this, the reader should look again at Figure 14-2.) But the theoretical pricing modelcannot differentiate between our two volatility scenarios. The model simply sees avolatility of 16.7% and assumes that all price changes are evenly distributed, andconsistent with this volatility, over the life of the option.
Suppose a trader were to buy a 105 straddle and assumed, correctly, a volatility of16.7%. If he bought and sold underlying contracts in order to remain delta neutral overthe life of the option, the model tells him, because it assumes a constant volatility of16.7%, that the profit or loss to the trader will be identical in either of our scenarios. Isthis likely to be the case?
Notice that the underlying market is moving through 105 both at the beginningand at the end of the options’ life. We can therefore consider the 105 call and put asbeing at-the-money during these periods. Since at-the-money options have the highestgamma, the 105 straddle will be a relatively high gamma position both early and late inthe spread’s life. Short-term at-the-money options always have greater gammas thanlong-term at-the-money options, so any increase in volatility which occurs during thelatter part of the option’s life, as it does in Figure 18-2b, will have a much greater impacton the position than volatility which occurs during the early part of the option’s life, asit does in Figure 18-2a.
Because the gamma of a position determines both the magnitude and frequency ofthe adjustment process, the 105 straddle will be much more valuable under themovement represented in Figure 18-2b. The movement back and forth through theModels and the Real World
391
 
 
115
Figure 18-2a
110
 
95392
 
 
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 Models and the Real World 393
exercise price late in the option’s life will require larger and larger adjustments, each ofwhich is favorable to the holder of the 105 straddle. The scenario in Figure 18-2a willalso require adjustments. But the adjustments now will be relatively small because themovement in the underlying contract is much less dramatic.
Through experience most traders have come to the conclusion that the order inwhich volatility occurs does matter. It matters especially for at-the money options,because they have the greatest gammas. Any period of high volatility which occurs nearexpiration will have a greater impact on an at-the-money option than the same highvolatility which occurs when the option has a great deal of time remaining to expiration.Consequently, even if one knows the actual volatility over the life of an option, a modelwill tend to undervalue at-the-money options in a rising volatility market, and overvalueat-the-money options in a falling volatility market.
We have only considered one alternative volatility scenario, where volatility iseither increasing or decreasing. But there are an infinite number of paths which volatilitymight follow over the life of an option. A trader might even assume that volatility is itselfrandom, and that predicting volatility with any degree of accuracy is not possible.Models which assume stochastic volatility do exist and might, under some conditions,be more suitable than a traditional pricing model. At the same time, such models addanother dimension of complexity to a trader’s life, and for this reason are not widelyused.
Some contracts, by their very nature, are known to change their volatility charac-teristics over time. Interest rate products, in particular, fall into this category. As a bondapproaches maturity, the price of the bond moves inexorably towards par. At maturity,regardless of interest rates, the bond will have a fixed and known value. Clearly, onecannot assume that the price of the bond follows a random walk through time. Even ifone assumes that interest rates move randomly, and that the volatility of interest ratesis constant, interest rate instruments will change their volatility over time becauseinstruments of different maturities have different sensitivities to changes in interestrates. If we take into consideration the fact that interest rates also vary for differentmaturities, a traditional Black-Scholes type model is obviously inadequate for theevaluation of such products. This has led to the development of special models toevaluate interest rate instruments.>
4. For amore detailed discussion see:Hull, John and White, Alan, “The Pricing of Options with Stochastic Volatilities,” Journal of Finance,Vol. 42, No. 2, June 1987, pages 281-300,Scott, Louis D., “Option Pricing when Variance Changes Randomly: Theory, Estimation, and Application,”Journal of Finance and Quantitative Analysis, No. 22, December 1987, pages 419-438,Wiggins, J.B., “Option Values under Stochastic Volatility: Theory and Empirical Results,” Journal ofFinancial Economics, No. 19, December 1987, pages 351-372.
5. See for example:Ho, Thomas, T.S.Y. and Lee, Sang-Bin, “Term Structure Movements and the Pricing of Interest RateContingent Claims,” Journal of Finance, Vol. 41, No. 5, December 1986, pages 1011-1029.Heath, David; Jarrow, Robert; and Morton, Andrew, “Bond Pricing and the Term Structure of InterestRates: A New Methodology,” Econometrica, Vol. 60, 1992, pages 77-105.394 Chapter 18
TRADING IS CONTINUOUS
A theoretician trying to develop a realistic price distribution for use in a theoreticalpricing model might first ask how the prices of an underlying contract change over time.Perhaps not all contracts change prices in the same way. Price changes might, forexample, follow a diffusion process {see Figure 18-3). In a diffusion process there aresmooth and continuous changes from one price to the next, with no gaps betweenconsecutive prices. Moreover, the longer the period of time between price readings, thegreater the dispersion in prices. An example of a typical diffusion process might be thetemperature readings in a specific location. While the temperature can change veryquickly, there will never be any gaps. If the temperature at one moment is 15° but atalater time is 18°, then at some intermediate time, even if only very briefly, thetemperature must have also been 16° and 17°. And there is likely to be a greatertemperature change over six weeks than over six days.
Prices might also follow a jump process. In a strict jump process, the price of acontract remains constant for a period of time, then instantaneously jumps to a newprice, where it again remains constant for some period of time. The way in which centralbanks set interest rates is typical of a jump process. In the United States, when theFederal Reserve sets the discount rate, it remains there until it is fixed at a new rate. Itremains at this new rate until the Federal Reserve again announces a change in thediscount rate. A jump process is a combination of fixed prices and instantaneous jumps.
Most theoretical pricing models assume that trading in the underlying contractfollows a diffusion process. Trading is assumed to be continuous, proceeding 24 hoursper day, seven days per week, without interruption. There can be no gaps in the priceof the underlying contract. If a contract trades at 46.05, and at some later time tradesat 46.08, then at some intermediate time it must have also traded, even if only briefly,at 46.06 and 46.07. If one were to draw a graph of price changes under a diffusionprocess, one would never lift the pencil from the paper.
A diffusion process is a convenient, but clearly inexact, approximation of howprices change in the real world. Exchange-traded contracts cannot follow a purediffusion process because exchanges are not open 24 hours per day. At the end of thetrading day a contract often closes at one price, and then opens the next day at a differentprice. This causes a price gap, something which a diffusion process does not permit.Even during normal trading hours prices may not follow a diffusion process. If majornews hits the market, the impact can be almost instantaneous, causing prices to gap upor down.
If a theoretical pricing model makes the assumption that prices follow a diffusionprocess, when in fact they don’t, how is this likely to affect values generated by themodel? Consider a situation where an underlying futures contract is trading at 100 anda trader feels that implied volatility in the option market is much too high. Because alloptions are overpriced, the trader decides to sell the 100 straddle. Unfortunately,immediately after he sells the straddle the market gaps up to 105. How will the gapaffect the trader’s position?Models and the Real World
 
