SKEWS
There are clearly real problems associated with the use of a traditional theoretical pricingmodel. Markets are not frictionless; prices do not always follow a diffusion process;volatility may vary over the life of an option; the real world may not look like a lognormaldistribution. With all these weaknesses, one might wonder whether theoretical pricingmodels have any practical value at all. In fact, most traders have found that pricingmodels, while clearly not perfect, are an invaluable tool for making decisions in theoption market. While models don’t always work perfectiy, they have proven to be farbetter than any other method of evaluating options.
Still, a trader who wants to make the best possible decisions cannot afford to ignorethe problems associated with a theoretical pricing model. Consequently, a trader whouses a pricing model might look for a way to reduce the potential errors resulting fromthese weaknesses. Initially, one might simply look for a better theoretical pricing model.If such a model exists, then it will certainly be worth replacing the old model with thenew one. But “better” is a relative term. A model might be better in the sense that itgives slightly more accurate theoretical values. But if the model is extremely complexand difficult to use, or if it requires additional inputs of which a trader cannot always becertain, then the model may merely substitute one set of problems for another. Giventhe fact that most traders are not theoreticians, a more realistic solution might be to usea less complex model and somehow fine-tune it so that it is consistent with the realitiesof the marketplace.
A trader trying to offset weaknesses in a pricing model might make the assumptionthat the marketplace is using the same model as the trader, and then ask how themarketplace is dealing with the weaknesses in the model. This is somewhat analogousto calculating implied volatility where we assumed that everyone was using the samemodel, that the price of the option was known, and that everyone agreed on all theinputs except volatility. From these assumptions we were able to determine theVolatility that the marketplace was implying to the underlying contract. Now we wantto take the same general approach, but ask what weaknesses the marketplace isimplying to the model. This is a more difficult question to answer.
Figure 18-7 shows the settlement prices and implied volatilities for Decemberoptions on German Government Bond (Bund) futures traded on the London Interna-tional Financial Futures and Options Exchange (LIFFE) on July 17, 1992. Since optionson LIFFE are settled like futures contracts, the effective interest rate for theoreticalcalculations is zero. As a result, there is no economic value to early exercise, and onecan calculate all implied volatilities using the Black-Scholes model.
By most standards German Government Bonds are a very low-volatility contract.Even so, the numbers do reflect many of the characteristics of option markets withwhich traders have become familiar. Note that the implied volatilities for calls and putsat the same exercise price are always the same. If this were not true, the prices wouldreflect an arbitrage opportunity. A trader could profitably execute a conversion if thecalls were overpriced with respect to the puts; a trader could profitably execute a reverseconversion if the puts were overpriced with respect to the calls. The fact that call and406 Chapter 18
 
