Deutschemarkvolatility chart in Figure 14-4, we might surmise that volatility has some trendingcharacteristics. From July 1989 through June 1990 there was a downward trend involatility. From July 1990 to April 1991 there was an upward trend. And from April1991 to October 1991 there was again a downward trend. Moreover, within thesemajor 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 pricecharts, and it would not be unreasonable to apply some of the same principles used intechnical analysis to volatility analysis. It is important to remember, however, that whileprice changes and volatility are related, they are not the same thing. If a trader tries toapply exactly the same rules of technical analysis to volatility analysis, he is likely to findthat in some cases the rules have no relevance, and that in other cases the rules mustbe modified to take into account the unique characteristics of volatility. Since the author
 
1. For amare detailed discussion of volatility cones, see:Burghardt, Galen and Lane, Morton; “How to Tell if Options are Cheap”; The Journal of PortfolioManagement, Winter 1990, pages 72-78.276
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claims no particular expertise in the area of technical analysis, the reader is left to hisown devices in this regard.
VOLATILITY FORECASTING
Given the volatility characteristics that we have identified, how might we go aboutmaking a volatility forecast? First we need some volatility data. Suppose we have thefollowing 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 dataavailable, how might we use it to make a forecast? One method might be to simply takethe 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 bereasonable to assume that some data is more important than other data? A trader mightassume, for example, that the more current the data, the greater its importance. Sincethe 24% volatility over the last 30 days is clearly more current than the other volatilitydata, perhaps 24% should play a greater role in our volatility forecast. We might, forexample, 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 recentdata.
Of course, if it is true that the more recent volatility over the last 30 days is moreimportant than the other data, it follows that the volatility over the last 60 days oughtto be more important than the volatility over the last 120 days and 250 days. It alsofollows that the volatility over the last 120 days must be more important than thevolatility over the last 250 days. We can factor this into our forecast by using a regressiveweighting, giving more distant volatility data progressively less weight in our forecast.For example, we might calculate:
(40% x 24%) + (30% x 20%} + (20% x 18%) + (10% x 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 itsimportance. Is this always true? If we are interested in evaluating short-term options, itmay be true that data which covers short periods of time is the most important. ButSuppose we are interested in evaluating very long-term options. Over long periods of280 Chapter 14
time the mean reverting characteristic of volatility is likely to reduce the importance ofany short-term fluctuations in volatility. In fact, over very long periods of time the mostreasonable 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 theamount of time remaining to expiration for the options in which we are interested.
Ina 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 mostimportant? In addition to the mean reverting characteristic, volatility also tends toexhibit serial correlatton. The volatility over any given period is likely to depend on, orcorrelate with, the volatility over the previous period, assuming that both periods coverthe 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 awayfrom 15%. We can again use the weather analogy from Chapter 4. If the hightemperature yesterday was 25°, and we had to guess what the high temperature todaywould be, a guess of 30° would make more sense than a guess of 50°. Once we realizethis, we might logically choose to give the greatest weight to the volatility data coveringa time period closest to the life of the options in which we are interested. That is, if weare trading very long-term options, the long-term data should get the most weight. If weare trading very short-term options, the short-term data should get the most weight. Andif we are trading intermediate-term options, the intermediate-term data should get themost weight.
Suppose we are interested in evaluating six-month options. How should we weightour data? Since 120 (trading) days is closest to six months, we can give the 120-day datathe greatest weight, and give other data correspondingly lesser weight:
(15% x 24%} + (25% x 20%) + (35% x 18%} + (25% x 18%} = 19.4%
Alternatively, if we are interested in 10-week options, we can give the greatestweight to the 60-day volatility data:
(25% x 24%) + (35% x 20%) + (25% x 18%) + (15% x 18%] = 20.2%
In the foregoing examples we used only four historical volatilities, but the morevolatility 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 ofthe volatility characteristics of an underlying instrument, but it will enable a trader tomore closely match historical volatilities to options with different amounts of time toexpiration. In our example we used historical volatilities over the last 60 days and 120days 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 a10-week period.
The method we have described is one which many traders intuitively use toforecast volatility. It depends on identifying the typical characteristics of volatility, andthen projecting a volatility over the forecasting period. Theoreticians have recently triedto take essentially the same approach to volatility forecasting, and this has led to thedevelopment of autoregressive conditional heteroskedasticity (ARCH) and generalizedVolatility Revisited 281
autoregressive conditional heteroskedasticity (GARCH) volatility models. A detaileddiscussion of such models is beyond the scope of this text, since they can be mathemati-cally complex and are not widely used amang traders. Nevertheless, an option tradershould be aware that these models do exist, and that they are simply an attempt to applythe mean reverting and serial correlation characteristics of volatility to volatility forecast-ing.”Thus far we have focused only on the historical volatility characteristics of theunderlying instrument in trying to forecast a volatility. Is there any other informationwhich might be useful? No individual trader can hope to know everything affectingprice changes in an underlying contract. Perhaps there are factors which could affectthe future volatility, but about which the trader is unaware. If one believes that suchinformation is likely to be reflected in the prices of the contracts being traded, one wayto ferret out additional volatility information is to look at the prices of options. In otherwords, a trader will want to look at the implied volatility in the marketplace to find theconsensus volatility. Once he has done this, he will want to use this information in anyvolatility forecast he intends to make.
How much weight should a trader give the implied volatility? Some traderssubscribe to the efficient market principle, and believe that the implied volatility isalways the best volatility forecast since it reflects all available information. Most tradersbelieve, 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 between25% and 75% in making a volatility forecast. How much depends on the trader’sconfidence in forecasting a volatility based on historical volatility data? If a trader feelsvery 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 volat