Figure 18-3
39S396 Chapter 18
A trader who is short straddles does not want to see a gap in the market. The factthat a gap has occurred will hurt the trader, but will it cause him only minor pain, orwill it destroy him? If the options are relatively long-term, say nine months, the gap inthe underlying price is unlikely to be the end of the world. After all, with nine monthsremaining to expiration the underlying market could easily fall back to 100. And whilethe gap has clearly hurt the trader, it is probably not disastrous.® But if the gap occurswith only a very short time remaining to expiration, say one day, the trader is now inasituation where he might very well be ruined. With only one day to expiration, there issimply not enough time for the market to retrace its movement. The 100 calls whichthe trader sold as part of his short straddle will act like short futures contracts. Whilethe spread began approximately delta neutral, after the gap the trader will find himselfnaked short deeply in-the-money calls, each with a delta of 100, in a market which hasjust made a giant leap upward. Figure 18-4 shows how prices might change with ninemonths versus one day remaining to expiration.
With nine months remaining to expiration the 100 straddle increased in value byapproximately 7% (14.04/13.18 = 1.07). But with one day remaining to expiration the100 straddle increased in value by 495% (5.00/.84 = 5.95), The gap was painful withnine months to expiration, but it was devastating with one day to expiration.
The reason that the gap caused such a dramatic change in the one-day straddle wasthat the delta of the 100 straddle was much more sensitive to a change in the price ofthe underlying contract. With nine months to expiration the 100 call and put have agamma of only 2.2. But with one day to expiration they have a gamma of 38.1. In thelatter situation, with such a high negative gamma, any movement will undoubtedly bevery painful. The trader could reduce some of the damage if he were able to adjust theposition by purchasing futures as the market moved upward. But a gap is an instanta-neous move; there is no opportunity to adjust. The very high gamma, combined withan inability to make any adjustment, is what makes the gap so devastating.
Options have the unique characteristic of automatically, and continuously, rehedg-ing themselves by changing their deltas as the price of the underlying contract changes.It is this characteristic which buyers of options are paying for. A trader who uses atheoretical pricing model attempts to take advantage of a mispriced option by estab-
Figure 18-4Implied Volatility = 20%; Interest Rate = 6%
9 months to expiration 1 day to expiration
Futures Futures Futures Futuresat 100 at 105 at 100 at 105100 call price 6.59 9.41 42 5.00100 put price 6.59 4.63 42 0100 straddle price 13.18 14,04 84 5.00
 