Figure 18-7
July 27, 1992; German Goverment Bond (Bund) FuturesDecember Futures Contract = 87.86; Days to Expiration = 119; Interest Rate = 0%
Exercise Cail Cail implied Put PutPrice Price Volatility Price impiledVolatility84.50 3.40 3.98 04 3.9885,00 2.92 3.82 .06 3.8285.50 2.46 3.76 10 3.7686.00 2.02 3.68 16 3.6886.50 1.64 3.62 25 3.6287.00 1.25 3.62 .39 3.6287.50 .93 3.58 ST 3.5888.00 67 3.58 81 3.5888.50 46 3.56 1100 3.5689.00 cc 3.60 1.45 3.6089.50 20 3.62 1.84 3.6290.00 13 3.69 2.27 3.6990.50 .08 3.73 2.72 3.7391,00 05 3.81 3.19 3.8191,50 .03 3.87 : 3.67 3.87
put implied volatilities all line up indicates an efficient market with respect to arbitragerelationships, !2
Unfortunately, the implied volatilities at different exercise prices do not line up, andthis presents a problem to someone who believes that the Black-Scholes model is onehundred percent efficient. Can those traders trading the 89 call or put really believe thatthe December futures contract will have a volatility of 3.60, while those trading the 85call or put believe the December contract will have a volatility of 3.82?
If we assume that the exercise price, time to expiration, underlying price, andinterest rate are known, the theoretical value of an option in a Black-Scholes world willdepend solely on the volatility of the underlying contract over the life of the option.There can be only one volatility over this period. It’s true we won’t know what thatvolatility is until expiration arrives, at which point we can look back in time to see whatthe volatility was. But the December Bund future can have only one volatility over itslife. Since all the December Bund options have the same underlying contract, it doesn’tmake sense for every exercise price to have a different implied volatility. A true adherentof the Black-Scholes model would plugin his best guess about volatility and begin sellingthose options which appeared to be overpriced and buying those options whichappeared to be underpriced. We might think of the marketplace as one big trader. If theactivity in the marketplace were a result of everyone believing in the efficiency of the
12.  Settiement prices, such as those in Figure 18-7, are intended to reflect an efficient market with respectto arbitrage relationships. During the trading day, because there is a bid/ask spread for each option,and because all trades do not take place simultaneously, markets may appear less efficient.Models and the Rea! World 407
Black-Scholes model, the selling of overpriced options and the buying of underpricedoptions would eventually cause the implied volatility of every option to line up exactly.Yet this almost never happens in any market.
The marketplace, like every individual trader, is trying to evaluate options asefficiently as possible, given all the information available. Whether one believes thatmarkets are efficient or not, one can certainly argue that the marketplace is at least tryingto be efficient. From the wide range of implied volatilities found in almost every optionmarket, we can reasonably infer that the marketplace does not think the Black-Scholesmodel is one hundred percent efficient. Unfortunately, trying to identify the source ofthe inefficiency may not be possible. It might have to do with the fact that markets arenot frictionless; or the fact that prices do not always follow a diffusion process; or thatvolatility may vary over the life of an option; or that the real world may not look like alognormal distribution. Whatever the reason, the marketplace believes that options atany moment are priced efficiently, even if those prices happen to differ from model-gen-erated values.
For traders using a theoretical pricing model, prices are often expressed in terms oftheir implied volatilities. A trader who wants to continue using a theoretical pricingmodel, but wants to use it ina way which is consistent with the way the marketplaceis apparently using the model, might look to the distribution of implied volatilities foradditional information. He might begin by plotting the implied volatilities acrossdifferent exercise prices, and then draw a graph that fits the data points. This has beendone in Figure 18-8a for December Bund options.
The resulting graph, usually referred to as the volatility skew, has a well definedshape. The low point of the graph is close to the price of the underlying contract (87.86),and the tails of the graph rise as we move to exercise prices further away from theunderlying price. One inference which most traders draw from this shape is that themarketplace appears to believe there is a better chance for a large price move in the realworld than is predicted by the Black-Scholes model. This is not unreasonable if we lookagain at Figures 18-5a, 18-5b, and 18-5c. The outliers seem to indicate that large movesoccur more frequently in the real world than is predicted by a normal distribution. It isthe possibility of these large moves which apparently causes the marketplace to inflatethe prices of options which are far out-of-the-money or deeply in-the-money.
We can also see that some options are clearly too expensive or too cheap withrespect to their neighbors. We may not know exactly what the 88.50 call is worth, butwe can see that it is clearly too cheap with respect to the 88 and 89 calls. In the sameway, we may not know what the 87 put is worth. But it is clearly too expensive withrespect to the 86.50 and 87,50 puts. In the former case, a trader might try to sell the88.50/89.00/89.50 butterfly. In the latter case, a trader might try to buy the86.50/87.00/87.50 butterfly. Whatever we think of the skew, there ought to be asmooth progression of implied volatilities from one exercise price to the next.
A trader using the Black-Scholes model might take the view that the volatility skewcontains useful information which can be incorporated into the decision-making proc-ess. While he might believe that options are generally either overpriced or underpriced,he might also believe that the volatility skew reflects the relative value of options. He408 Chapter 18
 