8. For simplicity, we assume that there is no change in implied volatility after the gap, Admittedly, this Isunlikely to be the case.Models and the Real World 397
lishing a delta neutral hedge against the underlying, and then manually performing therehedging process himself over the life of the option. This was the basis for the examplesin Chapter 5. If amodel assumes, as most do, that prices follow a diffusion process, themodel assumes that one can continuously maintain a delta neutral hedge. But when themarket gaps, the assumptions on which the model is based are violated. Consequently,the values generated by the model are rendered invalid. This problem extends to anyapplication which attempts to replicate option characteristics through a continuousrehedging in the underlying market. The proponents of portfolio insurance suffered theirgreatest setbacks on October 19 and 20, 1987, when the market made several large gapmoves. Because of the gaps, the portfolio insurers were unable to make continuous deltaadjustments to their positions. As a consequence, they found that the cost of protectionoffered by portfolio insurance was much greater than they had ever expected.
Since traditional pricing models make an unrealistic assumption about how priceschange, perhaps a model which makes a more realistic assumption would yield moreaccurate theoretical values. Theoreticians tend to agree that underlying contracts inmost markets follow a combination of both a diffusion process and a jump process. Mostof the time prices change smoothly and continuously, with no gaps. Every now andthen, however, a gap will occur, instantaneously sending the price to a new level. Fromthere prices will again follow a smooth diffusion process until another gap occurs. If onecould generate a underlying price distribution which resulted from this jump-diffusionprocess, and feed this distribution into a theoretical pricing model, the model shouldgive more accurate values.
A variation of the Black-Scholes model which assumes that the underlying contractfollows a jump-diffusion process has in fact been developed. If used correctly, thisJump-diffusion model does give values which are probably more accurate than tradi-tional Black-Scholes values.” Unfortunately, the model is considerably more complexmathematically than the traditional Black-Scholes model. Moreover, in addition to thefive customary inputs, the model also requires two new inputs: the average size of ajump in the underlying market, and the frequency with which such jumps are likely tooccur. Unless the user can accurately estimate these new inputs, the values generatedby a jump-diffusion model may be no better, and might be worse, than those generatedby a traditional model. Most traders take the view that whatever weaknesses areencountered in a traditional model can be best offset through intelligent decision makingbased on actual trading experience, rather than through the use of a more complexjump-diffusion model.
Since a gap in the market will have its greatest effect on a high gamma option, andsince at-the-money options close to expiration have the highest gamma, it is theseoptions which are likely to be mispriced by a traditional theoretical pricing model withits continuous diffusion process. The closer one gets to expiration, the more suspect
 
7, Information on the jump-diffusion model can be found in most advanced texts on option theory. Foradditional information see:Merton, Robert, “Option Pricing when Underlying Stock Returns are Discontinous,” Journal ofFinancial Economics, Volume 3, March 1976, pages 125-144.Beckers, Stan, “A Note on Estimating the Parameters in the Jump-Diffusion Model of Stock Returns,”Journal of Financial and Quantitative Analysis, March 1981, pages 127—140,398 Chapter 18
model-generated values become. Consequently, traders pay less and less attention tomodel-generated values as expiration approaches. It is common practice among floortraders on option exchanges to take with them to the trading floor sheets withtheoretical values to enable them to make markets which are consistent with atheoretical pricing model. But as expiration approaches traders are more likely to discardtheir sheets because the values on the sheets become less reliable. With only one or twodays to expiration, most traders simply make decisions based on experience andintuition. While this is perhaps unscientific, it sometimes makes better sense thantrading from model-generated values when the model is known to be incorrect.
AS a result of the gaps which occur in the real world, both a trader’s experienceand empirical evidence seem to indicate that a traditional model, with its built-indiffusion assumption, tends to undervalue at-the-money options as expiration ap-proaches, From a risk standpoint, this means that it can be very dangerous to sell a largenumber of at-the-money options close to expiration, since any gap in the underlyingmarket can have devastating results. New traders, in particular, are advised to avoidsuch positions. No risk manager will appreciate even experienced traders being shortlarge numbers of at-the-money options as expiration approaches.
Expiration Straddles
If it is dangerous to sell atthe-money options going into expiration, perhaps there issome sense in taking the opposite position by purchasing at-the-money options asexpiration approaches. This may seem to contradict conventional option wisdom,which contends that a trader should sell at-the-money options in order to take advantageof the rapid time decay associated with such options. What traders tend to forget is thatthere is always a tradeoff between risk and reward. If one sells at-the-money options,the reward may be an accelerated profit if the market doesn’t move (high positive theta).But the risk is an increased loss if the market does move (high negative gamma). Becausethe model does not know about the possibility of a gap in the underlying market, therisk is often greater than the reward. If one sells at-the-money options, the losses froman occasional gap will more than offset the profits resulting from increased time decay.An experienced trader will therefore tend to do the opposite of conventional wisdom.If conditions are right, he will tend to buy at-the-money options close to expiration.
This is not to suggest that every time expiration approaches, a trader should runout and buy at-the-money options. As with any strategy, conditions must make thestrategy attractive. But because so many traders are intent on selling time premium asexpiration approaches, it is often possible to find cheap at-the-money options. Forexample, suppose that with three days remaining to expiration the Black-Scholes modelgenerates a value for an at-the-money call of .50, but that it is priced at .45. What canwe say about this call? While we may not know the exact value, we can certainly saythat the call is likely to be worth more than .50, since there is always the possibility ofa gap in the market. Clearly, if the call is trading at .45, but has a value in excess of .50,it must be a good buy.
As with any strategy based on volatility, the trader who buys these calls will try toestablish a delta neutral position. Because of the synthetic relationship, if the calls areModels and the Real World 399
underpriced the puts at the same exercise price will also be underpriced. A logicalstrategy might be the purchase of at-the-money straddles. This enables a trader to buypoth underpriced calls and underpriced puts, and to profit if the underlying market gapseither up or down.
In theory all volatility strategies, including an expiration straddle, ought to beperiodically adjusted in order to remain delta neutral. However, with little timeremaining to expiration the model is not only unreliable with respect to theoreticalvalues, it is also unreliable with respect to deltas. Because it is impossible to say whatthe right delta is, it is also impossible to say what the correct adjustment is. For thisreason, traders who initiate expiration straddles often forget about trying to adjust, andsimply sit on the position to expiration. This may not be the theoretically correct wayto handle a volatility position. But it may be the practical choice given all the uncertain-ties associated with theoretical evaluation as expiration approaches.