Figure 18-8a4.00
 
3.955
 
 
3.904
 
3.864
 
380+
 
 
3.757
Implied Volatility
 
3.707
3.65
 
 
" 9.60
 
 
 
 
 
 
 
 
 
 
 
© 8.55 +t84.0 85.0 86.0 87.0 88.0 89.0 90.0 91.0 92.0
Exercise Price
might therefore look for a method of using the skew as part of his method of evaluatingoptions. Unfortunately, dealing with a volatility skew can be a problem because thereis no exact formula to use as a guide. Moreover, each trader tends to use the skewinaway that best serves his ultimate goal. A market maker might treat the skew one way,while a speculator or hedger treats the skew ina different way. Additionally, each tradermust determine the relative accuracy of the information reflected in the skew. A traderwho disagrees with a skew will find that some strategies will be more attractive thanothers.
For example, consider a trader who is active in the Bund option market, and whois trying to find a simple way to integrate a volatility skew into his theoretical pricingmodel. He wants his theoretical values to be consistent with the skew, but he also hasan opinion on volatility. What might the trader do? Going back to Figure 18-7, we cansee that the at-the-money implied volatility in the Bund option market is approximately3.58. Suppose a trader felt that implied volatility was too low, and that a morereasonable volatility was .25 percentage points higher, at 3.83. If the trader still believedthat the skew represented the relative value of options, he might raise the whole skew.25 volatility points and use these new volatilities to evaluate options at differentexercise prices. This would reflect his opinion that implied volatility was too low, butwould also take into consideration the relative value of options reflected by the skew.
Ina similar manner, the trader could also shift the skew right or left as the price ofthe underlying contract changed. If the underlying contract several weeks later were to
 Models and the Real World 409
move up 2.00, a trader could Keep the shape of the skew constant but shift the wholeskew up 2 points. At the same time, he could raise or lower the skew, using theat-the-money options as a reference point, to reflect his opinion that the impliedvolatility was either too high or too low. He could again use the resulting skew toevaluate options at different exercise prices, and from this information to decide on astrategy. The result of shifting a skew either up or down, or from side to side, is shownin Figure 18-8b.
Shifting the whole skew might be a reasonable approach if a trader believes thatthe skew will remain constant in the face of changing market conditions. But is thislikely? The implied volatilities at different exercise prices are likely to depend on howthe marketplace views the likelihood of large moves in the underlying market. But allmoves are relative, both with respect to the underlying price and to time. In relativeterms a 4 point move in the underlying contract is larger with the underlying contractat 80 (a 5% move) than with the underlying contract at 100 (a 4% move). In relativeterms a four-point move over a two-week period is larger than a four-point move overa two-month period.
The fact that the volatility skew is likely to change with changing market conditionsis born out in Figure 18-8c. The skew changed its shape, generally becoming moresevere, as time passed. It also changed its position as both the price of the underlyingcontract and implied volatility changed. This poses a problem for a trader trying toincorporate a volatility skew into a theoretical pricing model. Pricing models are not
Figure 18-8b
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.30 T r Tr4.207- X |skew with aiivolatilities raised .25 ,
410+ | ; + {—> aot on ina Oe 4= NA . c skew with underiying3 \ contract 2.00/highero yO> 3.90} — + |z Ty2 aa SL 4& So bee .= 3.80 ey L 1 +
3.70+-
3.604 rR ane ff
current skew an3.50+- + 4 t _ _84 85 86 87 88 89 90 91 92 93 94
Exercise Price410 Chapter 18
Figure 18-8c
 
 
 
 
 