Even if a trader carefully chooses his expiration straddles, the great majority of timeno gap will occur in the market. In any single case he is more likely to show a loss thana profit. A trader needs to remember that the primary concern is not the profit or lossfrom any single trade, but what happens in the long run. Going Dack to the rouletteexample in Chapter 3, a player who buys a number at a roulette table can expect to winonly once in 38 plays. But if the theoretical value of the bet is 95¢, and the player canbuy the bet for less than 95¢, he expects to be a winner in the long run. If he pays avery low price for the bet, say 50¢, he still expects to lose 37 times out of 38. But nowthe bet is very attractive because even if he only wins once in 38 plays, this will still beoften enough to more than offset the small losses he takes each time he loses. The samelogic is true of expiration straddies. A trader may lose several times before winning.When he does win, he can expect a return great enough to more than offset all theprevious losses.
Because a trader knows that he will lose much more often than he will win, a tradershould only invest an amount in expiration straddles which he is prepared to lose. Butwhen conditions are right, a trader ought to be willing to make the investment. Even ifhe loses several times in succession, in the long run he will encounter gaps in the market,or sudden increases in volatility, often enough to make such strategies profitable.
VOLATILITY IS INDEPENDENTOF THE PRICE OF THEUNDERLYING CONTRACT
When a trader feeds a volatility into a theoretical pricing model, the volatility defines aOne standard deviation price change at any time during the life of the option, regardlessof whether the underlying contract happens to be rising or falling in price. If a contractis currently at 100, and we assume a volatility of 20%, a one standard deviation pricechange is always based on this volatility of 20%. If at some later time during the life ofthe option the contract should move up to 150 or down to 50, 20% is still assumed tobe the operative volatility.400 Chapter 18
Unfortunately, this assumption seems to fly in the face of most traders’ experience.If one were to ask a stock index or bond trader whether his market were more volatilewhen rising or falling, he would probably say that it was more volatile when falling. Onthe other hand, if one were to ask an agricultural or precious metal trader the samequestion about his market, he would almost certainly give the opposite answer. Thosemarkets tend to be more volatile when rising. In other words, the volatility of a marketis not independent of the price of the underlying contract. On the contrary, the volatilityover time seems to be dependent on the direction of movement in the underlyingcontract. In some cases a trader expects the market to become more volatile if themovement is downward and less volatile if the movement is upward; in other cases atrader expects the market to become more volatile if the movement is upward and lessvolatile if the movement is downward.
Because volatility in some markets does seem to be dependent on the price of theunderlying contract, a further variation of the Black-Scholes model has been proposed.The constant-elastictty of variance, or CEV, model® is based on an assumed relationshipbetween volatility and the price level of the underlying contract. This relationshipdetermines the probability of price moves of various magnitudes at each moment intime. Price changes are still random under a CEV assumption, but the randomness varieswith the price of the underlying contract.
Like the jump-diffusion model, the CEV model is mathematically complex, andtherefore difficult for most traders to compute. And like the jump-diffusion model itrequires additional input in the form of a mathematical relationship between thevolatility and price movement in the underlying contract. Given these difficulties, theCEV model has not found wide acceptance among option traders.
Over small periods of time the percent price changes in anunderlying contract are normally distributed, resulting in alognormal distribution of underlying prices at expiration
Does the real world look like a lognormal distribution? One way to answer the questionis to ask how percent price changes over small periods of time are distributed. If thisdistribution is normal, then the prices at expiration are likely to be lognormallydistributed.
Figure 18-Sa is a frequency distribution of daily price changes in the S&P 500 stockindex from 1989 through 1993. Each bar represents the number of occurrences of agiven price change to the nearest 14 percent. As one would expect, most of the changesare relatively small and close to zero. As we move away from the zero in either direction,
8. For information on the CEV model see:Cox, John C.; and Ross, Stephen A.; “The Valuation of Options for Alternative Stochastic Processes”;The Journal of Financial Economics, Volume 3, March 1976; pages 145-166.Beckers, Stan; “The Constant Elasticity of Variance Model and Its Implications for Option Pricing”;Journal of Finance; June 1980; pages 661-673.Schroder, Mark; “Computing the Constant Elasticity of Variance Option Pricing Model”; Journal ofFinance; Vol. 44, No. 1; March 1989; pages 211-219.Models and the Real World 401
Figure 18-5a: S&P 500 Stock Price Index,Percent Daily Price Changes 1989-1993
 