 
6.00\45.504 \ é\ 5 weeks to expiration /\ Bund futures = 91.57 ff\ /1 ks t irati \Z weeks to expiration£ 5.00 Bund futures = 69.28 /3 \ /°>3.2a€ 4.507 ott7” 17 weeks to expiirationBund futures = 87,864.007 | ~~Ny 2 weeks to expiration~ Bund futures = 91.573.5077 T T T T T84 86 88 90 92 94 96
Exercise Price
 
only used to evaluate options under current conditions. They are also used to evaluaterisk under changing market conditions. If a trader wants to fully incorporate a skew intohis model, he needs to know how the skew is likely to change with changing marketconditions.
We noted that movement in the underlying contract is relative, both with respectto the current price of the underlying contract and time to expiration. If we are using atheoretical pricing model, we might ask how the model expresses movement in theunderlying contract. If we know this, we can determine how the model expresses therelationship between the price of the underlying contract and different exercise prices.This may enable us to determine how the skew will change.
In the Black-Scholes model, movement in the underlying contract is measured ona logarithmic scale,!3 and the relationship between the exercise price of an option andthe current underlying price is expressed as the logarithm of the exercise price dividedby underlying price. At the same time, movement over time is governed by a squareroot relationship, so that in the Black-Scholes model the relative amount of movementrequired to reach an exercise price is fully expressed as
[natural logarithm (exercise price / underlying price)] / square root (time)
13. For a more detailed discussion of the logarithmic function, and the relationship between time andvolatility, see Appendix B.Models and the Reai Worid 411
where time is expressed in terms of years. Since we are using the Black-Scholes modelto calculate the implied volatilities of the various options, and since the Black-Scholesmodel expresses exercise prices in the way we have just described, perhaps it makessense to express the skew exercise prices in the same way. This has been done in Figure18-8d.
With our new x-axis measurements the skews start to look very much alike. Westill need to take into consideration the fact that the skews occupy different positionsbecause they represent different volatilities. In order to generalize the volatility scale(the y-axis), we might consider expressing all volatilities for a given skew in terms of thevolatility of a theoretical at-the-money option. For example, going back to Figure 18-8a,we can see from the graph that with the underlying contract at 87.86, the impliedvolatility of an option with an exercise price of 87.86 (a theoretically at-the-moneyoption) would be approximately 3.57%. We can therefore express the volatility at eachexercise price as the difference between that exercise price’s volatility and 3.57. Usingthis approach, the 91 exercise price, with an implied volatility of 3.81 (the point wherethe graph crosses the 90 exercise price), would be assigned a value of
 
3.81 — 3.57 = .24The 86 exercise price, with an implied volatility of 3.68, would be assigned a value of3.68 — 3.57 = .11Figure 18-8d6.00 T\\ ,§.50-7 \ — a\ BALERS /\. /in ,5,00 \ 4
4.504
Implied Volatility
      
41 weeks to expirationBund futures = 89.23
2 weeks to expiration
 
 
 
 
4.007 Bund tutures = 91.57_17 weeks to expirationBund futures = 87.863,50 v =a Tt r0.45 0.10 -0.05 0.00 0.05 0.10 0.15
In(ex. price/fut. price)/sqrt(time)412 Chapter 18
This method is satisfactory when implied volatilities remain relatively constant. Butsuppose the implied volatility in the Bund option market were to double to 7.00%.Under these circumstances, the implied volatilities at each exercise price can also beexpected to double. The 90 exercise price, instead of having an implied volatility of3.81, will have an implied volatility of 7.62. And the 86 exercise price, instead of havingan implied volatility of 3.68, will have an implied volatility of 7.36. Since impliedvolatility can change in the marketplace, we need a method of relating the overallchange in implied volatility to the change in each exercise price’s implied volatility. Theeasiest way to do this is to express the implied volatility at each exercise price as apercent of the at-the-money implied volatility.
For example, with an at-the-money implied volatility of 3.57, the 90 exercise price,with an implied volatility of 3.81, would be assigned a value of
3.81 / 3.57 = 106.7%The 86 exercise price, with an implied volatility of 3.68, would be assigned a value of3.68 / 3.57 = 103.1%
Combining this method of expressing the implied volatility in terms of a percent ofat-the-money volatility, together with the logarithmic/time method of expressing theexercise price, yields the transformations shown in Figure 18-8e.
Figure 18-8e
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
140 I 7 |\ |\ \ i|\ !VTA4130 ~s \ s 2} weeks to expiration i ’2 \ Bund futures = 91,57 f95® *© 420 \—-7 ——8 \, 5 | o5 . :x 4> \ . 33 110;— po ~6 og: nN :> 11 weeks to expiration se ho 5 weeks to expiration3 Bund futures i= 89.23 “sb ey Bund futures = 91.572 ss ’QE SS400- = ca17 weeks to gxpirationBund futures = 87.8690 ™ t-0.15 -0.10 -0.05 0.00 0.05 0.40 0.15
In(ex. price/fut. price)/sqrt(time)Models and the Real World 413
With our new x-axis and y-axis the skews, while not identical, tend to look muchalike. If we were to make the assumption that the skew would look similar to those inFigure 18-8e under all conditions, we might think of the skew as an additional variablein a theoretical pricing model. Instead of the five usual variables (time to expiration,exercise price, underlying price, interest rates, volatility), we now have six (Figure18-9).
If we are going to feed the skew into the model as a variable, we need to do itinaway which the model understands. This means we need to find a formula by which toexpress the skew. While this may sound difficult, many skews can be expressed assimple equations. In Figure 18-10 a function has been graphed out which approximatesthe shape of the volatility skews in Figure 18-8. If we make the assumption that thisfunction gives a good approximation of the skew, we can use the function as thevolatility skew variable in our theoretical pricing model. To analyze the value of anoption position under changing assumptions about underlying price, time to expiration,and volatility, we need only express the volatility at each exercise price using thisfunction:
volatility at an exercise price =the at-the-money volatility times f(x)
where the exercise price, E, is always expressed in relation to its underlying price, U,and the amount of time to expiration, t:
In (E/U) / VE
There is no law that says a trader must agree with the skew implied in the optionmarket. A trader might believe that there is little chance of an unexpectedly large movein the underlying market. In this case, he can lower the wings of the skew. On the other
Figure 18-9
Time to ExpirationExercise Price
Underlying Price
a Theoretical > Theoreticala > Pricing Model Value
Interest RatesVolatility
Volatility Skew
 414 Chapter 18
 
 
 
 
  
 
 
 
 
Figure 18-10140 . ri i: ! || |\ | |1309 ~~ —|> \ 4 wings| intlatedQ \ 5 (better chanced of a large move)E \& \&= 120 465 ‘x= \.£ \3 410:o>3aE100wings| reduced(smaller change of a large move)| f(x) = 30* 1 a+4whereix = In(ex.price/fut.price)/sqrt(time)90-r
 
 
 
 
 
b-O.45 -0.10 -0.08 0.00 0.05 0.10 0.15In(ex. price/fut. price)/sqrt(time)
hand, if he believes that a there is a high probability of an unusually large move, he canraise the wings of the skew. This type of modification is also shown in Figure 18-10.
A market maker can also use the skew to adjust his bids and offers. If the marketmaker finds himself with a large positive gamma position and wants to sell options tocut back on the risk associated with the position, he can shift the skew slightlydownward. This will have the effect of lowering his bids and offers on all options. If themarket maker finds himself short a large number of out-of-the-money calls or puts, hecan adjust his skew by raising the wings. This will have the effect of raising his bids andoffers on out-of-the-money options.
The Bund options in our example form what is probably the most common skewshape, a smile which is approximately symmetrical. But this is not always the case.Skews can be symmetrical or asymmetrical, and they can reflect inflated or defiatedimplied volatilities across exercise prices, depending on how the marketplace sees thepossibility, and consequences, of various moves in the underlying market. In addition,in stock index markets, because it is not always possible to take advantage of a mispricedarbitrage relationship (see Chapter 15), calls and puts with the same exercise prices donot necessarily trade at the same implied volatility. For this reason, many traders in stockindex markets prefer to evaluate call and put skews separately.
As an example, Figures 18-11a and 18-11b represent the implied volatilities ofMarch 1993 OEX calls and puts with different amounts of time remaining to expira-Models and the Real World 415
Figure 18-11a: OEX March 1993 Calls
 