225
             
200-4
1754
Total number of days = 1265Largest upward move = +3,.73 %128-5 Largest downward move = -6.12 %Mean = +.044 %
Standard deviation = .795 %
100 Annualized volatility = 12.653 %Skewness = -.352Kurtosis = +4.343
normal distribution
Occurrences
 
 
 
we encounter fewer and fewer occurrences. The distribution certainly seems to havemany of the characteristics of a normal distribution. But is it really a normal distribution,and if it isn’t, how does it differ from such a distribution?
If the frequency distribution conforms exactly to a normal distribution, the tops ofthe bars should coincide exactly with a true normal distribution. To find out if this isthe case, the mean (+.044%) and standard deviation (.795%) have been calculated forall 1,265 daily price changes over the five-year period. From these numbers, a theoreti-cally exact normal distribution has been overlaid on our frequency chart. Note that theactual frequency distribution is similar to the normal distribution, but there are someclear differences. Because the bars representing the small price changes rise above thenormal distribution curve, there seem to be more small price changes than one wouldexpect from a true normal distribution. There are also several large price changes, oroutliers, which rise above the extreme tails of the normal distribution. These outliersseem to indicate that there are more large moves in our frequency distribution than onewould expect from a true normal distribution. Finally, in the midsections, between thepeak of the distribution and the extreme tails, there seem to be fewer occurrences thanone would expect.
One might at first surmise that differences in Figure 18-5a between the S&P 500frequency distribution and the true normal distribution are either unique to the S&P500, or an aberration of the five-year period in question. It turns out, however, that price402 Chapter 18
change distributions for almost all underlying markets exhibit characteristics which arevery similar to our S&P 500 distribution. There are usually more days with small moves,more days with large moves, and fewer days with intermediate moves than arepredicted by a true normal distribution. These differences between the real world anda theoretical distribution can also be seen in Figures 18-5b and 18-5c, the distributionof price changes for German Bonds and soybeans over the same five-year period.
SKEWNESS AND KURTOSIS
Distributions such as those in Figures 18-5a, 18-5b, and 18-5Sc are approximatelynormal, but still differ from a true normal distribution. If one is trying to make decisionsbased on the characteristics of a distribution, it might be useful to know how the actualdistribution differs from the normal. A perfectly normal distribution can be fullydescribed by its mean and standard deviation. But two other numbers, the skewnessand kurtosis, are often used to describe the extent of the difference between an actualfrequency distribution and a true normal distribution.
The skewness of a distribution (Figure 18-6a) can be thought of as the lopsidednessof the distribution, or the extent to which one tail is longer than the other tail.9 If adistribution is positively skewed, the right-hand tail is longer than the left-hand tail. If
Figure 18-5b: German Bonds, Percent Daily Price Changes 1989-1993
225
 
Total number of days = 1261Largest upward move = +2.323 %Largest downward move = -1.930 %Mean = +.0038 %
Standard deviation = .3265 %Annulaized volatility = 5.183 %Skewness = -.009
Kurtosis = 6.473
            
2007
17575
15075
125-4)
1007) normal distribution
Occurrences
754
50>
254
 
 
9. The calculation of skewness and kurtosis can be found in Appendix B.Models and the Real World
403
Figure 18-5c: Soybeans, Percent Daily Changes 1989-1993
 
150
425- Total number of days = 1262Largest upward move = +7.75 %Largest downward move = -7.12 %Mean = -.020 %Standard deviation = 1.202 %
1005 Annulaized volatility = 19.084 %Skewness = +.152Kurtosis = +4.606
754
normal distributionco : Pw
25>
Occurrences
 
Figure 18-6a: Skewness
     
 
+2 +4 +6
 
Positive Skewness(longer right tail)
 
Negative Skewness(longer left tail)
 
 
 404 Chapter 18
the distribution is negatively skewed, the left-hand tail is longer than the right-hand tail.A true normal distribution has a skewness of zero. The frequency distributions in Figures18-5a (S&P 500) and 18-5b (German Bonds) are slightly negatively skewed, while thedistribution in Figure 18-Sc (soybeans) is slightly positively skewed.
The kurtosis of a distribution (Figure 18-6b) is the extent to which the peak of thedistribution is unusually tall and pointed, or low and flat. A distribution with a positivekurtosis has a tall, pointed peak (/eptokurtic), while a distribution with a negativekurtosis has a low, flat peak (platykurtic). A perfectly normal distribution has a kurtosisof zero (mesokurtic).
Note that a distribution with a low standard deviation also has a pointed peak, justas a distribution with positive kurtosis. But a distribution with a low standard deviationalso has narrow tails, while a distribution with a positive kurtosis has long tails. Apositive kurtosis distribution is also narrower, or squeezed inward, in the midsectionbetween its peak and tails. The frequency distributions in Figures 18-5a, 18-Sb, and18-5c exhibit the same positive kurtosis, as do almost all underlying markets. They havehigher peaks (more days with small moves), more elongated tails (more days with bigmoves), and narrower midsections (fewer days with intermediate moves) than arepredicted by a true normal distribution.
Figure 18-6b: Kurtosis
 
Positive Kurtosis (leptokurtic)a —i \
Neutral Kurtosis (meésokurtic) I \
   
Negative Kurtosis (platykurtic)
 
 
 Models and the Real World 405
VOLATILITY