30 T 1
 
5 weeks to expirationOEX = 406.26
20 1 week to exairation
 
15+
 
 
\mplied Volatility (°%)
 
 
 
 
 
 
 
 
 
 
 
T a360 370 3B0 390 400 410 420 430Exercise Price
 
Figure 18-11b: OEX March 1993 Puts
 
  
  
 
 
 
 
 
 
 
 
 
 
 
30: : \| :' i !| | oe * i \ ;ppt --—----— { . Bweekstoexpiration (2 0GEX 16.26 (406.79) | 7 1| ( | !'_~ — | \ t 1 week to expiration$ SL | wt OEX = 412.67 (412.89)By BO foo 5 An= mo | |3 —_& S i2g Pa ~o mm -o Se ,‘a 154 ~ = — - £ 4t . o104 — fm —4 T 4WwW wel te |OEX = (396.64 (397.28)i |{ i$4 7 t +r q360 370 360 390 400 410 420 430
Exercise Price
 416 Chapter 18
tion.!2 Note the extreme inflation of downside exercise prices, both for calls and puts.While it may not be possible to determine exactly what is causing the skew to have thisparticular shape (it may indicate that the marketplace believes stock markets go downfaster than they go up—as witness October 1987—or it may reflect the fact that stockindex options are often used to hedge long stock portfolios), the marketplace seems tobe giving greater relative value to lower exercise prices. A trader who wanted to use themarketplace’s relative evaluation might try to integrate the skew into his theoreticalpricing model.
As before, we might try to generalize the skew by expressing each exercise pricein terms of a logarithmic relationship to the underlying index price. Stock index prices,like stock prices, are assumed to be lognormally distributed around the forward price ofthe index, i.e., the current index price plus carrying costs less dividends. It thereforemakes sense to express the exercise price in relation to the forward price rather than tothe index price:
in(exercise price/forward price) / Vtime
This has been done in Figures 18-11¢ and 18-11d. (The forward price of the OEX is inparenthesis.)
While there are clearly some differences between volatility skews in Figures 18-11¢and 18-11d, there is considerably more consistency than one would expect from Figures18-11a and 18-11b. If a trader wanted to find functions, one for calls and one for puts,by which the skews could be expressed, he could use these functions to analyze aposition under a wide variety of market conditions.
It is only in the last several years that traders have begun to appreciate theimportance of volatility skews, both as an evaluation tool and as a risk managementconsideration. This section is intended not as a solution to all the problems that tradersencounter in analyzing and using volatility skews, but as an introduction to how tradersapproach the problem. Each trader as he gains more experience must decide how bestto deal with the volatility skew, given the characteristics of the market in which he isdealing, as well as his ultimate goal in that market.
A FINAL THOUGHT
Because the use of a theoretical pricing model requires a trader to make so manydifferent decisions, both with respect to the inputs into the model and the degree towhich the assumptions on which the model is based are likely to be accurate, the newoption trader may feel that making the right decisions is simply a matter of luck. Whileluck undoubtedly plays a role in the short run, in the long run those traders who arewilling to put in the effort required to understand how models work always seem tocome out ahead. Experienced traders know that using a model, with all its problems, isstill the best way to evaluate options and manage risk.
12. Because of the possibility of early exercise, the Cox-Ross-Rubenstein model was used to calculate OEXimplied volatilities.Models and the Real World 417
Figure 18-11¢: OEX March 1993 Calls
 
 
 
 
 
 
         
 
 
 
 
 
 
 
 
 
 
 
220 ;\ ; 5 weeks t6 expiration i ' |\ ' OEX = 406,26 (406.79)' '200-}- —~ ene —-4 ;I. |i i |= := 180-+- ‘ +2 1 iE |o \ t (3 41 weeks to ekpiration1607 aan ee soeed tear 7:28)6 os. | |£= aa= 1407--— +-week-te-expiration—- —-5 OEX = 412.67 | (412.89)° i i> i i3 |120+ ———~ - 4E |j ' o: Lo4004 -~ a .